Ab initio study of deformed As, Sb, and Bi with an application to thin films

We present a comprehensive density-functional theory study of total energy and structural properties of As, Sb, and Bi in their A7 ground-state structure and in the bcc, fcc, and simple cubic (sc) modifications. We also investigate continuous structural transitions between these structures. The electronic structures and total energies are calculated both within the generalized gradient approximation (GGA) and local-density approximation (LDA) to the exchange-correlation energy as well as with and without inclusion of the spin-orbit coupling (SOC). The total energies of deformed structures are displayed in contour plots as functions of selected structural parameters and/or atomic volume; these plots are then used for understanding and interpreting structural parameters of As, Sb and Bi thin films on various substrates. Our calculated values of lattice parameters for (0001) thin films of Bi on Si(111) and Ge(111) substrates agree very well with available experimental data. In analogy with that, we suggest to investigate (0001) thin films of As on Ti(0001), Co(0001), Zn(0001) and Rh(111) substrates, of Sb on C(0001), Zn(0001), Al(111), Ag(111) and Au(111) substrates and of Bi on Co(0001), Al(111), Rh(111), Ba(111) and Pb(111) substrates. For these cases, we also predict the lattice parameters of the films. A large part of our results are theoretical predictions which may motivate experimentalists for a deeper study of these systems.

Figure 1

A7 structure corresponding to the parameters $c/a=3.11$ and $u=0.225.$ The parameters were chosen such as to demonstrate clearly the differences between a general A7 structure and the sc structure presented in Fig. 2. All three panels of this figure show the hexagonal envelope. The left panel also includes the rhombohedral cell (green) with rhombohedral angle ${\alpha }_{\mathrm{rh}}=49.8^{\circ }$ and the middle one shows an sc-like deformed body (blue) which, however, is not a rhomboeder in case of a general A7 structure. The stacking of the atomic planes (exhibited in the right panel) is A, B, and C intertwinned with atomic planes a, b, and c shifted by $2u{c}_{\mathrm{hex}}$. On the basis of that, the A7 structure can be represented by two fcc lattices shifted by $2u{c}_{\mathrm{hex}}$ along trigonal axis.

Figure 2

A7 structure with parameters corresponding to the sc structure, i.e., $c/a=\sqrt{6}$ and $u=0.25$ (see the description of Fig. 1 for further details). Let us note that the choice of $u=0.25$ implies that the atomic planes are equidistant and no bilayers are formed. The blue body in the middle panel represents a unit cell of the sc structure.

Figure 3

The $(r_{c/a},u)$ plane in $(V,r_{c/a},u)$ parameter space. The cubic structures, connected by the trigonal deformation path, are marked by larger squares and the A7 structure corresponding to the values of $c/a=3.11$ ($r_{c/a}\doteq 0.34$) and $u=0.225$ as in Fig. 1 is marked by a dot. The paths joining the equilibrium A7 structure with one of the cubic structures are discussed in detail in the text and, for Sb, exhibited also in Fig. 4. They are straight lines in the $(c/a,u)$ plane, but here they are represented by curves because we display them in the plane $(r_{c/a},a)$.

Figure 4

The behavior of the total energy of Sb in the $(c/a,u)$ plane, calculated within the LDA at the equilibrium atomic volume of the A7 structure, $V/V_{\exp }=0.977.$ Total energies of cubic structures correspond to stationary points and they are indicated by black squares lying at the vertical black lines. The fourth stationary point, the A7 structure, is indicated by single large blue dot. The saddle point close to the bcc region is marked by a star, a secondary minimum close to the fcc region is marked by a downward pointing triangle. (Let us note that this minimum is not fully evolved as, e.g., in the case shown in Fig. 24 from Appendix pp3). The deformation paths studied are indicated by curves shown by black squares (trigonal deformation path bcc-sc-fcc with $u=0.25$, further also BSF-lin path) and red triangles [two linear segments $u(c/a)$ bcc-A7 and A7-fcc used to create the BAF-lin path and one linear segment $u(c/a)$ A7-sc (AS-lin)]. Green circles show a cubic function $u(c/a)$ connecting the bcc, A7, and fcc structures (BAF-cub).

