An Examination of Monte-Carlo Implantation Profile Simulations and Their

Use in β-NMR

Submitted By: Amanda O’Halloran Program: BSc in Physics, Co-operative Education

Last Academic Semester Completed: 5 For: TRIUMF β-NMR Group

August 2006

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Table of Contents

Cover Page ........................................................................................................................... i Letter of Submission ........................................................................................................... ii Title Page ........................................................................................................................... iii Table of Contents............................................................................................................... iv List of Tables and Figures....................................................................................................v 1.0 Summary ........................................................................................................................1 2.0 Introduction....................................................................................................................2 2.1 β-NMR ...............................................................................................................2 2.2 TRIM.SP ............................................................................................................4 2.3 SRIM..................................................................................................................6 3.0 Method ...........................................................................................................................7 4.0 Analysis and Discussion ................................................................................................8 4.1 Comparison of Simulations with Data...............................................................8 4.1.1 TRIM.SP .............................................................................................9 4.1.2 SRIM.................................................................................................11 4.1.3 β-NMR Spectroscopy Data ...............................................................14 4.2 Identifying the Differences between TRIM.SP and SRIM..............................18 4.3 Studying the Effect of the Different Properties of Silver on Predicted Range Distributions.....................................................................................................21 4.4 Searching for Periodic Trends .........................................................................24 4.5 Exploring Layer Effects...................................................................................27 5.0 Conclusions..................................................................................................................30 6.0 References....................................................................................................................31

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List of Tables and Figures Figure 1: Setup for β-NMR Resonance Test .......................................................................3 Figure 2: TRIM.SP GUI ......................................................................................................5 Figure 3: SRIM GUI ............................................................................................................7 Figure 4: Stopping Distribution for 18 keV 8Li+ in Pt/MgO Film in TRIM.SP ..................9 Figure 5: Energy Dependence of 8Li+ Stopping Position from TRIM.SP .........................10 Figure 6: Stopping Distribution for 18 keV 8Li+ in Pt/MgO Film in SRIM ......................11 Figure 7: Energy Dependence of 8Li+ Stopping Position from SRIM...............................13 Figure 8: Comparison of SRIM and TRIM Predictions.....................................................13 Figure 9: Resonance Spectrum from β-NMR of 8Li+ into Pt/MgO Film at 28 keV ..........14 Figure 10: Energy Dependence of 8Li+ Stopping Position from β-NMR Data..................16 Figure 11: Comparison of TRIM.SP Energy Curves with Data ........................................17 Figure 12: Comparison of SRIM Energy Curves with Data..............................................18 Figure 13: Energy Dependence of 8Li+ Stopping Position in Ag/MgO Film ....................19 Figure 14: Energy Dependence of 8Li+ Stopping Position in Al/MgO Film .....................20 Figure 15: Energy Dependence of 8Li+ Stopping Position in Be/MgO Film.....................21 Figure 16: Stopping Distributions for Modified Silver Simulations .................................23 Figure 17: Dependence of Stopping Range on Atomic Density in Silver Simulations .....23 Figure 18: Dependence of Backscattered Component on Mass in Silver Simulations......24 Figure 19: Stopping Distributions for Elements from Row 4 of the Periodic Table .........25 Figure 20: Stopping Distributions for Elements from Group 1 of the Periodic Table ......26 Figure 21: Backscattered Components for Elements from Row 4 of the Periodic Table..26 Figure 22: Range Distribution for Au/Ag/Be/GaAs Film..................................................27 Figure 23: Range Distribution for Au/Ag/Be/MgO Film ..................................................28 Figure 24: Range Distribution for Au/Ag/B/Ag Film........................................................29 Table 1: Stopping Ranges for 8Li+ in Pt/MgO Film from TRIM.SP .................................10 Table 2: Stopping Ranges for 8Li+ in Pt/MgO Film from SRIM.......................................12 Table 3: Amplitudes of Pt and MgO Resonances from β-NMR........................................15 Table 4: T1 Values for Pt and MgO ...................................................................................16 Table 5: Data from Modified Silver Simulations ..............................................................22 Table 6: Backscattered Components of GaAs and MgO...................................................28

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1.0 Summary The implantation of 8Li ions into a sample of platinum on a magnesium oxide substrate was simulated using two different Monte-Carlo stopping range generators, SRIM and TRIM.SP. The results of these simulations were then compared to data taken on the Pt/MgO film using β-NMR spectroscopy. From this it was determined that the SRIM program seems to replicate the stopping behaviour of the 8Li+ probes used in β-NMR more accurately than TRIM.SP. After this conclusion was reached, a further investigation was taken into how the two simulations differ. It was determined that the primary difference between SRIM and TRIM.SP is their handling of backscattering. Next, several non-physical simulations were performed by SRIM in an attempt to reveal how individual properties of the sample each effect the stopping range distribution. Specifically, the changes in implantation depth and fraction of 8Li+ backscattered were tracked as the mass, atomic density, mass density of the virtual sample were modified. When the atomic number is held constant, the primary determinant of stopping range appears to be atomic density, while the mass of the target predicts the component of incident ions backscattered. A study of the fourth row of the periodic table was completed, in order to determine how the electronic stopping power varies as atomic number increases. The general trend uncovered was that as electron number increases across the periodic table, stopping power is increased. This trend was not found to apply as one moves down a column of the periodic table however. Finally, the effect of layering different materials in a sample film was examined. The treatment of backscattering by the simulations yielded unpredictable stopping distributions.

