Theory and simulation of photogeneration and transport in SiSiO_{ x } superlattice absorbers
 Urs Aeberhard^{1}Email author
DOI: 10.1186/1556276X6242
© Aeberhard; licensee Springer. 2011
Received: 19 September 2010
Accepted: 21 March 2011
Published: 21 March 2011
Abstract
SiSiO_{ x } superlattices are among the candidates that have been proposed as high band gap absorber material in allSi tandem solar cell devices. Owing to the large potential barriers for photoexited charge carriers, transport in these devices is restricted to quantumconfined superlattice states. As a consequence of the finite number of wells and large builtin fields, the electronic spectrum can deviate considerably from the minibands of a regular superlattice. In this article, a quantumkinetic theory based on the nonequilibrium 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 interwell transport mechanisms beyond the ballistic regime.
Introduction
SiSiO_{ x } superlattices have been proposed as candidates for the high band gap absorber component in allSi tandem solar cells [1, 2]. In these devices, photocurrent flow is enabled via the overlap of states in neighboring Si quantum wells separated by ultrathin 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 KronigPenney models for a regular, infinitely extended superlattice [3]. 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 WannierStark ladders. Furthermore, the charge carrier mobility, which has a crucial impact on the chargecollection efficiency in solar cells, depends on the dominant transport regime at given operating conditions, which may be described by miniband transport, sequential tunneling, or WannierStark hopping [4], 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 SiSiO_{ x } multilayer structure embedded in the intrinsic region of a pin diode, using a multiband effective mass approximation for the electronic structure and the nonequilibrium Green's function (NEGF) formalism for inelastic quantum transport, which permits to treat on equal footing both coherent and incoherent transport as well as phononassisted optical transitions at arbitrary internal fields and heterostructure potentials.
Theoretical model
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 phononassisted indirect transitions that dominate the photovoltaic processes in Sibased 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 carriercarrier 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 electronion potential [10].
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 contourordered operator average peculiar to nonequilibrium quantum statistical mechanics [15, 16] for arguments 1 = (r _{1}, t _{1}) with temporal components on the Keldysh contour [16].
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 selfenergy 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 offdiagonal terms in the band index, which means that only incoherent interband and subband polarization is considered [18].
Once the Green's functions and selfenergies have been determined via selfconsistent solution of Equations 4548 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 selfenergy from intraband scattering, e.g., via interaction with phonons, lowenergy 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 intervalley 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 selfenergies and Green's functions are determined selfconsistently as they must be done to guarantee current conservation, the Green's functions are related to the scattering selfenergies via the Dyson equation for the propagator and the Keldysh equation for the correlation functions as given in Equations 4548, 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 electronphoton scattering between the Γ bands are the solutions of Dyson equations with an intervalley phononscattering selfenergy and may thus contain contributions from the Xvalleys. In the same way, the Γ _{ c }Greens functions entering the electronphonon Γ _{ c } X scattering rate contain a photogenerated contribution. By this way, indirect, phononassisted optical transitions are enabled.
Implementation for SiSiO_{ x }superlattice absorbers
Electronic structure model
Interactions
with the Kane energy E _{P} ≈ 10 eV [25].
The parameters for intravalley scattering used in the numerical simulation are ρ = 2329 g/cm^{3}, c _{ s } = 9.04 × 10^{5} cm/s, and D _{ ac } = 8.9322 eV [14].
σ  Mode  ħ Ω_{ σ }(meV)  D _{ iv } K _{ σ }× 10^{8} (eV/cm)  Type 

(Γ  X)_{1}  LA  18.4  2.45   
(Γ  X)_{2}  TO  57.6  0.8   
(X  X)_{1}  TA  12.0  0.5  g 
(X  X)_{2}  LA  18.5  0.8  g 
(X  X)_{3}  LO  61.2  11  g 
(X  X)_{4}  TA  19.0  0.3  f 
(X  X)_{5}  LA  47.4  2.0  f 
(X  X)_{6}  TO  59.0  2.8  f 
For the numerical simulation, an optical deformation potential of D _{ op } = 10^{9} eV/cm and a constant phonon energy ħΩ _{op} = 60 meV are used.
Numerical results and discussion
Model system
Density of states
Generation and photocurrent spectrum
Conclusions
In this article, an adequate theoretical description of photogeneration and transport in SiSiO_{ 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 Sibased coupled quantum well structures, such as phononassisted optical transitions and fielddependent transport in superlattice states. Owing to the microscopic nature of the theory, energyresolved 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.
Abbreviations
 EMA:

effective mass approximation
 NEGF:

nonequilibrium Green's function.
Declarations
Acknowledgements
The financial support for this study was provided by the German Federal Ministry of Education and Research (BMBF) under Grant No. 03SF0352E.
Authors’ Affiliations
References
 Green MA: Potential for low dimensional structures in photovoltaics. J Mater Sci Eng 2000, 74(1–3):118–124. 10.1016/S09215107(99)005462View ArticleGoogle Scholar
 Green MA: Third generation photovoltaics: Ultrahigh conversion efficiency at low cost. Prog Photovolt: Res Appl 2001, 9: 123. 10.1002/pip.360View ArticleGoogle Scholar
 Jiang CW, Green MA: Silicon quantum dot superlattices: Modeling of energy bands, densities of states, and mobilities for silicon tandem solar cell applications. J Appl Phys 2006, 99(11):114902. 10.1063/1.2203394View ArticleGoogle Scholar
 Wacker A: Semiconductor superlattices: a model system for nonlinear transport. Phys Rep 2002, 357: 1. 10.1016/S03701573(01)000291View ArticleGoogle Scholar
 Aeberhard U, Morf R: Microscopic nonequilibrium theory of quantum well solar cells. Phys Rev B 2008, 77: 125. 343 343 10.1103/PhysRevB.77.125343View ArticleGoogle Scholar
 Aeberhard U: A Microscopic Theory of Quantum Well Photovoltaics. In Ph.D. thesis. ETH Zurich; 2008.Google Scholar
 Steiger S, Veprek RG, Witzigmann B: Electroluminescence from a quantumwell LED using NEGF. Proceedings  2009 13th International Workshop on Computational Electronics, IWCE 2009 2009.Google Scholar
 Steiger S: Modeling NanoLED. In Ph.D. thesis. ETH Zurich; 2009.Google Scholar
 Schäfer W, Wegener M: Semiconductor Optics and Transport Phenomena. Springer, Berlin; 2002.View ArticleGoogle Scholar
 Mahan GD: ManyParticle Physics. 2nd edition. Plenum, New York; 1990.View ArticleGoogle Scholar
 Kubis T, Yeh C, Vogl P, Benz A, Fasching G, Deutsch C: Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers. Phys Rev B 2009, 79: 195323. 10.1103/PhysRevB.79.195323View ArticleGoogle Scholar
 Lake R, Klimeck G, Bowen R, Jovanovic D: Single and multiband modelling of quantum electron transport through layered semiconductor devices. J Appl Phys 1997, 81: 7845. 10.1063/1.365394View ArticleGoogle Scholar
 Henrickson LE: Nonequilibrium photocurrent modeling in resonant tunneling photodetectors. J Appl Phys 2002, 91: 6273. 10.1063/1.1473677View ArticleGoogle Scholar
 Jin S: Modeling of Quantum Transport in NanoScale MOSFET Devices. In Ph.D. thesis. School of Electrical Engineering and Computer Science College of Engineering, Seoul National University; 2006.Google Scholar
 Kadanoff LP, Baym G: Quantum Statistical Mechanics. Benjamin, Reading, Mass; 1962.Google Scholar
 Keldysh L: Diagram technique for nonequilibrium processes. Sov PhysJETP 1965, 20: 1018.Google Scholar
 Henneberger K, Haug H: Nonlinear optics and transport in laserexcited semiconductors. Phys Rev B 1988, 38: 9759–9770. 10.1103/PhysRevB.38.9759View ArticleGoogle Scholar
 Pereira M, Henneberger K: Green's function theory for semiconductorquantumwell laser spectra. Phys Rev B 1996, 53: 16. 485 485 10.1103/PhysRevB.53.16485View ArticleGoogle Scholar
 Pereira M, Henneberger K: Microscopic theory for the influence of coulomb correlations in the lightemission properties of semiconductor quantum wells. Phys Rev B 1998, 58: 2064. 10.1103/PhysRevB.58.2064View ArticleGoogle Scholar
 In steady state, the Green's functions depend only on the difference τ = t  t' of the realtime variables, which is Fouriertransformed to energy
 For an explicit derivation of the contact selfenergy in the effectivemass tightbinding model, see, e.g., [12]
 The dimensions are those of a volume rate, = m ^{ 3 } s ^{ 1 }
 Jin S, Park YJ, Min HS: A threedimensional simulation of quantum transport in silicon nanowire transistor in the presence of electronphonon interactions. J Appl Phys 2006, 99: 123719. 10.1063/1.2206885View ArticleGoogle Scholar
 While the validity of the EMA for the oxide is debatable, moderate variations of the parameter should not have a strong impact on the results, since the dominant dependence is on the barrier energy
 Ridley BK: Quantum Processes in Semiconductors. Oxford Science Publications; 1993.Google Scholar
 Hamaguchi C: Basic Semiconductor Physics. Springer, Berlin; 2001.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.