Surface Photovoltage at Nanostructures on Si Surfaces - American

3530

J. Phys. Chem. C 2009, 113, 3530–3542

Surface Photovoltage at Nanostructures on Si Surfaces: Ab Initio Results Dmitri S. Kilin and David A. Micha* Quantum Theory Project, Departments of Chemistry and Physics, UniVersity of Florida, GainesVille, Florida 32611-8435 ReceiVed: October 8, 2008; ReVised Manuscript ReceiVed: December 9, 2008

Charge transfer photoinduced by steady light absorption on a silicon surface leads to formation of a surface photovoltage (SPV). The dependence of this voltage on the structure of surface adsorbates and on the wavelength of light is studied with a combination of ab initio electronic structure calculations and the reduced density matrix for the open excited system. Our derivations provide time averages of surface electric dipoles, which follow from a time-dependent density matrix (TDDM) treatment using a steady state solution for the TDDM equations of motion. Ab initio calculations have been carried out in a basis set of Kohn-Sham orbitals obtained by a density functional treatment using atomic pseudopotentials. Applications have been done to a H-terminated Si(111) surface and for adsorbed Ag, with surface coverage ranging from 0 to 3/24 of a monolayer. Calculations done also for amorphous Si agree with measured values of the SPV versus incident photon frequency for H-terminated a-Si. Surface adsorbates are found to enhance light absorption and facilitate electronic charge transfer at the surface. Specifically, Ag clusters add electronic states in the energy gap area, provide stronger absorption in the IR and visible spectral regions, and open up additional pathways for surface charge transfer. Our treatment can be implemented for a wide class of photoelectronic materials relevant to solar energy capture. I. Introduction Light absorbed at a semiconductor surface leads to electronic excitation and charge rearrangement. This creates a photovoltage characteristic of the surface properties, which varies with the wavelength of the exciting light. The photovoltage provides valuable insight into the optical properties of the surface, relevant to the primary events of photovoltaic phenomena,1,2 when photon energy first absorbed at a semiconductor surface is later converted into an electric current through charge separation. Most of the present ab initio computational efforts3-6 on photovoltaic effects have resulted in the prediction of photoinduced charge transfer on a (typically dye-sensitized) semiconductor surface. There are multiple advanced experimental techniques to measure the surface voltage created by light irradiation. In the present work, we develop a computational approach to describe such photoinduced voltage on a surface (SPV), with a novel approach based on the reduced density matrix for an open system. We develop a theoretical treatment starting from the atomic structure of the surface and a straightforward procedure to calculate the voltage between the surface and inner layers of an electronically excited semiconductor slab. This is done by converting ab initio electronic structure information about charge redistribution into results for surface electric dipoles, and into surface photovoltage. The essence of the photovoltaic process is surface charge rearrangement induced by steady light absorption. Therefore, it is important to investigate materials suitable for photovoltaic devices for their ability to undergo surface charge transfer. Our theoretical and computational treatment can be implemented for a wide class of photoelectronic materials relevant to solar energy capture. Here, we focus on Si surfaces and Ag cluster adsorbates on them. Silicon is a material of choice for photovoltaic applications * Corresponding author. E-mail: [email protected].

due to its long-term stability and a well-established technology for the preparation of Si wafers and Si thin films.7,8 In addition, the authors of ref 9 have recently reported a 7-16-fold enhancement of Si film light absorption due to deposition of silver nanoparticles in the vicinity of the surface, with stronger enhancement for thinner films. The size and shape of Ag nanoparticles are controlled by their interaction with a substrate or solvent.10 The basic concepts relevant to experimental measurement of SPV, such as the Kelvin probe, STM, and others, are covered in several reviews.11,12 A number of authors have reported the dependence of SPVs on the frequency of the incident photons for various forms of silicon surfaces. Specifically, for amorphous, H-terminated silicon (a-Si:H),13 an appreciable voltage signal starts at a photon energy 0.9 eV and, depending on the method of surface preparation, has a maximum between 1.7 and 2.4 eV, approaching a maximal voltage value of 160-210 mV. Results for the (111) surface of crystalline porous silicon have also been reported.14 Other types of Si(111) surfaces relevant to SPV are 7 × 715,16 and 2 × 117 surface reconstructions, as well as Si surfaces terminated by organic ligands. A typical surface used in experiment is Si(111), although, depending on the experimental conditions, reconstruction, capping, or the appearance of an oxide layer may occur. Here, we focus on the Si(111) surface terminated by hydrogen, which saturates dangling bonds and prevents reconstruction so that surface silicon atoms maintain crystalline symmetry. The deposition of a certain amount of Ag on the Si-surface is expected to affect both electronic structure and photophysical properties of the surface. According to ref 18, a fraction of a Ag monolayer deposited on a Si(111) surface tends to form Ag islands on the surface and to increase surface photovoltage for a given irradiation frequency. The ab initio theoretical description of surface photovoltage is currently in its initial stages of development. A rigorous

10.1021/jp808908x CCC: $40.75  2009 American Chemical Society Published on Web 02/11/2009

Surface Photovoltage at Nanostructures on Si Surfaces

J. Phys. Chem. C, Vol. 113, No. 9, 2009 3531

calculation of the electric dipole and voltage of a silicon surface treated with organic radicals is presented in refs 19, 20 for the ground electronic state, solving the Kohn-Sham equations using norm conserving pseudopotentials constructed within the local density approximation and using a plane wave basis. The photovoltage of a film of dye has been presented as a function of excitation frequency in recent experimental work,21 along with an analysis based on classical electrodynamics. The optical properties of a semiconductor surface can be described within the Kohn-Sham orbital (KSO) picture of density functional theory (DFT). The orbital representation is suitable for low dimensional structures such as quantum dots, quantum wires, and thin films, because their optical properties are controlled by quantum confinement.22,23 While more advanced approaches, such as the Bethe-Salpeter theory or DFT corrected for self-interaction, can produce better agreement with the experimental data for extended systems,24,25 they are more computationally demanding and cannot yet be applied to an electronically excited system of many atoms. The next section presents our ab initio treatment for the steady state optical excitation of surfaces and constructs solutions to the equations of motion of the reduced density matrix. The following section describes a multilayer slab model for Si(111):H and adsorbates on its surfaces, AgnSi(111):H. It contains results for crystalline and amorphous slabs, and a discussion of the effect of adsorbates on the distribution of electronic energy levels and on the SPVs. The conclusion section describes what is novel in our approach, its scope, and aspects that need further work. II. Ab Initio Density Matrix Treatment of Surface Photovoltages A. SPV from a Surface Dipole. The charge distribution of a slab electronically excited by steady light absorption changes from that of its ground state, to create a photoinduced polarization with a time-independent component.21 The polarization generates a homogeneous electric field, and, if the slab is brought into contact with an electric circuit, it leads to an electric potential. We start from the description of the photoinduced surface electric potential, often referred to as a surface photovoltage, based on an analysis of the electric dipole of the excited system. Consider a situation when a surface acquires an excess or deficit of electronic charge density. The electronic charge density is assumed to be homogeneously distributed along X and Y directions on the surface due to translational symmetry. The dependence of the electronic charge density on the Z coordinate perpendicular to the surface leads to an inhomogeneity with surface charge density per unit length across the surface cs(z) ) ∫∫Ac(x,y,z) dx dy, which becomes a source of an electric field εz ) cs(z)/(εrε0), perpendicular to the surface. The corresponding electric potential difference between charged layers is proportional to their average charge and to the distance between oppositely charged layers, and can be cast for a continuous charge distribution as,11

