2 Basic properties of nanoparticles-I

Dr. Anchal Srivastava

Contents of this Unit

1.  Introduction

2. Structure

2.1  Size dependence of properties

2.2 Crystal structures

2.3 Face centred cubic nanocrystals

3.  Energy bands

3.1 Classification of materials

3.2 Reciprocal space

4.  Summary

Learning Outcomes

• After studying this module, you shall be able to understand
• What is the meaning of word “nano”?
• Nanoclusters contain some finite numbers of nanoparticles called magic numbers. Understand the octahedron and tetrahedron sites.
• Two dimensional crystal structure and face centred cubic nanostructures Classification of materials based on their band gaps

1. INTRODUCTION

In the recent years nanotechnology has become one of the most important and exciting forefront fields in Physics, Chemistry, engineering and biology. The prefix nano in the word nanotechnology means a billionth (10-9). Nanotechnology deals with various structures of matter having dimensions of the order of a billionth of a meter. It is not clear when humans first began to take advantages of nanosized materials. It is known that in the fourth century A.D. Roman glassmakers were fabricating glasses containing nanosized metals. The potential importance of clusters was recognized by the Irish-born chemist Robert Boyle in his Sceptical Chymist published in 1661. Photography is an advanced and mature technology, developed in the eighteenth and nineteenth centuries, which depends on the production of silver nanoparticles sensitive to light. Richard Feynman was awarded the Nobel Prize in physics in 1965 for his contributions to quantum electrodynamics. Feynman, in 1960, presented a visionary and prophetic lecture at a meeting of the American Physical Society, entitled “There is Plenty of room at the Bottom,” where he speculated on the possibilities and potential of nanosized materials.

2. STRUCTURE

2.1    Size dependence of properties:

Many properties of solids depend on the size range over which they are measured. Microscopic details become averaged when investigating bulk materials. At the macro or large scale range ordinarily studied in traditional fields of physics such as mechanics, electricity and magnetism and optics, the sizes of the objects under study range from millimeter to kilometers. The properties that we associate with these materials are averaged properties, such as the density and elastic moduli in mechanics, the resistivity and magnetization in electricity and magnetism, and the dielectric constant in optics. However, when the properties are measured in micrometer and nanometer range, many properties of materials change, such as mechanical and ferroelectric, and ferromagnetic properties. Below the nanometer range, there is the atomic scale near 0.1 nm, followed by the nuclear scale near a fermi (10-15 m). Before going to the nanometer range, first we should understand the corresponding properties at the macroscopic and mesoscopic levels. Many of the nanostructure include the group IV elements such as Si or Ge, type III-V semiconducting compounds such as GaAs, or II-VI semiconducting materials such as CdS, so these materials will be used as example for bulk properties that become modified with incorporation into nanostructures.

2.2 Crystal structures:

In its solid form materials are characterized into two categories: a) Crystalline and b) amorphous. Crystalline materials have so called long range ordering as regularity can extend throughout the crystal while amorphous materials are short ranges. Examples of the crystalline materials are NaCl, KCl, etc. and glass, wax etc. are the examples of amorphous materials. Gases lack both long range and short range order.

A two dimensional crystal have five type lattice ordering a) square; b) primitive rectangular; c) centered rectangular; d) hexagonal and e) oblique types, which is shown in the following figure 1.1.

Figure 1.1: The five Bravais lattices that occur in two dimensions, with the unit cells indicated: (a) square; (b) hexagonal; (c) primitive rectangular; (d) centered rectangular; (e) oblique.

These arrangements are called Bravais lattices. The general or oblique Bravais lattice has two unequal lattice constants a ≠ b and an arbitrary angle θ between them. For the perpendicular case when θ = 90ͦ, the lattice becomes the rectangular type. For the special case a = b and θ = 60ͦ, the lattice is the hexagonal type formed equilateral triangles. Each lattice has a unit cell, indicated in the figures, which can replicate throughout the plane and generate the lattice.

In three dimensions there are three lattice constants, a, b and c and three angles: α between b and c; β between c and a; and γ between lattice constants a and b. There are 14 Bravais lattices, ranging from the lower symmetry triclinic type in which all three lattice constants and all three angles differ from each other (a ≠ b ≠ c and α ≠ β ≠ γ), to the highest symmetry cubic case in which all the lattice constants are equal and all the angles are 90ͦ (a = b =c andα = β = γ = 90ͦ).There are three Bravais lattices in the cubic system, namely, a primitive or simple cubic (SC) lattice in which the atoms occupy the eight corners of the cubic unit cell as shown in the fig 1.2a, a body centered cubic (BCC) lattice points occupied at the corners and in the center of the unit cell as indicated in fig. 1.2b, and a face centered cubic (FCC) Bravais lattice with atoms at the corners and in the centers of the faces, as shown in the fig.1.2c.

