First Solar, Perrysburg, OH, USA. Abstract â â In this work, we report on developing 1D reaction-diffusion solver to understand the kinetics of p-type.
One-Dimensional Reaction-Diffusion Simulation of Cu Migration in Polycrystalline CdTe Solar Cells D. Guo1, R. Akis1, D. Brinkman2, I. Sankin3, T. Fang3, D. Vasileska1 and C. Ringhofer2 1
School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, USA 2 School of Mathematical and Statistical Science, Arizona State University, Tempe, AZ, USA 3 First Solar, Perrysburg, OH, USA
Abstract — — In this work, we report on developing 1D reaction-diffusion solver to understand the kinetics of p-type doping formation in CdTe absorbers and to shine some light on underlying causes of metastabilities observed in CdTe PV devices. Evolution of intrinsic and Cu-related defects in CdTe solar cell has been studied in time-space domain self-consistently with free carrier transport and Poisson equation. Resulting device performance was simulated as a function of Cu diffusion anneal time showing pronounced effect the evolution of associated acceptor and donor states can cause on device characteristics. Although 1D simulation has intrinsic limitations when applied to poly-crystalline films, the results suggest strong potential of the approach in better understanding of the performance and metastabilities of CdTe photovoltaic device. Index Terms — semiconductor device reliability, numerical simulation, photovoltaic cells, cadmium compounds, copper.
I. INTRODUCTION It is well known that Cu plays an important role in CdTe solar cell by forming back contact and dopants. Small amount of Cu partially diffused into the CdTe absorber layer results in increased hole density, and improved open circuit voltage (Voc) [1]. Also, long term Cu migration in CdTe solar cells is related to the device degradation [2]. Excess Cu creates recombination centers that significantly reduce the fill factor (FF) and Voc [3]. In the process level, high accumulation of the Cu near the back contact can be explained by the Cd vacancies generated in the etching process and can be seen as the Cu2Te contact layer [4, 5]. Cu also diffuses into the CdTe layer and serves as both donors (Cui), acceptors (CuCd) [6, 7] and neutral defects (CuCd or CuCd-Cui complex) [8, 9]. This present work provides a diffusion-reaction model for Cu migration in CdTe solar cells that can be used to understand and potentially optimize Cu incorporation processes. This model will also be used to understand dark/light crossover and distortions in apparent quantum efficiency [1], related to defects in CdTe devices. II. REACTION DIFFUSION MODEL The general reaction-diffusion model had been developed long ago for impurities in silicon based devices[10]. This model calculates the diffusion, drift and reactions of different kinds of defects, gives time-dependent solutions of defect
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profile in semiconductor devices. Let us consider one of the major Cu defect reaction in CdTe, reaction B in Fig. 1, free Copper interstitials knock Cadmium out of its site and occupy the Cd site and Cd become free moving Cd interstitials:
Cui + CdCd ⇔ CuCd + Cdi
(1)
To build a general model describing this reaction, we first determine the reaction rates in the following form: R f = K f Cs ([Cui ] − [Cui ]* ) * * Rb = K b ([CuCd ][Cd i ] − [CuCd ] [Cd i ] )
(2)
In Eq. (2), subscripts f and b indicate the forward and the backward reaction, Cs is the atomic concentration in CdTe, [X] is the concentration of defect X, [X]* is the equilibrium concentration of defect X and Kf,b is the reaction constant. The reaction constant is related to the frequency of the collision of the reactants. In dilute approximation, the defect reaction rates Reaction A: Cui + VCd ⇔ Cu Cd Cu Cu Vacancy
R eaction B: Cu i + Cd Cd ⇔ C u Cd + Cd i Cd
Cu
Te
Cu Cu
Cu
Reaction C: Cd i + VCd ⇔ Cd Cd
Vacancy
Fig. 1. The reactions considered to describe Cu diffusion into CdTe in this work.
2011
could be treated in the same way as bi-molecular reactions [10], e.g.
