Plasmons in Sodium under Pressure: Increasing Departure from Nearly-Free-Electron Behavior
I. Loa, K. Syassen, G. Monaco, G. Vanko, M. Krisch, M. Hanfland
PPlasmons in Sodium under Pressure: Increasing Departure from Nearly-Free-Electron Behavior
I. Loa, ∗ K. Syassen, G. Monaco, G. Vank´o, † M. Krisch, and M. Hanfland Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstr. 1, 70569 Stuttgart, Germany European Synchrotron Radiation Facility, BP 220, 38043 Grenoble Cedex, France (Dated: October 30, 2018)We have measured plasmon energies in Na under high pressure up to 43 GPa using inelastic x-ray scattering(IXS). The momentum-resolved results show clear deviations, growing with increasing pressure, from the pre-dictions for a nearly-free electron metal. Plasmon energy calculations based on first-principles electronic bandstructures and a quasi-classical plasmon model allow us to identify a pressure-induced increase in the electron-ion interaction and associated changes in the electronic band structure as the origin of these deviations, ratherthan e ff ects of exchange and correlation. Additional IXS results obtained for K and Rb are addressed briefly. PACS numbers: 71.45.Gm, 62.50.-p, 78.70.Ck, 71.20.-b
Sodium, at ambient conditions, is one of the best manifes-tations of a “simple” or nearly-free-electron (NFE) metal [1].It is characterized by a single s -type valence electron, weakinteraction between the conduction electrons and the atomiccores (electron-ion interaction), and conduction band statesof sp orbital character. Na crystallizes in the high-symmetrybody-centered cubic (bcc) structure at pressures up to 65 GPa,where it transforms to face-centered cubic (fcc) [2]. Theproperties of Na change fundamentally under pressure in themegabar pressure range, where a series of phase transitionsinto lower-symmetry crystal structures has been predicted [3]and observed [4–6], accompanied by marked changes in itsoptical properties [4, 6, 7] and culminating in the formationof a non-metallic, visually transparent phase at ∼
200 GPa [8].A central question is how the transformation from a simplemetal to a semiconductor progresses, not only in Na, but alsoin other metals such as Li, which was reported to becomesemiconducting above 70 GPa [9]. As for Na, does it remainNFE-like in its bcc and fcc phases up to ∼
100 GPa [7] so thatthe non-NFE behavior starts only with the transitions into thelower-symmetry phases above 105 GPa, or are there signifi-cant precursors at lower pressure?To provide an answer, we measured and calculated the pres-sure dependence of Na plasmon energies. Plasmon excita-tions provide information on the collective electronic excita-tions in the form of longitudinal charge density waves at fi-nite wavevector, and they determine the optical response of ametal, specifically the plasma reflection edge. Plasmons havebeen studied for many years by electron energy loss spec-troscopy (EELS) at zero pressure (see for example [10, 11]and references therein), but this technique is not suitable forsamples enclosed in high-pressure cells. Mao et al. [12]have demonstrated the possibility of measuring plasmon ex-citations in Na under pressure using inelastic x-ray scattering(IXS), and they found their experimental results up to 2.7 GPato be in agreement with theoretical predictions.We report here detailed IXS results on the plasmon energydispersion in Na under pressure up to 43 GPa, correspondingto a 2.6-fold increase in density. Our results evidence a sig-nificant departure from the predictions for a NFE metal. Inorder to explain this discrepancy between theory and experi- ment, we also present plasmon energy calculations based onfirst-principles electronic band structures and a quasi-classicalplasmon model after Paasch and Grigoryan (
PG model ) [13].These calculations reconcile experiment and theory and allowus to identify changes in the electron-ion interaction as thedominant e ff ect, rather than changes in the electron-electroninteractions. Some experimental results are also reported forK and Rb.The theoretical description of plasmons in simple metals iswell established. The most commonly used approach startsfrom the free-electron (FE) gas and uses the Random PhaseApproximation (RPA) [14]. The plasmon energy dispersion E p ( q ) is then given by E p ( q ) = (cid:126) ω p + (cid:126) m α q with the plasmafrequency ω p = (cid:112) ne /(cid:15) (cid:15) s m and the dispersion coe ffi cient α FE = E F / (cid:126) ω p , where q is the plasmon momentum, m theelectron mass, n the electron density, (cid:15) s a dielectric constantdescribing the polarizability of the ionic cores ( (cid:15) s = E F = ( (cid:126) / m )(3 π n ) / is the Fermienergy. This relatively simple model works reasonably wellfor simple