A Study of Sheath in Argon Plasma ContainingSputtered Metal

T. Ibehej and R. HrachCharles University in Prague, Faculty of Mathematics and Physics,Prague, Czech Republic.

Abstract. We present results of a two-dimensional particle model of plasma–solidboundary. The model follows trajectories of charged species in plasma which formsheath and pre-sheath layers in the vicinity of a biased substrate. We studiedsheath parameters and particle transport through the sheath in high-density argonplasma containing sputtered metal ions. The plasma parameters were taken froma volume-averaged model of inductively coupled plasma during ionized physicalvapour deposition (IPVD) using magnetron sputtering. Our model simulated thecapability of IPVD to fill the high-aspect-ratio trenches and evaluated local fluxes,energy and angular distributions of ions impinging on the trench. We observed thatthe substrate bias influenced the energy of ions but not their flux. Therefore, thesequantities can be regulated independently by changing bias and magnetron power.

Introduction

There are three main approaches to simulate plasma — fluid, particle and hybrid simula-tions [Hrach, 2003]. The fluid models are less precise and hard to program, but much fasterthan the particle models. They are based on numerical solution of a system of several equations,e.g. Poisson’s equation, continuity equation, Maxwell’s equations, etc. The range of physicalconditions in which they are valid is restricted by the range of validity of all approximationsused in the equations. An example is the calculation of mobility and diffusion coefficient inHruby and Hrach [2010] which is not applicable for plasma pressures lower than approx. 10 Paand for distribution functions substantially different from Maxwell energy distribution. On theother hand, the particle models [Ibehej and Hrach, 2012] solve only two equations — Newton’sequation of motion and Poisson’s equation. The independent solution of both of these equationsis well-described in the literature. The main disadvantage of particle models is their high re-quirement for computing time. However, at this cost we obtain results with high level of detail,relatively easy programming and less restrictions for physical conditions.

As was mentioned above, the particle approach is applicable also under lower pressures.This is one of the reasons why we chose to simulate the process of Ionized Physical VapourDeposition (IPVD) using particle model. The deposition method is based on magnetron sput-tering and post-ionization of sputtered material by a high-density inductively coupled plasma.The ions continue to the substrate which is negatively biased. They are accelerated towards thesubstrate in the sheath and therefore they can penetrate deeper to high-aspect-ratio structuresof the substrate such as trenches or vias. The coating on these structures has better propertiesthan the one coated by traditional PVD.

Coating of a two-dimensional trench is the subject of study of our contribution. Thesimulation can predict the local ion flux along the boundary of trenches with different depthand width. It shows also the energy and angular distribution functions of impinging ions andparameters of the sheath. Our input data were taken from a volume-averaged model of argonplasma containing sputtered copper in cylindrical magnetron chamber.

Model description

Input data of the particle model were obtained from a global (volume-averaged) kineticcalculation. The calculation was based on the Hopwood and Qian [1995] paper which was

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WDS'13 Proceedings of Contributed Papers, Part II, 54–59, 2013. ISBN 978-80-7378-251-1 © MATFYZPRESS

IBEHEJ AND HRACH: SIMLUATION OF SHEATH IN AR/METAL PLASMA

devoted to the study of aluminium deposition. We reproduced the model and changed theinput to estimate the degree of ionization of copper atoms. The processes involved in the modelwere:

1. Electron collisions. Rate constants of all interactions were computed at runtime byintegrating cross section of the given interaction with Maxwell energy distribution ofelectrons according to equation (5).

Ar + eke−→ Ar∗ + e . . . [Phelps, 2013] (1)

Ar + eki,Ar−−−→ Ar+ + e . . . [Phelps, 2013] (2)

Ar∗ + eki,Ar∗−−−→ Ar+ + e . . . [Vriens, 1964] (3)

Cu + eki,Cu−−−→ Cu+ + e . . . [Bolorizadeh et al ., 1994] (4)

kx =

∞∫0

vσx(E)fTe(E)dE (5)

2. Penning ionization. The rate constant was taken directly from the Bogaerts and Gijbels[1998] paper.

Ar∗ + CukP−−→ Ar + Cu+ + e (6)

3. Ambipolar diffusion of Ar+ and Cu+. The time constants were computed fromstandard formula for ambipolar diffusion in cylindrical chamber:

τa,X+ =Λ2

Da,X+(7)

Da,X+ =kBTeµX+

e(8)

1

Λ2=

(πL

)2+

(2.405

R

)2

(9)

The mobilities µX+ of Ar+ in Ar and Cu+ in Ar were taken from Chanin and Biondi[1957].

