Mathematical model depicting the deposition kinetics process into rf-magnetron co- sputtering of strontium barium titanate thin films

a Q&P Consulting, Justo Sierra S/N, Col. Santa Bárbara, Azcapotzalco, Z.P. 02230, CDMX México. b Higher School of Mechanical and Electrical Engineering (ESIME-IPN-AZ), Azcapotzalco Unit, National Polytechnic Institute, Av. Granjas, Mexico City, Z.P.02250, México. c National Polytechnic Institute, CICATA-IPN Legaria Unit, Legaria 694 Avenue, Col. Irrigación, Miguel Hidalgo, Z.P. 11500 Mexico City, México. d Autonomous University of Guerrero, Faculty of Agricultural and Environmental Sciences. Master in Agricultural Sciences and Local Management. Tuxpan Unit, km 2.5 Carretera Iguala-Tuxpan, Iguala de la Independencia, Guerrero, México. Z.P.40101.


Introduction
Recent research has shown that in the BST thin films deposition process by means of RFMCS when complementary and simultaneous applied powers used in two magnetrons, Ba and Sr are depositions on the substrate following a mathematical sigmoidal form. That is, the concentration follows a sigmoidal S-profile equation as a function of the complementary applied power on nonstoichiometric targets of BaTiO 3 (BTO) and SrTiO 3 (STO) [1,2,3]. This sigmodal mathematical model and the fit with experimental data measured by EDS and XRD has been used to explain the chemical structure and some properties such as: the gap energy, the deposition rate, the crystal size, and the resistivity of the BST on quartz substrates for different temperatures. Also, the relationship with the amorphous or crystalline structure and with deposition areas rich in some of the cations of the BST solid solution (Ba-rich or Sr-rich) [1][2][3][4][5][6][7]. In other types of research, the Ba/Sr, (Ba+Sr)/Ti, Ba/Ti, Sr/Ti ratios have been scrupulously studied in relation to properties such as: electrical, dielectric, and optical. The way in which the change in the flow of gases (Ar/O 2 , O 2 /(Ar+O 2 ), (Ar+O 2 ) (and its relationship with the working pressure or total chamber gas pressure, or the pressure partial oxygen) in the RFMCS chamber influences physical parameters. The properties influence crystallinity, crystalline structure, oxygen vacancies, the deposition rate, the thickness, and the final stoichiometric values with the stoichiometric sum of Ba, Sr, Ti and O in the solid solution that makes up Ba x Sr 1-x TiO 3 [8][9][10][11][12][13]. This research work establishes a simple way of understanding the RFMCS depositions by means of sigmoidal profile between Ba and Sr, and the arithmetic relationship of the perovskites (ABO 3 : 1:1:3) that interact to establish mathematical models fitted in the relationships to experimental data of Ba/Sr, (Ba+Sr)/Ti, Ba/Ti, Sr/Ti. These mathematical expressions as a whole establish unpublished valid hypotheses in the interpretation and understanding of the deposition kinetics that include the deposition of Ti and O 2 in the RFMCS. The values of the experimental discrete points are obtained of [14] see Table II.

Materials, methods, and samples preparation
Every target was subjected during 15 minutes of a pre-sputtering process. When 120 W was applied to the STO target and simultaneously 0 W was applied to the BTO target then STO films (x=0) were obtained. The inverse procedure was used to obtain BTO thin films (x=1). To get values in the 0 ≤ x ≤ 1 interval for the BST films 15 W of difference was applied on STO and BTO targets in steps in a complementary form, always keeping the total RF-magnetron applied power to both targets at 120W. See Table 2.

