Photoabsorbance and Photovoltage of Crystalline and Amorphous

Article pubs.acs.org/JPCC

Photoabsorbance and Photovoltage of Crystalline and Amorphous Silicon Slabs with Silver Adsorbates Tijo Vazhappilly,† Dmitri S. Kilin,‡ and David A. Micha*,† †

Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, Florida 32611-8435, United States ‡ Department of Chemistry, University of South Dakota, Vermillion, South Dakota 57069, United States ABSTRACT: The optical properties of silicon surfaces are affected by their atomic structure and in particular by whether their lattice is crystalline or amorphous. Silver atoms adsorbed on the Si surface enhance the absorption of light and electronic charge transfer at the surface, and the size and shape of the adsorbed Ag clusters play a big role in the photovoltaic properties of Si. We have modeled the photoabsorbance and photovoltage of a nanostructured Si(111) surface with a slab terminated with hydrogen (H) atoms on both surfaces to compensate for dangling bonds, without and with a periodic lattice of adsorbed Ag cluster. Similar structures were also constructed with amorphous lattices to compare the properties of the structures. The optical properties of these structures are investigated using density functional theory to generate a basis set of orbitals and to construct equations of motion for a reduced density matrix from which properties have been obtained in a unified way. Density of electronic states, band gap, and intensity of light absorption with and without silver adsorbates are presented. Light absorbance and surface photovoltages have been calculated in terms of the reduced density matrix. The absorbance in the region around visible light and surface photovoltage (SPV) created by steady light absorption and charge redistribution are calculated for Si slabs containing one, three, or four adsorbed Ag atoms. The ratio of averaged values of absorption flux densities over photon energies in the IR and visible region generally show an increase in absorption with increasing size of a Ag cluster. The changes of absorbance due to silver adsorbates were not large but should be observable. Crystalline Si slabs absorb light mainly at high photon energies, while amorphous Si structures show broader absorption with less intensity. In the case of SPVs, we found that addition of silver adsorbates enhances the SPV of both c-Si and a-Si slabs with a very large increase for c-Si and smaller ones for a-Si. The a-Si structures also show broader SPV spectra compared to the corresponding c-Si structures.

1. INTRODUCTION When light shines on a semiconductor surface, electrons from the filled valence band (VB) get excited to the empty conduction band (CB) if enough energy is provided by the photons. This electronic excitation creates an empty (hole) state of the VB and forms an electron−hole pair in the semiconductor. The intensity of light absorption depends on the composition and atomic structure of the surface. Optical measurements in the visible to infrared (IR) range can reveal the electronic structure and nature of the photoexcitation at the surface. When the absorbed light is steady, an electronic surface dipole is formed which leads to a measurable surface photovoltage (SPV). Absorption of light and related phenomena are measured by different optical techniques. The absorption coefficient is related to the complex refractive index of the material and thus, to its dielectric constant. The absorption at semiconductors surfaces can be measured using spectrophotometry or reflectance spectroscopy.1 Recent study on c-Si wafers and thin films with silver nanoparticles on the surface shows an enhancement in the absorbance in the entire solar spectrum. A 7- to 16-fold enhancement is reported for the wafers and thin © 2012 American Chemical Society

silicon solar cells, respectively, at wavelengths close to the band gap of Si.2 Experimental studies on optical absorption of a-Si structures with silver nanostructures adsorbed show increased absorption compared to the pure a-Si.3 The size of the deposited silver clusters also plays a role in the optical properties. The enhanced photoabsorbance and surface photovoltage due to silver nanoparticles on different silicon surfaces are observed in experiments and discussed in refs 2, 3, and 13. The size of nanoparticles is significant in the measured properties. In ref 2, the silver nanoparticles are deposited as islands with thicknesses from 10 to 22 nm. In ref 3, the average sizes of silver islands are about 10−100 nm. In both cases, an increase in absorption is explained by the presence of localized surface plasmons. For the SPV measurements in ref 13, the deposited silver clusters have an average diameter of 6 nm. In contrast, in our present models, only a few Ag atoms are deposited on Si slab surfaces and localized surface plasmons are unlikely. Received: July 10, 2012 Revised: October 16, 2012 Published: October 24, 2012 25525

dx.doi.org/10.1021/jp306845g | J. Phys. Chem. C 2012, 116, 25525−25536

The Journal of Physical Chemistry C

Article

in our calculations provide not only the ground electronic state of the system, but also unoccupied states that play a role in photoexcitation and are employed in our calculations. Phenomena such as electronic rearrangement are described by changes in occupation numbers of KS orbitals. The dissipative dynamics of photoexcited electrons, due to electron−boson (phonon or exciton) coupling with the medium, is described by rates obtained from perturbation theory. There are more accurate theoretical treatments for the optical properties of semiconductors such as the Bethe− Salpeter (GW) theory or TD-DFT corrected for electron selfinteraction, but they are computationally expensive or not available for extended systems.19,20 The absorption of light in semiconductors creates excitons (electron−hole pairs) by electronic excitation from valence band to conduction band. Photoexcitations in the near-infrared (IR), visible, and nearultraviolet (UV) regions of the solar spectrum are important for the development of new photovoltaic materials, and computational tools can help to identify useful new materials. We have recently investigated theoretically the optical properties of systems composed of Ag clusters on Si surfaces using ab initio electronic structure calculations combined with a RDM treatment. Relating to light absorption properties, the trends in absorption spectra of only crystalline Si slabs with one and two layers having different silver clusters with one, two, three, and four atoms were investigated by DFT and TD-DFT methods.21 With larger Si slabs and an increasing number of silver atoms, the energy band gap of corresponding structures were found to be reduced and an increase of absorption was noted. This trend is qualitatively seen in both DFT treatments. We have also calculated the optical absorption of both c-Si and a-Si structures with silver adsorbates for thin two layer slabs using DFT and TD-DFT. The band gap was found to decrease with the addition of larger silver clusters. And light absorption was enhanced and red-shifted for both systems due to the presence of silver adsorbates.22 Relating to surface photovoltage, in our previous studies we have calculated the SPV for a pure c-Si slab with four and six layers.23 In our most recent study, the SPV for a c-Si slab with eight layers having no, one, and three silver atoms on both surfaces has been investigated.24 c-Si slabs with larger silver clusters show higher SPV and additional peaks at lower photon energies compared to the pure c-Si slab. The main goals of our present work are to study the optical properties of comparable pure c-Si and a-Si structures for a large slab with eight layers in the cases where silver atoms are deposited on one surface. For this purpose, we systematically investigate the density of electronic states, absorption spectra and SPV of the above-mentioned systems. We furthermore present here a unified density matrix treatment of light absorption and SPVs to compare the effects of structure and adsorbate coverage, and we extend previous calculations to larger Si slabs with adsorbed Ag clusters. The intensity of light absorption depends on the structure of the Si slab and Ag adsorbates. Our procedure starts with the calculation of electronic structure using DFT to obtain results for ground and excited electronic states, and to generate a basis set of Kohn−Sham (KS) orbitals to construct matrix elements of operators and the equation of motion for a reduced density matrix. Light absorption depends on structure and composition; it creates charge redistribution at the surface and bulk and changes of the surface dipole. We study both the surface

