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J. Phys. Chem. C 2011, 115, 770–775
Modeling the Photovoltage of Doped Si Surfaces Dmitri S. Kilin† and David A. Micha* Quantum Theory Project, Departments of Chemistry and of Physics, UniVersity of Florida, GainesVille, Florida 32611-8435, United States ReceiVed: NoVember 10, 2010
Doped Si(111) surfaces have been modeled with a Si slab, and their photovoltage has been calculated with a combination of ab initio electronic structure and density matrix treatments from the steady state solution of the electronic density matrix (DM) for electrons interacting with thermalized lattice vibrations. Densities of electronic states and photovoltage spectra of the silicon surfaces are drastically affected by the presence of p or n doping. We report results on the effect of surface doping by group III (B, Al, and Ga) and group V (N, P, and As) elements, of interest in the study of surface optical properties, for concentrations of doping atoms in the host lattice in the range of 0.5-1.5%, obtained from slab atomic models with several hundred atoms. Analysis of the results provides insight on trends relevant to the absorption of near IR, visible, and near UV light and to measurements of photovoltages and shows some of the trends found for doped bulk Si in experiments at lower dopant densities. 1. Introduction Solid crystalline silicon doped with elements of group III and V are important materials in microelectronics, optoelectronics, and photovoltaics, and many recent applications rely on the properties of thin layers of such materials.1 This contribution treats doping of Si near its crystal surface, using theoretical and computational methods to describe its optical properties and how they vary with the type of doping. The subject of doping in bulk crystalline Si (or c-Si) has long been considered with theoretical and experimental methods.2 Doping in bulk c-Si has been studied by theory,3,4 numerical simulations,5,6 and experimental measurements7,8 for bulk concentrations relevant to applications. The goal of the present work is instead to describe optical properties of doped Si surfaces, where the dopant concentration is larger, and how these properties vary with the type of doping. Doping at an interface with Si has been studied numerically with a focus on the doping-induced lattice distortion,9 and doping in nanoclusters of Si has also been studied by numerical simulations.10-13 Our recent related work has modeled Si surfaces with Si slabs, obtaining optical properties and photovoltages with a combined density matrix (DM)-ab initio electronic structure treatment, for c-Si and amorphous Si (or a-Si).14-21 In these papers, the photovoltage has been obtained from the steady state solution of the electronic DM for electrons interacting with thermalized lattice vibrations. These provide dissipative terms in the equation of motion (EOM) for the reduced DM describing relaxation rates. The steady state solutions of the EOM are obtained in a basis of Kohn-Sham orbitals (KSOs) generated by a DFT Hamiltonian solving the K-S equations with periodic boundary conditions and a basis set of plane waves. This approach is applied here to doped slabs, with dopants from group III and V elements in the Periodic Table. * To whom correspondence should be addressed. E-mail: micha@ qtp.ufl.edu. † Present address: Department of Chemistry, University of South Dakota, 414 E. Clark St., Vermillion, SD 57069.
Basic features of doping in our systems can be understood from simple hydrogenic models of the electrons or holes added to the structures of present interest. An atomic model of doping can be constructed for a perfect pure Si lattice if one replaces one lattice atom with an atom from groups III or V. The added electron or hole is localized around the dopant core, and its energy levels can be obtained as hydrogenic solutions for a single dopant.22 Hydrogenic energies Ekp-dop and Ekn-dop are extracted from experiment by measuring the absorption coefficient as a function of photon frequency in the near IR range, at low temperatures (4 K). Sharp peaks are obtained at frequencies in the IR region corresponding to energy separation between the ground state and the excited hydrogenic states. Such experimental methods are accurate for donors and acceptors in bulk Si and Ge.7,8 In what follows, we present results for densities of state (DOSs) and photovoltages for dopants B, Al, Ga, Si, N, P, and As in slabs of c-Si with 10 layers, with more details presented here for dopants Al and P, and compare trends in our results with those available in the literature from experiments in bulk systems with lower doping densities. 2. Model and Equations of Motion for the DM Our treatment starts from an atomic model of slabs with a chosen number of Si layers to construct a Hamiltonian with doping terms and EOMs for the electronic density operator including relaxation terms. The electronic structure is described at the density functional (DFT) level with a generalized gradient functional (PW91) and a large basis set of plane waves. This gives energies and shapes of KSOs as eigenfunctions of the K-S effective Hamiltonian FKS with eigenenergies εj and orbitals labeled as |j〉 ) |φj〉. The orbitals are generated with periodic boundary conditions suitable for supercells replicating the slabs. Details can be found in refs 14, 23, and 24. The DFT treatment that we have chosen has been considered in some detail in our previous calculations on Si slabs.15 It is adequate for intramolecular electron transfer of the type that we are considering and for our system, which is finite insofar in that it describes electronic states within a supercell in a slab.
