First-principles materials design of high-performing bulk photovoltaics with the LiNbO 3 structure
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug First-principles materials design of high-performing bulk photovoltaics with theLiNbO structure Steve M. Young
Center for Compuational Materials Science, United States Naval Research Laboratory, Washington, DC 20375, USA
Fan Zheng and Andrew M. Rappe
The Makineni Theoretical Laboratories, Department of Chemistry,University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA
The bulk photovoltaic effect is a long-known but poorly understood phenomenon. Recently,however, the multiferroic bismuth ferrite has been observed to produce strong photovoltaic responseto visible light, suggesting that the effect has been underexploited as well. Here we present threepolar oxides in the LiNbO structure that we predict to have band gaps in the 1-2 eV range and veryhigh bulk photovoltaic response: PbNiO , Mg / Zn / PbO , and LiBiO . All three have band gapsdetermined by cations with d s electronic configurations, leading to conduction bands composed ofcation s -orbitals and O p -orbitals. This both dramatically lowers the band gap and increases the bulkphotovoltaic response by as much as an order of magnitude over previous materials, demonstratingthe potential for high-performing bulk photovoltaics. Photovoltaic effects have long been observed in bulkpolar materials, especially ferroelectrics [1–4]. Knownas the bulk photovoltaic effect (BPVE), it appeared toderive from inversion symmetry breaking. Despite in-tense initial interest, early explorations revealed low en-ergy conversion efficiency, in part due to the high bandgaps of most known ferroelectrics. Additionally, despiteseveral proposed mechanisms, the physical origin of theBPVE was unclear [2, 5–8].However, recent emphasis on alternative energy tech-nologies and the observation of the effect in novel semi-conducting ferroelectrics (with band-gaps in the visiblerange) has renewed interest [9–14]. Several studies haveattempted to elucidate the various contributions to thephotovoltaic response – bulk or otherwise – in ferro-electrics [15–23]. In particular, bismuth ferrite (BFO) hasbeen found to generate significant bulk photocurrents;combined with its unusually low band gap of 2.7 eV, ithas attracted a great deal of attention for its potential inphotovoltaic applications [21, 23–31]. The understand-ing of the fundamental physics behind the effect has ad-vanced as well; recently we demonstrated that the BPVEcan be attributed to “shift currents”, and that the bulkphotocurrents may be calculated from first-principles.The ab initio calculation of the shift current and sub-sequent analysis yielded several chemical and structuralcriteria for optimizing the response. These criteria havebeen used previously to modify or identify existing mate-rials with enhanced response [14, 32–34]. In this work, weuse these insights to propose several candidate bulk pho-tovoltaics with calculated response as much as an orderof magnitude higher than well-known ferroelectrics, whilehaving band gaps in or slightly below the visible spec-trum. Our results demonstrate that bulk photovoltaicresponse can be much stronger than previously observed,supporting the possibility of materials suitable for appli-cation. There are two figures of merit for evaluating the BPVEin a material: the current density response to a spatiallyuniform electric field, and the Glass coefficient [3]. Thecurrent density response is given by the tensor J q ( ω ) = σ rsq ( ω ) E r ( ω ) E s ( ω ) σ rsq ( ω ) = e X n ′ ,n ′′ Z d k I rs ( n ′ , n ′′ , k ; ω ) R q ( n ′ , n ′′ , k )where E is the vector of the illumination field, and n ′ and n ′′ index bands. Letting f denote filling, χ the Berryconnection, and φ the phase of the transition dipole, theexpression I rs ( n ′ , n ′′ , k ; ω ) = π (cid:16) em ¯ hω (cid:17) ( f [ n ′′ k ] − f [ n ′ k ]) × h n ′ k | ˆ P r | n ′′ k i h n ′′ k | ˆ P s | n ′ k i× δ ( ω n ′′ ( k ) − ω n ′ ( k ) ± ω ) (1)describes the intensity of transitions, and R q ( n ′ , n ′′ , k ) = − ∂φ n ′ n ′′ ( k , k ) ∂k q − [ χ n ′′ q ( k ) − χ n ′ q ( k )](2)is