∆ECui ,Cd 4π (aCui + aCd ) ( DCui + DCdCd ) exp − K f = kT ΩCS (3) K = 4π (aCuCd + aCdi ) ( D + D ) exp − ∆ECuCd ,Cdi Cdi CuCd b ΩCS kT where aX represent capture radii of reactants, Ω is the volume of unit cell, DX is the diffusion constant of the defect and ∆Ex,y represent the energy barriers for the defects to collide with each other. Time-space evolution of point defects involved the major reaction in Eq. (1) could be described by the following set of reaction-diffusion equations [10]: dJ Cui d [Cui ] dt = − dx − R f + Rb (4) d [CuCd ] = R − R b f dt Note that in Eq. (4), diffusion of CuCd has been ignored due to very small diffusion coefficients. The fluxes JX in (4) result both from the diffusion due to concentration gradients and from the drift due to electrostatic field: J X = DX
d[ X ] + υX [ X ] dx
(5)
Assuming Boltzmann statistics (valid for diluted concentrations) and the charge θ carried by the defect, its drift velocity υ in electric field F could be found as µXF, where mobility µX could be expressed with diffusion coefficient using Einstein relation (for non-degenerate statistics):
Fig. 2. The formation energy of involved defects as a function of Fermi energy at both Cd-rich and Te-rich cases. The slope of the energy line gives the charge state of the defect. The transition energy level corresponds to the Fermi energy at which the slope changes. The line intersects at positions where Fermi energy might be pinned for equilibrium.
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DX (6) F kT Electric field F in the film is not only determined by the electron/hole concentrations, but also by the ionized defect distributions inside the grain and its boundaries (surfaces), and could be found by solving the Poisson equation:
υX = θ
d dV + − (7) ε sc = −q ( p − n + N D − N A ) dx dx In Eq. (7), V is the electrostatic potential, p stands for the hole concentration, n is the electron concentration, εs is the spatially varying dielectric constant of the film, and the doping concentration is calculated as: N D+ = [Cui + ] + 2[Cdi2 + ] (8) − 2− − − N A = [Cucd ] + 2[Vcd ] + [Vcd ] where the ionized defects concentration is determined by their formation energies [11] (shown in Fig. 2.) and total Cu concentration [12]: 0 ) / k BT ) Cs exp(−∆H f (CuCd 0 Cu = N [ ] Cd Cu fd − − g Cs exp(−∆H f (CuCd ) / k BT ) [Cu − ] = N Cu Cd fd (9) Cs exp(−∆H f (Cui0 ) / k BT ) 0 [Cui ] = N Cu fd + g Cs exp(−∆H f (Cui+ ) / k B T ) [Cui+ ] = NCu fd .
Note that g is the degeneracy factor and the denominator fd gives the total allowed states for Cu: − 0 ) ) −∆H f (CuCd −∆H f (CuCd ) + g − exp( ) f d = Cs (exp( k BT k BT (10) −∆H f (Cui0 ) −∆H f (Cui+ ) + + exp( ) + g exp( )) k BT k BT The equilibrium concentration employed for reaction rate calculation in Eq. (2) is the sum of the concentrations of different charge states: [Cui ]* = [Cui0 ] + [Cui+ ] (11) * 0 − [CuCd ] = [CuCd ] + [CuCd ] The VCd in Eq. (8) is the concentration of vacancies at Cd site. Its reactions with other species had to be included in the simulation for a realistic picture of copper defects in CdTe solar cells, as in Eq. (12): Cui + VCd = CuCd (12) Cd i + VCd = Cd Cd
The concentration of ionized vacancies and their reaction rates can be calculated similarly to Eqs. (2) and (3), and results in the following reaction-diffusion equation system in our simulator:
2012
dJ Cui d [Cui ] dt = − dx − RCui d [Cdi ] = − dJ Cdi − R Cdi dt dx (13) d [CuCd ] = − R CuCd dt d [V ] Cd = − RV Cd dt Similarly to the reaction-diffusion equations, drift-diffusion equations for free carriers are solved self-consistently with Poisson equation, giving accurate (quasi) Fermi levels in the device under variety of stress conditions for each time step of the defect migration. ∂n 1 ∂t = q ∇ ⋅ J n − U n ∂p = − 1 ∇ ⋅ J + U p p q ∂t
(14)
In Eq. (14), n, p stands for electrons and holes, U is the net generation-recombination rate and J is the carrier flux. During the device simulation process, ionizations of the defects are updated based on new formation energies obtained from new quasi Fermi level with each iteration in the Gummel cycle, as described in Eq. (9).