Figure 1.2. Unit cells of the three cubic Bravais lattices: (a) simple cubic (SC); (b) body centered cubic (BCC); (c) face centered cubic (FCC).

In two dimensions the most efficient way to pack identical circles (or spheres) is the equilateral triangle arrangement shown in fig. 1.3a, corresponding to the hexagonal Bravais lattice of fig. 1.1d. A second hexagonal layer of spheres can be placed on top of the first to form the most efficient packing of two layers, as shown in fig. 1.3b. For efficient packing, the third layer can be placed either above the first layer with an atom at the location indicated by T or in the third possible arrangement with an atom above the position marked by X in the figure 1.3b. In the first case a hexagonal lattice with a hexagonal close packed (HCP) structure is generated, and in the second case, a face centered cubic lattice results.

In the three dimensional case of close packed spheres there are spaces or sites between the spheres where smaller atoms can reside. The point marked by X on fig. 1.3b, called an octahedral site, is equidistant from the three spheres O below it and from the three spheres O above it. An atom A at this site has the local coordination AO6. The radius aoct, of this octahedral site is

where a is the lattice constants and a0 is the radius of the spheres. The number of octahedral sites is equal to the number of spheres. There are also smaller sites, called tetrahedral sites, labeled T in the figure that are equidistant from the nearest neighbor spheres, one below and three above, corresponding to AO4 for the local coordination. This is a smaller site since its radius aT is

Figure 1.3. Close packing of spheres on a flat surface: (a) for a monolayer; (b) with a second layer added. The circles of the second layer are drawn smaller for clarity. The location of an octahedral site is indicated by X and the position of a tetrahedral site is designated by T on panel (b).

There are twice as many tetrahedral sites as there are spheres in the structures. Many diatomic oxides and sulfides such as MgO, MgS, MnO and MnS have their larger oxygen or sulfur anions in a perfect FCC arrangement with the smaller metal cations located at octahedral sites. This is called NaCl lattice type, where we use the term anion for a negative ion and cation for a positive ion. The mineral spinel MgAl2O4 has a face centered arrangement of divalent oxygen O2- (radius 0.132 nm) with the Al3+ ions (radius 0.051 nm) occupying one half of the octahedral sites and Mg2+ (radius 0.066 nm) located in one-eighth of the tetrahedral sites in a regular manner.

2.3 Face centered cubic nanoparticles

Most metals in the solid state form close packed lattices; thus Ag, Al, Au, Cu, Co, Pb, Pt and Rh, as well as rare gases Ne, Kr and Xe are face centered cubic (FCC), and Mg, Nd, Os, Re, Ru, Y, and Zn are hexagonal close packed (HCP). A number of other metallic atoms crystallize in the not so closely packed body-centered cubic (BCC) lattice, and a few such as Cr, Li and Sr crystallize in all three structure types, depending on the temperature. An atom in each of the two close packed lattices has 12 nearest neighbors. Figure 1.4 shows the 12 neighbors that surround an atom (darkened circle) located in the center of a cube for a FCC lattice. Figure 1.5 shows the 14 sided polyhedron, called a dekatessarahedron, that is generated by connecting the atoms with planar faces. Sugano and Koizumi (1998) call this polyhedron a cuboctahedron. The three open circles at the upper right of fig. 1.5 are the three atoms in the top layer. The six, darkened circles plus an atom in the center of the cube of fig. 1.5 constitute the middle layer and the open circle at the lower left of fig. 1.4 is one of the three obscured atoms in the plane below the cluster. This 14 sided polyhedron has six square faces and eight equilateral triangles faces.

If another layer of 42 atoms is laid down around the 13 atom nanoparticles, one obtains a 55 atom nanoparticle with the same dekatessarahedron shape. Larger nanoparticles with the same polyhedron shape are obtained by adding more layers, and the sequence of numbers on the resulting particles, N = 1, 13, 55, 147, 309, 561…… which are listed in table 1.1, are called structural magic numbers.

Figure 1.4. Face centered cubic unit cell showing the 12 nearest neighbor atoms that surround the atom (darkened circle) in the center.