the expression for the “shift vector”, which describes adistance associated with the excited carrier [5, 35], anddepends on the differences in the wavefunction centers upto a unit cell, as provided by the Berry connections, andthe average separation in unit cells given by the transi-tion dipole phase derivative. Roughly speaking, the twoterms I and R can be thought of as giving the number ofcarriers excited and the velocity of those carriers. We em-phasize that this mechanism is profoundly different fromother photovoltaic effects; rather than relying on excita-tion of carriers which are then separated by an electricfield, the carriers are electron/hole pairs in coherent ex-cited states that possess intrinsic momentum of oppositesign. Crucially, this allows for arbitrarily high photovolt-ages; the Schockley-Queisser limit does not apply, a ma-jor advantage of BPVE. In particular, the open-circuitvoltage is determined by the competition between thephotocurrent and countervailing voltage-driven leakagethat depends on the overall resistance of the sample [36].This sample dependence prevents straightforward calcu-lation.Determining the total current in a sample is compli-cated by the attenuation of incident illumination as ittravels through the material. In the limit of a thick sam-ple that will completely absorb the illumination, the totalcurrent can be obtained from the Glass coefficient G ¯ J q ( ω ) = σ rrq ( ω ) α rr ( ω ) (cid:12)(cid:12) E r ( ω ) (cid:12)(cid:12) W = G rrq ( ω ) I r ( ω ) W (3)where α is the absorption coefficient, and W is the sam-ple width. Thus, the current density tensor and Glasscoefficient describe the response in the regimes of near-zero and near-infinite thickness, respectively. In practice,“infinite thickness” is on the order of microns, and totalphotocurrent is usually best described by the Glass coef-ficient.However, the Glass coefficient provides additional in-formation about the response. In the limit where ǫ i ≪ ǫ r , α ≈ ωcn ǫ i = (cid:16) em (cid:17) πǫ cn ¯ hω X n ′ ,n ′′ Z d k I rs ( n ′ , n ′′ , k ; ω )and the Glass coefficient becomes G rrq ( ω ) = 12 ǫ cn σ rrq ( ω ) α rr ( ω ) = e hω P n ′ ,n ′′ R d k I rr ( n ′ , n ′′ , k ; ω ) R q ( n ′ , n ′′ , k ) P n ′ n ′′ R d k I rr ( n ′ , n ′′ , k ; ω )The Glass coefficient is therefore closely related to theweighted average shift vector, allowing us to estimate thecontribution of both terms in the shift current expression.Shift current response was calculated as in Ref. [37],from wavefunctions generated using density functionaltheory, with the generalized gradient approximation(GGA) and optimized, norm conserving pseudopoten-tials [38, 39]. The presented results exclude spin-orbiteffects; calculations with and without spin-orbit were per-formed for both LiBiO Mg / Zn / PbO and were notfound to substantially influence the results. For BFO, aHubbard U of 5 eV was used for Fe 3 d , as in Ref. [40].For PbNiO , a Hubbard U of 4.6 eV was used for Ni 3 d ,as in Ref. [41]. QUANTUM ESPRESSO [42] was usedfor the electronic structure calculations, and OPIUM wasused to generate pseudopotentials. The Heyd-Scuseria-Ernzerhof (HSE) hybrid functional [43] was used to com-pute band gaps, as it is known to frequently producesignificantly more accurate values than GGA. These cal-culations were performed on 8 × × FIG. 1. The LiNbO (LNO) primitive cell, overlaid with thepseudo-cubic perovskite cell. The direction of the polar dis-tortion is indicated by the black arrow. × × , and computationally relaxed structures for theother materials. Structural relaxations and calculationsof the shift current were performed at the level of LDAand found to vary minimally from the GGA results; dueto the high expense of exact exchange calculations, thedense k-point grids required to converge shift current cal-culations cannot presently be obtained using HSE, andscissor corrections [44, 45] to the HSE gaps were appliedto account for the dependence of Glass coefficient on fre-quency.Previously, we revealed the dependence of shift vec-tor magnitude on the chemical and structural propertiesof materials. Large shift vectors were characterized byvalence and/or