TABLE I DEVICE PARAMETER AND MATERIAL CONSTANT EMPLOYED CdS CdTe Layer thickness (µm) 0.1 1.9 N: 1017 Doping (cm-3) NC 300K(cm-3) 2.62·1018 1.07·1018 NV 300K(cm-3) 1.72·1019 6.08·1018 Eg (eV) 2.38 1.46 χ (eV) 4.5 4.28 100 µn (cm2/Vs) 100 100 µp (cm2/Vs) 100 10-8 10-8 τn (s) τp (s) 10-8 10-8 9.0 ε/ε0 10.3 It is important to note that the approach designed and implemented enables us device-level simulations, such as J-V curve, C-V curve and spectral response test, based on profiles of point defects/complexes generalized by diffusion-reaction solver. Although SIMS profiles of different atoms can be achieved, it is still extremely difficult to tell Cui/CuCd from total Cu with equilibrium or steady state assumption, which most likely is not true in the reality as the operation environment of the solar cells changes continuously. Device performance is the only approach we could do to verify the accuracy of this defect simulation. III. SIMULATION RESULTS The initial Cui concentration was set to be its solubility limit (1015 to 1017 cm-3 in CdTe) [13] at the back contact (left) and 1cm-3 elsewhere. No flux boundary condition was applied to vacancies and substitutional defects. Small surface recombination was applied to the interstitials. Charge
Fig. 3. The general configuration of the simulation. The device simulation block is required to achieve convergence for each time step of the defect evolution.
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Fig. 4. Comparison between simulated Cu profile and experiment in poly-CdTe layer after 100oC and 10 hour back contact processing.
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Fig. 5. The defects concentration profile and corresponding band diagram of the material in one particular moment of the simulation. Due to the accumulation of CuCd and its ionized acceptor state, it behaves as p-type near the back contact.
Fig. 6. The defects concentration profile and corresponding band diagram of the CdTe solar cell for different copper anneal time at 220oC. Cdi and VCd are neglected in the figure due to their relative low concentrations.
neutrality was assumed at both boundaries for the device simulation block. The experimental Cu profile in Fig. 4 shows two exponential decay regions. The fast decay near contact (x0.2um is caused by copper’s fast diffusion through grain boundaries [13], which cannot be properly simulated in a onedimensional solver. Fig. 5 indicated the band diagram and defect profiles in one particular moment in the back contact processing. The ionized Cdi donors made the CdTe junction region n-type while CuCd acceptors dominated the back contact region. Similar simulation are performed for the doping profiles in a standard CdTe/CdS solar cell (as Tables 1 indicated) without massive surface vacancies at the back contact. Fig. 6 illustrates the gradual diffusion of Cui donors into the bulk, followed by the formation of CuCd acceptors, and corresponding band diagrams in a 220oC stress test. At the beginning, back contact region is n-type doped with the introduction of Cui donors. Part of the Cui transformed into CuCd and others continued diffusing into the bulk, made the entire CdTe layer n-type, with zero response for the solar spectrum for the beginning 160 seconds. The back contact region became p-type due to the formation of CuCd acceptors
roughly in 170 seconds, while the junction area remained ntype, achieving non-zero spectral response. Continuing the copper process resulted in increasing quantum efficiency for the entire solar spectrum as acceptors dominated the CdTe layer, which may not be caused by the formation of CuCd but by the back flow of Cui donors and Cdi donors under the influence of the built-in potential in the device introduced by p-type back contact. These continued movements of the defects could be responsible for the dark/light crossover commonly observed from CdTe solar cells.
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Fig. 7. Variations in spectral response with varying anneal time during the stress test.
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ACKNOWLEDGEMENT This work was supported by the Department of Energy under award number DE-EE0006344. REFERENCES
Fig. 8. Variations in simulated J-V characteristics with varying anneal time during the stress test.
