21 Semiconductor Nanoparticles 2

6.2 Electronic Structure of Semiconductor Nanoparticles

The electronic structure of semiconductor NPs strongly depends on size due to quantum confinement which can be understood via two different approaches. In the first approach, known as top-down method, the NP can be treated as a small piece of bulk three dimensional semiconductor crystal and electronic structure can be calculated by extending the band structure calculation employed for bulk crystal with appropriate boundary conditions. In second approach known as bottom up method, NP can be treated as giant molecule, built up by atom by atom and electronic structure is calculated by quantum chemical approach generally used for calculating the electronic structure for molecules.

6.2.1 The Top-Down Approach: Nanoparticle as a Small Crystal

In this approach, one starts from the band structure of the bulk semiconductor and the Bloch wave functions describing the bulk properties of the semiconductor. The wave functions of electrons (holes) in NPs are described by the Block wavefunction of the bulk crystal multiplied by an envelope function to take care the spatial confinement of the charge carriers (electrons and holes) and given by

Where Ψcrystal is the Block function for the bulk crystal and ψef is the envelope function due to spatial confinement. The envelope function env can be obtained from the solution of the Schrödinger equation for a particle in a spherical quantum well. The eigenfunctions can then best be described as the product of spherical harmonics, Ylm((θ,φ) and a radial Bessel function R(r)

The envelope function is similar to the  wave functions describing the electron in  a hydrogen atom in previous chapter. Only difference is the potential experienced by the electron,. In hydrogen atom, it is coulomb potential V(r) ~1/r due to proton in nucleous, whereas in NP   electrons experience a spherical 2 potential well of diameter d for which V(r)=V0 for r < d/2 and V(r)=0 elsewhere. The solutions of eigenvalues can be obtained by putting spherical well potential in the Schrödinger equation

Which gives the following solutions for the discrete energy-levels of a electron confined in a

where m* is the effective mass of electrons (holes) decided by the band structure of the semiconductor crystal and χnl are the roots of the Bessel function and depend on the principal quantum numbers n (=1,2,3,…) and azimuthal quantum number l (= 0,1,2,3…, corresponding to s, p, d,…, orbitals), as discussed in previous chapter. The columbic potential in hydrogen atom put a restriction on quantum no. l i.e. l ≤ n – 1, however constant potential in NP put no such restriction on quantum no l. Hence, the ground state of NP is n=1 and l=0 represented by 1S state, the second energy level has quantum numbers n = 1 and l = 1 represented by 1P and third level is described by n = 1, l = 2 represented by 1D, and the fourth level is a 2S level (n = 2, l = 0). NPs are referred as artificial atoms due to similarity between atomic orbitals and the envelope wave functions.

Figure 8 Schematic of the effect of quantum confinement on the electronic structure of a semiconductor. The arrows indicate the lowest energy absorption transition in between the discrete hole and electron states.

Figure 8 shows the energy level in NP due to quantum confinement. In top down approach the band gap of a NP is then the sum of the fundamental bulk band gap (Eg0 ) and the confinement energy of both electrons and holes,

In this analysis the coulombic interaction between electrons and holes is neglected under independent electron approximation. However, under this approximation it is not possible to have excitons in NP. Hence, this approximation is valid if the radius of NP is less than the Bohr radius of exciton, aB. In this case, the kinetic energy of the electrons (holes) is very high reducing the effect of Coulomb interaction and therefore, the electron and hole can be treated independently. This called is weak coupling regime. As the correlation in this case is not important, Coulomb term can be added as first order energy corrections. If the radius of NP is larger than aB, the exciton can be treated as hydrogen-like with proton replaced by hole confined in a spherical potential, and its discrete energy levels are given by an expression similar to Eq.9 by replacing the electron (or hole) effective mass by the exciton effective mass. This is called strong coupling regime. The shift of the energy levels in the weak confinement regime is much smaller than that in the strong confinement regime. If the Coulomb interaction is also taken into account, the band gap of a NP of radius d can be given by