10.1021/jp110756u 2011 American Chemical Society Published on Web 12/20/2010
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Furthermore, we use the generated electronic orbitals as a basis set and calculate properties as sums over states, so that the treatment does not demand accuracy for each orbital but only a reasonably complete basis set. Indicating the K-S effective Hamiltonian operator of pure Si with Fˆ0KS, the model Hamiltonian for the doped slab must KS derived for the electronic changes due to a include a term Fˆdop KS for the induced dopant in the host lattice and a term Fˆdist distortion of the lattice, so that
ˆ KS ˆ KS FˆKS ) FˆKS 0 + Fdop + Fdist
(1)
Further considering the formation of an excited state at a doped silicon surface continuously irradiated by light with an electric field E(t) ) E0 cos (Ωt) of frequency Ω and constant amplitude, consistent with an experimental setup, the total timeˆ · E(t), in terms of the dependent Hamiltonian is Fˆ ) FˆKS - D ˆ dipole operator D. The evolution of the electronic-reduced density operator can conveniently be described in a basis set of KSOs, with matrix elements Fjk satisfying an EOM with a dissipative term (dFjk/dt)diss that can be derived from electronic transition rates generated by lattice vibrations.15,25-28 It is convenient to transform the EOM of the DM going from the Schroedinger picture (SP) of quantum mechanics to the interaction picture (IP), to introduce the form F(I) jk and to simplify the new EOM by taking an average of its terms over times long as compared with Ω-1. This justifies dropping from the EOM rapidly oscillating terms with frequencies |ωjk| + Ω and rapidly varying dissipative terms. Returning to the SP and further introducing new (rotating frame) elements F˜ jk ) Fjk exp (isjkΩt), with sjk ) sign(εj - εk), leads to a simplification of the EOM for the RDM, which reads in the rotating frame15,25
averaging to zero to first order, but it gives to second order a contribution to the averaged dynamics. The square of the Vjk nonadiabatic coupling that appears in the rate κjk gets replaced by the semiclassical expression
|Vjk | 2 )
p2 Nt
∑ (|〈j| ∂t∂ |k〉|2)n
(4)
n
where the summation is over Nt time steps along a trajectory {qξ(t), pξ(t)} run for the lattice at thermal equilibrium, and we have used ∂/∂t ) ∑ξ(∂/∂qξ) · dqξ/dt. The initial values for the RDM correspond to the system initially at thermal equilibrium and then excited by light, during a very long time. The thermal equilibrium state in our electron system is specified by the Fermi-Dirac distribution Feq jk ) δjkfFD(εj; T) as applies to our system at equilibrium at temperature T before excitation. A system driven by steady light of a given frequency shows two competing tendencies that establish a steady state: Light promotes electronic population from the valence band (VB) to the conduction band (CB), and interaction with the medium returns the population back to the VB. Furthermore, interaction with the medium over long times damps rapidly oscillating coherences. In such cases, as long as the intensity of irradiation is kept constant, it is possible to search for solutions of the EOM ˜ ij(t) for times t . γ-1 with steady state values F˜ ss ij ) limtf∞F ij , which can be obtained imposing the condition ∂F˜ ij/∂t ) 0. In the limit of weak optical field intensity for which |gjk| , γjj, a Taylor series expansion of matrices leads to
