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Bonny
Dongre
*^{a},
Jesús
Carrete
^{a},
Shihao
Wen
^{b},
Jinlong
Ma
^{b},
Wu
Li
^{b},
Natalio
Mingo
^{c} and
Georg K. H.
Madsen
^{a}
^{a}Institute of Materials Chemistry, TU Wien, A-1060 Vienna, Austria. E-mail: bonny.dongre@tuwien.ac.at; Tel: +43 1 58801165307
^{b}Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
^{c}LITEN, CEA-Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France

Received
16th October 2019
, Accepted 5th December 2019

First published on 6th December 2019

The mechanisms causing the reduction in lattice thermal conductivity in highly P- and B-doped Si are looked into in detail. Scattering rates of phonons by point defects, as well as by electrons, are calculated from first principles. Lattice thermal conductivities are calculated considering these scattering mechanisms both individually and together. It is found that at low carrier concentrations and temperatures phonon scattering by electrons is dominant and can reproduce the experimental thermal conductivity reduction. However, at higher doping concentrations the scattering rates of phonons by point defects dominate the ones by electrons except for the lowest phonon frequencies. Consequently, phonon scattering by point defects contributes substantially to the thermal conductivity reduction in Si at defect concentrations above 10^{19} cm^{−3} even at room temperature. Only when, phonon scattering by both point defects and electrons are taken into account, excellent agreement is obtained with the experimental values at all temperatures.

Owing to its high abundance, non-toxicity, and ease of dopability, Si continues to be the linchpin of the semiconductor industry. Highly P- and B-doped Si are routinely used as source/drain materials in transistors to avoid unwanted Schottky junctions.^{3} Furthermore, highly-doped Si has also found usage in photovoltaics,^{4} microelectromechanical systems,^{5} and microelectronics,^{6,7} to cite a few applications. In thermoelectric applications the advantage of using highly-doped Si is twofold:^{8–10} the thermoelectric figure of merit is, on the one hand, proportional to the electronic power factor, which increases with increasing carrier concentration, and, on the other, inversely proportional to the thermal conductivity. A recent advancement in the synthesis of such highly-doped systems was achieved by Rohani et al.,^{11} who were able to dope Si nanoparticles with boron beyond its equilibrium solubility limit.

The lattice contribution to the thermal conductivity dominates in Si. Phonon scattering by point defects (PDPS) and by electrons (EPS), were identified as the two main contributors to the reduction in highly-doped Si.^{9,10,12,13} Zhu et al.^{10} reported a ≈ 36% reduction in in fine-grained, highly P-doped Si, and asserted that EPS is the major contributor to the reduction. In contrast, Ohishi et al.^{9} attributed the reduction in single-crystal, highly P- and B-doped Si solely to intrinsic anharmonic phonon–phonon scattering and PDPS.

The aforementioned disagreement highlights the problem in studying thermal transport when employing fitted models based on strongly-simplified assumptions about the underlying phonon band structures and scattering mechanisms. Important progress towards a more predictive treatment of in doped Si was made recently by Liao et al.,^{13} who performed an ab initio study of n- and p-doped Si and showed that, EPS at a carrier concentration of p ≈ 10^{21} cm^{−3} can result in a ≈ 45% reduction in at room temperature. The calculations reproduce how is lower in p-doped samples than in n-doped ones, in agreement with the experiments.^{14,15} However, they do not capture the magnitude of the reduction observed in B-doped p-type single-crystal Si, which at a doping level of 5 × 10^{20} cm^{−3} amounts to more than 70%.^{14}

In the present work, we investigate the precise mechanisms responsible for the reduction observed in highly-doped Si. We calculate by employing the Boltzmann transport equation (BTE) for phonons, using only inputs in the form of interatomic force constants (IFCs) and electron–phonon coupling (EPC) matrix elements obtained from density functional theory. We extend the earlier work on EPS^{13} and include also the PDPS from first principles. At high defect concentrations, we find that the PDPS rates dominate the EPS rates at all frequencies except the lowest ones and contribute substantially to the reduction at all temperatures. On the other hand, EPS dominates at low defect concentrations due to a fundamentally different frequency and concentration dependence. As a result, a correct quantitative prediction of the dependence on defect concentration and temperature is obtained only when both EPS and PDPS are taken into account.

