Charge carrier interaction with a purely electronic collective mode: Plasmarons and the infrared response of elemental bismuth Riccardo Tediosi,1 N. P. Armitage,1,2 E. Giannini,1 and D. van der Marel1 1D´epartement de Physique de la Mati`ere Condens´ee, Universit´e de Gen`eve, quai Ernest-Ansermet 24, CH1211 Gen`eve 4, Switzerland. 2Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA. 7 (Dated: February 5, 2008) 0 0 Wepresent adetailed optical studyof single crystalbismuth usinginfrared reflectivityand ellip- 2 sometry. Colossal changes in theplasmon frequency are observed as a function of temperature due tochargetransferbetween holeand electron Fermipockets. Intheoptical conductivity,an anoma- n loustemperaturedependentmid-infraredabsorptionfeatureisobserved. AnextendedDrudemodel a analysis reveals that it can be connected to a sharp upturnin the scattering rate, the frequency of J which exactly tracks the temperature dependent plasmon frequency. We interpret this absorption 8 and increased scattering as the first direct optical evidence for a charge carrier interaction with a 1 collective mode of purely electronic origin; here electron-plasmon scattering. The observation of a plasmaron as such is made possible only by the unique coincidence of various energy scales and ] l exceptional properties of semi-metal bismuth. e - r PACSnumbers: 71.45.-d,78.40.Kc,78.20.-e,78.30.-j t s . t a Elemental semi-metals, such as graphite and bismuth, strong electron-electron interactions due to the relative m are materials of much long term interest due to their scalesbetweenpotentialandkineticenergyatlowcharge - exceptional properties, including large magnetoresistive densities [14]. d andpressuredependenteffects[1,2,3]. Inthecaseofbis- Despite the scientific and technological interest in n muththesepropertiesderivefromitslowcarriernumber, o bismuth, its optical and infrared properties have been (≈ 10−5 electrons per atom), reduced effective masses c under-investigated. In this letter we present detailed [ (≈10−2 electron masses), small Fermi wavevector(≈40 nm), long mean free path (≈1 mm), and large high fre- temperature dependent optical measurements over the 1 full optical range from FIR to UV of single crystal bis- v quency dielectric constant (ǫ∞ ≈100). muth. We observe a narrow Drude peak which has a 7 A number of recent results are causing an increased plasma frequency value consistent with the low carrier 4 interest in these materials, both from the side of funda- number. Colossal changes in the plasma frequency are 4 1 mental solid-state physics as well as applications poten- observed as a function of temperature, due to charge 0 tial. Forinstanceafielddependentcrossoverreminiscent transfer between electron and hole pockets. We find an 7 ofthe2Dmetal/insulatortransitioninMOSFETs[4]has anomalousmid-infraredabsorptionintherealpartofthe 0 beenobservedinbothgraphiteandbismuth[5]. Isolated conductivity. An extended Drude model analysis reveals / t single layers of graphene have been shown to have novel thatthescatteringratehasanabruptonsetatatemper- a transport properties and an anomalous quantization of aturedependentenergyscalewhichisfoundtobealmost m the Quantum Hall effect resulting from their low carrier exactlycoincidentwiththeindependentlymeasuredplas- - d number and exceptionalzeromass Dirac cone dispersion mon energy. This is the first direct optical observation n relation[6,7]. Moreovertherecontinuestobe interestin of a strongly coupled electron-plasmon elementary exci- o bismuth for studies of quantum confinement [8]. On the tation - a plasmaron. c technical side, advances in film growth [9], anomalously : Single crystal bismuth was grown by a modified v long spin diffusion lengths and very large magnetoresis- Bridgman-Stockbargertechnique in averticalthree-zone i X tive response makes bismuth useful for possible incor- furnace. A silica tube was filled with ∼ 5 grams of porationin nanomagnetometers,magnetoopticaldevices r 99.999% pure Bi powder (Cerac) and sealed under vac- a and spintronics applications [10, 11, 12, 13]. uum. The ampoule was held vertically in a three-zone In principle transport phenomena in bismuth should furnace and annealed