F˜ jjss ) F˜ jjeq + γjj-1
∑ gjk(F˜ eqkk - F˜ jjeq)
(5)
k
F ˜˙ jj )
i 2
∑ Ωjk(F˜ kj - F˜ jk) - γjj(F˜ jj - F˜ jjeq)
(2)
k
F ˜˙ jk ) -i∆jkF˜ jk - iΩjk(F˜ kk - F˜ jj) - γjk(F˜ jk - F˜ jkeq), j * k (3) where γjj ) ∑k*jκjk, γjk ) ∑l(κjl + κlk)/2 + γ0jk are population and coherence relaxation rates and κjk ) (1/p2) |Vjk|2 J(ωjk) |fBE(ωjk, T)| is a state-to-state transition rate. Here, Ωjk ) -Djk · E0/p, and ∆jk ) Ω - pωjk stands for Rabi and detuning frequencies respectively, with pωjk ) εj - εk giving electronic transition energies. Furthermore, J(ω) ) Σξδ(ω - ωξ), fBE(ω), and T stand for the spectral density of phonons in modes ξ, their thermal distribution at vibrational frequencies ω, and temperature, respectively, while γ0jk stands for a pure dephasing component.27 Additional details are described in refs 25, 29, and 30. The electronic potential coupling Vjk appearing in the stateto-state rates arises from transitions induced on electronic states by their coupling to atomic vibrations. These are relatively slow and with small excitation energies, and we therefore proceed with a quantum-classical treatment where atomic motions are obtained from classical trajectories. The average over an initial thermal distribution of vibrations, with small excitation energies as compared with electronic transitions energies, allows us to approximate the square of the electron-phonon coupling potential energy with a classical average obtained from an ab initio molecular dynamics, which treat the vibrational motions as classical. This leads to an electron-phonon interaction
F˜ kjss ) iγjkΩjk(γjk2 + ∆jk2 )-1(F˜ ss ˜ jjss), k * j kk - F
(6)
with gjk(Ω) ) γjkΩ2jk[γ2jk + ∆jk(Ω)2]-1, and details are given in ref 15. These steady state populations provide properties such as the photoinduced steady state change of population distribution for the excited electronic system
∆nss(ε, Ω) ) nss(ε, Ω) - neq(ε) nss(ε, Ω) )
(7)
∑ δ(εj - ε)F˜ jjss(Ω)
(8)
∑ δ(εj - ε)F˜ jjeq
(9)
j
neq(ε) )
j
and the surface photovoltage (SPV) VSPV ) Vs(Ω) - Vs(0)14 constructed from
j s(Ω) ) [εrε0A]-1 V
{∑ σ,iσ
CσZiσ - e
}
∑ F˜ iiss(Ω)〈i|z|i〉 i
(10)
with A the surface area of the simulation supercell, Cσ and Ziσ the atomic core charge and position in the lattice, and 〈i|z|i〉 an j SPV(Ω) is obtained with average electronic position, so that V respect to the voltage of the nonexcited system. The specifics of electronic structure of doped slabs can be analyzed using density of states (DOS). DOS d(ε) ) ∑jδ(εj -
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Kilin and Micha
Figure 1. Schematic representation of the doped silicon band structure, showing the CB and VB and energy levels introduced by the dopants. The energies of HO and LU are indicated with arrows and labeled as HOIII, LUIII, HOIV, LUIV, and HOV, LUV, for group III ) B, Al, and Ga; group IV ) Si; and group V ) N, P, and As. Levels of filled orbitals are symbolized by solid lines, and levels of empty orbitals are symbolized by dashed lines. Dopant levels are labeled by the principal quantum number, n ) 1, 2, 3... according to the hydrogenic model. The bonding energy of an electron (or hole) at a dopant site is labeled as Eb.
