Introduction
In studies on unique properties of nanocrystals, the real crystal structure of an individual nano-grain is very important since a complete picture of the physical properties of nanocrystals requires information about their atomic structure. Such information was not really available until recently due to a lack of reliable information about the actual atomic arrangements at the surface and inside the nanocrystal grain. Our ultimate goal is to learn about the “whole” atomic structure of nanocrystals, i.e., about the atomic architecture of the surfaces terminating a single nanocrystalline grain and on the internal lattice underneath the surface, using CdSe as the model material.
Since the best available experimental techniques either lack the required resolution (HRTEM), or can only visualize atoms at the very surface (STM, AFM), modeling and computer simulation is practically the only way to obtain information on the atomic structure of nanocrystals with resolution required to examine the symmetry and short- and long-range atomic arrangements in the crystals.
The rearrangement of atoms at the surface occurs due to the environment of the atoms located at the crystal terminating planes, which is always different than that in the crystal volume. Recently, we presented the results of molecular dynamics (MD) simulations of models of CdSe crystal platelets about 6 nm thick with basal planes being the atomic planes with low Miller indices of (100), (110), or (111). We showed that the changes in the symmetry and the lengths of interatomic bonds at the surface layer have a pronounced effect on the positions of atoms underneath the surface. The corresponding changes in interatomic lattice distances are observed up to 3 nm below the surface, beyond which the parent (bulk) crystal lattice remains unaffected. This borderline depth can be called the relaxation length of the surface strain (RL). In crystals with sizes comparable to surface relaxation length the strains appearing at the end surfaces (their magnitude depending on the atomic planes terminating the crystal) will obviously interact with each other, i.e., the changes of positions of atoms underneath the terminating surfaces will overlap (Stelmakh et al.
2017). This effect can appear in nanocrystals with only a few nanometers in size. In such crystals, the atomic lattice throughout the entire grain volume is determined by the type of surfaces terminating the crystal volume and their spacing. There are multiple experimental reports on the size dependence of lattice parameters of semiconductor nanocrystals, including CdSe (Tolbert and Alivisatos
1995; Zhang et al.
2002; Masadeh et al.
2007). The proposed explanations usually recall surface tension (Zhang et al.
2002) and the resulting internal pressure as being responsible for the effect. In the present work, we show that the actual situation is by far more complex. The internal structure of the nanocrystals is neither uniform nor isotropic and the observed changes of the apparent average lattice parameters are caused by the crystal lattice rearrangement that originates at the surface and extends into the bulk of nanocrystals.
Summary and conclusions
Properties of a nanocrystal are determined by the atomic structure of both its interior and the surface. At the first approximation some properties of the material are referred to the crystal surface (e.g., adsorption), some to their volumetric part (e.g., optical), and others are a combination of both types of contribution (e.g., compressibility, thermal expansion). It is common to attempt to describe specific physical properties of nanocrystals as being scaled with their dimensions. This approach is correct only if one assumes that the structures of both the surface and of the bulk part of the nanocrystal are invariant with nanocrystal dimensions.
There were numerous approaches to learn about the atomic structure of nanocrystal surfaces using experimental techniques that directly probe interatomic distances like XANES (Hamad et al.
1999), EXAFS (Wu et al.
2007), and NMR (Berrettini et al.
2004). Other such efforts were undertaken through measurements of the elastic properties (Huxter et al.
2009) or studies on the effect of various ligands used for colloidal growth of CdSe nanocrystals on their shape (Nair et al.
2007). Some possible atomic arrangements existing at the surface were suggested to explain experimental observations.
Modeling, both ab initio and MD, has been frequently utilized to investigate CdSe clusters and small crystallites. Most attention have been paid to electronic and phonon structures and far less to the atomic architecture. Atomic positions analysis was limited to the atoms at the very surface and the information on the crystallographic orientation of the analyzed surface was ignored in most cases. Pokrant and Whaley (Pokrant and Whaley
1999) found that the surface Se atoms relax outwards while Cd atoms remain near their original positions. Puzder et al. (Puzder et al.
2004) as well as Botti and Marquez (Botti and Marques
2007) found that surface Cd atoms relax towards the bulk by approximately 0.7 Å, while Se atoms remain in place. Zhu et al. (Zhu et al.
2009) modeled (001) and (111) terminated flat CdSe slabs and found outwards relaxation of Se and inwards relaxation of Cd at (100) surface, while at the (111) surface, some Cd atoms moved in and others moved out. They also reported several specific dimer and tetramer atomic arrangements that appear to be stable on the analyzed surfaces. Only Cherian and Mahadevan (Cherian and Mahadevan
2008) reported gradual increase of the bond length between the center and the surface in small CdSe clusters. In general, available information on the lattice deformation in CdSe nanocrystals is scarce, inconclusive, and often contradictory. Change of the bond lengths and lattice rearrangement is always found at the top layer, but its influence on the underlying crystal lattice is hardly discussed.
Knowledge on structural processes that may occur at the grain surface is desired to verify the proposed models and correlate them with specific physical properties of nanocrystalline materials. This work shows a need for verification of “general considerations” of thermodynamic properties like the melting point temperature (Cherian and Mahadevan
2008) or the “internal pressure” (Fu et al.
2017) which are routinely considered as size-dependent properties of nanocrystal. Without taking into account a specific surface structure of individual nanocrystals, many properties of real nanomaterials cannot be satisfactorily explained.