Vs ) -[εrε0]-1

∫0w zcs(z) dz

(1)

where w, z, cs, ε0, and εr stand for the width of a photoactive layer, the distance perpendicular to the surface, the charge density per unit length across the surface, the dielectric constant of vacuum, and the dielectric coefficient of the photoactive medium, respectively. Note that this universal equation is valid for cs representing the ground or excited state of the semiconductor surface. The total charge density contains both ionic and

electronic contributions, so that c(z) ) cion(z) + cel(z), with cion(z) ) ∑σ,iσCσδ(Ziσ - z), and cel(z) ) -en(z). Here, symbols are Cσ for the ion charge, Ziσ for the z-component of the position vector Riσ, and iσ an index for each atom of type σ, with e the electron charge, and n(z) the electronic number density along z. Within DFT, we model the electron density as the sum of partial densities from Kohn-Sham orbitals. The electronic charge follows from the electronic density matrix of the whole system in the coordinate representation at each time t, as cel(r,t) ) -eF(r,r,t). Here, the electronic density matrix F(r,r′;t) ) ∑ijFij(t)φ*i (r)φj(r′) is expressed in a basis set of Kohn-Sham orbitals {φj(r)}. This leads to the following electric potential expression:

∫∫∫Vd3rz{cion(r) - cel(r)}

Vs ) [εrε0A]-1

) [εrε0A]-1

{∑ σ,iσ

CσZiσ - e

}

∑ Fij〈i|z|j〉 ij

(2)

abbreviating ∫-∞∞r dr φ*i (r)φj(r) )〈i|r|j〉 and using A for the surface area. The density matrix elements Fij depend on the frequency Ω of the incident light as shown below. B. Reduced Density Matrix in a Basis of Kohn-Sham Orbitals. Consider the formation of an excited state at a silicon surface continuously irradiated by light of frequency Ω, consistent with an experimental setup. Light promotes electrons up in energy from the valence band (VB) to the conduction band (CB), while relaxation processes lead to de-excitation back toward the ground electronic state. These processes are described within a reduced density matrix (RDM) treatment within standard approximations. Diagonal elements of the density matrix give state populations, and off-diagonal elements of density matrix correspond to quantum coherences. The evolution of electronic states of a model system is conveniently described by the equation of motion (EOM) in the Schro¨dinger picture of quantum mechanics:26-28

F˙ jk ) F )F ˆ

i p

ˆ KS

∑ (FjlFlk - FjlFlk) + (dFjk/dt)rel l

- D · E(t)

(3)

Equation 3 contains populations Fii and coherences Fij for Kohn-Sham orbitals (KSOs) of a system with periodic boundary conditions, given by φj(r) ) ∑|G| εj 2τ t-τ Ωτ

k

0 ) - i∆jkFjkss - iΩjk(F˜ ss ˜ jjss) - γjk(F˜ jkss - F˜ jkeq) kk - F

F ˜ii(t) ) F˜ iiss

Solving for off-diagonal elements, one obtains F˜ ss ˜ ss kj - F jk ) 2 2 -1 ss ss 2iγjkΩjk(γjk + ∆jk) (F˜ kk - F˜ jj ). Substitution of the last expression into the diagonal steady state equation yields

F˜ jjeq )

∑ (δjk + Mjk)F˜ sskk k

Mjk ) gjk/Γj - δjk

∑ gjl/Γj

{∑

V¯ s(Ω) ) [εrε0A]-1

l

gjk(Ω) ) γjkΩjk2 [γjk2 + ∆jk(Ω)2]-1

σ,iσ

These equations can be used to derive the steady state populations. In simplified notation, diagonal density matrix elements are represented by the column matrix P, and the transition intensities by the square transition matrix M. The linear algebraic equation for steady state populations can then be cast in the matrix form (I + M)Pss ) Peq, with the identity matrix I plus a small increment matrix M and a column matrices Peq of equilibrium populations. The solution for the steady state populations is given by Pss ) (I + M)-1Peq. In the limit of weak optical field intensity gjk , Γj, a Taylor expansion of matrices leads to (I + M)-1 ≈ I - M + O([M]2), so that an approximate solution for the steady state populations is Pss ) (I - M)Peq, and in detail

F˜ jjss ) Γj-1

∑ gjkF˜ eqkk + (1 - Γj-1∑ gjk)F˜ jjeq k

(7)

k

For the equilibrium population at temperature T ) 0 K, we have ˜ eq F˜ eq jj ) 0, j g LUMO, and F jj ) 1, j e HOMO, where the LUMO is the lowest unoccupied molecular orbital at the bottom of the VB while the HOMO is the highest occupied MO at the top of the VB. To evaluate the steady state populations, we neglect intraband population transition, because they are far off resonance with light of optical frequency Ω so that the gjk(Ω) are negligible for such transitions. This implies that

F˜ jjss ) Γj-1

HOMO

∑

gjk, j g LUMO

(8)

k)0

F˜ jjss ) 1 - Γj-1

∞

∑

-1 for all t > γ-1 ij > τ . Ω . This uses sinΩ(τ)/(Ωτ) = 0 where the average is performed over a long time τ as compared to light period Ω-1. Consequently, the average of fast oscillations gives vanishing off-diagonal DM elements in the expression for the SPV. This leads to a practical expression containing only diagonal elements of the steady state solution of EOM, representing the population of Kohn-Sham orbitals,

gjk, j e HOMO

(9)

k)LUMO

F˜ kjss ) iγjkΩjk(γjk2 + ∆jk2 )-1(F˜ ss ˜ jjss), k * j kk - F

(10)

