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The lattice kinetic Monte Carlo simulation (kMCS) was applied to study the boron diffusion in Si-SiGe beyond nanotechnology. Both the interstitialcy and kick-out mechanisms of boron diffusion were considered, including the effects of annealing temperatures, boron dopant concentrations, Ge compositions, and concentrations of Si self-interstitial defects (SiI). The effects on boron diffusion caused by single and double layer(s) of SiGe phase with different Ge contents and varying boron concentrations in double layers of SiGe phase were also simulated. The results show that boron diffusion in Si and between SiGe-Si both largely increase as the temperature or concentration of SiI increases, but the boron diffusion between SiGe-Si is much less than in Si. Increasing the Ge contents in SiGe alloy could retard boron diffusion heavily, while increasing the boron concentration on SiGe phase would enhance boron diffusion.

In situ heavily boron-doped Si_{1-x}Ge_{x} epitaxy that is thermally stable in source/drain regions is very essential for a p-type metal-oxide-semiconductor field effect transistor. As the Si-based device continue downscaling to sub-100 nm range, ultra shallow junctions for source/drain extension (SDE) could be required, therefore, the dopant diffusion in Si_{1-x}Ge_{x} plays an important role to form the ultra shallow pn junctions. It is known to boron atoms diffuse via self-interstitials or vacancies during epitaxy growth [_{I}) introduction [_{I}). In order to reduce boron TED, lots of methods have been proposed, such as rapid thermal annealing (RTA) and impurity doping (introducing N or P into Si substrate) [_{1-x}Ge_{x} alloys [_{1-x}Ge_{x} alloys largely decreased at Ge content up to 40% but increased again at high Ge levels were also reported [_{1-x}Ge_{x} alloy by DFT calculation and they pointed out that the retarded B diffusion could be attributed to the following reasons: 1) the presence of Ge increases the migration energy of a Si self-interstitial defect then reduces the numbers of Si self-interstitials available for B diffusion; and 2) increases the migration energy of the B interstitial [^{ }

The Boron diffusion between the source/drain and the n-well channel of p-type metal-oxide semiconductor (pMOS) which has drawn considerable attention is an important topic because owing to the electrical characteristics properties of n-type substrate could be degraded by boron diffusion., and then the efficiency of pMOS is reduced. In this study, the frameworks of two phases lattice kinetic Monte Carlo simulation (kMCS) were applied to characterize boron diffusion mechanisms between SiGe and Si. The boron atoms are put in a region of width of 54 nm in a lattice of 108 nm × 54 nm area for nano device applications. In the case of small lattice size, the concentration of Si_{I} is very low in our lattice kMCS, and hard to be regarded as independent species to consider (i.e. it is hard to create “one” Si_{I} in the simulation). In order to simulate the boron diffusion under this small size, hence, the “average” Si_{I}, therefore, was applied in place of the “dot” Si_{I}, the details were reported in Section 2. In addition to the effects of concentrations of Si_{I}, boron concentration, Ge composition, and annealing temperature were all considered in our simulations. The simulation details were presented in Section 2. Results and discussion were reported in Section 3. Conclusions were drawn in Section 4.

In order to simulate the boron diffusion between Si and SiGe, it is important to understand the basic diffusion mechanism. Up to now, The mechanisms of boron diffusion have been extensively studied. From the experimental or theoretical evidences, The Boron diffusion would follow two possible mechanisms, the interstitialcy and kick-out mechanisms [

In the B-doped Si or SiGe, several different charge states of species can be found. To purify the problem, only the important charged species for boron diffusion are considered in our simulation. The pair between boron and self-interstitial Si defect plays an important role for boron diffusion. After forming the

In this study, the lattice kinetic Monte Carlo simulation (kMCS) with Metropolis algorithm was performed to simulate the boron diffusion between Si and SiGe. The basic equation of Metropolis algorithm is shown as

where the v is attempt frequency, v_{0} is collision frequency and E is reaction or diffusion activation energy. To apply the kMCS in the simulation of the boron diffusion, the collision frequency and activation energy for each reaction or diffusion path must be determined. Jung et al. had been reported the activation energies of various elementary steps for the boron diffusion [^{−1} is used in our study. For the association reaction of _{s}^{−} and Si_{i}^{+2} must be through the diffusion of Si self-interstitial defect, the association activation energy can be regarded as the activation energy of Si self-interstitial defect diffusion.

