Tin-pest problem as a test of density functionals using high-throughput calculations

At ambient pressure tin transforms from its ground state, the semimetal $\alpha$-Sn (diamond structure), to metallic $\beta$-Sn at $13{}^{\circ }\mathrm{C}$ (286 K). There may be a further transition to a simple hexagonal phase, $\gamma$-Sn, above 450 K. These relatively low transition temperatures are due to the small energy differences between the structures, $\approx 20$ meV/atom between $\alpha$- and $\beta$-Sn, which makes tin an exceptionally sensitive test of the accuracy of density functionals and computational methods used in calculating electronic and vibrational energy, including zero-point energy. Here we use the high-throughput automatic-flow (AFLOW) method to study the energetics of tin in multiple structures using a variety of density functionals and examine the vibrational contributions to the free energy with the AFLOW Automatic Phonon Library (APL). We look at the successes and deficiencies of each functional. We also discuss the necessity of testing high-throughput calculations for convergence of systems with small energy differences.

Figure 1

The three low-energy structures of tin, drawn approximately to scale. Top: $\alpha$-Sn, Strukturbericht $A4$ (gray tin, diamond structure), AFLOW Designation A_cF8_227_a. Middle: $\beta$-Sn, Strukturbericht $A5$ (white tin), AFLOW Designation A_tI4_141_a. Bottom: simple hexagonal $\gamma$-Sn, Strukturbericht$A_f$, AFLOW Designation A_hP1_191_a. The conventional cells are shown for the cubic $\alpha$-Sn and tetragonal $\beta$-Sn. The $\gamma$-Sn figure contains three primitive cells to show the hexagonal structure.

Figure 2

Electronic density of states for $\alpha$-, $\beta$-, and $\gamma$-Sn computed using the LDA at the equilibrium structure of each phase found in Table 3.

Figure 3

Static lattice energy-volume curves for the tin structures discussed in Sec. 2 as predicted by AFLOW/VASP using the LDA functional. Structural notation is from Table 1. We plot $\mathrm{\Delta }\mathrm{U}$, the change in energy per atom compared to the equilibrium energy of the $\alpha$-Sn ($A4$) structure. The lines labeled “A5 Expt.” and “A4 Expt.” represent the experimental volume of $\beta$-Sn ($A5$) at 298 K [77] and $\alpha$-Sn ($A4$) at 90 K [74], respectively.

Figure 4

Electronic density of states for $\alpha$-, $\beta$-, and $\gamma$-Sn computed using the PBE functional at the equilibrium structure of each phase found in Table 3.

Figure 5

Static lattice energy-volume curves for the tin structures discussed in Sec. 2 as predicted by AFLOW/VASP using the PBE functional. The notation is identical to that in Fig. 3.

Figure 6

Electronic density of states for $\alpha$-, $\beta$-, and $\gamma$-Sn computed using the PBEsol functional at the equilibrium structure of each phase found in Table 3.

Figure 7

Static lattice energy-volume curves for the tin structures discussed in Sec. 2 as predicted by AFLOW/VASP using the PBEsol functional. The notation is identical to that in Fig. 3.

Figure 8

Electronic density of states for $\alpha$-, $\beta$-, and $\gamma$-Sn computed using the AM05 functional at the equilibrium structure of each phase found in Table 3.

Figure 9

Static lattice energy-volume curves for the tin structures discussed in Sec. 2 as predicted by AFLOW/VASP using the AM05 functional. The notation is identical to that in Fig. 3.

Figure 10

Electronic density of states for $\alpha$-, $\beta$-, and $\gamma$-Sn computed using the revTPSS functional at the equilibrium volume for each structure found in Table 3.

Figure 11

Static lattice energy-volume curves for most of the tin structures discussed in Sec. 2 as predicted by AFLOW/VASP using the revTPSS functional. The notation is identical to that in Fig. 3.