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2.0 Introduction 2.1 β-NMR In the years since its discovery, spectroscopic techniques using nuclear magnetic resonance have proven to be such powerful analysis tools that the acronym ‘NMR’ has become familiar to scientists in almost every field. Conventionally, the technique uses host nuclei, that is, nuclei that are already present in a sample, as probes of their own local electronic and magnetic environment. The ability to do this arises as a consequence of nuclei possessing spin (I), and therefore different spin states (mI) which consist of the values -I, -I+1, …, I. This spin gives rise to a nuclear magnetic moment which depends on the spin state of the nucleus (mI), its gyromagnetic ratio (γ) and Planck’s constant (ħ) according to the equation:

µz = ħγmI When placed in a magnetic field, the energy of a nuclear magnetic moment is given by:

E = -µzB = ħγmIB

There are 2I+1 possible values for mI so accordingly, in a non-zero magnetic field, the energy of the nucleus is split into 2I+1 Zeeman levels. The energy difference between these levels is:

∆E = ħγB Consequently, a photon of energy ħω will induce spin transitions, where ω is known as the Larmor frequency, and is equal to γB. In NMR spectroscopy, a large magnetic field is applied to the sample in order to polarize a fraction of the host nuclei. An RF coil is then used to apply a varying transverse field. By monitoring at what field strength and frequency the spin transitions are induced, scientists can determine ω, γ and ultimately B, the local magnetic field at the nuclei. While this technique has repeatedly proved its worth it does have some drawbacks. First, it requires the application of a large external magnetic field (~10T) to polarize the host nuclei. It is difficult to achieve a high level of polarization in a sample and requires very low temperatures. A large sample is therefore required to ensure an adequate number of individual polarized nuclei to create a signal. As such, NMR is a bulk probe of matter and is limited to a resolution of between 1mm3 and 1µm3. Beta-detected nuclear magnetic resonance (β-NMR) addresses some of these problems arising from the difficulty in polarizing host nuclei by using implanted, polarized, radioactive nuclei as the probes instead. At TRIUMF, 8Li+ was chosen as the probe. After the polarized 8Li+ is in the sample, a varying transverse RF field is applied to induce spin transitions, like in conventional NMR.

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Another difference between β-NMR and conventional NMR is the method of detection. In conventional NMR, an RF coil measures the magnetization in the sample and therefore the polarization. In β-NMR it is the products of nuclear decay that are detected. 8Li+ decays through anisotropic beta decay with a mean lifetime of 1.21 s. Two detectors, one located behind the sample and one in front of it, monitor the decay pattern as the RF frequency is varied. When the RF frequency is equal to the Larmor frequency, the 8Li+ undergoes spin transitions and looses its polarization, which is recorded as a loss of asymmetry between the counts in the two detectors. In these experiments, a beam of 8Li+ is continuously implanted while a continuous RF pulse is swept over the frequency range of interest. A consequence of this method is any signal detected is a combination of signals from ‘new’ 8Li+, which is very highly polarized and ‘old’ 8Li+, which may have lost its polarization. Data gathered must be weighted according to the half life of 8Li+ and the T1 relaxation time of 8Li+, which is a measure of how long the 8Li+ in a sample remains polarized before returning to its equilibrium state.

Figure 1: Setup for β-NMR Resonance Test

The major benefit to using β-NMR is the massive decrease in scale. While most conventional NMR requires on the order of 1018 spins to generate a signal, β-NMR requires significantly less (on the order of 108, a difference of ten orders of magnitude). With this decrease, the study of much smaller samples becomes possible. Furthermore, as β-NMR techniques use implanted probes, the magnetic and electronic environment studied can be chosen by controlling the placement of the 8Li+. This is done by placing the entire experimental apparatus at a higher potential, causing the positive radioactive ions to decrease in speed as they approach the sample. This allows the selective study of interfaces and surfaces on a nanometer scale.

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However, to fully exploit this benefit it becomes essential to have a complete understanding of where exactly the 8Li+ is in the sample. Several Monte-Carlo computer simulations for generating implantation profiles are already in existence, two of the more common being TRIM.SP and SRIM. Both simulations are variations of an original, now-outdated program known simply as TRIM, and as such were predicted to give identical stopping range distributions. The accuracy of these programs can be tested by comparison of the simulated distributions and distributions derived from data collected on a real sample using β-NMR techniques. 2.2 TRIM.SP Written in the early 1980’s by J.P. Biersack and W. Eckstein, TRIM.SP was the first Monte-Carlo computer simulation in the TRIM family to follow the paths and collisions of the recoil atoms as well as the projectiles. The acronym stands for ‘TRansport of Ions in Matter SPuttering’ as it was developed to deal primarily with sputtering and other surface effects. Like all other derivatives of TRIM, TRIM.SP uses the binomial collision approximation in its calculations. That is, it assumes that each individual collision can be treated simply as a two body problem, ignoring possible effects from nearby atoms. The justification for this assumption lies in the fact that the impact parameter, which is a measure of the distance at which the two particles can be considering to be interacting, is much smaller than the distance between individual atoms in the target. The impact parameter for each collision is determined randomly from between zero and a maximum value which is dependent on the atomic density of the sample. Between collisions, the projectile (or a recoil atom set in motion by a projectile atom) moves along a mean free path whose length is also dependent on the atomic density of the sample. This method was designed to ensure the maintenance of the proper atomic density in that it guarantees one possible target atom in each volume of inverse density. After the impact parameter has been determined, the scattering angle is calculated. TRIM.SP does this by using a formula referred to as “Biersack’s Magic Formula” or simply “magic”. The formula uses the randomly generated impact parameter, the screening length, and the geometry of the collision in centre of mass co-ordinates as dictated by the interaction potential function to produce a scattering angle. This angle is then used to calculate the energy transferred to the target atom. If that energy is greater than the binding energy, the target atom leaves its position and becomes a recoil atom. Both recoil atom and incident atom then move on to their next collision. Once the energy of a particle drops below a predetermined minimum, that particle is considered stopped and its final resting place recorded. Also, if at any point in time an atom leaves the target the program no longer follows its progression and simply counts it as backscattered or transmitted in the case of incident ions or sputtered in the case of target atoms. In reality the collisions between incident atoms and target atoms are not perfectly elastic. The atoms moving in a solid will lose energy due to interaction with the electrons in that solid. This energy loss can take many forms, including kinetic energy transfers to

