Theory and simulation of photogeneration and transport in Si-SiO x superlattice absorbers
© Aeberhard; licensee Springer. 2011
Received: 19 September 2010
Accepted: 21 March 2011
Published: 21 March 2011
Si-SiO x superlattices are among the candidates that have been proposed as high band gap absorber material in all-Si tandem solar cell devices. Owing to the large potential barriers for photoexited charge carriers, transport in these devices is restricted to quantum-confined superlattice states. As a consequence of the finite number of wells and large built-in fields, the electronic spectrum can deviate considerably from the minibands of a regular superlattice. In this article, a quantum-kinetic theory based on the non-equilibrium Green's function formalism for an effective mass Hamiltonian is used for investigating photogeneration and transport in such devices for arbitrary geometry and operating conditions. By including the coupling of electrons to both photons and phonons, the theory is able to provide a microscopic picture of indirect generation, carrier relaxation, and inter-well transport mechanisms beyond the ballistic regime.
Si-SiO x superlattices have been proposed as candidates for the high band gap absorber component in all-Si tandem solar cells [1, 2]. In these devices, photocurrent flow is enabled via the overlap of states in neighboring Si quantum wells separated by ultra-thin oxide layers, i.e., unlike in the case of an intermediate band solar cell, the superlattice states contribute to the optical transitions and, at the same time provide transport of photocarriers, which makes it necessary to control both the optical and the transport properties of the multilayer structure. To this end, a suitable theoretical picture of the optoelectronic processes in such type of structures is highly desirable.
There are several peculiar aspects of the device which require special consideration in the choice of an appropriate model. First of all, a microscopic model for the electronic structure is indispensable, since the relevant states are those of an array of strongly coupled quantum wells. In a standard approach, these states are described with simple Kronig-Penney models for a regular, infinitely extended superlattice . The superlattice dispersion obtained in this way can then be used for determining an effective density of states as well as the absorption coefficient to be used in macroscopic 1 D solar cell device simulators. However, depending on the internal field and the structural disorder, the heterostructure states may deviate considerably from regular minibands or can even form Wannier-Stark ladders. Furthermore, the charge carrier mobility, which has a crucial impact on the charge-collection efficiency in solar cells, depends on the dominant transport regime at given operating conditions, which may be described by miniband transport, sequential tunneling, or Wannier-Stark hopping , relying on processes that are not accessible to standard macroscopic transport models.
In this paper, the photovoltaic properties of quantum well superlattice absorbers are investigated numerically on the example of a Si-SiO x multilayer structure embedded in the intrinsic region of a p-i-n diode, using a multiband effective mass approximation for the electronic structure and the non-equilibrium Green's function (NEGF) formalism for inelastic quantum transport, which permits to treat on equal footing both coherent and incoherent transport as well as phonon-assisted optical transitions at arbitrary internal fields and heterostructure potentials.
In order to enable a sound theoretical description of the pivotal photovoltaic processes in semiconductor nanostructures, i.e., charge carrier generation, recombination and collection, both optical transitions and inelastic quantum transport are to be treated on equal footing within a consistent microscopic model. To this end, a theoretical framework based on the NEGF formalism was developed [5, 6] and applied to quantum well solar cell devices. In this article, we reformulate the theory for a multiband effective mass Hamiltonian, similar to [7, 8], and extend it to cover the phonon-assisted indirect transitions that dominate the photovoltaic processes in Si-based devices. Furthermore, in contrast to the former case, both photogeneration and transport processes take place within superlattice states, since escape of carriers to continuum states is not possible due to the large band offsets.
Hamiltonian and basis
consisting of the coupled systems of electrons (Ĥ e), photons (Ĥ γ), and phonons (Ĥ p). Since the focus is on the electronic device characteristics, only Ĥ e is considered here, however including all of the terms corresponding to coupling to the bosonic systems.
where V 0 is the heterostructure potential, and U is the Hartree term of the Coulomb interaction corresponding to the solution of Poisson's equation that considers carrier-carrier interactions (Ĥ ee) on a mean field level.
and V is the absorbing volume.
where r is the electron coordinate, and U Λ,Q are related to the Fourier coefficients of the electron-ion potential .
where ĉ †, and ĉ are single fermion creation and annihilation operators.