where Je−h represents the effective Coulomb interaction between the electron and the hole and Je-h=1.786e2/εr, ε is the dielectric constant of the NP. The terms Eepol and Ehpol give the self-polarization energies of the electron and hole and depend e2/r and on the dielectric constants of the NP and surrounding . The ERy* is the exciton Rydberg energy. Eq.11 describes the two most important consequences of quantum confinement: (i) the band gap of a semiconductor NP becomes larger with decreasing size and scales as d−2 if the Coulomb interaction is neglected, (ii) the discrete energy levels arise at the band-edges of both the conduction band and valence band. These two size-dependent effects are schematically shown in Figure 8 . which summarizes the fact that the optical band gap of QDs can be tuned by simply changing their size. Figure 9 shows the absorption spectra of different sized CdTe NPs that emit visible light at different wavelength.

Figure 9 Absorption spectra of colloidal suspensions of different sized CdTe nanoparticle.

6.2.2 The Bottom-Up Approach: Nanoparticle as Giant Molecule

Another method to explain the electronic properties of a NP is based on a bottom-up approach which treat NP as a giant molecule. As it is adopted in tight binding method or linear Combination of Atomic Orbitals (LCAO) method, the total wave function of the NP can be constructed from the superposition of individual atomic orbitals. The similar approach is generally adopted in calculating HOMO and LUMO of any molecule. As explained in previous chapter, the molecular orbital of simplest molecule such as H2 is superposition of atomic orbials of individual H atom and there will be two molecular orbitals, known as bonding and anti-bonding orbital. The molecular orbitals are occupied by electrons in such way that the total energy of the molecule is minimized and normally bonding orbitals are occupied with electrons and known as HOMO. In contrast, anti-bonding orbitals are unoccupied with electrons and known as LUMO. Exactly similar approach can be applied to NP to find out HOMO and LUMO of NP. Similar to large molecule, as the no of atoms increases in NP, the no of atomic orbital increases, leading to decreasing the HOMO-LUMO energy gap or band gap in NP. Generally, the density of molecular orbitals is maximum at intermediate energy due to Gaussian distribution of density of states, so there would fewer molecular levels at lower energy and higher energy. As the no of atoms increases in a giant molecule like NP, the molecular energy levels become closely spaced, so that HOMO and LUMO become quasi continuous, analogous to the conduction and valence bands as described in first approach for determining the electronic structure of NP. A semiconductor NP can be regarded as a giant molecule consisting of a few tens to a few thousand atoms, for example, a 1.5 nm diameter CdSe NP contains about 50 atoms, while a 10 nm NP consists of 104 atoms. As it is in case of top down approach, the electronic structure of NP will be characterized by energy bands with a large density of levels at intermediate energy values and discrete energy levels at low and high energy, near the band edges.

Figure 10 Schematically shown how the energy levels evolve a diatomic molecule (extreme left) to a bulk semiconductor (extreme right). Egnc and Eg0 indicate the energy gap between the HOMO and LUMO for a nanocrystal and bulk, respectively.

Figure 10 shows the evolution of the molecular orbitals that would be obtained from superposition of an increasingly larger number of identical atomic orbitals. In case of compound semiconductors (CdSe, CdTe, PbSe, InP) different types of atomic orbitals are combined. For example, the atomic structure of Cd and Te are [Kr]4d105s2 and [Kr] 4d105s25p4, respectively. In case of CdTe, the LUMO (conduction

band) is comprised of linear combinations of the (empty) 5s atomic orbitals of Cd2+ and HOMO (valence band) is comprised of p orbitals of Te2−. If CdTe is treated as a single molecule, the LUMO will be closer in energy to the 5s atomic orbital of Cd, while the HOMO will be closer in energy to the 5p atmic orbital of Te (Figure 11).

