Diffuse scattering in metallic tin polymorphs
Björn Wehinger, Alexeï Bosak, Giuseppe Piccolboni, Keith Refson, Dmitry Chernyshov, Alexandre Ivanov, Alexander Rumiantsev, Michael Krisch
DDiffuse scattering in metallic tin polymorphs
Bj¨orn Wehinger , Alexe¨ı Bosak , Giuseppe Piccolboni , KeithRefson , Dmitry Chernyshov , Alexandre Ivanov , AlexanderRumiantsev , and Michael Krisch European Synchrotron Radiation Facility, BP 220 F-38043 Grenoble Cedex 9,France STFC Rutherford Appleton Laboratory, Oxfordshire OX11 0QX, United Kingdom Swiss-Norwegian Beamlines at European Synchrotron Radiation Facility, Grenoble,France Institut Laue-Langevin, Grenoble, France Russian Academy of Sciences, Moscow, RussiaE-mail: [email protected]
Abstract.
The lattice dynamics of the metallic tin β and γ polymorphs hasbeen studied by a combination of diffuse scattering, inelastic x-ray scattering anddensity functional perturbation theory. The non-symmorphic space group of the β -tin structure results in unusual asymmetry of thermal diffuse scattering. Strongresemblance of the diffuse scattering intensity distribution in β and γ -tin wereobserved, reflecting the structural relationship between the two phases and revealingthe qualitative similarity of the underlying electronic potential. The strong influenceof the electron subsystem on inter-ionic interactions creates anomalies in the phonondispersion relations. All observed features are described in great detail by densityfunctional perturbation theory for both β - and γ -tin at arbitrary momentum transfers.The combined approach delivers thus a complete picture of the lattice dynamics inharmonic description. a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n iffuse scattering in metallic tin polymorphs
1. Introduction
Metallic tin crystallizes in a body-centred tetragonal lattice (space group I4 /amd) atambient conditions, known as white tin ( β − Sn). Despite the fact that the stabilityrange of white tin lies between 291 and ≈
450 K [1] it can be supercooled far below thetransition temperature maintaining the crystal structure. Below ≈ α − β phase transition in tin is possiblythe simplest and prototypical case of an entropy-driven structural transformationdetermined by the vibrational properties of the two phases [4].Alloying tin with indium results in a substitutionally disordered crystal with aprimitive hexagonal lattice containing one atom per unit cell [5], called γ -tin. Itis a convenient model system in the study of lattice dynamics and electron-phononinteractions [6], because its phonon dispersion relations consist only of acoustic branchesand it is stable at ambient conditions. The γ -phase of pure tin [1] is orthorhombic anddiffers thus slightly from the primitive hexagonal lattice. It is stable between ≈
450 Kand the melting point of tin (505 K).Diffuse scattering of white tin has a long history. First Laue photographs showing a”diffuse background with regions of maximum intensities” were published in 1943 [7] and”considered in the light of thermal theory” in 1946 [8]. Elastic constants were derivedfrom the diffuse features in 1955 [9]. Phonon dispersion relations have been largelystudied in the past, in particular by inelastic neutron scattering (INS) [3, 10, 11] anddensity functional perturbation theory [4]. The available data are nevertheless limitedto high-symmetry directions and the determination of eigenfrequencies. The rich Fermisurface of β − Sn [12] suggests a complex topology of electron-phonon interaction, studiedin [13].In this study we investigate the lattice dynamics of the metallic tin polymorphsemploying a combination of thermal diffuse x-ray scattering (TDS), inelastic x-rayscattering (IXS) and density functional perturbation theory (DFPT) in order to obtainthe full description of the lattice dynamics at arbitrary momentum transfer.
2. Experimental Details
The diffuse x-ray scattering experiment was conducted at the Swiss-NorwegianBeamlines at ESRF (BM01) and the ID29 ESRF beamline. Monochromatic X-rays iffuse scattering in metallic tin polymorphs × µ mcross section at room temperature. The sample was rotated orthogonal to the incomingbeam over an angular range of 360 ◦ and diffuse scattering pattern were recorded intransmission geometry. Preliminary experiments were performed with a mar345 imageplate detector. The follow-up experiments employed a single-photon-counting PILATUS6M pixel detector [14]. The diffuse scattering patterns were collected in shutterless modewith a fine angular slicing of 0.1 ◦ . The experimental set-up is documented elsewhere[15]. The orientation matrix and the geometry of the experiment were refined usingthe CrysAlis software package; 2D and 3D reconstructions were prepared using locallydeveloped software. The single crystal IXS study was carried out at beamline ID28 atthe ESRF. The spectrometer was operated at an incident energy of 17.794 keV, providingan energy resolution of 3.0 meV full-width-half-maximum. IXS scans were performedin transmission geometry along selected directions in reciprocal space. Further detailsof the experimental set-up can be found elsewhere [16].