Fig. 8 shows the typical variations seen in the CdTe currentvoltage (J-V) curves when the stress time of copper in the back contact is varied at elevated temperature (220oC). Consistent with the quantum efficiency trend depicted in Fig. 7, as more copper atoms occupying Cd sites in the CdTe layer, the short circuit current (JSC) get increased. The open circuit voltage (VOC) still get extra increment, even after the JSC reached stable, as the 200s and 240s curves indicated, probably leaded by the formation of abrupt p-n junction or the formation of better back contact. Due to the lack of carrier recombination mechanisms with Cu defects and deep-level traps, the flattening of the J-V curve [1, 4] cannot be achieved in this temperature elevated stress test simulation. IV. CONCLUSION The diffusion-reaction model has been applied to Cu in CdTe PV devices using available diffusivity and solubility data. The simulations agree quantitatively with the observations of Cu accumulation near the back contact and qualitatively with the trend for the change in device performance with respect to copper’s inclusion. As another commonly introduced extrinsic defects in CdTe solar cells, Cl and its related defects, Cli, ClTe and potentially Cl A-Center [14], should also be included in the reactiondiffusion simulations for better understanding of defects’ evolution in CdTe solar cells. The prototype simulator is general and can be used to suggest Cu process improvement and to study long term device performance change and metastabilities related to defects in the time range of weeks and months as well.
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[1] C. Corwine, A. Pudov, M. Gloeckler, S. Demtsu, and J. Sites, "Copper inclusion and migration from the back contact in CdTe solar cells," Solar Energy Materials and Solar Cells, vol. 82, pp. 481-489, 2004. [2] K. D. Dobson, I. Visoly-Fisher, G. Hodes, and D. Cahen, "Stability of CdTe/CdS thin-film solar cells," Solar Energy Materials and Solar Cells, vol. 62, pp. 295-325, 2000. [3] S. Demtsu, D. Albin, J. Sites, W. Metzger, and A. Duda, "Curelated recombination in CdS/CdTe solar cells," Thin Solid Films, vol. 516, pp. 2251-2254, 2008. [4] A. Pudov, M. Gloeckler, S. Demtsu, J. Sites, K. Barth, R. Enzenroth, et al., "Effect of back-contact copper concentration on CdTe cell operation," in Proceedings of the 29th IEEE Photovoltaic Specialists Conference, New Orleans, LA, 2002, pp. 760-763. [5] B. McCandless and J. Sites, "Chapter 14 of Handbook of Photovoltaic Science and Engineering. Chichester," ed: West Sussex, UK: John Wiley & Sons, 2011. [6] E. Kučys, J. Jerhot, K. Bertulis, and V. Bariss, "Copper impurity behaviour in CdTe films," Physica Status Solidi (a), vol. 59, pp. 91-99, 1980. [7] S.-H. Wei and S. Zhang, "Chemical trends of defect formation and doping limit in II-VI semiconductors: The case of CdTe," Physical Review B, vol. 66, 155211, 2002. [8] D. Krasikov, A. Knizhnik, B. Potapkin, S. Selezneva, and T. Sommerer, "First-principles-based analysis of the influence of Cu on CdTe electronic properties," Thin Solid Films, vol. 535, pp. 322-325, 2013. [9] J. Perrenoud, L. Kranz, C. Gretener, F. Pianezzi, S. Nishiwaki, S. Buecheler, et al., "A comprehensive picture of Cu doping in CdTe solar cells," Journal of Applied Physics, vol. 114, 174505, 2013. [10] P. M. Fahey, P. B. Griffin, and J. D. Plummer, "Point defects and dopant diffusion in silicon," Reviews of Modern Physics, vol. 61, pp. 289-384, 1989. [11] S. Zhang and J. E. Northrup, "Chemical potential dependence of defect formation energies in GaAs: Application to Ga selfdiffusion," Physical Review Letters, vol. 67, p. 2339, 1991. [12] J. Ma, S.-H. Wei, T. A. Gessert, and K. K. Chin, "Carrier density and compensation in semiconductors with multiple dopants and multiple transition energy levels: Case of Cu impurities in CdTe," Physical Review B, vol. 83, 245207, 2011. [13] E. Jones, N. Stewart, and J. Mullin, "The diffusion of copper in cadmium telluride," Journal of Crystal Growth, vol. 117, pp. 244-248, 1992. [14] D. Hofmann, P. Omling, H. Grimmeiss, B. Meyer, K. Benz, and D. Sinerius, "Identification of the chlorine A center in CdTe," Physical Review B, vol. 45, pp. 6247-6250, 1992.
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