conduction states that are both stronglyasymmetric and delocalized in the current direction [37].In this regard, many distorted perovskite ( AB O ) ferro-electrics are crippled by the presence of d cations en-closed in octahedral oxygen cages. The conduction bandedge is dominated by t g -like d states that are largelynonbonding. Coupled with the tendency for d states tolocalize, the result is that both shift vectors and tran-sition response are very weak near the band gap. Thedelocalized e g states are much higher in energy, effec-tively raising the energy threshold for significant BPVE.To overcome the weak BPVE response of d oxides,we investigated systems that involve both large distor-tions to oxygen cages, (increasing the bonding characterof any d states) as well as d cations with less localized s and/or p states near the band edge [32]. It has alreadybeen noted that d cations can dramatically improve theactivity of photocatalysts [50]. We found polar oxidestaking the LiNbO structure to be promising candidates, PbNiO Mg / Zn / PbO LiBiO a 5.63 ˚A 5.77 ˚A 5.67 ˚A α ◦ ◦ ◦ A (2a) (0.0, 0.0, 0.0) (0.0, 0.0, 0.0) (0.0, 0.0, 0.0) B (2a) (0.214, 0.214, 0.214) (0.216, 0.216, 0.216) (0.213, 0.213, 0.213)O (6b) (0.830, 0.098, 0.415) (0.794, 0.128, 0.390) (0.798, 0.122, 0.405)P 99 µ C/cm µ C/cm µ C/cm TABLE I. The structural data for the three compounds presented here. PbNiO and LiBiO are in space group R3c, whileMg / Zn / PbO is in R3. However, the deviations of the coordinates from the R3c positions are miniscule ( < . A position. Polarizations were determined based on anon-polar structure featuring the A -site atom coplanar with oxygen, and the B -site midway between oxygen planes [46–49]. eV xxZzzZ ( A / m ²) / ( W / m ²) - (a) eV xxZzzZ V c m - - (b) FIG. 2. The current density response for BFO (GGA+ U ) isshown in (a). The Glass coefficient of BFO appears in (b).Only the response in the direction of material polarizationis shown, for both perpendicular ( xxZ ) and parallel ( zzZ )light polarization. Dashed lines appear at benchmark valuesof current density and Glass coefficient chosen to representthe maximum response of these materials. with d s cations Pb and Bi . This structure canalso be obtained by distorting the perovskite structurerhombohedrally, and allowing polar distortions along andoxygen-cage rotations about h i . Notable ferroelectricswith this structure (but with d cations) include LiNbO (LNO) and BiFeO (BFO). LNO is known for its largenonlinear optical response, and, often doped with iron,it was one of the first materials in which the bulk photo-voltaic effect was observed and studied [2, 51, 52]. How-ever, its bulk band gap is well outside the visible spec-trum [53]. BFO has garnered much attention recently forits multiferroic behavior [54] and low band gap of about2.74 eV [55], which has led to explorations of its photo-voltaic response [21–24, 56, 57]. We have used BFO as abenchmars for the present study; as with the archetypalferroelectrics BaTiO and PbTiO , its LUMO is dom-inated by cation d -states and yields a very similar re-sponse magnitude.We consider only current response in the direction ofmaterial polarization for both perpendicular ( xxZ ) andparallel ( zzZ ) light polarization, as these are the onlytensor elements that can contribute to the response tounpolarized light. For ease of comparison, we mark base-line values reflecting the maximum response of our bench-mark, shown in Fig. 2, with a dashed, red line. These are,for the current density and Glass coefficient, respectively, 5 × − (A / m ) / (W / m ) and 5 × − cm/V.We have studied three materials taking the LNO struc-ture (Fig 1): PbNiO , Mg / Zn / PbO , and LiBiO .The first has been synthesized [58], and the latter twoare similar in composition to known materials. The struc-tural parameters and bulk polarizations are given in Ta-ble I. The distortion from cubic perovskite is sufficientlystrong that assignment of A - and B -sites is ambiguous;we have followed the assignment of Ref. [58] for PbNiO ,but note that treating Ni as the A -site (reversing theorientation), the Wyckoff position of oxygen becomes(0 . , . , . d cations, and large