Estimation of the Pressure-Distance Product for Thermalization in Sputtering forSome Selected Metal Atoms by Monte Carlo Simulation

Takeo NAKANO* and Shigeru BABA
Department of Materials and Life Science, Seikei University, Tokyo 180-8633, Japan
E-mail: nakano@st.seikei.ac.jp

Abstract
We propose a method to obtain the pressure-distance product (pd) for thermalization in the sputtering
process using a Monte Carlo (MC) simulation. Sputter-ejected atoms proceed forward with high energy,and fall into random motion finally. That is, after many MC trials the "average position" of atoms
reaches saturation. The thermalization distance d can be estimated from this saturation position. We
could obtain the pd product for Al, Cu, and Mo in argon atmosphere. The pd values agreed well withthe pressure dependence of the deposition profile observed experimentally.

Takeo Nakano

1

In sputter deposition, the transport process of sputtered atoms from the target to the substrate is known
to be strongly affected by gas pressure1). Initially, sputtered atoms have energy on the order of 1-10eV2), which is much greater than the kinetic energy of ambient gas atoms and molecules. At low gaspressures, sputtered atoms experience few collisions before they arrive at the substrate, so they retain
their high energy throughout the transport process. On the other hand, when gas pressure is high, sputtered atoms are decelerated many times by collisions with ambient gases until their speed distribution
reaches an equilibrium with the gases. This process is called “thermalization” of sputtered atoms. The
transport modes before and after thermalization are often called “ballistic” and “diffusive”, respectively,
and the mode transition caused by gas pressure gradually changes the deposition rate and the thickness
profile of the sputter deposition process. Gas pressure is also known to affect strongly on the microstructure of sputtered films3) because the energetic bombardment onto growing films4,5), significant in
the ballistic transport regime, is lost upon thermalization.
Thermalization is dominated by the amount of collision with gas atoms. Therefore, the product of
the gas pressure and the thermalization distance, called the “pd product,” should be a specific value
for sputtered atoms of a given element. This concept was discussed by Keller and Simons in their
theoretical consideration6), and it motivated the development of the Monte Carlo (MC) simulation of
the particle transport process in its early years7–10). A state-of-the-art MC simulation can be found in
the review by Depla and Leroy11).
The pd product is dependent on the target materials because the atomic size and mass affect theTakeo Nakano

2collision and scattering processes. The difference in the pd product between elements of different
atomic mass was reported to be the origin of the compositional difference between sputtered films andalloy targets such as AlCu and TiW12). If it is possible to obtain a direct measurement of the pd productfor various elements, such a method would be useful to the sputter deposition community.
In a previous study13), we experimentally evaluated the pressure dependence of the thickness profilein sputter deposition with three types of metal targets (Al, Cu, and Mo) with different atomic mass. We
prepared a holder equipped with three quartz crystal microbalance monitors (QCMs). Two QCMs werefacing the target - on the “confronted” face - and the third was located on the back face which washidden from the target. The holder was located 4 and 6 cm away from the target, and the pressure
dependence of the deposition rates was measured. On the confronted face, the deposition rate and thethickness uniformity were almost unchanged at lower gas pressures. However, the deposition rates
decreased and the thickness uniformity worsened with increasing the gas pressure beyond a certain
value. The transition occurred at around 1–5 Pa. On the hidden face QCM, the deposition reached its
maximum at a gas pressure of 1–2 Pa. Both the transition pressure on the confronted face and the peakpressure on the back face were lower for lighter elements. Our MC simulation could well explain theseexperimental results. Especially the deposition rate peak on the back face was reproduced quantitatively both in height and gas pressure. The experimentally observed phenomena could be ascribed tothe thermalization of the sputtered atoms, and the lighter elements were expected to have a smaller pd
product value.

