Electronic Electr onic and and La t La t t ice ice Polar a i r zat za ion n Effe f c e t c s on the the Band Band Structur

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1 Electronic and Lattice Polarization Effects on the Band Structure of Delafossite Transparent Conductive Oxides Fabio Trani 1 IDEA Virtual Lab (In Silico Development for emerging applications) Scuola Normale Superiore, Pisa, Italy [ 2 LPMCN (Laboratoire de Physique de la Matière Condensée et Nanostructures) CNRS & Université Claude Bernard Lyon 1, Villeurbanne, France [ lyon1.fr] lyon1.fr] E mail: fabio.trani@sns.it

2 In collaboration with Julien Vidal Laboratoire des Solides Irradiés, Ecole Polytecnique, Palaiseau, France now at NREL, Golden, Colorado Fbi Fabien Bruneval CEA, Service de Recherces de Métallurgie Physique, Gif sur Yvette, France Silvana Botti, Miguel A.L. Marques Laboratoire de Physique de la Matière Condensée et Nanostructures, CNRS & Université Claude Bernard Lyon 1, Villeurbanne, France

3 Outline Outline 1. Motivations p type conduction and p n homojunctions in delafossites interpretation of the results: the difficult comparison of experimental data and theoretical predictions 2. Methods GW 3. Results Band structures and band gaps of delafossite CuMO Conclusions

4 Motivations Thedelafossite crystal structure CuMO 2 Edge sharing MO 6 octahedral layers z CuO 2 dumbbell layers y x M is a group III element (Al, Ga, In, B)

5 Motivations A bit of literature: the first results 1997 Kawazoe et al. [Nature 389, 939 (1997)] P type electrical conduction in delafossite CuAlO 2 thin films 2001 Yanagi et al. [Applied Physics Letters 78, 1583 (2001)] Bipolarity in electrical conduction of delafossite CuInO Yanagi et al. [Solid State Comm. 121, 15 (2002)] All transparent oxide p n homojunction with delafossite CuInO 2. The situation coming from optical spectroscopy experiments on CuMO 2 thin films : low energy indirect gap, direct gap comparable to other TCOs. Indirect band gap (ev) Direct band gap (ev) CuAlO CuGaO CuInO Nie et al. [Phys. Rev. Lett. 88, (2002)] Simple interpretation of the band gap anomalies in delafossite TCOs: the band gaps are supposed to decrease on increasing the atomic number of M

6 Motivations The recent literature 2001 Yanagi et al. [Journal of Applied Physics 88, 4159 (2001)] Photoemission and inverse photoemission spectra for CuAlO 2, the fundamental band gap is about 3.5eV, inconsistent with the optically reported indirect gap 2002 Robertson et al. [Thin Solid Films 411, 96 (2002)] B3LYP calculations on delafossites, there is no trace of a low energy indirect bandgap 2006 Pellicer Porres et al. [Applied Physics Letters 88, (2006)] Single crystals instead of thin films, indirect gap at about 3eV in single crystals CuAlO Lasowsi et al. [Physical y Review B 79, (2009)] Large excitonic effects in CuMO 2, exciton binding energy at about 0.5eV Pellicer Porres et al. [Semicond. Sci. Technol. 24, (2009)] static dielectric constant measurement in delafossite CuAlO 2. Large polaron constant Snure and Tiwari [Applied Physics Letters 91, (2007)] First production of CuBO 2 dlf delafossite: large p type conductivity. it BUT the geometry was inconsistent (more than 10% off) with theoretical calculations [Scanlon et al. Chemistry of Materials 21, 4568 (2009)].

7 Motivations Motivations Summarizing: Delafossite CuMO 2 are technologically appealing for p n homojunctions, because they can serve as a transparent conductive layer for CuInSe 2 much used in solar cell technology. But they are very complex materials. They are characterized by Strong correlation effects (LDA band gaps are very far from the experiment) Strong excitonic effects Large electron phonon coupling Rich band structure: the transition at the direct band gap is optically forbidden It is unclear whether hth a low energy indirect gap exists it. Delafossite copper oxides are wide band gap semiconductors, or not? Lacing of Photoemission and Inverse photoemission measurements New experiments and theoretical investigation on CuBO 2 are required

8 The method The methods Scientific codes Abinit, VASP, Crystal Methods Density Functional Theory with LDA LDA+U Hybrid (B3LYP, PBE0, HSE03, HSE06) exchange correlation functionals Perturbative GW (calculated on top of LDA wavefunctions) SelfConsistent GW Pseudopotentials A particular care has been taen in the choice of the pseudopotentials, semicore states have been added to the Cu pseudopotentials. They strongly influence the GW energy levels.