above the melting point of Bi (T m ◦ be well described by the conventional theory of metals, = 271.4 C) for 10 hours before decreasing the temper- ◦ butdue toitsexceptionalparameters,therearesubstan- ature at a rate of 30 C/h, while keeping a tempera- ◦ tial departures from standard metallic behavior. For in- ture gradient of 10-15 C/cm. The crystals were cleaved stance, electronic energy scales, like the Fermi energy, from the as-grown boule along a plane perpendicular to are very low giving strongly temperature dependent ef- the trigonal direction [001] at LN temperatures. X-ray 2 fective masses and charge densities. Moreover, the ma- powder diffraction revealed that the mirror-like cleavage terial’s very low carrier density opens up the possibility surfaces were [110] planes perpendicular to the trigonal - at least in principle - for novel plasmonic effects and axis which were subsequently used as reflecting surfaces 2 Frequency (meV) 0 500 1000 1500 cm-12000 0 50 100 150 200 meV 0 20 40 60 80 ) 1.6 −1 (b) m 1.0 290 K 1c 1.2 1 230 K −Ω (c) 20 K 0.9 170 K 30 0.8 0.1 60 K 1 1 100 100 K y 0.8 100 K ω) ( 0.4 114700 KK vit 20 K s(1 0.0 223900 KK cti 0.7 25 e fl e 20 R 0.6 2 ε 15 0.5 10 (a) 0.4 -1 100 200 300 400 500 cm 0 200 400 600 800 20 30 40 50 60 meV -1 wavenumber (cm ) FIG. 2: (color online) (a) Optical conductivity derived from Kramer-Kronig analysis for selected temperatures. (b) The FIG. 1: (color online) Single-crystal bismuth reflectivity vs wavenumber(ω/2πc)fortemperatureT=290(red),230(or- imaginary dielectric constant ǫ2 over a much smaller energy range that emphasizes theremarkable prepeak structure. (c) ange), 170 (green), 100 (blue) and 20 K (cyan). The pro- nouncedshiftingoftheminimumintherange180−350cm−1 Log-Log scale plot of σ1(ω) for T =290 K, unitsare meV. ∗ represents the position of the screened plasma frequency ω . p −1 The small peak around 380 cm is an experimental artifact due to a small nonlinearity in a strong absorption of the in- tivity σˆ(ω,T) = σ +iσ . Note that although it does 1 2 terferometer’s beam splitter. not affect our conclusions either way, the 380 cm−1 ar- tifact has been removed from the data used to generate the plots in subsequent figures. The extended frequency for optical experiments. range and ellipsometry used in our experiment allows us We measuredthe DC resistivityandopticalspectrain to extractmore accurateparametervalues fromthe sub- the frequency range from 50 cm−1 (6.2 meV) to 30000 sequentanalysisthanwhathasbeenreportedinprevious cm−1 (3.8 eV) combining infrared (IR) reflectivity via work[16]wherejusttheFIRspectralrangewasanalyzed. FT spectroscopyandellipsometry in the VIS-UV energy InFig.2(a)theopticalconductivityσ (ω)ispresented 1 range. In the IR experiment the sample was mounted for corresponding data of Fig. 1 in the FIR-MIR range. ◦ in a quasi-normal incidence configuration (θinc = 11 ) The plot reveals two main features developing with de- and reflected signal intensity was recorded during slow creasing temperature; the first is a progressive narrow- temperature scans in the temperature range from 290 K ingofthe Drude peakfromanhalf-width-half-maximum downto20Kwitharesolutionofapproximately1K.The valueof42.3cm−1 atroomtemperaturedowntoavalue absolute value of the reflectivity R(ω,T) was calculated of3.3cm−1 at20K.Additionally,weobserveadramatic using a reference gold layer evaporated in situ on the appearance and strengthening of an absorption centered sample surface. around 700 cm−1 and characterizedby an onset approx- In Fig. 1 the reflectivity is presented for five selected imately around 350 cm−1 at room temperature which temperatures in the far-infrared spectral range. A spec- appears to shift downward as the temperature is low- tacular shift of the reflectivity edge from around 333 ered. In the inset to Fig 2 the conductivity is presented cm−1 at room temperature to a value of 164 cm−1 at 20 overthe full measurement range at roomtemperature; a K indicates a strong reduction of the plasmon frequency prominent peak is evident in the figure centered around with cooling. This derives from a change in density due 5500 cm−1 whose energy and temperature dependence to thermalchargetransfer