For n layers the numbers of atoms N in these FCC nanoparticles is given by the formula

Figure 1.5. Thirteen atom nanoparticle set in its FCC unit cell, showing the shape of the 14 sided polyhedron associated with the nanocluster. The three open circles at the upper right correspond to the atoms of the top layer, the six solid circles plus the atom (not pictured) in the center of the cube constitute the middle hexagonal layer and the open circles at the lower left corner of the cube is ine of the three atoms at the bottom of the cluster.

For each value of n, Table 1.1 lists the numbers on the surface, as well as the percentage of atoms on the surface. The table also lists the diameter of each nanoparticle, which is given by the expression (2n – 1) d, where d is the distance between the corners of nearest neighbor atoms and d = √2, where a is the

lattice constant. If the same procedure is used to construct nanoparticles with the hexagonal close packed structure that was discussed in the previous paragraph, a slightly different set of structural magic numbers is obtained, namely, 1, 13. 57, 153, 321, 581……

Table 1.1 Number of atoms (structural magic numbers) in rare gas or metallic nanoparticles with face-centered cubic close packed structures

*The diameters d in nanometers for some representative FCC atoms are Al 0.286, Ar 0.376, Au 0.288, Cu 0.256, Fe 0.248, Kr 0.400, Pb 0.350 and Pd 0.275.

Purely metallic FCC nanoparticles such as Au55 tend to be very reactive and have short lifetimes. They can be ligand-stabilized by adding atomic groups between their atoms and on their surfaces. The Au55 nanoparticles has been studied in the ligand-stabilized from Au55(PPh3)12Cl6 which has the diameter of ~1.4 nm, where PPh3 is an organic group. Further examples are the magic numbers nanoparticles Pt309(1, 10-phenantroline)36O30 and Pd561(1, 10-phenantroline)36O200.

The magic numbers that we have been discussing are called structural magic numbers because they arise from minimum-volume, maximum-density nano-particles that approximate a spherical shape, and have close packed structures characteristics of a bulk solid. These magic numbers take no account of the electronic structure of the constitute atoms in the nanoparticles. Sometimes the dominant factor in determining the minimum-energy structure of small nanoparticles is the interactions of the valence electrons of the constituents’ atoms with this potential, so that the electrons occupy orbital levels associated with this potential. Atomic cluster configurations in which these electrons fill closed shells are especially stable, and constitute electronic magic numbers.

When mass spectra were recorded for sodium nanoparticles NaN, it was found that mass peaks corresponding to the first 15 electronic magic numbers N = 3, 9, 20, 36, 61…. were observed for cluster sizes up to N = 1220 atoms (n = 15), and FCC structural magic numbers starting with N = 1415 for n = 8 were observed for larger sizes. The mass spectral data versus the cube root of the number of atoms N1/3 are plotted in figure 1.6, and it is clear that the lines from both sets of magic numbers are approximately equally spaced, with the spacing between the structural magic numbers about 2.6 times that between the electronic ones. This result provides evidence that small clusters tend to satisfy electronic criteria and large structures tend to be structurally determined.

Fig. 1.6 Dependence of the observed mass spectra lines from NaN, nanoparticles on the cube root N1/3 of the number of atoms N in the cluster. The lines are labeled with the index n of their electronic and structural magic numbers obtained from Martin et al (1990).

3. ENERGY BANDS

3.1 Classification of materials

When a solid is formed the energy levels of the atoms broaden and form bands with forbidden gaps between them. The electrons can have energy values that exist within one of the bands, but cannot have energies corresponding to values in the gaps between the bands. The lower energy bands due to the inner atomic levels are narrower and are all full of electrons, so they do not contribute to the electronic properties of a material. The outer or valence electrons that bond the crystal together occupy what is called a valence band. For an insulating material the valence band is full of electrons that cannot move since they are fixed in position in chemical bonds. There are no delocalized electrons to carry current, so the material is insulator. The conduction band is far above the valence band in energy, as shown in figure 1.7a, so it is not thermally accessible and remains essentially empty. In other words, the heat content of the insulating materials at room temperature T = 300 K is not sufficient to raise an appreciable number of electrons from the valence band to conduction band, so the number in the conduction band is negligible. Another way to express this is to say that the value of the band gap energy Eg far exceeds the value of kBT of the thermal energy, where kB is Boltzmann constant.

Figure 1.7. Energy bands of (a) an insulator, (b) an intrinsic semiconductor and (c) a conductor. The cross-hatching indicates the presence of electrons in the bands.

