Optical vs electronic gap of hafnia by ab initio Bethe-Salpeter equation
Benoît Sklénard, Alberto Dragoni, François Triozon, Valerio Olevano
OOptical vs electronic gap of hafnia by ab initio Bethe-Salpeter equation
Benoˆıt Skl´enard,
1, 2, a) Alberto Dragoni,
1, 2, 3
Fran¸cois Triozon,
1, 2 and Valerio Olevano
1, 3 Univ. Grenoble Alpes, F-38000 Grenoble, France CEA, LETI, MINATEC Campus, F-38054 Grenoble, France CNRS, Institut N´eel, F-38042 Grenoble, France (Dated: 7 November 2018)
We present first-principles many-body perturbation theory calculations of the quasiparticle electronic struc-ture and of the optical response of HfO polymorphs. We use the GW approximation including core electronsby the projector augmented wave (PAW) method and performing a quasiparticle self-consistency also onwavefunctions (QS GW ). In addition, we solve the Bethe-Salpeter equation on top of GW to calculate opticalproperties including excitonic effects. For monoclinic HfO we find a fundamental band gap of E g = 6 .
33 eV(with the direct band gap at E dg = 6 .
41 eV), and an exciton binding energy of 0.57 eV, which situates theoptical gap at E og = 5 .
85 eV. The latter is in the range of spectroscopic ellipsometry (SE) experimental esti-mates (5.5-6 eV), whereas our electronic band gap is well beyond experimental photoemission (PE) estimates( < GW works. Our calculated density of states and optical absorption spectra comparewell to raw PE and SE spectra. This suggests that our predictions of both optical and electronic gaps areclose to, or at least lower bounds of, the real values. Introduction
Hafnia (HfO ) is a transition metal ox-ide having attracted much attention due to its numeroustechnological applications, mainly related to its opticaland electrical insulating properties. It is used for opti-cal coatings in the near-ultraviolet (UV) to infrared (IR)wavelengths range, or as a high permittivity dielectric insubmicrometer silicon-based technologies. More recently,it is gaining interest as an insulating layer in resistive ran-dom access memories (ReRAM) which are a promisingcandidate for the next-generation nonvolatile memories.At ambient pressure, bulk HfO exists in three ther-modynamically stable crystalline polymorphs. At lowtemperature, the most stable phase exhibits a mono-clinic P /c symmetry (m-HfO ), and transforms intoa P /nmc tetragonal phase (t-HfO ) around 2000 K. At higher temperature, the tetragonal structure under-goes another phase transition to a
F m m cubic fluoritesymmetry (c-HfO ). In contrast to bulk samples, as-deposited HfO thin films are typically amorphous butcrystallize after anneal. After crystallization, the lowest-energy monoclinic phase is prevalent, but the presence oftetragonal and metastable orthorhombic phases have alsobeen observed.