Takeo Nakano3In this work, we propose a method to obtain the pd product by developing our Monte Carlo simulation. The model takes the ambient Ar gas motion into account14) and can handle the thermalization of
sputtered atoms seamlessly. Because of their high initial kinetic energy, sputtered atoms are expected
to proceed forward. After thermalization, however, they diffuse isotropically. Hence, if we trace many
atoms with MC trials and average their position, the “center-of-mass” (COM) of the atoms may reachsaturation with time. By multiplying the distance by the ambient gas pressure, we can determine thethermalization pd product. We applied this idea to Al, Cu, and Mo transport in an Ar atmosphere andobtained the resulting pd product values.
The MC framework was the same as in the previous study13). A full description of the simulation isprovided in one of the author’s doctoral dissertation15). The scattering between the sputtered atoms byAr gas atoms was treated with Born-Meyer type interatomic potential with parameters given by Abrahamson16). The ambient Ar gas pressure was varied between 1 and 10 Pa, and the gas temperature wasassumed to be 350 K. Atoms were injected into the free volume from the origin along the z axis andtheir motion, including the free flight and collision/scattering with argon, was recorded. Typically, 106
atoms were traced and processed in each condition.
To illustrate the behavior of sputtered atoms, we applied the MC code for a single initial energy case.
Figure 1 shows the simulation snapshots of real and velocity spaces for copper atoms with 5 eV ofinitial kinetic energy (which corresponds to a speed of 3900 m/s) in argon of 5 Pa and 350 K. As shownin Fig. 1 (left), the atoms started at the origin at 𝑡 = 0, proceeded in the 𝑧 > 0 direction in the 20 μs

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4snapshot, and diffused isotropically after that. At the Fig. 1 (right), it is also shown that the copper
atoms were decelerated and came to have an isotropic velocity distribution as time proceeds. Figure 2
shows the time dependence of the z component of the COM (a) and the time evolution of the speed
distribution of copper atoms (b). It shows clearly that the distance of the averaged position from theorigin saturates at around 50 μs. At that time, the speed distribution becomes Maxwellian, which indicates the thermalization of the atoms. The x and y components of the atoms’ COM were zero within
the MC randomness.
In the actual calculation of the pd products of metals, we determined the initial energy of atoms byutilizing Thomson’s energy distribution formula17), which describes the energy distribution of atoms
sputtered from a solid surface. The binding energy of the target material, a parameter of the formula,was assumed to be 3.39, 3.49, and 6.82 eV for Al, Cu, and Mo, respectively; these values were used in
the work by Yamamura and Tawara18). The primary (Ar) ion energy, another parameter in the formula,was assumed to be 400 eV. The gas temperature was assumed to be 350 K.
Figure 3 shows the pressure dependence of the saturation distance of Al, Cu, and Mo. As expected,
the distance was inversely proportional to the gas pressure, and the pd products were found to be 7.1,10.0, and 14.3 Pa cm for Al, Cu, and Mo, respectively. These values are on the same order with thosereported in previous studies; Keller and Simmons reported 8 Pa cm for Si by fitting their theoreticalformula to the experimentally obtained deposition rate6). The previous MC works7–10) did not directly
present a specific pd value for each case, but showed their calculation results as pd dependence of theTakeo Nakano

5average energy (or the energy distribution) of sputtered atoms. The pd at which the prominent deceleration of sputtered atoms occurred was around 10 Pa cm in their calculation.
Comparing our MC values with the experimental results13) described earlier, it is reasonable that thetransitions caused by the thermalization occurred at gas pressures of 1–5 Pa, where the target-to-substrate distance was 4–6 cm. The pd product in this study was smaller for lighter elements. In ballisticmode transport, where gas motion is negligible, atoms much heavier than gases are considered to go
further since they lose less kinetic energy in binary collisions. Light atoms also lose less energy incollisions, but the backscattering of the atoms of such element by gases is more effective on the reduction of the pd product.
The thermalization distance d is greater than the collision mean free path (MFP) 𝜆, because sputtered atoms do not completely lose their energy and forward momentum by a single collision event.
High energy atoms, such as sputtered atoms from the target, tend to have longer MFP because of theless effect of ambient gas motion19) and of effectively smaller cross section of the potential scattering11).
The "𝑝𝜆” can be calculated to be 0.8~1.7 Pa cm by applying energy dependent cross sections (seediscussion by Somekh7) for example). Therefore, sputtered atoms are considered to experience several
to 10 times of collisions before thermalization.
It should be noted that the obtained pd product is sensitive to the gas temperature. As a matter offact, when we assumed the gas temperature to be 400 K, the calculated pd product increased by ~14%
in comparison to the 350 K case. In actual sputter deposition, it is known that the gas density reduction

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6occurs locally due to heat flux from the target (“sputtering wind”)20,21). This is prominent at higher gas
pressures, so that slightly greater pd values should be employed in such situations.
In summary, we have developed a method to calculate the thermalization pd product in the sputter
deposition process using an MC framework where the thermal motion of gases is properly taken into
consideration. We obtained results of 7.1, 10.0, and 14.3 Pa cm for Al, Cu, and Mo, respectively, in thecase of Ar gas at a temperature of 350 K. The obtained values agree well with results from our previous
experiment in explaining the transition gas pressures and their element dependence.