metals such as Na and Al at ambient conditions[10, 11], but experimental dispersion coe ffi cients α tend to belower than the theoretical values, which has been attributedto electron exchange and correlation e ff ects [11]. Both theplasma frequency ω p and the dispersion coe ffi cient α dependon the electron density and can thus be tuned by the applica-tion of pressure; both are expected to increase with increasingpressure. The stability of bcc sodium over a large pressurerange of 0–65 GPa permits to generate an up to threefold in-crease in (electron) density without a structural transition.IXS experiments were performed on beamline ID16 at theESRF, Grenoble. Silicon crystal monochromators were usedto monochromatize the incident beam and to analyze the scat-tered radiation. The incident x-ray beam with a photon energyof 9.877 keV was focussed onto the polycrystalline samplein a high-pressure cell with a spot diameter of 100–200 µ m,depending on the sample size. IXS spectra of the samplesat room temperature were recorded in energy-scanning modewith an overall spectral resolution of 0.6 eV and a momentumresolution of 0.4 nm − . In most of the experiments, the sam-ples were pressurized in diamond anvil cells (DACs). Rhe- a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r I n t en s i t y ( c oun t s / s , s h i ft ed ) Energy Transfer (eV) q = 5 nm –1 Na P (GPa) (a) (b) Na P = 16 GPa I n t en s i t y ( c oun t s / s , s h i ft ed ) Energy Transfer (eV) q (nm –1 ) FIG. 1. IXS spectra of polycrystalline sodium pressurized in a dia-mond anvil cell. (a) Energy transfer spectra for a momentum transferof q = − and pressures of 3.5–43 GPa. (b) Energy transfer spec-tra of Na at 16 GPa and momentum transfers of 2–9 nm − . Verticalo ff sets are added for clarity. nium and stainless steel gaskets were used with initial thick-nesses of 50–100 µ m and hole diameters of 150–200 µ m. Theloading of distilled Na, K, and Rb into the pressure cells wascarried out in an argon atmosphere. Because of the softnessof these metals no pressure transmitting medium was added.Pressures were determined with the ruby method [15] or bymeasuring an x-ray powder di ff raction of the sample and us-ing its known equation of state [2]. In the experiments us-ing a DAC, the incoming x-ray beam passed through one dia-mond anvil onto the sample, and the scattered radiation wascollected through the opposing anvil (thickness ∼ ∼ q , isdue to plasmon and interband excitations in the diamond anvilthrough which the scattered radiation is detected [16].Plasmon energies and linewidths were determined by fittingthe spectra with a Gaussian peak for the plasmon line and apolynomial background. As all experiments were performedon polycrystalline samples, the reported plasmon energies aredirectional averages. The Na plasmon energies increase withincreasing pressure as illustrated in Fig. 2 for q = − .This is in qualitative agreement with the NFE picture, wherethe plasmon energies scale with the electron density.Figure 2 also shows results for K and Rb. These two metalsclearly do not follow the expectations for NFE metals. More- RbK P l a s m on E ne r g y ( e V ) Pressure (GPa) q = 5 nm –1 Na FIG. 2. Plasmon energies in Na, K, and Rb at q = − as a func-tion of pressure. Ambient-pressure EELS results [10] are indicatedby stars. For the Na data, di ff erent symbols distinguish di ff erent sam-ple loadings and pressure cells. Lines are guides to the eye. over, their plasmon linewidths increased and their plasmonintensities decreased rapidly with increasing pressure. Thesee ff ects are attributed to the pressure-driven s – d hybridizationof conduction band states, as is also evident from the opticalresponse of K and Rb under pressure [17, 18] (see also [19]).We will therefore focus on Na that could be studied over thelargest pressure range and which, at ambient conditions, is oneof the best manifestations of a nearly-free-electron metal.Figure 3(a) shows the measured plasmon dispersion rela-tions of Na at several pressures. The plasmon energies areplotted versus q because of the anticipated parabolic dis-persion relation, see the E p ( q ) relation given above. At thelowest pressure, 1 GPa, the Na plasmon dispersion measuredhere is indeed very close to parabolic. Results of an ambient-pressure EELS study [10] are included in Fig. 3(a) for com-parison. In the low- q region, the IXS and EELS data are rea-sonably consistent, but the EELS results exhibit some devia-tion from a parabolic dispersion. The o ff set, at low q , betweenthe IXS and EELS data is largely due to the pressure appliedin the IXS experiment. The 1-GPa plasmon dispersion mea-sured by IXS is described best by E (0) = (cid:126) ω p = . α = . α is lower than the FE / RPA value of α RPA = .