ε) follow directly from the K-S eigenenergies. Comparing the DOSs d0 and d for pure and doped Si slabs, it is possible to identify changes due to electronic contributions from dopants and those due to lattice distortion, with
d(ε) - d0(ε) ) ∆d(ε) ) ∆ddop + ∆ddist
(11)
The DOSs d0(ε) versus ε shows a null region typical of the gap between VB and CB of the Si crystal.14 The additional DOS due to a dopant, the larger change, can be interpreted with a simple hydrogenic model based on the effective (screened) potential energy around the dopant atom, so that ∆ddop = ∆dhydr. The dopant atomic core is charged and sets up a Coulomb KS = Uhydr(r) ) ce/(εSiε0r), with εSi the dielectric potential Fdop constant of the medium that adds to the pure Si electronic potential. Depending on the type of doping, an electron of effective mass m*e or a hole of effective mass m*h moves in the potential Uhydr(r), in hydrogenic orbitals of energy Ek. The k-th energy level for p doping appears above the VB edge EV as 2 ) EV + EHk m*/(m Ep-dop k h eεSi), while the k-th energy level for n ) EC - EHk m*/ doping appears below the CB edge EC as En-dop k e 2 ). Here, EHk ) -mec4e /(8h2ε02k2) stands for the hydrogen (meεSi atom energy level with principal number k, electron charge ce, and electron mass me, and the superscript H stands for “hydrogenic”. Figure 1 gives a schematic representation of the doped Si band structure, showing the CB and VB and energy levels introduced by the dopants. At a low concentration of doping, the band structures of pure and p- or n-doped Si are similar. The HO (highest occupied) and LU (lowest unoccupied) orbitals do not always coincide with the edges of CB and VB. The energies of HO and LU are indicated with arrows and labeled as HOIII, LUIII; HOIV, LUIV; and HOV, LUV so that a superscript labels the group of elements used for doping. These are groups III ) B, Al, and Ga, IV ) Si, and V ) N, P, and As. Levels of filled orbitals are symbolized by solid lines, and levels of empty orbitals are symbolized by dashed lines. Dopant levels are labeled by the principal quantum number, k ) 1, 2, 3... according to the hydrogenic model. The bonding energy of an electron (or hole) at a dopant site is labeled as Eb. For example, HOIII is occupied by an electron, while LUIII is unoccupied or, alterna-
Figure 2. Atomic model of a thin semiconductor film with a doping atom, constructed from 2Si119X1H24 with X ) B, Al, Ga, Si, N, P, and As. The model contains two mirror images, each with a dopant inside five monatomic layers, with 24 atoms per layer. The figure shows only one-half of the structure. The vertical axis z is perpendicular to the surface and is positive above the surface. Periodic boundary conditions are implied along x and y directions. Dangling surface bonds are compensated by hydrogenation.
tively, LUIII is occupied by a hole that upon excitation moves down into the VB. The presence of a p dopant changes DOS and populations in the VB, which acquires a sequence of hydrogenic levels. Ab initio calculations of energy levels are expected to give DOSs with a similar structure. 3. Results from Ab Initio Electronic Structure and the Steady State DM 3.1. DOSs and SPVs with Doping. The atomic model is presented in Figure 2, which displays one-half of the supercell. Calculations were done for the compositions 2Si119X1H24, where X ) B, Al, Ga, Si, N, P, and As, which contain two symmetric images rotated by 180°, each with five monatomic layers, 24 atoms per layer, and dangling surface bonds compensated by hydrogenation. The vertical axis z indicates 〈111〉 direction and is normal to the exposed (111) surface and is positive above the surface. Periodic boundary conditions are implied along x and y directions. DOS d(ε) and photoexcited population densities n(ε) are shown in Figure 3, where ε ) 0 indicates the electron orbital energy at the top of the VB. Panels a and b compare DOS including dopants d(ε) (lines) with equilibrium population distributions of doped Si neq(ε), (filled area) in models with p and n doping, for Al (a) and P (b), respectively. Arrows indicate the energy locations of doping orbitals. In the Al-doped p-Si, the edge of the VB