Ferromagnetic Compounds for High Efficiency Photovoltaic Conversion

week ending 5 JUNE 2009

PHYSICAL REVIEW LETTERS

PRL 102, 227204 (2009)

Ferromagnetic Compounds for High Efficiency Photovoltaic Conversion: The Case of AlP:Cr P. Olsson,1,2 C. Domain,1,2 and J.-F. Guillemoles1 1

IRDEP, UMR-7174 CNRS ENSCP EDF R&D, 6 quai Watier, F-78401 Chatou, France 2 EDF R&D, De´partement MMC, Les Renardie`res, F-77250 Moret sur Loing, France (Received 23 September 2008; published 5 June 2009)

The photovoltaic conversion efficiency in usual semiconductors is limited to 30% while thermodynamics sets an upper limit of above 70%. Here we show how efficiencies in the 50% range could be achieved using carefully chosen magnetic doping in wide gap semiconductors. To meet the requirement to obtain useful compounds we propose rules and a selection method based on ab initio calculations coupled with material efficiency predictions. As a result of an investigation over hundreds of compounds, AlP:Cr was found to be the most promising semiconductor. DOI: 10.1103/PhysRevLett.102.227204

PACS numbers: 75.50.Pp, 71.15.Mb, 71.20.b, 71.55.Eq

Dilute magnetic semiconductors (DMS) have emerged as promising compounds for spintronics, especially as some have been found ferromagnetic (FM) at room temperature [1,2]. Although spin-LED devices have been realized [3], the field of coupling optoelectronics with spintronics has been essentially left untouched. Specifically, the fact that spintronic materials could exhibit enhanced lifetimes of excited states, due to spin dependent transition selection rules, has not yet been considered. For photovoltaic conversion applications, long lifetimes of excited states are important as they improve the chance that the photogenerated carriers will be collected. Indeed, organic molecules display this property in the long lived triplet states that can be formed, opening the way to ground breaking applications in energy conversion. Notwithstanding stability issues, organic compounds suffer from important losses for energy conversion applications, because of low electron mobilities combined with significant atomic relaxation upon formation of the triplet state [4]. For this reason, an inorganic analog of long lifetime triplet states in molecules, however desirable, does not have an easy equivalent since one always finds free carriers of both spin states with which recombination is readily allowed. Thus, using FM DMS to mimic triplet states in bulk inorganic semiconductors [5] for photovoltaic conversion is very attractive. In DMS, the spin degeneracy of the bands is lifted, opening the way for configurations favorable for the operation of intermediate band (IB) devices (Fig. 1) where unwanted recombinations are impeded by spin selection rules or by low occupancy of states involved in the allowed recombinations. Previously, proposals have been made to better utilize the low energy part of the incident solar spectrum by using Intermediate Level Semiconductors (ILSC) either with localized states [6] or delocalized ones [7,8]. It has been shown that such concepts have efficiency expectations up to 49% in an optimal device [1], comparable to those of triple junction devices, at 51%. Indeed these ILSC also have three possible optical transitions that may be adjusted to optimally use the solar spectrum. Contrary to multijunction devices, 0031-9007=09=102(22)=227204(4)

only one absorber material is needed, a fact which limits optical losses, tunnel junction losses and would help to design simpler, possibly cheaper, devices. However, approaching those values would require rather low nonradiative recombination rates as compared to the radiative ones [6], a fact which has so far impeded the development of

E

E1

E′g

Eg

EF E2

n(↑)

DOS

n(↓)

FIG. 1 (color). Schematic density of states of a DMS (spin up and down) optimal for photovoltaic conversion. In most systems, the Fermi level is located in the middle of the 3d-induced states (generally t2g ). This level defines the optical threshold for a transition from the valence band (VB) or to the conduction band (CB). In the class of compounds considered, the VB has a p character and the CB an s character. The transitions from the VB to the t2g state have greater oscillator strengths than to those from t2g to the CB, and enable an optimal allocation of the photons when the t2g is located at 2=5 of Eg from the VB maximum.

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Ó 2009 The American Physical Society

PRL 102, 227204 (2009)


ILSC devices: without a specific mechanism to prevent them, nonradiative processes are far more efficient than radiative ones, especially those that involve the intermediate levels (IL). The here proposed DMS-based scheme with three absorption bands (Fig. 1) can provide a pathway for the preparation of solids with the desired optical properties by taking advantage of spin selection rules on interband transitions.