Mathematical Model
Since the values following the Boltzmann sigmoidal modified function, Equation (1) is proposed: The first and second derivatives of Equation (1) is expressed by the following Equations (2 and 3). (2) The first derivative has a maximum in the:

Results and discussion
In Figure 1 for simple mathematical analysis and a broader understanding, it is convenient to establish the stoichiometric chemical relationship of a perovskite denoted by ABO 3 , where A and B are the cations and oxygen is the anion. For our work, the cation A is formed by the Ba+Sr total atoms and cation B is the totality of Ti atoms, which is an ideal perovskite maintaining the following stoichiometric relationship with oxygen: ABO 3 =1:1:3 or 20:20:60 expressed as a percentage. Then, from the mathematical point of view, this stoichiometric relationship is explained by two sigmoidal equations and the addition of two lines with slope equal to zero, to understand the experimental design.
Then it establishes that Ba+Sr = 20%, Ti=20%, and O 2 =60%, therefore; where P 0 = 45W and α = 18.05 for calculating 20%Ba and P 0 = 75W y α = 18.05 for calculating 20%Sr, =RF applied power on magnetron in every experimental discrete point. According to the experimental mathematical model, the straight dashes indicate the limits or lines with zero slope where the BST elements should converge for the sum of Ba+Sr, Ti, and oxygen, respectively, to obtain the BST 100 . Note that in Figure 1, the increases and decreases of the relationships between the cations and anions of the experimental discrete points are indicated by asymmetric rectangles that indicate the variation between their deposition rates. These reasons are explained in relevant points and trends in detail, and which are explained in a general way from theory and what was found by other researchers, starting from the description of Figure 1 in combination with Figures 2, 3, and 4 of this manuscript. The Applied Power Combinations (APC) 0/120W (STO) and 120/0 (BTO) are analyzed below. It can be seen in Figure 1 that the applied power rates have a gap between the experimental values of the cations in relation to the anion, which is greater in the first applied power rate than in the second. In Figure 2b, the cations Ba and Sr for both applied powers remain below the expected percentage, but with very similar values in Figure 2c. Sr maintains in better equilibrium with Ti than Ba; in fact, the imbalance between Ba and Ti is the most pronounced with a value of 3.13% lower than expected. In Figures 2(e) and 2(f) it is expected that APC1 and APC9 coincide with zero because only Sr and Ba are deposited respectively. However, as can be seen, Sr has a better control in the deposition for all APC than the Ba. In Figure 3, the Sr/Ti rate is very close to the ideal value, and the Ba/Ti rate is the furthest. Furthermore, the (Ba+Ti)/O rate is closer to the ideal value of the applied mathematical model. In Figure 4, APC1 and APC9 are not shown for convenience of explanation of the BST. The cations and oxygen decrease and increase respectively from APC2 to APC4, which according to the experimental data and the S-profile of the BST deposition with a higher content of Sr reveal according to Figure 2a and 2d, to oxygen and cations above and below the expected percentage respectively. The (A+B)/O rate also remains below the ideal percentage expected according to Figure 3. The Ba/Sr rate in Figure 4 remains at very small values close to unity. APC 4, according to Figure 1 is the inflection point. The Ba/Sr=1 rate should take place and the value of Sr is the closest to 50% of the total percentage of cation A and where the gap between cations and anion is the largest. According to Figure 2a, oxygen has its maximum deviation value above the model and at the same time according to Figure 2b, the minimum value of Ba+Sr below the model. Furthermore, it is observed in Figure 2c that the difference ratio between the cations remains below the model from that point and is conserved in the other APC. According to Figure 2e the minimum value below the S-profile is the percentage of Ba and Sr very close to the expected value according to Figure 2f. According to Figure 3 the rate (Ba+Sr)/Ti obtains the minimum value below the mathematical model. From APC4 to APC8, the ratio between cations and oxygen tends to rise. This is the area where the substitution of Ba and Ti become predominant in the BST (see Figures 2d, 2e, and 3) while the Sr tends to stabilize with respect to the S-profile. In Figure 2a, the oxygen content begins to drop and reaches its lowest point, right at APC8, where the BST ends its deposition. In Figures 2c and 3, it is observed that the rate (Ba+Sr)/Ti remains below the expected value.

