4 Basic properties of nanoparticles – III

Dr. Anchal Srivastava

Contents of this Unit

1. Introduction

2. Potential wells

3. Partial confinement

4. Properties dependent on density of states

5. Exciton

6. Summary

Learning Outcomes

After studying this module, you shall be able to understand
Understand the different types of potential wells.
Energy and wave functions corresponding to the different types of quantum wells
Partial confinement and expressions for the number of particles and density of states for 0D, 1D, 2D and 3D.
Meaning of excitons.

1. INTRODUCTION

In the earlier two modules we have learnt about the nanoparticles, types of quantum confined structures etc. Nanostructures can be categorised mainly into quantum wells, quantum wires and quantum dots. We have also learnt about the size effect, changes in band gaps, variation in the number of surface atoms etc. Now in this module we will discuss about the change in the density of states and their corresponding effects.

2. POTENTIAL WELLS

In the previous section we have discussed the delocalization aspects of conduction electrons in a bulk metal. These electrons were referred as free electrons, but perhaps unconfined electrons would be a better word for them. This is because when the size of a conductor diminishes to the nano region, these electrons begin to experience the effect of confinement, meaning that their motion becomes limited by the physical size of the region or domain in which they move. The influence of electrostatic forces becomes more pronounced and the electrons become restricted by a potential barrier that must be overcome before they can move more freely. More explicitly, the electrons become sequestered in what is called potential well and enclosed region of negative energies. A simple model that exhibits the principal characteristics of such a potential well is a square well in which the boundary is very sharp or abrupt. Square well can exist in one, two, three and higher boundary for simplicity, we describe a one dimensional case.

Standard quantum mechanical texts show that for an infinitely deep square potential well of width ‘a’ in one dimension, the coordinate x has the range of values – 1/2 a ≤ x ≤ 1/2 a inside the well, and the energies there are given by the expressions

Which are plotted in figure 3.1, where E0 is the ground state energy and the quantum number n assumes the values n = 1, 2, 3…..The electrons that are present fill up the energy levels starting from the bottom, until all the available electrons are in place. An infinite square well has an infinite number of energy levels, with ever-widening spacing as the quantum number n increases. If the well is finite, then its quantized energies E, all lie below the corresponding infinite well energies, and there are only a limited number of them. Figure 3.2 illustrates the case for a finite well of potential depth V0 =7 E0 which has only three allowed energies. No matter how swallow the well, there is always at least one bound state E.

The electrons confined to the potential well move back and forth along the direction x, and the probability of finding an electron at a particular value of x is given by the square of the wave function │ψn(x)│2 for the particular level n where the electron is located. There are even and odd wave functions ψn(x) that alternate for the levels in the levels in the one-dimensional square well, and for the infinite square well we have the unnormalized expressions

Figure 3.1. Sketch of the wave functions for the four lowest energy levels (n = 1 – 4) of the one dimensional infinite square well. For each level the form of the wave function is given. On the left, and its parity (even or odd) is illustrated on the right.

Figure 3.2. Sketch of a one dimensional square well showing how the energy levels En of a finite well (right side, solid horizontal lines) lie below their well counterparts (left side dashed lines).

These wave functions are sketched in figure 3.4 for the infinite well. The property called parity is defined as even when ψn(-x) = ψn(x), and it is odd when ψn(-x) = – ψn(x).

Another important variety of potential well, is one with a curved cross section. For a circular cross section of radius ‘a’ in two dimensions, the potential is given by V = 0 in the range 0 ≤ ρ ≤ a, and has the value V0 at the top and outside, where ρ = (x² + y²)^1/2, and tan φ = y/x in polar coordinates. The particular finite well sketched in figure 3.3 has only three allowed energy levels with the values E1, E2 and E3. There is also a three dimensional analogue of the circular well in which the potential is zero for the radial coordinate r in the range 0 ≤ r ≤ a, and has the value V0 outside, where r = (x² + y² + z²)^1/2. Another type of commonly used potential well is the parabolic well, which is characterized by the potentials V(x) = ½ kx², V(ρ) = ½ kρ² and V(r) = ½ kr² in one, two and three dimensions, respectively and Figure 3.4 provides a sketch of the potential in the one dimensional case.