4. Diffusion of Ar∗ and deexcitation on the walls of the chamber. τd = Λ2/DAr∗. Thecoefficient DAr∗ was taken from Futch and Grant [1956]

5. Electron impact excitation of metastable Ar∗ and following resonant radiation to theground level. The electron — Ar∗ collision frequency was calculated from the formula

µm−e = 5× 10−8[e]T 1/2e (10)

where µm−e is in s−1, electron density [e] is in cm−3 and electron temperature Te in eV[Cohen, 1989]. The mean time of life of the Ar∗ metastable in relation to the resonantradiation (τr) was calculated using following formulae:

τr ∼1

µm−e+ τt (11)

τt ∼ τ∗k0π−1/2

[1.875

L

√ln(k0L) +

1.6

R

√ln(k0R)

]−1(12)

k0 = λ3[Ar]

8π

g2g1

(√π 〈v〉 τ∗

)−1(13)

where the radiative lifetime of the resonant transition τ∗ was equal to 1×10−8 s [Holstein,1947].

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IBEHEJ AND HRACH: SIMLUATION OF SHEATH IN AR/METAL PLASMA

A

B

C

2 cm

2 cm

0.4 mm

h

xy

s

2 mm

Figure 1. A diagram of the computational domain. A — source of particles, B — compu-tational domain, C — substrate, (x, y) — Cartesian coordinates, s — coordinate along thesubstrate boundary.

All of the above mentioned processes can be summarized to a set of balance equations forCu+ (14), Ar∗ (15) and Ar+ (16) which were solved together with the mass conservation law(17) and quasineutrality condition (18):

kP [Ar∗][Cu] + ki,Cu[e][Cu] = τ−1a,Cu+[Cu+] (14)

ke[e][Ar] = [Ar∗](ki,Ar∗[e] + kP [Cu] + τ−1d + τ−1r

)(15)

ki,Ar[Ar][e] + ki,Ar∗[Ar∗][e] = τ−1a,Ar+[Ar+] (16)

[Cu]0 = [Cu] + [Cu+] (17)

[e] = [Cu+] + [Ar+] (18)

For the given plasma density [e], total amount of sputtered copper in cm3 [Cu]0, argondensity [Ar], neutral atoms mean thermal velocity 〈v〉 and cylindrical chamber dimensions Rand L we obtain an estimation of electron temperature, degree of ionization of Ar and Cu atomsand density of argon metastable atoms.

Data obtained from the global model were used as an input of the particle simulationof sheath and pre-sheath region near the substrate with a trench. The dimensions of ourcomputational domain were 2 × 2 cm, trench width was set to 0.4 mm and its depth wasvariable. The domain is shown in Figure 1.

The motion of charged particles was determined by two factors:

1. Electrostatic force which originated from (a) negative substrate bias and (b) other chargedparticles. The influence of electrostatic force was simulated using the Particle in Cellmethod, Cloud in Cell algorithm [Hrach, 2003]. The Poisson’s equation applied on a netof 1000× 1000 nodes was solved by a sparse system solver Umfpack [Davis, 2004].

2. Collisions with neutral particles. The collisions were simulated as random processes usingan improved null-collision method [Roucka and Hrach, 2011]. This method do not neglectthe velocity of neutral collision targets and is also applicable for higher pressures.

The Newton’s equation of motion was integrated using Verlet’s algorithm and particles enteredthe area from a source of particles where we assumed Maxwell energy distribution.

Comparison of Ar/Al and Ar/Cu plasma

The global model was verified using the same input parameters and similar process param-eters as in Hopwood and Qian [1995] — i.e. by simulation of Ar inductively coupled plasma with

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IBEHEJ AND HRACH: SIMLUATION OF SHEATH IN AR/METAL PLASMA

Figure 2. Effects of addition of sputtered metal. [M]0 is total amount of sputtered metal —Cu or Al (see eq. 17). (a) electron temperature — a comparison of aluminum (taken fromHopwood and Qian [1995] and from our model) and copper (our model). (b) density of argonmetastable atoms and ions. (c) ion flux fraction and bulk ion fraction.

addition of defined amount of sputtered aluminium in a cylindrical magnetron chamber. Thanwe changed the process parameters related to aluminium to processes related to copper andcompared obtained results. The comparison of electron temperature, quenched by increasingamount of sputtered metal in plasma is shown in Figure 2a together with results taken fromthe original paper.