Some reports show that the effective range of association of

The effect of Si self-interstitial defect is large and cannot be ignored, because the diffusion of boron must be through the auxiliary of Si self-interstitial defect. Hence, in our simulation, the “average” Si self-interstitial defects instead of the “dot” Si self-interstitial defects. The concentration of Si self-interstitial defect for the perfect Si crystal is

where the E_{f} is the formation energy of Si_{i}.

Because the simulated lattice length is 0.27 nm, the equilibrium constant for Si_{i} formation in the perfect Si crystal is

In order to include the effect of “average” Si self-interstitial defect in the simulation, the rate equation of association reaction of

Because the activation energy of association reaction can determine by considering the diffusion of Si interstitial defect, the reaction constant k_{assoc} can be expressed as

where f is the Debye frequency and E_{diff} is the activation energy of self-interstitial defect diffusion.

Combining the Equations (7)-(9), the rate equation can be written as

Hence the accept frequency for

The concentration of Si self-interstitial defect is an important factor for boron diffusion. Here, the Si self-in- terstitial defect ratio, C_{I}, with respect to the concentration of perfect Si crystal was defined and can be considered as

The diffusion ratio (R_{D}) for boron is defined by Equation (13). The N_{out} is the number of boron atoms which diffuse out of original region (i.e. diffusion number of boron), N_{boron} is the number of total boron atoms.

In the SiGe phase, the effect of Ge for boron diffusion was needed to consider. Many reports show that Ge atom could retard boron diffusion since Ge atom can trap Si self-interstitial defects then reducing the number of Si self-interstitial defects to form the B_{s} – Si_{i} pair. [_{i} diffusion in SiGe phase are also increased about 0.05 eV than in Si phase. When the boron atom diffuses across different phases, such as from SiGe phase into Si phase, the activation energy of boron diffusion is the average activation energy of in SiGe phase and in Si phase, the equation can be shown as

where

Because in the average SiGe phase cannot show the effect on different Ge contents in SiGe phase, so this effect was needed to consider in our simulation. If boron atom diffuses from or into SiGe grid, the activation energy of boron was increased as same as in the SiGe phase. The numbers of first-neighborhood of Ge were also needed to be considered, as the surrounding Ge also affected the boron diffusion [

First, the boron atoms are randomly distributed in a region of width of 54 nm in a lattice of 108 nm × 54 nm as shown in ^{−3} in phase I and the diffusion time is 100 seconds in our simulation. In order to improve the accuracy of simulation and reduce the random error, 400 simulation runs were done for each condition to take the average. The effects of temperature and concentration of Si_{I} on boron diffusion were considered.

_{I} from C_{I} = 1 to C_{I} = 5 at 1300 K, the solid line is the average of totally initial conditions and the dotted line is the average of totally final conditions. It can be found that the initial conditions are not dependent on the temperature or concentrations of Si_{I}, but the final conditions are very different. The boron diffusion length can be to about 9 nm at 1400 K or 6 nm at high concentration of Si_{I} as C_{I} of 5. It also shows that the diffusion number and length of boron both increase when the temperature or concentration of Si_{I} increases. The average diffusion ratio and length of boron in Si phase at boron concentration of 2E19 cm^{−3} with respect to different temperature and concentrations of Si_{I} at 1300 K was collected in

It can be found that the average diffusion length is strongly dependent on temperature. The concentrations of Si_{I} was included in the simulation by controlling the accepted frequency of BI formation and listed in Equation (12), which the pre-exponential factor C_{I} was tuned to decide the concentration of Si_{I}. The results show that the enhancement of boron diffusion ratio is large when increasing the concentration of Si_{I}. As the concentration of