polar distortions. Furthermore, as seenin Fig. 3, all three have qualitatively similar band struc-tures, featuring highly dispersive conduction bands, incontrast to the usual case of d perovskite derivatives.As we will show, this arises due to unfilled s -like – ratherthan d -like – states composing the conduction band, andhas profound consequences for the bulk photovoltaic re-sponse.PbNiO has recently been synthesized [58] and ex-plored theoretically [41, 59]. Like BFO, it is antiferro-magnetic with weak spin-canting, and possesses an evenlarger polarization, calculated at 100 µ C/cm [41]. Itsband gap is even lower than BFO, with HSE predict-ing 1.2 eV [41]. In BFO, Bi has oxidation state 3+, sothat its 6 s orbital is filled, and the exchange splittingof Fe determines the gap. However, in PbNiO , Pb is4+, and its 6 s -states appear lower in energy than theNi exchange-split bands, resulting in a distinct electronicprofile. This can be clearly seen in the projected densityof states (Fig. 4(a)): the lowest conduction band is al-most entirely Pb 6 s and O 2 p states, while the d -statesonly appear in the valence band and higher in the con-duction manifold. While this serves to lower the bandgap dramatically, a further result of this is a Glass coeffi-cient (Fig. 4(d)) over an order of magnitude larger thanthe benchmark value. The current density is modest bycomparison, though it still exceeds the benchmark, indi-cating large shift vectors with relatively low absorption.HgPbO [60] and ZnSnO [61, 62] are known to take (a) (b) (c) FIG. 3. The band structures of (a) PbNiO , (b) Mg / Zn / PbO , and (c) LiBiO . Note the similar, highly dispersiveconduction band edges. -5 0 5 10 eV O ( p )Ni ( d ) (a) -5 0 5 10 eV O ( p ) Pb ( s )Pb ( p ) (b) ( A / m ²) / ( W / m ²) - xxZzzZ eV (c) eV xxZzzZ V c m - - (d) FIG. 4. (a) and (b) give the projected density of states forPbNiO . The unfilled half of e g of the high-spin d Ni ap-pears as a sharp peak above the unfilled Pb s -orbitals, whichhave strongly hybridized with oxygen p -orbitals, resulting ina low band gap (1.2eV in HSE [41]). This material has alarge (c) current density response ( ≈ × benchmark), and avery large (d) Glass coefficient( ≈ × benchmark). the ilmenite and LiNbO structures, respectively. How-ever, the first is metallic and the second has a high bandgap and only modest photovoltaic response. We first cal-culated the response of ZnPbO , but found it to be bor-derline metallic, despite promising response; to raise thegap, we substituted Mg for half of the Zn. Phonon cal-culations indicate that the structure is metastable. Onceagain, as seen in Fig. 5(b), hybridized Pb 6 s states com-pose the lowest unfilled band. The magnitude of theresponse is quite high, but the current is antiparallel tothe computed polarization. This is unlike most mate-rials, including our benchmark materials and the afore-mentioned PbNiO , however, we emphasize the ambi-guity of both polarization (the addition of polarization -5 0 5 10 eV O ( p ) Zn (d) (a) -5 0 5 10 eV O ( p ) Pb ( s )Pb ( p ) (b) -30-25-20-15-10-50 ( A / m ²) / ( W / m ²) xxZzzZ eV (c) - eV xxZzzZ V c m - - (d) FIG. 5. Orbital-projected densities of states forMg / Zn / PbO are shown in (a) and (b). The valence bandis formed almost entirely from oxygen p -orbitals, and the con-duction band is hybridized Pb 6 s and O 2 p -states. This re-sults in a low band gap (1.2 in HSE), (c) high current densityresponse ( ≈ × benchmark), and (d) a very large Glass coef-ficient ( ≈ × benchmark). quanta) and structure orientation (designation of A and B cations) for these materials. If we compare the two Pbcompounds with Pb as the B-site in both, not only arethe structures more similar, but their responses becomeparallel.LiBiO is known to exist in a structure with edge shar-ing oxygen octahedra [63]. However, our calculationsplace the LiNbO -type structure – which phonon anal-ysis reveals to be metastable – nearby in energy, at onlyabout 0.01 eV per atom higher; additionally, NaBiO isknown to take the closely-related ilmenite structure [63].In light of this, we consider it highly possible that theLiBiO can be synthesized in the LiNbO