EPAPS Document “CalculationDetails.doc” for Comment on “Self-Purification in
Semiconductor Nanocrystals”
Mao-Hua Du,1 Alexander L. Efros,1 David J. Norris,2 and Steven C. Erwin1
1
Center for Computational Materials Science, Naval Research Laboratory, Washington,
DC 20375
2
Department of Chemical Engineering & Materials Science, University of Minnesota,
Minneapolis, Minnesota 55455
This document provides a detailed description of the density-functional-theory
(DFT) calculations appearing in our Comment on “Self-Purification in
Semiconductor Nanocrystals” [G. M. Dalpian and J. R. Chelikowsky, Phys. Rev.
Lett. 96, 226802 (2006)]. Also included is a brief guide to the accompanying
EPAPS archive file “vaspfiles.tar.gz”, which contains input files needed to
confirm these calculations using the DFT program VASP.
1. Introduction
We describe here the general methods we used for performing the density-functionaltheory (DFT) calculations reported in our Comment. We passivated the nanocrystals
using hydrogen-like atoms with appropriate fractional charge, and computed the Mn
impurity formation energy relative to its value in bulk CdSe, which was represented by a
216-atom supercell. We performed complete structural relaxation in each pure and Mndoped nanocrystal, as well as in the bulk supercell, by relaxing the positions of all atoms
until the force on each atom was less than 0.05 eV/Å. To obtain the formation energies
shown in Fig. 1 we used the projector augmented-wave (PAW) method within VASP1,2
and the PBE generalized-gradient correction3. The valence wavefunctions were
expanded in a plane-wave basis with cutoff energy 274 eV. Convergence checks were
performed at cutoff energy 320 eV, resulting in numerically insignificant changes of the
relative Mn impurity formation energies. The Cd 4d electrons were treated explicitly as
valence electrons. Below we provide further details of our methods, and describe a
number of tests and checks that we performed on our results.
2. Surface passivation
To passivate the Cd and Se atoms on the surface of the nanocrystals we used PBE/PAW
potentials, supplied with VASP, representing hydrogen-like atoms having effective
atomic numbers Z=1.5 (to passivate Cd) and Z=0.5 (to passivate Se). After structural
relaxation, the resulting Cd-H1.5 bond lengths were 1.86 Å and the Se-H0.5 bond lengths
were 1.60 Å; these values varied by less than 1% among the different nanocrystals we
studied. The relaxed H1.5-Cd-H1.5 and H0.5-Se-H0.5 surface-bond angles varied from the
tetrahedral angle (109.47 by less than 6and the H1.5-Cd-Se and H0.5-Se-Cd surfacebond angles by less than 2 We found no deviations in these results for either pure or
Mn-doped nanocrystals, even for the smallest nanocrystal we studied, Cd13Se16. This
establishes that for the nanocrystals we studied—including the smallest—our surface
passivation leads to consistent and physically sensible trends in the relaxed nanocrystal
geometries.
To investigate further the extent to which different surface passivation techniques might
affect the calculated Mn impurity formation energies, we considered two possible effects:
differences in surface geometry (of unspecified origin), and differences in the
pseudopotentials defining the hydrogen-like atoms. We describe our tests below.
a. Differences in surface geometry. We considered as a simple test case the nanocrystal,
Cd43Se44, for which the relative Mn impurity formation energy shown in Fig. 1 (the fully
relaxed PAW/PBE result) was 0.01 eV. We recomputed the Mn impurity formation
energy for this nanocrystal using a frozen geometry with perfect tetrahedral angles, bulk
Cd-Se bond lengths, and arbitrarily chosen Cd-H1.5 and Se-H0.5 bond lengths of 1.5 Å.
For the reference bulk CdSe supercell no relaxation was performed. Because of the
artificial strain created by using these frozen coordinates, the total energy of each
separate calculation increased by a very large amount, ~27 eV. Nevertheless, when we
subtract the relevant total energies to obtain the formation energies the cancellation of
this strain energy is very systematic, and we again obtain 0.01 eV for the relative Mn
impurity formation energy. This result strongly suggests that variations in surface
geometry have no significant effect on the Mn impurity formation energy.
b. Differences in the pseudopotentials. We considered our smallest Cd13Se16 nanocrystal,
and recomputed the Mn impurity formation energy using full Z/r potentials to passivate
the surface atoms. This approach eliminates the arbitrariness in the construction of the
pseudopotential for the hydrogen-like atoms. (For this test we used the ABINIT code4,5,
which permits a wide variety of pseudopotentials to be used, including full potentials.)
The resulting relative Mn impurity formation energy was 0.02 eV. This result
demonstrates that our hydrogen-like pseudopotentials give results numerically
indistinguishable from full potentials.
3. Supercells for nanocrystals
Since VASP assumes a periodic system, one must use a supercell of sufficient size to
approximate an isolated nanocrystal. For the results shown in Fig. 1 we chose supercells
that maintain a separation of at least 8 Å between the outermost hydrogen-like atoms of
adjacent nanocrystals. We tested the adequacy of these supercells to ensure there were
no numerically significant interactions between nanocrystals.
4. Supercells to represent bulk CdSe
In order to provide a reference for the Mn impurity formation energy in the nanocrystals,
a supercell representing bulk CdSe must be used. To obtain the results in our Fig. 1 we
used a 216-atom supercell with converged k-point sampling. We checked the adequacy
of this supercell by comparing the resulting Mn formation energy to results obtained
using supercells with 128, 64, and 32 atoms, all with converged k-point sampling.
