- Email: [email protected]
GISAXS analysis of ion beam modified films and surfaces
Accepted Manuscript GISAXS analysis of ion beam modified films and surfaces Maja Buljan, Marko Karluˇsi´c, Nikolina Neki´c, Marko Jerˇcinovi´c, Iva Bogdanovi´c-Radovi´c, Sigrid Bernstorff, Nikola Radi´c, Igor Mekterovi´c PII: DOI: Reference:
S0010-4655(16)30317-4 http://dx.doi.org/10.1016/j.cpc.2016.10.011 COMPHY 6075
To appear in:
Computer Physics Communications
Received date: 2 June 2016 Revised date: 15 October 2016 Accepted date: 19 October 2016 Please cite this article as: M. Buljan, M. Karluˇsi´c, N. Neki´c, M. Jerˇcinovi´c, I. Bogdanovi´c-Radovi´c, S. Bernstorff, N. Radi´c, I. Mekterovi´c, GISAXS analysis of ion beam modified films and surfaces, Computer Physics Communications (2016), http://dx.doi.org/10.1016/j.cpc.2016.10.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
GISAXS analysis of ion beam modified films and surfaces
Maja Buljana*, Marko Karlušića, Nikolina Nekića, Marko Jerčinovića, Iva BogdanovićRadovića, Sigrid Bernstorffb, Nikola Radića and Igor Mekterovićc a
Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia
b
Elettra-Sincrotrone Trieste, 34149 Basovizza, Italy
c
Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia
Correspondence email: [email protected] Abstract Simple and efficient methods for the accurate structural characterization of ion-beam modified materials are important due to their interesting properties and many applications. Here we demonstrate the application of the Grazing incidence small angle X-ray scattering (GISAXS) method on the structural analysis of swift heavy ion-beam modified materials. Passing through a material, the accelerated ion usually modifies its surrounding causing the formation of nano-objects along its trajectory, called the ion track. We show that GISAXS can be used to determine the structural properties of the nano-objects formed along the ion tracks, as well as their ordering properties. We have developed theoretical models describing GISAXS intensity distributions of the systems having different ordering types. The efficiencies of the models are tested on experimental examples. Keywords GISAXS Modelling; Ion tracks; Ion beam irradiation; 1. Introduction Ion beam modification of materials is a powerful tool for the design of a materials structure. Very different material types and interesting structures are successfully produced using this technique [1-10]. More precisely, swift heavy ion beam irradiation can be used for the production of nano-objects (NO) with very different shapes starting from spherical to very elongated like nanorods or some very special nanoparticle shapes [11,12]. Another advanced feature of swift heavy ion-beam irradiation is the realization of regularly ordered nanostructures, due to their formation along the ion trajectories [13,14]. Using these 1
properties, strongly anisotropic materials and regularly corrugated materials surfaces may be 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
produced, which find many relevant applications in nanotechnology. The characterization of such features is usually performed by microscopy techniques like atomic force microscopy (AFM), scanning electron microscopy (SEM) or transmission electron microscopy (TEM). These techniques have many advantages, but they are timeconsuming and limited to a relatively small sampling area. GISAXS is a technique widely used for the characterisation of nanostructures and it is the ideal complementary technique to microscopy [15,16]. It provides data in the reciprocal space with excellent statistics. The measurements are very fast, non-destructive, and do not require a specific sample preparation. However, the main drawback for the application of GISAXS on the analysis of ion-beam modified materials is the fact that there is no suitable theory available for the interpretation of the measured data. More precisely, each specific structure should have its own model for the GISAXS description in order to enable a correct and accurate interpretation of the GISAXS measurements. In our previous research we have developed three models for the description of GISAXS from three-dimensional lattices of NOs [19]. The models are based on a paracrystal arrangement of NOs, and they are suitable for the description of 3D arrays of NOs formed in multilayer systems by self-assembled growth or by ion beam irradiation. However, these models are not suitable for the description of ion tracks formed on the film surfaces. Namely, the type of NO ordering depends on the way they are formed. The NOs formed along ion tracks have a specific ordering depending on the ion track properties, and they differ from the ordering properties of NOs in the multilayer systems mentioned above. In this paper we develop a theory that is suitable for the description and modelling of the GISAXS intensity from materials modified by swift heavy ion-beams. We limit our investigations to systems that consist of different NOs formed along the ion trajectories. Due to the many specific properties of these systems we develop three main models. The first model is intended for systems in which NOs form along non-correlated ion tracks. It is suitable for the description of single track properties and for low-ion doses [8, 17]. The second model is specialised for multiple ion tracks formed after single ion irradiation, which were observed very recently for the first time [18]. The last model describes systems formed by high-dose irradiation when the ion tracks overlap, so that a correlation between their distances can occur [17]. All models are supported by experimental examples demonstrating the applicability of these models to real systems. Finally, all models are incorporated in the
2
modelling platform GisaxStudio, [20] aimed for a simple and efficient analysis and modelling 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of GISAXS data. 2. Ion tracks - structural overview The formation of an ion track, i.e. of NOs along the ion trajectory, is a consequence of energy transfer mainly due to the electronic stopping mechanism. In that mechanism a swift heavy ion loses its energy by interaction with the electron cloud of the material atoms. The consequence of the interaction is formation of NOs along some part of the ion trajectories. Several examples of a NO formation on a material surface by ion irradiation are given in Figure 1.
Figure 1. GISAXS intensity maps and corresponding AFM images of Ge+ITO thin films irradiated under an angle of 1 deg (a) single ion tracks, 15 MeV Si ions, fluence F = 5×108 ions/cm2 (b) 15 MeV Si ions, F= 3.5×1010 ions/cm2 (c) double tracks, 92 MeV Xe ions, F=5×108 ions/cm2 (d) correlated tracks, 92 MeV Xe ions, F=2×109 ions/cm2
The structures visible in Fig. 1 are created by irradiation of Ge+ITO thin film using different ion types and ion fluences (F), and they clearly have different structural properties. The differences are clearly visible in both the AFM images and the GISAXS maps. Well separated and uncorrelated ion tracks form for low ion fluences, as shown in Fig 1(a). The GISAXS maps show the averaged information of many such isolated tracks. However, for larger ion fluences, several ions pass the same place on the surface, so the track properties are different than for low-dose irradiation. Depending on the material type the formation of spatially correlated hybrid ion tracks may occur, as illustrated in Figure 1(b). This occurs due to material redistribution caused by the irradiation, and it is a type of self-assembly process. Finally, very energetic ions can cause the formation of multiple parallel tracks after a single ion passage, as illustrated in Figure 1(c) 3
and (d). The separation and ordering properties of such tracks are different from the previous 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
cases, as they appear in groups which are usually not mutually correlated. Therefore, each of these systems needs a specialized model for the description of the NO arrangement and consequently for the modelling of the GISAXS intensity distributions. 2.1.
Ordering models
In this section, the proposed models for the description of NO ordering in different irradiated systems are presented. We assumed that the formed NOs are arranged in a special type of paracrystal lattice [19]. We describe this lattice by n basis vectors a1-an which define the ideal positions of the NOs in the lattice, and by the disorder parameters 1-n which describe the deviation of the NO positions from the ideal ones. The number of needed parameters depends on the ordering properties. The basis vectors are defined as follows: (1) (2) (3) For the description of the disorder, three main ordering types are used: long-range ordering (LRO), short range ordering (SRO) and random ordering (RO) models. The main difference between these ordering types is the following: the ideal positions of the NOs are predefined for the LRO case, the actual positions fluctuate around them. For the SRO, only the separation between the NOs is predefined but not their positions, so the deviation probability increases with the distance from a particular NO position. Finally, for the RO neither the positions nor the distances are predefined, both are random. The positions of the nth NO for these three ordering types for the one-dimensional case are given by: (4) (5) , where
(6)
is the deviation vector of the nth NO. A detailed description of these ordering types
may be found in Ref. [19]. When the positions of the NOs are described by some model and their shape is known, then the GISAXS intensity distribution is defined by the well-known formula [15,19]: 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(7) In this formula A is a constant, dq is the difference in the electron densities of the NO material and the surrounding matrix, R and R’ are the position vectors of the NOs, Q = Kf - Ki is the scattering vector (the difference of the wave vectors of the scattered and incident beams), q is the complex scattering vector corrected for refraction at the vacuum–substrate interface. (8) is the Fourier transformation of the shape function R(r) of a dot occurring in the position R; the shape function is unity in the NO volume and zero outside it. ti and tf are the Fresnel transmissivities of the substrate surface corresponding to the primary and scattered waves, respectively. The
brackets in equation (7) denote the averaging over the positions and
shapes of the NOs. In order to calculate this averaging, one has to assume how the NO sizes are connected with their positions. For the case of a three-dimensional system this formula reduces to [19]: ,
(9)
where N1-3 represent the number of NOs in the direction of the basis vector a1-3. The functions Gj(q) are structure factors that describe the arrangement of the NOs along a particular basis vector of the formed NO lattice. They are specific for each model that we develop and will be described in more details in the following text. The shape-related functions f1(q) and f2(q) are given by: (10) (11) In the following chapters we define the G1-G3 functions for the three respective models differing by arrangement type of the NOs within an ion trajectory.
