Self-diffusion In Silicon Research Articles

Atomic mechanisms of self-diffusion in amorphous silicon

Based on recent calculations of the self-diffusion (SD) coefficient in amorphous silicon (a-Si) by classical Molecular Dynamics simulation [Posselt et al., J. Appl. Phys. 131, 035102 (2022)], detailed investigations on atomic mechanisms are performed. For this purpose, two Stillinger–Weber-type potentials are used, one strongly overestimates the SD coefficient, while the other leads to values much closer to the experimental data. By taking into account the individual squared displacements (or diffusion lengths) of atoms, the diffusional and vibrational contributions to the total mean squared displacement can be determined separately. It is shown that the diffusional part is not directly correlated with the concentration of coordination defects. The time-dependent distribution of squared displacements of atoms indicates that in a-Si, a well-defined elemental diffusion length does not exist, in contrast to SD in the crystalline Si. The analysis of atoms with large squared displacements reveals that the mechanisms of SD in a-Si are characterized by complex rearrangements of bonds or exchanges of neighbors. These are mono- and bi-directional exchanges of neighbors and neighbor replacements. Exchanges or replacements may concern up to three neighbors and may occur in relatively short periods of some ps. Bi- or mono-directional exchange or replacement of one neighbor atom happens more frequently than processes including more neighbors. A comparison of results for the two interatomic potentials shows that an increased three-body parameter only slows down the migration but does not change the migration mechanisms fundamentally.

Atomistic simulations on the relationship between solid-phase epitaxial recrystallization and self-diffusion in amorphous silicon

Recent experimental results on self-diffusion (SD) in amorphous silicon (a-Si) [Kirschbaum et al., Phys. Rev. Lett. 120, 225902 (2018)] indicate that the atomic mechanism of this process is akin to that of solid-phase epitaxial recrystallization (SPER). In the present work, this relationship is investigated using classical molecular dynamics (MD) simulations with selected interatomic potentials. In the beginning, an overview of the status of the present knowledge on SPER and SD is given. Then, it is shown that the Stillinger–Weber (SW)-type and Tersoff (T)-type potentials considered yield structural data of a-Si, which are in rather good agreement with measurements. On the other hand, deviations are found for thermal properties. The results of partially extremely long MD calculations of SPER and SD yield that both processes can be described by a simple Arrhenius relation and that the activation enthalpies of SPER and SD are rather equal, which is in qualitative agreement with experiments. Obviously, the simulated atomic-level processes are very similar. However, for the known SW- and T-type interatomic potentials, a quantitative agreement with SPER and SD measurements cannot be found. This work demonstrates that significant improvements can be achieved if SW-type potentials with an increased value of the three-body parameter are used.

Interconnection of a thermodynamical model for point defect parameters in solids with the dynamical theory of diffusion

A multitude of recent studies of point defect parameters in solids including several defect processes in fluoride superionic conductors as well as self-diffusion and heterodiffusion in silicon have been found to obey the cBΩ thermodynamical model. This model is shown to be compatible with the dynamical theory of diffusion.

Self-Diffusion in Amorphous Silicon by Local Bond Rearrangements.

Experiments on self-diffusion in amorphous silicon (Si) were performed at temperatures between 460 to 600° C. The amorphous structure was prepared by Si ion implantation of single crystalline Si isotope multilayers epitaxially grown on a silicon-on-insulator wafer. The Si isotope profiles before and after annealing were determined by means of secondary ion mass spectrometry. Isothermal diffusion experiments reveal that structural relaxation does not cause any significant intermixing of the isotope interfaces whereas self-diffusion is significant before the structure recrystallizes. The temperature dependence of self-diffusion is described by an Arrhenius law with an activation enthalpy Q=(2.70±0.11) eV and preexponential factor D_{0}=(5.5_{-3.7}^{+11.1})×10^{-2} cm^{2} s^{-1}. Remarkably, Q equals the activation enthalpy of hydrogen diffusion in amorphous Si, the migration of bond defects determining boron diffusion, and the activation enthalpy of solid phase epitaxial recrystallization reported in the literature. This close agreement provides strong evidence that self-diffusion is mediated by local bond rearrangements rather than by the migration of extended defects as suggested by Strauß etal. (Phys. Rev. Lett. 116, 025901 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.025901).

