MODELING C-BN GROWTH WITH HIGH ION ENERGIES 105

0 10 20 30 40

Ion Energy [keV]

0 20 40 60 80 100

Depth [nm]

Boron Ion Range

Maximum of Boron Target Vacancies

Figure 4.6: Projected range of¹¹B ions implanted into c-BN as a function of ion energy (trian-gles). Circles represent the position of the maximum of the projected boron target vacancy profile as created by the incident ion beam. The data has been calculated by SRIM.

be estimated by using the available SIMS data in addition to SRIM simulations. A comparison of the measured and calculated profiles in figure 4.3 yields a diffusion length xthat is approx. given by the depth difference of vacancy and implantation profile. Figure 4.6 shows the projected range of ¹¹B ions implanted into c-BN as a function of ion energy (triangles), and the position of the maximum of the boron vacancy distribution (circles) as calculated by SRIM. Within the given energy range, both vacancy maximum position and ion range increase linear with the ion energy.

Thus, the distance between both profiles is proportional to the ion energy as well, and this value will be used as (energy-dependent) diffusion length x. However, it should be mentioned that the effective diffusion length could be larger than this assumed value. This is related to the fact that both SRIM and SIMS data only show the projected ion and vacancy distribution, but particle trajectories in a material have to be treated three-dimensionally. Hence, the calculated value xis most likely being underestimated.

2. Diffusion constant D₀ of a simple random walk diffu-sion mechanism.

Atomic diffusion processes are often discribed by us-ing random walk models, which treat atomic motion in a material as jumps of fixed length in specific di-rections, usually between different minima in a po-tential landscape (figure 4.7). In crystalline solids, these minima are represented by lattice sites or in-terstitial sites. Thermally activated, jumps occur at a rateν⁰, with a temperature dependence that obeys an Arrhenius law:

ν⁰ =ν₀exp(−W_m/k_BT), (4.10) where ν₀ is the attempt frequency, W_m is the migration energy, k_B the Boltzmann constant, andT is the temperature. If jumps can occur toZ neighboring sites, the total jump frequency is then given by

ν =Zν⁰. (4.11)

After a given timet, a particle was able to performN =νtjumps, and its trajectory is composed of a sequence of elementary jumps with average jump length a. The diffusion coefficient in three dimensions is then given by the Einstein-Smoluchowski relation

D= 1

6a²ν, (4.12)

which leads to a diffusion constant D₀ of D₀ = Z

6a²ν₀. (4.13)

The attempt frequency can be set to ν₀ = 10¹³ Hz, which is a typical phonon fre-quency in diamond-like materials [Hof98]. Cubic BN with its zinc-blende structure exhibits a variety of lattices sites that can be occupied by interstitial atoms, e.g.

tetrahedral, bond-center, or hexagonal sites (see e.g. ref. [Wah97] for an overview).

As the use of a simple random walk mechanism to calculate the diffusion constant is only a rough approximation in the first place, it will be assumed that atomic jumps

4.5 MODELING C-BN GROWTH WITH HIGH ION ENERGIES 107 occur mainly between tetrahedral sites. Thus, an interstitial atom has the possibil-ity to jump into 4 adjacent sites, which leads to a coordination number of Z = 4 and a jump length of the order of the c-BN bond length (a = 1.5 ˚A). Combining the above estimates yields a diffusion constant of D₀ ≈1.5×10⁻³ cm²/s.

3. Diffusion time t

The diffusion time t can be estimated when considering the deposition parameters used during film growth. As described in detail by Hofs¨aß et al. [Hof98], each ion impact during MSIB deposition can be considered as an individual event, well separated both in time and distance from preceding and following impacts. The affected projected area during a single ion impact has been estimated to 1–10 nm² in the energy range from 1 keV to 40 keV by using SRIM. With typical ion currents of 20 µA deposited on an area of about 2 cm², the time between subsequent ion impacts into the same target region calculates to t≈0.1 s. Therefore, the affected target volume can relax for at least 0.1 s before the next ion hits the same area.

When comparing this value to the typical duration until thermalization (≈ 1 ps)

Figure 4.8: 3-dimensional visualization of several ion impacts with Eion= 5 keV into c-BN as calculated by SRIM.

for an ion impact, it becomes obvious that the picture of individual ion impacts holds even for an ion flux of up to several A/cm².

However, it is difficult to give an exact value for the timet. This is mainly related to the fact that particles in a material move on three-dimensional trajectories, but the value for t has been derived by projecting the trajectories onto a two-dimensional plane. Figure 4.8 shows a number of SRIM simulated impacts of 5 keV ¹¹B⁺ ions into a c-BN target. Although all ions enter the material at the exact same spot, the affected target volumes do not necessarily overlap for each impact. Thus, the relaxation time t could be significantly larger than the assumed value of 0.l s. As pointed out by Lifshitz et al., the duration of this relaxation stage can be as large as t = 1 s [Lif90]. Nevertheless, for the following calculations, the diffusion time will be set to t = 0.1 s, and the influence of a larger value t will be discussed in a following paragraph.

4.5.3 Discussion for c-BN growth

By using the estimated values for diffusion lengthx(E), diffusion constantD₀, and diffusion time t, one can now calculate the critical temperatures required to main-tain c-BN growth at a given ion energy. In order to avoid defect accumulation, at least stationary equilibrium between defect production and annealing has to be established. This state is reached, when an interstitial atom is able to diffuse the distance x(E) within the time t = 0.1 s. Figure 4.9 presents the critical tempera-tures using the data from table 3.1. Those values have been fitted using equation (4.9), witht = 0.1 s,x=x(E), andD0 = 1.5×10⁻³ cm²/s as mentioned previously andWmas fit parameter. The fit has been included in the figure as indicated by the solid line and shows a good agreement with the experimentally gained data points.

A migration energy ofW_m = 0.5(1) eV has been extracted from the fit. For 5 keV ions, the model predicts c-BN growth to be possible even at room temperature, in agreement with the experimental results. Using 40 keV B⁺ ions (and 60 keV N⁺ ions, respectively) would require a substrate temperature of at least 400 K.

The use of larger diffusion times does not change the calculated migration energy significantly. Fort= 1 s for example, a value of W_m ≈0.55 eV is obtained. This is mainly related to the logarithmic dependency of temperatureT on diffusion timet