35. Towards higher pressures, the measured plas-mon dispersion remains approximately parabolic, deviationsbeing most notable at 16 GPa.Figure 3(b) shows that also the plasmon linewidth isstrongly pressure dependent. This e ff ect is tentatively at-tributed to a reduction of the plasmon lifetime due to decaysinvolving electron-hole excitations. A detailed study by, e.g.,time-dependent density functional theory could be a subjectfor future studies. For the remainder of this paper we focus onthe plasmon energies, assuming that self-energy e ff ects on theplasmon frequency can be neglected.The IXS results on the plasmon energies of Na as a func-tion of momentum and pressure allow us to test the validityof the FE / RPA description of Na and to assess the relative
43 GPa (a) Na P l a s m on E ne r g y ( e V ) q (nm –2 )
16 GPa8 GPa1 GPaEELS, 0 GPa (b) Na q = 5 nm –1 P l a s m on W i d t h ( e V ) Pressure (GPa) m ea s u r ed d e c o n v o l u t e d FIG. 3. (a) Plasmon dispersions E ( q ) of polycrystalline Na as afunction of pressure. Ambient-pressure electron-energy-loss spec-troscopy results by vom Felde et al. [10] are indicated by small opensymbols. (b) Plasmon linewidth of Na versus pressure. importance of band structure e ff ects, core polarizability, andexchange-correlation. Using E p ( q ) as given above and the ex-perimental equation of state of Na [2], the pressure depen-dence of the q = − plasmon was calculated within theFE / RPA framework as shown in Fig. 4. This FE / RPA esti-mate is significantly higher in energy than the experimentalvalues, and the deviation increases with increasing pressure.Inclusion of the core polarization ( (cid:15) s = .
16, Ref. [20]) leadsto a good agreement with the experiment at low pressure, buta major deviation between theory and experiment remains athigh pressure.The important observation here is the striking increase inthe deviation between theory and experiment with increas-ing pressure. In previous work on other metals, deviationsfrom the FE / RPA predictions were discussed in relation to ex-change and correlation e ff ects [10, 21], and a number of exten-sions of the RPA were proposed in this spirit (see [10, 21, 22]and references therein). Their main e ff ect is to reduce theplasmon dispersion coe ffi cient α , and their inclusion can im-prove the agreement between theory and experiment. How-ever, exchange-correlation e ff ects decrease with increasingelectron density, and they can thus be excluded as the originof the deviation between the ‘FE / RPA + core polarization’ re-sults and the experimental data in Fig. 4. More recent theoreti-cal studies have emphasized the importance of band-structuree ff ects on the plasmon properties, regarding both the energydispersion [13] and the plasmon linewidth [19]. The presentIXS results o ff er an opportunity to test these proposals.The PG plasmon model [13] adopted here is an extensionof the FE / RPA approach. This classical model is not ex-pected to describe the plasmon properties as accurately as,e.g., time-dependent density functional theory (DFT) [19],but it allows us better to understand the underlying physics.The electronic structure of the metal is described here by asingle isotropic conduction band with a quartic dispersion, E ( k ) = E k + E k . The E k term accounts for deviationsfrom the parabolic band shape of the free-electron gas. The F E / R P A + c o r e p o l . PG model P l a s m on E ne r g y ( e V ) Pressure (GPa) q = 5 nm –1 EELS, 0 GPa F E / R P A Na IXS