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electrons, excitation or ionization of target atoms and excitation, ionization or electron capture by projectile atoms. Many different ways of calculating this energy loss have been proposed, including considering the momentum transfer between overlapping electronic distributions and treating the electrons in the target as a free electron gas. The method employed by TRIM.SP however is based on the concept of effective charge and the scaling of experimentally determined stopping powers of hydrogen in various targets. A function predicting the form of the electronic stopping powers of hydrogen at various implantation energies was derived. It depends on the energy and mass of the projectile, but also contains eight separate constants which are completely target dependent. These constants arise from fitting the function to the experimental data available for hydrogen into monatomic targets of elements 1 through 92. Thus 92 separate values for each constant exist. For non-hydrogen projectiles, the functions are scaled by a value dependent on the projectile velocity, effective charge, electronic density as well as the Fermi velocity, Bohr velocity and Bohr radius. The version of TRIM.SP used for this project was specialized for the implantation of 8Li+, as that is the probe used in β-NMR. Therefore, when setting up a simulation it was only necessary to specify the composition of the target. This was done by inputting each layer of the target stoichiometrically. During a simulation, the individual target atom for each collision is determined based on the stoichiometric composition of the layer and the electronic stopping power used is the linear superposition of the stopping powers of the constituent atoms. The densities of each layer and implantation energy are also required input parameters.

Figure 2: TRIM.SP GUI

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Designed to take as little computation time as necessary, TRIM.SP does not contain a lot of extra features. One extremely convenient option that is present however, is the option to setup and run multiple simulations consecutively. Frequently one is interested at the stopping range of a sample at multiple energies. In TRIM.SP it is possible to define the target and then specify a range of energies for the program to simulate. It is also possible to define individual layers and then set a range of layer thicknesses for consecutive simulation. This allows a number of separate tests to be performed without the necessity of reconstructing the sample between them. 2.3 SRIM The version used in this exercise, SRIM 2003, is the latest in a long line of ion implantation Monte-Carlo simulations beginning with TRIM85 in 1985. This family of programs is also based off TRIM, the original simulation that laid the grounds for TRIM.SP, but was developed in a slightly different manner. The code is credited to J. F. Ziegler and J.P. Biersack with contributions from many others. The acronym SRIM stands for ‘Stopping and Range of Ions in Matter’ and was adopted in 1991 to differentiate between the Monte-Carlo simulator program TRIM.exe and the package of programs that constitute SRIM. Like TRIM.SP, SRIM uses the binomial collision approximation and a randomly determined impact parameter to characterize collisions; however the formula for the maximum possible impact parameter is different, as is the formula for mean path length. The two new formulas are necessarily density dependent and consistent with maintaining the proper atomic density, however SRIM is programmed to select the smallest impact parameter which occurs over the path length, since that leads to maximum deflection. Once the impact parameter has been determined, SRIM also uses the ‘magic’ formula to calculate the scattering angle and the energy transferred. The ‘magic’ formula for SRIM uses slightly different numbers than the one for TRIM.SP as the two interaction potential functions are not identical. SRIM also uses an updated database of constants for the electronic stopping power calculations. One of the major differences between SRIM and TRIM.SP is the level of control one can exercise over aspects of the SRIM code. For instance, it is possible to select the ion for implantation, as well as set its mass. Target selection also includes the option to modify atomic masses, atomic densities, or binding energies as the user wishes. A compound dictionary is included with the software so that a target may be specified by its common name opposed to its detailed chemical composition. SRIM is also very user friendly. Once a simulation is running it is possible to monitor the passage of every individual ion in the sample to ensure that the calculation is progressing as desired. Real time statistics keep the user apprised of the current average ion range, straggling, transverse ion range, transverse straggling as well as the number of ions backscattered and transmitted. SRIM also closely monitors the damage to the target, that is, the number of vacancies/recoils created along with how their energy is dispersed.

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Figure 3: SRIM GUI

A variety of more advanced modes exist for SRIM as well. Sputtering, for instance, is one of the possibilities. In these, the details of each individual collision can be recorded, but the user is warned that this data file can get quite big quite quickly. These additions are not without a price however, as calculation time increases with difficulty. 3.0 Method The sample chosen to test the accuracy of the TRIM.SP and SRIM programs was 50 nm of platinum on a substrate of magnesium oxide (MgO, ρ = 3.58 g/cm3). Resonance spectra for this sample using β-NMR were already available at 8 different implantation energies: 1 keV, 3 keV, 8 keV, 10.5 keV, 13 keV, 18 keV, 23 keV and 18 keV. To acquire these spectra, the sample was placed in the β-NMR spectroscope under a static magnetic field of 4.1 T at a temperature of 295 K. A transverse RF pulse was then applied to the sample at 25820 kHz. The asymmetry between the forwards and backwards detector was recorded by the monitoring software and the frequency of the RF pulse was increased by 250 Hz. This process was repeated until all frequencies up to 25850 kHz had been irradiated. The scanning of frequencies was performed twice, once with the 8Li+ polarized towards the forwards detector and once with it polarized towards the backwards detector. Therefore, there were two separate helicities for each implantation energy.