fromelectronto hole pockets. are compatible with the direct interband transition at The small feature present at 380 cm−1 at all temper- the L-symmetry point. atures is a non-linear effect of the detector as a result While the narrowing of the Drude peak is typical be- of absorption in the beamsplitter and is not an intrin- havior for a metal whose DC conductivity increases at sic feature of bismuth. The ellipsometry and IR data lower temperatures as a consequence of the reduction of werecombined using a Kramers-Kronigconsistentvaria- the scattering processes, the appearance of an MIR ab- tionalfittingprocedure[15]. Thisallowstheextractionof sorption is unusual. In fact, a closer look shows an even all the significant frequency and temperature dependent more interesting aspect, as shown in Fig. 2b where we optical properties like for instance, the complex conduc- plota greatlyexpandedview ofthe relatedquantity, the 3 imaginary dielectric constant ǫ = 4πσ /ω; the low en- 2 1 meV meV ergy onsetof this MIR absorptionfeature is preceded by 0 50 100 0 50 100 asmall,but distinctandtemperature dependentprepeak 400 1.2 absorption structure. 290 K 230 K In a previous study, the MIR absorption has been as- 170 K 1.0 signed to the threshold for direct interband transitions 300 100 K at the L point [16]. Certainly interband transitions play 20 K 0.8 ) a role in part of this energy range, but the energy scale -1m m of the onset and prepeak are not quite right for them to τ (c 200 0.6 */m be the entire contribution. The interband gap reported 1/ 0.4 at low temperature is 13.7 meV [3] and combined with 100 anL point FermienergyE , givesa minimum threshold F fordirectabsorptionofE +2E of67.1meV(540cm−1 0.2 c F (a) (b) ), which is much bigger than the onset. The discrepancy 0 0.0 even increases at low temperature where the onset falls 0 400 800 0 400 800 to125cm−1 . Itispossiblethattheregionoftheabsorp- cm-1 cm-1 tion onset is partially affected by an indirect interband phonon coupled process,as recently proposedfor a simi- FIG. 3: (color online) Extended-Drude analysis results: (a) lar absorption in bismuth nanowires [17]. Although this the low frequency scattering rate τ−1(ω) progressive falls as process has been investigated theoretically [18, 19] we the temperature is lowered. An approximately frequency in- observe that our experimental data deviate considerably dependentregionisinterruptedbyasharponsetinscattering. from the expectations, in particular in the region where (b)Theeffectivemassm∗(ω)/misflatasoverasimilarrange. the prepeak appears. In the rest of this work we will m is defined as the band-mass of the carrier for the specific thus concentrate on the explanation that such anomaly crystal orientation used. derives from an electron-plasmon interaction. In Fig 3, we analyze the complex conductivity data in terms of an extended Drude model. We extract the fre- frequencyofthisonsetor“kink”intheε2(ω)functionωτ quency dependent scattering rates τ−1(ω) and effective vs.ωp∗ asdefinedbythezerocrossingoftheexperimental masses m∗(ω) via the relations ε1(ω) and the onset is defined by the energy position of the prepeak’s half maximum on its low frequency side. τ−1(ω) = −(ωp2/ω)Im(ε(ω)−ε˜∞)−1 Based on this parametric plot in Fig. 4 we can conclu- m∗(ω)/me = −(ωp2/ω2)Re(ε(ω)−ε˜∞)−1 (1) ssicvaelelyfiodrenthtiefyotbhseerpvleadsmabasforrepqtuieonncpyraoscesesst.tinTghtehpeleansmerogny energychangesbyalmostafactoroftwooveralmostthe where ωp is the plasma frequency and ε˜∞ represents the entire frequency range, but continues to set the scale for temperaturedependentinterbandcontributiontothedi- the increased scattering throughout. electricconstant. We shouldstressherethe factthatthe extendedDrude modelis strictly validonly inthe region Due to its longitudinal character, direct excitation of whereinterbandtransitionsdonotplayamajorrole. We a plasmon by an incident electromagnetic wave is gener- see in Fig. 3 that the data is relatively well described at ally not possible. However, a number of scenarios may thelowestfrequencieswithintheusualDrudeframework existto induce anelectron-plasmoncoupling effect. Pre- wherethescatteringratesandmassesarefrequencyinde- viously, an