Figure 5

Total energy profiles of Sb along the trigonal deformation path (BSF-lin) calculated within the LDA at various fixed volumes $V/V_{\exp }$ (given in the upper left corner together with the corresponding colors). The insets show the behavior of $E\left(u\right)$ in the neighborhood of the cubic structures ($u=0.25$). The total energy is set to zero at the sc structure for all volumes considered. The units along the vertical axis are the same both in the main plot and the insets. The description of the vertical axis in the insets is omitted.

Figure 6

Total energy profile $E(c/a,u)$ (eV/at) of Bi calculated within the LDA at the equilibrium volume of the A7 structure $V/V_{\exp }=0.958$. Total energy minimum for the A7 structure is shifted to zero.

Figure 7

Total energy profile $E(c/a,u)$ (eV/at) of Bi calculated within the LDA+SOC approach at the equilibrium volume of the A7 structure $V/V_{\exp }=0.987$. Total energy minimum for the A7 structure is shifted to zero.

Figure 8

Total energy profiles of Bi along the trigonal deformation path (BSF-lin) calculated within the LDA+SOC at various fixed volumes $V/V_{\exp }$ (given in the upper left corner together with the corresponding colors). The insets show the behavior of $E\left(u\right)$ in the neighborhood of the cubic structures ($u=0.25$). The total energy is set to zero at the sc structure for all volumes considered. The units along the vertical axis are the same both in the main plot and the insets. The description of the vertical axis in the insets is omitted.

Figure 9

Radii of coordination shells of the A7 structures displayed as functions of $r_{c/a}$ for a fixed $u=0.215$ (corresponding to position of saddle point in case of Bi, calculated within the LDA+SOC approach at $V/V_{\exp }=1$). The vertical lines ordered from left to right show the $r_{c/a}$ of the bcc structure (black), of the “at $c/a=1.27$” structure of Ref [40] (red), of the saddle point structure (blue) with $c/a=1.57$ (1.28 in the “1-2-4” notation), and of the sc (black) and fcc structures (black). As $u\ne 0.25$, the black lines correspond to A7 structures at $u=0.215$ which are closest to bcc, sc and fcc structures; therefore, these structures are shown in parentheses in this figure. Let us note that the radius of the coordination shell represented by a green curve in the lower left corner of this figure attains a minimum at $r_{c/a}\approx$ 0.8 and increases again for higher values of $r_{c/a}$; no intersections have been found for $d_{nn}<3$ Å.

Figure 10

The same as Fig. 9 but for $u=0.25$ corresponding to the trigonally deformed cubic structures. Let us note that some of the shells from the previous figure coincide and there are more crossings at the $c/a$ values corresponding to the three cubic structures.

Figure 11

Analogue of Fig. 9 but radii are displayed as functions of $u$ for a fixed $c/a=1.57$ (corresponding to position of saddle point in case of Bi, LDA+SOC, $V/V_{\exp }=1$). We have indicated two values of $u$ by vertical lines—the saddle point (blue, left, $u=0.215$) and $u=1/4$ corresponding to a trigonally distorted cubic structures (black, right).

Figure 12

Total energy profile $E(c/a,V)$ (eV/at) of Bi calculated within the LDA+SOC approach along the BSF-lin path. Total energy minimum corresponding to the sc structure is shifted to zero. Let us note that the saddle points at $r_{c/a}$ equal approximately $-0.75$ and 0.77 do not correspond to the two structures discussed in Ref. [40].

Figure 13

Details of total energy profiles $E(c/a,V)$ (eV/at) of Bi along the BSF-lin path calculated within the LDA, LDA+SOC, GGA, and GGA+SOC approach (from left to right). The total energy minimum of the sc structure is shifted to zero in each segment. Vertical axis is the same for all segments of the plot and its description is given in the leftmost segment.

Figure 14

Total energy profile $E(c/a,V)$ (eV/at) of Bi calculated within the LDA+SOC approach along the BAF-lin path from the bcc to A7 structure. Total energy minimum of the lowest-energy A7 structure is shifted to zero.

Figure 15

The same as 14 but along the AS-lin path (from sc to A7 structure).

Figure 16

The same as 14 but along the BAF-lin path from A7 to fcc structure.

Figure 17

Total energy profile $E(c/a,V)$ (eV/at) of Bi calculated within the LDA+SOC approach along the BAF-cub path. Total energy minimum of the lowest-energy A7 structure is shifted to zero.