Takeo Nakano

7References1) S. M. Rossnagel: IEEE Trans. Plasma Sci. 18 (1990) 878.

2) H. Gnaser: in Sputtering by Particle Bombardment: Experiments and Computer Calculations fromThreshold to MeV Energies, ed. R. Behrisch and W. Eckstein (Springer, Berlin, 2007).

3) J. A. Thornton: J. Vac. Sci. Technol. A 4 (1986) 3059.

4) S. M. Rossnagel: J. Vac. Sci. Technol. A 7 (1989) 1025.

5) D. M. Mattox: J. Vac. Sci. Technol. A 7 (1989) 1105.

6) J. H. Keller and R. G. Simmons: IBM J. Res. Dev. 23 (1979) 24.

7) R. E. Somekh: J. Vac. Sci. Technol. A 2 (1984) 1285.

8) T. Motohiro and Y. Taga: Surf. Sci. 134 (1983) L494.

9) G. M. Turner, I. S. Falconer, B. W. James, and D. R. McKenzie: J. Appl. Phys. 65 (1989) 3671.

10) A. Bogaerts, M. van Straaten, and R. Gijbels: J. Appl. Phys. 77 (1995) 1868.

11) D. Depla and W. P. Leroy: Thin Solid Films 520 (2012) 6337.

12) S. M. Rossnagel, I. Yang, and J. J. Cuomo: Thin Solid Films 199 (1991) 59.

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813) T. Nakano and S. Baba: Vacuum 80 (2006) 647.

14) T. Nakano, I. Mori, and S. Baba: Appl. Surf. Sci. 113-114 (1997) 642.

15) T. Nakano: Dr. Thseis, Fuculty of Engineering, The University of Tokyo, Tokyo (2001).

16) A. A. Abrahamson: Phys. Rev. 178 (1969) 76.

17) M. Thompson: Vacuum 66 (2002) 99.

18) Y. Yamamura and H. Tawara: At. Data Nucl. Data Tables 62 (1996) 149.

19) J. Jeans: The Dynamical Theory of Gases (Dover, New York, 1954), 4th ed. .

20) S. M. Rossnagel: J. Vac. Sci. Technol. A 6 (1988) 19.

21) F. J. Jimenez and S. K. Dew: J. Vac. Sci. Technol. A 30 (2012) 041302.

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9Figure Captions

Fig. 1: The evolution of Cu atoms injected into the volume along the z axis at 𝑡 = 0 with 5 eV ofenergy. The argon gas pressure and temperature in the volume were 5 Pa and 350 K, respectively.
Snapshots of 5000 atoms in real space (left) and in velocity space (right) are given.

Fig. 2: (a) The time dependence of the z component of the “center-of-mass” of Cu atoms which wereinjected into the volume under the same conditions as in Fig. 1 (Cu initial energy: 5 eV, Ar gas pressure:
5 Pa, gas temperature: 350 K). The saturation length at longer periods is considered to give the thermalization distance. (b) The speed distribution of the Cu atoms. The black line denotes the Maxwellianfor Cu vapor at 350 K.

Fig. 3: Gas pressure dependence of the thermalization distance of Al, Cu, and Mo atoms. The Ar gastemperature was 350 K, and the initial energy of the injected atoms was in accordance with the Thompson’s formula with a primary ion energy of 400 eV.

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10Real Space

Velocity Space

5 μs20 μs

100 μsFig. 1

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11

(a)
2dN/dv (s/m)

Average of Z (cm)

10-2

10

10-6

10-5Time (s)1010-310-4

10-50-45 s
10 s
20 s
50 s
100 sM. dist(b)

1000

2000

Cu Atom Speed (m/s)

Fig. 2

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12Al ( 7.1 Pa cm)
Cu ( 10.0 Pa cm)
Mo ( 14.3 Pa cm)

dtherm (cm)101
1

Pressure (Pa)10

Fig. 3

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