23 Semiconductor Nanoparticles-4

6.5.2 Nanoparticle with single contact

In the absence of a applied bias, the isolated contact and NP are at the same potential and when the contact is connected with the NP, the combined system must be equilibrium having the same Fermi energy in both the contact and the NP. Since the Fermi levels change with the addition or removal of electron, equilibrium is restored by charge transfer between the metal contact and the molecule. As charges are transferred, the potential of the contact relative to the molecule changes resulting shift the relative vacuum energies. This is known as charging which also changes the Fermi levels as electrons fill some states and empty out some states. Both charge transfer induced effects, charging and state filling can be modeled by capacitors.

When there is no applied bias, the Fermi energy must be constant in the metal and the NP and charge starts flowing either from NP to metal contact or metal contact to NP as bias is applied between metal contact and NP. It is possible that only a fractional amount of charge is transferred depending on the strength of coupling between contact and NP. Let us assume  that a  fractional quantity δn electrons are transferred from the contact to the NP i.e. the wavefunction of the transferred electron will be represented by a mixed state including both the contact and the NP. Hence, some part of the electron still stays with the contact. However, when δn is +1 or -1 the LUMO or HOMO would be half full, respectively and the Fermi energy would lie in the LUMO, or HOMO, respectively. In general, the number of charges on the molecule is given by

where g(E) is the density of states in NP. As the charge transfer happens at EF, Eq… reduces to

As shown in Figure 27, the Fermi energy in NP changes by δn/g after the charge transfer. The filling of density of states can be modeled by the “quantum capacitance” which can be defined as

If the NP has a large density of states at the Fermi level, its quantum capacitance is large, hence more charge are to be transferred to shift the Fermi level.

Figure 27 Transferring from metal contact to NP changes the Fermi level in NP. The magnitude of the change is determined by the density of states at the Fermi level, and expressed in terms of a quantum capacitance.

In case of metal contact, quantum capacitance is infinite as there is a large density of states at the Fermi level, so a large number of electrons are to be transferred to shift its Fermi level. Essentially, the Fermi energy of the contact is pinned by the large density of states.

The transfer of charge establishes the equilibrium and reduces the number of electrons that are transferred after contact is made between contact and NP. The transfer of electrons from the contact to NP leaves a net positive charge on the contact and a net negative charge on the molecule and charging at the interface changes the potential of the NP relative to the contact. In this scenario, the contact and the NP can be considered as electrostatic capacitor CES which is different from quantum capacitance discussed before. As shown in Figure 28 , when charge is transferred at the interface, the capacitor is charged, a voltage is established and the NP changes potential which is known as charging energy and is responsible for a shift in the vacuum energy.

where V is the voltage across the capacitor. The change in potential due to transfer of charge of an amount

Figure 28 (a) A metallic contact and a NP can be modelled as two plates of a parallel capacitor. (b) When charge is transferred, this electrostatic capacitance determines the change in electrostatic potential, and hence the shift in the vacuum energy.

The charging energy can be very high due to very high electrostatic capacitance. In case of measurement of current transport through a single NP,  the  spacing between contact and NP could be very small (~ 1 nm), the charging energy can be more than    1V per electron.    By incorporating the contribution from quantum and electrostatic capacitances (Figure 29), the Fermi energy of the NP-contact EF can be related to Fermi energy of neutral NP EF0 and given by

Figure 29 Changes in energy level alignment when charge is transferred from the metal to NP. Charging of the NP corresponds to applying a voltage across an interfacial capacitor, thereby changing the potential of the NP leading to shift of the vacuum level NP’s location, shifting all the NP energy levels along with it. In addition, the transferred charge fills some previous empty states in the NP. Both effects change the Fermi energy in the molecule.

Figure 30 A small signal model for the metal-NP junction, including the effects of charging.

To calculate CES of spherical NP of radius R, the potential at a point r from the center of the sphere is

where r > R, ε is the dielectric constant of NP and Q is the net charge on the NP sphere and the potential of the sphere is V= Q/4πεR and the capacitance can be given by

Hence, in case of NP, capacitance scales with radius of NP. For example, the capacitance of a NP with a radius of R = 1nm is approximately CES = 10^-19 F and charging energy is then UC = 1.6eV per charge.