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The T1 tests were performed by implanting 8Li+ into a sample and monitoring how the polarization decays as time progressed. Again, this was done twice, with the 8Li+ polarized first towards the forwards detector and then towards the backwards detector, to yield two curves. This sample was then simulated at all integral implantation energies from 1 keV to 28 keV in both TRIM.SP and SRIM. Samples of 50 nm of aluminum, beryllium and silver on an MgO substrate were then simulated at implantation energies from 1 keV to 28 keV in both TRIM.SP and SRIM. This was done to further explore the difference between the two programs. Next, a pure silver sample was simulated in SRIM at an implantation energy of 10 keV. The atomic density and mass of the silver were modified to determine their effect on the range distribution and fraction of incident ions backscattered. SRIM simulations were also performed at an implantation energy of 10 keV on a pure sample of each of the elements in the fourth row of the periodic table and a selected number of elements from the other rows. This was done to search for periodic trends in stopping ranges and backscattering projections. Finally, different multi-layered samples were chosen to be simulated based on the results from the search for periodic trends to explore how passing through different layers affected the behaviour of the simulated 8Li+. Based on this methodology, the project can be organized into five separate sections:

1) Comparison of simulations with data, 2) Identifying the differences between TRIM.SP and SRIM, 3) Studying the effect of the different properties of silver on predicted range

distributions, 4) Searching for periodic trends, and 5) Exploring layering effects.

4.0 Analysis and Discussion Analysis of data was completed after each separate section so that the results from the earlier sections could be used in the later sections. 4.1 Comparison of Simulations with Data Some manipulation was necessary to transform the data given by β-NMR spectroscopy, the SRIM simulation and the TRIM.SP simulation into forms that were comparable. The operations were mostly very repetitive and several short C scripts were written to handle those aspects of the conversions.

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4.1.1 TRIM.SP Data from TRIM.SP is outputted in two different forms; one file which contains data for creating a stopping range distribution, and another which summarizes the final position of each incident ion. While setting up the target before the simulation commences the user is required to set an increment width. During runtime the target is divided into bins, each of width equal to the increment set. If no increment is set, the default bin width is 30 Å. The output file contains how many projectile atoms are contained within each bin. As such, a graph constructed from the data contained within that file gives a non-normalized stopping range distribution. The reason the stopping distribution is considered non-normalized is that it is dependent on the number of incident ions. From these stopping distributions the mean range and standard deviation of the range were calculated, along with the root-mean-square range.

Figure 4: Stopping Distribution for 18 keV 8Li+ in Pt/MgO Film by TRIM.SP

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Table 1: Stopping Ranges for 8Li+ in Pt/MgO Film from TRIM.SP

Implantation Energy (keV)

Mean Range(Å)

RMS Range(Å)

Standard Deviation (Å)

1 49 57 29 3 114 131 65 8 265 307 155

10.5 353 412 213 13 462 540 279 18 718 821 397 23 1008 1127 504 28 1298 1432 606

To normalize the distribution, the contents of each bin were transformed from a number of ions stopped within that bin to what fraction of the total ions stopped within that bin. This was a fairly simple calculation, as the number of incident ions is set by the user. Once the data was normalized the fraction of ions stopped in each layer was calculated. The simulations were designed so that the thickness of the substrate prevented any transmission of 8Li+, but was thin enough so as to not contain many empty bins. A C script was written to count the number of ions stopped in bins within the first 50 nm (i.e. within the platinum) and convert that to a fraction of the total number of ions. The number of ions backscattered was read from the summary file and similarly converted to a fraction of the total. The fraction stopped within the magnesium oxide substrate was taken to be the difference between the total and the sum of the fractions in the platinum and backscattered. Once this had been completed for each simulated energy, the energy dependence of the stopping layer of the 8Li+ could be plotted.

Figure 5: Energy Dependence of 8Li+ Stopping Position from TRIM.SP

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4.1.2 SRIM Once the SRIM simulation is completed, the user is provided with several different options for output. The output selected for this experiment was ion distribution. Like TRIM.SP, SRIM divides the sample into bins; however it does not allow you to select the bin size. In SRIM, the sample is automatically divided into 100 equal bins. The importance of minimizing the thickness of the substrate becomes apparent now as a method of obtaining the smallest bins possible. Unlike TRIM.SP, SRIM does not give the number of ions stopped in each bin. Instead it gives a distribution (in atoms/Å-ion) which is intended to be multiplied by a dose rate (in atoms/cm2) to get the number of ions stopped in that bin.

Number of Ions Stopped = Distribution × Dose Rate

This is again, clearly, a non-normalized range distribution, as it depends on the dose rate. From this data however, it is possible to acquire the mean range, standard deviation of the range and root-mean-square range as before.