(1) |

(2) |

(3) |

T = (1 − Vg^{+})^{−1}V. | (4) |

The matrix element of g^{+} projected on the atom pairs lη and l′η′ is given as:^{19}

(5) |

V = V_{M} + V_{K}, | (6) |

(7) |

(8) |

The EPC matrix element can be computed within density functional perturbation theory as:^{21}

(9) |

For the calculation of EPS rates, the EPC matrix elements are first computed on coarse grids and then interpolated to dense grids with the Wannier function interpolation method.^{31,32} The interpolations are performed using Quantum Espresso^{33} and the built-in EPW package^{34} with norm-conserving pseudopotentials. Likewise, both the LDA and PBE exchange and correlation functionals are considered. The initial k and q grids are both 6 × 6 × 6, which are interpolated to 35 × 35 × 35 meshes needed for the thermal conductivity calculations. The energy conservation δ-function is treated by Gaussian function with self-adaptive broadening parameters.^{21}

Finally, the bulk thermal conductivity is also calculated using the 35 × 35 × 35 q-point mesh with the ALMABTE^{35} package. Due to the dense q-mesh the thermal conductivity is converged down to 40 K.

Fig. 2 The low-frequency EPS rates for different concentrations as a function of the electronic density of states. The marker colors correspond to the doping levels in the inset. |

If we first consider the low frequency (ω < 12 rad ps^{−1}) behavior, the PDPS rates exhibit a simple Rayleigh ω^{4} behavior and the EPS rates for a given carrier concentration are close to being independent of ω. Plotting the EPS rates as a function of the electronic density of states (DOS), obtained at the electron chemical potential corresponding to a given doping concentration, shows an almost linear dependence (Fig. 2). The EPS rates thus behave in accordance with a simple τ^{−1} ∝ n(ε) model^{36} at low frequencies. With a n(ε) ∝ε^{1/2} behavior of the electronic DOS, the low-frequency EPS rates will scale approximately as n_{def}^{1/3} with the carrier/defect concentration as opposed to the linear scaling of the PDPS rates evident from eqn (3). These simple relations are in accordance with the expectation that EPS will dominate over PDPS at low temperatures and defect concentrations while PDPS will become increasingly important at high defect concentrations.

At the same time, it is clear from Fig. 1 that for ω > 12 rad ps^{−1} the calculated rates deviate substantially from the aforementioned simple relations. The cumulative plots in the respective top panels in Fig. 1 (blue curves) illustrate that modes with ω > 12 rad ps^{−1} carry about two-thirds of the heat at 300 K. Simply extrapolating the low frequency behavior would lead to a strong overestimation of the scattering. This is especially so for the EPS rates where a simple extrapolation would result in a strong overestimation of the predicted suppression. The cumulative plots in the top panels of Fig. 1 also show for PDPS (dotted lines), at a large defect concentration, that the contribution to for frequencies higher than 20 rads ps^{−1} is only ≈15% in case of B_{Si}^{(−1)} defect and ≈40% in case of P_{Si}^{(+1)}. In contrast, the EPS causes a majority reduction in by frequencies below 20 rads ps^{−1}, for both P- and B-doping.

Next, in Fig. 3, we look into the individual contributions from the EPS (dashed lines) and PDPS (dotted lines) to the room temperature thermal conductivity reduction for increasing doping concentrations and compare them to the experimental data from Slack.^{14} In accordance with the analysis of the scattering rates, EPS dominates the reduction of at low carrier concentrations. In the case of n-doped Si, EPS alone is almost enough to explain the experimental point at 2 × 10^{19} cm^{−3}. However, at higher carrier concentrations, EPS alone clearly underestimates both the absolute reduction of and the trend. Interestingly, PDPS captures the trend correctly for large defect concentrations, but also underestimates the absolute reduction. At a doping concentration of 10^{21} cm^{−3}, the EPS and PDPS contribute almost equally in reduction for B doping. Even though both EPS and PDPS contribute substantially to reduction in Si, neither can explain the absolute reduction in on its own. Only when both are taken into consideration in eqn (2) is the experimentally observed reduction of reproduced. This is shown by the black and red solid lines in Fig. 3. Apart from a slight underestimation of in case of B-doping, the calculated value of considering both EPS and PDPS agree very well with the experimental values available for concentrations ∼10^{19}–10^{21} cm^{−3} both in value and trend, Fig. 3.