The photoinduced changes in the surface voltage of semiconductors have been widely studied to understand the surface and bulk properties. The surface photovoltage (SPV) is measured as a function of applied photon energy in surface photovoltage spectroscopy. It provides useful information about electronic structure of semiconductors and insight on populations in the surface and bulk states. A SPV can be measured using a Kelvin probe by detecting the photoinduced work function changes of the semiconductor. Scanning tunneling microscopy has also been used to measure the SPV of Si(111)-(7 × 7) surface.4 Photoelectron microscopy is another powerful tool for SPV measurements. Further information about commonly used experimental techniques can be found elsewhere.5 Due to the importance of silicon solar cells, SPVs for different Si surfaces such as Si(111) surface,6,7 aSi/H,8,9 and porous Si10 are well studied. The photovoltaic properties of the semiconductor surface have been extensively studied and continue to be a timely topic. A critical need exists for new photovoltaic materials with higher efficiency for light capture and production of electronic currents. Both crystalline and amorphous forms of silicon are employed for solar cell,11 and understanding their properties continues to be a very active area of research. There have also been several studies in the literature that show that deposition of silver on silicon surfaces enhances the optical properties of silicon.2,3,12,13 The interaction of visible light with a semiconductor surface can be described in terms of time-dependent perturbation theory, introducing a suitable basis set of electronic states in a well established treatment,14 which must be combined with an statistical treatment of thermal effects. For nanostructured surfaces undergoing localized excitation, medium effects are important and can be treated by means of density matrix methods that combine quantum and statistical effects and incorporate dissipative electron dynamics. Here we are combining ab initio electronic structure calculations to generate a basis set of electronic orbitals with a reduced density matrix (RDM) treatment for steady state excitation by light absorption. Electronic structures are obtained from a density functional theory (DFT) suitable for extended systems with many atoms and capable of generating many excited orbitals with energies well above the excitation photon energies.15,16 Optical excitation in semiconductors can also be treated within a linear response theory using time-dependent DFT (TD-DFT) or Green’s function methods.16 However, in our present treatment of nanostructured surfaces with adsorbates in a periodic structure involving translation of an atomic supercell, it has been advantageous to generate orbitals at the DFT stage with periodic boundary conditions and then use a RDM to calculate several physical properties (absorption and SPV among them) in a unified way. Optical properties of a semiconductor surface can be studied within the Kohn−Sham DFT formalism implemented for large many-atom systems. This formalism is also good for the treatment of low dimensional structures such as quantum dots, quantum wires and thin films, and provides valuable information on quantum confinement effects to their optical properties.17,18 We use a large number of KS orbitals as an expansion basis set to solve equations of motion and for our calculations of optical properties. Details are given in section 2, where the equations of motion for the reduced density matrix are described in detail, and the solution needed to obtain stationary states is given. The large number of KS orbitals used 25526



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where εj is the energy of the jth KSO and δW is an energy level density that becomes a Dirac-delta function in the limit of no interactions with the medium. In the quantum mechanical treatment of localized electronic states embedded in a medium, the shift and broadening of their energy levels can be accounted by means of perturbation theory and are given by a Lorentzian function28 LW(ε) = W/[(W/2)2 + (ε − εj − Δε)2], where W is a level broadening. The term Δε describes the shift of an energy level and is omitted in the calculation of EDOS because it is negligible compared to most ε − εj differences. In our physical systems, localized electronic states are interacting with the vibrations of the lattice and with electronic density fluctuations of the medium, both of which can change the energy levels. We therefore model δW(ε) as a normalized Lorentzian

absorbance and the surface photovoltage created for a range of absorbed photon energies. The SPV has been measured experimentally for the related materials using a Kelvin Probe or STM.5,9 Upon photoexcitation, electronic wave function extends over to the vacuum (z-direction). This creates in our systems a negative charge outside the surface and a positive charge inside the surface, changing the surface dipole. Another contribution comes from the rearrangement of atoms at the surface (in the case of a-Si). The presence of adsorbates also changes the surface dipole, which depends on the partial charge transfer between the adsorbate and the semiconductor surface.5 The SPV can have a negative or positive sign depending on the wavelength of the absorbed light. In general, an electronic transition from a localized state at the adsorbates to a state at the conduction band edge creates a negative SPV while transitions from the valence band to an empty localized state cause a positive SPV. The paper is organized as follows. A steady state solution to the equations of motion of the reduced density matrix treatment for the electronically excited systems is first described. The next section explains the absorption of light at a semiconductor surface followed by the modeling of the surface photovoltage. In the following section, atomic structures for a silicon slab with and without silver adsorbates and computational details for the electronic structure calculations are given for both crystalline and amorphous structures. The results obtained for the above-mentioned optical properties are discussed followed by conclusions.

