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Physical modeling and implementation scheme of native defect diffusion and interdiffusion in SiGe heterostructures for atomistic process simulation
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Image of FIG. 1.
FIG. 1.

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 DCI in Silicon, Ref. 32 (circles) for DCV in Silicon, and Ref. 49 (squares) for DCV in Germanium.

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

(Color online) (a) Ge and (b) Si self-diffusivities in Si1−xGex alloys as a function of Ge fraction (DGe(x) and DSi(x), respectively) at different temperatures in intrinsic conditions. Symbols correspond to experimental values, obtained by Arrhenius interpolation from the available measurements. Experimental values for DGe are from Refs. 38 (triangles), 39 (circles), and 41 (squares), whereas, experimental values for DSi are from Refs. 33 (rhombus), 39 (circles), 41 (squares) and 42 (triangles). Lines correspond to the calculated values using the parameters of Table I.

Image of FIG. 4.
FIG. 4.

(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 Si1−xGex alloy.

Image of FIG. 5.
FIG. 5.

(Color online) Calculated (lines) and experimental (symbols) DGe/DSi ratio vs temperature in Si1−xGex 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).

Image of FIG. 6.
FIG. 6.

(Color online) Calculated composition dependences of point-defect transport capacities (DCV, dotted lines, and DCI, dashed lines) and self-diffusivities (DGe, red double lines, and DSi, red solid lines) in Si1−xGex alloys at 880 °C in intrinsic conditions for different substrates: (a) silicon, (b) Si0.7Ge0.3, and (c) Si0.44Ge0.56, implying different strain situations. As a reference, DCV and DCI for unstrained alloys are included (thin dotted and thin dashed lines, respectively).

Image of FIG. 7.
FIG. 7.

(Color online) Transport energy scheme for a point defect Aj 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 Aj in each material.

Image of FIG. 8.
FIG. 8.

Illustration of band alignment effects for a Si/Si1−xGex/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 (Ec and Ev, 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 ei is also shown. All the energies are referenced to the Fermi level eF. 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).

Image of FIG. 9.
FIG. 9.

(Color online) Experimental (Ref. 89) (solid lines) and simulated (circles) 880 °C, 90 min annealed profiles of Si/Si1−xGex/Si structures grown on three different relaxed Si1−yGey 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 DGe(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).

Image of FIG. 10.
FIG. 10.

Experimental (Ref. 10) (solid lines) and simulated (circles) 800 °C, 120 min annealed profiles of Si/Si1−zGez/Si0.44Ge0.56/Si1−zGez/Si0.69Ge0.31 heterostructures on relaxed Si0.69Ge0.31 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.


Generic image for table
Table I.

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 Si1−xGex alloys (bottom). For Si1−xGex 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/Cat ≈ 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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Physical modeling and implementation scheme of native defect diffusion and interdiffusion in SiGe heterostructures for atomistic process simulation