FIG. 4. Experimental pressure dependence of the directionally-averaged plasmon energy in Na at q = − (large symbols) andresults of the free-electron (FE) gas model, FE model with core polar-ization, and the PG model (solid line). The star indicates the ambient-pressure EELS result [10]. plasma frequency ω p and the plasmon dispersion coe ffi cient α can then be determined from E , E and the Fermi energy E F as described in detail in [13].Electronic structure calculations of Na were performed inthe framework of first-principles DFT, using the full-potentialL / APW + lo method [23, 24] and the Generalized Gradient Ap-proximation [25] as implemented in the WIEN2K code [26].The electronic band structure of bcc Na was calculated for aseries of volumes corresponding to the pressure range of 0–50 GPa [27]. The coe ffi cients E and E were determined forthree directions in the Brillouin zone, i.e. along [ ξ ξξ ξξξ ], and then averaged [28]. The only adjustable param-eter in the calculation of the plasma frequency and the disper-sion coe ffi cient is the static dielectric constant (cid:15) s that accountsfor the polarizability of the ionic cores. A value of (cid:15) s = . q = − plasmonin Na [28]. The computed results are in excellent agreementwith the IXS data. To trace the source of the di ff erence be-tween the FE / RPA results and those of the PG model, Table Isummarizes the plasma frequency (cid:126) ω p and the dispersion co- TABLE I. Experimental plasma frequencies ω p and dispersion coef-ficients α of Na as a function of pressure and comparison with theo-retical results. The experimental values were determined from linearfits to the E ( q ) data as shown in Fig. 3(a).experiment FE / RPA PG model P (cid:126) ω p α (cid:126) ω p α (cid:126) ω p α (GPa) (eV) (eV) (eV)1 . . . .
84 0 .
35 5 .
82 0 . . . . .
99 0 .
37 6 .
77 0 . . . . .
74 0 .
39 7 .
35 0 . . . . .
10 0 .
41 8 .
33 0 . e ffi cient α for the two models at selected pressures. At 1 GPa,the results of the NFE / RPA and the PG model are very similar,and the calculated plasma frequency agrees well with the ex-periment. As noted before [10], the calculated values of α are20–30% larger than in the experiment, and this is probably dueto exchange-correlation e ff ects not included here. The e ff ectof the non-parabolic contribution ( E <
0) is to reduce both (cid:126) ω p and α . In Na, this e ff ect is very small near ambient pres-sure, confirming the analysis by Paasch and Grigoryan [13].With increasing pressure, however, | E | increases and leads tosubstantial corrections [28]. At 40 GPa, it reduces the plasmafrequency by 8% and the dispersion coe ffi cient by 35% com-pared to NFE / RPA. Table I also shows that α decreases withincreasing pressure in the PG model, in contrast to the free-electron behavior. We would like to emphasize that it is thenon-parabolic contribution that causes the renormalization ofthe plasmon energies, even though the corrections to the bandenergies are less than 3% of the band width [28].As for the physical origin of these corrections, the bandstructure calculations show that pressure causes the band gapsat the N, P, and H points of the Brillouin zone to grow rela-tive to the width of the conduction band [28]. This evidencesa strengthening of the interaction between the valence elec-trons and the ionic cores (electron-ion interaction). In otherwords, Na becomes increasingly less free-electron like un-der compression. The quartic correction ( E <
0) corre-sponds to a lowering of the conduction band energies near theBrillouin zone boundary in comparison to the free-electroncase, and this distortion of the band structure [28] leads toa reduction of the Fermi velocity, v F = (1 / (cid:126) ) ∂ E ( k ) /∂ k | k F = (2 k F / (cid:126) )( E + E k F ), which is the key physical quantity thatdetermines the plasmon dispersion.In summary, we have performed inelastic x-ray scatteringexperiments to determine the e ff ect of pressure on the plasmonexcitations in the ‘simple metal’ sodium. While Na is consid-ered one of the best manifestations of a nearly-free-electronmetal at ambient conditions, our results evidence substantialand increasing deviations from the behavior of a NFE metal athigh pressure up to 43 GPa. This can be seen as an early pre-cursor of the fundamental changes in the electronic structureof Na at megabar pressures. The deviation from NFE behaviorcan in part