ρjk̇ = −

F̂ = F̂

i ℏ

KS

∑ (Fjlρlk l

⎛ dρjk ⎞ ⎟⎟ − ρjl Flk) + ⎜⎜ ⎝ dt ⎠diss

− D̂ ·,(t )

(3) (4)

,(t ) = , 0(eiΩt + e−iΩt )

(5)

where the dissipative rate arises from the exchange of energy between the localized states and vibrations plus electronic fluctuations in the medium, and is constructed here for long times when fast excitations have disappeared and one can use the Redfield form of the dissipative rate.25,29 The parameters used in the calculations are taken from either ab initio electronic structure calculations or from experimental values. The diagonal ρjj and off-diagonal ρjk elements of the RDM represent the electronic state populations and quantum coherences of the KSO, respectively. The energy of interaction with the incident light is given in the dipole approximation by the coupling of the electric dipole vector D̂ to the external electric field , (t) in eq 4, with amplitude dependent on the light frequency Ω. All matrix elements are constructed from the plane wave combinations in the KSOs, and in particular the transition dipole moment Djk for the j → k transition is given by Djk = ⟨j|D̂ |k⟩ =

∑ G, G ′< Gcut

C*j , G DG, G ′Ck , G ′ (6)

where the matrix element DG,G′ = −e∫ d3r eiG.r re−iG′.r. In our calculations, the dipole matrix elements have been re-expressed in terms of matrix elements of the momentum operator, ⟨j|p̂|k⟩ = iℏ∑G,G′ εk

(7)

In the rotating wave approximation (RWA), the steady state values for the diagonal elements of the reduced density matrix are given by24 dρ̃jk/dt = 0, with solutions for diagonal elements gjk , j ≥ LUMO

j,k

j,k

(8)

=2−

∑

gjk , j ≤ HOMO

k = LUMO

γ Ω2jk γ 2 + Δjk (Ω)2

(10)

with Δjk(Ω) = Ω − ωjk the detuning frequency, using ωjk = (εj − εk)/ℏ. The electric field strength , 0 is derived from the incident photon energy flux S as follows, S = cε0, 02/2. Here ε0 is the vacuum permittivity and c is the speed of light. The reduced density matrix has been introduced to distinguish our system of interest from its medium. In our model, the system includes active electrons moving in a lattice of atomic cores, and the medium involves lattice vibrations, electronic excitons, and passive electrons. The medium effects lead to a dissipative electron dynamics described by the formalism of RDMs25,29 and rates of dissipation are given in terms of energy width parameters within steady state solutions for the RDM in eq 10. 2.2. Light Absorbance. The expression for the absorption coefficient can be constructed from the steady state RDM elements as follows, using results from time-dependent perturbation theory or, more generally for a system in a steady state, from the relation between the dielectric function of frequency and the absorption coefficient. A generalized treatment for the absorption coefficient can be derived from RDM theory,33 and we present here some of the needed special results. The absorption coefficient α(ω) is related to the imaginary part ε″(ω) of the dielectric function by34 α(ω) =

ωε″(ω) 4π ω = ℏc n(ω) n(ω)c

× (ρkk ̃ ss − ρjjss̃ ),

εj > εk

(13a) (13b)

where the primed summation symbol means that j ≤ HOMO, k ≥ LUMO, fjk̅ is an oscillator strength per active electron for the j → k transition, ωjk is the frequency of the transition, and Ω is the frequency of the light, as before. The number of active electrons is defined here as Nel* = 2∑j,k ′ f jk/Nocc and is obtained from the number of occupied orbitals, Nocc, in the energy range chosen for the calculation of dynamic absorbance. The delta function δγ is a Lorentzian with a width given by the broadening of the spectral transition lines. Thus, the absorption density can be obtained from eq 13a from the oscillator strengths, transition energies, and photon frequency. The denominator in eq 13a provides the frequency dependence of this dynamic absorbance as the oscillator strength changes for each state-to-state transition. Because practical applications involve new materials for solar light absorption, we model the absorption of solar energy according to the blackbody radiation of the Sun. Thus, the dynamic absorbance is weighted with the blackbody flux distribution of photon energies f(Ω,T) at T = 5800 K, normalized to an incident photon flux of 1.0 kW/m2, to get the absorbed flux of light energy

(9)

where Γj is a depopulation rate and γ below is a decoherence rate. Here Ωjk is the Rabi frequency calculated from the applied electric field , 0 and transition dipole moments as Ωjk = −Djk × , 0/ℏ, and gjk (Ω) =

ℏγ /2 1 2 π [(ℏΔjk ) + (ℏγ /2)2 ]

fjk̅ = f jk /(Nel*/2)

∞

2Γ−j 1

(12)

= Σ′fjk̅ (ρjj − ρkk )

k=0

ρ̃jjss

|Djk |2

α̅(Ω) = Σ′fjk̅ (ρjj − ρkk )δγ(ℏΩ − ℏωjk)

HOMO

∑

3e 2ℏ

We report in what follows a dynamic absorbance in terms of a normalized oscillator strength,

31,32

ρ̃jjss = 2Γ−j 1

2meωjk

FT(Ω) = α̅(Ω)f (Ω, T )ℏΩ

(14a)

where the Planck blackbody distribution is given by34 f (Ω , T ) =

(ℏΩ)3 1 CT 2 3 2 π ℏ c exp(ℏΩ/kBT ) − 1

(14b)

Here kB is the Boltzmann constant, T is the temperature, c is the speed of light in vacuum, and CT is the flux normalization constant. The incident photon energy flux is given by S(Ω,T) = ℏΩf(Ω,T), from which we obtain the electric field amplitude for each frequency. 2.3. Surface Photovoltage. The steady absorption of light changes the surface dipole and creates an electric potential difference between the charged layers at the surface and the bulk. This electric potential difference is called as surface photovoltage for the photoexcited system. This potential difference depends upon the average charge on each layer and the distance between them which is defined as5