The potential efficiency of solar cells is derived from the photon and electron balance equations which are computed within the Boltzmann transport formalism, where additional selection rules based on electron spin have to be considered in the case of DMS. For the photons, the scattering integral is expressed in term of photon (f ) and electron (fn ) occupation factors as

X @f ð; q; rÞ þ crf ð; q; rÞ ¼ Mnn0 ffn ðE þ h; k þ qÞð1 fn0 ðE; kÞÞð1 þ f ð; q; rÞÞ @t n;n0 ;k fn0 ðE; kÞð1 fn ðE þ h; k þ qÞÞðf ð; q; rÞÞg; where is the photon helicity, h is the photon energy, q is the photon momentum, c is the photon velocity, is the electron spin, n and n0 are band indices for electrons, E is the electron energy, r is the electron position, k the electron momentum, Mn;n0 is the matrix element for an optical transition (intraband if n ¼ n0 or interband otherwise, Mn;n0 are independent of E and k). Note that ¼ 2, so that spin up electrons produce positive helicity photons (and conversely). To avoid being too restrictive on the device geometry, some assumptions regarding electron transport are generally made [1,8,9]. (i) Intraband carrier relaxation rates (carrier-carrier and carrier-phonon scattering) are assumed to be very fast, thus leading to a thermalized Fermi-Dirac carrier distribution in each band. When quasielastic, the spin flip is also assumed to be very fast. (ii) The carrier mobility is assumed to be very high in all bands, to the extent that the gradient of the carriers’ electrochemical potential is negligible in steady state, the case under consideration. (iii) For optimal efficiency, the samples are optically thick and absorption of photons occurs only at the highest possible transition energy [8]. In this case, the result is insensitive to the details of the transition matrix elements [1]. (iv) Photocarrier recombination is considered to occur predominantly via radiative processes (the so-called detailed balance [1]). The impact of nonradiative recombination can be evaluated without assuming a specific mechanism for it [1], but rather by assuming a given ratio of the radiative to nonradiative mechanism, valid for all transitions. Within these assumptions, describing the limiting optimal case, the photon balance, Eq. (1), suffices to compute the conversion efficiency: the output power of the device is the product of the current and the voltage, where the output voltage is simply given by the electrochemical potential difference between electrons in the CB and holes in the VB, obtained using hypothesis (ii), and the current is obtained using the detailed balance which relates the collected electron current in the external circuit to the absorbed photons [10]. In steady state, with electron distributions time and position independent, the numerical solution of Eq. (1) is straightforward [1,8]. The above assumptions are validated by the facts that photovoltaic devices made out of III-V compounds are

(1)

operating close to the radiative limit (over 90% radiative recombination [1,11], with a maximum reported 99.7% [12]), that charge carrier mobilities are indeed high enough, and that the quasi Fermi level can be considered flat in the device in the operation range of interest [1,13]. Absorption and collection of photogenerated carriers is also excellent in existing structures, with over 90% of the photons above the absorption threshold yielding collected charges in the best devices [14]. In the case of DMS, the helicity of photons had to be taken into account in the absorption selection rules and in the allocation of photons between the possible transitions. Full absorption of photons of both helicities was assumed (mirror at the back). The IL absorption threshold was assumed to be at the Fermi level. By doping, this threshold can be shifted to some extent, within the width of the band (Moss-Burstein shift) [15]. Within this framework, where only the physically limiting processes are taken into account [1,8,9], a realistic upper bound estimate of the efficiencies can be achieved without entering the specifics of the used materials. As a rule of thumb, present devices have achieved over 80% of the efficiency set by their physical limits [14]. In both IL type devices (DMS and ILSC), the sensitivity to precise energy level positions is not very large. This is illustrated by the contour plots of Fig. 2 where isoefficiency lines are plotted as a function of both the smallest and largest transition energies: efficiencies up to 52% are achievable with DMS cells (Table I). Remarkably, non ideal DMS compounds suffering from significant nonradiative recombination still offer excellent efficiency expectations. The efficiency decrease in an ILSC is very similar to that in a DMS but the possibility of attaining a high level of radiative recombination without a blocking mechanism is very low with the former. This is well known for the case of impurities [16] where the branching ratio of radiative to nonradiative recombination is in the 104 to 106 range, or worse. This ratio is also very unfavorable in state-of-the-art quantum dots [17]. Considering Figs. 1 and 2 and the physical guidelines for a photovoltaic semiconductor, we propose the following design criteria for suitable DMS compounds: (i) band gap (Eg ) in the 1.5–3.5 eV range, (ii) lift of CB minimum and VB maximum spin degeneracy by 0.1–0.2 eV for efficient

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PRL 102, 227204 (2009)

55 25

50

40

30

35

3

45 40

35

45

2

50

g

40

51

35

45

E (eV)

2.5

30 1.5 40

25 35

1

25

0.3

0.35

0.4

0.45

20

E /E 2

g

FIG. 2 (color). Contour plots of the conversion efficiency of an AM 1.5 (100 mW=cm2 ) illumination spectrum [28] by a DMS solar cell as a function of the E2 and Eg gaps. Contours are spaced by 1%. Under the same assumptions, a semiconductor in a standard photovoltaic device would be limited to 31% (with an optimal gap of 1.31 eV [1]).