The decrease of electron temperature can be explained by different ionization energies. Theionization energy of argon atoms is 15.8 eV and the energy needed for their excitation to thefirst excitation level is 11.6 eV. Both of these values are higher than the ionization energy ofcopper (7.7 eV) and aluminum (6.0 eV). Therefore, by adding more metal atoms, electrons inthe plasma are able to participate in inelastic interactions more often. For this reason, the effectis stronger in the case of aluminum. Due to the decrease of electron temperature, the numberdensities of both, Ar+ and Ar∗ are also decreasing. This is shown in Figure 2b. The addition ofmetal causes decrease of ion flux fraction — a ratio of ion flux Γ+ and neutral flux Γn computedby the formula

F =Γ+

Γn=

0.6[M+]√

kBTe

mi

0.25[M] 〈v〉. (19)

However, slower decrease of bulk ion fraction, also shown in Figure 2c, implies that the amountof both, aluminum and copper ions is increasing with the addition of sputtered metal. All theeffects connected with addition of sputtered metal shown in Figure 2 are stronger for Ar/Alplasma than for Ar/Cu plasma.

Different bias during trench filling

The final set of input parameters for the particle model is shown in Table 1.

Table 1. A set of input parame-ters of the particle model.

p 5.3 Pa[e] 3.5× 1011 cm−3

[e]:[Ar+]:[Cu+] 50:49:1Te 1.5 eV

The number density of neutral copper atoms was 4 orders of magnitude lower than argonnumber density. Therefore, our model contained collisions only with neutral argon and not with

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IBEHEJ AND HRACH: SIMLUATION OF SHEATH IN AR/METAL PLASMA

Figure 3. Effects of different bias. (a) Angular distribution of ions which hit the flat part ofthe substrate. (b) energy distribution of ions which hit the flat part of the substrate. (c) energydistribution of ions which hit the bottom of trench with aspect ratio of 2:1 and width of 0.4mm.

58

Figure 4. (a) Ion fluxes along the trench boundary for different aspect ratios of the trench.s axis is defined in Figure 1. (b) shape of the sheath — electrostatic potential with contourlines from −9 to −2 V, dashed line represents the trench boundary.

neutral copper. The results for different bias and for trench with aspect ratio of 2:1 are shownin Figure 3.

Higher substrate bias causes higher energy of impinging ions as well as narrower angulardistribution. The most probable energy of ions which hit the flat part of the substrate wasobserved to be approx. 3 eV lower than the applied bias for copper and 3.3 eV for argon. Lowerenergy of argon ions can be explained by higher cross section of their interactions with argonneutral atoms. Slightly higher energy of ions was observed at the bottom of the trench. We didnot observed any substantial change in fluxes along the trench caused by the change of bias.

Ion fluxes and sheath dimensions

Our model also showed local ion fluxes along the boundary of the trench. Figure 4a showsthe fluxes of both types of ions for different aspect ratios of the trench.

The maximum was observed at the opening corners of the trench and minimum at thebottom corners. For all of the studied aspect ratios the ions were able to reach the bottom, buttheir flux was decreasing with increasing depth. Figure 4b shows the shape of the sheath whichis relatively thin due to short Debye length of the plasma of ≈ 15 µm.

IBEHEJ AND HRACH: SIMLUATION OF SHEATH IN AR/METAL PLASMA

Conclusion

Two models of plasma used in IPVD of aluminum and copper were presented. The firstvolume-averaged model estimated the electron temperature and degree of ionization of argonand metal atoms for given physical conditions. The second particle model simulated a treatmentof a two-dimensional trench. It was shown that the models are capable of estimating sheathparameters, ion fluxes to the substrate and ion energy and angular distributions which are crucialparameters for the studied technological application. We also showed some effects connectedwith different substrate bias and different aspect ratio of the trench.

Acknowledgment. Authors acknowledge the support of the Grant Agency of Czech Republic(project P205/10/0979).

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