∑ |Djk|2 γ[γ 2 + Δ2jk ]−1 j,k

(11)

where n(ω) is the refraction index of the medium and c is the speed of light. This expression simplifies when the level populations are approximated by zero (unoccupied) or two (occupied), and the refraction index is taken to be constant in a region of frequencies, which we have chosen to do here. The strength of the electronic transition is expressed as usual in the literature in terms of the dimensionless oscillator strength dependent on transition frequency and the transition dipole moment for that transition, averaged over orientations of the electric field, as in

VS = −[εrε0A]−1

∫0

w

zcS(z)dz

(15)

where w, z, cs, ε0, and εr are the width of the photoactive layer, distance perpendicular to the surface, the charge density per unit length across the surface, vacuum permittivity, and relative permittivity of the charged medium. The permittivity is introduced to include the polarization of atomic cores in the medium and the density changes of passive electrons there, both of which are not part of our chosen primary system. 25528



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Figure 1. c-Si(111)/H with eight layers, a-Si/H structure, pyramidal Ag4 cluster on c-Si(111)/H (Ag4 (pyr)Si192H21), and planar Ag4 cluster on cSi(111)/H (Ag4(plr)Si192H20) geometries are shown from left to right, respectively.

Figure 2. Top panels show EDOS of pure c-Si and a-Si slabs plotted versus (Fermi energy shifted) orbital energies. Electronic density of states with no, one, three, and four silver atoms compared to no silver adsorbates for the c-Si and a-Si are shown in the bottom side of left and right panels respectively. The Lorentzian level broadening with fwhm, W = 20 meV is employed here.

3. ATOMIC MODELS OF THE NANOSTRUCTURED SURFACES We model a Si slab with a supercell and periodic boundary conditions (PBC) on X and Y directions. We also use PBC in the Z direction with added vacuum space to avoid interactions between periodic images. Si(111) surfaces were terminated by 24 hydrogen (H) atoms to compensate the dangling bonds. In the case of Si192H24 containing eight layers (Si8L), 192 Si atoms were arranged in eight layers of size 4 × 6 with 24 H atoms on lower and upper surfaces of the slab. In the case of AgnSi/H structures, silver atoms are deposited on the top layer of the Si surface where n = 0, 1, 3, or 4. Ag3Si192H21 (Ag3Si8L) has three Ag atoms on the upper surface replacing three H atoms. For Ag4 cluster, both planar (plr) and pyramidal (pyr) structures are introduced. For the corresponding amorphous structures, the modeling of a-Si is done through a procedure with molecular dynamics (MD), which starts the simulation with cSi brought to high temperatures, followed by simulated thermal quenching using ab initio MD calculations.33 In the case of a-Si with Ag atoms, a-Ag3Si8L is taken as the reference structure. For the optimization of other a-AgnSi/H structures, we start

Surface photovoltage is calculated from steady state solutions to the time dependent density matrix (TDDM) equations of motion,23 including dissipation due to lattice vibrations. Thus, the time evolution of surface dipoles are treated by TDDM in a basis set of KSO. The photoinduced surface potential is given by the nuclear and electronic dipoles at the surface Vs(Ω) = [εrε0A]−1 {∑ CσZiσ − e ∑ ρjjss̃ (Ω)⟨j|z|j⟩} σ , iσ

i

(16)

where εr is the relative permittivity for silicon (11.8), ε0 is the vacuum permittivity, A is surface area of the silicon slab, and e is the electron charge. Cσ, Ziσ, and iσ are the nuclear charge of the atom σ, z-component of the position vector of the atom σ, and index of each atom type σ, respectively. z is the electronic distance perpendicular to the surface and ρ̃ssjj is the steady state population of the orbital j. The SPV is calculated as the change in voltage in the presence of light with respect to the ground state, that is, nonexcited system. VSPV = Vs(Ω) − Vs(0)

(17) 25529



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Table 1. Energy for the State-to-State Transition with Maximum Absorbance and Corresponding Oscillator Strength Defined in Eq 13b for c-Si and a-Si with Ag Adsorbates system

energy at maximum absorbance (eV)

oscillator strength at maximum absorbance

system

energy at maximum absorbance (eV)

oscillator strength at maximum absorbance

c-Si192H24 c-Ag1Si192H23 c-Ag3Si192H21 c-Ag4Si192H20(plr) c-Ag4Si192H21(pyr)

2.70 2.71 2.70 2.71 2.70

1.74 1.77 1.71 1.85 1.88

a-Si192H24 a-Ag1Si192H23 a-Ag3Si192H21 a-Ag4Si192H20(plr) a-Ag4Si192H21(pyr)

0.57 0.56 0.59 0.56 0.58

0.19 0.13 0.17 0.15 0.18

Figure 3. Dynamic absorption flux densities FT(Ω) of the pure Si slabs are given in the top panels. Change in absorption flux density with one, three and four Ag atoms compared to no silver adsorbates for the c-Si and a-Si are shown in the left and right lower panels, respectively. The used line width of 150 meV corresponds to a decay time of 27 fs.