6.5.3 Nanoparticle positioned between source and drain electrodes

When NP is placed between two electrodes source and drain, to determine the current through the NP, the potential profile for an arbitrary distribution of charges has to be calculated. Under the approximation that the electric field between two contact is uniform, the source and drain contacts can be modeled as a parallel plate capacitor with area A which is the cross sectional area of each contact and d is the separation between the two contacts. This approximation is valid if A >> d. The source and drain capacitances at a distance z from the source are given by

The potential can be shown vary as well  linearly as expected for a uniform electric field.

Figure 31 A uniform electric field between the source and drain metal contacts yields a linearly varying potential.

NP with two metallic contacts is shown in Figure 32. At equilibrium, μ1 = EF = μ2, no current flows through the NP. When a potential is applied between the source and drain contacts, Fermi level of one contact will be shifted with respect to the other, so that

There are two effects on the NP

(i)    The electrostatic effect: The energy levels within the NP will move up or down relative to the contacts.

(ii)    The charging effect: As the NP is out of equilibrium, a current will flow resulting change in the charge in NP. The amount of charge will increase if current flows through the LUMO, or decrease if current flows through the HOMO (Figure 32).

Figure 32 At equilibrium no current flows through the NP.

However, these effects are correlated, because the movement of the energy levels with respect to the contact energy levels changes the amount of charge transferred to the NP by the contacts. And the charging energy due to the charge transfer in turn changes the potential of the molecule.

6.5.4 Electrostatics: The Capacitive Divider Model of Potential

In this model,  contact/NP/Contact    device can be modelled by a NP   linked to the source and drain contacts by two capacitors, CS and CD, respectively. If the NP is placed at the middle of two contacts, the CS ~ CD. If the NP is nearer to source but far from the drain, in ther words if NP strongly couple to source compared to drain CS >> CD and vice versa.

Figure 33 Two single NP two terminal devices with potential profiles in the NP. (a) symmetric contacts, (b) asymmetric contacts. The voltage in the center of the NP is determined by the voltage division factor, η. It can be obtained by from a voltage divider constructed from capacitors.

 Figure 33 shows the potential profile under these two situations of different coupling of NP with source. Thus, an applied voltage, V, shifts the chemical potentials of both the source and drain contacts

If  the Fermi energy of the NP  at equilibrium is taken as  reference,  EF = 0

Now we can define a voltage division factor, η which provides the dropped between NP and the source contact

The voltage division factor, η determines whether conduction occurs through the HOMO or the LUMO. If η = 0 (Figure 34), then the energy levels in NP are fixed with respect to the source contact and as the potential of the drain is increased or decreased with respect to source, conduction occurs through the HOMO or LUMO, respectively. Hence the current-voltage characteristic of this device will exhibit a gap around zero bias that corresponds to the HOMO-LUMO gap of NP. If η = 0.5 (Figure 35), then irrespective of whether the bias is positive or negative, current always flows through the energy level closest to the Fermi energy and the gap around zero bias is more than HOMO-LUMO gap.

Figure 34 The voltage division factor is crucial in determining the conduction level in a single molecule device. In this example, when η = 0, conduction always occurs through the HOMO when the applied bias is positive, and through the LUMO when the applied bias is negative. The conductance gap is determined by the HOMO-LUMO separation

Figure 35 When η = 0.5, conduction always occurs through the NP energy level closer to the Fermi Energy. In this example that is the HOMO, irrespective of the polarity of the applied bias.

6.6 Growth Mechanism of Semiconductor Nanoparticles

6.6.1 Survey of Synthesis Methods

Methods for preparing colloidal NPs can be divided into two categories depending on the nature of the solvent and temperature of growth. NPs prepared by wet chemistry uses polar solvents such as water or methanol at low temperature. On the other hand, organometallic synthesis uses non-polar solvents such as trioctylphosphine oxide (TOPO) at relatively higher temperature. Within each category, there are several variations depending on the use of precursors, solvents, and special additives, as well as temperature and pressures. Out of these methods, organometallic routes generally produce better quality NPs characterized with narrow photoluminescence emission peaks. In wet chemistry routes, the synthesis temperature is limited by the boiling point of the solvent, typically less than 100oC, so the relatively low growth temperatures produces NPs with higher defect density resulting poor optical properties and wider particle size distributions.

