Ab initio Description of Optoelectronic Properties at Defective Interfaces in Solar Cells
Philippe Czaja, Massimo Celino, Simone Giusepponi, Michele Gusso, Urs Aeberhard
AAb initio Description of OptoelectronicProperties at Defective Interfaces in Solar Cells
Philippe Czaja , Massimo Celino , Simone Giusepponi , Michele Gusso , andUrs Aeberhard IEK-5 Photovoltaik, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany, [email protected] ENEA, C.R. Casaccia, 00123 Rome, Italy ENEA, C.R. Brindisi, 72100 Brindisi, Italy
Abstract.
In order to optimize the optoelectronic properties of novelsolar cell architectures, such as the amorphous-crystalline interface insilicon heterojunction devices, we calculate and analyze the local mi-croscopic structure at this interface and in bulk a-Si:H, in particularwith respect to the impact of material inhomogeneities. The microscopicinformation is used to extract macroscopic material properties, and toidentify localized defect states, which govern the recombination prop-erties encoded in quantities such as capture cross sections used in theShockley-Read-Hall theory. To this end, atomic configurations for a-Si:Hand a-Si:H/c-Si interfaces are generated using molecular dynamics. Den-sity functional theory calculations are then applied to these configura-tions in order to obtain the electronic wave functions. These are analyzedand characterized with respect to their localization and their contribu-tion to the (local) density of states. GW calculations are performed forthe a-Si:H configuration in order to obtain a quasi-particle corrected ab-sorption spectrum. The results suggest that the quasi-particle correctionscan be approximated through a scissors shift of the Kohn-Sham energies.
Keywords: amorphous silicon, molecular dynamics, electronic struc-ture, optical properties
The silicon hetero-junction (SHJ) technology holds the current efficiency recordof 26.33% for silicon-based single junction solar cells [1] and shows great potentialto become a future industrial standard for high-efficiency crystalline silicon (c-Si)cells.One of the key elements of this technology is the passivation of interface de-fects by thin layers of hydrogenated amorphous silicon (a-Si:H), and the physicalprocesses at the so-formed c-Si/a-Si:H interface largely influence the macroscopiccharacteristics of the cell. In particular the cell performance depends criticallyon the optimization of transport and the minimization of recombination acrossthe interface, which requires a profound understanding of the underlying mecha-nisms. Special regard has to be given to the role of localized tail and defect states a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov P. Czaja et al. in a-Si:H and at the interface, which behave substantially different from bulkstates and thus prohibit a treatment in terms of bulk semiconductor physics. Anaccurate and physically meaningful description of the local microscopic structureis therefore an essential step in understanding and predicting the macroscopic de-vice characteristics, which gave rise to a growing interest in ab initio approaches[2,3,4].In our investigation presented here, we use ab initio molecular dynamics togenerate atomic configurations of defective a-Si:H and c-Si/a-Si:H interfaces, andsubsequently perform electronic structure calculations to obtain and characterizethe electronic states. The electronic structure at the interface is analyzed withrespect to the existence of localized defect states which have an impact on thedevice performance due to their role as recombination centers in non-radiativerecombination [5]. The density of these defect states is an important parameter inthe Shockley-Read-Hall model for calculating capture cross sections, and shouldtherefore attain realistic values in the generated structures. The states of thebulk a-Si:H are further used for calculating the absorption coefficient from abinitio, which is a first step towards linking the global device characteristics tothe local microstructure in a comprehensive multi-scale simulation approach [6].As the optical properties of any materials depend crucially on their band gapthis quantity is of essential importance for obtaining physically relevant results.Unfortunately the independent-particle approximation, which is at the heart ofstandard first-principles methods, is unable to correctly predict its value [7],which is why so-called quasi-particle corrections [8] need to be applied. Theexact calculation of these corrections is however computationally expensive, aheuristic approach – termed scissors shift (SS) [9] – , where the electron energiesare simply shifted to fit the experimental band gap, is therefore often favored.Since a distinct experimental value of the band gap of a-Si:H does however notexist, a set of shifting parameters can only be determined from a quasi-particlecalculation. In this paper we present the results of such a calculation for ana-Si:H configuration.