structure.As shown in Fig. 6(a) and Fig. 6(b), the electronic -5 0 5 10 eV O ( p ) Li ( p ) (a) -5 0 5 10 eV O ( p ) Bi ( s )Bi ( p ) (b) - ( A / m ²) / ( W / m ²) - eV xxZzzZ (c) -505101520 xxZzzZ eV V c m - - (d) FIG. 6. The density of states for LiBiO , shown in (a)and (b), is dominated by bismuth and oxygen. The band gapis set by transitions from O 2 p to hybridized Bi 6 s states.The band gap is modest (1.7eV in HSE). The current densityresponse, shown in (c) is quite high ( ≈ × benchmark), witha large (d) Glass coefficient ( ≈ × benchmark), indicatingstrong absorption in addition to long shift vectors.Direct gap Max. G Max. σ × eV × − cm/V × − / m W / m BiFeO [33] 1.1 [64] 98 30PbNiO / Zn / PbO and otherrecently proposed bulk photovoltaics (with scissor correctedresponses) for comparison. structure is very similar to the previous two materials. Aswith Pb-containing compounds, the low-lying hybridizedBi s states form the lowest unfilled bands, though theBi s proportion is lower than that of Pb s in the afore-mentioned materials. Possibly as a consequence, the dis-persion of the conduction band is reduced compared toPbNiO and Mg / Zn / PbO (Fig. 3), and the BPVEresponse is somewhat different: while the Glass coeffi-cient is not as large as for the two lead-containing ma-terials, the photocurrent density is higher, indicating in-creased absorption. Additionally, the band gap is larger,with HSE predicting 1 . . withthat of the two lead compounds: the former has a notably distinct response, especially near the band edge. Thiscan be attributed to the difference in valence band char-acter; in PbNiO and Mg / Zn / PbO the valence bandedge contains considerable density from the secondarycation filled d -states. This alters the character of thewavefunctions and improves delocalization and responsemagnitude by sharing density, as opposed to lithium’salmost completely ionic character. This suggests that in-clusion of an appropriate dopant with higher electroneg-ativity may allow for significant tuning of the responsein LiBiO .We have proposed several polar oxides in the LiNbO structure with strong computed BPVE response and lowband gaps, summarized in Table II. The compositions,featuring Pb or Bi cations, were chosen for the ab-sence of d -states at the band edge. Instead, these mate-rials have conduction bands formed by low-lying s -stateshybridized with oxygen p -states. In addition to creatingsignificantly lower band gaps, this makes for large, diffuseorbitals and strongly delocalized states; combined withlarge polar distortions, they effect significant shift cur-rent response that is over an order of magnitude higherthan that previously observed, and roughly double thebest performing materials previously proposed. Giventhe minimal contributions from the other cations, thepossibility of tuning the response via composition with-out altering its fundamental character is strongly sug-gested. Moreover, in combination with recent demonstra-tions that careful device construction can dramaticallyimprove BPVE performance [13, 23, 36], these results in-dicate that BPVE can be much stronger than previouslythought, bolstering hopes that the phenomenon can besuccessfully exploited.S. M. Y. was supported by the Department of En-ergy Office of Basic Energy Sciences, under grant numberDE-FG02-07ER46431, and a National Research Coun-cil Research Associateship Award at the US Naval Re-search Laboratory. F.Z. was supported by the Officeof Naval Research under Grant No. N00014-11-1-0664.A.M.R was supported by the Office of Naval Researchunder grant N00014-12-1-1033. Computational supportwas provided by the HPCMO of the DoD and NERSC ofthe DOE. [1] A. G. Chynoweth, Phys. Rev. , 705 (1956).[2] F. S. Chen, J. Appl. Phys. , 3389 (1969).[3] A. M. Glass, D. von der Linde, and T. J. Negran, Appl.Phys. Lett. , 233 (1974).[4] V. M. Fridkin, Crystallog. Rep. , 654 (2001).[5] R. von Baltz and W. Kraut, Phys. Rev. B , 5590(1981).[6] B. I. Sturman and V. M. Fridkin, The Photovoltaic andPhotorefractive Effects in Noncentrosymmetric Materi-als , edited by G. W. Taylor, Ferroelectricity and Related
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