Figure 6: Stopping Distribution for 18 keV 8Li+ in Pt/MgO film in SRIM

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Table 2: Stopping Ranges for 8Li+ in Pt/MgO film from SRIM Implantation Energy

(keV) Mean Range

(Å) RMS Range

(Å) Standard Deviation

(Å) 1 56 64 30 3 113 128 61 8 230 263 127

10.5 290 335 167 13 360 419 216 18 525 612 316 23 718 825 407 28 937 1055 484

Normalization of the SRIM data is slightly more involved than for the data simulated by TRIM.SP. First the total distribution over the entire sample is calculated by adding the contributions from each bin. The total number of ions implanted in the sample is calculated based on the difference between how many ions were incident to the surface, a number set by the user, and how many ions were backscattered, which is provided in the output file by SRIM. From these numbers, the dose rate used is calculated using the following formula.

Number of Ions Stopped in Sample Dose Rate = Distribution Over Entire Sample Once the dose rate is known, the distribution of the backscattered atoms is easily calculated by rearranging the equation for number of ions stopped. The total distribution is then simply the sum of the distribution backscattered and the distribution found in the sample. This total can then be used to normalize the other distributions by transforming them into fractions of a total. Similar to the analysis performed on the TRIM.SP data, the energy dependence of the fraction of 8Li+ backscattered and in the platinum and magnesium oxide was graphed.

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Figure 7: Energy Dependence of 8Li+ Stopping Position from SRIM

This graph was then compared directly to the normalized distribution graph for the TRIM.SP energy dependence.

Figure 8: Comparison of TRIM.SP and SRIM Predictions

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The difference in the two predictions is noticeable. For all simulated energies, TRIM.SP predicts a higher fraction of total incident atoms to be backscattered, a higher fraction to be located within the magnesium oxide substrate and a lower fraction found within the platinum. These results had to be compared to the data taken on the Pt/MgO film using β-NMR spectroscopy in order to determine which better represented the physical behaviour of the 8Li+. 4.1.3 β-NMR Spectroscopy Data Resonance data was available for the platinum on magnesium oxide sample at eight different implantation energies: 1 keV, 3 keV, 8 keV, 10.5 keV, 13 keV, 18 keV, 23 keV and 28 keV. The two helicities, from the two separate polarizations of the 8Li, were combined to give an average asymmetry curve and to minimize any possible dependence on the equipment.

Figure 9: β-NMR Resonance Spectrum for Implantation of 8Li+ into Pt/MgO Film at 28 keV

The first peak in the resonance spectrum represents a signal from the 8Li+ in the platinum layer, while the second peak represents a signal from the 8Li+ in the magnesium oxide. The two peaks were fit to lorentzian functions of the form:

AMP × WID2y = 4(x – RES)2 + WID2 + BCK

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Where AMP is the amplitude of the peak, WID is the full width of the peak at half its maximum value, RES is the frequency location of the peak and BCK corrects for a nonzero baseline. This fitting was performed using bnmrfit, a physica script designed specifically for the fitting of β-NMR spectra. The amplitudes for each data set are tabulated below.

Table 3: Amplitudes of Pt and MgO Resonances from β-NMR Implantation Energy

(keV) Amplitude of Pt Fit Amplitude of MgO Fit

1 0.0537 ± 0.0012 0 3 0.0640 ± 0.0014 0 8 0.0658 ± 0.0012 0

10.5 0.0700 ± 0.0013 0.0094 ± 0.0016 13 0.0561 ± 0.0011 0.0247 ± 0.0015 18 0.0391 ± 0.0010 0.0591 ± 0.0024 23 0.0297 ± 0.0008 0.0706 ± 0.0021 28 0.0248 ± 0.0006 0.0807 ± 0.0021

The amplitudes of the peaks, minus the baseline correction, were chosen to be representative of the amount of 8Li+ contained within that layer. This is due to the fact that the sum of the amplitudes in the platinum and the magnesium oxide is close to the maximum difference between the two helicities. A further correction is required however before these measurements of the energy dependence of the position of 8Li+ ions are compared to the simulated data from TRIM.SP and SRIM. The signal measured in the sample comes from the decay of 8Li+ which is being continuously implanted into the sample while the test runs. With a half life (τ) of 838 ms, the majority of the signal will obviously come from ‘new’ 8Li+, but a portion of it will be from the ‘old’ 8Li+ which may no longer be polarized. The time it takes polarized 8Li+ to lose its polarization is given by the T1 time. The asymmetry at a given time, A(t), is given by the formula:

A(t) = Aoe-t/T

In this formula, Ao is the asymmetry that would be measured if there was no loss of polarization and t represents the time since implantation. However, a correction by this factor only would be too large. That is because most of the signal measured is generated by ‘newer’ 8Li+. Furthermore, the data collected during a resonance test is time averaged, so the formula must be likewise time averaged. When all of this is taken into consideration, the formula for the measured asymmetry, Am(t), becomes:

1Am(t) = τ × ∫A(t)e-t/τdt

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The formula, once the integral is solved, can be rearranged to find the asymmetry that would be obtained without polarization loss from the measured asymmetry:

Am(t) × (T1 + τ) Ao = T1

To correct the measured amplitude by this amount, it is necessary to know the T1value for 8Li+ in each layer. Unfortunately, at the time the resonance spectra were taken, T1 measurements were not made. Similar samples had been tested before however, and suitable replacements were found. The T1for platinum was taken from a sample of platinum on magnesium oxide at 287 K in a static field of 4.1 T. The implantation energy was low enough (8 keV) that the vast majority of the signal can be considered to be from the platinum. The T1 for the magnesium oxide was taken from a pure sample of magnesium oxide at 210 K in a static field of 4.1 T. As with the resonance spectra, the two helicities for each T1 test were combined to give an average curve. That curve was fitted in bnmrfit to an exponential decay formula of the form y = A1e-λt, where A1 is the initial amplitude and λ is the inverse of T1.

