Table Of ContentTransferable Pair Potentials for CdS and ZnS Crystals
Michael Grünwald,1 Phillip L. Geissler,2 and Eran Rabani3
1)Department of Chemistry, University of California, Berkeley, California 94720
2)Department of Chemistry, University of California, and Lawrence Berkeley National Laboratories, Berkeley,
California 94720
3)School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978,
Israel
(Dated: January 17, 2012)
A set of interatomic pair potentials is developed for CdS and ZnS crystals. We show that a simple energy
function, which has been used to describe the properties of CdSe [J. Chem. Phys. 116, 258 (2002)], can be
parametrized to accurately describe the lattice and elastic constants, and phonon dispersion relations of bulk
2 CdSandZnSinthewurtziteandrocksaltcrystalstructures. Thepredictedcoexistencepressureofthewurtzite
1
androcksaltstructures,aswellastheequationofstateareingoodagreementwithexperimentalobservations.
0
These new pair potentials enable the study of a wide range of processes in bulk and nanocrystalline II-VI
2
semiconductor materials.
n
a
J I. INTRODUCTION catalyticdevices.25 Thepotentialsaredesignedtorepro-
4 ducethebulklatticeandelasticconstantsoftherelevant
1 crystalstructures,aswellasphonondispersionrelations.
Manyimportantprocessesinsolidstatematerials,like
the melting transition,1,2 structural transformations,3,4 They are specifically constructed to be compatible with
]
i or diffusion of impurities and defects5 require atomistic each other and with the existing model for CdSe16 and
c
thereforealsoenablesimulationsofmixturesofthethree
s resolution in space and time for a comprehensive under-
- standing of the underlying mechanism. Despite major compounds.
l
r advances in electron microscopy,6 experiments can only The paper is organized as follows: In Section II, we
t discusstheconstructionofthepairpotentialsandspecify
m provideacoarse-grainedviewofsuchprocesses. Molecu-
their parameters. In Section III, we apply the models to
lar dynamics computer simulations can in principle pro-
.
at videthenecessarymicroscopicperspective,buttheirpre- calculatebulkenthalpiesasafunctionofpressureandthe
equations of state for CdSe, CdS and ZnS and compare
m dictivepowerdependsonthereliabilityandfeasibilityof
the predictions to experimental results. Discussion and
available models.
-
d Methodsbasedonfirstprinciplesthatretainadescrip- conclusions are given in Section IV.
n tionoftheelectronicstateofthesystempotentiallyoffer
o
the highest accuracy. Because of their high computa-
c II. THE PAIR POTENTIAL
tional demand, however, they are not currently suited
[
for in-depth studies of systems involving more than a
1 We use the simple model developed for CdSe16 as a
fewhundredatoms,ortimescaleslongerthanafewtens
v template to also describe CdS and ZnS. The two-body
of picoseconds. Classical interaction potentials, parame-
2
interatomic potential consists of a long range Coulomb
terized to reproduce emergent properties of the modeled
9
part and a short range part which is represented by a
9 material,offeracompromisebetweenaccuracyandcom-
Lennard-Jones (LJ) form:
2 putational speed. Potentials of different functional form
01. awniddelcyomdipslpeaxriatytehcahveembiceaelnadnedveplhoypseidcafloprrmopaetretriieasl,srwanitgh- V = qiqj +4(cid:15) (cid:40)(cid:18)σij(cid:19)12−(cid:18)σij(cid:19)6(cid:41), (1)
2 ing from water7 to gold8 and biopolymers.9 Depending ij rij ij rij rij
1 on the properties studied, agreement with experiment is
: usually good, in some cases rivaling or even besting that where the indexes i and j refer to Cd, Zn, S, and Se
v
i of affordable ab initio methods. atoms. To facilitate transferability and reduce the num-
X Sparkedbyacomprehensiveexperimentalstudyofthe ber of parameters, we use standard combining rules for
√
r structural changes occurring in CdSe nanocrystals un- interactions of unlike atom types, namely (cid:15)ij = (cid:15)i(cid:15)j
a
der pressure,3,10–15 a simple pair potential has been de- and σ = 1(σ +σ ).
