PBS: FROM SOLIDS TO CLUSTERS

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1 PBS: FROM SOLIDS TO CLUSTERS E. HOFFMANN AND P. ENTEL Theoretische Tieftemperaturphysik Gerhard-Mercator-Universität Duisburg, Lotharstraße Duisburg, Germany Semiconducting nanocrystallites like PbS exhibit electronic and optical properties greatly differing from those observed in the bulk material due to quantum size effects. By decreasing the diameter of PbS nanoparticles to about 1 nm the optical band gap increases significantly by a factor of 10 with respect to the bulk material. In this paper we investigate with ab initio methods the dependence of the optical band gap on the size of the system. 1 Introduction The fundamental band gap of semiconductors like galena (PbS) is very sensitive against finite size effects. By varying the size and shape from bulk material to nanoparticles it is possible to change the optical band gap E g from about ev up to 2.8 or 5.2 ev. 1,2 Therefore, it is possible to build optical sensors with adjustable properties. Galena is an ionic bounded system which crystallizes in the fcc based B1 structure. It is a narrow gap semiconductor with extrema of conduction and valence bands at the L point of the Brillouin zone. The fundamental band gap is a direct gap having at 4.2 K a width of ev which increases up to 0.41 ev at room temperature. 3 If the size of PbS particles is reduced the absorption spectrum shows two effects. First the wedges of the absorption spectrum are smeared out and show long tails near the absorption edge 1,2,4-6 (see Fig. 1) which could be due to defect states, particle size distributions or indirect transitions. In order to determine the band gap energy, the transitions can be characterized by the Tauc 5 relation σ E = A (E E g ) m, where σ is the absorption coefficient and m=1/2, 2, 3/2 or 3 for allowed direct, allowed indirect, forbidden direct and forbidden indirect electronic transitions. Second the band gap shows a hyperbolic dependence on the particle size (see Fig. 2). Wang et al. 2 reported that E g increases up to 2.32 ev for particles with a diameter of 0.24 nm. Thielsch et al. 1 found a band gap maximum of 5.2 ev for a particle size of 1 nm. Additionally they found a photoluminescence transition with an energy of 2.85 ev for particles less than 2 nm in size. The smallest possible PbS particle is a dimer for which the lowest allowed dipole transition has the energy ev (Table 1). hoffmann: submitted to World Scientific on November 2,

2 ² 14 3x10 5.0E+04 <13A (x0.1) 14 2x10 cv (PbS) Absorption Coeff., 1/cm (αε) [ev²/cm²] 14 2x x A Bulk 13 5x10 125A E A Wavelength (nm) Photon Energy [ev] 5 6 Figure 1. Left panel: The absorption spectra of PbS nanoparticles in a polymer film (see Wang et al. 2 ) Right panel: Tauc s plots for the direct transition of PbS nanoparticles in a SiO 2 matrix, after Thielsch et al. 1 With empirical models based on the exciton model, it is possible to describe the band gap behavior of nanoparticles. The models are based on the single-particle approximation and describe the electronic states in the nanoparticle as electrons and holes in a box of a given size (see Fig. 2). The only input required is then the effective mass of the valence electrons and holes. Wang 2 assumes an infinite potential and obtained a quantitatively good description for particles larger than 80 Å in diameter. Nosaka 9 found an improved agreement for particles down to 2 Å if the potential well of the exciton is reduced to 4.5 ev. In this short communication we investigate in how for the band gap behavior of PbS nanoparticles can be described by ab initio methods. We have performed density functional calculations for the bulk system, a monolayer, Transition E (ev) forbidden X A X B X C X D X E X F Table 1. Electronic excitation spectrum of PbS dimers from the ground state after Joos and Saur. 7 The lowest transition is forbidden for dipole radiation, see Wang. 2 hoffmann: submitted to World Scientific on November 2,

