Computational Materials Science 50 (2010) 356–362
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Computational Materials Science
journal homepage: www.elsevier.com/locate/commatsci
Surface stability of potassium nitrate (KNO3) from density functional theory
O.M. Løvvik a,b,⇑, T.L. Jensen c,⇑⇑, J.F. Moxnes c, O. Swang a,d, E. Unneberg c
a
SINTEF Materials and Chemistry, PO Box 124, Blindern, NO-0314 Oslo, Norway
Department of Physics, University of Oslo, PO Box 1048, Blindern, N-0316 Oslo, Norway
c
Norwegian Defence Research Establishment, PO Box 25, NO-2027 Kjeller, Norway
d
Department of Chemistry, University of Oslo, PO Box 1033, Blindern, N-0315 Oslo, Norway
b
a r t i c l e
i n f o
Article history:
Received 7 April 2010
Received in revised form 25 August 2010
Accepted 26 August 2010
Keywords:
Potassium nitrate
Surface energy
Density functional theory
a b s t r a c t
Potassium nitrate has been studied by accurate DFT calculations. The bulk crystal structure and electronic
structure were calculated and compared to previous studies. In addition, the surface stability of various
faces was quantified, confirming that the {0 0 1} face has the lowest surface energy of 0.19 Jm 2. Other
surfaces terminated by nitrate ions exhibited reconstructions upon relaxation, rotating the ions into an
orientation parallel to the surface plane.
Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction
Potassium nitrate (KNO3) has many applications, e.g. as a fertilizer, as an oxidizing agent in pyrotechnics, and in food preservation. The metastable c phase has been extensively studied due to
its ferroelectric properties [1–5]. It has, however, been difficult to
take any practical advantage of this, due to the lack of stability
[6–9].
KNO3 crystallizes at room temperature into the a phase (also
called phase II). For a long time it was believed that this phase
belonged to the space group Pmcn (orthorhombic, with Z = 4)
[10–12]. However, a neutron diffraction study by Adiwidjaja and
Pohl demonstrated that the crystal structure actually consists of
a 2 2 1 supercell of the Pmcn unit cell, with Z = 16 and within
the space group Cmc21 [13]. The crystal structure transforms into
the b phase (I) at around 128 °C when heated, and passes through
the ferroelectric c phase (III) between 124 and 110 °C upon
cooling. A number of phases also exists at lower temperatures
[14,15]. Transitions between these phases have been extensively
studied by a variety of techniques [7,16–41].
The growth of KNO3 crystals has been examined by several
authors [42–46], and it has been shown that the {0 0 1} face has
the lowest surface energy [43]. Other crystal faces have also been
observed, including the {0 1 0}, {1 0 0}, {1 1 0}, {0 1 1}, {0 2 1},
⇑ Corresponding author at: SINTEF Materials and Chemistry, PO Box 124,
Blindern, NO-0314 Oslo, Norway.
⇑⇑ Corresponding author.
E-mail addresses: ole.martin.lovvik@sintef.no (O.M. Løvvik), tomas-lunde.jensen@ffi.no (T.L. Jensen).
0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.commatsci.2010.08.027
{0 1 2}, and {1 1 1} faces. An overview of the possibilities of different kinds of growth is given by Rolfs et al. [43].
Previous theoretical studies on this system have focused on the
ferroelectric properties of the c phase [5,47,48] and on phase transitions [30,49]. The Hartree–Fock study by Aydinol et al. also presented calculations of bonding characteristics and the structural
stability of various phases of KNO3 [5]. The present contribution
has a slightly different focus. We report a detailed computational
study of the bulk electronic structure of the a phase of KNO3, followed by an investigation of the stability of various surfaces. All
calculations are performed within the scope of density functional
theory (DFT) using a plane wave description of the electron density
and periodic boundary conditions.
2. Methodology
Calculations were performed within density functional theory
(DFT) as implemented in the Vienna ab initio simulation package
(VASP) [50,51], using the PBE generalized gradient approximation
(GGA) density functional [52]. The projector augmented wave
(PAW) method [53] was used to represent the electron density.
This is a generalization of the linearized augmented plane wave
(LAPW) and the pseudopotential (PP) methods, with reliability
comparable to that of LAPW and efficiency close to that of PP methods [53]. All calculations were spin unrestricted, allowing for spin
polarization. Despite this, no net spin density was seen for any of
the systems described herein.