be attributed to the polarizability of the ionic cores,but to a larger extent it is caused by pressure-induced changesin the electronic band structure. Plasmon energies determinedon the basis of electronic band structure calculations and thequasi-classical PG model are in excellent agreement with theIXS results. They show that the electron-ion interaction inNa increases with pressure and leads to the renormalizationof the plasmon energies via a modification of the electronicband structure. As for the heavy alkali metals under pressure,band structure e ff ects can be expected to be even more impor-tant due to the pressure-driven hybridization of the valence s orbitals with d states, as discussed before [13]. We observeda weak pressure dependence of the plasmon frequencies ofK and Rb combined with fast broadening of their plasmonresonances under pressure. These results may aid the inter- pretation of plasmon dispersions of the heavy alkali metals atambient pressure. As for bulk Li metal, the quite detailed pre-dictions on the collective electronic response under pressure[29, 30] still await a related experimental investigation.We thank F. K¨ogel (MPI-FKF, Stuttgart) for providing thedistilled metals used in this study. G. V. was supported by theHungarian Scientific Research Fund (contract No. K72597). ∗ Corresponding author: E-mail [email protected]; Present ad-dress: SUPA, School of Physics and Astronomy, Centre forScience at Extreme Conditions, The University of Edinburgh,United Kingdom. † Present address: KFKI Research Institute for Particle and Nu-clear Physics, PO Box 49, H-1525 Budapest, Hungary.[1] E. Wigner and F. Seitz, Phys. Rev. , 509 (1934).[2] M. Hanfland, I. Loa, and K. Syassen, Phys. Rev. B , 184109(2002).[3] J. B. Neaton and N. W. Ashcroft, Phys. Rev. Lett. , 2830(2001); N. E. Christensen and D. L. Novikov, Solid State Com-mun. , 477 (2001).[4] M. Hanfland et al. , Sodium at megabar pressures , Poster at2002 High Pressure Gordon Conference.[5] E. Gregoryanz et al. , Science , 1054 (2008).[6] L. F. Lundegaard et al. , Phys. Rev. B , 064105 (2009).[7] A. Lazicki et al. , Proc. Nat. Acad. Sci. USA , 6525 (2009).[8] Y. Ma et al. , Nature , 182 (2009).[9] T. Matsuoka and K. Shimizu, Nature , 186 (2009).[10] A. vom Felde, J. Spr¨osser-Prou, and J. Fink, Phys. Rev. B ,10181 (1989).[11] J. Spr¨osser-Prou, A. vom Felde, and J. Fink, Phys. Rev. B ,5799 (1989).[12] H.-K. Mao, C. Kao, and R. J. Hemley, J. Phys.: Condens. Mat-ter , 7847 (2001).[13] G. Paasch and V. G. Grigoryan, Ukr. J. Phys. , 1480 (1999).[14] D. Pines, Elementary excitations in solids (W.A. Benjamin,New York, 1964).[15] G. J. Piermarini, S. Block, J. D. Barnett, and R. A. Forman, J.Appl. Phys. , 2774 (1975); H. K. Mao, J. Xu, and P. M. Bell,J. Geophys. Res. , 4673 (1986).[16] S. Waidmann et al. , Phys. Rev. B , 10149 (2000).[17] H. Tups, K. Takemura, and K. Syassen, Phys. Rev. Lett. ,1776 (1982).[18] K. Takemura and K. Syassen, Phys. Rev. B , 1193 (1983).[19] W. Ku and A. G. Eguiluz, Phys. Rev. Lett. , 2350 (1999).[20] R. Nieminen and M. Puska, Physica Scripta , 952 (1982).[21] L. Serra et al. , Phys. Rev. B , 1492 (1991).[22] P. Vashishta and K. S. Singwi, Phys. Rev. B , 875 (1972).[23] E. Sj¨osted, L. Nordstr¨om, and D. J. Singh, Solid State Commun. , 15 (2000).[24] G. K. H. Madsen et al. , Phys. Rev. B , 195134 (2001).[25] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. ,3865 (1996).[26] P. Blaha et al. , WIEN2k , An Augmented Plane Wave + Lo-cal Orbitals Program for Calculating Crystal Properties (K.Schwarz, Techn. Universit¨at Wien, Austria, 2001).[27] Computational details: sphere size R MT = . ff defined by R MT × max( k n ) = k -points (190 in the IBZ).[28] See EPAPS Document No. ***** for additional details. For more information on EPAPS, seehttp: // / pubservs / epaps.html.[29] A. Rodriguez-Prieto, V. M. Silkin, A. Bergara, and P. M. Echenique, New J. Phys. , 053035 (2008).[30] I. Errea et al. , Phys. Rev. B81