Ab initio thermal transport in compound semiconductors

We use a recently developed ab initio approach to calculate the lattice thermal conductivities of compound semiconductors. An exact numerical solution of the phonon Boltzmann transport equation is implemented, which uses harmonic and anharmonic interatomic force constants determined from density functional theory as inputs. We discuss the method for calculating the anharmonic interatomic force constants in some detail, and we describe their role in providing accurate thermal conductivities in a range of systems. This first-principles approach obtains good agreement with experimental results for well-characterized systems (Si, Ge, and GaAs). We determine the intrinsic upper bound to the thermal conductivities of cubic aluminum-V, gallium-V, and indium-V compounds as limited by anharmonic phonon scattering. The effects of phonon-isotope scattering on the thermal conductivities are examined in these materials and compared to available experimental data. We also obtain the lattice thermal conductivities of other technologically important materials, AlN and SiC. For most materials, good agreement with the experimental lattice thermal conductivities for naturally occurring isotopic compositions is found. We show that the overall frequency scale of the acoustic phonons and the size of the gap between acoustic and optic phonons play important roles in determining the lattice thermal conductivity of each system. The first-principles approach used here can provide quantitative predictions of thermal transport in a wide range of systems.

Figure 1

Calculated ${\kappa }_{\mathrm{L}}$ vs temperature for Si with experimental data for isotopically enriched Si [circles (Ref. 42)] and for naturally occurring Si concentrations [triangles (Ref. 42) and squares (Refs. 43 and 44)]. Solid curves correspond to calculated ${\kappa }_{\mathrm{pure}}$ and dashed curves correspond to ${\kappa }_{\mathrm{natural}}$. ${\kappa }_{\mathrm{L}}$ was determined with anharmonic IFCs calculated in a 64 atom supercell (black curves) and a 216 atom supercell (red curves). The calculated ${\kappa }_{\mathrm{natural}}$ for a 64 atom supercell without enforcing TI is given by the green dashed curve.

Figure 2

Calculated ${\kappa }_{\mathrm{L}}$ vs temperature for Ge with experimental data for isotopically enriched Ge (circles) and for naturally occurring Ge concentrations (triangles) (Ref. 46). ${\kappa }_{\mathrm{L}}$ was determined with TI enforced via the ${\chi }^2$ minimization procedure (red curves), the Lagrange multiplier method (black curves), and the simple ASR method (purple curves). ${\kappa }_{\mathrm{pure}}$ is given by solid curves and ${\kappa }_{\mathrm{natural}}$ is given by dashed curves. The green dashed curve corresponds to ${\kappa }_{\mathrm{natural}}$ without TI conditions enforced.

Figure 3

Calculated ${\kappa }_{\mathrm{natural}}$ vs temperature for GaAs with experimental data for isotopically enriched GaAs [green circles (Ref. 47) and black circles (Ref. 48)] and for naturally occurring Ga concentrations [green triangles (Ref. 47) and black triangles (Ref. 48)]. The calculated curves included second (green), third (solid black), fourth (red), and fifth (blue) nearest neighbors for determining the anharmonic IFCs. The dashed black curve gives ${\kappa }_{\mathrm{natural}}$ for GaAs using IFCs determined from DFPT to seventh nearest neighbors and with long range Coulomb interactions included (described in text).

Figure 4

Calculated ${\kappa }_{\mathrm{pure}}$ vs temperature for AlP (solid green curve), AlAs (solid red curve), and AlSb (solid black curve). Experimental data for ${\kappa }_{\mathrm{L}}$ of AlP are solid green circles (Ref. 51) and for ${\kappa }_{\mathrm{L}}$ of AlSb are solid (Ref. 51) and open (Ref. 52) black circles. An experimental data point for ${\kappa }_{\mathrm{L}}$ of AlAs at $T=$ 300 K (Ref. 53) (not shown) lies directly on the solid green circle. Calculated ${\kappa }_{\mathrm{natural}}$ for AlSb is given by the dashed black curve. The acoustic-only contribution to ${\kappa }_{\mathrm{natural}}$ for AlSb is given by the dotted black curve. The inset gives ${\kappa }_{\mathrm{pure}}$ (black curve), the acoustic contribution to ${\kappa }_{\mathrm{pure}}$ (red curve), and the optic contribution to ${\kappa }_{\mathrm{pure}}$ (green curve) vs temperature for AlSb.

Figure 5

Calculated ${\kappa }_{\mathrm{L}}$ vs temperature for GaSb (black curve), GaAs (green curve), GaP (red curves), and cubic GaN (blue curves). The solid curves give ${\kappa }_{\mathrm{pure}}$ and the dashed curves give ${\kappa }_{\mathrm{natural}}$. Experimental data is shown for ${\kappa }_{\mathrm{L}}$ of GaSb [black circles (Ref. 56)], GaAs [green circles (Ref. 47) and green triangles (Ref. 48)], and GaP [red circles (Ref. 51) and red triangles (Ref. 52)].

Figure 6

Calculated isotope effect $P$ vs temperature for cubic GaN (solid black curve), wurtzite GaN (dashed black curve), GaP (green curve), GaSb (orange curve), diamond (gold curve), Ge (red curve), AlSb (brown curve), and 3C-SiC (blue curve).