DFT calculations were also performed applying the ab initio simulation package Quantum Espresso (QE) v4.0.3 [67], using the BLYP
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O.M. Løvvik et al. / Computational Materials Science 50 (2010) 356–362
-117.8
-116.8
Fig. 1(b). The total energy convergence with respect to Ec is even
worse for the norm-conserving potentials. We have plotted the
behaviour for two different choices of the cut-off energy for charge
Echarge; Echarge = 4 Ec (default) and Echarge = 2 Ec. Proper convergence
has not been found in any of the cases, even when Ec is increased to
more than 2000 eV (note the different scales on the axes of Fig. 1
(a) and (c)).
This means that the convergence criterion we proposed (1 meV
change of total energy/50 eV increase of Ec) may be too strict. After
all, we are usually interested in relative energies, not the absolute
total energy. To test another option, we plotted the calculated pressure of the experimental cell (which is non-zero because the applied density functional fails to predict the exact magnitude of
the lattice parameters) as a function of Ec. The calculated pressure
is an important quantity when performing automatic relaxation of
the unit cell size (and shape), which can be used to search for unknown crystal structures [54]. For VASP, the pressure is apparently
already converged (changes less than 1 kbar when Ec is increased
by 50 eV) at 500 eV for the soft potentials (Fig. 1(b)). When using
hard potentials, similar convergence is reached at a cut-off of
950 eV. In the case of QE using norm-conserving pseudopotentials,
we see that a cut-off of 1360 eV is necessary to obtain proper convergence of the pressure (Fig. 1(d)). In this case we can clearly see
that using Echarge = 2 Ec is necessary to obtain convergence; when
the default value of Echarge = 4 Ec is used, the pressure seems to diverge when Ec is increased further.
Another important set of parameters is the calculated forces,
which are used when optimizing the ionic positions. We checked
numeric convergence of the forces with respect to Ec in a similar
vein as above. The convergence criterion was now that the force
changes should be less than 0.05 eV/Å when Ec is increased with
50 eV. In the case of VASP we found converged of the forces for
Ec = 450 and 600 eV for the soft and hard potentials, respectively.
QE exhibited converged forces at Ec = 1224 eV when Echarge = 2 Ec;
when Echarge = 4 Ec the forces were converged at 1496 eV.
(b)
Soft
Hard
-118
-117
-118.2
-117.2
400
900
Calculated pressure (kbar)
(a)
Total energy (eV)
functional and norm-conserving pseudopotentials. The QE calculations allowed no spin polarization.
The density of k points in the reciprocal space integration was
always kept below 0.2 per Å 1. As an example, a gamma-centred
4 2 4 mesh was used for the Pmcn bulk unit cell. Convergence
tests up to an 8 6 8 mesh confirmed that the uncertainty in total energy resulting from the chosen sampling was below 1 meV
per unit cell. For slab calculations, only the gamma point was used
in the vacuum direction. The tetrahedron method with Blöchl corrections was used to smear partial occupancies near the Fermi level. The criterion for self-consistence was a change in total
electronic energy of less than 10 6 eV between consecutive electron density iterations. The QE calculations used the same density
of k points.
Computer codes for plane-wave calculations have often built in
default values for energy cut-off. For KNO3, the default value would
be 400 eV for the standard (soft) potentials in VASP. However, we
expected an unusually strong perturbation of the N and/or O atoms
upon formation of nitrate groups, and therefore conducted convergence tests of the total energy as a function of the kinetic energy
cut-off Ec. Indeed, as seen in Fig. 1(a), the total energy changes dramatically with increasing Ec. The total energy is not properly converged (defined here as changing by less than 1 meV when
increasing Ec by 50 eV) until Ec = 1250 eV. Even at this point the total energy continues to drop, and is 2.5 meV lower when
Ec = 1500 eV. This indicates the existence of electron density
changes uncommonly close to the atomic nuclei, and we found it
necessary to repeat the test with harder potentials (smaller frozen
core and higher plane wave cut-offs). These results are also shown
in Fig. 1(a), and again we see that the default Ec (which is much
higher (750 eV) for the hard potentials) is clearly insufficient to
achieve convergence. This time, however, proper convergence is
obtained at 950 eV, and the change in total energy is less than
0.1 meV when increasing Ec further to 1100 eV. We investigated
the same convergence behaviour for QE, this is shown in
60
40
20
Hard
0
400
(d)
-6420
4x
-6422
2x
-6424
-6426
400
900
1400
Energy cut-off (eV)
600
800
1000
Energy cut-off (eV)
1900
Calculated pressure (kbar)
Total energy (eV)
Energy cut-off (eV)
(c)
Soft
200
100
4x
2x
0
-100
-200
400
900
1400
Energy cut-off (eV)
Fig. 1. Convergence of the calculated electronic total energy (a and c) and the calculated pressure (b and d) as a function of the energy cut-off of the kinetic energy plane wave
expansion. The upper panels present VASP calculations with blue diamonds denoting standard (soft) potentials, and red squares denoting potentials with smaller frozen core
and higher cut-offs (hard). The lower panels present data from QE calculations, with two different values of the charge density energy cut-off (augmentation cut-off); 4 and 2
times that of the kinetic energy cut-off, represented by blue diamonds and red squares, respectively. Lines are drawn as guides to the eye only. (For interpretation of the
references to colour in this figure legend, the reader is referred to the web version of this article.)