The ab initio PWscf (Plane-Wave Self-Consistent Field) code of the QuantumESPRESSO suite is used [10,11] to perform Born-Oppenheimer Molecular Dy-namics (BOMD) simulations of the a-Si:H and the a-Si:H/c-Si interface. PWscfperforms many different kinds of self-consistent calculations of electronic struc-ture properties within Density-Functional Theory (DFT) [12,13], using a plane-wave (PW) basis set and pseudopotentials (PP). We use the Si and H ultrasoftpseudopotentials with Perdew-Burke-Ernzerhof (PBE) [14] approximant GGAexchange-correlation potential, available in the Quantum ESPRESSO library[11]. To mimic infinitely extended systems, a supercell approach with periodicboundary conditions (PBC) is used. ptoelectronic properties at interfaces from ab initio 3
To generate an a-Si:H system, a random starting configuration is producedwith a percentage of H atoms of about 11%, which is the nominal concentrationset in experimental materials optimized for PV performance [15]. Initially, asmall system of 64 Si + 8 H atoms in a cubic supercell with size L= 11.06˚A (the volume is chosen to fix the density to the experimental value of 2.214g/cm [16]) is used to perform calculations with a wide range of quench rates.This is due to the fact that the resulting amorphous configuration is largelydependent on the quench rate used to produce the amorphous structure fromthe melt configuration. Experimental results indicate that the amorphous phasecontains a very low number of defects and that the majority of Si atoms havecoordination four. To this end we select a small amorphous configuration (Fig.1) that minimizes both the total value of defects and the deviation from thefourfold coordination of the Si atoms. Then, this configuration, is used as startingconfiguration for a BOMD simulation on the electronic ground state at constantvolume and constant temperature for 6.5 ps, controlling the ionic temperature(T = 300 K) by using an Andersen thermostat [17].Fig. 1: Snapshot of the a-Si:H in the simulation box. Hydrogen atoms and bondswith Silicon atoms are blue, Silicon atoms and their bonds are yellow.The final configuration is then used to produce a large system by replicat-ing it in all directions. The resulting large system is thus composed of 512 Si+ 64 H atoms and has a size of L=22.12 ˚A. Due to the high computationalcosts required by PWscf, BOMD simulations on this large system are performedwith the Quickstep code of the CP2K suite [18]. CP2K is a quantum chemistryand solid state physics software package that can perform atomistic simulationswith different modelling methods (such as DFT) using a mixed Gaussian andplane wave approach. Norm conserving Goedecker-Tetter-Hutter pseudopoten-tials with PBE exchange-correlation and an optimized TZV2P gaussian basis setare used [19,20,21]. Self consistency at each MD step is achieved using the orbitaltransformation method [22]. An annealing process from T = 300 K up to T =600 K, and then back to T = 300 K for 60 ps was then used to thermalize thewhole atomic configuration, and minimize the defects at the internal interfaces.After the annealing, a simulation run at T = 300 K was performed for about 20ps. P. Czaja et al.
The a-Si:H/c-Si interface is built by putting nearby two free surfaces obtainedcutting both the crystalline silicon and the hydrogenated amorphous silicon. Therelaxed p(2 ×
1) symmetric reconstruction of the Si(001) surface constitutes thec-Si side of the interface. It is formed by 192 Si atoms: 12 layers of silicon eachof them with 16 atoms. The a-Si:H side of the system is generated using asimulated-annealing quench-from-the-melt simulation protocol and is composedof 128 Si atoms and 16 H atoms. A void region of about 10 ˚Ais added to suppressthe interaction between the external surfaces due to PBC. This distance waschecked by convergence tests. The total length of the system is L z =38.70 ˚A,while in the x and y direction the system has L x = L y = 15.48 ˚A. Total energycalculations of the system at different distances between c-Si and a-Si:H, wereperformed to find the interface configuration corresponding to the lowest totalenergy. The configurations were built moving rigidly by hand the a-Si:H partand keeping fixed the c-Si one. (a)(b) Fig. 2: Snapshots of the a-Si:H/c-Si interface in the simulation box. The structureis infinitely extended in both x and y directions. A void region is considered tosuppress the interaction between the external surfaces due to periodic boundaryconditions. Free surfaces and a-Si:H/c-Si interface are perpendicular to the yaxis. Hydrogen atoms are blue, silicon atoms are dark yellow in the c-Si partand light yellow in the a-Si:H part. (a) Initial configuration. (b) Configurationat 35 ps of the MD simulation. The Si atoms near the interface have moved toform bonds between the c-Si and the a-Si:H layer. ptoelectronic properties at interfaces from ab initio 5
The interface shown in Fig. 2a, is used as starting configuration for MDsimulation on the electronic ground state at constant volume and constant tem-perature (NVT). The ionic temperature is fixed at T = 300 K and is controlledusing an Andersen thermostat [17]. The first four layers of c-Si atoms on the leftare kept fixed to impose a bulk like behavior to the crystalline silicon part of thesystem. The MD simulation is performed for more than 35 ps, the initial partof the simulation (20 ps) was used to thermalize the system and reach a stableconfiguration. Figure 2b displays the configuration of the a-Si:H/c-Si interfaceat 35 ps.