3 0 6 5 V = 3.6 ev 1d 1p 1d 1p 1f 2s 1d 1p 1f 2s 1d 1p Bandgap Energy / ev 4 b a R = 1.0 nm R = 1.5 nm R = 2.0 nm diameter / nm Figure 2. Left panel: Energy levels and wavefunctions of electrons having an effective mass of 0.19 m e in a spherical well of finite depth. The arrows show the energy levels for V 0 =, see Nosaka 9 for details. Right panel: Band gap energy of PbS nanoparticles calculated with the finite depth potential model, 9 a:v 0 =, b: V 0 =4.5 ev. The dashed curve is calculated for the hyperbolic band model and the squares indicate the experimental results reported by Wang et al. 2 and clusters with 8 and 18 atoms, and also for the dimer system. Nowadays density functional theory is a well established method which allows to calculate ground state properties of atoms, molecules, semiconductors and metals with high precision. For the investigation of band gaps the errors are much larger and usually of the order of 50% or more. 11 The reason for this large deviations is due to the fact that the Kohn-Sham eigenvalues are not the excitation energies of the electronic system, 12 nevertheless the agreement between the experimental and theoretical band structure inside the valence and conduction band is usually good. The description of band gaps can be improved by self interaction corrections (SIC) 13 or local mass approximations (LMA) Method of Calculation All calculations presented here have been done in the framework of density functional theory employing the Vienna ab initio Simulation Package (VASP). 15,16 This method uses a plane wave basis set and ultrasoft Vanderbilt pseudopotentials, 17 allowing a smaller energy cutoff than in the case of normconserving pseudopotentials. Scalar relativistic effects are taken into account when generating the pseudopotentials. The exchange and correlation effects are treated in the local density approximation with the parameterization of Perdew and Zunger. 13 Because of the basis set the program uses hoffmann: submitted to World Scientific on November 2,

4 periodic boundary conditions. For the plane wave expansion we use a cutoff energy of 198 ev. The Kohn-Sham Hamiltonian is solved via iterative matrix diagonalization schemes. 16 Low dimensional structures like monolayers or clusters are calculated by introducing vacuum layers in the unit cell. We usually used 10 Å vacuum, the use of 20 Å does not change the results. For the bulk and monolayer calculation we used the tetrahedron method with Blöchl correction, 18 while the dimer and cluster calculations are performed using the Γ point only. In this case we used a Gaussian smearing of the density of states with σ =0.1eV. The eight atom cluster was initialized by using a simple cubic basis of the B1 structure: (0, 0, 0), ( 1 2, 1 2, 0), (0, 1 2, 1 2 ), ( 1 2, 0, 1 2 ); ( 1 2, 0, 0), ( 1 2, 0, 0), (0, 0, 1 2 ), ( 1 2, 1 2, 1 2 ). The initial eighteen atom cluster was constructed by doubling the lattice constant in each direction. All structures have been fully relaxed in the selfconsistent cycle leading to no significant deviations from the B1 positions. The partial density of states have been calculated by projections the wavefunctions onto a radial basis set up to l max = 2. As mentioned above the fundamental gap of PbS is related to a direct transition, therefore, we investigated only the density of states. 3 Results and Discussion For solid galena in the B1 structure we found a lattice constant of 5.86 Å which is in good agreement with the experimental value of 5.94 Å at room temperature. For the band gap we obtained E g =0.58 ev which is by a factor of two larger than the experimental low temperature value of ev. We would like to point out that in contrast to this the ab initio calculations of Wang et al. 2 yields metallic behavior. The reason is most probably not the failure of density functional theory, instead we believe that the main source of error is the use of only radial symmetric basis functions, because the packing System d (Å) E g (ev) E F (ev) Bulk Monolayer Cluster (18) Cluster (8) Dimer , Table 2. Ground state properties of Galena for different structures; d denotes the next neighbor distance, E g the band gap derived from the density of states and E F the Fermi energy with respect to the vacuum. hoffmann: submitted to World Scientific on November 2,

5 d=2.7 A d=2.5 A Energy (ev) d=3.5 A d=3.05 A Energy (ev) Figure 3. Density of states of solid PbS in the B1 structure. d indicates the nearest neighbor distance, for small d Galena becomes metallic. Zero energy denotes the Fermi level. hoffmann: submitted to World Scientific on November 2,

6 Dimer Cluster (8 Atoms) Energy (ev) Monolayer Bulk Energy (ev) Figure 4. Density of states of PbS in reduced dimensions. All structures are fully relaxed, but there are no visible deviations from the B1 structure. hoffmann: submitted to World Scientific on November 2,