compared to the corresponding c-Si systems due to the energy level spread created by disorder in the structure. The addition of Ag atoms reduces the band gap of the silicon slabs. Ag adsorbates also add new levels near the gap region, as shown in Figure 2. These findings are in good agreement with our previous results on some of the same systems.21,22,24 The EDOS for pure Si slabs (c-Si8L and a-Si8L) are plotted together with changes in EDOS when Ag atoms are adsorbed, but they are shown with different vertical scales. A closer inspection of the top of the VB shows that the planar Ag4 cluster has additional levels near the gap region compared to Ag3 cluster, which in turn has more levels there than one Ag adsorbate. A pyramidal Ag4 cluster also has additional electronic states in EDOS, but this is less pronounced than for the planar Ag4 cluster. The case of Ag adsorbates on a-Si8L shows similar trend. A clear difference is the decrease of the energy gap going from c-Si to a-Si. 4.2. Absorption Spectra. The oscillator strength for each transition from VB to CB is calculated using eq 13b. Electronic levels in the energy range EHOMO − 4 eV to ELUMO + 4 eV are included for the absorbance calculation. We present the absorbance per active electron for each system in our selected photon energy range. The values for N*el from the oscillator strengths for c-AgnSi/H structures with n = 0, 1, 3, 4 (plr) and 4 (pyr) are 8.642, 8.493, 8.196, 8.087, and 7.986, respectively. Similarly, the number of active electrons for our a-AgnSi/H structures with n = 0, 1, 3, 4 (plr), and 4 (pyr) are 7.043, 7.122, 6.875, 6.899, and 7.398, respectively. The state-to-state transitions with highest oscillator strength and corresponding energy are given in Table 1 for each system. For all c-AgnSi/H

with a-Ag3Si8L structure by freezing all the atoms except the Ag atoms and their neighboring Si atoms. The structures obtained this way and used in our calculations are shown in Figure 1. Our structures are optimized using DFT with a planewave basis set implemented in the VASP software package. We used convergence criteria for atomic forces so that at their stable or metastable equilibria, they are smaller than 0.001 eV/ Å for the c-Si systems and 0.01 eV/Å for the amorphous systems. The optimized geometries obtained from ab initio calculations were used to compute the electronic energy levels and KSOs. Transition dipole moments were calculated for the full slab using the full basis set. More details on the electronic structure optimizations for the crystalline structures are given in our previous studies on similar systems.21,24

4. RESULTS AND DISCUSSION OF OPTICAL PROPERTIES 4.1. Electronic Density of States. The EDOS is calculated using eq 2 with a Lorentzian level broadening shape with W = 20 meV. This value is taken from our previous ab initio molecular dynamics calculations giving the average value of energy level fluctuations from ab initio dynamics in c-Si slabs,35 and from additional calculations recently done for Ag adsorbed on Si. The calculated EDOS versus the KS orbital energy shifts with respect to the Fermi energy of each system are shown in Figure 2 for our c-Si and a-Si systems. The Fermi energy (E(Fermi)) is defined as the midpoint of HOMO and LUMO energy levels. The pure c-Si8L slab has a band gap (EHOMO − ELUMO) of 0.8438 eV, while pure a-Si8L has a smaller band gap of 0.3313 eV. The a-Si systems have generally a lower band gap 25530



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Table 2. Ratios of Averaged Absorption Flux Densities for c-Si and a-Si with Ag Adsorbates to the Corresponding Values for Pure Si Structures, within the [0.5,3.0] eV Range, are Seen To Be Generally Higher as Ag is Adsorbed c-Si192H24 1.0000 a-Si192H24 1.0000

c-Ag1Si192H23 1.0095 a-Ag1Si192H23 0.9948

c-Ag3Si192H21 1.0865 a-Ag3Si192H21 1.0646

c-Ag4Si192H20(plr) 1.1270 a-Ag4Si192H20(plr) 1.0697

c-Ag4Si192H21(pyr) 1.0919 a-Ag4Si192H21(pyr) 0.9801

Figure 4. Scaled SPV with no silver, and one, three, and four silver atoms for the c-Si are shown in the left panel, and for the a-Si are shown in the right panel, for slabs scaled to a thickness of 1.0 mm. A depopulation rate Γ corresponding to 4.13 meV (decay time of 1 ps) is employed here.

are available, to our knowledge, for c-Si to compare these features with. The intensity of absorption is smaller for a-Si slabs. These features can be understood from the characteristics of transitions in the a-Si systems, which have transitions at lower energies due to smaller band gaps and have lower oscillator strengths. The ratio of averaged values F̅T(Ω) of the absorption flux, over the [0.5,3.0] eV range, are given in Table 2. One can see that F̅T(Ω) generally increases by several percent with the addition of Ag atoms, by roughly the same values in both crystalline and amorphous AgnSi/H structures. 4.3. Surface Photovoltage. The SPVs have been calculated for steady incident photons exciting electrons using eq 16. The required state occupation values obtained from the diagonal elements of the RDM have been obtained from a depopulation rate Γ corresponding to a fwhm of 4.13 meV (depopulation time of 1 ps)35,36 and a decoherence rate γ corresponding to 150 meV20 (decoherence time of 27 fs), taken from the literature for similar systems. The average position ⟨j|z|j⟩ of electrons in the z direction for each KSO is obtained from ab initio electronic structure calculations and the partial charge analysis of orbitals. Electronic levels in the energy range EHOMO − 3 eV to ELUMO + 3 eV are included in the SPV calculation. The SPV is calculated in the IR and visible region of the electromagnetic spectrum up to a photon energy maximum of 3 eV. The diagonal elements of the density matrix are calculated using eqs 8 and 9. The potential energy due to surface dipoles for excited and nonexcited Si slabs is calculated using eq 16. The surface area A of our Si slab is 154.524 Å2. As before for the absorbance, the blackbody distribution of photon energies at 5800 K normalized to 1.0 kW/m2 incident photon flux is employed for the SPV calculation. The distance between two Si layers in our c-Si slab with eight layers is 2.86 Å, from which we can scale the system to a thicker slab with many layers. The SPV reported here are linearly scaled for a 1.0 mm thick Si slab (equivalent to one containing 3496503 layers giving a scaling factor of 437063) and shown in the graphs in Figure 4.

