Supporting Information for: First-principles design and analysis of an ...
Supporting Information for: First-principles design and analysis of an efficient, Pb-free ferroelectric photovoltaic absorber derived from ZnSnO3 Brian Kolb∗ and Alexie M. Kolpak∗ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139 E-mail: [email protected]; [email protected]
ZnSnO3 — Response to Epitaxial Strain As mentioned in the main text, the band gap of LN-ZnSnO3 is particularly sensitive to epitaxial strain. The band gap and polarization as a function of epitaxial strain is shown in Figure S1. In the figure, band gaps were computed within the G0W0 approximation and polarization using the Berry phase formalism of the modern theory of polarization. 1 With each in-plane strain the system was fully relaxed in the out-of-plane (ˆz) direction. Not that, with compressive strain, the band gap quickly saturates as the Sn s, O p-state combinations that comprise the conduction band edge at equilibrium move energetically above states comprised largely of Zn d-states, which are less affected by strain. With tensile strain, on the other hand, the band gap decreases monotonically with strain, as the band edge drops in energy due to an overall increase in volume. ∗ To
whom correspondence should be addressed
1
Figure S1: Band gap and polarization of LN-ZnSnO3 as a function of epitaxial strain. Strain value is given in percent relative to the equilibrium structure, with positive values indicating tensile strain and negative indicating compressive strain. Full relaxation is allowed in the zˆ-direction at each inplane strain. The change in band gap with strain seen in Figure S1 is common among ferroelectric oxides, but the magnitudes depend quite strongly on the nature of the conduction band edge. As described in the paper, the observed effect is only half this strong in BaTiO3 and nearly non-existent in PbTiO3 . It is the delocalized Sn s-states that are responsible for the large magnitude of the effect seen here, and the reason for the extreme change in band gap with sulfur substitution described in the main text.
Structure of LN-ZnSnS3 The structure of LN-ZnSnS3 primitive cell is rhombohedral, with a = b = c = 6.76 Å and α = β =
γ = 55.79◦. The atomic positions, in lattice coordinates, are given in Table S1.
2
Table S1: Lattice coordinates of atoms in the primitive cell of LN-ZnSnS3 . Atom
x
y
z
Zn
0.71554 0.71554 0.71554
Zn
0.21549 0.21549 0.21549
Sn
0.99788 0.99788 0.99788
Sn
0.49788 0.49788 0.49788
S
0.78870 0.39408 0.10547
S
0.10547 0.78870 0.39408
S
0.39408 0.10547 0.78870
S
0.89399 0.28854 0.60557
S
0.28854 0.60557 0.89399
S
0.60557 0.89399 0.28854
Electronic properties The properties in this section were calculated in VASP using perturbation theory (i.e. LEPSILON=.TRUE.) including local field effects at the DFT level. Table S2 gives the dielectric tensor (ε ) and Born effective charges (Zχ∗ ) for ZnSnO3 and ZnSnS3 .
3
Table S2: Static, ion-clamped dielectric constant (ε ) and Born effective charge (Zx∗ ) tensors for ZnSnO3 and ZnSnS3 . The Born effective charges for each species are averaged over all atoms of the given type. Property
ZnSnO3 " 4.357 0 0 " 2.919 −0.001 0 " 4.140 0 0
ε
∗ ZZn
∗ ZSn
∗ ZO/S
ZnSnS3
# 0 0 4.357 0 0 4.180 # 0.001 0 2.913 0 0 2.694 # 0 0 4.139 0 0 4.165
" 10.209 0 0 " 3.042 0 0 " 4.519 0 0 " −1.452 0 0
" # −1.449 −0.002 0 0.002 −1.452 0 0 0 −1.395
# 0 0 10.209 0 0 10.318 # 0 0 3.063 0 0 2.919 # 0 0 4.505 0 0 4.772 # 0 0 −1.460 0 0 −1.494
Band Structures Band structures were calculated by first computing the G0W0 eigenvalues on a regular grid of kpoints, then performing a Wannier-function interpolation over the prescribed path using the Wannier90 code. 2 Since interpolation can add noise to band structures, the location of the valence band edge was checked and confirmed with extremely careful DFT calculations. 10
4
6
Band Energy (eV)
Band Energy (eV)
8
4 2 0
2
0
-2
-2 -4 A
Γ
L
P
Z
A
P QF B
(a)
Γ
L
P
Z
P QF B
(b)
Figure S2: Band structures for (a) ZnSnO3 and (b) ZnSnS3 as calculated using GW and Wannier interpolation. The point labeled A is [δ δ¯ 0] with δ = 0.368. 4