9 The method Band gap anomaly LDA band structure and the band gap anomaly for CuMO 2. Indirect gap Direct gap gp Optical gap Both hindirect and direct fundamental lband gaps decrease on increasing i the M atomic number. BUT the optical band gaps, according to symmetry, are at L, and increase. Picture taen from Nie et al. Phys. Rev. Lett. 88, (2002)]

10 The method TheKohn Sham band gap underestimation van Schilfgaarde, Kotani, and Faleev, PRL 96, (2006)

11 The method Hybrid functionals Semi empiricalhybrid empirical exchange correlation correlation functionals contain a combination of standard functionals (LDA, GGA) and Hartree Foc exchange. There are many functionals existing in literature. The problem is that their accuracy is system dependent. Some of them are more accurate for wide bandgap materials, other functionals only wor for narrow gap semiconductors. Some functionals, lie B3LYP, or HSE06, give accurate band gaps in many structures. For instance, I used B3LYP to perform band structure calculations of oxygen vacanciesinin SnO2, obtaining a nice agreement with experiments[1,2]. However, they have to be used very carefully. The best thing is using them as as a starting point to perform G 0W 0 calculations, instead of LDA. [1] Trani F., et al. Phys. Rev. B 77, (2008) [2] Lettieri S., et al. Jour. Chem. Phys. 129, (2008)

12 The method The GW method Σ = igw Self energy: non local, non Hermitian, frequency dependent operator W = ε 1 v Screened potential: it requires the calculation of the dielectric matrix Ingredients: Kohn Sham Green s function RPA Dielectric matrix G 0 ( q ) 1 ε,ω GG' L. Hedin, Phys. Rev. 139 (1965).

13 The method Perturbative GW Kohn Sham equation: Quasiparticle equation: H 0 QP [ H + v ( r) ] ϕ ( r) = ε ϕ ( r) 0 xc KS ( r) + dr' Σ( r, r', ω = E ) φ ( r' ) E φ ( r) φ = Quasiparticleenergies energies within firstorder perturbativecorrection with QP KS QP KS QP QP Σ = igw E QP ε KS = ϕ KS Σ ( E ) QP vxc ϕks = Z ϕks Σ( ε KS ) vxc ϕks Basic assumption: φqp ϕ KS This scheme is called perturbative GW, in short G 0 W 0. M.S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986)

14 The method Perturbative GW In many materials, perturbative GW correct the LDA band gaps, and gives reasonable agreement with experimental data. In most metal oxides, however, quasiparticle wavefunctions are badly reproduced by LDA wavefunctions, so the GW correction is too small. van Schilfgaarde, Kotani, and Faleev, PRL 96, (2006)

15 The method SelfConsistent GW A recipe to perform SelfConsistent t GW has been given by Faleev and van Schilfgaarde[1,2]. It is based on a procedure to mae the SelfEnergy static and Hermitian. We used an approach, with neglect of dynamical correlation effects (COHSEX, Coulomb Hole and Static Screened Exchange approximation), while the last iteration is done using full GW[3]. Our approach is a perturbative GW on top of SelfConsistent Cohsex. It has been checed giving gexcellent results for many compounds, for instance CuIn(S,Se) 2, Cu 2 O, VO 2. [1] S.V. Faleev, Phys. Rev. Lett. 93, (2004) [2] M. van Schilfgaarde, et al. Phys. Rev. Lett. 96, (2006) [3] F. Bruneval et al. Phys. Rev. Lett. 97, (2006)

16 A tric to get converged results The method The method A tric to get converged results A b ttl f t d l i l i th l l ti f it d t t A bottlenec for most models involving the calculation of excited states: the convergence with respect to the number of empty states N b. Several approaches have been proposed in literature. The calculation of the polarizability in reciprocal space requires a sum over the states. The results are strongly dependent on the number of empty states N b included in the calculation ( ) ( ) ( ) ( ) ( ) Ω = < η ε ε ω η ε ε ω ω i i M M N P j i i j N j N i N ij ij v b v q q GG' G' q G q q 1 1 2, * T t d lt f ll dth t i d i Rf[1] To get converged results, we followed the tric proposed in Ref [1]. The idea is that of replacing the energy level of an empty state above N b by a constant parameter, and thus getting a simple expression for the error introduced by the use of a small number of empty states The next expression was easily implemented in abinit ( ) ( ) ( ) ( ) ( ) Ω = Δ b N j N i ij ij i j j M M j e j i i N r G) (G' GG' G' q G q q * , η ε ε ω η ε ε ω ω a small number of empty states. The next expression was easily implemented in abinit [1] F. Bruneval and X. Gonze, Phys. Rev. B 78, (2008) v N j