ij 2 i j
veloped16 and successfully applied to reveal the mech- The parameters q , (cid:15) , and σ were obtained by fit-
i i i
anisms of structural rearrangements in both bulk and ting the lattice and elastic constants, and phonon dis-
nanocrystalline CdSe.4,17–24 However, simulation stud- persion relations of bulk CdS and ZnS in three crystal
ies of processes in many other semiconductor materials, structures: wurtzite, zinc-blende and rocksalt. As an
or in multi-component systems like core/shell crystals additional constraint, the energy difference between the
or seeded nanorods, have been precluded by the lack of wurtzite and rocksalt structure at zero pressure was fit-
available models. Here we present a set of model po- tedtoab initiocalculations.26,27 Thiswasdonetoensure
tentialsforcrystallineCdSandZnS,twosemiconductors that the wurtzite structure is the more stable structure
with potential use in various light harvesting and opto- atlowpressures. Thefittingcalculationswereperformed
2
q(e) σ(Å) (cid:15)/k (K)
B 15
Zn 1.18 0.02 17998.4
S -1.18 4.90 16.5 Zn-S
Cd 1.18 1.98 16.8 10 Zn-Zn
Se -1.18 5.24 14.9 S-S
V) 5
Table I. Potential parameters defining the interatomic inter- (e
actions in ZnS, CdS and CdSe. r)
V(ij 0
at 0 K,althoughthe experimentaldatawere obtainedat -5
finite temperatures.
Toobtaintransferablepotentialsandensureneutrality
-10
of the modeled materials, we fixed the magnitude of all
ion charges to that of the original CdSe model, i. e.,
10 Cd-Se
|q |=1.18e. Wethenproceededinthefollowingway. To
i Cd-Cd
obtainamodelforCdScompatiblewiththatofCdSe,we Se-Se
)
relaxedtheLJparametersforsulfur,keepingparameters V 5
e
for Cd fixed. In the second step, to arrive at a model ) (
r
for ZnS, we likewise relaxed the LJ parameters for Zn, V(ij 0
keeping S fixed. Thus, a total of four parameters (σ ,
Zn
σ , (cid:15) , and (cid:15) ) was determined with the fitting.
S Zn S -5
We used a relaxed fitting procedure similar to the one
discussed in Refs. 28 and 29. This procedure is substan-
-10
tially more expensive computationally than the conven- 0 2 4 6 8 10
tionalfitting. However,itallowsamuchhigherqualityof
r (Å)
fittingwhichisrequiredtoproperlyreproducethestruc-
turalpropertiesofCdSe. Intherelaxedfittingprocedure,
theerrorwasdefinedontheresidualofthestructuraland
Figure1. InteratomicpairpotentialsforZn-Zn,S-S,andZn-
dynamical properties of the optimized configurations of S (top panel) and Cd-Cd, Se-Se and Cd-Se (bottom panel).
thedifferentcrystalphasesratherthanontheexperimen- Open circles in the upper panel show the pure Coulomb re-
tal observed structures. Namely, the configurations and pulsion term.