2.2.
Simulation of GISAXS intensity distributions
2.2.1. Model 1: Uncorrelated ion tracks The model for the uncorrelated ion tracks is illustrated in Figure 2. It assumes the existence of uncorrelated chains of N1 NOs directed along the ion track direction (defined as x direction), 5
with the period between the NOs given by the basis vector a1 (Eq. (1)). The NOs are assumed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
to have a LRO along the y direction (their y positions are predefined by the ion trajectory which is assumed along the x direction), while they have a SRO along the x direction (no predefined positions, but there is some characteristic separation between them). There are N2 such chains (tracks), and they have random starting points (origins) defined by the position where an ion hits the surface. Therefore, the position of a NO (Rn1,n2) is defined by two indices; n1 that denotes the position of a NO within the chain and n2 denoting the position of the origin of the chain (indicated by red circles in Fig. 2). (12) In this model, Dn2RO defines the origin of each ion track, while defines the position of the NOs within the ion track. The correlation functions for Model 1 (G1-3(M1)), needed for the calculation of the GISAXS intensity by Eq. (9), are given by: ,
where, while
, and
,
(13)
;
.
Figure 2. Model 1: Uncorrelated ion tracks. It describes the ordering of NOs within the surface tracks. The starting position of each track (denoted by red circles) is random. Parameters: the separation between the islands in the track (|a1|, number of islands in a single track (N1), disorder 6
properties (x,y,z) and island size properties. is the standard deviation of the normal distribution of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the NO position deviations from the ideal position. Low values of the disorder parameters (1x and 1y) are taken for the drawing of this scheme.
Using the model described above, we have simulated GISAXS maps using different characteristic parameters. The parameters are given in Table 1, except for the parameters indicated in the GISAXS simulations. The shape of the NOs is assumed to be ellipsoidal with the characteristic radii Rx, Ry, Rz in x, y and z directions respectively. The distribution of the NO radii is assumed to be a Gamma distribution with standard deviation R. We start with the dependence of the GISAXS intensity distributions on the angle between the ion tracks (assumed along the x direction) and the probing x-ray beam. This angle is defined by the measurement conditions and indicated by . The dependence of GISAXS on is shown in Fig. 3. As expected, the intensity distribution significantly changes with . When the x-ray beam is parallel to the x direction (=0 deg, Fig. 3(a)), mainly the shape function of the NOs is seen in the GISAXS map. However, even a small deviation from the =0 angle induces a huge change in the GISAXS maps and an elongated feature similar to a tail appears (Fig. 3(b)). Increasing the angle up to 30 deg, the correlation of the NO positions within the tracks starts to have a significant contribution to the GISAXS intensity. Consequently, two strong lateral sheets appear in the GISAXS maps. It is important to note that we have assumed relatively small deviations from the ideal NOs positions, so these peaks are therefore very strong. In reality this is usually not the case. These sheets dominate the GISAXS maps for higher angles. Table 1. Parameters of the track properties used for modelling of GISAXS intensity distributions. All values except N1 (dimensionless) are in nm. par
|a1|
N1
x
y
z
Rx,y
Rz
R
20
10
1
0.1
0.5
1.5
2.0
1
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 3. Dependence of the GISAXS intensity distribution on the angle between the probing x-rays and the ion track direction (denoted by ).
The dependence of the GISAXS maps on the disorder of the NO positions within the ion traces, i.e. along the x direction (see Fig. 2 parameter 1x) is illustrated in Fig. 4. The dependence is shown for two characteristic angles (=30, 90 deg). For both angles it is clear that an increase of the disorder of the NO positions within the ion tracks causes a broadening of the characteristic sheets and a decrease in their intensity.
8
Figure 4. Dependence of the GISAXS intensity distribution on the disorder in the NOs position along 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the x direction (1x) within the ion tracks. The dependence for the two angles between the probing beam and ion tracks direction (=30, 90 deg) are shown.
For the GISAXS analysis the parameter 1y is also very important, describing a deviation of the NO positions on the surface in direction perpendicular to the ion track (y direction). The dependence for two characteristic angles (=5, 30 deg) are shown in Fig. 5. This type of disorder is assumed to have a long range ordering, so an increase of the disorder parameter causes only a decrease in the intensity of the characteristic intensity sheets (not their widths).
Figure 5. Influence of 1y on the GISAXS intensity maps. The dependence for two angles (=5, 30 deg) between probing beam and ion tracks are shown.
Finally, the dependence of the GISAXS maps on the number of the NOs in the track (N1) is presented in Fig. 6. The intensity distributions for =5 deg depend strongly on N1. The characteristic tail becomes stronger and narrower with increasing the number of NOs. However, the effect is not so pronounced for higher angles, as is visible for the case of =30 deg. There is also no dependence for =0 deg, therefore it is not shown.
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 6. Dependence of GISAXS on the number of NOs (N1) in the track. The dependencies for the two angles =5, 30 deg are shown. 2.2.2. Example 1: Uncorrelated ion tracks
Here we analyse an experimental example for the GISAXS intensity distribution from uncorrelated ion tracks. The tracks are produced by irradiation with low-fluence 23 MeV I6+ ions of a TiO2 surface. Details are given in Ref. [18]. The AFM images of the material surface are shown in Fig. 7(a), (b). The isolated tracks with well-defined NOs are clearly visible there. The GISAXS maps of these films (=0 deg, and =5 deg) and the corresponding simulations obtained by fitting of the data using Model 1, are shown in Figs. 7(c)-(f). The map of the film measured for =90 (not shown here) shows very weak vertical sheets at Qy= 0.21 nm-1, corresponding to the separation between the NOs within the track of |a1|=29±1 nm. The results of the fitting are given in Table 2. The formed NOs are found to have a core/shell structure with the centre of the core shifted with respect to the shell centre by a vector d. The NOs radii in the surface (x-y) plane are denoted by Rxy, while the vertical radius is Rz. Multiplying the number of NOs in a track (N1) and their distance |a1|, we get the average track length. The values of the deviation parameters x-z provide an estimation of the regularity in the NO ordering. While x is relatively large (10±3 nm), y and z are much smaller (1.7±0.2 1 nm and 0.26±0.06 nm, respectively). The same follows from the AFM measurements; the NOs appear in nearly straight lines along the ion trajectories (low y deviations), while their arrangement within each track (x direction) is not so regular. It is important to note that the GISAXS analysis provides significantly more structural parameters 10
than the AFM measurements. In addition these values are obtained from a very large sampling 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
volume.
Figure 7. (a),(b) AFM image of an TiO2 surface irradiated by 23 MeV I6+ ions, 50 ion tracks/m2 [18]. (c), (d) Measured GISAXS map and the corresponding simulation, respectively, for =0 deg. (e), (f) Measured and simulated GISAXS maps for =5 deg. Table 2. Parameters of the ion tracks obtained by GISAXS analysis. All parameter values except N1(dimensionless) are in nm. |a1|
N1
x
y
z
Rx,y
Rz
Rcore
R
|d|
GISAXS
29±1
6±3
10±3
2.2±0.2
0.26±0.06
1.9±0.1
1.9±0.1
1.6±0.1
1.1±0.2
0.39±0.02
AFM
25±5
6±3
-
-
-
-
1.5±0.5
-
-
-
2.2.3. Uncorrelated multiple ion tracks Recently, the formation of multiple tracks after a single ion passage was observed [17]. That means the formation of two or more parallel chains of NOs along a single ion trajectory. Here we develop a model that describes the GISAXS intensity distributions from such structures.
11
The model that shows the arrangement of the NOs within the double tracks can be seen 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
schematically in Fig. 8. We assume the formation of N2 parallel chains of NOs after a single ion passage (N2=2 in Fig. 8). The arrangement of the NOs within each chain is determined by the vector a1, as in Model 1. The origin of the nth chain has the position determined by the vector n2a2 from the origin of the ion track (it is assumed random). So, the chains are separated by |a2|. Finally, the origin of each track is assumed random, and the total number of tracks is N3. The disorder of the NOs within the chains is the same as for Model 1. For the disorder related to the basis vector a2 we assume that the chains from each track show LRO in the x direction, i.e. the starting points of the chains in the same track can fluctuate around predefined (origin of the track) positions. The ordering of the chains along the y direction is assumed to be SRO, because there are no predefined positions in that direction. The deviations in the z direction are assumed to be LRO, i.e. the positions are predefined by the surface plane. Therefore, the position of the NO defined by the indices n1-n3 (Rn1,n2,n3) is given by: (14) The correlation functions needed for the calculation of the GISAXS intensity are the following: (15) (16)
where
,
,
,
while G3(M2)=N3. Using these equations and Eq. (9), we have performed a series of simulations to illustrate the effects of the different disorder parameters and measurement conditions on the GISAXS intensity distributions. The constant parameters used for the modelling are given in Table 3, while the tuned parameter is shown in the corresponding simulation.