structures the maximum oscillator strength is around 2.70 eV. For a-AgnSi/H structures, the energies of maximum absorbance are around 0.56 eV. This value is lower than the c-Si systems due to the smaller band gap of a-Si systems. The highest oscillator strengths for our a-Si systems are an order of magnitude less than for the c-Si systems. This might be due to the reduced overlap of initial and final orbital wave functions in a-Si systems. Higher oscillator strengths for c-AgnSi/H structures compared to the corresponding a-AgnSi/H structures are in good agreement with our previous calculations for Ag adsorbates on c-Si and a-Si surfaces.22 Absorption spectra of the Si slabs with Ag atoms are obtained from light absorbance calculated from oscillator strengths and transition energies. The light absorbance α̅ (Ω) of the system is calculated from eq 13a with a Lorentzian line shape where the decoherence rate γ corresponds to a width of 150 meV (decay time of 27 fs).20 The intensities at different photon energies are weighted with the blackbody solar spectrum at 5800 K. The resulting photon energy flux per unit energy FT(Ω) of the pure Si slabs, in units of s−1 Å−2, is given in the top panels of Figure 3. Absorption spectra for AgnSi/H structures are plotted as the difference in FT(Ω) values from those of pure Si slabs. These changes in absorption flux density with respect to the absorption density for pure Si slabs (c-Si8L and a-Si8L) are shown in the lower panels of Figure 3. In the case of c-AgnSi/H structures, addition of Ag atoms increase the intensity of absorption at lower photon energies. A similar trend is seen in the case of a-AgnSi/H structures. Comparing values for pure cSi8L and a-Si8L, a-Si shows a broader absorption in the entire energy range while c-Si shows sharp peaks at higher photon energies. The sharp increase in flux density at about 2.70 eV for c-AgnSi/H is mainly due to the increase of oscillator strengths to higher photon energies when one Ag atom is added, compared to the pure c-Si slab. This increase remains and dominates the absorption change even when more Ag atoms are added. Similarly, there is a decrease of oscillator strength and absorption decrease because of that, at about 2.68 eV for cSi slabs with one or several Ag atoms. No experimental results 25531



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Table 3. Averaged SPV in mV (from Figure 4) for c-Si and a-Si with Ag Adsorbates within the [0.5,3.0] eV Rangea c-Si192H24 −1.87 × 10−5 a-Si192H24 −1.92 × 10−2 a

c-Ag1Si192H23 1.24 × 10−3 a-Ag1Si192H23 −1.88 × 10−2

c-Ag3Si192H21 1.52 × 10−3 a-Ag3Si192H21 −2.82 × 10−2

c-Ag4Si192H20(plr) 4.60 × 10−3 a-Ag4Si192H20(plr) −4.56 × 10−2

c-Ag4Si192H21(pyr) 6.08 × 10−2 a-Ag4Si192H21(pyr) −4.24 × 10−2

The population relaxation rate Γ corresponding to 4.13 meV is employed here.

Table 4. SPV Features from Figure 4 at Chosen Maxima and Minima Energies are Listed Below for c-Si and a-Si with Ag Adsorbates in the [0.5,3.0] eV Rangea E (eV)

SPV (mV)

E (eV)

c-Si192H24

a

2.67 2.97

3.82 × 10−5 −2.26 × 10−4

1.71 2.51 2.78

a-Si192H24 −4.60 × 10−2 −1.16 × 10−2 −1.76 × 10−2

SPV (mV)

E (eV)

c-Ag1Si192H23 −2.50 × 10−3 −2.44 × 10−3 1.61 × 10−2 a-Ag1Si192H23 1.48 −3.16 × 10−2 2.4 −1.23 × 10−2 2.88 −2.52 × 10−2 1.73 2.23 2.97

SPV (mV)

E (eV)

SPV (mV)

c-Ag3Si192H21

c-Ag4Si192H20(plr)

−1.88 × 10−2 −1.65 × 10−2 4.36 × 10−2 a-Ag3Si192H21 1.54 −4.67 × 10−2 1.90 −3.14 × 10−2 2.18 −3.98 × 10−2

1.15 −3.29 × 10−2 1.58 −2.73 × 10−2 2.72 8.36 × 10−2 a-Ag4Si192H20(plr) 1.49 −6.31 × 10−2 1.84 −6.13 × 10−2 2.1 −6.55 × 10−2

1.16 1.61 2.95

E (eV)

SPV (mV)

c-Ag4Si192H21(pyr) 2.68

2.23 × 10−1

a-Ag4Si192H21(pyr) 0.59 6.92 × 10−3 1.70 −7.97 × 10−2 1.92 −7.25 × 10−2

The population relaxation rate Γ corresponding to 4.13 meV is employed here.

previous calculated and experimental results is possible to ascertain trends as one goes from crystalline to amorphous structures. The averaged SPV, V̅ SPV, within the [0.5,3.0] eV range is given in Table 3, where similarities and differences between c-Si and a-Si are clearly seen. The V̅ SPV is very small for a pure cSi8L slab due to the symmetry of the structure, as expected, but disorder in the structures with a-Si give a SPV different from zero even for the pure a-Si slab. The ratios of SPVs for slabs with adsorbates show a very large increase with Ag-cluster size for c-Si and also a large increase for a-Si. A closer inspection of Figure 4 reveals that there are prominent local maxima and minima SPV features at certain photon energies. SPVs and corresponding photon energies for these features are given in Table 4. One can see that as the number of Ag adsorbates increases there are larger values in the lower energy region for c-AgnSi/H structures with planar geometry. In the case of a-AgnSi/H structures, this trend is more noticeable for all structures. These characteristics of SPV have been further investigated considering the mode of electronic excitation from VB to CB, as found from population changes upon excitation, and from the shape of involved KS orbitals. The SPV for c-Si192H24 shows two distinct peaks at 2.67 and 2.97 eV photon energies. The population analyses at these two photon energies show that electrons are transferred from VB to CB. Both SPV result from the many KSOs involved in the electron transfer process as shown in Figure 5 (top panel), where population changes are shown for each KSO number. Electrons are mainly excited from the KSOs close to the top of the VB. A similar analysis for c-Ag1Si192H23 shows three distinct peaks at 1.73, 2.23, and 2.97 eV photon energies. One can see that there is a prominent additional peak at a lower photon energy. Again, population analyses at these photon energies in Figure 5 (middle panel) show electronic excitations involving the participation of many KSOs. c-Ag3Si192H21 shows three distinct peaks at 1.16, 1.61, and 2.95 eV photon energies. There is a considerable red shift of SPV peaks compared to the pure cSi slab. The population transfer from VB KSOs to CB KSOs is shown in Figure 5 (bottom panel). The photoinduced electron excitation at 1.16 eV reveals that there is one prominent