Figure 12

Electronic density of states for $\alpha$-, $\beta$-, $\gamma$-Sn computed using the SCAN functional at the equilibrium volume for each structure found in Table 3. We also include the DOS of simple cubic tin at its SCAN equilibrium structure, $a=3.0432\AA$.

Figure 13

Static lattice energy-volume curves for the tin structures discussed in Sec. 2 as predicted by AFLOW/VASP using the SCAN functional. The notation is identical to that in Fig. 3. Structures not shown ($A1,A2,A3,A6$) are above the $\mathrm{\Delta }U=150$ meV/atom limit of the graph.

Figure 14

Phonon spectra of $\gamma$-Sn using the PBE functional at equilibrium as given in Table 2. The solid line shows the results for the 125 atom supercell, while the dashed line shows the 216 atom supercell. The high-symmetry paths through the face-centered cubic Brillouin zone are defined by Setyawan and Curtarolo [80].

Figure 15

Phonon density of states for the same cell described in Fig. 14. The solid line shows the results for the 125 atom supercell, while the dashed line shows the 216 atom supercell. The density of phonon modes is normalized for one atom, so the area under each curve is equal to three.

Figure 16

Vibrational free energy (5) for the same cell described in Fig. 14. The left axis shows $F_{\mathrm{ph}}$ for the 126 atom supercell (solid line) and the 216 cell (dashed line). The dotted line is the difference in the vibrational free energy between the cells, with the scale on the right.

Figure 17

Vibrational free energy (5) for $\alpha$-Sn using the PBE functional at equilibrium (Table 2). The left axis shows $F_{\mathrm{ph}}$ for the 128 atom supercell (solid line) and the 250 atom supercell (the barely distinguishable dashed line). The dotted line is the difference between the cells, with the scale on the right.

Figure 18

Vibrational free energy (5) for $\beta$-Sn using the PBE functional at equilibrium (Table 2). The left axis shows $F_{\mathrm{ph}}$ for the 180 atom supercell (solid line) and the 448 atom supercell. The dotted line is the difference between the cells, with the scale on the right.

Figure 19

Phonon spectra of $\alpha$-Sn at the experimental volume. The solid lines are the frequencies predicted using the APL module of AFLOW with the PBE functional, and the dashed lines are the predicted frequencies for the SCAN functional. The circles are the frequencies measured by Price et al. [74] at 90 K. The calculations use the supercell described in Sec. 4. The high-symmetry paths through the face-centered cubic Brillouin zone are defined by Setyawan and Curtarolo [80]. Note that Price et al. only determined the frequencies of one of the transverse branches along the $\mathrm{\Gamma }\text{−}K$ and $U\text{−}X$ directions (the $\mathrm{\Sigma }$ line).

Figure 20

Phonon spectra of $\beta$-Sn at the experimental volume. The solid lines are the frequencies predicted using the APL module of AFLOW with the PBE functional, and the dashed lines are the predicted frequencies for the SCAN functional. In both cases the volume is held fixed, but the value of $c/a$ is chosen to minimize the total energy. The calculations use the supercell described in Sec. 4. The circles are the frequencies measured by Price [76] at 300 K and Rowe et al. [75] at 296 K. The high-symmetry paths through the body-centered tetragonal Brillouin zone are defined by Setyawan and Curtarolo [80]. Note that Price only determined the frequencies of one acoustic and one optic transverse branch along the $\mathrm{\Gamma }\text{−}X$ ($\mathrm{\Delta }$) line. Data points were obtained from the references using the Engauge Digitizer [72].

Figure 21

Phonon spectra of computational $\gamma$-Sn and experimental ${\mathrm{Sn}}_{0.8}{\mathrm{In}}_{0.2}$ at the experimental volume. The solid lines are the frequencies predicted using the APL module of AFLOW with the PBE functional, and the dashed lines are the predicted frequencies for the SCAN functional. In both cases the volume is held fixed, but the value of $c/a$ is chosen to minimize the total energy. The calculations use the supercell described in Sec. 4. The circles are data taken by Ivanov et al. [14] for ${\mathrm{Sn}}_{0.8}{\mathrm{In}}_{0.2}$ at room temperature. The high-symmetry paths through the simple hexagonal Brillouin zone are defined by Setyawan and Curtarolo [80]. Data points were obtained from the references using the Engauge Digitizer [72].