Phase I into phase II | Average diffusion ratio (%) | Average diffusion length (nm) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1300 K | 1325 K | 1350 K | 1375 K | 1400 K | 1300 K | 1325 K | 1350 K | 1375 K | 1400 K | |

Si→Si | 0.94 | 3.33 | 4.63 | 5.72 | 7.57 | 1.06 | 1.65 | 2.27 | 2.77 | 3.42 |

SiGe→Si | 0.15 | 0.94 | 1.17 | 1.83 | 2.36 | 1.02 | 1.50 | 2.23 | 2.68 | 3.70 |

Phase I into phase II | Average diffusion ratio (%) | Average diffusion length (nm) | ||||||||

C_{I} = 1 | C_{I} = 2 | C_{I} = 3 | C_{I} = 4 | C_{I} = 5 | C_{I} = 1 | C_{I} = 2 | C_{I} = 3 | C_{I} = 4 | C_{I} = 5 | |

Si→Si | 0.94 | 1.63 | 1.58 | 1.86 | 1.91 | 1.06 | 1.51 | 1.70 | 1.98 | 2.22 |

SiGe→Si | 0.15 | 0.19 | 0.22 | 0.45 | 0.63 | 1.02 | 0.89 | 1.77 | 1.79 | 1.75 |

Si→SiGe | 1.24 | 1.76 | 2.37 | 2.44 | 2.62 | 0.57 | 0.67 | 0.81 | 0.86 | 1.07 |

SiGe→SiGe | 0.31 | 0.51 | 0.68 | 0.83 | 0.89 | 0.43 | 0.66 | 0.76 | 1.04 | 1.13 |

Si_{I} increases, the values of average diffusion ratio are 0.94, 1.63, 1.58, 1.86 and 1.91% and the values of average diffusion length are 1.06, 1.51, 1.70, 1.98, and 2.22 nm with respect to C_{I} = 1, 2, 3, 4 and 5 respectively. It again shows the average diffusion ratio and length both largely increase when the temperature or concentration of Si_{I} increases. To compare the effect between the temperature and the concentration of Si_{I}, it shows that the effect of temperature on the boron diffusion is larger than that of concentration of Si_{I}.

By changing the phase I in ^{−3} in phase I and the diffusion time is 100 seconds in our simulation. Again, the effects of temperature and concentration of Si_{I} on boron diffusion were considered.

It can be found that the diffusion lengths of boron largely increase when the temperature increases. The average diffusion ratio and length of boron from SiGe phase into Si phase at concentration of boron = 2E19 cm^{3} with respect to different temperature and concentrations of Si_{I} at 1300 K was collected in _{I} increases from C_{I} = 1 to C_{I} = 5, and (b) different concentration of Si self-interstitial defects from 1.02 to 1.75 nm. Again, it shows that the effect of temperature on the boron diffusion is larger than that of at 1300K, the average diffusion ratio increases from 0.15% to 0.63% and the average diffusion length increases concentration of Si_{I}.

_{I}. The average diffusion ratio and length generally increase when the temperature or concentration of Si_{I} increases. As can be seen from

By varying the Ge content from 0% to 50% in SiGe phase of single layer in phase I as shown in ^{−3} and 1300 K. As can be seen in

The effect on boron diffusion caused by double layers of SiGe phase with different composition of Ge as shown in

of boron atoms which diffuse into phase III and IV were too low, so the exchange of boron atoms between phase III and IV were neglected that can be concluded the boron diffusion into phase III(IV) was all from phase I(II). 1000 simulation runs were done for each condition to take the average. ^{−3} and 1300 K at different combination of Ge composition on two layers of SiGe phase. As can be seen from _{15%} (top) + Ge_{30%} (bottom), the average diffusion ratio was slightly reduced comparing with that for homogenous SiGe configuration of Ge_{26%}. It also can be found that the exchange of boron atoms between phase I and phase II in the homogenous SiGe of Ge_{26%} and Ge_{30%} is nearly zero. But in nonhomogenous SiGe phase of Ge_{15%} (top) + Ge_{30%} (bottom), it has some boron atoms diffused from phase I into phase II since the Ge content between these two phases is different.