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O.M. Løvvik et al. / Computational Materials Science 50 (2010) 356–362
Table 1
Lattice constants, volume (V) and interatomic distances from experiment and DFT calculations using soft and hard PBE potentials with VASP as well as norm-conserving BLYPpotentials with QE The difference between the hard and soft potentials is the size of the core region, where the electron density is kept fixed. The energy cut-offs Ec are 400, 755.6,
950, and 1360 eV, respectively – the two former are default values for the respective potentials, while the latter are the converged values for the hard PBE and norm-conserving
BLYP potentials. The number of formula units Z is 16 for the Cmc21 unit cell and 4 for the Pmcn unit cell. The minimum and maximum interatomic distances refer to the first
coordination sphere – this includes three O atoms for N–O and nine O atoms for K–O.
Ec (eV)
Potential
Z
a (Å)
b (Å)
c (Å)
V (Å3)
N–O min. (Å)
N–O max. (Å)
K–O min. (Å)
K–O max. (Å)
Expt. (Ref. [13])
VASP PAW
VASP PAW
VASP PAW
QE norm-cons
10.825
18.351
6.435
1278.3
1.23
1.26
2.81
2.95
400
PBE soft
16
10.58
17.87
6.00
1134
1.27
1.27
2.67
2.82
756
PBE hard
16
10.56
17.80
6.16
1159
1.26
1.26
2.70
2.83
950
PBE hard
16
10.97
18.57
6.56
1337
1.26
1.27
2.88
2.96
1360
BLYP
4
11.45
19.40
7.01
1558
1.28
1.29
3.03
3.13
It is necessary to use the hard version of the potentials for K, N,
and O atoms in VASP to obtain a reliable calculated pressure. However, the largest effects are seen when going from the soft to hard
version for oxygen; this suggests that oxygen has the largest electronic changes in the core region.
Relaxations of bulk unit cells were performed by simultaneously optimizing atomic positions, unit cell size, and shape. A
quasi–Newtonian formalism using the residual minimization
scheme with direct inversion in the iterative subspace was used
to minimize the forces. The convergence criterion for equilibrium
was that all forces should be lower than 0.05 eV/Å.
Most of the test calculations (finding convergence etc.) were
performed using the Pmcn unit cell, rather than the four times larger Cmc21 structure. This saved much computational effort, but
should pose no problems when it comes to reliability of the results,
since the local bonding environment is very similar in the two
structures. Also, the difference in calculated total energy per unit
cell between the two models is within the error bar of the calculations (1 meV/unit cell), and the relaxed cell parameters were also
very similar when using the different cells (differences were less
than 0.3%). As an aside, we note that since our calculations are performed at 0 K, the crystal structure at room temperature will not
necessarily coincide with the computed results, even assuming
that the numerical model otherwise is perfect. An interesting future study would be to compare the energy of different KNO3 isomorphs as a function of temperature from phonon calculations.