Table 4: T1 values for Pt and MgO T1 for Pt = (1.646 ± 0.0016) s

T1 for MgO = (1.032 ± 0.002) s The corrected amplitudes representing the concentration of 8Li+ in the platinum layer and the magnesium oxide layer were then graphed based upon their energy dependence as the results from the TRIM.SP and SRIM simulations were.

Figure 10: Energy Dependence of 8Li+ Stopping Position from β-NMR Data

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It was at this stage that a minor complication arose. From the method of data collection used, there was no way of determining either the fraction of 8Li+ backscattered, or what the full asymmetry should have been. As such, there was no readily apparent way to normalize the curves for comparison to the simulations. The decision was made to leave these parameters free and simply find the best possible fit to each simulation individually. This fitting was not done mathematically, but instead was estimated by sight in order to give a rough estimation of which simulation was performing better, that is, which simulation was producing 8Li+ position energy dependencies that were more consistent with the shape of the 8Li+ position energy dependency displayed by the data.

Figure 11: Comparison of TRIM.SP Energy Curves with Data

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Figure 12: Comparison of SRIM Energy Curves with Data

From this evidence, it was concluded that the SRIM simulation appeared to produce more realistic predictions for the position of 8Li+ within the sample. This conclusion comes from the fact that the SRIM curves appear to fit the curves made by the data better, especially at higher energies. The TRIM.SP curves appear to give a similar fit for lower energies, but clearly deviate from the data curves at higher energies. The higher energy results can be considered to be more accurate, as at high energies a clear signal can be seen in the β-NMR resonance spectrum from both the platinum and the magnesium oxide, while at lower energies only a weak signal from the platinum is visible. 4.2 Identifying the Differences between TRIM.SP and SRIM From the previous section it was determined that the largest difference between the TRIM.SP and SRIM simulations lay in how each handled backscattering. Backscattering occurs when the calculated scattering angle is such that the incident ion is reflected back from the surface of the sample. The scattering angle is calculated from the randomly determined impact parameter, the interaction potential and the screening length, which in turn depend on the mass, charge and density of the target as well as the kinetic energy of the projectile. Therefore the fraction of 8Li+ backscattered is predicted to increase with each of the target variables and increase as the implantation energy decreases. As platinum is very dense (the third densest element after iridium and osmium), as well as relatively massive and high in charge, it was assumed that it has a very large backscattering component. This assumption was supported by the results from both TRIM.SP and SRIM, which put the fraction of incident 8Li+ backscattered at the highest

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implantation energy (28 keV) to be 28% and 24% respectively. At the lowest implantation energy, these values increased to 54% backscattered in TRIM.SP and 45% backscattered in SRIM. Three further simulations were performed in each TRIM.SP and SRIM on samples with a similar structure to the Pt/MgO film; however, the platinum was replaced with an element less likely to induce backscattering. By comparing the results of these simulations, it could be determined if there were any other large differences between the predictions given by TRIM.SP and SRIM. The three samples simulated were: 50 nm of beryllium on a magnesium oxide substrate, 50 nm of aluminum on a magnesium oxide substrate and 50 nm of silver on a magnesium oxide substrate. The results obtained from TRIM.SP and SRIM were normalized following the same procedure described in the previous section.

Figure 13: Energy Dependence of 8Li+ Stopping Position in Ag/MgO Film

The predictions from TRIM.SP and SRIM for the silver on magnesium oxide sample are very similar to those given for the platinum on magnesium oxide sample. Again, the largest difference appears to be in the backscattering component. It can be seen that TRIM.SP consistently predicts a higher fraction of 8Li+ backscattered and a lower fraction of 8Li+ in the platinum. The curves for the component stopped in the magnesium oxide appear to be more similar than either the curves for the component stopped in the

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silver or the curves for the component backscattered. TRIM.SP appears to be giving consistently higher scattering angles, such that low energy particles near the surface are backscattered out of the sample opposed to remaining in the silver. Higher scattering angles would also place more low energy particles in a backscattering position (i.e. near the surface) as they could lose their energy deeper in the sample before being reflected back.

Figure 14: Energy Dependence of 8Li+ Stopping Position on Al/MgO Film

The backscattered component predicted by both TRIM.SP and SRIM for 8Li+ incident on aluminum is vastly reduced from that predicted for 8Li+ incident on both platinum and silver. This reduction is attributed to the decrease in charge, mass and density from platinum and silver to aluminum. Although TRIM.SP still predicts a slightly higher fraction backscattered, the fractions predicted by both simulations to be in the aluminum and magnesium oxide are nearly identical. In a situation without a lot of backscattering, it appears that TRIM.SP and SRIM give the same results.