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 8. Model 2: Uncorrelated multiple ion tracks. It describes the arrangement of NOs formed within multiple tracks after single ion passage. Each track consists of N2 parallel chains (in this case N2=2) of NOs, and each chain is shifted by the vector a2 from the previous one. Each chain consists of N1 NOs with the periodicity given by the vector a1. The origin of each track is assumed random. The deviation parameter 2x,y describes the fluctuation of the entire chain from the ideal position given by the basis vectors a1 and a2. The deviation parameters describing the disorder within each chain are the same as in Model 1.
The dependence of the GISAXS intensity distributions calculated with this model on the angle is shown in Fig. 9. For =0, characteristic lateral maxima appear that are the consequence of the existence of multiple tracks. Their distance is inversely related to the distance between the chains in a multiple track. These characteristic maxima disappear for larger values of the angle . Table 3. Parameters of the track properties used for the modelling of GISAXS intensity distributions with Model 2. The exceptions are the tuned parameters which are indicated in the GISAXS simulations. All values except N1, N2 (dimensionless) are given in nm. par.
|a1|
|a2|
N1
N2
x
y
x
y
z
z
Rx,y
Rz
R
20
15
10
2
2.5
1.0
1.0
1.0
0.1
0.1
1.5
2.0
1.1
13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 9. Dependence of GISAXS intensity distributions from the multiple ion tracks on the angle between the probing beam and the ion track direction.
The influence of the disorder parameter 2y (deviations of the chain y position within the track) is demonstrated in Fig. 10. As expected, the characteristic sheets visible for =0 deg, broaden and decrease in intensity with increasing disorder. A similar effect, but much less prominent, is visible for =5 deg.
Figure 10. Dependence of the GISAXS intensity distributions on the disorder parameter 2y for the model describing multiple ion tracks. The cases =0 deg and =5 deg are shown.
14
Finally, we test the influence of the disorder existing in the chain x-positions on the GISAXS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
maps. As we mentioned above the chains can fluctuate around the origin of the track positions. This type of disorder is described by the parameter 2x, and its effect on the GISAXS intensity maps is shown in Fig. 11. This parameter has no influence on GISAXS for =0, however it has an influence for all other values of the angle . In Fig. 11 are shown two cases differing by the number of parallel NO chains within the track (N2).
Figure 11. Dependence of GISAXS intensity distributions on 2x. Two different numbers of parallel NO chains within the track (N2=2 and 8) are used for the simulations.
2.2.4. Example 2: Uncorrelated multiple ion tracks This example demonstrates the analysis of GISAXS maps originating from double tracks produced by low-fluence irradiation of Ge+ITO film with 92 MeV Xe ions [18]. The AFM images of the tracks are visible in Fig. 12(a),(b). Two parallel chains of NOs exist in the each track.
15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 12. (a),(b) AFM images of the ion tracks consisting two parallel chains of NOs formed in the irradiated Ge+ITO film [18]. The maximum of the height scale is 9 nm. Measured GISAXS maps of the tracks and corresponding simulations with: =0 deg (c),(d) and 5 deg (e),(f).
In the GISAXS map of the irradiated Ge+ITO surface, shown in Fig. 12(c) (probing angle =0 deg), two lateral maxima are well visible close to Qy=0.36 nm-1. They are due to the double track formation. They also exist in the simulation of the experimental measurement obtained by fitting, shown in Fig. 12(d).These maxima disappear already for =5 deg (Fig. 12 (e)) in accordance with the simulations from Fig. 9. All parameters obtained by numerical analysis of the GISAXS measurements are shown in Table 4. We see that the parameters obtained by GISAXS agree well with the AFM measurements. However, GISAXS provides significantly more parameters. The deviation parameters are in qualitative agreement with the AFM images and with the data obtained for the previous example. The disorder parameter y (describing the fluctuation of NOs within the chains) is larger than for the Example 1, which is obvious if we compare the AFM images. It is interesting to note that the vertical deviation parameter z, describing the fluctuation of the entire chain within the track, is larger than z which is related to the fluctuation of the individual NOs within each chain. Table 4. Parameters of the track properties obtained by modelling of the experimental GISAXS intensity shown in Fig. 12. The parameters obtained from the AFM image are shown for the comparison. par
|a1|
|a2|
N1
N2
x
y
x
y
16
z
z
Rx
Ry
Rz
R
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
GISAXS
26
12
22
2
12
2.5
1.8
2.6
0.10
2.0
2.9
2.5
2.8
1.1
AFM
28±5
14±1
20±8
2
-
-
-
-
-
-
-
-
3±1
-
2.2.5. Model 3: Correlated ion tracks This model describes the arrangement of NOs formed after irradiation of surface with higher ion fluences leading to an overlapping of the ion tracks. The overlapping in some cases causes the self-assembly of the surface material, so the ion tracks have some characteristic distances. The formation of single ion tracks is assumed for this model. The arrangement of the NOs for this model is illustrated in Fig. 13. The NOs are arranged within single tracks (with period a1) as for the Models 1 and 2, while the periodicity of the tracks in y direction is given by the vector a2, as for the previous model. We assume that there are N2 such tracks in the y direction. Finally, we assume that the single ion tracks make long chains along the x direction, consisting of single tracks repeating with the vector a3=N1 a1 (set along the x direction). So, the entire surface is covered by ion tracks (see the AFM image and the model shown in Fig. 13). There are N3 tracks in the x direction, so the total length of one track chain is |a3|N3. The disorder related to the basis vector a1 is the same as for the previous models. The disorders of the track positions related to the vector a2 (2x and 2y) both show SRO for this model, while the corresponding x-related deviations had LRO for the Model 2. The reason is that there are no predefined origins of the tracks for this Model. The deviations from the periodicity a3 along the x and y directions are described by the parameters 3x and 3y, respectively. In real systems these parameters are usually large because there are no physical reasons for a strong correlation along this basis vector. However it is possible that some external parameters like steps on the surface or surface patterning cause such behaviour. 1-3z all are assumed to have LRO due to the assumption that the tracks are formed on a flat surface.
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 13. Model 3: Correlated ion tracks. The model describes the arrangement of NOs within ion tracks that have correlated distances, for example due to some kind of self-assembly process. The NOs with the period a1 are arranged in single ion tracks along the x axis. The single ion tracks have the periodicity a2 in y direction and a3in the x direction.
So, the positions of the NOs on the surface are described by:
(17) (18) (19)
where
,
,
.
These equations are used for the calculation of GISAXS simulations by Eq. (9). The constant parameters used for the simulations made with this model are given in Table 5, while the parameter that is tuned is shown in the corresponding simulation. The angle dependence of the GISAXS intensity distributions from the correlated ion tracks is given in Fig. 14. Their GISAXS maps show lateral peaks for =0 deg, similar to the ones in Model 2. However that is not the case for the random tracks shown in Fig. 3. The main difference between Models 2 and 3 is in the assumed ordering of the chains in the multiple
18
tracks: Model 2 assumes LRO, while Model 3 assumes SRO. The maps of higher angles 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
also differ from the uncorrelated ion tracks. The main reason is that relatively small disorder parameters 2,3x,y are taken for the simulations. For that case, the chains of the tracks are relatively long and well-ordered, so the characteristic tails are much narrower than for the uncorrelated tracks. Table 5. Parameters of the track properties used for modelling of GISAXS intensity distributions. All values except N1 – N3 (dimensionless) are given in nm. par
|a1|
|a2| N1
N2
20
15
100 5
10
N3
x
y
x
y
x
y
1.0
0.1
1.0
1.5
12.5 2.5
z
Rx,y
Rz
R
0.5
1.5
2.0
1.1
Figure 14. The dependence of GISAXS intensity distributions on the angle between the X-ray probing beam and the track direction is shown for the correlated ion tracks. The simulations are obtained using Eqs. (9), (18) and (19).
The dependence of the GISAXS on parameter 2y is demonstrated in Fig. 15. This parameter describes the disorder in the separation of ion tracks. When increasing this parameter, the characteristic lateral sheets for =0 deg broaden and decrease in intensity. A similar effect was observed for Model 2. For the =5 deg angle the characteristic tail also broadens, but the effect is significantly less pronounced, as is visible for the angle =0 deg.
19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 15. Influence of 2y on the GISAXS intensity distributions of correlated ion tracks. The dependences for =0 deg and 5 deg are shown.
The parameter 3x describing the fluctuation of the track-chains origins in x direction is actually not very physical, but can be important in some cases. The effect of this parameter on the GISAXS intensity distributions is shown in Fig. 16. From the simulations, it follows that the parameter has a significant effect only on small-period oscillations.
Figure 16. Influence of 3x on the GISAXS intensity distributions for the angles =30 deg and 90 deg.