around ε ) 0.0 eV is underpopulated, with neq(ε) < d(ε) providing hole population, while in the P-doped n-Si, the edge of the CB around ε ) 0.8 eV is overpopulated, neq(ε) > 0 providing electron population. Panels c and d give isocontours of the population changes ∆nss(ε,Ω), see eq 5, of photoexcited steady states relative to the thermal equilibrium populations, calculated for p and n doping. Here, red-, green-, and blue-colored areas label the distribution ∆nss for gain, no change, and loss, respectively, as compared to the thermal populations; red areas can be understood as relating to electrons, and blue ones can be understood as relating to holes. Photons
Modeling the Photovoltage of Doped Si Surfaces
Figure 3. (a and b) Comparison of DOS (lines) with equilibrium population distribution (filled area) in models with p and n doping for Al (a) and P (b), respectively. Here, ε ) 0 indicates the VB maximum. Arrows indicate doping orbitals. (c and d) Isocontours of the population ∆nss(ε, Ω), see eq 8, giving the steady state DOS changes from photoexcitation, calculated for the Si surface with p and n doping for photon energy pΩ. Here, red-, green-, and blue-colored areas label the distribution ∆nss for gain, no change, and loss, respectively, relative to the equilibrium distribution; red areas can be understood as relating to electrons, and blue ones to holes. Photons at energies below pΩ < 1 eV induce intraband transitions: In p-Si, electrons are excited from deep in the VB (blue spot at ε ) -1 eV) to the top of the VB (red spot at ε ) 0 eV), while in n-Si electrons are excited from the bottom of the CB (blue spot at ε ) 1 eV) to higher orbitals of the CB (red spot at ε ) 2 eV). Photons at energies above pΩ > 1 eV induce interband transitions: In both p-Si and n-Si, an electron is excited from the VB (blue area at ε < 0 eV) to the CB (red area at ε > 1 eV).
of energies below pΩ < 1 eV induce intraband transitions. For p-Si, electrons are excited from deep in the VB (blue spot at orbital energy ε ) -1 eV) to the top of the VB (red spot at ε ) 0 eV), while in n-Si, electrons are excited from the bottom of the CB (blue spot at ε ) 1 eV) to higher orbitals of the CB (red spot at 2 eV). Photons at energies around pΩ ) 2 eV induce interband transitions: In both p-Si and n-Si, an electron is excited from the VB (blue area at ε < 0 eV) to the CB (red area at ε > 1 eV). Interestingly, photons of higher energy pΩ induce creation of e-h pairs from a wider range of initial energy levels. The new states are added to the ones for pure Si, which are slightly shifted due to lattice distortion. Therefore, there are new conduction states in the thermal equilibrium populations depending on the nature of the doping and reasons to expect drastic difference in optical absorption and photovoltage formation. For transitions at pΩ > 1 eV, all occupied orbitals are in the VB, while all of the unoccupied orbitals are in the CB. Such interband transitions are available for all models considered here: n-doped Si (III), pure Si (IV), and p-doped Si (V). For pΩ < 1 eV, there appear additional intraband transitions, which involve orbitals contributed by doping and are available only in doped Si. Therefore, the rate of intraband transitions scales as the concentration of doping atoms. Figure 4 shows our calculated SPV spectra from our doped surface model and also experimental data31 on a doped bulk Si crystal surface. Measured SPVs are shown as a function of excitation frequency for n-Si (+ symbols and short dashes) and p-Si (× symbols and long dashes). Lines correspond to our calculations. Theory and experiment are not strictly comparable because our surface calculations have been done for much higher doping density, but some insight can be anyway extracted from a comparison. A high concentration of doping atoms is present in our slabs, of 1 per 120 host atoms. In experimental samples,
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Figure 4. SPV as a function of photon excitation energy for Al-doped p-Si (× symbols and dashed line) and P-doped n-Si (+ symbols and solid line). Lines correspond to our calculations. Symbols to experimental results of ref 31.