spin polarization, (iii) existence of a FM state with, ideally, a Curie temperature TC greater than 300 K, (iv) existence of a spin polarized IB at a suitable energy (about 2=5 of the gap), with a high absorption coefficient for all transitions, (v) an E0 g gap smaller than the Eg gap. As noted before [17], efficient absorption on both transitions with the IB as the initial or final state will also require partially occupied intermediate states. The electronic structure of the ILSC has been studied in the framework of density functional theory (DFT). The total energy differences between FM and anti-FM configurations have been calculated as well as approximate determinations of the positions of the IB states. As is well known, the band gaps predicted by DFT are systematically too small. As such, the IB position of the 3d states as predicted by DFT is put in question. In ordinary semiconducting systems one can correct the DFT bandgap by applying the GW method [18] or by using exact exchange [19] or self-energy corrected potentials [20]. However, for simple systems, such as Si and Ge, the introduction of a

TABLE 1. Efficiencies and optimal values of absorption thresholds of DMS and ILSC solar cells for a given branching ratio of radiative to nonradiative recombination (Rr =Rnr ). The same branching ratio was assumed for all interband transitions. Type

Efficiency (%)

Eg (eV)

E2 (eV)

Eg E0g (eV)

DMS ILSC DMS ILSC DMS ILSC

52.0 49.5 42.1 39.5 34.2 31.0

2.338 2.433 2.341 2.481 2.678 2.863

0.931 0.933 0.934 0.953 1.14 1.15

0.301 0 0.424 0 0.649 0

Rr =Rnr 1 1 103 103 106 106


constant, k-independent shift (scissor operator) of the bands above the Fermi level that expands the bandgap without further affecting the band-structure can very well approximate the GW correction of the band-structure [18]. In the DMS case the scissor operator cuts can a priori be placed above and/or below the IB. In comparison with experimental results [21,22] it was clearly seen that the band gap correction should be fully applied above the Fermi level and thus the IB [16]. The DFT calculations were performed using projector augmented waves implemented in the VASP code [23] using both the spin polarized local density approximation and the generalized gradient approximation with the parameterization by Perdew and Wang [24]. Supercells of 48 to 64 atoms with periodic boundary conditions were studied with regular MonkhorstPack grids of 96 to 64 k-points in the Brillouin zone. The ionic positions were allowed to relax while the volume was held fixed at the theoretical equilibrium lattice parameter. Using Metropolis Monte Carlo simulations on an Ising spin system, one can from the exchange interaction determine approximate Curie or Ne´el temperatures of a given supercell configuration [5]. The supercells for the compounds with randomly dispersed Cr were constructed using special quasirandom structures [25]. For the more thorough study of AlP:Cr, supercell size effects were investigated by inserting several dopant atoms into larger cells; see Fig. 3. With the above design criteria in mind, the corresponding fraction of tetrahedraly coordinated semiconductors was searched for candidates. For binaries, growth induced compressive strain can be used to lift the degeneracy of the top of the VB to enable an efficient spin polarization, while selected ternaries such as I-III-VI2 exhibit naturally this feature [3,26]. To find FM candidates, doping with transition metals of the 3d row was explored ab initio. Of hundreds of simulated compounds (II-VI, III-V, and I-III-VI2 with Eg > 1:5 eV), almost half were found to be FM stable, however weakly, and among those the search was narrowed in order to find the ideal type of electronic structure. Among all studied compounds only 2% were found to satisfy all criteria. This justifies the ab initio screening of complex functional compounds having a long list of requirements to meet. The complete list of studied compounds cannot be discussed within the scope of this Letter. Here we present the results of Cr doping in the binary zinc-blende AlP: an indirect semiconductor with an almost optimal bandgap of 2.43 eV. Substitution by Cr on the Al site gives rise to a near perfect electronic structure, from the point of view of the design rules. The IB is present in only one spin channel and is there placed at 2=5 of the gap. In the other spin channel, the gap is somewhat closed by the spin split d levels that end up at the conduction band minimum. The band edge separations, as given in Fig. 1, are: E1 ¼ 1:33 eV, E2 ¼ 0:52 eV and Eg E0g ¼ 0:08 eV. An optimal conversion efficiency of 50% is calculated for AlP doped with 8% Cr on the cationic sites, only a few percent under the theoretical maximum conver-

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Institute, contract EP-P23606/C11428. The DFT computations were carried out on the EDF Blue-Gene/L parallel calculator.

