3 Basic properties of nanoparticles – II

Dr. Anchal Srivastava

Contents of this Unit

1. Introduction

2.Size and dimensionality effect

2.1 Size effect

2.2 Conduction electrons and dimensionality

3.  Fermi gas and density of states

4.  Summary

Learning Outcomes

• After studying this module, you shall be able to understand
• Quantum confinement of electrons in one, two and three dimensions. What are quantum well, quantum wire and quantum dots?
• What is quantization?
• Femi gas and change in density of states for quantum well, quantum wire and quantum dot.

1. INTRODUCTION

When the size or dimension of a material is continuously reduced from a large or macroscopic size, such as a meter or centimetre, to a very small size, the properties remains the same at first, then changes begin to occur, until finally when the size drops below 100 nm, dramatic changes in properties can occur. If one dimension is reduced to the nano range while the other two dimensions remains large, then we obtain a structure known as a quantum well. If two dimensions are reduced and one remains large, the resulting structure is referred as a quantum wire. The extreme case of this process of size reduction in which all three dimensions reach the low nanometre range is called a quantum dot. The word quantum is associated with these three types of nanostructures because the changes in properties arise from the quantum mechanical nature of physics in the domain of the ultra-small. Figure 2.1 illustrates these processes of diminishing the size for the case of rectilinear geometry. In this module we probe the dimensionality effect that occur when one, two or all three dimensions becomes small.

Figure 2.1. Progressive generation of rectangular & curvilinear nanostructures.

2. SIZE AND DIMENSIONALITY EFFECT

2.1    Size effect:

Nanostructures can be prepared by using top-down or bottom-up approaches. If we select the type III-IV semiconductor GaAs as a typical material, the lattice constant is a = 0.565 nm, and the volume of the unit cell is 0.180 nm3. The unit cell contains four Ga and four As atoms. Each of these atoms lies on a face centered cubic (FCC) lattice and the two lattices are displaced with respect to each other by the amount 1/4, 1/4, 1/4 along the unit cell body diagonal. This puts each Ga atom in the center of a tetrahedron of As atoms corresponding to the grouping GaAs4, and each atom arsenic has a corresponding configuration AsGa4. There are about 22 of each atom type per cubic nanometer and a cube shaped quantum dot 10 nm on a side contains 5.56 X 103 unit cells.

The question arises as to how many of the atoms are on the surface and it will be helpful to have a mathematical expression for this in terms of the size of the particle with the zinc blende structure of GaAs which has the shape of cube. If the initial cube is taken in the form of figure 2.2 and nanostructures containing n3 of these unit cells are built up, then it can be shown that the number of atoms Ns on the surface, total number of atoms NT, and the size or dimension d of the cube are given by

Here a = 0.565 nm is the lattice constant of GaAs. These equations, (2.1) – (2.3), represent a cubic GaAs nanoparticle with its faces in the x-y, y-z and z-x planes, respectively. Table 2.1 Ns, NT, d and the fraction of atoms on the surface for small n is one of the principal factors that differentiates properties of nanostructures from those of the bulk material. An analogous table could easily be constructed for cylindrical quantum structures of the types illustrated in figure 2.1.

Figure 2.2 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 corresponding to the atoms of the top layer of the nanoparticles, the six solid circles plus the atoms (not shown) in the center of the cube constitute the middle hexagonal layer of that figure, and the open circle at the lower left corner of the cube is one the three atoms at the bottom.

Table 2.1 Number of atoms on the surface Ns, number in the volume Nv and percentage of atoms Ns/Nv on the surface of a nanoparticles

A charge carrier in a conductor or semiconductor has its forward motion in an applied electric field periodically interrupted by scattering off phonons and defects. An electron or hole moving with a drift velocity v will, on average, experiences a scattering event every τ seconds, and travel a distance l called mean free path between collisions, where

l = vτ

(2.4)

This is called interband scattering because the charge carrier remains in the same band after scattering, such as the valence band in the case of holes. Mean free paths in metals depend strongly on the impurity content, and in ordinary metals typical values might be in the low nanometer range, perhaps from 2 to 50 nm. In very pure samples they will, of course, be much longer. The resistivity of a polycrystalline conductor or semiconductor composed of microcrystallite with diameters significantly greater than the mean free path resembles that of a network of interconnected resistors, but when the microcrystallite dimensions approach or become less than l, the resistivity depends mainly on scattering off boundaries between crystallites. Both types of metallic nanostructures are common.