20
Finally, the parameter 3y describes the disorder of the single tracks within the track chain in y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
direction. It has a small effect on the GISAXS intensity distributions for =0 deg. However, its effect is significant for the angle =5 deg. The characteristic tail visible for =5 deg is very sensitive to the length and width of the ion track. Thus, if 3y is small, then the single ion tracks actually make a long chain (they are repeating with the period a3, see Fig. 8) causing the tail to be strong and narrow. We had a similar behaviour for a large number N1 of NOs in a single track (Fig. 6). Increasing the disorder, the individual tracks are moved left or right from the ‘ideal’ centre position, so the chain is not continuous and regular any more.
Figure 17. Influence of 3y on the GISAXS intensity distributions for =0 deg and 5 deg.
2.2.6. Example 3: Correlated ion tracks Herein we apply Model 3 in order to determinate the structural parameters of correlated ion tracks formed by irradiation. An example showing the AFM images of such a system obtained by irradiation of a TiO2 surface with 23 MeV I6+ ions (900 ion tracks/m2), [18] is demonstrated in Figs. 18(a),(b). A characteristic distance of about 25 nm between the ion tracks is found from the analysis of these AFM images. These correlations are manifested in the measured GISAXS map (Fig. 18(c)) as two broad sheets (indicated by arrows in Fig. 18(c)). The results of the GISAXS analysis for this case are given in Table 6. We see that the parameters obtained by the numerical analysis are again in good agreement with the AFM results. Additionally, we get many more parameters than for the shown AFM measurements. 21
We could get some of them by careful analysis of many AFM images. However, some 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
parameters like the NO lateral sizes or their sizes if they are buried below the surface are accessible only by GISAXS.
Figure 18. (a),(b) AFM image of aTiO2 surface irradiated by 23 MeV I6+ ions, 900 ion tracks/m2 [18]. The maximum of the scale is 4.6 nm. (c) Measured GISAXS map and (d) corresponding simulation for =0 deg. (e), (f) Measured and simulated GISAXS maps for =5 deg.
Table 6. Parameters of the track properties for Example 3 obtained by the analysis of the measured GISAXS maps. The parameters found from the AFM images are shown for the comparison. All values except N1 – N3 (dimensionless) are given in nm. par
|a1|
|a2|
N1
N2
N3
x
y
x
y
x y z
Rx,y
Rz
Rcore
R
|d|
GISAXS
29
22
6
100
5
10
1.7
15
16
6
4
0.5
2.3
1.7
1.5
1.0
0.6
AFM
-
24±2
-
-
-
-
-
-
-
-
-
-
-
2.2±0.2
-
-
-
3. Discussion, reliability, limitations Using the models given above it is possible to get the structural parameters of the NOs formed by ion beam irradiation within the ion tracks. However, it is important to note, that the fitting procedure should be performed with care because many parameters are used in the fit. Due to
22
that fact it is possible to get wrong parameter values and still a relatively good fit. To avoid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
such situation, it is very useful to have some ‘initial’ parameters for the fit set (estimate) the initial parameter values well, and to constrain the parameter values during the fit within to some reasonable limits. It is also necessary to know at least roughly which structure is expected, including the arrangement type and maybe the shape of the NOs, and their sizes. For the same reasons, it is recommended to perform additional microscopy measurements for at least one sample from each series to check the fitting results. In some cases (like for the formation of Ge nanoparticles in some matrix), the atomic composition determination can help checking the obtained parameters. Namely, the percentage of Ge can be estimated from the size and arrangement properties obtained by fitting and compared with the measured value. To our experience a non-correct value of one parameter always appears in combination with non-correct value of some other parameter. Also, we are working on an open-source application that will use a number of optimization algorithms in order to optimally fit the model(s), and the fit stability will be a part of our future research [20]. Finally, it should be mentioned that the central part of the GISAXS map close to Qy=0 cannot be simulated accurately due to the model limitations. Namely, the contributions originating from the shape of the entire assumed NO lattice (which is defined by basis vectors a1-a3), are significant for small Qy values, and they are not present in reality. 4. Conclusions We have investigated GISAXS intensity distributions from three different types of ion tracks formed on the material surfaces. The theoretical models for their description have been presented and a series of simulations are demonstrated showing the effects of different disorder parameters on GISAXS intensity distributions. Each model is accompanied by an experimental example that demonstrates the efficiency of the developed models. We believe that the models will be useful for the successful characterization of different ion tracks by the GISAXS technique.
Acknowledgements The authors acknowledge J. Erceg for the assistance in the sample preparation. MB, NR acknowledge the Croatian Science Foundation (pr. No. 2334). Support by the Croatian Centre of Excellence for Advanced Materials and Sensing Devices and Calipso for GISAXS
23
measurements (pr. No. 20145403) are acknowledged. The assistance of H. Lebius and B. Ban 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
d’Etat during the ion irradiation in GANIL, as well as assistance of M. Schleberger with the AFM measurements, is gratefully acknowledged. References [1] Akcöltekin E., Peters T., Meyer R., Duvenbeck A., Klusmann M., Monnet I., Lebius H., Schleberger M. (2007) Nat. Nanotech. 2 290-294 [2] Kluth P., Schnohr C. S., Pakarinen O.H., Djurabekova F., Sprouster D. J., Giulian R., Ridgway M. C., Byrne A. P., Trautmann C., Cookson D. J., Nordlund K., Toulemonde M. (2008) Phys. Rev. Lett. 101 175503 [3] Karlušić M., Akcöltekin S., Osmani O., Monnet I., Lebius H., Jakšić M., Schleberger M. (2010) New J. Phys. 12 043009 [4] Aumayr F., Facsko S., El-Said A. S., Trautmann C., Schleberger M. (2011) J. Phys.: Condens. Matter 23 393001 [5] Akcöltekin S., Bukowska H., Peters T., Osmani O., Monnet I., Alzaher I., Ban-d’Etat B., Lebius H., Schleberger M. (2011) Apl. Phys. Lett. 98 103103 [6] Ridgway M. C., Bierschenk T., Giulian R., Afra B., Rodriguez M.D., Araujo L. L., Byrne A.P., Kirby N., Pakarinen O.H., Djurabekova F., Nordlund K., Schleberger M., Osmani O., Medvedev N., Rethfeld B., Kluth P. (2013) Phys. Rev. Lett. 110 245502 [7] Ochedowski O., Osmani O., Schade M., Bussmann B. K., Ban-d’Etat B., Lebius H., Schleberger M. (2014) Nat. Comm. 5 3913 [8] Karlušić M., Kozubek R., Lebius H., Ban-d’Etat B., Wilhelm R. A., Buljan M., Siketić Z., Scholz F., Meisch T., Jakšić M., Bernstorff S., Schleberger M., Šantić B. (2015) J. Phys. D: Appl. Phys. 48 325304 [9] Ochedowski O., Lehtinen O., Kaiser U., Turchanin A., Ban-d`Etat B., Lebius H., Karlušić M., Jakšić M., Schleberger M. (2015) Nanotechnology 26 465302 [10] Papaléo R. M., Thomaz R., Gutierres L. I., de Menezes V. M., Severin D., Trautmann C., Tramontina D., Bringa E. M., Grande P. L. (2015) Phys. Rev. Lett. 114 118302
24
[11] Ridgway M.C., Giulian R., Sprouster D.J., Kluth P., Araujo L.L., Llewellyn D.J., Byrne 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
A.P., Kremer F., Fichtner P.F.P., Rizza G., Amekura H., Toulemonde M., (2011) Phys. Rev. Lett. 106 095505 [12] Ridgway M.C., Djurabekova F., Nordlund K., (2015) Curr. Opin. Solid State Mat. Sci. 19 29 [13] Buljan M., Bogdanović-Radović I., Karlušić M., Desnica U. V., Dražić G., Radić N., Dubček P., Salamon K., Bernstorff S., Holý V. (2009) Appl. Phys. Lett. 95 063104 [14] Bogdanović-Radović I., Buljan M., Karlušić M., Skukan N., Božičević I., Jakšić M., Radić N., Dražić G., Bernstorff S. (2010) Phys. Rev. B 86 165316 [15] Renaud, J., Lazzari, R. & Leroy, F. (2009). Surf. Sci. Rep. 64, 255–380. [16] Babonneau, D. (2010). J. Appl. Cryst. 43, 929–936. [17] Karlušić M., Buljan M., Lebius H., Ban d’Etat B., Šantić B., Bogdanović-Radović I., Radić. N., Jakšić M., Bernstorff S., Schleberger M., (2016), in preparation [18] Karlušić M., Bernstorff S., Siketić Z., Šantić B., Bogdanović-Radović I., Jakšić M., Schleberger M., Buljan M., (2016), Formation of swift heavy ion tracks on rutile TiO2 (001) surface J. Appl. Cryst., under review [19] M. Buljan, N. Radić, S. Bernstorff, G. Dražić, I. Bogdanović-Radović, V. Holý, Acta Cryst. A 2012, 68, 124-138. [20] Mekterović I., Buljan M., GisaxStudio, http://homer.zpr.fer.hr/gisaxstudio, last accessed: 2016-05-31
25
S0010-4655(16)30317-4 http://dx.doi.org/10.1016/j.cpc.2016.10.011 COMPHY 6075
To appear in:
Computer Physics Communications
Received date: 2 June 2016 Revised date: 15 October 2016 Accepted date: 19 October 2016 Please cite this article as: M. Buljan, M. Karluˇsi´c, N. Neki´c, M. Jerˇcinovi´c, I. Bogdanovi´c-Radovi´c, S. Bernstorff, N. Radi´c, I. Mekterovi´c, GISAXS analysis of ion beam modified films and surfaces, Computer Physics Communications (2016), http://dx.doi.org/10.1016/j.cpc.2016.10.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
GISAXS analysis of ion beam modified films and surfaces
Maja Buljana*, Marko Karlušića, Nikolina Nekića, Marko Jerčinovića, Iva BogdanovićRadovića, Sigrid Bernstorffb, Nikola Radića and Igor Mekterovićc a
Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia
b
Elettra-Sincrotrone Trieste, 34149 Basovizza, Italy
c
Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia
Correspondence email: [email protected] Abstract Simple and efficient methods for the accurate structural characterization of ion-beam modified materials are important due to their interesting properties and many applications. Here we demonstrate the application of the Grazing incidence small angle X-ray scattering (GISAXS) method on the structural analysis of swift heavy ion-beam modified materials. Passing through a material, the accelerated ion usually modifies its surrounding causing the formation of nano-objects along its trajectory, called the ion track. We show that GISAXS can be used to determine the structural properties of the nano-objects formed along the ion tracks, as well as their ordering properties. We have developed theoretical models describing GISAXS intensity distributions of the systems having different ordering types. The efficiencies of the models are tested on experimental examples. Keywords GISAXS Modelling; Ion tracks; Ion beam irradiation; 1. Introduction Ion beam modification of materials is a powerful tool for the design of a materials structure. Very different material types and interesting structures are successfully produced using this technique [1-10]. More precisely, swift heavy ion beam irradiation can be used for the production of nano-objects (NO) with very different shapes starting from spherical to very elongated like nanorods or some very special nanoparticle shapes [11,12]. Another advanced feature of swift heavy ion-beam irradiation is the realization of regularly ordered nanostructures, due to their formation along the ion trajectories [13,14]. Using these 1
properties, strongly anisotropic materials and regularly corrugated materials surfaces may be 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
produced, which find many relevant applications in nanotechnology. The characterization of such features is usually performed by microscopy techniques like atomic force microscopy (AFM), scanning electron microscopy (SEM) or transmission electron microscopy (TEM). These techniques have many advantages, but they are timeconsuming and limited to a relatively small sampling area. GISAXS is a technique widely used for the characterisation of nanostructures and it is the ideal complementary technique to microscopy [15,16]. It provides data in the reciprocal space with excellent statistics. The measurements are very fast, non-destructive, and do not require a specific sample preparation. However, the main drawback for the application of GISAXS on the analysis of ion-beam modified materials is the fact that there is no suitable theory available for the interpretation of the measured data. More precisely, each specific structure should have its own model for the GISAXS description in order to enable a correct and accurate interpretation of the GISAXS measurements. In our previous research we have developed three models for the description of GISAXS from three-dimensional lattices of NOs [19]. The models are based on a paracrystal arrangement of NOs, and they are suitable for the description of 3D arrays of NOs formed in multilayer systems by self-assembled growth or by ion beam irradiation. However, these models are not suitable for the description of ion tracks formed on the film surfaces. Namely, the type of NO ordering depends on the way they are formed. The NOs formed along ion tracks have a specific ordering depending on the ion track properties, and they differ from the ordering properties of NOs in the multilayer systems mentioned above. In this paper we develop a theory that is suitable for the description and modelling of the GISAXS intensity from materials modified by swift heavy ion-beams. We limit our investigations to systems that consist of different NOs formed along the ion trajectories. Due to the many specific properties of these systems we develop three main models. The first model is intended for systems in which NOs form along non-correlated ion tracks. It is suitable for the description of single track properties and for low-ion doses [8, 17]. The second model is specialised for multiple ion tracks formed after single ion irradiation, which were observed very recently for the first time [18]. The last model describes systems formed by high-dose irradiation when the ion tracks overlap, so that a correlation between their distances can occur [17]. All models are supported by experimental examples demonstrating the applicability of these models to real systems. Finally, all models are incorporated in the
2
modelling platform GisaxStudio, [20] aimed for a simple and efficient analysis and modelling 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
of GISAXS data. 2. Ion tracks - structural overview The formation of an ion track, i.e. of NOs along the ion trajectory, is a consequence of energy transfer mainly due to the electronic stopping mechanism. In that mechanism a swift heavy ion loses its energy by interaction with the electron cloud of the material atoms. The consequence of the interaction is formation of NOs along some part of the ion trajectories. Several examples of a NO formation on a material surface by ion irradiation are given in Figure 1.
Figure 1. GISAXS intensity maps and corresponding AFM images of Ge+ITO thin films irradiated under an angle of 1 deg (a) single ion tracks, 15 MeV Si ions, fluence F = 5×108 ions/cm2 (b) 15 MeV Si ions, F= 3.5×1010 ions/cm2 (c) double tracks, 92 MeV Xe ions, F=5×108 ions/cm2 (d) correlated tracks, 92 MeV Xe ions, F=2×109 ions/cm2
The structures visible in Fig. 1 are created by irradiation of Ge+ITO thin film using different ion types and ion fluences (F), and they clearly have different structural properties. The differences are clearly visible in both the AFM images and the GISAXS maps. Well separated and uncorrelated ion tracks form for low ion fluences, as shown in Fig 1(a). The GISAXS maps show the averaged information of many such isolated tracks. However, for larger ion fluences, several ions pass the same place on the surface, so the track properties are different than for low-dose irradiation. Depending on the material type the formation of spatially correlated hybrid ion tracks may occur, as illustrated in Figure 1(b). This occurs due to material redistribution caused by the irradiation, and it is a type of self-assembly process. Finally, very energetic ions can cause the formation of multiple parallel tracks after a single ion passage, as illustrated in Figure 1(c) 3
and (d). The separation and ordering properties of such tracks are different from the previous 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
cases, as they appear in groups which are usually not mutually correlated. Therefore, each of these systems needs a specialized model for the description of the NO arrangement and consequently for the modelling of the GISAXS intensity distributions. 2.1.
Ordering models
In this section, the proposed models for the description of NO ordering in different irradiated systems are presented. We assumed that the formed NOs are arranged in a special type of paracrystal lattice [19]. We describe this lattice by n basis vectors a1-an which define the ideal positions of the NOs in the lattice, and by the disorder parameters 1-n which describe the deviation of the NO positions from the ideal ones. The number of needed parameters depends on the ordering properties. The basis vectors are defined as follows: (1) (2) (3) For the description of the disorder, three main ordering types are used: long-range ordering (LRO), short range ordering (SRO) and random ordering (RO) models. The main difference between these ordering types is the following: the ideal positions of the NOs are predefined for the LRO case, the actual positions fluctuate around them. For the SRO, only the separation between the NOs is predefined but not their positions, so the deviation probability increases with the distance from a particular NO position. Finally, for the RO neither the positions nor the distances are predefined, both are random. The positions of the nth NO for these three ordering types for the one-dimensional case are given by: (4) (5) , where
(6)
is the deviation vector of the nth NO. A detailed description of these ordering types
may be found in Ref. [19]. When the positions of the NOs are described by some model and their shape is known, then the GISAXS intensity distribution is defined by the well-known formula [15,19]: 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(7) In this formula A is a constant, dq is the difference in the electron densities of the NO material and the surrounding matrix, R and R’ are the position vectors of the NOs, Q = Kf - Ki is the scattering vector (the difference of the wave vectors of the scattered and incident beams), q is the complex scattering vector corrected for refraction at the vacuum–substrate interface. (8) is the Fourier transformation of the shape function R(r) of a dot occurring in the position R; the shape function is unity in the NO volume and zero outside it. ti and tf are the Fresnel transmissivities of the substrate surface corresponding to the primary and scattered waves, respectively. The
brackets in equation (7) denote the averaging over the positions and
shapes of the NOs. In order to calculate this averaging, one has to assume how the NO sizes are connected with their positions. For the case of a three-dimensional system this formula reduces to [19]: ,
(9)
where N1-3 represent the number of NOs in the direction of the basis vector a1-3. The functions Gj(q) are structure factors that describe the arrangement of the NOs along a particular basis vector of the formed NO lattice. They are specific for each model that we develop and will be described in more details in the following text. The shape-related functions f1(q) and f2(q) are given by: (10) (11) In the following chapters we define the G1-G3 functions for the three respective models differing by arrangement type of the NOs within an ion trajectory.