The steady state matrix elements are therefore given by superpositions of many Lorentzians due to the high density of electronic transitions, so that only the envelopes of the Lorentzians are needed. This allows for convenient parametrizations as discussed in what follows. Note that values of RDM elements depend on frequency of the incident field Ω, and that offdiagonal elements F˜ ss jk are higher order in the field strength. Returning back from the rotating frame, one recovers the time dependence of DM elements in the Schroedinger picture, Fij(t) ) F˜ ij(t) exp(-iΩt), εi > εj and Fii(t) ) F˜ ii(t). However, because SPV measurements provide time-independent values, we perform a local time average over a time period τ . Ω-1 as follows:

CσZiσ - e

}

∑ F˜ iiss(Ω)〈i|z|i〉 i

(11)

which is used in the following calculations of the SPV given by VSPV ) Vs(Ω) - Vs(0), that is, obtained with respect to the voltage of the nonexcited system. Here, A stands for the surface area of a simulation cell. Note that VSPV is linear in the light energy flux J ) c|E|2/2, with c as the speed of light. A quantitative criterion of applicability of our linear regime formula is 0 < Fii < 1. For higher field intensities or for large values of transition dipoles, one requires a higher order solution, above the linear regime. In the applications that follow, the light flux falls within the linear regime. III. Results on Surface Photovoltage and Electronic Excitations A. Slab Model for Si(111):H and AgnSi(111):H. Our numerical approach employs a density functional theory implemented in a code with a basis set of augmented plane-waves and atomic pseudopotentials.38 For the crystal surfaces with periodic atomic structure of present interest, our atomic models periodically replicate along the surface X and Y directions. In the Z-direction, we also use periodic boundary conditions but with a vacuum layer above the surface. We model the Si(111)-surface with a slab containing 192 Si atoms arranged in 4 layers of size 4 × 6, and 24 hydrogen atoms that compensate dangling bonds on upper and lower (111) surfaces. A larger cell with 432 Si atoms in 18 layers is used for verification of results. The edges of the slab supercell do not coincide with the crystallographic axes of Si. Instead, the z-axis of the slab supercell is aligned along the 〈111〉 direction of the crystal, perpendicular to the Si(111) surface. The simulations were performed with the VASP software package,38 using a basis of 106 augmented plane waves, and periodic boundary conditions. The core electrons were modeled using the Vanderbilt pseudopotentials,39 while the valence electrons were treated explicitly. The PW91/GGA density functional40 with a generalized gradient approximation was used to account for electron exchange and correlation effects. The electronic structure was converged to the 0.0001 eV tolerance limit. A vacuum layer of at least 7 Å was added above the upper surface of the slab to avoid spurious interactions between the periodic images of the system. In our present calculations, we introduce average relaxation and dephasing rates by the following parametrization of eq 8 for the light-induced values of populations, Γj-1∑kgjk ) Γ-1kΩ2jkγ{∆2jk + γ2}-1. Here, Γ ) ∑jΓj/Nst is an average population relaxation rate where the sum extends over the Nst states that fall within the range of electronic energies εF - ∆ε e ε e εF + ∆ε with ∆ε ) 2.50 eV, while γ is an average dephasing rate defined by the equation. The values of Ωij and ∆jk follow from our calculations of energy levels and transition

3534 J. Phys. Chem. C, Vol. 113, No. 9, 2009 dipoles, and from an electric field chosen to give a laser power of 0.1-1.0 W per cm2. Dephasing in semiconductor materials occurs at ultrafast time scale of the order of ten femtoseconds,41 as a result of interaction between electronic states and molecular vibrations with high density of vibrational states. Therefore, in this contribution, we use an estimated value of pγ ) 150 meV24 (corresponding to a dephasing time of 27 fs). The population relaxation times Γ-1 at silicon surfaces are much longer and j range from a few to hundreds of picoseconds,42 and in this contribution we use an estimated average value of pΓ ) 0.15 meV (for a population relaxation time of 27 ps). We have calculated SPVs according to eq 11 with parameter values εr ) 11.8 (silicon), ε0 ) 1/4π (in atomic units). For the chosen simulation cell, the surface area is A ) 154.525 Å2, providing the following numerical value of the conversion factor between dipole and voltage 1/(εrε0A) ) 0.098901 V/(Å e), with e as the electron charge. We have also averaged our calculated values of the voltage for the upper and lower surfaces of the model slab to obtain smooth change of densities. Our calculations give SPVs and also the shape of KSOs describing the electron and hole states created by light absorption. These provide insight on photoinduced electronic rearrangement and can be used to classify the types of electronic rearrangements by considering whether there is more or less localization of electronic charge densities as a result of light absorption. Each electron-hole excitation has an energy and an oscillator strength that can be estimated from our calculations. In a first approximation, excitation energies are given by differences in orbital energies if the energy of the electron-hole interaction is neglected. This is valid when at least one of the orbitals in the electron-hole pair has a delocalized character. The e-h interaction is strong when excitons are formed, but this does not happen for slab thickness smaller than the exciton Bohr radius for this material, which is about 4.9 nm.43 Thicker slabs are expected to develop excitons that require a description of electronic interactions as provided, for example, by the TDDFT or GW treatments. A qualitative analysis as provided in what follows appears justified for small-size slabs and for amorphous structures that supress electronic correlation. B. Results for Si(111):H. The atomic conformation of the surfaces was energy optimized, and it was found that hydrogen coverage prevented surface reconstruction. The interatomic Si-Si distance slightly decreases by about 0.002 ( 0.001 Å between the first and second layers as compared to the inner region of the slab. The shape of the slab is shown in Figure 1A, and the density of electronic states for the atomic models is shown in Figure 1B versus orbital energies referred to the Fermi energy EFermi, taken here to equal the middle point energy between the HOMO and the LUMO. Our calculations can be compared to features known for the bulk Si crystal, which we briefly mention for comparison purposes. Crystalline silicon is an indirect band gap semiconductor, insofar as the active optical transition and band gap between extrema of CB and VB energy levels versus wavevector k correspond to different points in momentum space. Specifically, a strong optical transition occurs at the band Γ-point, k ) 0 with ∆k ) 0, while weaker transitions correspond to ∆k * 0 with momentum transfer in 〈100〉 and 〈111〉 directions toward X- and L-points, respectively. The density of electronic states (DOS) for the atomic model of Si(111):H with 8 and 12 layers are shown in Figure 1B versus orbital energies referred to the Fermi energy EFermi. States below the Fermi energy (the middle point between the highest occupied and lowest unoccupied levels) represent the valence band (VB),