Figure 7

Calculated ${\kappa }_{\mathrm{L}}$ vs temperature for the indium-V compounds with experimental data for InP [black circles (Ref. 59), squares (Ref. 60), diamond (Ref. 61), and triangle (Ref. 62)], InAs [red circles (Ref. 63), squares (Ref. 64), diamond (Ref. 65), and triangles (Ref. 66)], and InSb [blue circles (Ref. 56), squares (Ref. 67), and triangles (Ref. 68)]. The dashed black curve corresponds to ${\kappa }_{\mathrm{natural}}$ for InP with harmonic and anharmonic IFCs determined using the lattice constant from energy minimization, $a_{\mathrm{InP}}=$ 5.79 Å. The solid black curve corresponds to ${\kappa }_{\mathrm{natural}}$ using the corrected lattice constant, $a_{\mathrm{InP}}=$ 5.87 Å. Calculated ${\kappa }_{\mathrm{natural}}$ of InAs is given by the dashed [$a_{\mathrm{InAs}}=$ 5.97 Å (energy minimization)] and solid [$a_{\mathrm{InAs}}=6.03$ Å (corrected)] red curves and ${\kappa }_{\mathrm{natural}}$ of InSb is given by dashed [$a_{\mathrm{InSb}}=$ 6.39 Å (energy minimization)] and solid [$a_{\mathrm{InSb}}=$ 6.45 Å (corrected)] blue curves.

Figure 8

Calculated ${\kappa }_{\mathrm{pure}}$ (solid black curve) and ${\kappa }_{\mathrm{natural}}$ (dashed black curve) for 3C-SiC with experimental data given by circles (Ref. 69). The calculated in-plane and out-of-plane ($c$-axis) ${\kappa }_{\mathrm{pure}}$ for AlN are given by the red and green curves, respectively, with experimental data for $c$-axis ${\kappa }_{\mathrm{L}}$ (Ref. 70).

Figure 9

Calculated phonon dispersion for Si in the indicated high symmetry directions (black curves). The pseudopotential used in the LDA/DFPT calculation was Si.pz-vbc.UPF (Ref. 34). Experimental data are given by black circles (Ref. 71).

Figure 10

Calculated phonon dispersion for Ge in the high symmetry directions (black curves). The pseudopotential used in the LDA/DFPT calculation was Ge.pz-bhs.UPF (Ref. 34). Experimental data are given by black circles (Ref. 71).

Figure 11

Calculated phonon dispersion for AlP in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Al.pz-vbc.UPF and P.pz-bhs.UPF (Ref. 34). Experimental data are given by black circles (Ref. 72).

Figure 12

Calculated phonon dispersion for AlAs in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Al.pz-vbc.UPF and As.pz-bhs.UPF (Ref. 34). Experimental data are given by black circles (Ref. 73).

Figure 13

Calculated phonon dispersion for AlSb in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Al.pz-vbc.UPF and Sb.pz-bhs.UPF (Ref. 34). Experimental data are given by black circles (Ref. 74).

Figure 14

Calculated phonon dispersion for GaP in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Ga.pz-bhs.UPF and P.pz-bhs.UPF (Ref. 34). Experimental data are given by black circles (Ref. 75).

Figure 15

Calculated phonon dispersion for GaAs in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Ga.pz-bhs.UPF and As.pz-bhs.UPF (Ref. 34). Experimental data are given by black circles (Ref. 76).

Figure 16

Calculated phonon dispersion for GaSb in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Ga.pz-bhs.UPF and Sb.pz-bhs.UPF (Ref. 34). Experimental data are given by black circles (Ref. 77).

Figure 17

Calculated phonon dispersion for cubic GaN in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Ga.pz-bhs.UPF and N.pz-vbc.UPF (Ref. 34). We note that the calculated dispersion was obtained using a 1$\%$ increase to the lattice constant from energy minimization as discussed in Ref. 16.

Figure 18

Calculated phonon dispersion for InP in the high symmetry directions. Red curves give dispersion using a lattice constant determined by energy minimization and black curves give dispersion using this lattice constant with a 1.5$\%$ increase. The pseudopotentials used in the LDA/DFPT calculation were In.pz-n-bhs.UPF and P.pz-bhs.UPF (Ref. 34). We note that the indium pseudopotential included a core correction. Experimental data are given by black circles (Ref. 78).

Figure 19

Calculated phonon dispersion for InAs in the high symmetry directions. Red curves give dispersion using a lattice constant determined by energy minimization and black curves give dispersion using this lattice constant with a 1$\%$ increase. The pseudopotentials used in the LDA/DFPT calculation were In.pz-n-bhs.UPF and As.pz-bhs.UPF (Ref. 34). We note that the indium pseudopotential included a core correction. Experimental data are given by black circles (Ref. 79) and black squares (Ref. 80).

Figure 20

Calculated phonon dispersion for InSb in the high symmetry directions. Red curves give dispersion using a lattice constant determined by energy minimization and black curves give dispersion using this lattice constant with a 1$\%$ increase. The pseudopotentials used in the LDA/DFPT calculation were In.pz-n-bhs.UPF and Sb.pz-bhs.UPF (Ref. 34). We note that the indium pseudopotential included a core correction. Experimental data are given by black circles (Ref. 81).

Figure 21

Calculated phonon dispersion for cubic 3C-SiC in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Si.pz-vbc.UPF and C.pz-vbc.UPF (Ref. 34). Experimental data are given by black circles (Ref. 82).

Figure 22

Calculated phonon dispersion for wurtzite AlN in the high symmetry directions (black curves). The pseudopotentials used in the LDA/DFPT calculation were Al.pz-vbc.UPF and N.pz-vbc.UPF (Ref. 34). Experimental data are given by black circles (Ref. 83).