The monoclinic is therefore the refer-ence phase and we will mainly focus on it, except whenthe other phases are explicitly mentioned.Several experimental techniques have been employedto characterize HfO thin films. On the one hand, bothX-ray (XPS) and ultraviolet (UPS) photoemission spec-troscopy (PES), and inverse photoemission (IPS) havebeen used to study the electronic structure. On theother hand, X-ray or optical absorption, spectro-scopic ellipsometry (SE), and electron energy-loss spectroscopy (EELS) have been used to studythe optical and dielectric properties. By using lineariza-tion and extrapolation techniques over measured spec- a) Electronic mail: [email protected] tra, these experiments extracted gap values ranging from5.1 to 5.95 eV . Surprisingly, the ranges for optical(5.1–5.95 eV) and electronic gaps (5.7–5.86 eV) over-lap, making unclear the distinction between them.The electronic structure of HfO polymorphs has alsobeen studied theoretically. First works on m-HfO ,by using density functional theory (DFT) in the localdensity (LDA) or generalized gradient approximation(GGA) , found band gaps (3.8–4.0 eV) underestimatedby 30% with respect to experimental data. Most recentworks, by using advanced semi-empirical functionals likeTBmBJ or many-body perturbation theory (MBPT)in the GW approximation , found band gaps in therange 5.7–5.9 eV and reconciled a good agreement withexperimental data. These studies do not account for ex-citonic effects but agree well with optical gaps derivedfrom SE and EELS.In this work we revisit the situation. We calculate theelectronic structure in the framework of MBPT withinthe GW approximation, also including core electronsby the projector-augmented wave (PAW) method andapplying self-consistency on wavefunctions (QS GW ). On top of QS GW , we perform Bethe-Salpeter equation(BSE) calculations of the optical gap and spectra in-cluding electron-hole interaction (excitonic effects). Fur-thermore, we perform a careful convergence study of ourresults (See supplementary material). Finally, instead ofcomparing our gaps with experimental values, we com-pare our DOS and optical absorption directly with theraw measured spectra. Our results indicate 5.85 eV (theenergy of the first exciton) as a lower bound for the them-HfO optical gap. This is still in the range of the ex-perimentally derived optical gaps. On the other hand,the comparison of our DOS with PES spectra clearly in-dicate 6.33 eV as a lower bound for the m-HfO electronicband gap. Our BSE calculation indicates the first peakof optical absorption as due to an exciton whose bindingenergy is 0.57 eV. a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov B D Y64202468 E n e r g y ( e V )
400 500 600 700Energy cutoff (eV)5.705.755.80 Q P b a n d g a p ( e V ) b Q P b a n d g a p ( e V ) FIG. 1. Left: Calculated band structure of monoclinic HfO within the DFT-PBE (dotted line) and QS GW (plain redline) approximations. The QS GW band structure has beeninterpolated using MLWF. PBE and GW Fermi energies arealigned at zero. Top right: convergence of the QP gap at Γ atthe G W level as a function of plane wave energy cutoff, withN b fixed to the maximum number of plane waves. Bottomright: convergence of the QP gap at Γ at the G W level asa function of the number of empty bands included (N b ) witha 500 eV plane wave energy cutoff. Computational details Ab initio calculations based ondensity functional theory in the LDA or PBE ap-proximations are carried out using the vasp code. The core-valence interaction are described with PAWdatasets including the semicore 5 s and 5 p states for Hf.Electron wave functions are expanded in a plane wavesbasis set with kinetic energy cutoff of 500 eV, and theBrillouin zone is sampled using 4 × ×
4, 6 × × × × phases, respectively.Many-body effects are accounted for by computing thequasiparticle (QP) energies at the G W , GW (self-consistency only on the eigenvalues) and QS GW level ontop of DFT but fixing W at W . Indeed, fixing W hasbeen shown to improve the agreement with experimentalband gaps than using self-consistent W . In contrast to G W and GW , QS GW allows to reduce the influence ofthe DFT starting point (LDA vs. PBE) on the electronicstructure. The cutoff for response function is taken to be333 eV and about 500 empty bands per formula unitand 100 frequency grid points are needed to obtain con-verged band gaps within 0.1 eV (see supplementary ma-terial for details). QP band structures are interpolatedusing maximally localized Wannier functions (MLWF)with the Wannier90 code. To determine the opticalproperties, the Bethe-Salpeter equation (BSE) is solvedon top of GW using the Tamm-Dancoff approximation. Band gap and electronic structure