Kilin and Micha

Figure 1. (A) Atomic slab model of a crystalline Si(111):H surface, showing the first layers. (B) Electronic density of states normalized to the number of atoms for slabs with 12 (solid) and 18 (dashes) monatomic layers. States below the Fermi energy (the middle point between highest occupied and lowest unoccupied levels) represent the valence band (VB), while states above the Fermi energy represent the conduction band (CB). The first three inflection points in the CB correspond to subbands with minima at momentum space points labeled as X, L, and Γ (see text). (C) Band gap as a function of the Si slab thickness (crosses), fit to a model of an electron in a box with Coulombic potentials (short dashes) and also fit to a screening potential model (long dashes). Asymptotic values of band gap are shown by thin dashes.

while states above the Fermi energy represent the conduction band (CB). The first three inflection points in the CB correspond to

Surface Photovoltage at Nanostructures on Si Surfaces subbands with minima at momentum space points labeled as X, L, and Γ. The calculated density of states can be interpreted in the following way: A first local maximum of the DOS in the VB, at energies below EFermi, corresponds to hole subbands with different values of the hole effective mass at k ) 0. In the CB, at energies above EFermi, the first DOS maximum corresponds to a subband with a CB minimum at the X-point, a second maximum corresponds to a subband with a minimum at the L-point, and a third maximum at about 2 eV from EFermi corresponds to a minimum at the Γ point. The calculated DOS can be noticeably affected by spatial confinement in the 〈111〉 direction. Our model suggests there is a noticeable optical activity for transitions at or above excitation energies of 2 eV. Also, for crystalline Si without adsorbates, electronic transitions ifj involving states i at the top of the VB and j at the bottom of the CB have small Dij and contribute neither to the absorption spectrum nor to the SPV because they do not conserve momentum k. The energy band gap has been calculated for slabs of the same surface area but various thickness for a sequence of 4, 6, 8, 10, 12, and 18 layers, and varies as shown in Figure 1C. In a H-terminated slab, all surface localized states are very far from the band gap region of the DOS and do not affect the dependence of Egap on slab thickness. The gap should agree with the bulk value as slab thickness approaches infinity, but for a finite slab the features of the band gap follow from quantum confinement due to the finite slab on the thickness. The calculated dependence of the band gap Egap slab (R) ) slab thickness R can be fitted to an empirical equation E gap bulk + K/R2 - U/R, with K ) 70 eV Å2 and U ) 0.27 eV Å, in Egap accordance with an effective mass model for electrons and holes.44 This empirical formula works well for eight and more layers. For the whole range of studied slab thickness, one gets a better fit with an alternative empirical formula acounting for dielectric screening scr bulk (R) ) Egap + R exp{-βR}, R ) 0.72 in the exponential form Egap -1 eV, β ) 0.081 Å . Upon increase of the number of layers, the band gap approaches the value calculated for bulk silicon with a model consisting of a cell of 144 atoms and periodic boundary conditions, giving the value Ebulk gap ) 0.7392 eV, as shown in Figure bulk value is less than the experimental band gap of 1C. This Egap crystalline silicon (1.1 eV)45 because of known shortcomings of the DFT for an infinite system. The difference is expected to decrease when DFT is applied to our finite slabs, and our estimated error for the gap values in our slabs is 0.10 eV. Higher accuracy for the band gap value can be expected using, for example, TDDFT with hybrid functionals46 or the scissor operator technique. Alternatively, the interpolation formulas above could be used with the experimental value of the band gap, but here we keep our calculated values obtained for the slab models. We have recently published some of our calculations of SPV for crystalline Si (c-Si).31 The results provide the correct sign of the photovoltage and qualitatively correct positions of peaks in the dependence with photon energy. Here, we add new results for an amorphous form of Si (a-Si:H) as modeled by a sample prepared by simulated annealing of c-Si started above the melting temperature and cooled to temperatures around 200-300 K. We present results of a single “snapshot” atomic configuration in Figure 2, leaving a systematic study with an ensemble average for later consideration. Note that an average over ensemble of N random realizations of a model with M atoms will approach the results of a single realization with M × N atoms, as Nf∞. Therefore, a single snapshot of our large system provides a credible description of general features of an amorphous phase of Si. The obtained structure is presented in Figure 2A for a-Si:H. Preliminary calculations show that our results for a-Si are stable with respect to variation of the number of atoms. We have also checked that


Figure 2. Electronic structure of atomic slab models of a Si(111):H surface. (A) Snapshot view of the structure of amorphous Si:H. (B) Electronic density of states for three atomic models: a-Si:H with as many atoms as eight monatomic Si-layers (long green dashes), c-Si:H with eight monatomic Si-layers (solid line), and c-Si:H with 18 monatomic Si-layers (short blue dashes). Bands are schematically labeled as valence band (VB), conduction band (CB), doubly occupied defect (D-), and unoccupied defect (D+). (C) Surface photovoltages for the amorphous Si surface, calculated (long green dashes) and experimental13 (red crosses).

different snapshots of the same model at different times yield small fluctuations of DOS. Most of the inner Si atoms approximately preserve bulk symmetry on short distance, that is, have four neighbors arranged in tetrahedral coordination, while most of the surface atoms are passivated by hydrogen atoms. This geometry is one of the possible local minima. Repeating the annealing