17 The method A tric to get converged results Not just a technical point. Without any trics the convergence is very slow, not reached with 1000 empty bands, and it leads to an underestimation of the band gaps. Using a value of 9.5Ha, the results are independent of the number of bands, so we could do our calculations using only 200 bands. This value is the same for all delafossite CuMO 2. F. Trani, J. Vidal, S. Botti, and M.A.L. Marques, Phys. Rev. B 82, (2010)

18 Results Results: CuAlO 2 band structure Indirect gap Direct band gap F. Trani, et al. Phys. Rev. B 82, (2010)

19 Results Results: CuAlO 2 band structure Upon going from LDA to G 0 W 0 to scgw, there is an upshift of the conduction band at Gamma with respect to L. Within scgw, the minimum of the conduction band is at L, the direct and indirect band gaps are almost identical Our results were confirmed by a recent, independent d calculation l reported by Christensen et al. Phys. Rev. B 81, (2010) F. Trani, et al. Phys. Rev. B 82, (2010)

20 Results Results: CuAlO 2 band structure Hybrids (HSE03) show a large difference with scgw in the conduction band, while the valence bands are very accurate. scgw corrections are strongly dependent. d The usual operation of adding a constant correction, is forbidden in delafossites. J. Vidal, et al. Phys. Rev. Lett. 104, (2010)

21 The method Lattice polarization Accordingto the experimental data, the polaron constant in CuAlO2is α 1 In delafossites we have non negligible contribution of the lattice polarization to the electronic screening. Ltti Lattice polarization effects can be included dwithin GW, simply adding the ionic i contribution to the dielectric matrix in the calculation of the screened potential W [1]. ( q, ω) ( q ) lat = δgg' + 4πPGG' ω + 4π PGG' ( q,ω) ε, GG' Using the long wavelength contribution of the lattice polarizability And the Lyddane Sachs Teller relationship for the static case, a simple expression for the lattice polarizability is derived from static dielectric constant ε 0 low frequency electronic constant ε zone center optical frequencies ω, LO ω TO [1] Bechstedt et al. Phys. Rev. B 72, (2005) P lat 00 ( ) q 0,ω

22 Results Results: CuAlO 2 band structure Be careful: the experiments come from optical measurements, and also include excitonic binding energy, estimated about 0.5eV. The agreement of LDA+U and HSE06 hybrid functional to the experiment is casual. scgw shows that t the band gaps are much higher in energy, the agreement with the experiment is due to the addition of lattice polarization effects. J. Vidal, et al. Phys. Rev. Lett. 104, (2010)

23 Results Results: CuAlO 2 band structure On increasing the complexity of the method (LDA+U >hybrid functionals >GW) the direct to indirect gap difference (green) becomes smaller and smaller There is no trace of a low energy indirect tbandgap J. Vidal, et al. Phys. Rev. Lett. 104, (2010)

24 Results Results: CuInO 2 band structure The fundamental direct gap gpis at Γ, but it is optically forbidden The direct gap at L is optically active Just lie CuAlO2, the difference with the experiment is about 1.7eV, liely due to excitonic (0.5eV) and polaronic (1.2eV) effects. There is no trace of a low energy indirect bandgap.

25 Results Results for CuMO 2 band gaps a LDA optimized geometry b Experimental geometry The direct to indirect band gap increases on increasing the M atomic number. Apart from the CuBO 2, scgw adds to the LDA direct band gap a constant amount of about 25eV 2.5eV.

26 Conclusions Conclusions scgw calculations show that the band gaps of delafossite CuMO 2 are very wide. Lattice polarization effects are very important, and their inclusion leads to a nice agreement with experimental data. There is no trace of a low energy indirect band gap. LDA+U and hybrid functionals must be used with care. The convergence with the number of empty bands has to be carefully checed. The GW corrections are dependent. Scissor operator is forbidden in delafossites.