Effective Mass The effective mass can be defined as:
∗
m =
h¯ 2 ∂ 2E ∂~k2
(1)
This quantity was calculated in Q UANTUM E SPRESSO after an initial self-consistent calculation by computing eigenvalues non-self-consistently for k-points along a line through the band edge in a chosen direction and using quadratic interpolation to find the second derivative at the extremum. Figure 4 of the main text was generated by performing this calculation for 3600 directions spanning 0 ≤ θ < 2π and 0 ≤ φ ≤ π2 , with θ being the azimuthal angle from the positive x-axis and φ measuring the angle down from the positive z-axis. As mentioned in the main text, the valence band edge in ZnSnO3 occurs at Γ. This is not true ¯ direction. Nevertheless, to see the direct of the sulfide, where the band edge shifts toward the [110] effect of sulfur substitution on effective mass, Table S3 gives the effective masses of heavy and light holes at the Γ point in ZnSnS3 . These values show a marked reduction from those in Table 1 of the main text for the hole effective masses in ZnSnO3 . Table S3: Effective mass of heavy holes (m∗hh ) and light holes (m∗lh ) at the Γ-point of ZnSnS3 . We stress that this is not the valence band edge for this system, and these values are provided only for comparison with those of ZnSnO3 given in Table 1 of the main text. direction
m∗hh
m∗lh
x
1.242
0.644
y
1.124
0.682
z
15.362 12.328
Phonons Figure S3 shows the phonon spectrum of LN-ZnSnS3 throughout the Brillouin zone. Phonons were calculated in Q UANTUM E SPRESSO using density functional perturbation theory. Reciprocal 5
space was sampled with a 6 × 6 × 6 grid of q-points and a phonon band structure and density of states were computed via Fourier interpolation.
-1
Wavenumber (cm )
300
200
100
0 Γ
Z
P
BQ F
Γ
DOS
Figure S3: Phonon band structure and density-of-states for LN-ZnSnS3 .
References (1) Resta, R.; Vanderbilt, D. Physics of Ferroelectrics; Topics in Applied Physics; Springer Berlin Heidelberg, 2007; Vol. 105; pp 31–68. (2) Mostofi, A. A.; Yates, J. R.; Lee, Y.-S.; Souza, I.; Vanderbilt, D.; Marzari, N. Comput. Phys. Commun. 2008, 178, 685 – 699.
6
ZnSnO3 — Response to Epitaxial Strain As mentioned in the main text, the band gap of LN-ZnSnO3 is particularly sensitive to epitaxial strain. The band gap and polarization as a function of epitaxial strain is shown in Figure S1. In the figure, band gaps were computed within the G0W0 approximation and polarization using the Berry phase formalism of the modern theory of polarization. 1 With each in-plane strain the system was fully relaxed in the out-of-plane (ˆz) direction. Not that, with compressive strain, the band gap quickly saturates as the Sn s, O p-state combinations that comprise the conduction band edge at equilibrium move energetically above states comprised largely of Zn d-states, which are less affected by strain. With tensile strain, on the other hand, the band gap decreases monotonically with strain, as the band edge drops in energy due to an overall increase in volume. ∗ To
whom correspondence should be addressed
1
Figure S1: Band gap and polarization of LN-ZnSnO3 as a function of epitaxial strain. Strain value is given in percent relative to the equilibrium structure, with positive values indicating tensile strain and negative indicating compressive strain. Full relaxation is allowed in the zˆ-direction at each inplane strain. The change in band gap with strain seen in Figure S1 is common among ferroelectric oxides, but the magnitudes depend quite strongly on the nature of the conduction band edge. As described in the paper, the observed effect is only half this strong in BaTiO3 and nearly non-existent in PbTiO3 . It is the delocalized Sn s-states that are responsible for the large magnitude of the effect seen here, and the reason for the extreme change in band gap with sulfur substitution described in the main text.
Structure of LN-ZnSnS3 The structure of LN-ZnSnS3 primitive cell is rhombohedral, with a = b = c = 6.76 Å and α = β =
γ = 55.79◦. The atomic positions, in lattice coordinates, are given in Table S1.
