Supplementary Material for
Tuning the Electronic Structure of II-VI and III-V
Semiconductors with Biaxial Strain
Shenyuan Yang, David Prendergast, and Jeffrey B. Neaton
Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road,
Berkeley, California 94720
Energy band gaps calculated from DFT and GW
Table SI summarizes our calculation results for equilibrium CdSe and GaN. One-shot
PBE-G0W0 calculations always underestimate the band gap by 0.5-0.6 eV using the
experimental geometry. After updating the eigenvalues in G for four iterations, the
partially self-consistent GW0 approximation yields excellent agreement with experiments,
consistent with previous calculations.19 The success of GW0 may result from a
cancellation of errors, e.g., too small DFT band gaps and neglect of electron-hole
interaction [Ref.A]. On the other hand, fully self-consistent GW has a tendency to
overcorrect the band gaps due the neglect of electron-hole interaction [Refs.A,B]. For
instance, the band gap of CdSe from self-consistent GW is 0.25 eV larger than the
experiment gap. The band gaps predicted by hybrid functionals are much larger than the
PBE gaps but still smaller than experimental measurements. We notice that the hybrid
functionals yield excellent dielectric constants due to a good cancellation of errors
[Ref.A]. However, GW approximation based on hybrid functionals always overcorrects
the band gaps.
In Table SI, we compare the band gaps of the experimental and computed PBE lattice
parameters. Due to the volume expansion, the band gaps of PBE structures decrease
compared to the band gaps of experimental structures. As a result, the band gaps from
G0W0 approximation based on HSE03 exhibit a better agreement with the experimental
band gaps.
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The trends and magnitudes of band gap changes
Although PBE predicts correct trends of the changes in the band gap, the magnitudes
differ substantially by as much as hundreds of meV. As seen from Fig. 1(b), the decrease
of band gap of CdS from PBE saturates at larger tensile strain (>7%), since the PBE gap
at equilibrium is too small and the strained CdS has to maintain a finite band gap as the
atoms are pulled apart from each other. The underestimation of the band gap changes
under tensile strain has also been observed in CdSe, CdTe, and GaN (see Fig. S1). In AlN,
the changes in the band gap from PBE agree well with other calculations (Fig. 1(a)),
since the PBE gap is large (4.06 eV) and PBE has a good description of the simple sp
compounds without d states. On the other hand, the magnitudes of the band gap changes
under compressive strain are similar for all the considered compounds (Fig. 1). However,
due to the underestimation of band gap at equilibrium, PBE predicts the compressed
compounds to be metallic at smaller compressive strain compared to other calculations.
References:
[Ref.A] M. Shishkin, M. Marsman, and G. Kresse, Phys. Rev. Lett. 99, 26403 (2007).
[Ref.B] M. van Schilfgaarde, T. Kotani, and S. Faleev, Phys. Rev. Lett. 96, 226402
(2006).
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CdSe
Exp. latt.
PBE Latt.
Exp. latt.
PBE Latt.
a(Å)
4.30
4.395
3.18
3.216
c(Å)
7.01
7.178
5.17
5.240
PBE
0.702
0.548
1.988
1.739
PBE+G0W0
1.562
1.349
3.222
2.907
PBE+GW0
1.713
1.494
3.435
3.110
HSE03
1.471
1.296
3.051
2.770
HSE03+G0W0
1.975
1.778
3.738
3.423
HSE06
1.652
1.477
3.282
2.997
HSE06+G0W0
2.060
1.864
3.836
3.512
Exp.
ε∞(HSE03)
ε∞(Exp.)
Table SI
GaN
1.80
5.964
3.503
6.609
6.1
4.815
4.945
4.86
The band gaps (in eV) of CdSe and GaN from different levels of calculations
at the PBE and experimental lattice parameters, compared to experimental band gaps.
The dielectric constants from HSE03 are also compared to experimental values.
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Fig. S1 Energy band gaps of wurtzite (a) GaN, (b) CdSe, and (c) CdTe as a function of
biaxial strain in the (0001) plane computed with different levels of calculations.
Fig. S2 The axial bond length d1 and non-axial bond length d2 of (a) GaN and (b) CdSe
as a function of biaxial strain in the (0001) plane.
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