^{1,a)}, R. Pinacho

^{1}, M. Jaraiz

^{1}and J. E. Rubio

^{1}

### Abstract

In order to simulate the diffusion kinetics during thermal treatments in SiGe heterostructures, a physically-based atomistic model including chemical and strain effects has been developed and implemented into a nonlattice atomistic kinetic monte carlo (KMC) framework. This model is based on the description of transport capacities of native point defects(interstitials and vacancies) with different charge states in SiGe alloys in the whole composition range. Lattice atom diffusivities have been formulated in terms of point defect transport, taking into account the different probability to move Si and Ge atoms. Strain effects have been assessed for biaxial geometries including strain-induced anisotropic diffusion, as well as charge effects due to strain-induced modifications of the electronic properties. Si-Ge interdiffusion in heterostructures has been analyzed from an atomistic perspective. A limited set of physical parameters have been defined, being consistent with previously reported *ab initio* calculations and experiments. The model has been implemented into a nonlattice KMC simulator and the relevant implementation details and algorithms are described. In particular, an efficient point defect mediated Si-Ge exchange algorithm for interdiffusion is reported. A representative set of simulated interdiffusion profiles are shown, exhibiting good agreement with experiments.

This work has been supported by the Spanish Government (Grant No. TEC2008-05301) and by the Castilla y Leon regional Government (Grant No. VA062A08).

I. INTRODUCTION

II. PHYSICAL MODELS

A. Point-defect diffusion in unstrained SiGe alloys

B. Self-diffusion in unstrained SiGe alloys

C. Parameter set for unstrained SiGe alloys

D. Strain effects on point defectdiffusion in homogeneous SiGe alloys

E. Point defectdiffusion in SiGe heterostructures

F. Lattice atom-diffusion in SiGe heterostructures and graded alloys

III. ATOMISTIC IMPLEMENTATION

A. Implementation scheme

B. Point-defect diffusion implementation

C. Si-Ge interdiffusion algorithm

IV. SIMULATION RESULTS

V. CONCLUSIONS

### Key Topics

- Germanium
- 140.0
- Point defects
- 46.0
- Diffusion
- 39.0
- Heterojunctions
- 25.0
- Chemical interdiffusion
- 22.0

## Figures

Calculated (lines) and experimental (symbols) transport capacities of vacancies, , and self-interstitials, , in pure silicon and germanium in intrinsic conditions vs the inverse of temperature. Solid lines and white symbols correspond to , whereas dashed lines and black symbols correspond to . The experimental data are extracted from Refs. 31 (rhombus) and 46 (triangles) for DC_{I} in Silicon, Ref. 32 (circles) for DC_{V} in Silicon, and Ref. 49 (squares) for DC_{V} in Germanium.

Calculated (lines) and experimental (symbols) transport capacities of vacancies, , and self-interstitials, , in pure silicon and germanium in intrinsic conditions vs the inverse of temperature. Solid lines and white symbols correspond to , whereas dashed lines and black symbols correspond to . The experimental data are extracted from Refs. 31 (rhombus) and 46 (triangles) for DC_{I} in Silicon, Ref. 32 (circles) for DC_{V} in Silicon, and Ref. 49 (squares) for DC_{V} in Germanium.

(Color online) Experimental and calculated Si and Ge diffusivities in silicon and germanium crystals in intrinsic material under equilibrium conditions vs the inverse of temperature. Calculated self-diffusivities (Si in silicon or Ge in germanium) are represented by solid lines, whereas, calculated impurity diffusivities (Ge in silicon or Si in germanium) are represented by dashed lines. Experimental data are from Refs. 35 (filled red rhombus), 36 (filled red squares), 38 (filled red triangles) and 39 (filled red circles) for self-diffusivity in germanium; 39 (empty black circles) and 42 (empty black squares) for diffusion of Si atoms in germanium; 39 (empty red circles), 40 (empty red triangles) and 48 (empty red rhombus) for the diffusivity of Ge atoms in silicon; and 33 (filled black rhombus), 34 (filled black circles), and 37 (filled black triangles) for self-diffusivity in Silicon.

(Color online) Experimental and calculated Si and Ge diffusivities in silicon and germanium crystals in intrinsic material under equilibrium conditions vs the inverse of temperature. Calculated self-diffusivities (Si in silicon or Ge in germanium) are represented by solid lines, whereas, calculated impurity diffusivities (Ge in silicon or Si in germanium) are represented by dashed lines. Experimental data are from Refs. 35 (filled red rhombus), 36 (filled red squares), 38 (filled red triangles) and 39 (filled red circles) for self-diffusivity in germanium; 39 (empty black circles) and 42 (empty black squares) for diffusion of Si atoms in germanium; 39 (empty red circles), 40 (empty red triangles) and 48 (empty red rhombus) for the diffusivity of Ge atoms in silicon; and 33 (filled black rhombus), 34 (filled black circles), and 37 (filled black triangles) for self-diffusivity in Silicon.