3. Calculation
First-principles lattice dynamics calculations were performed with the CASTEP package[17, 18] using the DFPT solver for metallic systems [19] at 0K. The local densityapproximation (LDA) and general gradient approximation (GGA) within the densityfunctional theory formalism were used as implemented with a plane wave basis setand norm-conserving pseudopotentials. For the exchange correlation functional thePerdew and Zunger parametrization [20] of the numerical results of Ceperley and Alder[21] were used in LDA and the density-gradient expansion for exchange in solids andsurfaces (PBEsol functional) [22] in GGA. The self-consistent electronic minimizationwas performed with density mixing in the Pulay scheme and the occupancies weresmeared out by a Gaussian function of 0.1 eV full-width-half-maximum. The Sn pseudo-potential was of the optimized norm conserving type generated using the Vanderbiltscheme with a single projector for each of the 5s and 5p electrons, with a cut-offradius of r c = 1 . a . The pseudopotentials for the LDA and PBEsol calculationswere created using the CASTEP on-the-fly technology, created separately for eachexchange and correlation functional and carefully tested for transferability ‡ . Numericalapproximations were chosen to achieve convergence to a tolerance of < − eV/˚A for ‡ The CASTEP on-the-fly stings used in this work are 2 | . | . | . | . | . | . | . | . | . | . | . | . | . | .
05) in PBEsol. iffuse scattering in metallic tin polymorphs Table 1.
Lattice constants of β -Sn. LDA PBEsol Experiment [33]a = b 5.755 ˚A 5.808 ˚A 5.831 ˚Ac 3.114 ˚A 3.144 ˚A 3.182 ˚Ainternal forces which required a plane wave cut-off of 380 eV and 24 × ×
24 Monkhorst-Pack grid sampling of the first Brillouin zone.The structure optimization was performed using the Broyden-Fletcher-Goldfarb-Shannon method [23] by varying lattice and internal parameters. The equilibriumlattice constants of β -Sn as obtained in LDA and PBEsol are reported in Table 1.The cell parameters agree within 2.2 % with the experimental values in LDA and 1.2% in PBEsol. Phonon frequencies and eigenvectors were computed by perturbationcalculations in harmonic approximation on a 8 × × < W j, q = a | ∂ω j, q ∂ q | ∆ q (1)proportional to the band derivatives, where a is a dimensionless constant of the orderof unity.The lattice dynamics calculation for γ -tin was performed in LDA with the sameparameters and pseudopotential as used for β -Sn. The primitive hexagonal structure wasimposed for the unit cell containing one Sn atom. The optimized cell parameters werea = b = 3.1667 ˚A and c = 2.9722 ˚A, in agreement within 1.4 % with the experimentalvalues (a = b = 3.213 ˚A and c = 2.999 ˚A [26]). TDS and IXS intensities were calculatedfollowing the previously established formalism [16, 27, 28], assuming the validity of bothharmonic and adiabatic approximation. The scattering intensities were calculated in iffuse scattering in metallic tin polymorphs I ( Q , ¯ hω ) = ¯ hN I e (cid:88) j ω Q ,j coth (cid:16) ¯ hω Q ,j k B T (cid:17) | (cid:88) s f s √ m s e − M s ( Qe Q ,j,s ) e − i Ql s | δ (¯ hω − ∆¯ hω ) , (2)where Q denotes the scattering vector, ω the phonon frequency, e the phononeigenvector, j the phonon branch index, N the number of unit cells, I e the intensityfrom single electron scattering [29], k B the Boltzmann constant, T the temperature, f the scattering factor of atom s with mass m and atomic basis vector l . The DebyeWaller factor M is calculated by summation over a fine wavevector ( k ) grid, M s = 14 m s (cid:88) k ,j ¯ hN ω k ,j coth (cid:32) ¯ hω k ,j k B T (cid:33) | Qe k ,j,s | . (3)
4. Results and Discussion