2
Table S1: Lattice coordinates of atoms in the primitive cell of LN-ZnSnS3 . Atom
x
y
z
Zn
0.71554 0.71554 0.71554
Zn
0.21549 0.21549 0.21549
Sn
0.99788 0.99788 0.99788
Sn
0.49788 0.49788 0.49788
S
0.78870 0.39408 0.10547
S
0.10547 0.78870 0.39408
S
0.39408 0.10547 0.78870
S
0.89399 0.28854 0.60557
S
0.28854 0.60557 0.89399
S
0.60557 0.89399 0.28854
Electronic properties The properties in this section were calculated in VASP using perturbation theory (i.e. LEPSILON=.TRUE.) including local field effects at the DFT level. Table S2 gives the dielectric tensor (ε ) and Born effective charges (Zχ∗ ) for ZnSnO3 and ZnSnS3 .
3
Table S2: Static, ion-clamped dielectric constant (ε ) and Born effective charge (Zx∗ ) tensors for ZnSnO3 and ZnSnS3 . The Born effective charges for each species are averaged over all atoms of the given type. Property
ZnSnO3 " 4.357 0 0 " 2.919 −0.001 0 " 4.140 0 0
ε
∗ ZZn
∗ ZSn
∗ ZO/S
ZnSnS3
# 0 0 4.357 0 0 4.180 # 0.001 0 2.913 0 0 2.694 # 0 0 4.139 0 0 4.165
" 10.209 0 0 " 3.042 0 0 " 4.519 0 0 " −1.452 0 0
" # −1.449 −0.002 0 0.002 −1.452 0 0 0 −1.395
# 0 0 10.209 0 0 10.318 # 0 0 3.063 0 0 2.919 # 0 0 4.505 0 0 4.772 # 0 0 −1.460 0 0 −1.494
Band Structures Band structures were calculated by first computing the G0W0 eigenvalues on a regular grid of kpoints, then performing a Wannier-function interpolation over the prescribed path using the Wannier90 code. 2 Since interpolation can add noise to band structures, the location of the valence band edge was checked and confirmed with extremely careful DFT calculations. 10
4
6
Band Energy (eV)
Band Energy (eV)
8
4 2 0
2
0
-2
-2 -4 A
Γ
L
P
Z
A
P QF B
(a)
Γ
L
P
Z
P QF B
(b)
Figure S2: Band structures for (a) ZnSnO3 and (b) ZnSnS3 as calculated using GW and Wannier interpolation. The point labeled A is [δ δ¯ 0] with δ = 0.368. 4
Effective Mass The effective mass can be defined as:
∗
m =
h¯ 2 ∂ 2E ∂~k2
(1)
This quantity was calculated in Q UANTUM E SPRESSO after an initial self-consistent calculation by computing eigenvalues non-self-consistently for k-points along a line through the band edge in a chosen direction and using quadratic interpolation to find the second derivative at the extremum. Figure 4 of the main text was generated by performing this calculation for 3600 directions spanning 0 ≤ θ < 2π and 0 ≤ φ ≤ π2 , with θ being the azimuthal angle from the positive x-axis and φ measuring the angle down from the positive z-axis. As mentioned in the main text, the valence band edge in ZnSnO3 occurs at Γ. This is not true ¯ direction. Nevertheless, to see the direct of the sulfide, where the band edge shifts toward the [110] effect of sulfur substitution on effective mass, Table S3 gives the effective masses of heavy and light holes at the Γ point in ZnSnS3 . These values show a marked reduction from those in Table 1 of the main text for the hole effective masses in ZnSnO3 . Table S3: Effective mass of heavy holes (m∗hh ) and light holes (m∗lh ) at the Γ-point of ZnSnS3 . We stress that this is not the valence band edge for this system, and these values are provided only for comparison with those of ZnSnO3 given in Table 1 of the main text. direction
m∗hh
m∗lh
x
1.242
0.644
y
1.124
0.682
z
15.362 12.328
Phonons Figure S3 shows the phonon spectrum of LN-ZnSnS3 throughout the Brillouin zone. Phonons were calculated in Q UANTUM E SPRESSO using density functional perturbation theory. Reciprocal 5
space was sampled with a 6 × 6 × 6 grid of q-points and a phonon band structure and density of states were computed via Fourier interpolation.
-1
Wavenumber (cm )
300
200
100
0 Γ
Z
P
BQ F
Γ
DOS
Figure S3: Phonon band structure and density-of-states for LN-ZnSnS3 .
References (1) Resta, R.; Vanderbilt, D. Physics of Ferroelectrics; Topics in Applied Physics; Springer Berlin Heidelberg, 2007; Vol. 105; pp 31–68. (2) Mostofi, A. A.; Yates, J. R.; Lee, Y.-S.; Souza, I.; Vanderbilt, D.; Marzari, N. Comput. Phys. Commun. 2008, 178, 685 – 699.
6