Self-diffusion in crystalline silicon: A single diffusion activation enthalpy down to755∘C

Self-diffusion in silicon and the contribution of vacancies and self-interstitials have been controversially discussed for 50 yr. Most recent results show that the intrinsic silicon self-diffusion coefficient deviates from an Arrhenius-type, single exponential function for temperatures below $950{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$ [Y. Shimizu, M. Uematsu, and K. M. Itoh, Phys. Rev. Lett. 98, 095901 (2007); R. Kube, H. Bracht, E. H\uger, H. Schmidt, J. L. Hansen, A. N. Larsen, J. W. Ager, E. E. Haller, T. Geue, and J. Stahn, Phys. Rev. B 88, 085206 (2013)]. This led us to propose temperature-dependent thermodynamic properties of vacancies in order to achieve full consistency to vacancy-mediated dopant diffusion in silicon. Concepts of temperature-dependent properties of native defects or distinct forms of defects with different formation entropies suggested by Cowern et al. [N. E. B. Cowern, S. Simdyankin, C. Ahn, N. S. Bennett, J. P. Goss, J.-M. Hartmann, A. Pakfar, S. Hamm, J. Valentin, E. Napolitani, D. De Salvador, E. Bruno, and S. Mirabella, Phys. Rev. Lett. 110, 155501 (2013)] question the present understanding on atomic transport in semiconductors. To verify these concepts, additional self-diffusion experiments under particular gettering conditions were performed. As a result, silicon self-diffusion was found to be accurately described by one single diffusion activation enthalpy of $(4.73\ifmmode\pm\else\textpm\fi{}0.02)$ eV down to $755{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$. This provides full consistency to dopant diffusion without claiming native-defect concepts that were originally proposed by Seeger and Chik in 1968 [A. Seeger and K. P. Chik, Phys. Stat. Sol. 29, 455 (1968)] and confirms most recent density functional theory calculations on the activation energy of self-diffusion via vacancies and self-interstitials. Overall, this unravels the old debate of self-diffusion in silicon with the supposed intrinsic temperature dependence.

A thermodynamic approach to self-diffusion in silicon: Evidence of a single diffusion mechanism?

The self-diffusion in silicon is investigated in terms of the cBΩ thermodynamic model, which connects point defect parameters with the bulk elastic and expansion properties. The curved Arrhenius plot of the recently reported experimental data on self-diffusion in Si by Kube et al. [Phys. Rev. B 88 (2013) 085206] and by Bracht [Physica B 376–377 (2006) 11] which is associated with two mechanisms (vacancies and self-interstitials) can be explained, if the non-linear anharmonic behavior of the isothermal bulk modulus is considered over the entire temperature range of the available experimental data. The latter suggests the existence of a single diffusion mechanism, in contrast to experimental evidence, which in the frame of the cBΩ model exhibits temperature dependent thermodynamic properties. Indeed, the calculated point defect parameters (activation enthalpy, activation entropy and activation Gibbs free energy) exhibit considerable temperature dependence, with the activation enthalpy varying in a non-linear way from 3.6 eV to 4.9 eV, in agreement with the interpretation of the aforementioned experimental data.

Short range atomic migration in amorphous silicon

Experiments on self-diffusion in amorphous silicon between 400 and 500 °C are presented, which were carried out by neutron reflectometry in combination with 29Si/natSi isotope multilayers. Short range diffusion is detected on a length scale of about 2 nm, while long range diffusion is absent. Diffusivities are in the order of 10−19–10−20 m2/s and decrease with increasing annealing time, reaching an undetectable low value for long annealing times. This behavior is strongly correlated to structural relaxation and can be explained as a result of point defect annihilation. Diffusivities for short annealing times of 60 s follow the Arrhenius law with an activation enthalpy of (0.74 ± 0.21) eV, which is interpreted as the activation enthalpy of Si migration.