Figure 11 LCAO diagram of a single CdTe unit or hypothetical CdTe molecule. The HOMO and LUMO of CdTe molecule is obtained by bonding and antibonding superposition of Cd ([Kr]4d105s2 ) and Te ([Kr] 4d105s25p4) atomic orbitals.

Figure 12 LCAO representation of    the     σ-molecular      orbital levels   in   a   one   dimensional CdSe      NP. The sinusoidal functions describe the phase of the atomic orbitals.

As we keep adding CdTe single unit one by one to build CdTe NP, LUMO or conduction band will be constructed pregressivle by superposition of 5S orbitals of Cd and HOMO or valence will be constrctued by superposition of 5p orbitals. The sinusoidal function that describes the phase of each molecular orbital is  plotted in Figure 12.  As the molecular orbital is the superposition of atomic orbitals, so it is so it is important to look into the energetically most favourable combination. In CdSe. this happens when the phase of each p-orbital changes sign at each selenium site. The VB and CB consist of the linear combination of the Se 4p orbitals which are all in phase. Similarly and linear combinations of the 5s- orbitals of Cd, respectively. The MO with the lowest energy and highest energy are the linear combination of s-orbitals that are all in phase and all the s orbitals out of phase, respectively. CdSe is a direct band gap semiconductor, hence the fundamental band gap of CdSe is located at the Γ-point in the Brillouin zone. This is consistent with all the p orbitals being in phase for the highest lying valence band level or HOMO, which can be described by the sinusoidal function with the longest wavelength or smallest k. On the other hand, the lowest valence band level has the maximum k-value of π/a, which corresponds to all the p-orbitals changing phase at each selenium site and can be described by the sinusoidal function with the smallest wavelength or highest k. To bring the analogy with top-down method, the periodic function in the Bloch-equation can be seen as the atomic orbitals; which is periodic over the lattice constant a because at each (selenium) site there is an identical atomic (p) orbital. Only the phase changes, which is described by the plane wave. A similar analysis can be carried out for the conduction band. The MO with the lowest energy is the linear combination of s-orbitals that are all in phase. The number of anti-bonding s-orbitals increases with increasing number of phase-flips, and the conduction band level with the highest energy consists of a MO with only anti-bonding s-orbitals (k = π/a). Though this 1D analysis is an oversimplification of the NP electronic structure, but it is intuitive and useful to understand the optical properties of NP. The discrete energy levels of the NPs can be related to different MOs, such as those shown in Figure 12. The atomic-like symmetry of the band-edge levels (1S, 1P, etc.) is not apparent from the sinusoidal functions in the top-down method.

6.3 Optical transitions in semiconductor nanoparticles

6.3.1 Effect of quantum confinement on optical transitions in nanoparticles

The effect of quantum confinement in NPs is best exemplified in optical transitions as observed in their absorption and photoluminescence spectra. This is illustrated in Figure 13

Figure 13(Left) size-dependent evolution of absorption spectra of InP NPs. (Right) TEM images of the InP NPs.

which also shows the TEM images smallest (~2nm) and largest (~7nm) NPs. In the strong quantum confinement regime (<2-3nm) the energy spacing between the discrete levels in NPs is in the order of hundreds of meV, and therefore optical transitions between these levels can be clearly resolved in the optical absorption spectrum. However, in weak confinement regime the energy spacing is typically order of magnitude less than in strong confinement regime and  it becomes difficult to resolve the  transitions in  absorption or photoluminescence spectra.  As discussed before, the lowest energy optical transition can be assigned to the 1Sh to 1Se-level, the second transition to the 1Ph to 1Pe level, which are shown in Figure 14.

Figure 14 (a) Three lowest electron (1Se, 1Pe, 1De ) and hole (1Sh, 1Ph, 1Dh ) energy levels in a semiconductor NP. The corresponding wavefunctions are also shown and allowed optical transitions are shown by the arrows. (b) Corresponding transitions in the absorption spectrum of colloidal CdTe NP.