FIG. 3 (color online). Magnetic stability (squares, right-hand scale) versus the open fraction of the gap (dashed line, left-hand scale) as functions of the Cr concentration for compounds with randomly dispersed dopants. The open gap fraction decreases with increasing dopant concentration. The dotted line marks the practical limit for the open gap fraction. At larger coverage the risk of band overlap between the spin channels becomes significant.

sion efficiency. The solute Cr produces an IB having direct transitions to both the CB and VB [16]. The compound is FM stable and has an estimated Curie temperature of about 160 K. The FM interaction between Cr atoms is longranged and particularly strong when the dopants are separated by cations, due to anisotropic p-d hybridization as in GaAs doped with Mn [27]. Thus a directed growth pattern could increase the FM stability. The exchange coupling is also increasing with concentration; see Fig. 3. At the same time, a doping level higher than 10% broadens the IB prohibitively for practical photovoltaic applications. The Curie temperature here presented correspond well with those found in the literature [2]. Indeed, alloys of GaAlP: Cr have been produced that exhibit large TC [29]. Game-changing pathways to attain highly efficient photovoltaic energy conversion are an active area of research with concepts, but not yet any real materials, showing potential for conversion efficiencies beyond 50%. The model here developed has well proven its usefulness since only one compound in 50 fits the basic requirements. Having defect levels or bands within the bandgap, a quite common feature, is unlikely by itself to provide the hoped for efficiency enhancement, but in favorable conditions, FM versions could. The dilute Cr doped AlP compound that has been shown to fulfill the criteria for optimal photovoltaic conversion may be one of them. With the rapid progresses of semiconductors for both spintronics and wide gap LED’s, suitable materials for high efficiency photovoltaics may be at hand, if looked for. The authors acknowledge the support by the French National Research Agency in project THRI-PV, contract ANR-PSPV-014-003, and by the Electric Power Research

[1] M. A. Green, Third Generation Photovoltaics: Advanced Solar Electricity Generation (Springer, Berlin, 2003). [2] T. Dietl et al., Science 287, 1019 (2000). [3] R. Fielderling et al., Nature Mater. 402, 787 (1999). [4] J. Guillemoles et al., J. Phys. Chem. A 106, 11 354 (2002). [5] J. M. Raulot, C. Domain, and J. F. Guillemoles, Phys. Rev. B 71, 035203 (2005). [6] M. J. Keevers and M. A. Green, J. Appl. Phys. 75, 4022 (1994). [7] S. Kettemann and J. F. Guillemoles, in Proceedings of the 13th Photovolt. Sol. Energy Conference, Nice (H. S. Stevens & Associates, Nice, 1995). [8] A. Luque and A. Marti, Phys. Rev. Lett. 78, 5014 (1997). [9] W. Shockley and H. J. Queisser, J. Appl. Phys. 32, 510 (1961). [10] In the radiative limit, the absorbed minus emitted photon balance is equal to the number of electrons flowing in the external circuit, and is simply obtained using assumption (iv) otherwise. [11] K. W. J. Barnham and G. Duggan, J. Appl. Phys. 67, 3490 (1990). [12] I. Schnitzer et al., Appl. Phys. Lett. 63, 2174 (1993). [13] P. Wurfel, Physics of Solar Cells (Wiley-VCH, Weinheim, 2005). [14] M. A. Green et al., Prog. Photovoltaics 14, 455 (2006). [15] A. Titov et al., Phys. Rev. B 72, 115209 (2005). [16] See EPAPS Document No. E-PRLTAO-102-070923 for discussion on the radiative to nonradiative recombination ratio, justification of the use of the scissor operator to correct the conduction band position, and details on the ab initio calculations on AlP:Cr. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html. [17] A. Luque et al., J. Appl. Phys. 99, 094503 (2006). [18] L. Hedin, Phys. Rev. 139, A796 (1965). [19] M. Sta¨dele et al., Phys. Rev. Lett. 79, 2089 (1997). [20] T. C. Schulthess et al., Nature Mater. 4, 838 (2005). [21] T. Dietl, F. Matsukura, and H. Ohno, Phys. Rev. B 66, 033203 (2002). [22] P. Mahadevan and A. Zunger, Phys. Rev. B 69, 115211 (2004). [23] G. Kresse and J. Furthmu¨ller, Comput. Mater. Sci. 6, 15 (1996). [24] J. P. Perdew et al., Phys. Rev. B 46, 6671 (1992). [25] A. Zunger et al., Phys. Rev. Lett. 65, 353 (1990). [26] A. Janotti and S.-H. Wei, Appl. Phys. Lett. 81, 3957 (2002). [27] B. Sanyal, O. Bengone, and S. Mirbt, Phys. Rev. B 68, 205210 (2003). [28] K. Emery, in Handbook of Photovoltaic Science and Engineering, edited by A. H. Luque, S. Hegedus (John Wiley & Sons, New York, 2003); http://rredc.nrel.gov/ solar/standards/am1.5/. [29] M. E. Overberg et al., Solid-State Electron. 47, 1549 (2003).

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