The explicit form of the interaction term still depends via U Λ,Q on the specific phonon modes considered and will be detailed in the section on the model implementation.
Green's functions, self energies, and quantum kinetic equations
where 〈...〉 C denotes the contour-ordered operator average peculiar to non-equilibrium quantum statistical mechanics [15, 16] for arguments 1 = (r 1, t 1) with temporal components on the Keldysh contour .
where is the speed of the light in the active medium. The modal photon flux, in turn, is given by the modal intensity of the EM field as .
Since the principal value integral in the expression for the retarded self-energy corresponds to the real part of the latter and thus to the renormalization of the electronic structure, which is both small and irrelevant for the photovoltaic performance, it is neglected in the numerical implementation. A further approximation is made by neglecting the off-diagonal terms in the band index, which means that only incoherent interband and sub-band polarization is considered .
Once the Green's functions and self-energies have been determined via self-consistent solution of Equations 45-48 and 49, 50, they can be directly used for expressing the physical quantities that characterize the system, such as charge carrier and current densities as well as the rates for the different scattering processes.
Microscopic optoelectronic conservation laws and scattering rates
with units [R] = s -1. If we are interested in the interband scattering rate, then we can neglect in Equation 54 the contributions to the self-energy from intraband scattering, e.g., via interaction with phonons, low-energy photons (free carrier absorption), or ionized impurities, since they cancel upon energy integration over the band. Since inequivalent conduction band valleys may be described by different bands, the corresponding inter-valley scattering process has also an interband character with a non vanishing rate, as long as only one of the valleys is considered in the rate evaluation. Furthermore, if self-energies and Green's functions are determined self-consistently as they must be done to guarantee current conservation, the Green's functions are related to the scattering self-energies via the Dyson equation for the propagator and the Keldysh equation for the correlation functions as given in Equations 45-48, and will thus be modified because of the intraband scattering. In the present case of indirect optical transitions, the Greens functions entering the rate for electron-photon scattering between the Γ bands are the solutions of Dyson equations with an intervalley phonon-scattering self-energy and may thus contain contributions from the X-valleys. In the same way, the Γ c Greens functions entering the electron-phonon Γ c - X scattering rate contain a photogenerated contribution. By this way, indirect, phonon-assisted optical transitions are enabled.
Implementation for Si-SiO x superlattice absorbers
Electronic structure model
with the Kane energy E P ≈ 10 eV .
The parameters for intravalley scattering used in the numerical simulation are ρ = 2329 g/cm3, c s = 9.04 × 105 cm/s, and D ac = 8.9322 eV .
For the numerical simulation, an optical deformation potential of D op = 109 eV/cm and a constant phonon energy ħΩ op = 60 meV are used.
Numerical results and discussion
Density of states
Generation and photocurrent spectrum
In this article, an adequate theoretical description of photogeneration and transport in Si-SiO x superlattice absorbers was presented. Based on quantum kinetic theory, the formalism allows a unified approach to both quantum optics and inelastic quantum transport and is thus able to capture pivotal features of photogeneration and photocarrier extraction in Si-based coupled quantum well structures, such as phonon-assisted optical transitions and field-dependent transport in superlattice states. Owing to the microscopic nature of the theory, energy-resolved information can be obtained, such as the spectra for photogeneration rate and photocurrent density, which shows that in the case of high internal fields, excess charge is transported via sequential tunneling in the miniband where it is generated.
effective mass approximation
non-equilibrium Green's function.
The financial support for this study was provided by the German Federal Ministry of Education and Research (BMBF) under Grant No. 03SF0352E.
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