Self-Diffusion in Amorphous Silicon.

The present Letter reports on self-diffusion in amorphous silicon. Experiments were done on ^{29}Si/^{nat}Si heterostructures using neutron reflectometry and secondary ion mass spectrometry. The diffusivities follow the Arrhenius law in the temperature range between 550 and 700 °C with an activation energy of (4.4±0.3) eV. In comparison with single crystalline silicon the diffusivities are tremendously higher by 5 orders of magnitude at about 700 °C, which can be interpreted as the consequence of a high diffusion entropy.

Neutron Reflectometry for the Investigation of Self-Diffusion in Amorphous Silicon

We present experiments based on neutron reflectometry in combination with 29Si/natSi isotope multilayers in order to investigate the self-diffusion in amorphous silicon. Such experiments allow the detection of diffusion processes in the amorphous state on length scales below 10 nm. First results at 650 °C show a continuous decrease of the artificial Bragg peak produced by the multilayer, corresponding to a diffusivity of (1.1 ± 0.4) x 10-20 m2/s on a length scale of 2 - 7 nm. The diffusivity is not time-dependent for annealing times between 3 min and 1 h. Compared to recent measurements in silicon single crystals by the same method, the diffusivity is higher by a factor of about 105.

Comment on “Contributions of vacancies and self-interstitials to self-diffusion in silicon under thermal equilibrium and nonequilibrium conditions”

Comment on “Contributions of vacancies and self-interstitials to self-diffusion in silicon under thermal equilibrium and nonequilibrium conditions”

Reply to “Comment on ‘Contributions of vacancies and self-interstitials to self-diffusion in silicon under thermal equilibrium and nonequilibrium conditions’ ”

Suezawa et al. [Phys. Rev. B 90, 117201 (2014)] claim in their Comment that the data reported by Shimizu et al. [ Phys. Rev. Lett. 98, 095901 (2007)] and Kube et al. [Phys. Rev. B 88, 085206 (2013)] on silicon self-diffusion for temperatures between 900 and 735 \ifmmode^\circ\else\textdegree\fi{}C are affected by carbon and vacancy clusters and, accordingly, do not reflect self-diffusion under thermal equilibrium conditions. We demonstrate in our Reply that an impact of carbon on self-diffusion can definitely be excluded. In addition it is rather unlikely that the self-diffusion data reported by Shimizu et al. [Phys. Rev. Lett. 98, 095901 (2007)] and Kube et al. [Phys. Rev. B 88, 085206 (2013)] are affected by the dissolution of vacancy clusters since strong differences exist not only in the preparation of the samples used for the experiments, but also in the time of diffusion. Finally, the vacancy formation enthalpy deduced by Suezawa et al. [J. Appl. Phys. 110, 083531 (2011)] from quenching experiments is consistent with the value obtained from the temperature dependence of the vacancy formation enthalpy reported by Kube et al. [Phys. Rev. B 88, 085206 (2013)]. Overall we conclude that the quenching experiments of Suezawa et al. [J. Appl. Phys. 110, 083531 (2011)] cannot disprove the interpretation of the low-temperature self-diffusion data reported by Shimizu et al. [ Phys. Rev. Lett. 98, 095901 (2007)] and Kube et al. [Phys. Rev. B 88, 085206 (2013)].