3536 J. Phys. Chem. C, Vol. 113, No. 9, 2009 simulation would result in a different structure at a different local minimum but with similar physical properties. The density of states of the a-Si is presented in Figure 2B and has many features in common with the DOS of c-Si. The inflection points of the CB appear at approximately the same energies in a-Si:H and c-Si:H, because the energies of the KSOs are determined by local symmetry among neighboring atoms, which is approximately preserved at short range. Noticeably, the amorphous slab shows a finite DOS at EFermi. The amorphous form yields additional electronic states in the valence band, and the band gap approaches zero in a-Si, due to the appearance of multiple electron traps at the surface and of inner hydrogen impurities.47 The CB contains additional states near the edge, and a decrease of the DOS at higher energies corresponds to optically active KSOs in the vicinity of the Γ point in k-space. Because the new surface states have higher density in the VB region and lower density in the CB region, one can expect that excitation of an electron from VB to CB will promote charge transfer between surface and inner layers of the slab in a stronger fashion, as compared to the c-Si case. Indeed, the calculated SPV of a-Si presented in Figure 2C supports these expectations. The SPV spectrum of a-Si shows three main features as compared to the c-Si case: (i) the SPV spectrum shifts toward the IR range; (ii) all peaks become broader; and (iii) there is better overall agreement with experiment. One should note that the SPV dependence on the light frequency Ω is insensitive to variations of (50% in the value of decoherence and depopulation rates because in our system transition lines are very close in energy, so that their envelopes remain unchanged. Our numerical results are compared to the experimental ones13 for the a-Si:H used as a reference and are presented in Figure 2C, for an electromagnetic field of flux 1.0 W/cm2. General features of both graphs are in qualitative agreement. Although one cannot expect close agreement, the features of both SPV spectra clearly correspond to each other. This agreement can be attributed to the fact that in the a-Si:H there are more additional states in the VB as compared to c-Si:H. Upon excitation, a-Si:H contains more holes on the surface corresponding to electron density migration inside the slab, and therefore a negative sign of photovoltage. The feature of our calculated SPV at about 2 eV is barely visible in the experimental data, and we expect it would decrease after an average over multiple snapshots of the amorphous surface. C. Results for AgnSi(111):H. Optical properties of Ag clusters adsorbed on Si surfaces follow from the electronic structure of the atomic models. Densities of electronic states of our atomic models of a Si surface with 0-, 1-, and 3-atom adsorbates involve increments at specific energies as shown in Figure 3. Arrows indicate energy values where Ag1 and Ag3 adsorbates contribute to the total DOS. Noticeably, the models with adsorbed silver show slight increments of electronic DOS at energies corresponding to Ag-localized KSOs. Ag-localized states are labeled as 1, 2 in the CB and as 1′, 2′ in the VB. According to expectations, a single Ag adsorbate yields fewer states and provides fewer changes in DOS as compared to the Ag3 planar cluster. Specifically, KSOs at energies labeled 2′ and 2 appear for both Ag1 and Ag3 adsorbates, while additional states 1′ and 1 appear only for the Ag3 adsorbate, so that the energy difference between Ag-localized KSOs decreases with the size of the adsorbate. An interesting feature is that the peak at the edge of the CB appears only in a model with a 3-atom silver cluster. This feature may be responsible for a positive photovoltage at the corresponding transition energy. We also expect that for clusters of larger size the Ag-localized orbitals

Kilin and Micha

Figure 3. Electronic structure of atomic models of Agn adsorbed on the Si(111):H surface. (A) Structure of a slab model with a Ag3 planar cluster. (B) Electronic density of states. Solid red line, Si:H without adsorbate; long green dashes, single Ag atom; short blue dashes, planar Ag3. (C) Differences in electronic density of states. Long green dashes, difference between DOS of Ag1Si:H and Si:H; short blue dashes, difference between DOS of Ag3Si:H and Si:H. Energies are labeled as 1, 2 for CB and as 1′, 2′ for the VB. The Ag-localized states in the CB are much closer to the band gap and to the band edge than those of the VB. This suggests that low-energy excitations involve Si-localized states in the VB and a Ag-localized state in the CB, corresponding to migration of charge toward the surface and positive sign of the photovoltage.

of both CB and VB will approach each other, yielding zero band gap for a large enough metal cluster. The SPV for these atomic models has been calculated using eqs 8, 9, and 11. We combine the SPV results for crystalline Si with various adsorbates in a single picture. Calculated SPVs are presented in Figure 4A. The flux of the light field is taken to be 0.1 W/cm2, which is lower than those in the previous figure, and of the order of sunlight flux at sea level. This keeps our model treatment within a linear regime and is consistent with our equations.



Figure 4. (A) Surface photovoltage of atomic slab models of Si(111):H surface with 0-, 1-, and 3-Ag atoms adsorbed, represented by long dash (green), full (red), and short dash (blue) lines, respectively, for slabs with eight monatomic layers of Si. Extrema of the SPV are labeled as (a)-(c), (a′)-(d′), and (a′′)-(c′′) for 0-, 1-, and 3-adsorbed atoms of Ag, respectively. (B)-(D) Photoinduced change of population of KS orbitals of a model with eight monolayers of Si. Orbitals belonging to the CB or VB have positive or negative indices, respectively. Populations are shown for transition energies corresponding to features (a)-(d), (a′)-(d′), and (a′′)-(c′′) as labeled. For each orbital, the length of a bar shows the population gain (for i > 0) or loss (for i e 0). (B) No adsorbates. Arrows point to a representative pair of orbitals (HOMO-1 f LUMO+21 and HOMO-2 f LUMO+27), which exchange electron population at photon energies of 1.59 eV (a) and 1.90 eV (b), respectively. (C) One adsorbed Ag atom. Arrows point to the representative pairs of orbitals (HOMO f LUMO+11 and HOMO-3 f LUMO+19), which exchange population at photon energies of 1.35 eV (a′) and 1.75 eV (b′), respectively. (D) Ag3 cluster adsobate. Arrows point to a representative pair of orbitals (HOMO f LUMO and HOMO - 4 f LUMO), which exchange population at photon energies of 0.74 eV (a′′) and 1.14 eV (b′′), respectively.

The first observation is that the maximum value of the photovoltage rises in a nonlinear fashion with the number of Ag-atoms. The second observation is that the presence of metallic adsorbates tends to increase the positive value of the photovoltage. For a single silver atom adsorbate, there are two trends: (i) A global minimum at 2.7 eV shifts to the red as compared to no adsorbate. (ii) A new maximum appears at the edge of the IR/visible boundary. The value of this maximum increases with the number of adsorbed Ag atoms. To understand these important features, we analyze the photoinduced change of populations of KSOs as functions of the size of adsorbates, as presented in Figure 4B-D. A general observation is that the presence of Ag adsorbates gives photoinduced population changes at smaller energies, closer to the band gap. In all three systems, a photoexcitation involves the HOMO or the next nearest orbitals in the VB. Yet for the CB the lowest transitions for 0-, 1-, 3-atom adsorbates involve the following orbitals of CB: LUMO+21, LUMO+11, LUMO, respectively. Evidently, the presence of Ag adsorbates introduces optical transitions at lower transition energies. Each pair of orbitals, whose populations are affected by photexcitation as presented in Figure 4B-D, contribute to the net SPV. The amount of this contribution depends on whether a given orbital’s electron density in the pair protrudes in or out of the silicon surface. Analyses of the lowest photoexcitated pairs of orbitals