General explanation with the help of the mathematical model
This section explains in a technical way the whole trends previously described. The combination of the total working pressure, the applied power on two magnetrons, and the sputtering yield of the Ar + ions with the Ar/O 2 =90/10 rate of our experiment is enough to achieve that the oxygen is integrated into the film as it happens in the APC where the applied power is greater in the STO target. This also obtains the closest values of the cations A and B of the ideal perovskite. Furthermore, it does not work where the APC are applied with the greatest applied power to the BTO target, where the values of cations A and B and oxygen are more dispersed than the ideal values. When the atomic mass of the target is greater, an Ar + ion hits the target with greater energy, so in the case of the STO and BTO targets the atomic masses are 183.490 g / mol and 233.192 g / mol, respectively. Therefore, as is expected Ar + ions impact the BTO target with greater kinetic energy. The differences found in the percentages of cations and oxygen in the change between APC, may be due to the fact, that there are greater dissociations of the covalent bonds between Ti and oxygen by means of the impact of more energetic ions where APC is greater on the applied power on BTO target following the principle, XY + e → (XY -). Under low-pressure conditions, the above process is often followed by dissociation of the excited complex: (XY -) → X + Y -. This dissociative attachment process is accepted as the major formation mechanism of negative ions in the low-pressure plasma, producing greater erosion and target surface oxidation [15]. Negative oxygen ions are generated when highly oxidative targets are bombarded. This intense sputtering yield of ions is considered the result of highly oxidative targets that are generated in the cathode sheath and plasma circumcising edges. [16,17]. Therefore, the formation of Oions that collide mostly with the Ar + ions, preventing their arrival to the substrate, is very likely. In addition, the Oions increase their kinetic energy when the applied power on target increases, thus allowing more collisions with ions Ar + [18]. In contrast, Ti ions and molecules with positive dipole moment travel without any restriction to the substrate for film deposition to take place. The collisions are between negative and positive O ions at the cathode sheath threshold and the plasma boundary. At the same time, inelastic collisions are likely in APC with higher applied power on BTO target, allowing the implantation of Ar ions and, on the contrary, inelastic collisions are greater in APC when the applied power increases on the STO target, allowing physical momentum transmission. 6 The proposed mathematical model develops the following expressions:  verified in Equation 6. It is well known that when the applied power increases on the target of BTO, and decreases on the STO target, after an inflection point in the APC, the sputtering yield of Ba increases due to the difference in atomic masses in the system (Ba =137.33u, Sr = 87.62 u, Ti = 47.867 u, O 2 = 15.999 u). Hence it follows that the Ba/Sr rate has this trend. Figure 5. The mathematical model proposed develops the following expressions: respectively. These results are very consistent between the proposed model, and the stoichiometric measurements method were performed with XRD reported in [2]. Figure 8, it can be noted that the profile that is drawn for x' vs x, that is, the rate of change in the content of x in the BST, is parabolic, and its minimum value coincides with the deposition of STO and BTO, where the sputtering yield is maximum. In Figure 9, an asymmetric sinusoidal profile is drawn if one starts from the point where the ratio Ba/Sr=1 and that reflects the asymmetry of the APC proposed for the experiment and which can show the speed-up rate of ions and molecules towards the substrate.

Conclusions
The mathematical model proposed to describe the kinetics of the BST/nichrome deposition of the interval 0 ≤ x ≤ 1 in Ba and Sr, establishes that there is a direct relationship between the content of cations and oxygen as a function of the RFMCS applied power. The knowledge of the exponential profiles of the Ba/Sr and sigmoid rate of Ba/Ti and Sr/Ti, and the described relationship of the kinetics of ions and molecules can contribute to establishing the setup of working conditions in a complementary way to other mathematical models that start from what happens in the RFMCS. This does not take into account what happens in the data obtained from the content of the cations and anions of the perovskite in the substrate. Among these conditions, the Ar/O 2 rate plays an important role to control, in a simple but effective way, the content of cations A and B, and of the oxygen anion in perovskite and perhaps in general in quaternary materials where the target is highly oxidative.