We use density functional theory (DFT) [12,13] with periodic boundary condi-tions to self-consistently calculate the electronic structure of the a-Si:H and theinterface configurations described above. The interface configuration is enclosedin a super cell that includes an additional vacuum layer to avoid self-interaction.All calculations are done with the PW-PP code Quantum ESPRESSO [10,11]using the PBE-GGA exchange-correlation functional [14]. For the c-Si/a-Si:Hinterface a k-point grid of size 4x4x1 and a plane-wave cut-off of 28 Ry is used,for the a-Si:H a 4x4x4 (72 atom configuration) and a 2x2x2 grid (576 atomconfiguration) respectively, together with a cut-off energy of 52 Ry.Subsequent to the electronic structure calculation the wave functions andelectronic density of states (DOS) of the c-Si/a-Si:H interface are analyzed toobtain information about its local microscopic properties, which are relevant forthe mesoscopic dynamics and macroscopic device characteristics. In particularthe wave function localization is analyzed qualitatively and quantitatively toallow for the distinction of localized states and the identification of their origins.In combination with the local DOS the contribution of dangling bonds andinterfaces to the important mid-gap states can be determined.As a quantitative measure for the localization of the wave function ψ we usethe spread S , which is calculated as the square root of the variance of | ψ | withrespect to the super cell: S z = (cid:114) (cid:16) (cid:104) z (cid:105) − (cid:104) z (cid:105) (cid:17) = √ · (cid:118)(cid:117)(cid:117)(cid:117)(cid:116)(cid:90) Ω d r | ψ ( r ) | z − (cid:90) Ω d r | ψ ( r ) | z ,where we assume that ψ is normalized. It can be easily seen that a maximallylocalized ψ (i.e., a delta function) gives S z = 0, whereas a wave function thatis maximally delocalized over the super cell (i.e., a plane wave) will result in S z = L , where L is the length of the super cell. This explains the factor √
12 in thedefinition. The integration volume Ω is naturally chosen such that the boundarieslie inside the vacuum layer where ψ ≈
0, such that shifting the integrationvolume does not affect S . This also provides an unambiguous definition of thewave function center (cid:104) z (cid:105) = (cid:90) Ω d r | ψ ( r ) | z , P. Czaja et al. which can be interpreted as the position where the wave function is local-ized. This definition allows us to identify localized states (i.e., states with smallspread), and to locate them both in real and in energy space.In order to relate the electronic properties of the interface to the atomicstructure, and in particular investigate the effect of structural defects, we usethe electron localization function (ELF) [23], which enables us to determine thecoordination of each atom, and to identify dangling bonds and weakly bondatoms. For that purpose the ELF is computed along the axes between neighbor-ing atoms, where it shows a characteristic behavior for covalent bonds [24]. Thisis performed with the Quantum ESPRESSO package.
The calculation of the absorption coefficient for the 72-atom a-Si:H configura-tion is carried out within the random phase approximation (RPA) [25] as im-plemented in the BerkeleyGW code [26], using the non-interacting Kohn-Shamstates on a 2x2x2 k-point grid. The same code is used for calculating the quasi-particle (QP) corrections to the Kohn-Sham energies with the GW formalism[27]. In order to reduce the computational costs we perform a single-shot G W calculation together with the plasmon-pole approximation [28,29,30]. This hasthe advantage of requiring the dielectric tensor (cid:15) ( ω ) only in the static limit ω →
0, as opposed to a full-frequency calculation, while offering similar accu-racy for many semiconductors, including c-Si [31]. The band gap is convergedwith respect to the cut-off energy E (cid:15)cut used in the calculation of (cid:15) , for whichwe find a value of 10 Ry, and with respect to the number of unoccupied bands N (cid:15) bands and N Σ bands included in the calculation of (cid:15) and of the self energy Σ . Wefind that a large number of roughly 3000 bands is needed to reach convergenceof both quantities (Fig. 3). The absorption coefficient is recalculated with thecorrected energies E QP , and compared to calculations where scissors shifts withdifferent sets of parameters are used. These parameters are obtained by applyinga linear fit E QPv/c = a v/c · E v/c + b v/c , where E v/c are the uncorrected energies,both to the valence and the conduction band. The absorption calculation forthe 576-atom configuration is carried out on a 2x2x2 k-point grid as well, usinguncorrected and scissors shift energies. Figure 4 shows the quasi-particle corrected electron energies as obtained fromthe GW calculation for the 72-atom a-Si:H structure described above. The re-sults show that the effect of the corrections consists mainly in a spreading ofvalence and conduction band by approximately 0 .