thelatticeconstantsofeachcrystalphasewerequenched
using the conjugate gradient algorithm, and the afore-
mentioned properties were calculated and compared to dispersionrelationsofwurtziteZnS,CdSandCdSealong
the experimental values for the quenched structures. In the ΓA direction is shown in Fig. 2. The experimental
alltheresultsreportedherewehaveusedEwaldsumsto results were obtained from inelastic neutron scattering33
evaluatetheelectrostaticinteractions,30 withapartition- on 116CdSe and Raman scattering for ZnS.32 For CdS,
ingparameterbetweenthetwospaceschosentominimize we have compared our results with calculations provid-
the computational effort.31 The Lennard-Jones part of inggoodagreementwithinfraredabsorption.34 Thesplit
the potential was cut at half the box length (≈10 Å). betweentheE andB attheΓ-pointindicatestheionic
2 1l
The final parameter values resulting from the fit (in- nature of the material.35 The overall frequencies in ZnS
cluding those for CdSe from Ref. 16) are summarized in are higher due to the lighter masses of the atoms com-
Table I. Consistent with their negative charge, the van pared to CdS and CdSe. The TA and LA branches are
der Waals (vdW) radii of the anions S and Se are sig- back-folded into the lower E and B branches, respec-
2 1l
nificantly larger than those of the cations. In agreement tively. Similar back-folding occurs for the A , B and
1 1h
with the corresponding ionic radii, the vdW radius of S the upper E and E branches. As can be clearly seen in
2 1
is smaller than that of Se and the radius of Zn is smaller thefigure,oursimplifiedmodelcapturestheback-folding
than that of Cd. For Zn we find that the best fit is in all branches. The overall agreement between the cal-
obtained for a nearly vanishing vdW radius and a large culationsandtheexperimentalresultsisreasonablegiven
value of (cid:15). These particular values are a result of the thesimpleformofthepotential(cf. Eq.(1)). Themodel
constraints imposed on the charge of the ions and the performsslightlybetterforZnSandismoreaccuratefor
sequence of fitting steps. Because of the small value of the lower frequency branches. The agreement can likely
σ, the forces between Zn atoms at relevant distances are be improved using a polarizable model, as is well known
governed by the Coulomb term only; the large value of (cid:15) for alkali halides.34,36 We note that even better agree-
setstheZn-Sbondlength. Aplotoftheinteratomicpair ment with experiment has been achieved with ab initio
potentials is shown in Fig. 1. methods33,37.
The accuracy of the model in reproducing the phonon In Table II we compare the lattice and elastic con-
3
10 CdS Wurtzite Zinc-blende Rocksalt
Hz) 8 CdS ac 46..1661 ((46..1730))3388,,3399 5.83 (5.82)40 5.43 (5.44)41
T
uency ( 46 CCC111123 1310557...893(((559210...107))) 9392..69 9563..97
q C 144.3 (93.8)
e 2 33
Fr C44 19.1 (15.0) 41.3 53.8
0 C 20.1 (16.3)
66
B 54.0 (65.0) 55.1 (55.0) 68.1
12 ZnS
)
z
H 10 ZnS Wurtzite Zinc-blende Rocksalt
(T 8 a 3.89 (3.85)26,42 5.48 (5.41)26,42 5.20 (5.06)26,42
ncy 6 c 6.26 (6.29)26,42
ue 4 C11 161.4 (131.2) 150.1 (94.2) 109.7
q C 53.8 (66.3) 51.4 (56.8) 88.0
e 12
Fr 2 C13 28.2 (50.9)
0 C33 213.1 (140.8)
C 32.4 (28.6) 62.2 (43.6) 88.1
44
) 7 A1 CdSe C66 53.8 (32.4)
Hz 6 B E B 82.3 (82.1) 84.3 (69.3) 94.6
T 5 1h 2
(
cy 4 B E1 CdSe Wurtzite Zinc-blende Rocksalt
en 3 1l a 4.37 (4.30)43 6.13 (6.08)43 5.74 (5.71)13
u
q 2 E LA c 6.97 (7.01)43
Fre 1 2 C11 87.2 (74.6) 82.4 73.2
0 TA C12 29.1 (46.1) 26.6 45.3
0 0.1 0.2 0.3 0.4 0.5
C 13.3 (39.4)
13
Wave vector (00x ) C 118.0 (81.7)
33
C 15.9 (13.0) 33.4 45.3
44
C 29.1 (14.3)
66
Figure2. PhonondispersionrelationsofwurtziteCdS(upper B 44.1 (53.4) 45.1 54.6
panel), ZnS (middle panel), and CdSe (lower panel) along
the ΓA direction. The filled diamonds represent literature
results.32–34 TableII.Calculatedlatticeconstants(inÅ),elasticconstants
(inGPa),andbulkmodulus(inGPa)ofCdS,ZnSandCdSe
for three crystal structures. Experimental results are given
stants calculated using our model with the correspond- in parenthesis when available. (Elastic constants and bulk
moduli were taken from Refs. 42, 44–46.)