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Figure 15: Energy Dependence of 8Li+ Stopping Position in Be/MgO Film

Despite the almost complete lack of backscattering (the backscattering curves are not visible on the scale presented) there is still a noticeable difference between the fractions of 8Li+ predicted to stop in the beryllium and the magnesium oxide by TRIM.SP and SRIM. Contrary to previous results however, in this situation SRIM predicts a higher fraction stopped in the magnesium oxide and a lower fraction stopped in the beryllium than TRIM.SP does. At the time the initial analysis was performed this appeared to be an anomaly; however a possible explanation is presented in section 4.5 once layer effects have been discussed. 4.3 Studying the Effect of the Different Properties of Silver on Predicted Range

Distributions The control over parameters allowed by the SRIM simulation allows an investigation into how those parameters each effect the predictions made by the program. Taking one element, silver, and modifying its mass, mass density and atomic density demonstrates how its stopping power depends on each of those factors individually. Any attempt to find the relation by examining different elements is complicated by the changing constants in the electronic stopping powers. It is important to recognize the relationship between the three factors available to the user for manipulation. Mass is the mass of a single atom of the target material. Changing the

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mass in SRIM does not change the number of neutrons or protons located in the nucleus of a target material atom, and hence does not change the charge of the target atom. In all cases, SRIM does not ask for a physical justification for the changes implemented by the user, and hence some clearly unphysical options are available, and indeed were tested. Mass density describes how the mass is distributed in the sample, while atomic density describes how the atoms are distributed. The three variables are related by the formula:

Mass Density Atomic Density = Mass × Mu The constant, Mu, is the conversion factor between grams and atomic mass units (amu) and is equal to 1 amu or 1.66 × 10-27 kg. Due to this relationship, only two variables may have their values selected, while the third must depend on those selections. Eight separate simulations were performed; four with the mass changing and the mass density remaining constant and four with the mass density changing and the mass remaining constant. One simulation in which nothing was changed was also performed as a control.

Table 5: Data from Modified Silver Simulations Material Mass

(amu) Mass

Density (g/cm3)

Atomic Density

(atoms/cm3)

Mean Range

(Å)

RMS Range

(Å)

Standard Deviation

(Å)

Percent Backscattered

Control 107.87 10.47 5.846×1022 333 375 172 20.56 ½ Mass 53.93 10.47 11.69×1022 153 171 76 14.48 ¼ Mass 26.97 10.47 23.38×1022 73 80 33 7.72 2×Mass 215.74 10.47 2.923×1022 719 810 373 24.88 4×Mass 431.47 10.47 1.461×1022 1528 1724 798 26.92 ½ Density 107.87 5.24 2.923×1022 655 739 342 20.32 ¼ Density 107.87 2.62 1.461×1022 1313 1481 685 20.46 2×Density 107.87 20.94 11.69×1022 172 193 86 20.56 4×Density 107.87 41.89 23.38×1022 91 101 43 20.90 In terms of the effect on the atomic density, halving the mass and doubling the mass density are the same operation; hence the simulations were run at four different atomic densities. Halving the mass and doubling the mass density appear to affect the mean stopping range of the silver in the same manner as well, although they effect the range distribution differently.

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Figure 16: Stopping Distributions for Modified Silver Simulations

Within the silver simulations, that is, when the charge of the target remains constant, the mean stopping range appears to be completely dependent on the atomic density.

Figure 17: Dependence of Stopping Range on Atomic Density in Silver Simulations

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Despite having identical mean stopping ranges, simulations with equal atomic densities do not have range distributions which are identical. The difference lies in the amount of 8Li+ predicted to enter into the sample. The amount of incident ions backscattered showed a heavy dependence on the mass of the target atoms.

Figure 18: Dependence of Backscattered Component on Mass in Silver Simulations

While the simulations for the silver with its mass halved and its density doubled may have the same atomic densities, the more massive test will backscatter more of the incident ions and therefore have less 8Li+ in the sample and shorter range distribution peaks. This is visible in Figure 16 above. 4.4 Searching for Periodic Trends The study of the modified silver helped explained how mass and atomic density were related to stopping range and stopping distributions in the absence of any change in atomic number and any change in the electronic stopping power. To investigate how these two properties modify the conclusions of the previous section, a simulation was performed on every element in the fourth row of the periodic table. To eliminate effects from factors previously investigated, the mass, atomic density and mass density of each element was kept constant. As such, only the atomic number of the target and the electronic stopping power factors were changing between simulations. The stopping ranges predicted are as follows.

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Figure 19: Stopping Distributions for Element from Row 4 of the Periodic Table

The stopping range distributions appear to be broken into three bands, as marked on the graph. The first band, with taller narrower distribution shapes and the smallest stopping range, contains elements 32 through 36, that is germanium, arsenic, selenium, bromine and krypton. All are non-metals, with the exception of germanium which is classified as a poor metal. The second band, with the average distribution shape and stopping range, contains elements 20 through 31, which is all of the transition metals in the fourth row plus calcium, an alkaline earth metal and gallium, a poor metal on either end. The third band contains only element 19, the alkali metal potassium. It has the widest range distribution and the largest stopping range. The general trend across all three bands is that increasing the atomic number, and thereby increasing the number of electrons, increases the electronic stopping power. This pattern, however, does not translate to increased electronic stopping power as a column of the periodic table is followed downwards. All of the alkali metals have stopping ranges much longer than any of the other groups.

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Figure 20: Stopping Distributions from Group 1 of Periodic Table

Obviously the structure of the electrons must have be taken into consideration when the electronic stopping powers were determined. The complexity of this relationship is hinted at by the eight fitting coefficients required in the formula for each element. The fraction of 8Li+ backscattered in each case was also recorded. The percent backscattered for each element is shown below, divided into the same three groups as described above.