In the case of a semiconductor the gap between the valence band and conduction bands is much less as shown in figure 1.7b, so Eg is closer to thermal energy kBT, and the heat content of the materials at room temperature can bring about the thermal excitation of some electrons from the valence band to the conduction band where they carry current. The density of electrons reaching the conduction band by this thermal excitation process is relatively low, but by no means negligible, so the electrical conductivity is small; hence the term semiconducting. A material of this type is called intrinsic semiconductor. A semiconductor can be doped with donor atoms that give electrons to the conduction band where they can carry current. The material can also be doped with acceptor atoms that obtain electrons from the valence band and leave behind positive charges called holes that can also carry currents. The energy levels of these donors and acceptors lie in the energy gap, as shown in figure 1.8. The former produces n-type, that is negative charge or electron conductivity and the later produces p-type, that is positive type or hole conductivity. These two types of conductivity in semiconductors are temperature dependent, as is the intrinsic semi conductivity.

A conductor is a material with a full valence band and a conduction band partly full with delocalized conduction electrons that are efficient in carrying electric current. The positively charged metal ions at the lattice sites have given up their electrons to the conduction band, and constitute a background of positive charge for the delocalized electrons. Figure 1.7 shows the energy bands for this case.

 

Figure 1.8. Sketch of the forbidden energy gap showing acceptor levels the typical distance

the top of the valence band, donor levels the typical distance D below the bottom of the conduction band, and deep trap levels nearer to the center of the gap. The value of the thermal energy kBT is indicated on the right.

In actual crystals the energy bands are much more complicated than in suggested by the sketches of figure 1.7, with the bands depending on the direction in the lattice.

3.2 Reciprocal space

In above sections we discussed the structures of different types of crystals in ordinary or coordinate space. These provided us with the positions of the atoms in the lattice. To treat the motion of conduction electrons, it is necessary to consider a different type of space that is mathematically called a dual space relative to the coordinate space. This dual or reciprocal space arises in quantum mechanics and a brief qualitative description of it is presented here. The basic relationship between the frequency f = ω/2π, the wavelength λ, and the velocity v of a wave is λf = v. it is convenient to define the wave vector k = 2π/λ to give f = (k/2π)v. For a matter wave, or the wave associated with conduction electrons, the momentum p = mv of an electron of mass m is given by p = (h/2π)k, where Planck’s constant h is universal constant of physics. Thus for the simple case momentum is proportional to the wavelength with the units of the reciprocal length, or reciprocal meters. We can define a reciprocal space called k-space to describe the motion of electrons.

If a one dimensional crystal has a lattice constant ‘a’ and a length that we take to be L = 10a, then the atoms will be present along a line or positions x = 0, a, 2a, 3a……10a = L. the corresponding wave vector k will assume the values k =  2π/L, 4π/L……20π/L = 2π/a. We see that the smallest value of k is 2π/L, and the largest value is 2π/a. the unit cell in this one dimensional coordinate space has the length a, and the important characteristic cell in reciprocal space, called Brillouin zone, has the value 2π/a. the electron sites within the Brillouin zone are at the reciprocal lattice points k = 2πn/L, where for our example n = 1, 2, ….10, and k = 2π/a at the Brillouin zone boundary where n = 10.

For a rectangular direct lattice in two dimensions with coordinates x and y, and lattice constants a and b, the reciprocal space is also two dimensional with the wave vectors kx and ky. By analogy with the direct lattice case, the Brillouin zone in this two dimensional reciprocal space has the length 2π/a and width 2π/b, as shown sketched in figure 1.9. The extension to three dimensions is straight forward. It is important to keep in mind that the kx is proportional to momentum px of the conduction electron in the x direction and similarly for the relationship between ky and py.

Figure 1.9. Sketch of (a) unit cell in two dimensional x and y coordinate space and (b) corresponding Brillouin zone in reciprocal space kx, ky for a rectangular Bravais lattice.

4. SUMMARY

• In this module you study
• Properties of the materials dependents upon their sizes.
• Crystalline materials have long range ordering and amorphous materials have short range ordering.
• There is only five crystal structure for 2 dimensional structures.
• Generally, crystals have close packed structures such as FCC or HCP.
• Nanostructures exist in the form of nanoclusters in which the number of nanoparticles is governed by magic numbers.
• Reciprocal space
• Classification of materials based on their band gap.

Suggested Readings

• C. Kittel, Introduction to Solid State Physics, 7th Ed., Wiley New York, 1996
• Charles P. Poole & Frank J. Ownes, Introduction to nanotechnology, Wiley Publishers, New York, USA.