for our atomic models with 0-, 1-, and 3-atom adsorbates are presented in Figures 5, 6, and 7, respectively. D. Classification of Excitation Features. Figure 5 presents pairs of orbitals for no adsorbate on the Si:H surface that contribute to the features (a) and (b) of Figure 4A. According to the effective mass model,44 the KSO can be represented as a product of a lattice periodic function and an envelope function. The envelope factor appears due to the spatial constraints imposed by the surfaces of the slab, which yield restrictions to the allowed values of the wavevector k. It describes quantization in the direction of confinement and is similar to the eigenstates of the particle in a box of the relevant shape, in our case, a 1D box for a slab. We have found in our calculations that orbitals localized on Si atoms develop such a spatial pattern of electron density, with a progression of charge density maxima and minima associated with quantization in the direction perpendicular to the slab surface. For a slab with NL layers separated by a distance a and confined by surfaces of planes (111) perpendicular to a unit vector nz, the envelope functions Φm show m ) 1,2,3... lobes, and the KSOs φi(r) can be interpreted as a product of two factors:

φi(r) ) uA,m(r)Φm(r)

(12)

where the lattice periodic function uA,m(r) and slow changing envelope function Φm(r) obey the following properties regarding

3538 J. Phys. Chem. C, Vol. 113, No. 9, 2009

Kilin and Micha

a translation r ) µa + r′, with a ) anz and µ ) (1, (2,..., along lattice vector a: uAm(µa + r′) ) uAm(r′) and Φm(µa + r′) ≈ Φm(µa), when |r′| < |a|. This gives φi(r) ) uA,m(r′)Φm(µa) for orbital i ) (A,m). Approximating envelope functions as Φn(X,Y,Z) ≈ √2 ⁄L sin(πnZ/L), with L ) aNL, and with i ) (A,m) and j ) (B,n), the integration

∫ d3rφi*(r)rφj(r)

Dij ) - e ) -e

∑ ∫V d3r′φi*(µa + r)(µa + r)φj(µa + r) (13) µ

c

where Vc is a cell volume, yields the transition dipole bulk Dij ) DAm,Bn = DAB δmn + ∆Am,Bn

eL (-1)|m-n| - 1 nz 2π |m - n|2

(14) with ∆Am,Bn ) d an overlap integral, up to a phase factor. Only the first term is taken into account in a standard effective mass theory without confinement. However, additional transitions are activated by quantum confinement, as described by the second term, and the additional term in the transition dipole is nonzero for |m - n| ) 1, 3, 5, etc., so the dipole matrix elements for orbitals m ) n ( 1 have maximal values. The first term imposes selection 3

ru*AmuBn

rules coinciding with those for bulk phase of Si, typically requiring conservation of the k-vector in initial and final states. The second term allows transitions between VB and CB KSOs with nonequal indices m, n of the envelopes. Figure 5 shows two pairs of transitions, with envelopes displaying different number of lobes. A first pair, with a transition from a VB KSO orbital with a one-lobe envelope to a CB KSO with two-lobes envelope, is found in Figure 5A-C. A second pair, from a VB KSO with two-lobes envelope to a CB KSO with three-lobes envelope, is seen in Figure 5D-F. The spatial properties of the envelope functions of the KSOs determine the sign of the photovoltage. For the transition from the one-lobe envelope to the two-lobe envelope, charge density migrates from the inner layers of the slab toward outer layers providing the positive sign of the SPV as one can see in feature (a) of Figure 4A. By contrast, the transition from the two-lobe envelope to the three-lobe envelope is associated with the charge density migration inward of the slab and yields negative contribution to the SPV, as seen in feature (b) of Figure 4A. The presence of Ag adsorbates adds additional transitions, which do not obey the above-mentioned selection rules. These additional transitions often have higher transition probability and provide migration of a larger amount of charge density as compared to the Si:H-specific transitions. Figure 6

Figure 5. Active orbitals of featured excitations of an eight layer (192 atoms) Si:H-slab showing typical charge density redistributions due to photoexcitation. Panels A,B represent the electron density of orbitals HOMO-1 and LUMO+21 with isosurfaces (gray) in three dimensions. Red, green spheres stand for Si-, H-atoms. Panel C gives charge densities of HOMO-1 (solid red) and LUMO+21 (green dashes) as functions of the z-coordinate (vertical axis), resulting from integration over x- and y-coordinates. Panels D-F show the same for orbitals HOMO-2 and LUMO+27. Charge densities of these orbitals display quantization in the (111) direction for the finite slab. (A) shows the upper half of a one-lobe envelope function, (B) and (D) show upper halves of the two-lobe envelope functions for different symmetry of the atomic part of the wave function, and (E) shows the upper half of a three-lobe envelope function. For interpretation of the color code, the reader is referred to the web version of this Article.



Figure 6. Active orbitals of a single Ag adsorbed on the 192-atoms Si slab showing typical charge density redistributions due to photoexcitation. Panels A,B represent the electron density of orbitals HOMO and LUMO+11 with isosurfaces (gray) in three dimensions. Red, green, blue spheres stand for Si-, H-, Ag-atoms. Panel C gives charge densities of HOMO (solid red) and LUMO+11 (green dashes) as functions of the z-coordinate (vertical axis), resulting from integration over x- and y-coordinates. Panels D-F show the same for orbitals HOMO-3 and LUMO+19. An electronic part of the transition (A)-(C) shows localization of charge density in the vicinity of the Ag cluster and a hybridization between surface and adsorbed Ag cluster. The slab orbital shows d-symmetry for the Ag atomic orbital and p-symmetry for the Si component. There is a nodal plane between the Ag atomic orbital and a neighboring Si atomic orbital reminding of the antibondig character of this slab orbital. For interpretation of the color code, the reader is referred to the web version of this Article.

illustrates active KSOs for one Ag adsorbate and the SPV features (a′) and (b′) in Figure 4A. The first transition involves a one-lobe envelope in the VB and a Ag-localized orbital in the CB. This corresponds to a photoinduced charge density migration outward of the Si slab surface and produces a positive contribution to the SPV at location (a′) in Figure 4A. The second transition shown in Figure 6 is similar to a situation with no surface adsorbates and shows transition from a KSO with a two-lobe envelope to a KSO with a one-lobe enevelope. This yields charge density migration inward of the Si slab and a negative sign contribution to the net SPV, as in feature (b′) in Figure 4A. For AgSi(111):H, both Si-localized and Ag-localized KSOs contribute to the formation of the resulting SPV signal. For Ag3Si(111):H, with a higher percentage of surface coverage by Ag, the majority of photoactive KSO-pairs have at least one Ag-localized or hybridized orbital. Figure 7 illustrates features