The energy of a two dimensional infinite rectangular potential well

Depends on two quantum numbers, nx = 0,1,2……. and ny = 0,1,2……, where n² = ??² + ??² . This means that the lowest energy state E1 = E0 has two possibilities, namely nx = 0, ny = 1 and nx = 1, ny = 0, so the total degeneracy (including spin direction) is 4. The energy state E₅ = 25 E₀ has more possibilities since it can have for example, n_x = 0, n_y = 5, or n_x = 3, n_y = 4, and so on, so its degeneracy is 8.

Figure 3.3. Sketch of a two dimensional finite potential well with crystalline geometry and three energy levels.

Figure 3.4. Sketch of a one dimensional parabolic potential well showing the positions of the four energy levels.

3. PARTIAL CONFINEMENT

In the previous section we examined the confinement of electrons in various dimensions, and we found that it always leads to a qualitatively similar spectrum of discrete energies. This is true for a broad class of potential wells, irrespective of their dimensionality. We also examined, in previous module, the Fermi gas model for delocalized electrons in these same dimensions and found that the model leads to energies and densities of states that differ quite significantly from each other. This means that many electronic and other properties of metals and semiconductors change dramatically when the dimensionality changes. Some nanostructures of technological interest exhibit both potential well confinement and Fermi gas delocalization, confinement in one or two dimensions, and delocalization on two or one dimensions, so it will be instructive to show how these strikingly different behaviours coexist.

In a three-dimensional Fermi sphere the energy varies from E = 0 at the origin to E = EF at the Fermi surface, and similarly for the one- and two dimensional analogues. When there is confinement in one or two directions, the conduction electrons will distribute themselves along the corresponding potential well levels that lie below the Fermi level along confinement coordinate directions, in accordance with their respective degeneracies di and for each the electrons will delocalized in the remaining dimensions by populating Fermi gas levels in the delocalization direction of the reciprocal lattice. Table 3.1 lists the formulas or the energy dependence of the number of electrons N(E) for quantum dots that exhibit total confinement, quantum wires and quantum well, which involve partial confinement and bulk material, where there is no confinement. The density of states formulas D(E) for these four cases are listed in the table 3.1. The summations in these expressions are over the various confinement well levels i.

Figure 3.5 shows plots of the energy dependence N(E) and density of states D(E) for the four types of nanostructures listed in table 3.1. We see that the number of electrons N(E) increase with the energy E, so the four nanostructure types vary only qualitavily from each other. However, it is the density of states D(E) that determines the various electronic and other properties, and these differ dramatically for each of the three nanostructure types. This means that the nature of the dimensionality and of the confinement associates with a particular nanostructure have a pronounced effect on its properties. These considerations can be used to predict properties of nanostructures and one can also identify types of nanostructures from their properties.

Table 3.1: Number of electrons N(E) and density of states D(E) = dN(E)/dE as a function of the energy quantum wires. Quantum wells and bulk for electrons delocalized/confined in quantum wires. Quantum wells and bulk , materials

4. PROPERTIES DEPENDENT ON DENSITY OF STATES

We have discussed the density of states D(E) of conduction electrons, and have shown that it is strongly affected by the dimensionality of a material. Phonons or quantized lattice vibrations also have a density of states DPH(E) that depends upon the dimensionality and like its electronic counterparts, it influences some properties of solids, but our principal interest is in the density of states D(E) of electrons.

The specific heat of a solid C is the amount of heat that must be added to it to raise its temperature by one degree Celsius. The main contribution to this heat is the amount that exists lattice vibrations and this depends on phonon density of states D_PH(E). At low temperatures there is also a contribution to the specific heat Cel of a conductor arising from the conduction electrons and this depends on the electronic density of states at the Fermi level.

where k_B is the Boltzmann constant

The susceptibility χ = M/H of a magnetic material is a measure of the magnetization M or magnetic moment per unit volume that is induced in the material by the application of and applied magnetic field H. The component of the susceptibility arising from the conduction electrons, called the Pauli susceptibility, is given by the expression χel = µ2 D_(EF), where µB is the unit magnetic moment called the Bohr magneton, and is hence χel characterized by its proportionality to the electronic density of states D(E) at the Fermi level, and its lack of dependence on the temperature.