By changing the boron concentration, one can study the boron concentration effect on boron diffusion. ^{−3} in top layer of Ge_{15%} and from 1.5E19, 2E19 to 4E19 cm^{−3} in bottom layer of Ge_{30%} on SiGe phase at 1300 K. As shown in ^{−3} and 4E19 cm^{−3}, but do not effectively change the average diffusion length. It also can be found the exchange of boron atoms between phase I and phase II in homogenous boron concentration is smaller than in nonhomogeneous boron concentration. It can be concluded that the boron atoms would diffuse from high boron concentration to low boron concentration on double layers of SiGe phase.

By increasing the boron concentration from 2E19, 4E19 to 8E19 cm^{−3} in the top layer of SiGe phase and keeping the boron concentration at 2E19 cm^{−3} in the bottom layer of SiGe phase, the results show that the increase of boron concentration in top layer would increase the total average diffusion number from 0.41, 0.52 to 0.79, respectively. It can be found that the increase of total average diffusion number can be attributed to the increase of average diffusion number in phase III. Since the boron concentration in the top layer of SiGe phase increases twice from 2E19 cm^{−3} to 4E19 cm^{−3}, the average diffusion numbers also increase about twice from 0.13 to 0.26 in phase III, but do not effectively change in phase IV. It also can be found the quantity of boron exchange between phase I and phase II largely increase from 0.02, 0.40 to 1.11 when the boron concentration increases in the top layer of SiGe phase.

By keeping the boron concentration at 4E19 cm^{−3} in the top layer of SiGe phase and changing the boron concentration from 1.5E19, 2E19 to 4E19 cm^{−3} in the bottom layer of SiGe phase, the results show the increase of B concentration in bottom layer would strongly increase the total average diffusion number from 0.47, 0.52 to 0.89, but the quantity of boron exchange between phase I and phase II is decreased from 0.50, 0.40 to 0.09, respectively. The decrease of boron exchange can be attributed to the boron concentration in bottom layer more and more close to the top layer. The increase of total average diffusion number can be contributed to the increase of average diffusion number in phase IV, as the boron concentration in the bottom layer of SiGe phase increases twice from 2E19 cm^{−3} to 4E19 cm^{−3}, the average diffusion number also increases about twice from 0.26 to 0.62 in phase IV.

Composition of Ge (%) | No. boron | Average diffusion ratio (%) | Average diffusion length (nm) | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Phase I | Phase II | Total | Phase III | Phase IV | Phase I to II | |||||

15 | 55 (16 + 39) | 0.82 (0.45) | 0.28 (0.16) | 0.54 (0.29) | −0.08 (−0.04) | 1.11 | ||||

26 | 0.78 (0.43) | 0.21 (0.11) | 0.57 (0.31) | 0. (0.) | 1.09 | |||||

30 | 0.72 (0.40) | 0.20 (0.11) | 0.52 (0.29) | 0. (0.) | 1.13 | |||||

15 | 30 | 0.74 (0.41) | 0.24 (0.13) | 0.50 (0.28) | 0.05 (0.02) | 1.10 | ||||

Concentration of B (cm_{−3}) | Number of B (I + II) | Average diffusion ratio (%) | Average diffusion length (nm) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Phase I | Phase II | Total | Phase III | Phase IV | Phase I to II | ||||||