(a′′) and (b′′) of the SPV signal presented in Figure 4A. Both transitions start from VB KSO with a one-lobe envelope and end up in a Ag3-localized KSO at the edge of the CB. The envelope functions of VB orbitals from each pair coincide. The only difference between them is the symmetry of the atomic part of the wave function, as is clearly seen in Figure 7. Both orbital pairs provide charge density migration outward of the Si slab surface, and, therefore, they both provide positive contribution to the net SPV signal in agreement with features (a′′) and (b′′) in Figure 4A. The larger cluster acts in three ways: (i) it creates new optically active states yielding transitions at lower photon energy; (ii) it increases the number of carriers, because it shifts EFermi; and (iii) it facilitates charge rearrangement on the surface. As a result, the global maximum of the SPV gets enhanced with the size of the cluster.

3540 J. Phys. Chem. C, Vol. 113, No. 9, 2009

Kilin and Micha

Figure 7. Active orbitals of of the Ag3 cluster adsorbed on the 192-atom Si slab showing typical charge density redistributions due to photoexcitation. Panels A,B represent the electron density of orbitals HOMO and LUMO with isosurfaces (gray) in three dimensions. Red, green, blue spheres stand for Si-, H-, Ag-atoms. Panel C gives charge densities of HOMO (solid red) and LUMO (green dashes) as functions of the z-coordinate (vertical axis), resulting from integration over x- and y-coordinates. Panels D-F show the same for orbitals HOMO-4 and LUMO. The slab orbitals show localization of charge density in the vicinity of the Ag cluster and a hybridization between surface and adsorbed Ag cluster. For interpretation of the color code, the reader is referred to the web version of this Article.

IV. Conclusions We have presented a new theoretical and computational treatment of surface photovoltages induced by charge transfer at semiconductor surfaces. Our method combines a density matrix formalism and ab initio electronic structure calculations, in a procedure that uses the time average of reduced density matrix elements and incorporation of relaxation phenomena. The method has been implemented with ab initio electronic structure results, which start from the known atomic conformation of a surface and provide a basis set of Kohn-Sham orbitals and their energies, and transition electric dipoles for excited electronic states. It allows calculation of important experimental observables such as the surface photovoltage and electronic charge redistribution at a semiconductor surface.

Here, we have applied the method to amorphous and crystalline silicon chemically prevented from surface reconstruction by H-adsorption. Our calculations reproduce features of measured photovoltage for a-Si. Specifically, it allows identification of photon energies at which there appear maximal values of photovoltage for H-terminated amorphous Si. This provides credibility to the calculated results for Si with adsorbed Ag atoms, for which no experimental values are available. Results of our calculations agree with experimentally observed trends for silicon surface. The SPV spectrum of a-Si shows three main features as compared to those of c-Si: (i) The SPV spectrum shifts toward the IR range; (ii) all peaks become broader; and (iii) there is better correspondence to experiment. Our calculated values reproduce features of experimental results for photovoltage versus light wavelength.

Surface Photovoltage at Nanostructures on Si Surfaces Our calculations predict trends regarding the influence of surface adsorbates on features of photovoltage spectra. One of the significant findings of this work is to reveal the role of varying coverage of the Si surface by Ag atoms, as a result of varying size of adsorbed Ag clusters. Specifically, increased Ag coverage on the Si surface promotes optical transitions at lower photon energy and facilitates charge transfer beyond the geometrical boundary of the surface. All of these conclusions are supported by analyses of the spatial distribution of Kohn-Sham orbitals whose population is affected by optical excitation. These reveal patterns of charge distributions, showing whether excited orbitals are localized, and what their patterns are inside a slab lattice. The underlying consequences of photoinduced surface charge transfer have been analyzed and understood within concepts of charge rearrangement and spatial confinement, introducing the role of orbital envelopes within a slab. A limitation of the present treatment is that specific dephasing and depopulation rates have not been calculated. These depend on the coupling of electronic and nuclear motions and could be calculated with an extension of the present procedure, allowing for atomic motions and (nonadiabatic) interaction with electrons48-50 driven by electron-phonon couplings. Such a treatment would take into account (i) the redistribution of electronic populationovervariousexcitedstatesasaresultofelectron-phonon interaction; (ii) vibrational reorganization as photoinduced polarization changes the equilibrium position of the medium; and (iii) decrease of the Coulomb interaction of electrons and holes as a result of the surrounding medium polarization. Another aspect requiring consideration is whether functionals such as GGA, which miss the long-range 1/r potential on electrons, are suitable for studies of electronic rearrangement and charge separation. Our approach has been to use the basis set of Kohn-Sham orbitals to expand the reduced density operator and to derive coupled equations of motion. This provides insight on electronic charge distributions and rearrangement upon excitation, but we do not give significance to individual elements of the density matrix. Instead, these are used to calculate average properties such as the surface electric dipole. The equations of motion contain relaxation times to provide the main features of the dissipative interaction with a medium at given temperature. This basis set approach would allow us to account for delayed dissipation with memory terms, and with charge separation in models involving coupling of many-electron state. These aspects could be incorporated in a more detailed density matrix treatment and goes beyond the present work. An alternative we briefly mentioned in subsection III.B is to use TDDFT, or the GW method. Another would be to employ a more advanced density functional theory, which would account for relaxation and correct long-range electron potentials. This would also allow for excitonic effects but is clearly beyond our present aims. We have performed a systematic study of trends of silicon’s photophysical properties as its surface is altered by adsorbed metallic clusters. It appears that, in general, metallic cluster adsorbates create new optically active states in the density of states yielding optical transitions at lower photon energy as compared to pure Si. As a consequence, it can be concluded that metal cluster adsorbates enhance light absorption and increase charge transfer at the surface. The treatment we have developed can be applied to other materials of interest in photovoltaic applications.