7 fraction of the B1 structure is only Finally Mian et al. 6 reported an energy gap of 6.1 ev from Hartree-Fock calculations. The experimental pressure dependence of the band gap at room temperature is -9.1 µev/bar. Assuming that the dependence is linear in pressure this, yields a critical pressure for the semiconductor-metal transition of 4 GPa. This is in remarkable good agreement with the theoretical value of 45 GPa. In contrast to the experimental spectra of PbS (Table 1), where six transitions between 1.8 ev and 3.7 ev exist, we found only two eigenvalues (E = 2.8, 5.5 ev) between the Fermi and vacuum level. We believe that the main reason for this is the neglect of spin-orbit coupling. Because of the high atomic number of lead (82) relativistic effects should be noticeable. In our calculations (Fig. 4) the states at the Fermi level and the first excited states corresponds to threefold degenerate molecular orbitals. If we average the experimental values in order to cancel the splitting, we obtain E g =2.66 ev, which is in good agreement with the gap energy E g =2.8eV obtained from the nonrelativistic calculation. If we increase the number of atoms to 8, which form the corner of a cube, the lattice constant is increased from 2.3 Å to 2.46 Å, but the value of the band gap does practically not change (see Table 2). In the case of the 18- atom cluster the band gap decreases to 1.5 ev. The decrease of the band gap with increasing particle size is stronger than observed in experiments (Fig. 2), where such a small band gap is expected for a particle diameter of about 40 Å. Besides of nanoparticles thin films are also technologically important. Therefore, we have also investigated a free PbS layer. With the experimental bulk lattice constant the band gap amounts to 0.93 ev. If we relax the lattice constant to minimize the total energy, the lattice constant shrinks by about 7 % and the system becomes metallic. This agrees well with the pressure dependence of the band gap of the bulk material. From this calculation it seems to be possible to prepare metallic thin PbS films on a suitable substrate. 4 Conclusions The electronic band gap of crystallized galena was calculated with a deviation of 0.3 ev with respect to the experimental value. The excitation spectra of PbS shows that relativistic corrections are not negligible. The result for the band gap in the atomic limit is in good agreement with the results of Wang et al., but there is a discrepancy with the results of Thielsch et al. who found a band gap of 5.5 ev, which is in reasonable agreement with the semiempirical exciton model. In our calculation we found a second state of 5.5 ev above hoffmann: submitted to World Scientific on November 2,

8 the Fermi level, which is in agreement with the result of Thielsch. In order to identify this energy difference with the band gap, the lower energy excitations must be forbidden. This could only be verified by analyzing the dipole radiation matrix elements. Another reason for the discrepancies between the two experiments may be the different methods of preparation the samples. In one experiment the nanoparticles are embedded in a polymer and in the other in SiO 2. Whether the different environment of the nanoparticle is responsible for the differences have to be investigated experimentally with accompanying theoretical calculations. References 1. R. Thielsch, T. Böhme, R. Reiche, D. Schläfer, H.-D. Bauer, and H. Böttcher, Nano Structured Materi. 10, 131 (1998). 2. Y. Wang, A. Suna, W. Mahler, and R. Kasowski, J. Chem. Phys. 87, 7315 (1987). 3. Nimitz, Landolt-Börnstein, Vol III/17f (1983). 4. K. Kotani, J. of Non-Crystalline Solids 126, 87 (1990). 5. J. Tauc and A. Menth, J. of Non-Crystalline Solids 8-11, 569 (1972). 6. M. Mian, N.M. Harrison, V.R. Saunders, W.R. Flavell, Chem. Phys. Lett. 257, 627 (1996). 7. Joos, A. Saur, Landolt-Börnstein, Vol I/3 (1951). 8. A.E. Sandström, Handbuch der Physik XXX, 78, (1951). 9. Y. Nosaka, J. Phys. Chem. 95, 5054 (1991). 10. O.B. Maksimenko, A.S. Mishchenko, Solid State Commun. 92, 797 (1994). 11. D. Vogel, P. Krüger, and J. Pollmann, Phys. Rev. B 54, 5495 (1996). 12. L.J. Sham and A. Schüter, Phys. Rev. Lett. 51, 1888 (1983). 13. J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 14. G.E. Engel and W.E. Pickett, Phys. Rev. B 54, 8420 (1996). 15. G. Kresse and J. Hafner, Phys. Rev. B 54, (1994). 16. G. Kresse and J. Furtmüller, Phys. Rev. Lett. 45, 566 (1996) and references therein. 17. D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). 18. P.E. Blöchl, O. Jepsen, and O.K. Anderson, Phys. Rev. B 49, (1994). hoffmann: submitted to World Scientific on November 2,