this concentration is instead about 10-7, and the intraband contribution vanishes. There are three criteria for comparison of theory and experiments: overall sign of the photovoltage, its value at low excitation energies, pΩ < 1 eV, and its value at high excitation energies, pΩ > 1 eV. The sign of the SPV can be understood by analysis of excitations in the optical range pΩ > 1 eV, which induce interband transitions. For p-Si, we have transitions from orbitals n ) 2, 3 introduced by impurities but excluding n ) 1 since this orbital is unoccupied. Hence, for p-Si, an electron is transferred from an occupied localized impurity orbital in the VB with n g 2 to a delocalized orbital in the CB. This creates an electric field directed inward of the slab and a negative photovoltage. For n-Si, the electron is instead promoted from a delocalized and occupied orbital in the VB to the impuritylocalized orbital n g 2 in the CB. This creates an electric field directed outward of the slab and a positive photovoltage. The behavior appears to apply to both surface and bulk doping, with a change between p and n doping, reversing the sign of the photovoltage. At pΩ < 1 eV, the experimental VSPV ) 0, while our results show VSPV ) (30 mV. Our nonzero values can be attributed to the intraband transitions and high concentration of doping atoms, 1 per 120 host atoms. In experimental samples, this concentration is instead about 10-7, and the intraband contribution vanishes. The difference in values at pΩ > 1.2 eV is likely related to the fact that the photoinduced electric dipole is proportional to the distance over which the electric charge is transferred. In our calculations, the proximity of the doping site to the surface and the small size of the model slab restrict the length of the electron transfer. In experimental samples with low concentration, the dopants are instead free to move large distances and create larger values of dipole and voltage. Additional insight can be gained by observing the shape of dopant orbitals in the slab. 3.2. Shape of Dopant Orbitals. The SPV is contributed by orbitals φi, which can be symbolically split into host orbitals and dopant orbitals with large excitation probability into other orbitals φj as indicated by large transition dipole Dij and large values of the electronic position 〈i|z|i〉. We have selected the orbitals that contribute the most to the SPV to display their shapes. The shapes of dopant orbitals can be extracted from the ab initio calculations, with some of their densities shown in Figure 5. First row panels correspond to CB orbitals, second
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Kilin and Micha TABLE 1: Structural and Electronic Properties for a Model of Composition 2X1Si139H12, with Doping Elements, X ) B, Al, Ga, Si, N, P, and Asa group III B dSi-X (Å) Egap (eV) Etheo (eV)c b Ehydr (eV) b e pωexp b (eV)
Al
group IV Ga
Si
group V N
2.086 2.424 2.413 2.364 2.032 0.760 0.7311 0.7125 0.8103b 0.5153 0.0626 0.0668 0.0640 0.5153d 0.0580 0.0454 0.0708
P
As
2.356 0.7792 0.1973 0.1120 0.04557
2.437 0.7524 0.1738 0.05317
a
Figure 5. Selected densities of K-S orbitals actively involved in photoexcitations and SPV creation. Panels a-c show density isocontours for a slab with dopant Al, and panel c gives electronic density profiles. Here, LUIII is the ground state density of a hole. Panels d-f show density isocontours and profiles with no doping. Panels g-i show density isocontours and profiles for a slab with dopant P. The electron density HOV is one of two nearly degenerate orbitals. First row panels a, d, and g show CB isocontours in space, and second row panels b, e, and h correspond to VB isocontours. Lower row panels c, f, and i show electron densities of KSOs for the VB (red lines) and CB (green lines) as functions of distance in the 〈111〉 direction, normal to the exposed (111) surface.