We have previously modeled the SPV for a c-AgnSi/H slab with a Ag cluster on both upper and lower surfaces in ref 24 for n = 0, 1, and 3 at each surface. The results were averages for upper and lower surfaces of the slab obtained by splitting it into two equal halves. In the present study the Ag cluster was instead located only on one surface in a model that allows us to describe the adsorbate effects at a slab with more Si layers. It was found that the SPVs for c-AgnSi/H structures increase with the number of Ag adsorbates on the Si surface. Also, they display SPV values at lower photon energies compared to the pure c-Si slab. This is in good agreement with our previous SPV calculations on c-AgnSi/H structures with Ag adsorbates on both surfaces of the silicon slab.24 The SPV for pure c-Si in Figure 4 is very small and can be considered to be zero within the accuracy of our calculations, in agreement with what is expected for the symmetric slab structure. The addition of a Ag cluster increases the SPV from that of pure c-Si by 2 orders of magnitude. The highest SPV value increases with the addition of Ag atoms to c-Si as also observed in ref 24. For a Ag3 cluster adsorbed on a c-Si surface, we see a Ag-induced SPV peak at 1.15 eV in current and previous studies. Here we also show results for new structures with Ag4 adsorbates, giving even higher SPVs. The sign for the SPV in the previous study is different from our current results due to a different sign convention used in the present study. The present studies allow us to directly compare c-Si with the new a-Si results done under the same model conditions. In the case of a-AgnSi/H structures, the SPVs show broad peaks in the entire photon energy range. The addition of Ag atoms leads to a moderate increase in the SPV values. The less prominent effect of Ag addition in a-AgnSi/H structures is due to the disorder in the system, which changes both level populations and electronic dipoles. The SPV for pure a-Si has features similar to our previous findings.23,24 The SPV increases in the IR region and broader SPV peaks are found there. The sign of the SPV as well as the overall shape of the peaks also match well with the experimental SPV measured for hydrogenated aSi.9 The a-Si structure used here is independently optimized and is one of the many structures possible at the different local minima reached by the simulated annealing procedure. Thus, only a qualitative comparison with the SPV spectrum of 25532



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population transfer from valence KSOs to CB KSOs at these transitions and show them in Figure 7 (top panel). The photoinduced electron excitation at 1.15 eV reveals that there is one prominent transition from HOMO-4(412) to LUMO(417). The partial charge distributions of these two orbitals are shown in Figure 8. As before, a partial charge analysis shows that upon photoexcitation the electronic charge moves from the Si slab to the surface and Ag cluster. The photoexcitations at 1.58 and 2.72 eV photon energies involve many transitions from VB to CB, each with population changes giving SPVs at those energies. Finally, c-Ag4Si192H21(pyr) shows a distinct peak at the 2.68 eV photon energy. The photoinduced population changes for VB KSOs and CB KSOs are shown in Figure 7 (bottom panel). The photoinduced electron excitation reveals that many KSO in the VB undergo the population change, while a few KSOs in the CB gain population and contribute to the SPV value at 2.68 eV. A parallel analysis has been undertaken for the amorphous structures for comparison purposes. To begin, a-Si192H24 shows three distinct peaks at 1.71 eV, 2.51 and 2.78 eV photon energies. The population transfer from VB KSOs to CB KSOs are shown in Figure 9 (top panel) as changes in the population of KSOs. The photoinduced electron excitation at all these photon energies shows that many KSOs in VB and CB participated in these transitions. a-Ag3Si192H21 shows three distinct peaks at 1.54, 1.90, and 2.18 eV photon energies. As in the case of c-Ag3Si192H21, SPV peaks are red-shifted here as well, compared to the pure a-Si. The photoinduced population change for VB KSOs and CB KSOs are shown in Figure 9 (bottom panel). There are many KSOs below HOMO and above LUMO involved in the photoexcitation at these photon energies. Compared to the pure a-Si structure, the population change for each KSO is different for a-Ag3Si192H21, although the gross features are similar for both systems. Further comparison of c-Si and a-Si population changes shows that the c-Si values have more pronounced changes with photon energies, which can be expected due to spatial confinement of KSOs in the regular crystal structures.

5. CONCLUSIONS We have presented results on the optical properties of semiconductor silicon surfaces without and with silver adsorbates, based on a unified theoretical treatment with the reduced density matrix. We have obtained results for both c-Si and a-Si slabs with the same number of Si atoms and the same computational procedure to allow direct comparison of results for both structural forms of Si. This has been done for two important properties of surfaces, which are accessible to experimental measurements: the photoabsorbance and the surface photovoltage. We found that addition of Ag adsorbates enhances the SPV of both c-Si and a-Si slabs with very large changes for c-Si and smaller ones for a-Si. The changes of absorbance values were not so large but should be detectable. These differences could be understood in terms of changes in density of states and changes in level populations. Because Ag clusters add levels near the gap region of the pure slabs, density of states available for the optical excitations are higher. Further, Ag adsorbates decrease the HOMO−LUMO gap and allow absorption of lower photon energies. The ratio of averaged absorption flux densities over photon energies in the IR and visible region generally show an increase in absorption as the size of a Ag cluster increases. Results have been presented for an incoming photon flux corresponding to the solar radiation

Figure 5. Population changes in each KSO due to excitation from VB to CB are plotted by setting the HOMO label number to zero. Negative sign for the population change shows loss of population while positive sign shows gain in population for each KSO. Photoinduced population change of orbital occupations are shown for the c-Si192H24 slab (top panel) at 2.67 and 2.97 eV photon energies, for c-Ag1Si192H23 slab (middle panel) at 1.73 eV, 2.23 and 2.97 eV photon energies, and for the c-Ag3Si192H21 slab (bottom panel) at 1.16, 1.61, and 2.95 eV photon energies.