Figure 18

Total energy profile $E(c/a,V)$ (eV/at) of Bi calculated within the LDA+SOC approach along the BAF-cub path. Total energy minimum of the lowest-energy A7 structure is shifted to zero. The straight lines correspond (from left to right) to the substrates Al(111), Ba(111), Co(0001), Si(111) (Ref. [51]), Pb(111), Si(111) (Ref. [7]), Rh(111), Si(111) (Ref. [8]), Si(111) (Ref. [12]), Ge(111) (Ref. [13]), Si(111) (Ref. [11]), and Si(111) (Ref. [9]). The black straight lines in the plot correspond to experimental data (Table 5), the blue lines represent the cases given in Table 6 selected for our predictions. Energy minima along all of the straight lines are denoted by blue crosses. As the straight lines are very close together, we give a more detailed representation of the comparison of our results with experiment and our theoretical predictions in Figs. 21 and 22.

Figure 19

The same as in Fig. 18 but for Sb instead of Bi. The straight lines correspond (from left to right) to the substrates Ag(111), Au(111), C(0001), Rh(111), Al(111), and Zn(0001). The blue crosses represent the predicted configurations of the Sb films on these substrates; the corresponding numerical values of the structural parameters $c/a$ and $u$ are given in Table 6. As the straight lines are very close together, we give a more detailed representation of our theoretical predictions in Fig. 22.

Figure 20

The same as in Fig. 18 but for As instead of Bi. The straight lines correspond (from left to right) to the substrates Ti(0001), Rh(111), Co(0001), and Zn(0001). The blue crosses represent the predicted configurations of the As films on these substrates; the corresponding numerical values of the structural parameters $c/a$ and $u$ are given in Table 6. As the straight lines are very close together, we give a more detailed representation of our theoretical predictions in Fig. 22.

Figure 21

Detailed comparison of our results with experimental data from Table 5. This figure is an enlarged part of Fig. 18 with the theoretical predictions left out. The experimental points are denoted by red stars with error bars whenever available, our calculated results are marked by blue crosses. The straight lines correspond (from left to right) to the substrates Si(111) (Ref. [51]), Si(111) (Ref. [7]), Si(111) (Ref. [8]), Si(111) (Ref. [12]), Ge(111) (Ref. [13]), Si(111) (Ref. [11]), and Si(111) (Ref. [9]).

Figure 22

The predicted configurations of As, Sb, and Bi (from left to right) thin films on selected substrates. The figures represent details of the neighborhood of the A7 structure from Figs. 20, 19, and 18, respectively. The straight lines correspond (from left to right) to the substrates: Ti(0001), Rh(111), Co(0001), and Zn(0001) in case of As; Ag(111), Au(111), C(0001), Rh(111), Al(111), and Zn(0001) in case of Sb; Al(111), Ba(111), Co(0001), Pb(111), and Rh(111) in case of Bi. The predicted configurations are marked by crosses and the corresponding numerical values of the structural parameters $c/a$ and $u$ are given in Table 6. Vertical axis is the same for all segments of the plot and its description is given in the leftmost segment.

Figure 23

Details of total energy profiles $E(c/a,u)$ (eV/at) for Bi calculated within the LDA+SOC approach. This set of regions near $r_{c/a}=-1$ (bcc structure) corresponds to $V/V_{\exp }$ equal to 0.958, 0.987, 1.000, 1.020 and 1.040 (left to right). For each segment, the zero of the total energy corresponds to the A7 structure of the segment. The coordinates of the saddle point A7 structures are given in Table 4. Vertical axis is the same for all segments of the plot and its description is given in leftmost segment.

Figure 24

Total energy profile $E(c/a,u)$ (eV/at) for Bi in the neighborhood of the fcc structure within the LDA (no SOC) at atomic volumes $V/V_{\exp }$ equal to 0.958, 0.987, 1.000, 1.020, and 1.040 (left to right). Total energy minimum for the A7 structure (not shown in the figure) is shifted to zero. The energy difference between neighboring contours is 0.01 eV/atom. Vertical axis is the same for all segments of the plot and its description is given in the leftmost segment.

Figure 25

The same as in Fig. 24 but for the LDA+SOC calculation. Vertical axis is the same for all segments of the plot and its description is given in the leftmost segment.

Figure 26

The same as in Fig. 24 but for As. Total energies are calculated within the LDA for atomic volumes $V/V_{\exp }$ equal to 0.900, 0.940, 0.960, 0.980, and 1.000 (left to right). Vertical axis is the same for all segments of the plot and its description is given in the leftmost segment.