When a good conductor such as aluminium is bombarded by fast electrons with just enough energy to remove an electron from a particular Al inner-core energy level, the vacant level left behind constitutes a hole in the inner core band. An electron from the conduction band of the aluminium can fall into vacant inner-core level to occupy it, with the simultaneous emission of an X ray in the process. The intensity of the emitted X radiation is proportional to the density of states of the conduction electrons because the number of electrons with each particular energy that jumps down to fill the hole is proportional to D(E). Therefore a plot of the emitted X–ray intensity versus the X-ray energy E has a shape very similar to a plot of D(E) versus E. These emitted X-rays for aluminium are in the energy range from 56 to 77 eV.

Some other properties and experiments that depend on the density of states and can provide information on it are photoemission spectroscopy, Seebeck effect (thermopower) measurements, the concentrations of electrons and holes in semiconductors, optical absorption, determined of the dielectric constant, the Fermi contact term in nuclear magnetic resonance (NMR), the de Haas-Van Alphen effect, the superconducting energy gap and Josephson junction tunnelling in the superconductors. It would take us too far afield to discuss any of these topics. Experimental measurements of these various properties permit us to determine the form of the density of states D(E), both at the Fermi level EF and over a broad range of temperature.

Figure 3.5. Schematic representation of density of states for 3D, 2D, 1D and 0D materials.

5. EXCITONS

Excitons are a common occurrence in semiconductors. When an atom at a lattice site loses an electron, the atom acquires a positive charge that is call hole. If the hole remains localized at the lattice, and the detached negative electron remains on its neighbourhood, it will be attracted to the positively charged hole through the Coulomb interaction and can become bound to form a hydrogen-type atom.

Technically it is called Mott-Wannier type of exciton. The Coulomb force of attraction between two charges Qe = -e and Qh = +e separated by a distance r is given by F = -ke²/εr², where e is the electronic charge, k is a universal constant, and ε is the dielectric constant of the medium. The exciton has a Rydberg series of energies and a radius given by aeff = 0.0529 (ε/ε0) / (m*/m0), where ε/ε0 is the ratio of the dielectric constant of the medium to that of the free space, and m*/m0 is the ratio of the effective mass of the exciton to that of a free electron. Using the dielectric constant and electron effective mass values respectively, we obtain for GaAs

E = 5.2 eV and a_eff = 10.4 nm (3.7)

showing that the exciton has a radius comparable to the dimension of a typical nanostructure.

The exciton radius can be taken as an index of the extent of confinement experienced by nanoparticle. Two limiting regions of confinement can be identified in the basis of the ratio of the dimension d of the nanoparticle to the exciton radius aeff namely, the weak-confinement regime with d > aeff (but not d >> aeff) and the strong confinement regime d < aeff. The more extended limit d >> aeff corresponds to no confinement. Under weak-confinement conditions the exciton can undergo unrestricted translational motion, just as in the bulk material but for strong confinement this translation motion becomes restricted. There is an increase in the spatial overlap of the electron and hole wave functions with decreasing particle size, and this has the effect of enhancing the electron hole interaction. As a result, the energy splitting becomes grater between the radiative and nonradiative exciton states. An optical index of the confinement is the blue shift (shift to higher frequencies) of the optical absorption edge and the exciton energy with decreasing nanoparticle size. Another result of the confinement is the appearance at room temperature of excitonic features in the absorption spectra that are observed only at low temperatures in the bulk materials.

6. SUMMARY

In this module, you study
About the quantum well: types, wave functions and correspondingly energy expressions. Partial confinement & number of electrons and density of states for 0D, 1D, 2D and 3D materials.
About the properties dependent upon the density of states. Excitons

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Learn More:

1.Introduction to Solid State Physics by C. Kittel.

2.Solid State Physics by S.O. Pillai

3. Introduction to Nanotechnology by C. Poole & F. Ownes