Various types of defects in a lattice can interrupt the forward motion of conduction electrons, and limit the mean free path. Examples of zero dimensional defects are missing atoms called vacancies, and extra atoms called interstitial atoms located between st and at d lattice sites. A vacancy-interstitial pair is called a Frenkel defects. An example of a one dimensional dislocation is a lattice defect at an edge, or a partial line of missing atoms. Common two-dimensional defects are a boundary between grains, and a stacking fault arising from a sudden change in the stacking arrangement of closed packed planes. A vacant space called a pore, a cluster of vacancies, and a precipitate of a second phase are three dimensional defects. All of these can bring about the scattering of electrons, and thereby limit the electrical conductivity. Some nanostructures are too small to have internal defects.

Another size effect arises from the level doping of a semiconductor. For typical doping levels of 10¹⁴ to 10¹⁸ donors/cm³ a quantum-dot cube 100 nm on a side would have, on the average, from 10^-1 to 10³ conduction electrons. The former figure of 10^{– 1} electrons per cubic centimeter means that on the average only 1 quantum dot in 10 will have one of these electrons. A smaller quantum-dot cube only 10 nm on a side would have, on the average 1 electron for the 1018 doping level. And be very unlikely to have any conduction electrons for the 1014 doping level. A similar in table 2.2 demonstrates that these quantum structures are typically characterized by very small numbers or concentrations of electrons that can carry current. These results in the phenomenon of single-electron tunneling and the Coulomb blockade discussed in next module.

Table 2.2 Conduction electron content of smaller size (on left) and larger size (on right) quantum structure containing donor concentrations of 10¹⁴ – 10¹⁸ cm^-3

2.2 Conduction electron and dimensionality:

We are accustomed to studying electronic systems that exist in three dimensions, and are large or macroscopic in size. In this case the conduction electrons are delocalized, and move freely throughout the entire conducting medium such as a copper wire. It is clear that all the wire dimensions are very large compared to the distances between atoms. The situation changes when one or more dimensions of the copper become so small that it approaches several times the spacing between the atoms in the lattice. When this occurs, the delocalization in impeded, and the electrons experience confinement. For example, consider a flat plate of copper that is 10 cm long, 10 cm wide and only 3.6 nm thick. This thickness corresponds to the length of only 10 unit cells, which means that 20% of the atoms are in unit cells at the surface of the copper. The conduction electrons would be delocalized in the plane of the plate, but confined in the narrow dimension, a configuration referred to as a quantum well. A quantum wire is a structure such as a copper wire that is long in one dimension, but has a nanometer size as its diameter. The electrons are delocalized and move freely along the wire, but are confined in the transverse directions. Finally, a quantum dot, which might have the shape of a tiny cube, a short cylinder, or a sphere with low nanometer dimension, exhibit confinement in all three spatial dimensions, so there is no delocalization.

3. FERMI GAS AND DENSITY OF STATES

Many of the properties of good conductors of electricity are explained by the assumption that the valence electrons of a metal dissociates themselves from their atoms and become delocalized conduction electrons that move freely through the background of positive ions such as Na+ of Ag+. On the average they travel a mean free path distance l between collisions, as mentioned in above section. These electrons act like a gas called a Fermi gas in their ability to move with varying kinetic energy, E=1/2 mv²=P^2/2m, where m is the mass of the electron, v is its speed or velocity, and p = mv is its momentum. This model provides a good explanation of Ohm’s law, whereby the voltage V and current

I are proportional to each other through the resistance R, that is, V = IR.