26 eV. This suggests that thecostly GW calculation can be substituted by a simple scissors shift in furthercalculations of a-Si:H structures. The choice of the right set of parameters de-pends on the energy range of interest. By applying a linear fit in the energy ptoelectronic properties at interfaces from ab initio 7 e rr o r [ m e V ] N bands εΣ Fig. 3: Convergence of band gap with respect to the number of bands includedin the calculation of (cid:15) and Σ respectively.range from − a v = 1 . b v = − .
097 eV, a c = 1 . b c = − .
228 eV. -6-4-2 0 2 4 6-6 -4 -2 0 2 4 6E g = 0.29 eVE gQP = 0.56 eV E Q P [ e V ] E [eV]
Fig. 4: Quasi-particle corrected vs. uncorrected electron energies. E g refers hereto the energy difference between the lowest unoccupied and the highest occupiedstate.Figure 5 shows the imaginary part of the dielectric function and the absorp-tion coefficient calculated within the independent-particle approximation, thatis, with the uncorrected Kohn-Sham energies, the GW approximation, and thescissors-shift approximation. The GW correction modifies the absorption spec-trum only in terms of a shift and a slight stretch. This correction can be verywell approximated by a scissors shift with the parameters given above, whichreproduce almost exactly the GW absorption spectrum.Using the scissors shift approximation enables us to calculate a quasi-particle-corrected absorption spectrum also for the 576-atom structure, which is shown P. Czaja et al. ε E [eV] IPSSGW
0 1 2 3 4 5 α [ / c m ] E [eV] IPSSGW
Fig. 5: Imaginary part of dielectric function (left) and absorption coefficient(right) for the 72-atom configuration, calculated with uncorrected states (IP),and with quasiparticle corrected states in GW and scissors shift (SS) approxi-mation. ε E [eV] IPSS
0 1 2 3 4 5 α [ / c m ] E [eV] IPSS
Fig. 6: Imaginary part of dielectric function (left) and absorption coefficient(right) for the 576-atom configuration, calculated with uncorrected states (IP),and with quasiparticle corrected states in scissors shift (SS) approximation.in Fig. 6. Comparison of the spectra for the two different configurations showsan increase of the optical band gap in the larger structure, along with a decreaseof the sup-gap absorption peaks. Even though this represents an improvement,the band gap is still small compared to the experimental value of 1 . In Fig. 7 the spread in z-direction (i.e., in growth direction) S z is shown as afunction of the wave function energy together with the total DOS around theFermi energy for the above described interface configuration. The figure showsthat there is a dense distribution of strongly localized states inside the c-Si bandgap, which can be clearly distinguished from the more extended tail and bulkstates. ptoelectronic properties at interfaces from ab initio 9 D O S [ / e V ] S p r ead S z [ Å ] E [eV] SpreadDOS
Fig. 7: DOS and wave function spread (in direction perpendicular to the inter-face) in the energy region around the Fermi energy at 0 eV.The origin of these states can be investigated further by looking at the localDOS and the wave function centers as shown in Fig. 8. In the top subfigure,the layer-resolved DOS is displayed as a function of the z-coordinate, which isobtained by integrating the local DOS over layers parallel to the interface. Thefigure shows that near the interface, which is marked by the dotted line, the bandgap starts filling up with states, and completely vanishes in the a-Si:H region.That these mid-gap states are indeed localized can be seen in the bottom figure,where S z is plotted as a function of the energy and the z-component of the centerof the wave functions. Each dot marks the energy and the position of one wavefunction, that is, where along the z-direction it is centered. The color of each dotrepresents the spread of the wave function. This representation indicates thatthe contribution to the mid-gap states comes mainly from localized states in thea-Si:H layer, whereas the interface region hardly contributes at all.The emergence of localized states in the a-Si:H region can be better un-derstood in terms of the atomic structure. For that purpose all the bonds areanalyzed by means of the ELF in order to identify dangling and weak bonds. Thisis shown exemplarily in Fig. 9 for a three-fold bond Si atom. By investigatingthe ELF between this atom and its nearest neighbors one can clearly distinguishone H bond, two Si bonds, and one dangling Si bond. Applying this analysisto all atoms yields a coordination map as shown in Fig. 10. This reveals thatthere is a large number of low-coordinated atoms in the a-Si:H layer whereas theatoms at the interface itself (represented by a dotted line) are mostly four-foldcoordinated. While supporting the conclusions from the localization analysis,this result also indicates that the quality of the amorphous layer is rather poor.In fact the defect density is of the order of 10 / cm , and thus four ordersof magnitude higher than the defect density measured experimentally for thina-Si:H films [32], which explains the high DOS inside the band gap. −3−2−1 0 1 2 3 E [ e V ] D O S [ / e V ] E [ e V ]
12 14 16 18 20 22 24 26 28 30 S p r ead S z [ Å ] Fig. 8: Top: Local DOS integrated over layers parallel to the interface as a func-tion of the z-coordinate. Bottom: Wave function spread in z-direction. Each dotmarks the energy and the position of the center of one wave function, whereasthe color represents its spread. The dotted line shows the approximate positionof the interface.