Formation and migration energies of the vacancy in Si calculated using the HSE06 range-separated hybrid functional

To overcome deficiencies of conventional density functional theory (DFT) utilizing the standard approximation for the exchange-correlation, the revised Heyd-Scuseria-Ernzerhof screened hybrid functional (HSE06) has been used for calculating the formation and migration energies of the vacancy in Si. It is demonstrated that the hybrid approach gives a much more accurate electronic description of the bulk and the vacancy. The correct description of the band gap and the donor transition levels obtained with the HSE06 functional builds confidence in predictions of the vacancy acceptor states. The calculated migration energies of the vacancy with different charge states agree well with low-temperature annealing measurements and, together with formation energies, provide an excellent estimate of the activation energy of vacancy-mediated self-diffusion in silicon.

Contributions of vacancies and self-interstitials to self-diffusion in silicon under thermal equilibrium and nonequilibrium conditions

Since many years, the contribution of vacancies ($V$) and self-interstitials ($I$) to silicon (Si) self-diffusion is a matter of debate. Native defects and their interaction among themselves and with foreign atoms influence the processes taking place during device fabrication, starting with the growth of Si single crystals and ending with doping of nanosized electronic devices. Considering this relevance, it is remarkable that present data about the properties of native point defects in Si are still limited and controversial. This work reports experiments on self-diffusion in Si for temperatures between 650${\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}$C and 960${\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}$C to verify recent results of Shimizu et al. [Phys. Rev. Lett. 98, 095901 (2007)] that give rise to inconsistencies in $V$-mediated self- and dopant diffusion. Two different structures of isotopically controlled epitaxial layers of Si are used for the diffusion study. One structure consisting of 20 bilayers of ${}^{29}$Si/${}^{28}$Si was grown by molecular beam epitaxy (MBE). The other structure with a ${}^{28}$Si layer sandwiched between natural Si was grown by means of chemical vapor deposition. Self-diffusion in (${}^{29}$Si/${}^{28}$Si)${}_{20}$ multilayers (ML) was analyzed by means of secondary ion mass spectrometry (SIMS) and neutron reflectometry, whereas self-diffusion in ${}^{\mathrm{nat}}$Si/${}^{28}$Si/${}^{\mathrm{nat}}$Si sandwich (SW) structures was measured with SIMS only. Analysis of the experimental profiles reveals an enhanced self-diffusion in ML compared to SW structures. The enhanced diffusion is ascribed to the dissolution of $V$- and $I$-related defect clusters grown-in during MBE. On the other hand, self-diffusion in the SW structures accurately confirms the data of Shimizu et al. that are considered to represent data for thermal equilibrium conditions. The temperature dependence of self-diffusion is described by $V$- and $I$-mediated contributions with temperature-dependent thermodynamic properties of $V$. This interpretation can solve the inconsistency between self- and dopant diffusion in Si, but further experiments are required to verify this concept.

Range-Separated Approach to the RPA Correlation Applied to the van der Waals Bond and to Diffusion of Defects

The random-phase approximation (RPA) is a promising approximation to the exchange-correlation energy of density functional theory, since it contains the van der Waals (vdW) interaction and yields a potential with the correct band gap. However, its calculation is computationally very demanding. We apply a range-separation concept to RPA and demonstrate how it drastically speeds up the calculations without loss of accuracy. The scheme is then successfully applied to a layered system subjected to weak vdW attraction and is used to address the controversy of the self-diffusion in silicon. We calculate the formation and migration energies of self-interstitials and vacancies taking into account atomic relaxations. The obtained activation energies deviate significantly from the earlier calculations and challenge some of the experimental interpretations: the diffusion of vacancies and interstitials has almost the same activation energy.

Measurement of photostimulated self-diffusion in silicon

Photostimulated diffusion within solid semiconductors has been examined for many years, but its existence above room temperature has not been unambiguously confirmed. Here, diffusion rates for silicon self-diffusion are shown to change by factors of up to 25 in response to optical fluxes on the order of 1 W/cm2. Results depend on doping type; the rates of both interstitial formation and migration are affected in the case of n-type material. A model based on photostimulated changes in defect charge state explains the primary results, and the basic outlines should apply to a wide variety of semiconductors.