2E19 | 2E19 | 55 (16 + 39) | 0.74 | (0.41) | 0.24 | (0.13) | 0.50 | (0.28) | 0.05 | (0.02) | 1.10 |

4E19 | 1.5E19 | 62 (31 + 31) | 0.75 | (0.47) | 0.39 | (0.24) | 0.36 | (0.23) | 0.81 | (0.50) | 1.12 |

4E19 | 2E19 | 70 (31 + 39) | 0.74 | (0.52) | 0.37 | (0.26) | 0.37 | (0.26) | 0.57 | (0.40) | 1.13 |

2E19 | 4E19 | 95 (16 + 79) | 0.73 | (0.69) | 0.15 | (0.14) | 0.58 | (0.55) | −0.25 | (−0.24) | 1.14 |

8E19 | 2E19 | 102 (63 + 39) | 0.78 | (0.79) | 0.49 | (0.49) | 0.29 | (0.30) | 1.09 | (1.11) | 1.16 |

4E19 | 4E19 | 110 (31 + 79) | 0.81 | (0.89) | 0.25 | (0.27) | 0.56 | (0.62) | 0.08 | (0.09) | 1.15 |

Condition | Average diffusion ratio (%) | Average diffusion length (nm) | |||
---|---|---|---|---|---|

Total | Phase III | Phase IV | Phase I to II | ||

1150 K | 0.03 (0.02) | 0.01 (0.01) | 0.02 (0.01) | 0.01 (0.01) | 0.28 |

1200 K | 0.11 (0.08) | 0.05 (0.04) | 0.06 (0.04) | 0.06 (0.04) | 0.40 |

1250 K | 0.37 (0.26) | 0.16 (0.11) | 0.21 (0.15) | 0.24 (0.17) | 0.57 |

1300 K | 0.74 (0.52) | 0.37 (0.26) | 0.37 (0.26) | 0.57 (0.40) | 1.13 |

1350 K | 1.71 (1.19) | 0.75 (0.52) | 0.96 (0.67) | 1.48 (1.03) | 2.16 |

C_{I} = 1 | 0.74 (0.52) | 0.37 (0.26) | 0.37 (0.26) | 0.57 (0.40) | 1.13 |

C_{I} = 2 | 1.11 (0.78) | 0.53 (0.37) | 0.58 (0.41) | 0.83 (0.58) | 1.51 |

C_{I} = 3 | 1.39 (0.97) | 0.72 (0.50) | 0.67 (0.47) | 1.10 (0.77) | 1.82 |

C_{I} = 4 | 1.48 (1.04) | 0.70 (0.49) | 0.79 (0.55) | 1.16 (0.81) | 2.17 |

C_{I} = 5 | 1.69 (1.18) | 0.70 (0.49) | 0.99 (0.69) | 1.49 (1.04) | 2.24 |

centration = 4E19 cm^{−3} on top layer of Ge_{15%} and boron concentration = 2E19 cm^{−3} on bottom layer of Ge_{30%} to Si phase with respect to different temperature and concentrations of Si_{I} at 1300 K. The diffusion time is 100 seconds and 400 simulation runs were done for each condition to take the average. When the temperature or concentrations of Si_{I} increases, the average diffusion ratio of boron in phase III, phase IV and between phase I and phase II all increase, and the average diffusion length also increase largely.

First, the effects of temperature and concentration of Si_{I} on boron diffusion in pure Si phase were considered. The results show the boron diffusion ratio and length both largely increase as the temperature or concentration of Si_{I} increases. It also shows the effect of temperature on the boron diffusion is larger than that of concentration of Si_{I}. For the boron diffusion from SiGe into Si phase, similar trends were found for the effects of temperature and concentration of Si_{I}, but the boron diffusion largely decreases compared to that in pure Si. The results of boron diffusion from SiGe phase at different compositions of Ge into Si phase show that the increase of Ge content could effectively decrease the boron diffusion, but the diffusion length is nearly the same. The effect on boron diffusion caused by double layers of SiGe phase with different Ge contents was also simulated. The results show the boron diffusion in nonhomogenous SiGe phase is slightly smaller than in homogenous SiGe phase. By fixing the Ge content of 15% on top layer and 30% on bottom layer of SiGe phase and varying the boron concentration on top- or bottom-layer in SiGe phase, the results show that the boron diffusion increases as the total boron concentration increases, while the exchange of boron atoms between these two layers would increase as the difference of boron concentration increases.

Financial assistance from National Science Council, Taiwan is gratefully acknowledged. We are also grateful to the National Center for High-performance Computing for computer time and facilities.