2.2.
Simulation of GISAXS intensity distributions
2.2.1. Model 1: Uncorrelated ion tracks The model for the uncorrelated ion tracks is illustrated in Figure 2. It assumes the existence of uncorrelated chains of N1 NOs directed along the ion track direction (defined as x direction), 5
with the period between the NOs given by the basis vector a1 (Eq. (1)). The NOs are assumed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
to have a LRO along the y direction (their y positions are predefined by the ion trajectory which is assumed along the x direction), while they have a SRO along the x direction (no predefined positions, but there is some characteristic separation between them). There are N2 such chains (tracks), and they have random starting points (origins) defined by the position where an ion hits the surface. Therefore, the position of a NO (Rn1,n2) is defined by two indices; n1 that denotes the position of a NO within the chain and n2 denoting the position of the origin of the chain (indicated by red circles in Fig. 2). (12) In this model, Dn2RO defines the origin of each ion track, while defines the position of the NOs within the ion track. The correlation functions for Model 1 (G1-3(M1)), needed for the calculation of the GISAXS intensity by Eq. (9), are given by: ,
where, while
, and
,
(13)
;
.
Figure 2. Model 1: Uncorrelated ion tracks. It describes the ordering of NOs within the surface tracks. The starting position of each track (denoted by red circles) is random. Parameters: the separation between the islands in the track (|a1|, number of islands in a single track (N1), disorder 6
properties (x,y,z) and island size properties. is the standard deviation of the normal distribution of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the NO position deviations from the ideal position. Low values of the disorder parameters (1x and 1y) are taken for the drawing of this scheme.
Using the model described above, we have simulated GISAXS maps using different characteristic parameters. The parameters are given in Table 1, except for the parameters indicated in the GISAXS simulations. The shape of the NOs is assumed to be ellipsoidal with the characteristic radii Rx, Ry, Rz in x, y and z directions respectively. The distribution of the NO radii is assumed to be a Gamma distribution with standard deviation R. We start with the dependence of the GISAXS intensity distributions on the angle between the ion tracks (assumed along the x direction) and the probing x-ray beam. This angle is defined by the measurement conditions and indicated by . The dependence of GISAXS on is shown in Fig. 3. As expected, the intensity distribution significantly changes with . When the x-ray beam is parallel to the x direction (=0 deg, Fig. 3(a)), mainly the shape function of the NOs is seen in the GISAXS map. However, even a small deviation from the =0 angle induces a huge change in the GISAXS maps and an elongated feature similar to a tail appears (Fig. 3(b)). Increasing the angle up to 30 deg, the correlation of the NO positions within the tracks starts to have a significant contribution to the GISAXS intensity. Consequently, two strong lateral sheets appear in the GISAXS maps. It is important to note that we have assumed relatively small deviations from the ideal NOs positions, so these peaks are therefore very strong. In reality this is usually not the case. These sheets dominate the GISAXS maps for higher angles. Table 1. Parameters of the track properties used for modelling of GISAXS intensity distributions. All values except N1 (dimensionless) are in nm. par
|a1|
N1
x
y
z
Rx,y
Rz
R
20
10
1
0.1
0.5
1.5
2.0
1
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 3. Dependence of the GISAXS intensity distribution on the angle between the probing x-rays and the ion track direction (denoted by ).
The dependence of the GISAXS maps on the disorder of the NO positions within the ion traces, i.e. along the x direction (see Fig. 2 parameter 1x) is illustrated in Fig. 4. The dependence is shown for two characteristic angles (=30, 90 deg). For both angles it is clear that an increase of the disorder of the NO positions within the ion tracks causes a broadening of the characteristic sheets and a decrease in their intensity.
8
Figure 4. Dependence of the GISAXS intensity distribution on the disorder in the NOs position along 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
the x direction (1x) within the ion tracks. The dependence for the two angles between the probing beam and ion tracks direction (=30, 90 deg) are shown.
For the GISAXS analysis the parameter 1y is also very important, describing a deviation of the NO positions on the surface in direction perpendicular to the ion track (y direction). The dependence for two characteristic angles (=5, 30 deg) are shown in Fig. 5. This type of disorder is assumed to have a long range ordering, so an increase of the disorder parameter causes only a decrease in the intensity of the characteristic intensity sheets (not their widths).
Figure 5. Influence of 1y on the GISAXS intensity maps. The dependence for two angles (=5, 30 deg) between probing beam and ion tracks are shown.
Finally, the dependence of the GISAXS maps on the number of the NOs in the track (N1) is presented in Fig. 6. The intensity distributions for =5 deg depend strongly on N1. The characteristic tail becomes stronger and narrower with increasing the number of NOs. However, the effect is not so pronounced for higher angles, as is visible for the case of =30 deg. There is also no dependence for =0 deg, therefore it is not shown.
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 6. Dependence of GISAXS on the number of NOs (N1) in the track. The dependencies for the two angles =5, 30 deg are shown. 2.2.2. Example 1: Uncorrelated ion tracks
Here we analyse an experimental example for the GISAXS intensity distribution from uncorrelated ion tracks. The tracks are produced by irradiation with low-fluence 23 MeV I6+ ions of a TiO2 surface. Details are given in Ref. [18]. The AFM images of the material surface are shown in Fig. 7(a), (b). The isolated tracks with well-defined NOs are clearly visible there. The GISAXS maps of these films (=0 deg, and =5 deg) and the corresponding simulations obtained by fitting of the data using Model 1, are shown in Figs. 7(c)-(f). The map of the film measured for =90 (not shown here) shows very weak vertical sheets at Qy= 0.21 nm-1, corresponding to the separation between the NOs within the track of |a1|=29±1 nm. The results of the fitting are given in Table 2. The formed NOs are found to have a core/shell structure with the centre of the core shifted with respect to the shell centre by a vector d. The NOs radii in the surface (x-y) plane are denoted by Rxy, while the vertical radius is Rz. Multiplying the number of NOs in a track (N1) and their distance |a1|, we get the average track length. The values of the deviation parameters x-z provide an estimation of the regularity in the NO ordering. While x is relatively large (10±3 nm), y and z are much smaller (1.7±0.2 1 nm and 0.26±0.06 nm, respectively). The same follows from the AFM measurements; the NOs appear in nearly straight lines along the ion trajectories (low y deviations), while their arrangement within each track (x direction) is not so regular. It is important to note that the GISAXS analysis provides significantly more structural parameters 10
than the AFM measurements. In addition these values are obtained from a very large sampling 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
volume.
Figure 7. (a),(b) AFM image of an TiO2 surface irradiated by 23 MeV I6+ ions, 50 ion tracks/m2 [18]. (c), (d) Measured GISAXS map and the corresponding simulation, respectively, for =0 deg. (e), (f) Measured and simulated GISAXS maps for =5 deg. Table 2. Parameters of the ion tracks obtained by GISAXS analysis. All parameter values except N1(dimensionless) are in nm. |a1|
N1
x
y
z
Rx,y
Rz
Rcore
R
|d|
GISAXS
29±1
6±3
10±3
2.2±0.2
0.26±0.06
1.9±0.1
1.9±0.1
1.6±0.1
1.1±0.2
0.39±0.02
AFM
25±5
6±3
-
-
-
-
1.5±0.5
-
-
-
2.2.3. Uncorrelated multiple ion tracks Recently, the formation of multiple tracks after a single ion passage was observed [17]. That means the formation of two or more parallel chains of NOs along a single ion trajectory. Here we develop a model that describes the GISAXS intensity distributions from such structures.
11
The model that shows the arrangement of the NOs within the double tracks can be seen 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
schematically in Fig. 8. We assume the formation of N2 parallel chains of NOs after a single ion passage (N2=2 in Fig. 8). The arrangement of the NOs within each chain is determined by the vector a1, as in Model 1. The origin of the nth chain has the position determined by the vector n2a2 from the origin of the ion track (it is assumed random). So, the chains are separated by |a2|. Finally, the origin of each track is assumed random, and the total number of tracks is N3. The disorder of the NOs within the chains is the same as for Model 1. For the disorder related to the basis vector a2 we assume that the chains from each track show LRO in the x direction, i.e. the starting points of the chains in the same track can fluctuate around predefined (origin of the track) positions. The ordering of the chains along the y direction is assumed to be SRO, because there are no predefined positions in that direction. The deviations in the z direction are assumed to be LRO, i.e. the positions are predefined by the surface plane. Therefore, the position of the NO defined by the indices n1-n3 (Rn1,n2,n3) is given by: (14) The correlation functions needed for the calculation of the GISAXS intensity are the following: (15) (16)
where
,
,
,
while G3(M2)=N3. Using these equations and Eq. (9), we have performed a series of simulations to illustrate the effects of the different disorder parameters and measurement conditions on the GISAXS intensity distributions. The constant parameters used for the modelling are given in Table 3, while the tuned parameter is shown in the corresponding simulation.