In Fig. 1 we re-port the electronic band structure of m-HfO , as cal-culated within DFT in the PBE approximation andwithin QS GW . In the monoclinic crystal structure HfO presents a direct band gap at B , whereas the fundamental minimum band gap is indirect at Γ → B . Between DFTand the GW approximation there is some rearrangementof the bands, but the most important effect is a shiftof both valence and conduction bands which increasesband gaps. This is also what we found for the cubic andtetragonal phases.In Table I we report the fundamental minimum gapsfor all phases and approximations considered in thiswork, and we compare them to previous theoretical worksand to experiments. Regarding DFT results, our PAWLDA and PBE gaps are more in agreement with the FP-LAPW (full potential linearized augmented plane wave)LDA and PBE gaps of, respectively, Ref. 31 and 29(both are all-electron calculations), than with the norm-conserving pseudopotentials plane waves (NCPP PW)LDA gap of Ref. 30. On the other hand, both our G W and GW band gaps are systematically larger thanthe ones of Refs. 30 and 31. We remark however that TABLE I. Fundamental minimum band gap for the cubic( X → X ), tetragonal ( Z → Γ), and monoclinic (Γ → B ) HfO phases, as calculated in various approximations. Ref. 30 useda norm-conserving pseudopotentials plane waves approach,whereas Ref. 31 and Ref. 29 used FP-LAPW. The determi-nation of the band gap from PES/IPS experiments relies onpost-processing data analysis to remove tails due to impuri-ties and fits: for the XPS+IPS experiment of Ref. 10 we quotetwo estimates removing or not the effect of tails.method cubic tetragonal monoclinicThis workDFT-LDA 3.68 4.41 3.93LDA+ G W GW GW G W GW GW G W GW G W GW D O S ( a r b i t r a r y un i t s ) monoclinic HfO UPS+IPSXPS+IPSQS GW Conduction band edge
FIG. 2. Density of states (DOS) of monoclinic HfO in the QS GW approximation compared to UPS+IPS andXPS+IPS spectra. The QS GW DOS has been interpolatedusing MLWF on a 40 × ×
40 k-point grid and convolutedwith a gaussian broadening of 0.7 eV. Experimental spectrahave been aligned at the valence band-edge (the zero of the en-ergy for the theoretical DOS), and the conduction IPS and va-lence PS independent measurements have been rescaled sepa-rately to match the height of the theoretical DOS. The errorin this procedure can be estimated by the maximum deviationamong valence band-edges in the steepest rising linear region:50 meV, which is less than our theoretical error bar. our PAW G W corrections, ∆ E GWg = E GWg − E DFT g ,are closer to the NCPP PW ones than to the FP-LAPW ones, which are 0.3 to 0.6 eV lower. Differ-ences between Ref. 30 and our G W gaps could thenbe explained by the different starting DFT gap. Appli-cation of self-consistency only on eigenvalues, using the GW approach, further increases the gap by ∼ ∼ using theQS GW approach, removes any influence of the LDA orPBE starting point in our study, reducing the gap dif-ference to less than 0.1 eV, a residual due to the use ofdifferent relaxed LDA and PBE atomic structures.In the following we consider only PBE relaxed atomicstructure, the closest to the experiment. Our GW andQS GW calculations systematically yield larger band gapsthan previous theoretical studies. For the monoclinicphase, our values are 0.4 ∼ GW estimates of Refs. 30 and 31 and the TB-mBJ-origof Ref. 29. The latter are in very good agreement with the5.7 and 5.86 eV band gap values determined from pho-toemission experiments, whereas our QS GW gap of6.33 eV appears as a large overestimation. However, asdiscussed in Secs III.C and F of Ref. 10, there is someuncertainty in this determination of band gaps by theconventional method of linear extrapolation of photoe-mission band edges to the background intensity, due tothe presence of band-tail and defects in the vicinity ofthe valence band maximum and conduction band mini-mum. For this reason we prefer to directly compare the experimental PES+IPS spectra to our calculated DOS(Fig. 2). This comparison was already suggested in thesame Ref. 10 to provide a safer estimate of the real bandgap. By using a DFT-LDA DOS and evaluating the scis-sor operator shift to make theoretical and experimentalDOS coincide, they arrived to an estimate of 6.7 eV forthe m-HfO band gap . Our QS GW DOS favorablycompares with photoemission spectra, especially on theshape, even though we have not taken into account ex-trinsic and finite state effects which are evident whencomparing XPS with UPS shapes. As it can be estimatedby the deviation of theory and experiment in the conduc-tion edge (Fig. 2 inset), our QS GW band gap of 6.33 eV isstill an underestimation of about 0.2 eV of the real bandgap. Our more prudent conclusion is that the real bandgap of monoclinic HfO is E g > .