For the abovementioned reasons, it was decided to base all surface calculations on the Pmcn unit cell as well. Lattice constants
were fixed at the bulk relaxed values, and positions were relaxed
starting from the bulk relaxed positions. Surface energies were
converged within 1 meV with a vacuum layer of 10 Å, which were
used for initial calculations, screening surface energies for a number of different surfaces. In the calculations reported here, a vacuum layer of 16 Å was used to separate the slabs. Since the
Cmc21 unit cell is a 2 2 1 supercell of the Pmcn unit cell, many
of the surface indices are the same for the two models. Whenever
there is room for doubt, it is specified for which crystal structure
the index is valid. The number of atomic layers should be large enough to ensure bulk-like conditions in the middle of the slab without compromising numerical efficiency (one layer is here defined
to consist of one formula unit). We found that six layers are sufficient to obtain converged atomic displacements after relaxation
(less than 0.01 Å displacements in the mid layer). No atoms were
kept fixed, as tests revealed that neither fixing a layer in the middle
(to ensure bulk conditions) nor fixing a layer at one of the surfaces
(to mimic bulk continuation) led to significant changes in energy or
geometry compared to allowing all atoms to move freely.
3. Results and discussion
As mentioned in the methodology section, it was important to
use significantly increased cut-off energies to achieve reliable calculated pressures. It is interesting to see how this influences the
predicted bulk crystal structure when using the experimental
structure as input. Lattice constants of the relaxed crystal structure
of KNO3 are given in Table 1 for different choices of potentials and
Fig. 2. The PBE relaxed crystal structure of KNO3 in the Cmc21 crystal structure using a ball- and stick model seen along the c axis (a) and along the a axis (b). K, N, and O
atoms are drawn as grey, blue, and red balls. In (c), the coordination of K by nine oxygen atoms is outlined with a polyhedron. The oxygen atoms are then drawn small when
belonging to the first coordination sphere of the K atom, and large otherwise. (For interpretation of the references to colour in this figure legend, the reader is referred to the
web version of this article.)
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O.M. Løvvik et al. / Computational Materials Science 50 (2010) 356–362
(a)
(b)
O
0
O
O
N
N
O
O
7.4
0
O
1
Fig. 3. Contour plot of the valence electron density (a) and the electron localization function (b) in a plane intersecting an NO3 unit. The electron density is plotted in units of
e/Å3, and the scales are logarithmic.
energy cut-offs, compared to the experimental structure reported
by Adiwidjaja and Pohl [13]. The relaxed crystal structure using
the hard potential in VASP with the highest cut-off is shown in
Fig. 2.
These results illustrate quite clearly the importance of using a
high enough cut-off for the relaxation: while the default (non-converged) cut-off of the hard potentials (755.6 eV) leads to predicted
lattice constants between 2.4% and 4.2% lower than experiment,
the converged cut-off (950 eV) implies lattice constants between
1.2 and 1.9% higher than experiment. The numerical error resulting
from using the default cut-off is thus up to 6% for the lattice constants, and more than 13% for the predicted volume. The case is
even worse when using the soft potentials; the predicted lattice
constants are then off by up to 8.5% and the volume by more than
15% from the converged values. This clearly demonstrates the
necessity to use increased cut-off energies when performing automatic relaxation; an increase of approximately 30% is sufficient to
approach reliable forces (see Fig. 1(b)), which is in accordance with
general recommendations in the VASP guide.
When the norm-conserving BLYP potentials are used in the
relaxation, the lattice parameters are up to 9% higher than experiment, and the volume is 22% higher than experiment. This deviation is much larger than normally seen in band-structure
calculations. To ensure that the automatic relaxation was not to
blame, we performed a manual optimization of the cell volume
keeping the relative lattice constants fixed. This gave an optimized
cell volume within 0.1% of that obtained by the automatic relaxation, so we believe that the results presented in Table 1 are representative for the BLYP potential. The BLYP potentials are fitted to a
set of atomic and molecular test systems, and this could be the
explanation for being less suited to extended systems. Also, a recent test of various DFT potentials concluded that the largest deviation of the enthalpy of formation from the experimental one was
found for NO2 in the case of BLYP [55]. The predicted enthalpy of
formation was significantly lower than the experimental one, indicating that BLYP is underbinding the N–O system. This is consistent
with the significant over-estimation of the unit cell volume found
in the present study.
It is important to note that the converged values of the PBE predicted lattice constants are larger than the experimental ones, even
if the temperature was higher in the experiments (room temperature) than in our calculations (in principle 0 K). If temperature effects were taken into account, the predicted lattice constants
would most probably be even larger, particularly since it is known
that the aragonite structure can exhibit large anisotropy of the
thermal expansion [30]. The difference between predicted and
measured lattice constants would thus be more than 2%. We believe that except for the temperature, the PBE functional is the only
significant remaining source of errors in the VASP calculations.