Correlation factors for interstitial-mediated self-diffusion in the diamond lattice: Kinetic lattice Monte Carlo approach

We have performed extensive analysis of the correlation factors for interstitial-mediated self-diffusion via various possible mechanisms and hopping networks in the diamond lattice using the kinetic lattice Monte Carlo approach. The correlation factor for the kick-out mechanism in the tetrahedral hopping network is calculated to be 0.73, in agreement with previous results; and the value for the hexagonal hopping network is 0.47 for the dominant mechanism. For the mechanism where a split interstitial is stable (``stable-split'' mechanism), the correlation factor for the tetrahedral network stays the same while that for the hexagonal network increases to 0.62. We then performed simulations for the diffusion process of silicon involving multiple mechanisms. The choice of mechanisms is justified by ab initio calculations. We conclude that unlike vacancy diffusion, interstitial self-diffusion has a temperature-dependent correlation factor. This conclusion holds in general for diffusion processes involving multiple mechanisms with different activation energies. The correlation factor obtained from ab initio results for interstitial-mediated self-diffusion in silicon at 1000--1100 \ifmmode^\circ\else\textdegree\fi{}C is 0.64--0.80, compared to the value of 0.6 extracted from the experiment.

Ab initiostudy of self-diffusion in silicon over a wide temperature range: Point defect states and migration mechanisms

We identified the states of intrinsic point defects underlying the self-diffusion in Si and clarified the change of dominant diffusion mechanism responsible for the self-diffusion over a wide temperature range using ab initio method. We presented a reliable self-diffusion model that the mechanisms of vacancies and self-interstitials dominate below and above 1220 K, respectively. Our calculations provided a clear picture of Si self-diffusion at lower and higher temperature ascribed to single point defects rather than extended defects. The calculations also provided valuable information on the energy levels and the thermal equilibrium concentrations of point defects, which are highly controversial in experimental reports due to detection limits.

Strain effect on self-diffusion in silicon: Numerical study

We present here a numerical study of the self-diffusion mechanisms in silicon using a semiempirical Stillinger-Weber potential to calculate formation and migration energies and entropies. We find that self-diffusion in bulk Si is mediated by vacancies at low temperature, but that interstitials play a more and more important role when temperature increases, in agreement with recent experimental data. This behavior is shown to strongly evolve under biaxial strain $(ϵ)$ which simulates the effect of epitaxial growth of a Si thin film. Our methodology allows us to classify vacancy vs interstitial self-diffusion within a $(T,ϵ)$ diagram, which reveals a transition from vacancy toward interstitial diffusion at low temperature beyond a critical tensile strain which corresponds to Si/Ge size mismatch.

Maximum a posteriori estimation of activation energies that control silicon self-diffusion

Self-diffusion in crystalline silicon is controlled by a network of elementary steps whose activation energies are important to know in a variety of applications in microelectronic fabrication. The present work employs maximum a posteriori (MAP) estimation to improve existing values for these activation energies, based on self-diffusion data collected at different values of the loss rates for interstitial atoms to the surface. Parameter sensitivity analysis shows that for high surface loss fluxes, the energy for exchange between an interstitial and the lattice plays the leading role in determining the shape of diffusion profiles. At low surface loss fluxes, the dissociation energy of large-atom clusters plays a more important role. Subsequent MAP analysis provides significantly improved values for these parameters.

The vacancy in silicon: Identical diffusion properties at cryogenic and elevated temperatures

The vacancy contribution to self-diffusion in silicon, recently revealed in the study by Shimitzu et al. [Phys. Rev. Lett. 98, 095901 (2007)], is analyzed to demonstrate that the pre-exponential term, as well as the enthalpy term, matches accurately to that of the isolated neutral vacancy, as determined many years earlier at cryogenic temperatures by electron paramagnetic resonance and deep level transient spectroscopy. The persistent suggestion of an “extended” vacancy with very different migrational properties at elevated temperatures is therefore incorrect. There is only one vacancy.