row to VB orbitals, and third row to profiles of electron densities (green for CB and red for VB) along the z direction. For pure Si, host orbitals |φhost〉 are delocalized as seen in the middle row Figure 5d,e. There, the lowest orbital of the CB is given by LUIV, and the highest orbital of the VB coincides with HOIV and HOIV + 1, which are degenerate and have the same spatial symmetry. There are two progressions of Si-host orbitals HOIV - i and LUIV + j. Because of quantum confinement, the spatial distribution of densities of KSOs is given by an envelope function with an increasing number of nodes as their energies go up. For undoped Si, both HOIV and LUIV have no nodes in the envelope, as illustrated in Figure 5d-f. Consider next orbitals for aluminum-doped p-Si, shown in Figure 5a-c. The orbitals are partly Si-host orbitals |φhost〉 and partly dopant-localized ones, |φdop〉. Here, LUIII is above the edge of the VB and largely a localized |φdop〉-orbital corresponding to a hole ground state. The band edge of the CB is represented by LUIII + 1 (solid green line in Figure 5c) and has a shape similar to LUIV. The envelope function of LUIII (red line in Figure 5c) is localized in the vicinity of the impurity atom, in agreement with a hydrogenic model.22 Consider finally orbitals of phosphorus-doped n-Si, in Figure 5g-i. The band edge of the CB for P-doped n-Si starts with the occupied orbital HOV in Figure 5g (and solid green line in Figure 5i). Figure 5h shows the band edge of the VB as HOV - 1 (short red dashes in Figure 5i), which is degenerate with HOV - 2, with both displaying complementary spatial symmetry, and an envelope with one maximum and no nodes. Orbitals in CBV (a CB with group V dopants) have unique features due to a contribution by a doping |φdop〉, but the structure of VBV (not shown) is very similar to the structure of VBIV since both are composed of Si-host orbitals |φhost〉. The dopant densities versus distance z from LUIII for a hole in its ground state and from HOV for an electron in its ground state provide some information on the size of corresponding hydrogenic orbitals. The dopant orbital φdop,k(r) can be ap0 (r) and proximately factorized into a Bloch wave function uk0
Our results are compared to experiment and to basic estimates within a hydrogenic model for electrons and holes. b The exp experimental value for bulk Si is Egap ) 1.12 eV. c Site bonding energies are defined as Ep-doped ) ε εVB and En-doped ) -εdop + b dop b εCB. d Results for a deep trap, instead of the present shallow one. e Result for low concentration in bulk. 0 (r)Fk(r).2,3 Furthermore, the an envelope Fk(r), as φdop(r) ) uk0 envelope for the ground state k ) 1 is of the form F1(r) ) π-1/2a*-3/2 exp(-r/a*), with a* the effective Bohr radius. Fitting of the densities of LUIII and HOV to this expression gives our = 8 Å and a*(DFT) = 4 Å, which are in a estimates a*(DFT) e h qualitative agreement with hydrogenic model estimates from ) 10.76 Å, and a*(exp) = 5.58 Å.3,32 experiment, a*(exp) e h Alternatively, these radii can be used to calculate electron and hole binding energies to compare with our calculated values in Table 1, from independent values of the dielectric constant and effective masses. Using εSi ) 11.4,33 m*e ) 1.08m0 and m*h ) 0.56m0,33 and the values a*e ) 5.58 Å and a*h ) 10.76 Å, the electron and hole excitation energies are known to be |En-dop k - EV| ) 0.058 eV and are shown in EC| ) 0.112 eV and |Ep-dop k Table 1 together with our calculated values for several dopants. Again, there is qualitative agreement, giving confidence to our calculations. More accurately, the hydrogenic orbitals are distorted by the crystal environment. The shape and energies of dopant orbitals are affected by a ligand field of tetrahedral symmetry.34 The p-orbitals of dopants split into orbitals of A and T characters. For type III dopants, there is only one p-electron, and the analysis is straightforward. For type V dopants, the π3 configuration can also be analyzed to provide a fine structure of levels. This is however beyond our present studies, which are meant to provide average features of light absorption. In addition to the orbitals selected for display in Figure 5, our ab initio calculations have provided the other orbitals generate for the CB and VB with and without dopants. Their shapes provide some guidance on the type of transitions induced by light and what orbitals contribute most to the SPV through the magnitude of transition dipoles, which appear in Fssjk(Ω). This can be done with reference to the shapes of localized dopant orbitals |φdop〉 and delocalized host orbitals |φhost〉. The spatial overlap 〈φdop|φhost〉 relating to dopant-to-host dipole transitions is noticeably smaller than those of host-to-host orbitals ′ 〉 or dopant-to-dopant overlaps 〈φdop|φdop ′ 〉, so that 〈φhost|φhost photoexcitations into orbitals of similar shape are more likely. The magnitude of the SPV depends in addition on the average electronic dipole and can have substantial contributions from both dopant and host orbitals.