Therefore, the DFT prediction of lattice constants is not so good
for this compound, for any of the methods studied. The PBE potential is otherwise known to perform very well when it comes to prediction of lattice constants, but the present results suggest that at
least in some cases this could be due to cancellation of errors. We
have performed additional tests with VASP to see if this large discrepancy is due to the PBE potential, details are not presented here.
Changing to the PW91 potential did not make a significant contribution to the overall picture; this also exhibited painstakingly slow
convergence and clear over-estimation of the lattice constants. We
will discuss possible reasons for the difficulties of predicting lattice
constants in this system in the following.
It is interesting to see how the choice of potential and cut-off
influences the interatomic distances. We can see from Table 1 that
the N–O distances are relatively unaffected by this choice, with all
the potentials predicting slightly too high values. The largest difference is seen in the K–O distances, which can serve to explain the
spread in predicted lattice constants. The potentials with default
cut-off predict significantly diminished K–O distances compared
to the experimental ones, while the hard potential with converged
cut-off predict a K–O distance 2.3% larger than the experimental
value. The latter difference is exactly the same as for the N–O distance when using the hard potential with converged cut-off. Since
there is no direct bond between K and N, the K–N distances reflect
the sum of the K–O and O–N distances. Thus, the poor performance
of the soft potentials mainly influences the K–O bond lengths, predicting a too close contact.
Some insight into why the soft potentials are insufficient for the
system under study can be achieved by considering the electronic
density in the plane defined by a nitrate ion. Fig. 3 shows how the
absence of atomic symmetry stretches far into the core region of
both nitrogen and oxygen atoms. This indicates a significant overlap of the N valence charge with the O core charge (and vice versa),
increasing the importance of the core region for the bonding
behaviour. The electron localization function (ELF) is also shown
in Fig. 3. The ELF is related to the kinetic energy of the electrons,
and can thus give a graphic picture of the chemical bonds [56].
The ELF confirms that significant parts of the N–O bond are located
in the middle between the atoms, indicating a mainly covalent
bond.
There are no such bonds between K and O, reflecting the strong
ionic character of the interaction between K+ and NO3 . There are
suggestions of directional bonds towards K around the O atoms,
but clearly centred at the O sites (they are seen as the strong red
areas in two directions, perpendicular to the N–O bond, from the
O atoms in Fig. 3 (b)). This can be seen more clearly in Fig. 4, where
the valence charge density is plotted as lines between nearest
O.M. Løvvik et al. / Computational Materials Science 50 (2010) 356–362
Valence charge density (e/Å3)
360
7
6
K-O
5
N-O
4
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
Normalized interatomic distance
Fig. 4. The valence charge density in e/Å3 between N and O (red solid curve) and
between K and O (blue dashed curve) as a function of the interatomic distance. The
distance has been normalized for ease of comparison. Oxygen is placed at relative
distance 1 in both cases. (For interpretation of the references to colour in this figure
legend, the reader is referred to the web version of this article.)
neighbour N–O and K–O pairs. It can be clearly seen how there remains a significant valence charge between N and O, while it vanishes in the middle between K and O. It is also notable how far into
the core of the oxygen atom the bond is reaching in the case of K–
O; this is probably why a small core region and high energy plane
waves are needed to describe this bond properly. We may regard
the potassium atom as a spherical, rather hard cation polarizing
the nitrate anion.
The calculated density of states (DOS) of bulk KNO3 with the
PBE functional is shown in Fig. 5. The PBE–GGA calculated band
gap is 3.0 eV. Since GGA tends to significantly underestimate the
band gap, we expect that the real band gap is somewhat higher;
strongly suggesting that KNO3 is an insulator. Indeed, an experimental band gap of 6–7 eV has been reported [57]. Three distinct
regions are clearly visible in the valence region. Between approximately 8 and 7 eV a relatively narrow section is seen with very
similar shape for all the three elements; the N states are of purely p
character, the O states are both s and p, while the K states exhibit
both s, p, and d character. These features thus suggest both N–O
and K–O interactions. The next filled region is between 2.5 and
1.5 eV, and is dominated by pure O-p states, as well as more K
states with s, p, and d character. This indicates an interaction between K and O. The last valence region lies between 0.5 and
0 eV, and contains O-p states as well as K-p and d states. The narrow and non-connected shape of the DOS confirms the picture of
an ionic solid with strong N–O and K–O interactions at low
energies.