(Color online) (a) Ge and (b) Si self-diffusivities in Si_{1−x}Ge_{x} alloys as a function of Ge fraction (D_{Ge}(x) and D_{Si}(x), respectively) at different temperatures in intrinsic conditions. Symbols correspond to experimental values, obtained by Arrhenius interpolation from the available measurements. Experimental values for D_{Ge} are from Refs. 38 (triangles), 39 (circles), and 41 (squares), whereas, experimental values for D_{Si} are from Refs. 33 (rhombus), 39 (circles), 41 (squares) and 42 (triangles). Lines correspond to the calculated values using the parameters of Table I.

(Color online) (a) Ge and (b) Si self-diffusivities in Si_{1−x}Ge_{x} alloys as a function of Ge fraction (D_{Ge}(x) and D_{Si}(x), respectively) at different temperatures in intrinsic conditions. Symbols correspond to experimental values, obtained by Arrhenius interpolation from the available measurements. Experimental values for D_{Ge} are from Refs. 38 (triangles), 39 (circles), and 41 (squares), whereas, experimental values for D_{Si} are from Refs. 33 (rhombus), 39 (circles), 41 (squares) and 42 (triangles). Lines correspond to the calculated values using the parameters of Table I.

(Color online) Calculated relative contribution of the different point-defect charge states to the total transport capacity in intrinsic conditions at 800 °C as a function of the Ge fraction of the Si_{1−x}Ge_{x} alloy.

(Color online) Calculated relative contribution of the different point-defect charge states to the total transport capacity in intrinsic conditions at 800 °C as a function of the Ge fraction of the Si_{1−x}Ge_{x} alloy.

(Color online) Calculated (lines) and experimental (symbols) D_{Ge}/D_{Si} ratio vs temperature in Si_{1−x}Ge_{x} alloys for three different Ge contents: x = 0.05, x = 0.45, and x = 1. Experimental data are extracted from Ref. 41 (for x = 0.05 and x = 0.45) and for the combination of results of Refs. 38 and 42 (for x = 1).

(Color online) Calculated (lines) and experimental (symbols) D_{Ge}/D_{Si} ratio vs temperature in Si_{1−x}Ge_{x} alloys for three different Ge contents: x = 0.05, x = 0.45, and x = 1. Experimental data are extracted from Ref. 41 (for x = 0.05 and x = 0.45) and for the combination of results of Refs. 38 and 42 (for x = 1).

(Color online) Calculated composition dependences of point-defect transport capacities (DC_{V}, dotted lines, and DC_{I}, dashed lines) and self-diffusivities (D_{Ge}, red double lines, and D_{Si}, red solid lines) in Si_{1−x}Ge_{x} alloys at 880 °C in intrinsic conditions for different substrates: (a) silicon, (b) Si_{0.7}Ge_{0.3}, and (c) Si_{0.44}Ge_{0.56,} implying different strain situations. As a reference, DC_{V} and DC_{I} for unstrained alloys are included (thin dotted and thin dashed lines, respectively).

(Color online) Calculated composition dependences of point-defect transport capacities (DC_{V}, dotted lines, and DC_{I}, dashed lines) and self-diffusivities (D_{Ge}, red double lines, and D_{Si}, red solid lines) in Si_{1−x}Ge_{x} alloys at 880 °C in intrinsic conditions for different substrates: (a) silicon, (b) Si_{0.7}Ge_{0.3}, and (c) Si_{0.44}Ge_{0.56,} implying different strain situations. As a reference, DC_{V} and DC_{I} for unstrained alloys are included (thin dotted and thin dashed lines, respectively).

(Color online) Transport energy scheme for a point defect A^{j} jumping along the growth direction from material 1 to material 2, with a frequency , having to surmount an energy barrier () due to its different transport capacities in each material. in each material involves chemical, elastic and electrostatic contributions (see Eq. (19)). represents the migration energy of A^{j} in each material.

(Color online) Transport energy scheme for a point defect A^{j} jumping along the growth direction from material 1 to material 2, with a frequency , having to surmount an energy barrier () due to its different transport capacities in each material. in each material involves chemical, elastic and electrostatic contributions (see Eq. (19)). represents the migration energy of A^{j} in each material.

Illustration of band alignment effects for a Si/Si_{1−x}Ge_{x}/Si undoped heterostructure on a relaxed silicon substrate. The Ge fraction profile is shown in the bottom-left graph and the temperature is 800 °C. The energies of the conduction and valence band edges (E_{c} and E_{v}, respectively) as a function of depth are displayed (a) without taking into account band alignment conditions and (b) enforcing the band offsets calculated from Ref. 66. The intrinsic level e_{i} is also shown. All the energies are referenced to the Fermi level e_{F}. The relative modifications of the transport capacities due to band alignment for charge states j =± 1 and ± 2 are shown in (c). These modifications correspond to the ratio between the values of for panels (b) and (a) and they are calculated using Eq. (10).