ing experimental values. Note that the results for CdSe
differ slightly from our original report.16 This deviation
isaconsequenceofusingasmallertoleranceforthemin-
imization, which was considered excessively cumbersome with the experimental values (which show some spread
at the time the CdSe parameters were generated. The from one report to the other). Moreover, the values of
calculated lattice constants are within 1% of the experi- the bulk modulus are comparable in accuracy to those
mental values, except for the case of the rocksalt struc- obtained by ab-initio methods.26,27
ture in ZnS, where the error is 3%. The agreement be-
tweenthecalculatedelasticconstantsandtheexperimen-
talvaluesisqualitative,withsmallerrorsinC11 andC44. III. PHASE DIAGRAM AND EQUATION OF STATE
Tocomparethebulkmodulusobtainedfromourmodel
with experimental results we have used the well known
To test the accuracy of our models on quantities not
relation between the bulk modulus and the elastic con-
directlyusedinthefittingprocedure,wecalculatedcoex-
stants:
istencepressuresforthewurtziteandrocksaltstructures
at T = 300 K, as well as equations of state for all three
C (C +C )−2C2
B = 33 11 12 13 , (2) crystal structures.
C +C +2C −4C
11 12 33 13 InFig.3weplottheenthalpyasafunctionofpressure
for hexagonal symmetry and for bulk ZnS, CdS, and CdSe. The pressure was varied
between0and15GPausingtheconstantpressureMonte
B =[(C +C +C )/3+2C ]/3, (3) Carlo simulation technique.30 For each crystal structure,
11 22 33 12
we have used a periodically replicated simulation box
for cubic symmetry. The calculated bulk moduli for of more than 400 atoms, and averaged the results over
wurtziteZnS,CdSandCdSeareinreasonableagreement 50,000 Monte Carlo cycles. Each cycle on average con-
4
-4.75 1
ZnS
CdSe Wurtzite 0.95
V) -5 Rocksalt
e
m ( CdS V00.9
per ato -5.25 ZnS V/0.85
py -5.5 0.8
al
Enth -5.75 1 CdS
0.95
-6
0 2 4 6 8 10 12 00.9
V
Pressure (GPa)
/
V
0.85
Figure 3. Enthalpies as a function of pressure of bulk ZnS
0.8
(upper panel), CdS (middle panel), and CdSe (lower panel),
at a temperature of 300 K. The solid and dashed lines show 1
results obtained for wurtzite and rocksalt crystal structures, CdSe
respectively. The vertical dotted lines mark points of equal Wurtzite
Rocksalt
enthalpies and approximately correspond to thermodynamic 0.9
Experimental
wurtzitetorocksalttransitionpressuresintherespectivema- V0
terials. V/
0.8
sistedofoneattempteddisplacementmoveforallatoms,
andoneattemptedchangeofthesimulationboxvolume. 0.7
-1 0 1
10 10 10
Approximatingthetruecoexistencepressurebypoints
pressure (GPa)
of equal enthalpy, we find that the phase transforma-
tion from wurtzite to the rocksalt structure occurs at ≈
2.4GPa, ≈1.6GPa, and≈11.4GPaforCdSe, CdSand
Figure 4. Volume asa function ofpressure for bulk ZnS(up-
ZnS, respectively. These values agree well with experi-
perpanel),CdS(middlepanel),andCdSe(lowerpanel). The
mentallyobservedtransitionpressuresof≈2.5GPa,13,47
solid and dotted lines are the calculated results for wurtzite
≈ 2.5−3.2 GPa,48–51 and ≈ 12 GPa,52 for CdSe, CdS, and rocksalt crystal structures, respectively. V is the unit
0
and ZnS, respectively. In all three materials, the zinc- cell volume of the wurtzite structure at zero pressure. Filled
blendecrystalstructureisnotstableinthepressurerange circlesshowexperimentalresults.13,52,53 (NotethattheCdSe
studied here; its enthalpy (not shown) is slightly higher data were obtained in experiments on 45 Å diameter CdSe
than the corresponding wurtzite enthalpy. nanocrystals.13)
In Fig. 4 we plot the equation of state (volume as a
function of pressure) for all three materials. We find
excellentagreementwithexperimentsonCdSe,13CdS,53 We have calculated the transition pressure of the
and ZnS.52 wurtzite to rocksalt transformation, as well as equations
ofstateatroomtemperatureforallthreematerials. Our
resultsareingoodagreementwithexperiments,thusver-
IV. CONCLUSIONS ifying the accuracy and practicability of the pair poten-
tial on quantities not used in the fitting procedure.