33 eV, and probably E g = 6 . foundwhen modifying the TB-mBJ functional (“semi” versionin Table I) to target the experimental DOS. The QS GW approximation has been reported to systematically over-estimate band gaps in all studied materials. In our case,for m-HfO , the close agreement between QS GW and ex-perimental spectra may be due to fortuitous error can-cellation with other effects not taken into account, suchas electron-phonon, and both single-particle (e.g. spin-orbit) or many-body (e.g. Breit interaction) relativisticcorrections. Optical gap and spectra
In Table II we report all theDFT and GW direct band gaps, and we add the opticalgaps calculated by solving the BSE on top of QS GW . TABLE II. Direct electronic band gap (DFT or GW ) andoptical gap (BSE first exciton eigenvalue energy) for the cubic( X → X ), tetragonal (Γ → Γ), and monoclinic ( B → B )HfO phases. We report also the exciton binding energy equalto the difference between the QS GW direct band gap andthe BSE optical gap, E excb = E dg − E og . For m-HfO , BSEgap agrees well with SE (spectroscopic ellipsometry) opticalonset. Energies are in eV.method cubic tetragonal monoclinicThis workDFT-LDA 3.68 4.58 4.03LDA+ G W GW GW G W GW GW GW +BSE 5.57 6.53 5.85 E exc b O p t i c a l a b s o r p t i o n () st exciton peak monoclinic HfO SE ALDSE MOCVDQS GW +BSEQS GW +RPA FIG. 3. Imaginary part of the dielectric function ε aver-aged over the three monoclinic HfO lattice directions andconvoluted with a gaussian broadening of 0.2 eV, comparedto ellipsometry spectra (SE) . We also report the first exciton binding energy, definedas the difference between the direct band gap energy andthe energy of the first exciton, E exc b = E dg − E og , foundto be 0.57 eV for m-HfO . Direct band gap and opticalgap are significantly different in m-HfO . The simulatedoptical gap can now be compared with the measured one,e.g., in optical or X-ray absorption, spectroscopic ellip-sometry (SE), or energy-loss (EELS). Refs. 7 and 9 re-ported values derived from SE spectra of 5.6-5.8 eV and5.5-6.0 eV, respectively. The uncertainty is due, as forthe band edges in the density of states, to the method(e.g. Tauc-Lorentz) used to linearly extrapolate to thebackground. However, with respect to photoemission,optical experiments are less affected by defect, surface,interface or substrate effects and more sensitive to thebulk. We remark that now our QS GW +BSE optical gapis in the range of the experimental reports. Neverthe-less, we again prefer to compare the raw SE spectrato our calculated optical absorption, ε ( ω ) (see Fig. 3).The QS GW +BSE dielectric function significantly im-proves the lower level of approximation QS GW +RPA,and achieves a very good agreement with SE spectra.BSE introduces electron-hole interaction effects and givesrise to the exciton which is to be identified with the firstpeak of the BSE spectrum, absent in the RPA. Neverthe-less we remark a 0.1 ∼ k -point sampling extrapo-lation to zero . Nevertheless, like the band gap, also ouroptical gap suffer an underestimation, so that they haveboth to be regarded as lower bounds of the real values.The nature of this small peak at the optical onset ob-served in SE spectra of crystalline samples has been at-tributed to different causes. By combining X-ray absorp-tion (XAS), X-ray diffraction (XRD) and SE techniques,Hill et al. found that this feature could be intrinsic tothe monoclinic phase. According to our analysis, this is monoclinic HfO xyz cubic HfO tetragonal HfO x / yz mono. y tetra. x / y cubic Energy [eV] O p t i c a l a b s o r p t i o n () FIG. 4. Imaginary part of the dielectric function ε for thethree HfO phases and all, (100), (010), and (001) polariza-tion directions. The arrow indicates the only non-dark boundexciton in the (010) polarization of m-HfO . a real bound exciton peak, as correctly interpreted inRefs. 15 and 29, and not a defect state, as interpreted byNguyen et al. . The exciton is present only in the y po-larization (see Fig. 4), confirming the unusual anisotropyin the dielectric properties of m-HfO . We foundan exciton also in c-HfO and t-HfO , but their oscillatorstrength is zero or almost, so that they are dark excitonsnot detectable in SE spectra. Hence SE spectra can beused to characterize the HfO monoclinic phase with re-spect to all other phases by simply detecting the presenceor absence of the 5.85 eV exciton peak. Conclusions
In this work we combine QS GW calcu-lations to compute the electronic structure of HfO withBSE to compute optical spectra. We compare our calcu-lated DOS and optical absorption with raw, as-acquired,experimental spectra measured for the monoclinic phase.Our calculated electronic band gap ( E g = 6 .