Phonon dispersion relations for β -tin along selected high symmetry directions asobtained from LDA and PBEsol calculations are presented in Fig. 1 and comparedto experimental results from IXS and to previously published results [6, 10, 11]. Bothexperimental and calculated dispersion relations show several anomalies, due to thecomplex electronic structure with long range force constants and the interplay ofelectrons and phonons. Some anomalies are indicated in Fig. 1. The influence ofthe applied sum rule on the phonon branches and the anomalies was carefully tested.The transformation was found to have minimal impact on the optical branches andthe anomalies, but created an artefact close to the M point, labelled G in Fig 1.Despite the fact that the lattice constants within the LDA are underestimated by thecalculation, we note a good agreement for the acoustic phonon branches and the phononanomalies. The dispersion relations are in close agreement with previous calculationsby Pavone et al. [4]. Some of the experimentally observed anomalies are betterreproduced by the present calculation. In particular the anomalies labelled A - F aremore accurately described. The highest energy optical mode shows several anomaliesin both experiment and our calculation whereas the same branch is almost completelyflat in the previous calculation. The anomalies are in fact sensitive to the Fermi surfacewhich is described more accurately in the present calculation due to a finer k-pointsampling and a smaller smearing width of the occupancies. We are therefore confidentthat the present calculation accounts sufficiently well for the electron-ion interaction. Wenote a slight over-estimation of the highest optical branch and a slight underestimation ofthe transverse acoustic branch in the Γ-H-M direction - the intersection of two equivalent iffuse scattering in metallic tin polymorphs g( ω )H M X05101520 E ne r g y B[ m e V ] b) H M X05101520 E ne r g y B[ m e V ] [0B0B1] [0B1B0][1B1B0][1B0B0]H PH’ MXMM MH’H [0B0B1] [1B0B0] AD F GE Δ Δ Δ Δ Δ Δ Σ Σ Σ Σ Σ Σ V V V V Λ Λ Λ Λ a) B C
Figure 1. (colour online) Dispersion relations of β -tin along the indicated highsymmetry directions. The calculations (solid lines - a) LDA and b) PBEsol) arecompared to experimental values from IXS measurements at 300K (circles), INS at300K (squares) [13] and (+) [10] and INS at 110K ( (cid:63) ) [3]. The dispersion relations alongthe Γ-X direction are labelled according to the symmetry classification proposed in [34].The differences between the experimental data sets are due to different experimentalconditions: Data were taken at different temperatures, with different statistics andresolution in momentum and energy transfer. Note the pronounced anomalies in thedispersion relations (arrows). The labelled anomalies are discussed in the text. Thephonon density of states computed in LDA as well as the first Brillouin zone and asection of the H0L plane in reciprocal space are shown on the right. iffuse scattering in metallic tin polymorphs Figure 2. a) - e) Experimental diffuse scattering (left part of individual panels) andcalculated (right part of individual panels) TDS intensity distribution of β -tin in theindicated reciprocal space sections. Note the almost forbidden reflections in a), c)and d) (arrows labelled F), visible due to the electron density asymmetry [32]. f)Experimental 3D isosurface of TDS in gray scale denoting the distance from (0 0 0). mirror planes. A phonon with wave vector in this direction is purely longitudinal ortransverse and the transverse modes are doubly degenerate. The acoustic ∆ branchalong the Γ-X direction (see Fig. 1) is slightly softer than experimentally observed. Thereproducibility of the optical phonon branches is improved in the PBEsol calculation butthe acoustic branch along Γ-H-M and the acoustic ∆ branch along Γ-X are significantlysofter. The low-energy branches are particular sensitive to numerical and physical errorsin ab initio lattice dynamics calculations. The case of tin is in particular difficult dueto the complex electronic structure and soft character of its crystalline form. Despitethe fact that the calculations are not exact, we note a good agreement in the shape ofthe phonon anomalies in both approximations. In the following the results of the LDAcalculation were used because this study focuses on the low-energy phonons.The phonon density of states as obtained using the adaptive broadening schemeis shown beside the dispersion relation in Fig 1. Adaptive broadening results in a wellconverged curve which is smooth at energies with highly dispersing phonon branches iffuse scattering in metallic tin polymorphs × ×