We have developed a set of transferable pair poten- Asafinalnote,wepointoutthatthesimplefunctional
tials for CdS and ZnS whose form is similar to that form of the potential naturally limits the portability of
used for CdSe.16 The model consists of positively and our model. In particular, we have tested its accuracy
negatively charged ions (Cd/Zn and S/Se, respectively) in reproducing the lattice constants of ZnSe, a material
which interact via a Coulomb potential, supplemented whose properties were not included in the fit but which
by short-range repulsion terms and van der Waals at- can be easily modeled using the parameters for Zn and
tractive terms. In order to be able to model alloys and Se. Wefindthatdeviationsfromexperimentalvaluesare
hetero-structures of CdSe/CdS/ZnS, we used standard ontheorderof5%,considerablylargerthanfortheother
combining rules for the cross terms and fixed the mag- materials.
nitude of the charges to the value obtained for CdSe,16 The current work extends our previous work on CdSe
thereby reducing the total number of fitting parameters andprovidesabasisforthestudyofstructuralproperties
to 4. The parameters were fitted to reproduce lattice and dynamical processes of a larger variety of physically
and elastic constants, and phonon dispersion relations of interesting materials. These include phase transforma-
wurtzite, zinc-blende and rocksalt crystals structures. tion in core-shell structures, alloys which are important
5
for suppression of the Auger process,54,55 seeded rods, 24R.Martonák,TheEuropeanPhysicalJournalB79,241(2011).
and more. 25T. Trindade, P. O’Brien, and N. Pickett, Chem. of Mater. 13,
3843(2001).
26M.Durandurdu,J.Phys.Chem.Solids70,645(2009).
27J. J. Tan, Y. Li, and G. F. Ji, ACTA Physica Polonica A 120,
V. ACKNOWLEDGMENTS
501(2011).
28J.D.Gale,Phil.Mag.B73,3(1996).
MG was supported by the Austrian Science Fund 29J.D.Gale,J.Chem.Soc.FaradayTrans.93,629(1997).
30D. Frenkel and B. Smit, Understanding Molecular Simulation
(FWF) under Grant number J3106-N16. ER thanks the
(AcademicPress,NewYork,2002).
FP7 Marie Curie IOF project HJSC and the Miller In-
31M.P.AllenandD.J.Tildesley,Computer simulation of liquids
stitute for Basic Research in Science at UC Berkeley for
(OxfordUniversityPress,Oxford,1987).
financial support via a Visiting Miller Professorship. 32Y.C.Cheng,C.Q.Jin,F.Gao,X.L.Wu,W.Zhong,S.H.Li,
andP.K.Chu,J.Appl.Phys.106,123505(2009).
33F. Widulle, S. Kramp, N. M. Pyka, A. Göbel, T. Ruf, A. De-
bernardi, R. Lauck, and M. Cardona, Physica B 263-264, 448
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