33 eV) issignificantly larger than the values obtained in previoustheoretical and experimental studies. However the di-rect comparison between QS GW DOS and experimentalspectra shows a good agreement and even indicates thatour band gap value slightly underestimates by ∼ . Supplementary Material
See supplementary materialfor information about the relaxed structures and conver-gence studies of GW and BSE calculations. Acknowledgements
Part of the calculations were runon TGCC/Curie using allocations from GENCI. cubic tetragonal monoclinicPBE LDA PBE LDA PBE LDAa (˚A) 5.082 4.994 3.594 3.533 5.145 5.048b (˚A) – – – – 5.206 5.142c (˚A) – – 5.225 5.076 5.326 5.206 β ( ◦ ) – – – – 99.63 99.53TABLE III. Structural parameters of cubic (spacegroup F m m ), tetragonal (spacegroup P /nmc ) and monoclinic (spacegroup P /c ) phases of HfO . Appendix A: Supplementary material
Structural parameters
All our calculations are basedon density functional theory within local density ap-proximation (LDA) or generalized gradient approxi-mation (GGA) with the parametrization of Perdew-Burke-Ernzerhof (PBE) in the framework of plane waveprojector-augmented wave (PAW) method as imple-mented in the vasp code.
The DFT calculations areperformed using the version 5.4 of LDA and PBE PAWpotentials of vasp (Hf sv GW and O GW for Hf and O,respectively).We study cubic (spacegroup
F m m ), tetragonal(spacegroup P /nmc ) and monoclinic (spacegroup P /c ) phases of HfO . Each structure is relaxed untilthe maximum residual forces are less than 10 − eV/˚A.For the relaxation we use a fine k mesh of 12 × × × × × × One-shot G W convergence We carefully examinethe convergence of our GW calculations in order toachieve QP band gap values converged within 0.1 eV.Indeed, to calculate the response function and the corre- b Q P c o rr e c t i o n CBMVBM
FIG. 5. QP corrections of the Kohn-Sham eigenvalues as afunction of the inverse of the number of empty bands N b (ornumber of plane waves N pw ). The dotted lines show the linearextrapolation of QP corrections (see text for details). lation part of self-energy, a summation over empty statesis required and quasi particle (QP) energies exhibit a veryslow convergence with respect to the number of virtualorbitals. Furthermore, the number of empty bands N b ,the corresponding orbital basis set N pw (controlled by theplane waves cutoff E pw ) and the size of the response func-tion basis set N χ pw (controlled by the plane waves cutoffE χ pw ) have to be increased simultaneously . Therefore,in our convergence study we fix E χ pw = 2 / E pw and usethe complete plane waves basis set (i.e. N b = N pw ) foreach considered E pw . We consider the monoclinic phaseof HfO with E pw ranging from 350 eV to 700 eV. Afrequency grid with 100 frequency points is used to rep-resent the polarizability and a 4 × × k -meshis used to sample the Brillouin zone. Fig. 5 shows the QPcorrections (∆ (cid:15) ) of the Kohn-Sham eigenvalues as a func-tion of 1/N b (which equals 1/N pw ) for a G W calcula-tion on top of PBE (a similar behavior is observed for anLDA starting point). The dotted lines show the linear ex-trapolation of QP corrections (∆ (cid:15) ( N b ) = A/N b +∆ (cid:15) ( ∞ ))where only the values corresponding to E pw ≥