Table 1 lists the computational costs of typical sets of calculations with convergedparameters for all three structures considered in this paper. As self-consistentfield (scf) calculations with DFT scale with O ( N log N ) in the number of atoms,the computational costs increase by two orders of magnitude when going from 72to 576 atoms. This is however only the theoretical scaling behavior and does notyet take into account I/O and communication costs, as well as the non-ideal scal-ing with respect to the number of cores. These effects become visible especiallyin the non-self consistent (nscf) calculation of unoccupied bands, where largermatrices have to be handled. Altogether the numbers indicate that the currentlimit for performing DFT calculations with conventional plane-wave approacheson reasonable time-scales is of the order of a few thousand atoms.Regarding GW, a full calculation for the small structure requires about 2400core-h, making it obvious that GW calculations for the larger systems are cur- ptoelectronic properties at interfaces from ab initio 11 Fig. 9: Left: 3-fold bond atom at the a-Si:H/c-Si interface (purple) and its threebonding partners (magenta). Right: ELF between the atom shown on the leftand its four nearest neighbors. The orange curve represents a bond with anH atom, the blue and purple curve represent Si-Si bonds, and the green curverepresents a dangling bond.
0 5 10 15 20 25z [Å] 0 2 4 6 8 10 12 14 y [ Å ] C oo r d i na t i on Fig. 10: Coordination numbers for all atoms in the a-Si:H/c-Si configuration.rently out of range. The most costly part here is the DFT calculation of a largenumber of unoccupied bands, which are needed exclusively in the GW calcula-tions and not for any other of the calculations we performed. In addition it hasto be pointed out once again that the numbers given here refer to an alreadyconverged calculation. The convergence process itself is computationally muchmore challenging, which is due to the fact that three interdependent parametershave to be converged simultaneously, resulting in a total cost of about 80000core-h.For the large a-Si:H and the interface configuration the vast majority ofthe computational time is spent on the BOMD simulations which is due tothe fact that the electronic ground state is computed at every time step by an scf calculation. The high computational demand for the interface generationmotivated the use of the CP2K Quickstep code for the large a-Si:H system,which allowed us to reduce the computational cost by using a mixed Gaussianand plane wave approach.
Calculation Computational costs [core-h]a-Si:H (72) a-Si:H (576) a-Si:H/c-SiMD 2300 190000 220000(6.5 ps) (80.0 ps) (35.0 ps)DFT scf 20 1940 200nscf 15 5450 750GW unoccupied bands 1800 (cid:15) Σ Table 1: Computational costs for typical sets of calculations for all three struc-tures investigated within this work. MD and DFT calculations were done usingQuantum ESPRESSO, except for the 576-atom system, where CP2K was used.GW and absorption calculations were done using BerkeleyGW. For the MD cal-culations also the simulation time is provided in brackets.
We presented an ab initio description of the atomic and electronic properties ofthe a-Si:H/c-Si interface, which is at the heart of the technologically relevantsilicon-heterojunction solar cells. We introduced and applied different methodsfor analyzing the electronic structure, in particular with respect to the role ofdefects and localized states, which have an influence on the cell performance vianon-radiative recombination.Furthermore, we generated configurations of a-Si:H and calculated the elec-tronic structure, including GW corrections. As a first step towards the extractionof macroscopic material properties from the local microscopic structure for usein multiscale models of solar cells, we calculated the absorption spectrum of ana-Si:H structure. As an important result we found that the expensive GW cor-rections can be replaced by a linear approximation, which makes calculations forlarger – and, thus, physically more representative – configurations possible.
Acknowledgments.
This project has received funding from the EuropeanCommission Horizon 2020 research and innovation program under grant agree-ment No. 676629. The authors gratefully acknowledge the computing time granted ptoelectronic properties at interfaces from ab initio 13 on the supercomputer JURECA [33] at J¨ulich Supercomputing Centre (JSC) andon the supercomputer CRESCO [34] on the ENEA-GRID infrastructure.
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