We now turn to the calculations of surface energies. The surface
energy of a particular face can be calculated by inserting a vacuum
layer between two opposite faces; the difference in total energy
between the resulting slab and the bulk is then equal to twice
the surface energy.
The surface energy is converged already when the size of the
vacuum layer is 10 Å, but we have nevertheless used vacuum layers of 16 Å between the slabs in order to manipulate surfaces in a
later study without the risk of interaction between images. In calculations of surface energies of metals, it is important to use an
alternative way of defining the bulk reference energy to avoid linear divergence of the surface energy as the number of layers is increased [58]. One typical way to obtain this is to calculate the total
energy of a series of slabs with increasing number of layers, and
use the (interpolated) difference in energy between such slabs as
the bulk reference energy [58]. It has later been shown that this
is not so important for insulators, for which only a few layers usually are sufficient to obtain convergent surface energies, and the
Fig. 5. The calculated local and total density of states (DOS) calculated by VASP. The
s-projected (filled blue), p-projected (solid black) and d-projected (dotted red) DOS
are shown for K, N, and O, as well as the total DOS for KNO3 (filled green). The
energy is measured in eV relative to the Fermi level, and the DOS units are states/
(atomunit cell volume) for the local DOS and states/unit cell volume for the total
DOS. (For color interpretation given in this figure legend the reader is referred to see
the web version of this article.)
true bulk energy can be used as the bulk reference energy [59].
We have found this to be the case for KNO3, and in the data in Table
2 show that the three methods give surface energies differing by
only a few percent. All other surface energies are thus calculated
using the true bulk energy as the reference energy.
Whether to relax the fractional coordinates for the whole slab
or only parts of it, is also a matter of choice. It is sometimes chosen
to keep one or more layers at one side of the slab frozen; this side
then represents a continuation into the bulk geometry [60]. This
approach makes it possible to induce an unphysical polarization,
and is often avoided for dielectric materials. Another option is to
keep one or more layers frozen in the middle of the slab to ensure
bulk geometry. We have compared these options with calculations
allowing all atoms to relax. No significant differences were found,
except for movements of the atoms that were kept fixed. Even the
neighbour atoms of the fixed ones changed very similarly in all the
three cases. We have therefore chosen to relax all atoms in the
Table 2
Surface energies (in J/m2) for different faces of KNO3. Relaxed and unrelaxed slabs are
compared in some cases. For the {0 0 1} face, two different bulk reference energies
were tested: The true bulk energy (‘‘Bulk”) and an interpolated energy representing
the middle layer of the slab for an increasing number of slab layers (‘‘Slab series”; see
text and Ref. [58] for details). Some of the faces can be chosen to be terminated by
either NO3 (‘‘NO3 top”) or K (‘‘K top”). The {1 1 2} face of the Cmc21 unit cell
corresponds to the {1 1 1} face of the smaller Pmcn unit cell. Similarly, the {1 1 4} face
corresponds to the {1 1 2} face. The results marked with an asterisk are calculated by
QE, other results are by VASP.
Face
{0 0 1}
{0 0 1}
{0 1 0}
{0 1 0}
{1 0 0}
{1 1 0}
{1 1 0}
{1 1 2}
{1 1 4}
{0 0 1}
Bulk
NO3 top
K top
NO3 top
K top
NO3 top
K top
NO3 top*
Slab series
Unrelaxed
Relaxed
Unrelaxed
Relaxed
0.235
0.671
0.894
0.410
0.593
0.862
0.410
1.020
0.658
0.148
0.199
0.544
0.666
0.344
0.250
0.297
0.337
0.219
0.662
0.193
O.M. Løvvik et al. / Computational Materials Science 50 (2010) 356–362
remaining calculations. The change in surface energy when going
from the unrelaxed to the relaxed slab varies from around 10% to
65%, see Table 2. This reflects the very different degree of surface
reconstruction for the different surfaces, as illustrated in Fig. 6,
where side-views of some of the surfaces of this study are shown.