The most popular aqueous method is arrested precipitation, which has been used to synthesize a wide range of semiconductor NPs, such as CdS, CdSe, CdTe, and HgTe. For example, in this method cadmium perchlorate is dissolved in water and hydrogen selenide gas is passed through the solution and then double replacement reaction yields CdSe NPs in solution. After that, to make the solution alkaline NaOH is added and additives such as phosphates, amines, and thiols are added to protect the NPs. For better crystalinity and lower density of defects, solvothermal processes in which pressure vessels are used to elevate the reaction temperature for better crystallinity. Whereas, in a nonpolar variation of the solvothermal process, dodecanethiol capping ligand, cadmium stearate, and selenium metal are combined and heated under pressure in tetralin which converts to naphthalene, producing a necessary hydrogen selenide precursor and the resulting CdSe NPs suspend in toluene and other nonpolar solvents. Using this method NPs as low as 2-3 nm in diameter can be obtained.

The most popular organometallic routes were initially developed for CdSe NPs and then extended to CdTe. In this method either dimethyl cadmium or cadmium oxide first dissolved in TOPO above 300 C and Se metal dissolves in TOPO at room temperature. Then Se solution is injected into the hot Cd solution and reaction typically proceeds at temperatures below 300 C. Large variations of organometallic based synthesis have been developed using different Cd precursors. However, if the bond in cadmium precursors is too strong, then NPs growth is not possible. The Cd-ligand complex must be less stable than CdSe in order to grow CdSe or CdTe NPs.

6.6.2 Nucleation

Any growth process starts with nucleation. It is required to assemble few atoms through a density fluctuation of the medium to a small crystal that is thermodynamically stable, so that does not decay to free atoms or ions even if the density fluctuation ceases to exist. Thermodynamically, few atoms to a nucleated configuration can be understood as the overcoming of a barrier. In the colloidal synthesis of NP, to start with we have two phases: solution phase, in which the atoms or molecules are dispersed in the solution and crystalline phase, in which the atoms are arranged with a symmetry. At constant temperature and constant pressure, the nucleation in a solution is driven by the difference in the free energy between the two phases. Under certain approximation, the driving forces responsible for nucleation can be reduced to two, (i) the gain in the chemical potential and (ii) the increase of the total surface energy. The gain in chemical potential is the energy released due to formation of the bonds in the growing crystal and the gain in surface energy that takes into account the correction for the incomplete saturation of the surface bonds. The change in total free energy due to formation of a spherical nucleus consisting of n atoms can be given by

where n is the number of atoms in NP, μc and μs are the chemical potentials of the crystalline phase and the solution phase, respectively, r is the radius of the nucleus and σ the surface tension. In Eq.46, the surface term arises due to the difference between nanomaterials and bulk materials. Usually, the bulk material is dominated by volume effects and thus the surface energy term in Eq.46 can be neglected, whereas in NP surface term is the most important. According to drop model of nulcleation, the surface tension σ is can be assumed to be constant for any size and morphology of the crystal. Qualitatively, one can understand the size effect considering surface tension as a result of the interaction between the surface atoms and the atoms inside the bulk. It is evident that this interaction is actually weaker for smaller NP as there are no long-range interactions. It can be shown assuming a Lennard-Jones interaction between the atoms the total surface energy of a cluster of 13 atoms (as shown in Figure 36) is reduced by 15% with respect to the total surface energy of a flat surface.

Figure 36 Different clusters formed by addition of closed shells in FCC structure. Each of the clusters represents the core of the subsequent cluster. Starting from a single atom, cubo octahedra of 13, 55 and 147 Au atoms can be build. The percentage of surface atoms is far above 50% for all structures

In case of spherical NPs, we can neglect any variation of the surface tension , i.e. NPs without facets. In this case the number of atoms n in Eq. 46 can be expressed in terms of the radius r of the NP, density of atoms in NP ds in NP,

The chemical potential of the system when atoms are in solution is less than that of an atoms in a crystal and the minimum of the free energy is obtained when all atoms are unbound, so no stable crystals are formed. However,  we are interested in the opposite case with the chemical potential of 16 atoms in solution being more than that of bound atoms, so the first term becomes negative, resulting free energy being a maximum for a certain critical radius rc, at which a nucleation barrier is imposed, as shown in Figure 37. In case of small nuclei the surface energy term dominates the free energy, whereas NP with radius larger than rc the growth is driven by the gain in chemical potential. The height of the nucleation barrier controls the rate at which crystals nucleate .