48 Monkhorst-Pack grid was sufficient for convergence with achoice of a = 1 .
0. The applied first order adaptive smearing should be reliable when theenergy spacing is gradient dependent. Similar to Ref. [25] we find that it works ratherwell even near critical points and the sharp features from Van Hove singularities are welldescribed. We note, that the phonon density of states at very low energies is slightlyaffected by the underestimation of the ∆ acoustic branch. At energies between 8 and9 meV we find a feature in the phonon density of states which has no correlation withanomalies in the dispersion relations along high symmetry directions and must thereforeoriginate from critical points located elsewhere.The classical methods in the study of lattice dynamics, such as INS and IXS areflux-limited, consequently the measurements are time consuming. We therefore use TDSwhich allows a rapid and detailed exploration of extended regions in reciprocal spaceand the identification of characteristic features in the lattice dynamics. Reciprocalspace sections and a three-dimensional isosurface of diffuse scattering as obtained fromexperiment and calculated TDS intensity distributions are shown in Fig. 2. Correctionsfor polarisation and projection [30], and the Laue symmetry of the system were applied.All shapes of diffuse features are remarkably well reproduced by the calculation inharmonic approximation. This implies that higher order scattering processes andanharmonic effects at room temperature are much less pronounced than previouslythought [31]. The pronounced elastic anisotropy is reflected by the butterfly shape ofTDS in the vicinity of the Brillouin zone centers Γ in the HK0 plane. The very differentsound velocities result in a large contrast in TDS intensities close to Γ for differentdirections. In fact, the TDS intensities scale ≈ /ω close to Γ. Non-typical diffusefeatures are observed in the HK2n+1 and HK2n reciprocal space sections:(i) An asymmetry in diffuse scattering of individual features is observed in theHK2n+1 reciprocal space sections. It is most pronounced around the (211) reflection,see Fig. 3. The HK2n+1 pattern as a whole are symmetric in agreement with the Lauesymmetry of the system. The asymmetry of the diffuse scattering in the vicinity of the(211) reflections is further investigated by IXS measured at selected reduced momentumtransfer q -values along the asymmetric TDS profile. IXS spectra are reported in Fig. 4.Comparing the two pairs of experimental spectra at wave vectors with equivalent q oneobserves a difference in the integrated intensities corresponding to TDS and an energy iffuse scattering in metallic tin polymorphs I n t en s i t y [ a r b . un i t s ] Figure 3.
Experimental diffuse scattering of β -tin crystal in the vicinity of the 211reflection in the HK1 plane. shift of the main excitation. The calculation shows that the experimentally observedexcitation contains the contribution of both ∆ and ∆ acoustic branches. The drasticchange of spectral weight between the two branches leads to an energy shift of theenvelope function. The ∆ optic branch, which is almost completely suppressed on oneside, shows the same particularity.(ii) A cross-like feature is observed around the almost forbidden reflections in theHK2 planes (these reflections become visible due to the asymmetry in the electrondensity distribution [32]). IXS is used to clarify the nature of the cross-like TDS feature.IXS scans along [ ξ ξ
2] are summarized in an intensity map in Fig. 5. The inelasticintensity close to (0 0 2) is dominated by the acoustic ∆ branch with a significantcontribution of the optic ∆ branch, determined from the experiment at (0.1 0.1 2)to be 10.8 %. The intensity close to (1 1 2) is dominated by the acoustic ∆ branchwith vanishing contribution of the optic ∆ branch. The intensity along [ ξ ξ
2] is suppressed. The diffuse features in the HK2n reciprocal space sections remainsymmetric.The measured phonon energies and IXS intensities are in good agreement withthe calculation in the illustrated direction. The three-dimensional isosurface of diffusescattering intensities, depicted in Fig. 2 f), allows one to identify the shape of diffusefeatures. We note plate-like, elongated cross-like and asymmetric shapes. The topologyof the diffuse scattering is in fact quite complex and its investigation requires a finesampling of 3D reciprocal space. The inspection of only a few planes in reciprocal spacemay provide an incomplete picture, the authors of a previous study [31] could only iffuse scattering in metallic tin polymorphs I n t en s i t y V[ a r b .V un i t s ] a)V(2.15V0.85V1.0)0100020003000400050006000 I n t en s i t y V[ a r b .V un i t s ] c)V(1.85V1.15V1.0)0100200300400 I n t en s i t y V[ a r b .V un i t s ] e)V(2.2V0.8V1.0)10 5 0 5 10EnergyVtransferV[m eV]050010001500200025003000 I n t en s i t y V[ a r b .V un i t s ] g)V(1.8V1.2V1.0) 0100200300400500600700800 b)V(2.15V0.85V1.0)0100020003000400050006000 d)V(1.85V1.15V1.0)0100200300400 f)V(2.2V0.8V1.0)10 5 0 5 10EnergyVtransferV[m eV]050010001500200025003000 h)V(1.8V1.2V1.0) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Figure 4.