Scaling the size of the NP with the exciton Bohr radius aB is the most convenient length scale to determine the effect of quantum confinement on the properties of semiconductor NPs. The onset of

Figure 15 Absorption spectra of three sets of different sized CdS, CdSe, and CdTe nanoparticles.

quantum confinement effects will occur at different size NP for different semiconductors, as the aB varies widely within the semiconductor materials depending on effective mass of electrons and holes and dielectric constant of the semiconductors. Through effective mass, aB and the band gap Eg are correlated, materials with narrower Eg have larger aB , for example Eg and aB are 0.26 eV and 46 nm for PbSe, 1.75 eV and 4.9 nm for CdSe, and 3.7 eV and 1.5 nm for ZnS, respectively. In case of insulators, Eg > 5 eV, excitons are strongly localized with aB <1 nm, so NPs with 10-20 atoms may be required to observe the effect of quantum confinement on optical transitions.

6.3.2 Absorption and photoluminescence spectra of semiconductor nanoparticle

Figure 15 shows a set of absorption spectra of three important semiconductors CdS, CdSe, and CdTe based NPs of different sizes. Due to size quantization, the absorption edge shifts towards shorter wavelengths in comparison to the corresponding bulk values (480 nm for CdS, 674 nm for CdSe, and 817 nm for CdTe). It is clear that excitonic transition is clearly observed in all the samples, but in smaller sized NPs, there are two peaks which establish that smallest sized NPs indeed behave like molecule in which absorption spectra is always double peaked due to Davydov splilliting. Using these three semiconductor NPs, optical transitions over a range of 300nm to 600nm can be covered. Even a wider range can be covered using CdTe-HgTe systems. Figure 16 shows a set of photoluminescence spectra of CdTe, CdxHgl-xTe and HgTe NPs. It is interesting to note that optical transitions for entire spectral spectral range from 500 to 2000 nm can be obtained by varying the size and composition of the NPs. In particular, the luminescence spectra of HgTe nanocrystals of different sizes cover the spectral region between 1000 and 1800 nm making them potential candidates for application as ultra-broadband optical amplifiers in telecommunications systems. Further the quantum efficiency of luminescence of CdTe, CdxHgl-xTe and HgTe NPs is quite high (40-50%) making it most attractive for different applications.

Figure 16 Photoluminescence spectra of  CdTe, CdxHg1-xTe, and HgTe NPs.

As we have discussed before, after absorption of a photon by a semiconductor NP an exciton is created, with bound state characterized with the discrete energy levels located within the conduction and valence bands. If the energy of the photon is larger than the band gap of NP, a hot exciton is formed, in which the electron and hole occupy excited states. In such a case, the electron and hole will quickly (<1 ps) relax to their ground states by means of a cascade of intraband non-radiative transitions, mostly through by dissipating excess energy as heat. These nonradiative transitions are not restricted by the selection rules, hence take place very fast, either by coupling with phonons or by Auger scattering. Once the exciton has reached its ground state, further relaxation can take place by electron-hole recombination, the excited electron returns to the valence band and recombine with hole releasing exciton energy radiatively or non-radiatively. In radiative recombination, photons with energy same as that of energy of exciton are emitted. In the emission spectra of NPs the direct radiative recombination of the exciton is observed as a well-defined peak with energy close to the lowest energy absorption transition. The effect of NP size on the emission spectra is same as that on absorption spectra. The peak widths of emission spectra depend on the size and shape distribution of the ensemble of NPs.