Figure 37 Sketch of the potential landscape for the nucleation. Only at small values of the radius r in Eq. 47 the r2 of the surface energy term outcompetes the r 3 contribution of the chemical potential, so that a barrier is imposed at the critical size rc

6.6.3 Growth

The growth of NPs is actually the  process of the deposition of monomers onto the growing NP happens in  two steps. First the monomers have to be transported towards the surface of the NP  and in 17 a second process they have to react with the NP surface. The first process is controlled by diffusion and thus the rate of the first process is determined by the diffusion constant D, whereas the speed of the second process is dominated by rate of reaction between free monomers and the NP surface.Let us view take mechanistic view of the growth in which the growth rate  ̇= dr/dt of a crystal of radius r depends only on the rate at which monomers are incorporated into the crystal. The rate of growth will be proportional to the time derivative of the number of monomers n which go through the two processes: diffusion and reaction. We can write the growth rate  ̇as

In this equation dm denotes the density of monomers in the crystal.

In a typical solution or colloidal synthesis of NP , an excess of free monomers is injected to initiate the growth process. Due to high concentration monomers initially, the diffusion process dominates. As monomers are available whenever there is a free site for their incorporation into a growing NP, the incorporation rate ṅdepends only on the rate of the reaction. This rate is proportional to the surface area of the crystal. Therefore the growth rate  ̇in Eq. 48 is independent of the radius of the NP. This growth regime is called reaction controlled growth and it is important only at very high concentrations of monomers. In this regime the width of the size distribution ∆r does not vary with time. Only the relative width ∆ / ̅decreases with time ( ̅ denotes the mean radius of the crystals). When the concentration of monomers is partially depleted and the growth rate is dictated by the rate at which monomers reach the surface of the crystal. The concentration gradient decides the flux J of monomers towards a growing crystal. The surface of a NP represents a sink for free monomers. In this model the monomer concentration is assumed to be constant on any sphere of radius x (greater than r) around the crystal. On the surface of the crystal the flux J is equal to the incorporation rate   ̇ of monomers:

j( x = r ) =

For any radius greater than r the flux through a spherical surface is determined by Fick’s law of diffusion which is given by

Here D is the diffusion constant. In a steady state, this flux J is independent of the distance x:

The profile of the concentration can be calculated from this differential equation and using Eq.50
on can calculate the flux J. Eq. 51 can be solved under certain boundary conditions which could be (i) monomer concentration Ci on the surface of a NP and (ii) the monomer concentration Cb in the bulk of the solution. With these boundary conditions the general form of the concentration profile around a crystal is derived asFinally, with this result the growth rate of a crystal in Eq.48 reads:

To this point an infinite stability of the NPs is assumed. This assumption is challenged by the Gibbs-Thompson effect which introduces a competing effect to the growth. As the size of the NPs reduces, their vapour pressure increases. Hence the monomers “evaporate” into solution more easily from smaller crystals than from larger ones. This can be understood on a molecular level when considering the higher curvature of smaller crystals. The surface atoms are more exposed to the surrounding due to the increased surface curvature and at the same time experience a weaker binding strength to the smaller crystal core. Experimentally this effect is seen in the lower melting temperature of NPs compared to bulk. With the help of the Gibbs-Thompson equation the vapour pressures of a crystal of radius r can be calculated. Through the general gas equation these vapour pressures can be expressed as the concentrations of monomers in the vicinity of the surface and given by

In this equation ∞is the vapour pressure of a flat surface, is the surface tension. The radius of a crystal in equilibrium with the concentration of monomers in the bulk is introduced as the critical size ∗ of the growth process

with these two quantities the growth rate from Eq.55 can be expressed as

The critical size ∗ is characterised by a zero growth rate. For crystals smaller than ∗ the growth rate is negative, the dissociation of monomers is more important than the supply of fresh monomers, and therefore these crystals melt. Here it becomes evident that the quantity r is actually equal to the critical size rc that characterizes the position of the energy barrier in the nucleation event.