Experimental IXS spectra (left panels) of β -tin on different momentumtransfers in the HK1 plane. The reduced momentum transfer q of panels a) and c)is equivalent, the same holds for panels e) and g). The peak position of the envelopefunction of the two acoustic branches is indicated by vertical lines in the experimentalspectra. The inelastic contribution of the different branches (vertical lines) as obtainedfrom the DFPT calculation and its convolution with the experimental resolution areshown for the corresponding momentum transfers in the right panels. The vertical linesare scaled by a factor 1/2 in respect to the convoluted spectra for best visualisation. iffuse scattering in metallic tin polymorphs Figure 5. (colour online) IXS intensity maps from (002) to (112) as obtainedfrom experiment (upper panel) and calculation (lower panel). The experimental mapconsists of eight IXS spectra with a q -spacing of 0.1 r.l.u and 0.7 meV energy step,linearly interpolated to a 200 ×
85 grid. The calculated IXS intensity is convolutedwith the experimental resolution function of 3.0 meV full-width-half-maximum. Thedispersion of the different branches is plotted as lines. identify rod-like features.The study on β -tin was extended to γ -tin. Also the γ phase exhibits a well definedequilibrium structure with a total static energy which is 1.85 meV per atom higher thanin β -tin. Taking into account the zero-point contributions to the internal energy, the β -phase results to be more stable than the γ -phase by 0.78 meV at 0 K. The obtaineddispersion relations along selected high symmetry directions and phonon density ofstates are presented in Fig. 6 and compared to experimental results from INS [6].We note, that the calculation describes the experimental results very well, includingthe phonon anomalies. Reciprocal space sections of diffuse scattering as obtainedfrom experiment and calculated TDS are confronted to the results of β -tin in Fig.7. The diffuse scattering in γ -tin is almost perfectly reproduced by the calculation.The similarity of the compared TDS intensity distributions of β and γ -tin can be iffuse scattering in metallic tin polymorphs Γ K M Γ A05101520 E n e r g y [ m e V ] ω ) 05101520 Figure 6.
Phonon dispersion relations along the indicated high symmetry directionsand the phonon density of states of γ -tin. The calculation (solid lines, pure Sn) iscompared to experimental values of a Sn . In . single crystal from [6] ( (cid:63) ). The labelledanomalies in the dispersion relation are discussed in the text. appreciated. It reflects the symmetry relation of the two phases which have commonsubgroups, consequently some symmetry elements are retained at the γ - β transition [5].The vectorial relationship between the two structures is given by a β b β c β ≈ − a γ b γ c γ , (4)where the vectors a , b and c denote the unit cell vectors with corresponding indices forthe β and γ phase. The Γ-H-M direction in β -tin for instance corresponds to the Γ-K-Mdirection in γ -tin. We note, that the anomalies in the dispersion relation labelled B andC (see Fig. 6) appear at the same position as in β -tin, see Fig. 1. The change in slopelabelled A is less pronounced in γ -tin, resulting in a smoother intensity distribution ofdiffuse scattering. The structure of the two phases is different, the momentum transferdependency of the underlying electronic potential is, however, similar. In fact theinteratomic distances and force constants are very comparable in the two structures.A strong asymmetry in diffuse scattering as observed in the HK2n+1 reciprocal spacesections in β -tin is not present in the γ -phase. The particularity in diffuse scattering in β -tin is thus a symmetry related feature. iffuse scattering in metallic tin polymorphs Figure 7.
Experimental diffuse scattering (left part of individual panels) andcalculated (right part of individual panels) TDS intensity distribution of β -tin (a) and(c) and γ -tin (b) and (d) in the indicated reciprocal space sections. The experimentalTDS intensity distribution of γ -tin was obtained from a Sn . In . single crystal,whereas the calculated one results from pure Sn.
5. Conclusions
This work demonstrates that the combination of TDS, IXS and ab initio latticedynamics calculation provides an optimized strategy in the study of lattice dynamics.Characteristic and anomalous features can be identified by the inspection of TDSintensity distribution in reciprocal space. Selected features were further investigated bymomentum resolved IXS and confronted to the calculation. We show with experimentalevidence that first principles calculations give a precise and detailed quantitativedescription of the lattice dynamics of the metallic tin β and γ polymorphs. Theexperimentally observed anomalies in the phonon dispersions are well reproduced by iffuse scattering in metallic tin polymorphs β -tin which is relatedto the non-symmorphic structure. The comparison of TDS from β - and γ -tin reveals astrong resemblance and reflects the symmetry relation between the two structures anda strong similarity of the underlying potential. Acknowledgments
The authors would like to thank Daniele de Sanctis for scientific support in the diffusescattering experiment and Alessandro Mirone and Ga¨el Goret for help in developing thesoftware for the data analysis.
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