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 8. Model 2: Uncorrelated multiple ion tracks. It describes the arrangement of NOs formed within multiple tracks after single ion passage. Each track consists of N2 parallel chains (in this case N2=2) of NOs, and each chain is shifted by the vector a2 from the previous one. Each chain consists of N1 NOs with the periodicity given by the vector a1. The origin of each track is assumed random. The deviation parameter 2x,y describes the fluctuation of the entire chain from the ideal position given by the basis vectors a1 and a2. The deviation parameters describing the disorder within each chain are the same as in Model 1.
The dependence of the GISAXS intensity distributions calculated with this model on the angle is shown in Fig. 9. For =0, characteristic lateral maxima appear that are the consequence of the existence of multiple tracks. Their distance is inversely related to the distance between the chains in a multiple track. These characteristic maxima disappear for larger values of the angle . Table 3. Parameters of the track properties used for the modelling of GISAXS intensity distributions with Model 2. The exceptions are the tuned parameters which are indicated in the GISAXS simulations. All values except N1, N2 (dimensionless) are given in nm. par.
|a1|
|a2|
N1
N2
x
y
x
y
z
z
Rx,y
Rz
R
20
15
10
2
2.5
1.0
1.0
1.0
0.1
0.1
1.5
2.0
1.1
13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 9. Dependence of GISAXS intensity distributions from the multiple ion tracks on the angle between the probing beam and the ion track direction.
The influence of the disorder parameter 2y (deviations of the chain y position within the track) is demonstrated in Fig. 10. As expected, the characteristic sheets visible for =0 deg, broaden and decrease in intensity with increasing disorder. A similar effect, but much less prominent, is visible for =5 deg.
Figure 10. Dependence of the GISAXS intensity distributions on the disorder parameter 2y for the model describing multiple ion tracks. The cases =0 deg and =5 deg are shown.
14
Finally, we test the influence of the disorder existing in the chain x-positions on the GISAXS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
maps. As we mentioned above the chains can fluctuate around the origin of the track positions. This type of disorder is described by the parameter 2x, and its effect on the GISAXS intensity maps is shown in Fig. 11. This parameter has no influence on GISAXS for =0, however it has an influence for all other values of the angle . In Fig. 11 are shown two cases differing by the number of parallel NO chains within the track (N2).
Figure 11. Dependence of GISAXS intensity distributions on 2x. Two different numbers of parallel NO chains within the track (N2=2 and 8) are used for the simulations.
2.2.4. Example 2: Uncorrelated multiple ion tracks This example demonstrates the analysis of GISAXS maps originating from double tracks produced by low-fluence irradiation of Ge+ITO film with 92 MeV Xe ions [18]. The AFM images of the tracks are visible in Fig. 12(a),(b). Two parallel chains of NOs exist in the each track.
15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 12. (a),(b) AFM images of the ion tracks consisting two parallel chains of NOs formed in the irradiated Ge+ITO film [18]. The maximum of the height scale is 9 nm. Measured GISAXS maps of the tracks and corresponding simulations with: =0 deg (c),(d) and 5 deg (e),(f).
In the GISAXS map of the irradiated Ge+ITO surface, shown in Fig. 12(c) (probing angle =0 deg), two lateral maxima are well visible close to Qy=0.36 nm-1. They are due to the double track formation. They also exist in the simulation of the experimental measurement obtained by fitting, shown in Fig. 12(d).These maxima disappear already for =5 deg (Fig. 12 (e)) in accordance with the simulations from Fig. 9. All parameters obtained by numerical analysis of the GISAXS measurements are shown in Table 4. We see that the parameters obtained by GISAXS agree well with the AFM measurements. However, GISAXS provides significantly more parameters. The deviation parameters are in qualitative agreement with the AFM images and with the data obtained for the previous example. The disorder parameter y (describing the fluctuation of NOs within the chains) is larger than for the Example 1, which is obvious if we compare the AFM images. It is interesting to note that the vertical deviation parameter z, describing the fluctuation of the entire chain within the track, is larger than z which is related to the fluctuation of the individual NOs within each chain. Table 4. Parameters of the track properties obtained by modelling of the experimental GISAXS intensity shown in Fig. 12. The parameters obtained from the AFM image are shown for the comparison. par
|a1|
|a2|
N1
N2
x
y
x
y
16
z
z
Rx
Ry
Rz
R
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
GISAXS
26
12
22
2
12
2.5
1.8
2.6
0.10
2.0
2.9
2.5
2.8
1.1
AFM
28±5
14±1
20±8
2
-
-
-
-
-
-
-
-
3±1
-
2.2.5. Model 3: Correlated ion tracks This model describes the arrangement of NOs formed after irradiation of surface with higher ion fluences leading to an overlapping of the ion tracks. The overlapping in some cases causes the self-assembly of the surface material, so the ion tracks have some characteristic distances. The formation of single ion tracks is assumed for this model. The arrangement of the NOs for this model is illustrated in Fig. 13. The NOs are arranged within single tracks (with period a1) as for the Models 1 and 2, while the periodicity of the tracks in y direction is given by the vector a2, as for the previous model. We assume that there are N2 such tracks in the y direction. Finally, we assume that the single ion tracks make long chains along the x direction, consisting of single tracks repeating with the vector a3=N1 a1 (set along the x direction). So, the entire surface is covered by ion tracks (see the AFM image and the model shown in Fig. 13). There are N3 tracks in the x direction, so the total length of one track chain is |a3|N3. The disorder related to the basis vector a1 is the same as for the previous models. The disorders of the track positions related to the vector a2 (2x and 2y) both show SRO for this model, while the corresponding x-related deviations had LRO for the Model 2. The reason is that there are no predefined origins of the tracks for this Model. The deviations from the periodicity a3 along the x and y directions are described by the parameters 3x and 3y, respectively. In real systems these parameters are usually large because there are no physical reasons for a strong correlation along this basis vector. However it is possible that some external parameters like steps on the surface or surface patterning cause such behaviour. 1-3z all are assumed to have LRO due to the assumption that the tracks are formed on a flat surface.
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 13. Model 3: Correlated ion tracks. The model describes the arrangement of NOs within ion tracks that have correlated distances, for example due to some kind of self-assembly process. The NOs with the period a1 are arranged in single ion tracks along the x axis. The single ion tracks have the periodicity a2 in y direction and a3in the x direction.
So, the positions of the NOs on the surface are described by:
(17) (18) (19)
where
,
,
.
These equations are used for the calculation of GISAXS simulations by Eq. (9). The constant parameters used for the simulations made with this model are given in Table 5, while the parameter that is tuned is shown in the corresponding simulation. The angle dependence of the GISAXS intensity distributions from the correlated ion tracks is given in Fig. 14. Their GISAXS maps show lateral peaks for =0 deg, similar to the ones in Model 2. However that is not the case for the random tracks shown in Fig. 3. The main difference between Models 2 and 3 is in the assumed ordering of the chains in the multiple
18
tracks: Model 2 assumes LRO, while Model 3 assumes SRO. The maps of higher angles 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
also differ from the uncorrelated ion tracks. The main reason is that relatively small disorder parameters 2,3x,y are taken for the simulations. For that case, the chains of the tracks are relatively long and well-ordered, so the characteristic tails are much narrower than for the uncorrelated tracks. Table 5. Parameters of the track properties used for modelling of GISAXS intensity distributions. All values except N1 – N3 (dimensionless) are given in nm. par
|a1|
|a2| N1
N2
20
15
100 5
10
N3
x
y
x
y
x
y
1.0
0.1
1.0
1.5
12.5 2.5
z
Rx,y
Rz
R
0.5
1.5
2.0
1.1
Figure 14. The dependence of GISAXS intensity distributions on the angle between the X-ray probing beam and the track direction is shown for the correlated ion tracks. The simulations are obtained using Eqs. (9), (18) and (19).
The dependence of the GISAXS on parameter 2y is demonstrated in Fig. 15. This parameter describes the disorder in the separation of ion tracks. When increasing this parameter, the characteristic lateral sheets for =0 deg broaden and decrease in intensity. A similar effect was observed for Model 2. For the =5 deg angle the characteristic tail also broadens, but the effect is significantly less pronounced, as is visible for the angle =0 deg.
19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Figure 15. Influence of 2y on the GISAXS intensity distributions of correlated ion tracks. The dependences for =0 deg and 5 deg are shown.
The parameter 3x describing the fluctuation of the track-chains origins in x direction is actually not very physical, but can be important in some cases. The effect of this parameter on the GISAXS intensity distributions is shown in Fig. 16. From the simulations, it follows that the parameter has a significant effect only on small-period oscillations.
Figure 16. Influence of 3x on the GISAXS intensity distributions for the angles =30 deg and 90 deg.