transition from HOMO-4(number 407) to LUMO(412). The partial charge distributions of these two orbitals are shown in Figure 6. A partial charge analysis adding electronic charges in each layer along the z-axis perpendicular to the surface shows that upon photoexcitation the electronic charge moves from the Si slab to the surface and Ag cluster. The photoexcitations at 1.61 and 2.95 eV photon energies involve again many electronic transitions from VB to CB. The SPV for c-Ag4Si192H20(plr) shows three distinct peaks at 1.15, 1.58, and 2.72 eV photon energies. We analyze the 25533



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Figure 6. Partial charge analysis of certain KSOs for the c-Ag3Si192H21 slab. Here, 407 is the HOMO-4 KSO and 412 is the LUMO for the system. One can see on the right, giving the electronic density along the z-axis, that an excitation from HOMO-4 to LUMO brings the electronic charge to the surface and Ag cluster.

clusters (n = 0, 1, 3, 4), the reduced enhancement in the absorbance can be understood simply in terms of the present smaller ratio of the number of Ag atoms to Si atoms. The averaged absorption flux densities for c-Si and a-Si with silver adsorbates generally show an enhancement compared to the pure Si slabs. The magnitude of enhancement is somewhat smaller in our model calculations compared to the experimental values mentioned in the Introduction.2,3 This is expected due to the small size of the silver clusters used in our model compared to the experimental clusters. The c-Si slabs show larger SPV at our high photon energies, while addition of Ag adsorbates increases SPV at lower energies. There is a noticeable difference in the SPV for planar and pyramidal Ag4 clusters on c-Si slab. The c-Ag4Si:H (plr) slab shows SPV values at lower photon frequencies which are absent in c-Ag4Si/H (pyr). The SPV for a-AgnSi/H structures shows similar trends as in the absorbances i.e., broader distributions. In the case of a-Si, addition of silver again shows increase in SPV from one Ag to four Ag. Because of the disorder in the different a-Si structures with and without Ag, they do not show a clear pattern of maxima and minima. The SPV we calculate for a pure a-Si slab shows very good agreement with previous theoretical24 and experimental9 results. The diameter of Ag nanoparticles in the mentioned experiments2,3,13 is in the 6 to 100 nm range, large enough to involve many electrons undergoing density fluctuations or plasmons. In contrast, in our c-AgnSi192H24 model, only a few Ag atoms (n = 0−4) are deposited on the Si(111) surface and localized surface plasmons are unlikely. The optical enhancement for the case of a few adsorbed Ag atoms is mainly due to the excitation of the deposited Ag atoms as a molecular group. As the number of Ag atoms is increased in the adsorbed clusters, it is likely that plasmonic excitation may play a significant role in light absorption.37,38 Several improvements could be introduced to improve the accuracy and generality of our treatment. More accurate DFT exchange-correlation functionals can be used to improve the treatment of electronic self-interaction39,40 and long-range interactions, some of which we have recently explored.41 An alternative is to obtain one-electron properties from a manyelectron theory within the GW method,42,43 which would however be computationally very demanding for our large and amorphous systems. In our present treatment, exciton effects and coupling to lattice vibrations are given by energy widths, which can be extracted from experimental relaxation rates.

Figure 7. Photoinduced population change for the c-Ag4Si192H20(plr) slab (top panel) at 1.15, 1.58, and 2.72 eV photon energies, and for the c-Ag4Si192H21(pyr) slab (bottom panel) at the 2.68 eV photon energy. Population changes in each KSO going from VB to CB are plotted by setting the HOMO label number to zero. A negative sign for the population change shows loss of population, while a positive sign shows gain in population for each KSO.

spectrum, with a weighted black body distribution of photon energies, but the analyses of results based on atomic structure and orbital distributions and shapes should be generally applicable to other distributions. The c-AgnSi/H structures show light absorbance mainly at high photon energies while addition of Ag atoms increase the absorption at lower photon energies. The a-AgnSi/H structures show broadened and weaker absorption in the calculated photon energy range compared to the corresponding c-AgnSi/ H structures. Thus, the structure of the Si slabs and silver clusters play an important role in the absorption. Our calculated features of optical absorption spectra agree qualitatively with our previous results published on some similar systems with fewer Si layers, done with TD-DFT. They showed larger absorption after Ag adsorbates were added. Due to the larger Si slab size employed in the current study with the same Agn 25534



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Figure 8. Partial charge analysis of certain KSOs for the c-Ag4Si192H20(plr) slab. Here 412 is the HOMO-4 KSO and 417 is LUMO for the system. One can see that an excitation from HOMO-4 to LUMO brings more electronic charge to the surface and Ag cluster for Ag4 as compared with Figure 6 for the Ag3 cluster.

insight on differences to be expected between crystalline and amorphous Si, with and without adsorbed metal clusters.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]fl.edu. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has been partly supported by the National Science Foundation Grant NSF CHE 1011967. Computing support was provided by the High Performance Computing facility of the University of Florida.

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REFERENCES

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Figure 9. Photoinduced population change for the a-Si192H24 structure (top panel) at 1.71, 2.51, and 2.78 eV photon energies, and for the aAg3Si192H21 structure (bottom panel) at 1.54, 1.90, and 2.18 eV photon energies. Population changes in each KSO due to excitation from VB to CB are plotted by setting the HOMO label number to zero. Negative sign for the population change shows loss of population, while positive sign shows gain in population for each KSO.

These effects could in principle be computed for our systems introducing coupled equations of motion for boson operators corresponding to excitons or phonons,44 an interesting possibility however beyond the present aims. Overall, the theory and computational procedures we have presented provide theoretical and computational tools for the comparison of optical properties of several materials, such as 25535



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Photoabsorbance and Photovoltage of Crystalline and Amorphous

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