The general dependence of the growth rate on the radius of the crystal is illustrated in Figure 38. It is interesting to note the presence of a maximum at a radius of 2 ∗ . If all crystals present in the solution have a radius larger than this value, the smallest crystals grow fastest, and therefore the size distribution becomes narrower over time. The value of ∗ depends mainly on the overall concentration of free monomers, but also on the reaction temperature and on the surface tension . During the run of the reaction the concentration of monomers decreases and the critical size shifts to higher values. If the critical size is sufficiently small the system is said to be in the narrowing or size focussing regime. During the run of the synthesis the critical size increases and as soon as the size of maximal growth rate, 2 ∗ , has reached a value situated in the lower end of the size distribution of the nanocrystals, the system enters in the broadening regime. Ultimately, when ∗is larger than the radius of the smallest nanocrystals

Figure 38 Growth rate dr/dt of the NPs in units of the critical size ∗. As examples two size distributions and their development with time are shown. In the broadening regime, the smallest particles are larger than ∗. Therefore the mean size of the particles still increases. The situation would be different if the size distribution would comprise nanocrystals smaller than ∗

present, the system enters into the Ostwald ripening regime, which is characterized by a large broadening of the size distribution and    more importantly a decrease of the total concentration of the NPs. The smallest NPs  melt to free monomers that are incorporated into the larger NPs.

As  the nucleation event has a strong influence on the size distribution of the final sample, the effect of the size focusing is limited by the broadness of the initial size distribution. In growth of NPs, it is desirable that the nucleation event  be finished  before the system enters into the diffusion controlled growth regime. The sharpness  of this transition from one regime to another is extremely important for size selection and  size distribution of NPs. If  the nucleation event extends  for a long time, the NPs that nucleated first have grown considerably already, resulting in a broad size distribution. In that case the effect of the size focusing might not be sufficient to obtain a reasonably narrow distribution at the desired average size of the NPs.

In the case of fluorescent nanocrystals, the width of the fluorescence spectrum is a good indicator for the quality of the size distribution. Samples of CdSe or CdTe nanocrystals generally have a band linewidth of   30 nm or less, which in this case corresponds to  100 meV. In Fig. 4 an example for the focussing of the size distribution is displayed. In this example the synthesis was carried out under conditions that favour size focussing. The nanocrystals were synthesised at a very high temperature, which reduces the nucleation event to a very short time span. Also, the concentration of monomers was sufficiently high to prevent the system to enter into the broadening regime.

Figure 39 Effect of size focussing in the synthesis of CdSe NPs. The synthesis was performed at high temperature (3700 C). Spectra were taken every 20 s during the run. The first spectrum (leftmost) is broad due to wide size distribution, whereas the last spectrum (rightmost is very narrow with a

In someFWHMsynthesis,of28 nmitis. required to use the reservoir of monomers resulting broadening of the fluorescence band. In these cases the system can be maintained in focussing regime by repeated injections of fresh monomers. Experimentally it is not difficult to obtain a perfect size distribution.

Nevertheless, size distribution can be improved substantially by  size-selective precipitation after the synthesis is completed. In this process, a non-solvent is added slowly in a controlled manner. For example, if  polar solvent methanol  is added slowly to  NPs  dissolved in a non-,polar solvent such as chloroform or toluene leads to  precipitation of the NPs. As the  larger particles become unstable in the solution at lower concentrations the smaller particles, NPs with larger diameter precipitate first. This process has been successfully applied for the synthesis of CdTe and CdSe. In Figure 40 an example of a size- selective precipitation is shown.

Figure 40 Effect of the size-selective precipitation. By careful addition of a non-solvent to samples of CdSe NPs with a bimodal (left) or simply broadened (right) size distribution the largest particles, i.e. those with an emission at higher wavelength, can be precipitated by centrifugation while the smaller particles remain in solution. The upper spectra show the fluorescence of the initial samples, whereas the lower spectra show the fluorescence of the supernatants.

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