Illustration of band alignment effects for a Si/Si_{1−x}Ge_{x}/Si undoped heterostructure on a relaxed silicon substrate. The Ge fraction profile is shown in the bottom-left graph and the temperature is 800 °C. The energies of the conduction and valence band edges (E_{c} and E_{v}, respectively) as a function of depth are displayed (a) without taking into account band alignment conditions and (b) enforcing the band offsets calculated from Ref. 66. The intrinsic level e_{i} is also shown. All the energies are referenced to the Fermi level e_{F}. The relative modifications of the transport capacities due to band alignment for charge states j =± 1 and ± 2 are shown in (c). These modifications correspond to the ratio between the values of for panels (b) and (a) and they are calculated using Eq. (10).

(Color online) Experimental (Ref. 89) (solid lines) and simulated (circles) 880 °C, 90 min annealed profiles of Si/Si_{1−x}Ge_{x}/Si structures grown on three different relaxed Si_{1−y}Ge_{y} pseudosubstrates with: (a) y = 0, (b) y = 0.3, and (c) y = 0.56. As-grown profiles are also included in (a)–(c) (dashed lines). Panel (d) displays the calculated Ge diffusivity D_{Ge}(x) for the strain situations of panels (a) (blue solid line), (b) (green dashed line), and (c) (red dashed-dotted line), as well as for the unstrained case (black dotted line).

(Color online) Experimental (Ref. 89) (solid lines) and simulated (circles) 880 °C, 90 min annealed profiles of Si/Si_{1−x}Ge_{x}/Si structures grown on three different relaxed Si_{1−y}Ge_{y} pseudosubstrates with: (a) y = 0, (b) y = 0.3, and (c) y = 0.56. As-grown profiles are also included in (a)–(c) (dashed lines). Panel (d) displays the calculated Ge diffusivity D_{Ge}(x) for the strain situations of panels (a) (blue solid line), (b) (green dashed line), and (c) (red dashed-dotted line), as well as for the unstrained case (black dotted line).

Experimental (Ref. 10) (solid lines) and simulated (circles) 800 °C, 120 min annealed profiles of Si/Si_{1−z}Ge_{z}/Si_{0.44}Ge_{0.56}/Si_{1−z}Ge_{z}/Si_{0.69}Ge_{0.31} heterostructures on relaxed Si_{0.69}Ge_{0.31} pseudosubstrates, for three different values of the Ge composition z of the intermediate layer: (a) z = 0, (b) z = 0.17, (c) z = 0.32, and (d) z = 0.45. As-grown profiles have been included (dashed lines) for comparison.

Experimental (Ref. 10) (solid lines) and simulated (circles) 800 °C, 120 min annealed profiles of Si/Si_{1−z}Ge_{z}/Si_{0.44}Ge_{0.56}/Si_{1−z}Ge_{z}/Si_{0.69}Ge_{0.31} heterostructures on relaxed Si_{0.69}Ge_{0.31} pseudosubstrates, for three different values of the Ge composition z of the intermediate layer: (a) z = 0, (b) z = 0.17, (c) z = 0.32, and (d) z = 0.45. As-grown profiles have been included (dashed lines) for comparison.

## Tables

Parameters used for native point defect diffusion, self-diffusion and biaxial strain effects in pure silicon (top), pure germanium (middle) and the whole composition range of Si_{1−x}Ge_{x} alloys (bottom). For Si_{1−x}Ge_{x} alloys, the energies and the logarithm of the prefactors are assumed to scale linearly with x, except for those cases in which the second derivative is indicated. All the electronic levels in the Table are referred to the valence band edge and the reported values correspond to the low temperature limit (T = 0). The effective volumes are given in the table in units relative to the atom volume Ω (Ω = 1/C_{at} ≈ 0.2 nm^{−3}). A detailed discussion on the procedure followed for the calibration can be found in the supplemental documentation (Ref. 66).

Parameters used for native point defect diffusion, self-diffusion and biaxial strain effects in pure silicon (top), pure germanium (middle) and the whole composition range of Si_{1−x}Ge_{x} alloys (bottom). For Si_{1−x}Ge_{x} alloys, the energies and the logarithm of the prefactors are assumed to scale linearly with x, except for those cases in which the second derivative is indicated. All the electronic levels in the Table are referred to the valence band edge and the reported values correspond to the low temperature limit (T = 0). The effective volumes are given in the table in units relative to the atom volume Ω (Ω = 1/C_{at} ≈ 0.2 nm^{−3}). A detailed discussion on the procedure followed for the calibration can be found in the supplemental documentation (Ref. 66).

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