500 eV areincluded in the fit. The extrapolated QP corrections ofthe valence band maximum (VBM) at Γ and conductionband minimum (CBM) at B are respectively -1.33 eV and0.51 eV giving an extrapolated band gap of 5.85 eV. Our GW LDA+ GW Q P b a n d g a p ( e V ) FIG. 6. Convergence of QP band gap as a function of self-consistent GW iterations at GW (top panel) and QS GW (bottom panel) level. DFT Convergence parameters QP energiesxc functional core elect. k mesh N b Dyn. Scr. E g (eV) ∆ E GWg (eV)This work G W PBE PAW 4 × × GW PBE PAW 4 × × G W LDA NCPP 4 × × GW G W LDA FP-LAPW 2 × × GW E g ) and GW corrections (∆ E GWg = E GWg − E DFT g ) from different GW calculationson monoclinic HfO . The calculations differ by the level of self–consistency, starting mean-field theory and GW convergenceparameters. The convergence parameters include: the k mesh, the number of empty bands (N b ) and the method to describedynamical screening (Dyn. Scr.). convergence study suggests that a 500 eV plane wavescutoff (3603 plane waves) allows to achieve a band gapconverged within 80 meV. For this basis set, the numberof empty bands can be decreased to 2000 without deteri-orating the convergence (see bottom right panel of Fig. 1in the main text).We also check the influence of the number of frequencypoints and k mesh. We find that increasing the frequencygrid from 100 to 200 points only changes the QP bandgap by 15 meV but tend to compensate the error donedue to incompleteness of the plane waves basis set. Whenusing a finer k-point sampling of 6 × ×
6, change in theQP band gap is below 1 meV.Our convergence study shows that the numerical con-vergence of our calculated QP band gaps is below80 meV. In table IV, we summarize the convergence pa-rameters used in our work with those from previous the-oretical studies. O p t i c a l a b s o r p t i o n () FIG. 7. Imaginary part of the dielectric function ε for themonoclinic phase of HfO calculated on top of PBE+G W for 4 × × × × k mesh. A Gaussian broadeningof 0.1 eV is used. Self-consistent GW convergence Self-consistent GW calculations are carried out with the same parametersas for G W . For GW and QS GW calculations, we in-clude respectively 256 and 512 QP energies in the self-consistent procedure for monoclinic HfO . In the caseof GW , 4 self-consistent iterations are enough to getconverged band gap. For QS GW , 8 self-consistent GW iterations allowed to get converged QP energies and bandgaps within 5 meV as shown in Fig. 6. Bethe-Salpeter equation (BSE) convergence
The ex-citonic properties are determined by solving the Bethe-Salpeter equation (BSE) within the Tamm-Dancoff ap-proximation on top of the GW quasiparticle band struc- D O S ( a r b i t r a r y un i t s ) monoclinic HfO UPS+IPSXPS+IPS G W GW QS GW FIG. 8. Density of states (DOS) of monoclinic HfO inthe G W , GW , and QS GW approximations compared toUPS+IPS and XPS+IPS spectra. The theoretical DOShave been interpolated using MLWF on a 40 × ×
40 k-pointgrid and convoluted with a gaussian broadening of 0.7 eV.Experimental spectra have been aligned at the valence band-edge (the zero of the energy for all theoretical DOS), and theconduction IPS and valence PS independent measurementshave been rescaled separately to match the height of the the-oretical DOS. ture. BSE calculations usually require fine k -point sam-pling to converge exciton spectra. However such cal-culations are computationally very expensive for both GW and BSE. In the case of monoclinic HfO we testthe k -point convergence of the BSE calculation on topof G W using 4 × × × × ε (we ignore the polarization dependence and assume ε =( ε xx + ε yy + ε zz ) / k -meshes give very similar spectra. Morequantitatively, the exciton binding energies of 4 × × × × k -mesh could be carried out using a model BSE (mBSE)scheme or interpolation techniques but have not beenconsidered in this work. DOS
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