Compared to the 216-atom supercell, the variations in the absolute Mn formation energy
were less than 0.01 eV. This result demonstrates that all of these supercells are adequate
for calculating the Mn impurity formation energy in bulk CdSe with an accuracy of 0.01
eV.
We also examined the impact of a finite concentration of Mn impurities (due to
the use of a supercell) on the lattice constant of the supercell. We focused on our
smallest supercell (32 atoms) because this effect will be maximal there. With one Mn in
the supercell, the optimized lattice constant is reduced by 0.7% from its original, undoped
2
value. The energy gain resulting from this optimization was only 0.03 eV. This result
demonstrates that optimization of the lattice constant for the Mn-doped supercell has a
negligible effect on the Mn impurity formation energy.
5. Exchange-correlation potential
To investigate the effect of the exchange-correlation potential, and associated changes in
bond lengths, on Mn impurity formation energies, we repeated some of our DFT/PBE
calculations using instead the local-density approximation (LDA). The theoretical lattice
constant of bulk zinc-blende CdSe differs by 3% between the two methods: 6.21 Å for
PBE, and 6.03 Å for LDA. In this test calculation we recomputed the relative Mn
impurity formation energy for the Cd13Se16 nanocrystal using the ideal geometry (which
was shown above to give the Mn relative impurity formation energy within 0.01 of its
fully relaxed value) based on the LDA lattice constant. The resulting relative Mn
impurity formation energy was within 0.01 eV of the value obtained with PBE at the
different lattice constant.
6. Pseudopotential scheme
We used the PAW method in most of our calculations to represent the electron-ion
interactions. This method is generally considered more accurate than either ultrasoft
pseudopotentials or Troullier-Martins soft pseudopotentials.
To check for any
dependence of our results on the choice of pseudopotential, we repeated one calculation
of the relative Mn impurity formation energy using ultrasoft pseudopotentials (within
VASP) and Troullier-Martins pseudopotentials (within ABINIT). The results for the Mn
impurity formation energy in an ideal Cd13Se16 nanocrystal were within 0.01 eV of our
PAW results. Since we obtained very similar results using different pseudopotentials, as
well as entirely different DFT codes, we conclude that our VASP PAW results are
correct.
7. Accompanying VASP files
Accompanying this EPAPS document is an archive (compressed tar file) containing all
the necessary files for running a VASP calculation to compute the fully converged
relative Mn impurity formation energy within LDA in one of the four nanocrystals,
Cd43Se44. Using these files one will find the relative Mn impurity formation energy to be
nearly zero (0.04 eV). These same files can also be used, with appropriate rescaling of
the lattice constant, to reproduce the PBE result (+0.01 eV) for the nanocrystal with
diameter 1.62 nm shown in Fig. 1 of our Comment. These results for LDA and PBE are
in excellent agreement with each other, to within the computational uncertainty expected
in such calculations.
The accompanying tar file contains a README file explaining the VASP files. It is
reproduced here:
3
This tar file contains 4 directories, with all the VASP files needed
to compute the relative Mn impurity formation energy in one of
nanocrystals we studied. These directories and the jobs are:
directory
--------cd43se44
mn1cd42se44
job
--87-atom passivated nanocrystal
The above, with 1 substitutional Mn atom
cd32se32
mn1cd31se32
64-atom simple cubic supercell
The above, with 1 substitutional Mn atom
PAW potentials constructed within LDA ("pot-PAW") were used
throughout. At the cutoff energy we used (270 eV), the equilibrium
CdSe lattice constant is 6.03 Angstroms. The ordering of the PAW
potentials for each run is found in the ".alt" batch script files.
The coordinate ("poscar") files represent completely relaxed
geometries. The residual Cartesian forces are listed next to the
coordinates. The largest total residual force on any atom is 0.003
eV/Angstrom.
By running these jobs the following total energies and Mn impurity
formation energies, in eV, will be obtained:
cd43se44
-471.076
mn1cd42se44
-478.516
------------------------------=> Mn formation energy
-7.440
cd32se32
-215.656
mn1cd31se32
-223.052
------------------------------=> Mn formation energy
-7.396
Therefore, the Mn impurity formation energy in this nanocrystal,
relative to its value the supercell, is -0.044 eV.
As a further test, we checked for any effect of strain on the 64-atom
supercell resulting from Mn substitution. In the VASP output file
("OUTCAR") for job "mn1cd31se32" the reported external pressure is
-18 kB. This number is almost entirely dominated by Pulay stress,
and is not a reliable guide to minimizing the total energy in this
cell. Instead, we directly minimized the total energy by varying the
supercell lattice constant, rescaling all the coordinates, and
re-relaxing the internal coordinates.
The resulting new equilibrium lattice constant was 6.015 Angstroms,
just 0.25 percent smaller than in pure CdSe. The resulting energy
gain was only 0.018 eV. These numbers are, as expected, even smaller
than our earlier results for the 32-atom supercell (0.70 percent
reduction, and 0.030 eV energy gain). We conclude that changes in the
supercell lattice constant are numerically insignificant on the scale
of interest to us.
4
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4
The ABINIT code is a common project of the Université Catholique de Louvain, Corning Incorporated,
and other contributors (URL http://www.abinit.org).
5
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Materials Science 25, 478 (2002).
2
5