It is clear from Fig. 6 that the reconstruction of the NO3 terminated {1 1 0} surface involves rotating the NO3 units into the surface plane. Thus, the O atoms initially pointing out of the surface
move closer to neighbouring K+ ions, which leads to electrostatic
stabilization. This does not happen in the {010} face covered by
K+ ions, as the surface is covered by spherically symmetric ions.
We note that the most stable face – {0 0 1} – is the only face with
the NO3 units already oriented parallel to the surface. This compares favourably with experiments, which have shown that the
{0 0 1} face is indeed the most stable one. The calculated surface
energy of 0.19 Jm 2 is very similar to that of other ionic crystals,
e.g. 0.16 Jm 2 for NaCl [61,62] and 0.17 Jm 2 for Li2CO3 [63].
It is in many cases possible to create different surface terminations for a specific face. As an example, the {1 1 0} face can have at
least three different terminations, even when enforcing stoichiometry. The structure shown in Fig. 6 is terminated by NO3 on both
sides of the slab – this is called ‘‘NO3 top” in Table 2 Alternatively,
the lower two NO3 units may be moved to the top of the slab,
yielding a slab with two different terminations – NO3 on one side
and K+ on the other. (This would create an unphysical dipole moment, and has not been studied here.) If the lower pair of K+ ions
(one is hidden behind the other) is also moved from bottom to
top, the slab is terminated by K+ ions on both sides. This is called
‘‘K top” in Table 2.
Reconstructions play different roles for different surface terminations. This can be illustrated by the three various situations in
Fig. 6. In the case of the {0 0 1} face, there is little energy to be
gained by reconstruction, since the NO3 ions are already parallel
to the surface. Neither does reconstruction take place for the K terminated {0 1 0} face; in this case electrostatic interaction energy
can only be slightly increased by reconstruction. This changes,
however, when moving to the NO3 covered {1 1 0} face. The bulk
cut structure exhibits oxygen ions pointing directly out of the surface plane, which obviously is an expensive construction as the
surface energy is relatively high for this surface before relaxation.
361
After relaxation, however, the surface energy is 65% lower, due to
the reconstruction of NO3 ions. This brings the NO3-covered
{1 1 0} face lower in energy than the K-covered analogue. The
opposite was the case before relaxation, emphasizing the importance of the reconstruction.
4. Conclusions
Density functional calculations of KNO3 have been performed
within the generalized gradient (PBE) approximation using hard
PAW potentials and norm-conserving potential. Thorough investigations of convergence demonstrated that very high cut-off energies were required to obtain reliable energies, forces, and
pressure. This tendency proved particularly strong for the default
(‘‘soft”) potentials, and harder versions were required for proper
convergence. We attribute this to the unusual bonding within
the nitrate group, with electronic changes taking place far into
the core region of the oxygen atoms. Analysis of the electron density and electron localization functions gives a picture of strongly
ionic interactions between K+ and NO3 , and strongly covalent
bonds between N and O.
Standard potentials predict lattice constants up to 8.5% higher
for the VASP calculations and 9% for the QE calculations, compared
to the results found using the proper choice of parameters. The
converged calculations predict slightly elongated lattice constants
(up to 2%) compared to experimental values.
Various crystal faces were constructed using a slab structure
with vacuum between the slabs. Relaxation of the slabs demonstrated that NO3 units with oxygen pointing out of the surface
plane reconstructed upon relaxation, bringing oxygen into the surface plane. The surface with the lowest surface energy was the
{0 0 1} face, in which the NO3 ions are parallel to the surface already in the bulk cut structure. The surface energy was found to
be 0.19 Jm 2.
Acknowledgements
A generous grant of computer time from the Norwegian Metacenter for Computing Science (NOTUR) and Norwegian Defence Research Establishment are gratefully acknowledged.
References
Bulk cut
001
010
110
Relaxed
Fig. 6. Some of the faces of this study, seen from the side. The slabs are bulk cut (not
relaxed, upper row) and relaxed (lower row). The {0 0 1} (NO3 top), {0 1 0} (K top),
and {1 1 0} (NO3 top) faces are shown from left to right. Note the orientation of the
NO3 ions (blue and red balls), which in the case of the {1 1 0} face is rotated parallel
to the surface plane upon relaxation. (For interpretation of the references to colour
in this figure legend, the reader is referred to the web version of this article.)
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