Figure 21: Backscattered Components for Elements from Row 4 of the Periodic Table

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Across each group the fraction backscattered increases, with two exceptions; element 27, cobalt, in the second group and element 33, arsenic, in the third group. The scattering angle is dependent on the interaction potential between the incoming ion and the stationary atom. The interaction potential changes with the charge of the target nuclei and how their electrons screen that charge. Thus, the possible values for the scattering angle have a convoluted dependence on the atomic number and the electronic distribution. 4.5 Exploring layer effects While examining how changes to different factors influence the stopping behaviour of different materials it seemed simplest to work with samples containing only one type of element. Many interesting phenomenon can be observed in the SRIM simulation however, once samples are allowed to have multiple layers of different materials. The transition from one layer to the next is often accompanied by a dip in the range distribution predicted by the simulation. SRIM’s accompanying documentation states that this anomaly arises from the sharp cut-off between layers. The program makes no attempt to smoothly transition from the environment inside one layer to the environment within another. Hence, as the particle crosses the boundary, a number of constants in the formulas describing its motion change instantaneously. This unphysical approximation causes the unpredictable shape of the range distribution where layers end. One multi-layer sample that had been previously tested in TRIM.SP to reveal interesting results was 40 Å of gold on 200 Å of silver on 140 Å of beryllium on a substrate of gallium arsenide (GaAs, ρ = 5.32 g/cm3). The SRIM simulated range distribution for this sample is presented here.

Figure 22: Range Distribution for Au/Ag/Be/GaAs Film

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As the majority of the 8Li+ appears to have stopped in the beryllium layer, it would be easy to naively assume that beryllium has a very high stopping power. This option was briefly explored and subsequently disproved. While beryllium does possess a high atomic density (12.34 × 1022 atoms/cm3), which was shown in section 4.3 to predict shorter stopping ranges, it did not exhibit this behaviour in section 4.2 where most of the 8Li+ passed right thought the beryllium and into the magnesium oxide substrate beneath. The sample was recreated with the gallium arsenide replaced by magnesium oxide.

Figure 23: Range Distribution for Au/Ag/Be/MgO Film

In this simulation the stopping range distribution does not possess the sharp peak in the beryllium layer. Therefore, it can be assumed that it is not the beryllium causing the large disturbance in the distribution. Two quick simulations on pure samples of gallium arsenide and magnesium oxide yielded the result that gallium arsenide has a much higher backscattered component than magnesium oxide.

Table 6: Backscattered Components of GaAs and MgO Backscattered Component of GaAs = 15.82 % Backscattered Component of MgO = 1.89 %

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In section 4.2 it was also shown that silver has a rather large backscattered component and beryllium has almost non-existent backscattering. The possibility that the 8Li+ was backscattering from the gallium arsenide back into the silver was then proposed. Furthermore, if the 8Li+ backscattered from the gallium arsenide layer all the way back through the beryllium layer to the silver, it had a relatively high probability of backscattering off the silver to remain in the beryllium. This would account for the high fraction of 8Li+ stopped in the beryllium layer. As beryllium has practically no backscattering, the chance of an 8Li+ ion not being able to re-enter the beryllium is extremely low. To test this hypothesis, a new sample was created, 40 Å of gold on 200 Å on 140 Å of boron on a substrate of silver. Boron, like beryllium, has a very low backscattered fraction and silver was chosen as the substrate for its proven high backscattered fraction.

Figure 24: Range Distribution for Au/Ag/B/Ag Film

The same pattern present in the beryllium on gallium arsenide sample is present in the boron on silver sample. The supposition that the 8Li+ gets ‘trapped’ between two heavily backscattering layers appears to be correct. Furthermore, this theory also explains the behaviour of the beryllium on magnesium oxide sample in section 4.2. Magnesium oxide has a relatively low backscattered component; lower than silver and aluminum. 8Li+ ions are therefore more likely to

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remain in the magnesium oxide instead of backscattering back into either the silver or the aluminum. Beryllium, however, has an even lower backscattered component than magnesium oxide. Therefore, the possibility of 8Li+ backscattering back into the beryllium is available. The slightly higher number of 8Li+ stopped in the beryllium in the SRIM simulation can be attributed to this phenomenon. 5.0 Conclusions Monte-Carlo simulations of range distributions are an invaluable tool to the field of β-NMR spectroscopy. As β-NMR experiments become more and more refined, knowledge of the location of the β-decaying probes becomes even more essential. Furthermore, simulations can give researchers insight into what implantation energies would be most interesting. Computer simulations also provide an effective method to test the limits of the equations currently used to describe the behaviour of atoms moving in solids. Discovery of odd phenomenon, like the ‘trapping’ of 8Li+ probes between the layers of silver and gallium arsenide, leads to questions of whether the formulas need revising or whether this is actually representative of what would happen were this sample to be tested using β-NMR spectroscopy. From this exercise, it appears that SRIM is currently the more accurate of the two simulations tested, however further comparison to experimental results is required to strengthen this conclusion.

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6.0 References Biersack, J.P. and W. Eckstein. Sputtering Studies with the Monte Carlo Program

TRIM.SP. Appl. Phys. A 34, 73-94 (1984). Blinder, S. M. Introduction to Quantum Mechanics in Chemistry, Materials Science and

Biology. Elsevier Inc., Burlington: 2004. Eckstein, Wolfgan. Computer Simulation of Ion-Solid Interactions. Springer-Verlag,

Berlin: 1991. Ziegler, J. F. SRIM Instruction Manual. 2003. Ziegler, J. F., J.P. Biersack and U. Littmark. The Stopping and Range of Ions in Solids.

Pergamon Press, Inc., Toronto: 1985.

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