Figure 22

Phonon density of states for $\alpha$- (solid line), $\beta$- (dashed line) and $\gamma$-Sn (dotted line) at the equilibrium configurations from Table 3. Top panel: PBE functional. Bottom panel: SCAN functional. We show the number of phonon modes per atom, thus the area under each curve is three.

Figure 23

The primitive-cell volume of $\alpha$-Sn as a function of temperature (left axis, red circles) and the linear expansion coefficient $\alpha$ (right axis, blue triangles) calculated by APL using the PBE density functional. We also plot the experimental data found in Touloukian et al. [83] (black diamonds).

Figure 24

The primitive-cell volume of $\alpha$-Sn as a function of temperature (left axis, red circles) and the linear expansion coefficient $\alpha$ (right axis, blue triangles) calculated by APL using the SCAN density functional. We also plot the experimental data found in Touloukian et al. [83] (black diamonds).

Figure 25

The primitive-cell volume of $\beta$-Sn as a function of temperature (left axis, red circles) and the linear expansion coefficient $\alpha$ (right axis, blue triangles) calculated by APL using the PBE density functional. We also plot the value of $\alpha$(T) found in Touloukian et al. [83] (solid black diamonds) and measured by Deshpande and Sirdeshmukh [77] (open black diamonds).

Figure 26

The primitive-cell volume of $\beta$-Sn as a function of temperature (left axis, red circles) and the linear expansion coefficient $\alpha$ (right axis, blue triangles) calculated by APL using the SCAN density functional. We also plot the value of $\alpha$(T) found in Touloukian et al. [83] (solid black diamonds) and measured by Deshpande and Sirdeshmukh [77] (open black diamonds).

Figure 27

The primitive-cell volume of $\gamma$-Sn as a function of temperature (left axis, red circles) and the linear expansion coefficient $\alpha$ (right axis, blue triangles) calculated by APL using the PBE density functional.

Figure 28

The primitive-cell volume of $\gamma$-Sn as a function of temperature (left axis, red circles) and the linear expansion coefficient $\alpha$ (right axis, blue triangles) calculated by APL using the SCAN density functional.

Figure 29

Free energy predicted by the PBE functional [(6), (7)] as a function of temperature for $\alpha$-Sn (solid line), $\beta$-Sn (dashed line), and simple hexagonal $\gamma$-Sn (dotted line). The vertical lines show the experimental $\alpha \text{−}\beta$ phase transition at 286 K [10] and $\beta \text{−}\gamma$ transition at 450 K [12]. Top: the free energy of each phase shifted so that the energy of the $\alpha$ phase is zero at $T=0$. Bottom: the difference in energy of the $\beta$ and $\gamma$ phases with respect to $\alpha$-Sn.

Figure 30

Free energy predicted by the SCAN functional [(6), (7)] as a function of temperature for $\alpha$-Sn (solid line), $\beta$-Sn (dashed line), simple hexagonal $\gamma$-Sn (dotted line), and simple cubic tin (dash-dot line). The vertical lines show the experimental $\alpha \text{−}\beta$ phase transitions at 286 K [10] and 450 K [12]. Top: the free energy of each phase shifted so that the energy of the $\alpha$ phase is zero at $T=0$. Bottom: the difference in energy of the $\beta$- and $\gamma$-, and simple cubic phases with respect to $\alpha$-Sn.

Figure 31

The approximate contribution (16) electrons to the free energy of $\alpha$-Sn (solid line), $\beta$-Sn (dashed line), $\gamma$-Sn (dotted line), and simple cubic tin (dash-dot line) at the equilibrium volume of each phase using the SCAN functional.