In a quantum mechanical description the component of the electron’s momentum along the x-direction px has the value px = 2  ℎ kx where h is the Planck’s constant and the quantity kx is the x component of the wave vector k. Each particular electron has unique kx, ky and kz values and these form a lattice in k space which is also called reciprocal space. At the temperature of absolute zero, the electrons of the Fermi gas occupy all the lattice points in reciprocal space out to a distance kF from the origin k = 0, corresponding to a value of the energy called the Fermi energy EF, which is given by

We assume that the sample is a cube of side L, so its volume V in ordinary coordinate space is V = L³. The distance between two adjacent electrons space is 2π/L, and at the temperature of absolute zero all the conduction electron are equally spread out inside a sphere of radius kF, and of volume 4/3πK^3_F in k space. This equal density is plotted in figure 3.1a for the temperature absolute zero of 0K, and in figure 3.1b we see that deviations from equal density occur near the Fermi energy EF level at higher temperatures.

The number of conduction electrons with a particular energy depends on the value of the energy and also on the dimensionality of the space. This is because in one dimension the size of the Fermi region containing electrons has the length 2kF, in two dimensions it has the area of the Fermi circle π 2, and in three dimensions it has the volume of the Fermi sphere 43 π 3. These expressions are listed in column of the table 3.1. If we divide each of these Fermi regions by the size of the corresponding k-space unit cell listed in column 2 of this table, and make use of equation 3.1 to eliminate kF, we obtain the dependence of the number of electrons N on the energy E given on the left side of the table 3.2, and shown in figure 3.2. The slopes of the lines N(E) is shown figure 3.2, provide the density of states D(E), which is defined more precisely by the ,mathematical derivative D(E) = , corresponding to the expression dN = D(E)/dE. This means that the numbers of electrons dN with an energy E within the narrow range of energy dE = E2 – E, is proportional to the density of states at that value of energy. The resulting formulas for D(E) for the various dimensions are listed in the middle column of table 3.2 and are shown in figure 3.3. We see that the density of states decreases with increasing energy for one dimension is constant for two dimensions and increases with increasing energy for three dimensions. Thus the density of states has quite a different behaviour for the three cases. These equations and plots of the density of states are very important in determining electrical, thermal and other properties of metals and semiconductors, and make it clear why features can be so dependent on the dimensionality.

Table 3.1 properties of coordinated and k space in one, two and three dimensions

Figure 3.1. Fermi-Dirac distribution function f(E), indicating equal density in k-space, plotted for the

temperature (a) T = 0 and (b) 0 < T << TF.

Table 3.2. Number of electrons N and density of states D(E) = dN(E)/dE as a function of the energy E for conduction electrons delocalized in one, two and three spatial dimensions

Figure 3.2. Numbers of electrons N(E) plotted as a function of the energy E for conduction electrons delocalized in one (quantum wire), two (quantum well), and three dimensions (bulk material).

Figure 3.3. Density of states D(E) = dN(E)/dE plotted as a function of the energy E for conduction electrons delocalized in one (Q-wire), two (Q-well) and three (bulk) dimensions.

4. SUMMARY

From the above module we learn that size reduction of material leads to the dramatic change in its properties. Size reduction in one dimension leads to quantum well structures, in which electrons are confined in one direction and free to move in two directions. In the same way, when the size is reduced in two dimensions we have nanostructure called nanowires. In nanowires electrons are free to move in one direction while confined in to two directions. Finally the maximum limit of size reduction in to three dimensions leads to the nanostructures called quantum dots having confinement of electrons in all the three directions. We have also concluded that as the size of the particles becomes smaller and smaller the number of atoms on the surface increases, eg. For 100 particles the percentage of atoms on the surface is 1.5 while it increases to 51.1 in case of 2 atoms per particle. A particle is said to be quantum confined when the size of the particle becomes less or comparable to its conduction electron’s mean free path. Another size effect comes from the doping as the reduction in size of the particles leads to the reduction of the number of electrons in the particles and hence a change in its electrical properties.