20
Finally, the parameter 3y describes the disorder of the single tracks within the track chain in y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
direction. It has a small effect on the GISAXS intensity distributions for =0 deg. However, its effect is significant for the angle =5 deg. The characteristic tail visible for =5 deg is very sensitive to the length and width of the ion track. Thus, if 3y is small, then the single ion tracks actually make a long chain (they are repeating with the period a3, see Fig. 8) causing the tail to be strong and narrow. We had a similar behaviour for a large number N1 of NOs in a single track (Fig. 6). Increasing the disorder, the individual tracks are moved left or right from the ‘ideal’ centre position, so the chain is not continuous and regular any more.
Figure 17. Influence of 3y on the GISAXS intensity distributions for =0 deg and 5 deg.
2.2.6. Example 3: Correlated ion tracks Herein we apply Model 3 in order to determinate the structural parameters of correlated ion tracks formed by irradiation. An example showing the AFM images of such a system obtained by irradiation of a TiO2 surface with 23 MeV I6+ ions (900 ion tracks/m2), [18] is demonstrated in Figs. 18(a),(b). A characteristic distance of about 25 nm between the ion tracks is found from the analysis of these AFM images. These correlations are manifested in the measured GISAXS map (Fig. 18(c)) as two broad sheets (indicated by arrows in Fig. 18(c)). The results of the GISAXS analysis for this case are given in Table 6. We see that the parameters obtained by the numerical analysis are again in good agreement with the AFM results. Additionally, we get many more parameters than for the shown AFM measurements. 21
We could get some of them by careful analysis of many AFM images. However, some 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
parameters like the NO lateral sizes or their sizes if they are buried below the surface are accessible only by GISAXS.
Figure 18. (a),(b) AFM image of aTiO2 surface irradiated by 23 MeV I6+ ions, 900 ion tracks/m2 [18]. The maximum of the scale is 4.6 nm. (c) Measured GISAXS map and (d) corresponding simulation for =0 deg. (e), (f) Measured and simulated GISAXS maps for =5 deg.
Table 6. Parameters of the track properties for Example 3 obtained by the analysis of the measured GISAXS maps. The parameters found from the AFM images are shown for the comparison. All values except N1 – N3 (dimensionless) are given in nm. par
|a1|
|a2|
N1
N2
N3
x
y
x
y
x y z
Rx,y
Rz
Rcore
R
|d|
GISAXS
29
22
6
100
5
10
1.7
15
16
6
4
0.5
2.3
1.7
1.5
1.0
0.6
AFM
-
24±2
-
-
-
-
-
-
-
-
-
-
-
2.2±0.2
-
-
-
3. Discussion, reliability, limitations Using the models given above it is possible to get the structural parameters of the NOs formed by ion beam irradiation within the ion tracks. However, it is important to note, that the fitting procedure should be performed with care because many parameters are used in the fit. Due to
22
that fact it is possible to get wrong parameter values and still a relatively good fit. To avoid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
such situation, it is very useful to have some ‘initial’ parameters for the fit set (estimate) the initial parameter values well, and to constrain the parameter values during the fit within to some reasonable limits. It is also necessary to know at least roughly which structure is expected, including the arrangement type and maybe the shape of the NOs, and their sizes. For the same reasons, it is recommended to perform additional microscopy measurements for at least one sample from each series to check the fitting results. In some cases (like for the formation of Ge nanoparticles in some matrix), the atomic composition determination can help checking the obtained parameters. Namely, the percentage of Ge can be estimated from the size and arrangement properties obtained by fitting and compared with the measured value. To our experience a non-correct value of one parameter always appears in combination with non-correct value of some other parameter. Also, we are working on an open-source application that will use a number of optimization algorithms in order to optimally fit the model(s), and the fit stability will be a part of our future research [20]. Finally, it should be mentioned that the central part of the GISAXS map close to Qy=0 cannot be simulated accurately due to the model limitations. Namely, the contributions originating from the shape of the entire assumed NO lattice (which is defined by basis vectors a1-a3), are significant for small Qy values, and they are not present in reality. 4. Conclusions We have investigated GISAXS intensity distributions from three different types of ion tracks formed on the material surfaces. The theoretical models for their description have been presented and a series of simulations are demonstrated showing the effects of different disorder parameters on GISAXS intensity distributions. Each model is accompanied by an experimental example that demonstrates the efficiency of the developed models. We believe that the models will be useful for the successful characterization of different ion tracks by the GISAXS technique.
Acknowledgements The authors acknowledge J. Erceg for the assistance in the sample preparation. MB, NR acknowledge the Croatian Science Foundation (pr. No. 2334). Support by the Croatian Centre of Excellence for Advanced Materials and Sensing Devices and Calipso for GISAXS
23
measurements (pr. No. 20145403) are acknowledged. The assistance of H. Lebius and B. Ban 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
d’Etat during the ion irradiation in GANIL, as well as assistance of M. Schleberger with the AFM measurements, is gratefully acknowledged. References [1] Akcöltekin E., Peters T., Meyer R., Duvenbeck A., Klusmann M., Monnet I., Lebius H., Schleberger M. (2007) Nat. Nanotech. 2 290-294 [2] Kluth P., Schnohr C. S., Pakarinen O.H., Djurabekova F., Sprouster D. J., Giulian R., Ridgway M. C., Byrne A. P., Trautmann C., Cookson D. J., Nordlund K., Toulemonde M. (2008) Phys. Rev. Lett. 101 175503 [3] Karlušić M., Akcöltekin S., Osmani O., Monnet I., Lebius H., Jakšić M., Schleberger M. (2010) New J. Phys. 12 043009 [4] Aumayr F., Facsko S., El-Said A. S., Trautmann C., Schleberger M. (2011) J. Phys.: Condens. Matter 23 393001 [5] Akcöltekin S., Bukowska H., Peters T., Osmani O., Monnet I., Alzaher I., Ban-d’Etat B., Lebius H., Schleberger M. (2011) Apl. Phys. Lett. 98 103103 [6] Ridgway M. C., Bierschenk T., Giulian R., Afra B., Rodriguez M.D., Araujo L. L., Byrne A.P., Kirby N., Pakarinen O.H., Djurabekova F., Nordlund K., Schleberger M., Osmani O., Medvedev N., Rethfeld B., Kluth P. (2013) Phys. Rev. Lett. 110 245502 [7] Ochedowski O., Osmani O., Schade M., Bussmann B. K., Ban-d’Etat B., Lebius H., Schleberger M. (2014) Nat. Comm. 5 3913 [8] Karlušić M., Kozubek R., Lebius H., Ban-d’Etat B., Wilhelm R. A., Buljan M., Siketić Z., Scholz F., Meisch T., Jakšić M., Bernstorff S., Schleberger M., Šantić B. (2015) J. Phys. D: Appl. Phys. 48 325304 [9] Ochedowski O., Lehtinen O., Kaiser U., Turchanin A., Ban-d`Etat B., Lebius H., Karlušić M., Jakšić M., Schleberger M. (2015) Nanotechnology 26 465302 [10] Papaléo R. M., Thomaz R., Gutierres L. I., de Menezes V. M., Severin D., Trautmann C., Tramontina D., Bringa E. M., Grande P. L. (2015) Phys. Rev. Lett. 114 118302
24
[11] Ridgway M.C., Giulian R., Sprouster D.J., Kluth P., Araujo L.L., Llewellyn D.J., Byrne 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
A.P., Kremer F., Fichtner P.F.P., Rizza G., Amekura H., Toulemonde M., (2011) Phys. Rev. Lett. 106 095505 [12] Ridgway M.C., Djurabekova F., Nordlund K., (2015) Curr. Opin. Solid State Mat. Sci. 19 29 [13] Buljan M., Bogdanović-Radović I., Karlušić M., Desnica U. V., Dražić G., Radić N., Dubček P., Salamon K., Bernstorff S., Holý V. (2009) Appl. Phys. Lett. 95 063104 [14] Bogdanović-Radović I., Buljan M., Karlušić M., Skukan N., Božičević I., Jakšić M., Radić N., Dražić G., Bernstorff S. (2010) Phys. Rev. B 86 165316 [15] Renaud, J., Lazzari, R. & Leroy, F. (2009). Surf. Sci. Rep. 64, 255–380. [16] Babonneau, D. (2010). J. Appl. Cryst. 43, 929–936. [17] Karlušić M., Buljan M., Lebius H., Ban d’Etat B., Šantić B., Bogdanović-Radović I., Radić. N., Jakšić M., Bernstorff S., Schleberger M., (2016), in preparation [18] Karlušić M., Bernstorff S., Siketić Z., Šantić B., Bogdanović-Radović I., Jakšić M., Schleberger M., Buljan M., (2016), Formation of swift heavy ion tracks on rutile TiO2 (001) surface J. Appl. Cryst., under review [19] M. Buljan, N. Radić, S. Bernstorff, G. Dražić, I. Bogdanović-Radović, V. Holý, Acta Cryst. A 2012, 68, 124-138. [20] Mekterović I., Buljan M., GisaxStudio, http://homer.zpr.fer.hr/gisaxstudio, last accessed: 2016-05-31
25