It should be noted that the emission transition in which electrons falls back from excited state to ground state is just the reverse of the lowest energy absorption transition in which electrons is photoexcited from ground state to one of the excited states . Therefore, the selections rules discussed above for absorption also apply for emission. In case of emission transitions, there is the mixing of the excited state orbital with the ground state orbital. This mixing is induced by the oscillating field light used for photoexcitation which is resonant with the transition and induce the mixing resulting in radiative recombination which is called stimulated emission. However, emission happens in all kind of fluroscence systems without the stimulus of an external oscillating field and the radiative transition is called spontaneous emission, where mixing of the excited and ground state is induced by so-called vacuum modes or vacuum fluctuations. The rate of spontaneous emission is not only dependent on the magnitude of the transition dipole moment μif between initial and final states and the density of optical modes that can be coupled to the transition. The density of optical increases with frequency. The radiative decay rate Γrad for spontaneous emission in an two-level atom system can be deduced from Fermi’s Golden Rule 

Γrad  depends on (i) atomic orbitals through μif   and   (ii) EM field   through density of optical modes.

Here. n isthe refractive index ε0  is the permittivity of free space.

All optical transitions at the simplest level can be modeled as a two level system. In optical transitions for a two level system, an electron is excited from a ground state orbital |1> to an excited state |2>. The excitation process can be represented by the mixing of the ground state and excited state orbitals into a mixed state ψ=c1|1>+c2|2>. The oscillating electric field of electromagnetic wave (photexcitation ) induces this mixing of the two orbitals and a polarization which can only happen if  dipole moment is induced in the mixed state, and if the frequency due to the energy difference between ground and excited states matches with the frequency of the EM field.

6.3.3 Dipole moment and optical transition in nanoparticles

Figure 17 Schematic representation of the linear combination of s and p orbitals. When two s or two p orbitals are mixed (left), a symmetric wave function results without a dipole moment. Mixing of an s and p orbital (middle and right) results in an asymmetric wave function with a dipole moment.

Figure 17 schematically shows what happens when different orbitals (s or p or both) with different phases are mixed. A symmetric mixed state results without a dipole moment when the two orbitals with the same parity are mixed, so mixing of these orbitals cannot be induced by an oscillating electric field. On the other hand, an antisymmetric mixed state results with a dipole moment when the two orbitals with the different parity are mixed. In this case mixing of these orbitals can be induced by an oscillating electric field. In summary, optical transitions in atoms or molecules or NPs can only occur between orbitals with a different parity ( s →p, p →s, or p →d). This is the origin of the parity selection rule (Δl = ±1). In case of NPs, the bottom-up approach is most convenient to analyse the optical transitions and corresponding selection rules. Electrons in a NP occupy molecular orbitals, so the optical transitions two MOs have to be considered in NP. The lowest energy interband transition is between the lowest hole level (1Sh) and lowest electron level (1Se) as shown in

As shown in Figure 18, in case of CdSe NP, this involves a transition from a MO formed by the linear combination of Se 4p-orbitals to an MO formed by the linear combination of Cd 5s orbitals.

Figure 18 Schematic representations of the individual dipoles and net dipole moments that arise when VB atomic orbitals are mixed with CB orbitals, for three different interband transitions in a CdSe NP. The direction of the individual dipole can be deduced from Figure

17. When the individual dipoles add up to a net dipole, the optical transition is parity-allowed. If they cancel out, the transition is parity-forbidden.

Within each unit cell, mixing of these states yields a dipole moment which is indicated by the small arrows in Figure 18. In this case, as all atomic orbitals are in phase, all the dipoles of the individual unit cells point in the same direction which results an asymmetric electron density distribution in the NP when the 1Se and 1Pe levels are mixed, giving rise to a net dipole, which makes the transition parity-allowed.

Summary:

• Due to quantum confinement which can be understood via two different approaches i.e. top-down method and bottom-up method.
• The HOMO and LUMO energy bands of CdTe nanoparticles have been studied in detail.
• We have discussed effect of quantum confinement on optical transitions in nanoparticles
• Absorption and photoluminescence spectra of semiconductor nanoparticles has been discussed.
• We studied the dipole moment and optical transitions in nanoparticles.