As a result of crystal truncation (surface formation), some of the bonds of the surface atoms get broken and the remaining ones change the length of their bonds in order to accommodate the new environment that leads to a formation of strain. Relaxation of the surface strains is necessary to establish a new equilibrium state corresponding to the minimum free energy of the whole ensemble of atoms constituting the crystal. It is realized through (i) changes of the nearest neighbor coordination (i.e., formation of new bonds), (ii) an appearance of disordering, and (iii) changes in the lengths of the existing bonds.
In a thick CdSe platelet, a surface terminating the volume changes the parent crystal lattice underneath down to the depth of 2–3 nm (Stelmakh et al.
2017). The bond length changes at the surface planes and between atomic planes underneath it are typically between 0.2 and 0.5%, but in some cases, they reach even 1%. The strongest strains occur at the surface and they gradually diminish with an increase in the distance from the surface until the parent (perfect) lattice is fully recovered.
In the models examined in the present work, only one dimension was in the nano-range, which allowed to determine what kind of deformation different truncating surfaces introduce into the crystal lattice. In real 3-D nanocrystals, the strains originating at different surfaces interfere with each other. In this work, we examined and quantified the effect of interference of strains originating at two opposite parallel surfaces of the same type terminating a platelet-like crystal. We showed that the observed structural changes concern not only the lengths of interatomic bonds, but they may also lead to a loss of the long-range order at the end atomic layer. It may also result in a reduction of the symmetry of the crystal lattice. The opposite surfaces interfere with each other when the sum of their surface strain relaxation lengths is larger than the distance between them. In a nanocrystal, the whole volume is affected by surface strains, regardless of which of the four types of surface examined in this work confines the crystal volume.
In the models examined in the present work only one dimension was in the nano-range, which allowed to determine what kind of deformation different truncating surfaces introduce into the crystal lattice. In real 3-D nanocrystals, the strains originating at different surfaces interfere with each other. In this work, we examined and quantified the effect of interference of strains originating at two opposite parallel surfaces of the same type terminating a platelet-like crystal. We showed that the observed structural changes concern not only the lengths of interatomic bonds, but they may also lead to a loss of the long-range order at the end atomic layer. It may also result in reduction of the symmetry of the crystal lattice. The opposite surfaces interfere with each other when the sum of their surface strain relaxation lengths is larger than the distance between them. In a nanocrystal, the whole volume is affected by surface strains, regardless of which of the four types of surface examined in this work confines the crystal volume.
The strain relaxation processes occur differently in structures with different surfaces and they have different influences on the inner part of the crystal lattice:
-
Symmetry:
(1)
At the (100) surface layer, the long-range order is lost. The fourfold arrangement of atoms in the subsequent layers turns into orthorhombic-type arrangement and, thus, the whole lattice becomes rhombohedral.
(2)
At the (110) layer, the symmetry of the original atomic arrangement is preserved, but in-plane dimensions are changed. As a result, the lattice underneath the surface is also deformed, and the cubic symmetry of the whole volumetric part of the grain is reduced to orthorhombic-like lattice.
(3)
At the (111)A surface, the original threefold symmetry of the end atomic layer is preserved. The threefold symmetry is preserved also in the whole crystal volume, although the c/a ratio and the relative shift of Cd and Se sub-lattices, u, are changed relative to the original hcp lattice.
(4)
At the (111)B terminal, atomic layer the long-range order is lost and so is the trigonal symmetry. The next two atomic layers underneath the disordered surface are strongly deformed. In the subsequent layers, the threefold symmetry is recovered, although both the c/a ratio and the u parameters are changed relative to the perfect hcp reference lattice.
-
Disordering:
A positional disordering (deviation of equilibrium atomic positions from perfectly periodic arrangement) is present in all models under examination. It is always the largest at the surface layer and decreases with an increase in the distance from the surface.
An appearance of the strongest disordering is accompanied by a loss of long-range order that occurs at (100) and (111)B surfaces. These surfaces show the atomic order which is intermediate between crystalline and amorphous-like structures. The disordering that is present at the (100) surface has only a small effect on the disorder appearing in next layers, but disordering at the (111)B surface has a strong effect on individual hexagonal layers in the entire grain volume.
A local change of interatomic intra-lattice distances (e.g., an expansion) always leads to a reciprocal “response” of the crystal lattice (a compression) that compensates the strains associated with the primary changes. Our present results show that an appearance of disordering is an alternate mechanism of strain relaxation.
Note that strains are related to intra-lattice bonds r
2, which decide on the lattice expansion or compression, while any changes in bonds between sub-lattices have no direct effect on the lattice density.
The largest changes in intra-lattice bond lengths are observed in the models where the smallest disordering is observed. In models with (110) and (111)A surfaces, the long-range order is well preserved and the changes in the lengths of interatomic bonds are the largest, up to 1% at the surface (Figs.
10,
11,
12,
13). The strongest disordering occurs in the model with the (111)B surfaces where local changes in bond lengths are only about 0.2–0.3% (Figs.
14 and
15). Similarly, in models with the (100) surfaces where the long-range order is lost at the terminal atomic layer, only small changes in the lengths of the interatomic bonds within Cd and Se sub-lattices are observed (Figs.
8 and
9).
MD calculations reported here were performed with the assumption that the CdSe nanocrystals are not passivated, i.e., they placed in a vacuum. Other types of environment, gases, liquids, organic ligands, or a solid coating may have a strong effect on the surface and consequently on the bulk structure. That might have a pronounced effect on the internal structure of materials and, consequently, affect their physical properties. Calculations of the effects of the CdSe environment (adsorbates, shells) on the nanocrystal structure are currently under investigation.