Fig. 3 Comparison of the reduction in caused by EPS and PDPS individually and combined together vs. increasing doping concentration. The filled squares are from the experimental data in ref. 14. |

Besides the work of Slack et al.^{14} used for Fig. 3, there is a general lack of systematic experimental data on the thermal conductivity of single-crystal Si with varying doping concentrations. In order to gain further confidence in our results, we compare them to the more recent experimental data from ref. 15, which were measured on single-crystal Si films. Even at low defect concentrations, n_{def} = 10^{17}–10^{18} cm^{−3}, these samples exhibit a substantially lower than the bulk samples.^{15} However, a good agreement with the experimental curves can be obtained by adding a simple boundary scattering term, 1/τ^{B}_{iq} = |v_{iq}|/L with L = 10 µm, to eqn (2) to emulate the effect of a film, as seen in Fig. 4 (purple line). Adding now the effect of the PDPS and EPS, we calculate the variation of as a function of temperature for both B- and P-doped Si films, shown in Fig. 4. For B-doping, when we include the PDPS and EPS along with the boundary scattering, we see that there is only a slight reduction from the purple line for the 10^{18} cm^{−3} doping level (dashed red lines). Nevertheless, this results in an excellent agreement with the experimental data at that concentration (red circles), throughout the temperature range. Keeping the boundary scattering constant, when the doping concentration is increased to 10^{19} cm^{−3}, a large reduction in is observed (solid red line) which also agrees well with the experimental data (red circles). Similarly, for the P-doped calculations, we obtain an excellent agreement with the experiments except for a slight underestimation in at temperatures below 80 K for 10^{18} cm^{−3} doped case. However, this is still under the uncertainties in experimental data.

Fig. 4
vs. temperature curves for the B(red)- and P(black)-doped Si considering the three-phonon, PDPS, EPS, and boundary scattering (at 10 µm). The open triangles and circles are the experimental data obtained from ref. 15. The solid and dashed (black and red) lines correspond to the calculations done at experimental carrier concentrations. |

We then make predictions for the highly-doped cases (10^{20} and 10^{21} cm^{−3}) as there is no experimental thermal conductivity data available for such high doping levels. These are shown in Fig. 5. At room temperature, we find more than 60% reduction as compared to the bulk value at 10^{20} cm^{−3} doping level and 90% at 10^{21} cm^{−3} for B doping, and 40% and 80% for P doping, respectively. Fig. 5 also shows the individual contributions to the reduction by EPS and PDPS in the case of B doping at a doping concentration of 10^{21} cm^{−3}, revealing their characteristic effects on the thermal conductivity over the temperature range. The reduction in caused by PDPS overtakes the one by EPS at ≈250 K. This behavior highlights the importance of PDPS at high doping concentrations in Si and also emphasizes that, at temperatures higher than 300 K, PDPS is the most dominant scattering mechanism besides the intrinsic anharmonic scattering. We have also performed calculations including an L = 10 µm boundary scattering term, however at such high doping concentrations the EPS and PDPS dominate the boundary scattering throughout the temperature range and only a small effect at very low temperatures was found on the calculated . This observation is also in agreement with Fu et al.^{37} who observed significant contribution of EPS towards room-temperature reduction in highly-doped Si nanowires of 1 µm diameter.

Fig. 5 Prediction of for highly P- and B-doped Si. Also shown is the comparison of EPS and PDPS to reduction considered at a concentration of 10^{21} cm^{−3}. |

It is important to note how the present case is very different from our previous work on SiC.^{38,39} In the present work, we observe a significant contribution of both EPS and PDPS to of a highly-doped system; whereas in our previous work we observed that PDPS was sufficient on its own to correctly predict the of B-doped cubic SiC owing to the resonant phonon scattering that boron causes.^{38,39} The resonant scattering was at least one to two orders of magnitude higher than that caused by other defects and resulted in a drastic reduction (approximately two orders of magnitude at room temperature) in the thermal conductivity even at relatively modest defect concentrations (≈10^{20} cm^{−3}). Boron does not cause resonant scattering in Si and therefore the contribution of both EPS and PDPS are comparable.

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