J. Phys. Chem. C, Vol. 113, No. 9, 2009 3541 Acknowledgment. This material is based upon work supported by grant CHE-0607913 from the National Science Foundation. References and Notes (1) Shah, A.; Torres, P.; Tscharner, R.; Wyrsch, N.; Keppner, H. Science 1999, 285, 692. (2) Chapin, D. M.; Fuller, C. S.; Pearson, G. L. J. Appl. Phys. 1954, 25, 676. (3) Wang, L. X.; Willig, F.; May, V. J. Chem. Phys. 2006, 124, 014712. (4) Stier, W.; Prezhdo, O. V. J. Phys. Chem. A 2002, 106, 8047. (5) Rego, L. G. C.; Batista, V. S. J. Am. Chem. Soc. 2003, 125, 7989. (6) Thoss, M.; Kondov, I.; Wang, H. Chem. Phys. 2004, 304, 169. (7) Green, M. A. Sol. Energy 2004, 76, 3. (8) Goetzberger, A.; Hebling, C.; Schock, H. W. Mater. Sci. Eng., R 2003, 40, 1. (9) Pillai, S.; Catchpole, K. R.; Trupke, T.; Green, M. A. J. Appl. Phys. 2007, 101, 093105. (10) Kilin, D. S.; Prezhdo, O. V.; Xia, Y. N. Chem. Phys. Lett. 2008, 458, 113. (11) Kronik, L.; Shapiro, Y. Surf. Sci. Rep. 1999, 37, 1. (12) Haase, G. Int. ReV. Phys. Chem. 2001, 19, 247. (13) Fefer, E.; Shapira, Y.; Balberg, I. Appl. Phys. Lett. 1995, 67, 371– 373. (14) Burstein, L.; Shapira, Y.; Partee, J.; Shinar, J.; Lubianiker, Y.; Balberg, I. Phys. ReV. B 1997, 55, R1930–R1933. (15) Takayanagi, K.; Tanishiro, Y.; Kajiyama, K. J. Vac. Sci. Technol., B 1986, 4, 1074–1078. (16) Sˇtich, I.; Payne, M. C.; King-Smith, R. D.; Lin, J. S.; Clarke, L. J. Phys. ReV. Lett. 1992, 68, 1351–1354. (17) Marsi, M.; Couprie, M. E.; Nahon, L.; Garzella, D.; Hara, T.; Bakker, R.; Billardon, M.; Delboulbe’, A.; Indlekofer, G.; Taleb-Ibrahimi, A. Appl. Phys. Lett. 1996, 70, 895. (18) Cahill, D. G.; Hamers, R. J. Phys. ReV. B 1991, 44, 1387. (19) Natan, A.; Zidon, Y.; Shapira, Y.; Kronik, L. Phys. ReV. B 2006, 73, 193310. (20) Deutsch, D.; Natan, A.; Shapira, Y.; Kronik, L. J. Am. Chem. Soc. 2007, 129, 2989. (21) Krasnoslobodtsev, A.; Smirnov, S. J. Phys. Chem. B 2006, 110, 17931–17940. (22) Patel, B. K.; Rath, S.; Sahu, S. N. Physica E 2007, 33, 268. (23) Nozik, A. J. Annu. ReV. Phys. Chem. 2001, 52, 193. (24) Rohlfing, M.; Louie, S. G. Phys. ReV. B 2000, 62, 4927. (25) Bylaska, E. J.; Tsemekhman, K.; Gao, F. Phys. Scr., T 2006, 124, 86. (26) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford Univ. Press: Oxford, 1995; Chapter 3. (27) May, V.; Ku¨hn, O. Charge and Energy Transfer Dynamics in Molecular Systems; Wiley-VCH: Berlin, 2000; Chapter 3. (28) Micha, D. A.; Santana, A.; Salam, A. J. Chem. Phys. 2002, 116, 5173. (29) Martin, R. M. Electronic Structure: Basic Theory and Practical Methods; Cambridge University Press: Cambridge, MA, 2004; Chapters 12 and 13. (30) Kilin, D. S.; Tsemekhman, K.; Prezhdo, O. V.; Zenkevich, E. I.; von Borczyskowski, C. J. Photochem. Photobiol., A 2007, 190, 342. (31) Kilin, D.; Micha, D. A. Chem. Phys. Lett. 2008, 461, 266. (32) Ramakrishna, S.; Willig, F.; May, V. Phys. ReV. B 2000, 62, R16330. (33) Schatz, G. C.; Ratner, M. A. Quantum Mechanics in Chemistry; Prentice-Hall: Englewood Cliffs, 1993; Chapter 11. (34) Sargent, M., III; Scully, M. O.; Lamb, W. E. Laser Physics; Addison-Wesley: London, England, 1974. (35) Cohen-Tannoudji, C.; Dupont-Roc, J.; Grynberg, G. Atom-Photon Interactions; Wiley: New York, 1992; Chapter V. (36) Apanasevich, P. A.; Kilin, S. Y.; Nizovtsev, A. P.; Onishchenko, N. S. J. Opt. Soc. Am. B 1986, 3, 587. (37) Kilin, D.; Kleinekatho¨fer, U.; Schreiber, M. J. Phys. Chem. A 2000, 104, 5413. (38) Kresse, G.; Furtmu¨ller, J. Phys. ReV. B 1996, 54, 11169. (39) Vanderbilt, D. Phys. ReV. B 1990, 41, 7892. (40) Perdew, J. P.; Wang, Y. Phys. ReV. B 1992, 45, 13244. (41) Kamisaka, H.; Kilina, S. V.; Yamashita, K.; Prezhdo, O. V. Nano Lett. 2006, 6, 2295. (42) Weinelt, M.; Kutschera, M.; Fauster, T.; Rohlfing, M. Phys. ReV. Lett. 2004, 92, 126801. (43) Beard, M. C.; Knutsen, K. P.; Yu, P. R.; Luther, J. M.; Song, Q.; Metzger, W. K.; Ellingson, R. J.; Nozik, A. J. Nano Lett. 2007, 7, 2506.

3542 J. Phys. Chem. C, Vol. 113, No. 9, 2009 (44) Efros, A. L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D. J.; Bawendi, M. Phys. ReV. B 1996, 54, 4843. (45) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Saunders: New York, 1976. (46) Kilina, S.; Tretiak, S.; Ivanov, S.; Klimov, V. I. submitted to J. Am. Chem. Soc. (47) Vardeny, Z.; Tauc, J. Phys. ReV. Lett. 1985, 54, 1844–1847.

Kilin and Micha (48) Tully, J. C. J. Chem. Phys. 1990, 93, 1061. (49) Micha, D. A. J. Phys. Chem. A 1999, 103, 7562. (50) Duncan, W. R.; Craig, C. F.; Prezhdo, O. V. J. Am. Chem. Soc. 2007, 129, 8528.

JP808908X

Surface Photovoltage at Nanostructures on Si Surfaces - American

Recommend Documents