4. Conclusions The present work has relied on our previous theory of SPVs, extended to include doping in Si slabs. This has provided what we think is the first atomistic description of surface doping, starting from the atomic composition of our systems and using
Modeling the Photovoltage of Doped Si Surfaces an ab initio treatment of electronic structure. Our treatment shows features of the doped materials similar to the ones usually discussed in terms of energy band theory and the appearance of additional doping levels above the VB and below the CB. However, in addition, it provides a more quantitative description in terms of calculated DOSs and populations. Our calculations also give some justification for simple hydrogenic models of energy levels, as shown in Table 1, and provide the shape of doping orbitals from realistic atomic structure calculations. An analysis of our electronic populations as functions of orbital energies and of photon excitation energies for Al and P doping has furnished a detailed picture of interband and intraband transitions. In particular, it shows that transitions induced by more energetic photons are spread over a larger range of energy levels. The combination of electronic structure calculations with a DM treatment of the steady state induced by light absorption has provided a method suitable for calculation of photovoltages at the doped surfaces. Our calculations give insight on the effect of doping on the magnitude and sign of photovoltages. Comparisons with experiments are not strictly possible because our surface doping involves larger dopant concentrations than the experimental samples with bulk doping of Si. In addition, doping close to the surface creates electronic states distorted by confinement in a slab. It has been nevertheless instructive to compare with doped bulk Si, and we have found that the sign of the photovoltages and the slope of their changes with photon excitation energy are similar. Our calculations go beyond what has been known about doping at surfaces. In particular, they show that confinement effects within slabs are important, and they suggest that experimental studies of doped slabs of varying thickness could provide information on doping populations and light absorption at surfaces. Also, our earlier work comparing photovoltages for crystalline and amorphous slabs has shown that the slab structure has large effects on the wavelength and intensity of light absorption,15,18 and this suggests additional studies for amorphous Si with dopants, with the same theoretical methods. Finally, the methods that we have used are applicable to semiconductor slabs containing both p or n type dopants and adsorbates. It should be possible to control the transfer of electronic charge at the surface by using the change of photovoltage sign with p and n nature of dopants. This could be helpful for improving photovoltaic materials. Acknowledgment. This work has been partly supported by NSF Grants CHE-0607913 and CHE-1011967 and by the Dreyfus Foundation. The High Performance Computing Center of the University of Florida has also provided partial support. We thank C. Stanton, S. Kilina, M. Mavros, J. Ramirez, and D. Arlund for discussions. References and Notes (1) Fan, Q.; Chen, C.; Liao, X.; Xiang, X.; Cao, X.; Ingler, W.; Adiga, N.; Deng, X. M. Spectroscopic aspects of front transparent conductive films for a-Si thin film solar cells. J. Appl. Phys. 2010, 107, 034505. (2) Jones, W.; March, N. H. Theoretical Solid State Physics; Dover Publications: New York, 1985. (3) Pantelides, S. T. The electronic structure of impurities and other point defects in semiconductors. ReV. Mod. Phys. 1978, 50, 797–858. (4) Ramdas, A. K.; Rodriguez, S. Spectroscopy of the solid-state analogs of the hydrogen atoms donors and acceptors in semiconductors. Rep. Prog. Phys. 1981, 44, 1297–1387. (5) Harrison, P. Quantum Wells, Wires, and Dots, 3rd ed.; Wiley: New York, 2009; Chapter 5.
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