explanation for the appearance of the MIR pendent over roughly the same frequency interval. How- peak has been given in terms of an impurity-mediated ever we observe a sharp onset in the scattering rate at electron-plasmon coupling [21, 22, 23, 24]. It was pro- a well-definedtemperature dependentenergyscale. This posed that enhanced electron-charged impurity scatter- reflects the MIR absorption pointed out previously and ing is found near ωp due to the divergence of 1/ǫ. Al- makes τ−1(ω) deviate from the constant value expected though our data are in qualitative agreement with such from a simple Drude model. a scenario, the model used by Gerlach et al. [22] has a As noted earlier, according to band structure param- quantitativeagreementonlybyconsideringanexception- eters [20] the position of the scattering onset ω is too allylargechargedimpurityconcentration(N =1.5×1019 τ high to derive from a direct interband process. In fact, cm−3). This is approximately two orders of magnitude the temperature dependence of this absorption’s onset greater than the carrier concentration itself and implies and prepeak is reminiscent of the large temperature de- a charged impurity concentrationof 1 partin 104, which pendenceoftheplasmafrequencyitself. Weobservethat we consider to be unrealistic considering the high purity the onset almost exactly tracks the independently mea- of our samples. sured plasma frequency as a function of temperature, as In contrast, we propose that we are observing the ex- showninthe parametricplotFig.4(a)whereweplotthe citation of plasmons via a decay channel of the excited 4 electron-hole pairs. This interaction is essentially identi- ω* (cm-1) P caltothatconsideredinthecontextofelectron-phonon 160 180 200 220 240 260 [25]orelectron-magnoninteractions[26]. Suchaninter- 260 (a) action may in fact be captured within the same Holstein 240 Hamiltonian that is used to describe the electron - lon- 1) 100 K -m 220 200 K gitudinal phonon coupling to treat polarons and so this c collectiveexcitationhasbeencalledaplasmaron[27,28] ω (τ200 20 K 150 K 180 intheoreticaltreatments. Suchanexcitationisonlypos- 160 sible optically in a system where translationalsymmetry 80 3 − (b) 20 K (Exp. Data) hasbeenbrokenfirstby,forinstance,Umklappscattering 10 60 Fit or disorder that moves oscillatorstrength to a frequency ω)] PPllaassmmoanro Pneak region near the plasmon energy. In normal metals such 1( 40 − ε aninteractioniscompletelyunobservableastheplasmon m[ 20 energy scales are many orders of magnitude higher than −I 0 transport ones. This is, to the best of our knowledge, 50 100 150 200 250 300 the first unambiguous observationof charge carrierscat- -1 wavenumber (cm ) teringwithacollectivebosonicmodeofpurelyelectronic origin. ∗ Thepossibilityalsoexiststhatweareobservingnotan FIG. 4: (color online) (a) A parametric plot ωτ vs.ωp shows aslopeof1supportingthehypothesisofanelectron-plasmon electron-hole decay channel, but instead a direct 3 body interaction. (b)The20Kelectronenergyloss(EEL)function excitation of an electron-hole pair and a plasmon with (red) presentsthemain plasmon peak and a plasmaron peak a net momentum q ∼ 0. Such processes are possible by appearing as a shoulderof themain one. going beyond the usual RPA and considering electron- electron interactions (electron-plasmon in the present case) mediated by the crystal potential [14]. In the case sponds exactly to the position of the absorption feature ofatranslationalinvariantsystem,thestandardLandau- seen in ε2(ω) thus demonstrating its admixture of longi- Fermi liquid treatment holds since the photon-induced tudinal plasmonic character. electron-hole pair are momentum conserving processes. In conclusion we have made the first optical observa- Introducing symmetry breaking terms, like strong Umk- tion of an electron-plasmon interaction - a plasmaron. lappscattering ordisorder,makessuch3 body processes Thisobservationismadepossibleonlybythe lowcarrier possible. Experimentally this effect may be directly ob- density in Bi, a verylargeǫ∞, anda lack ofoptically ac- servableusingopticalspectroscopysincethelongitudinal tive phonons. This work raises various questions about collectivemodewouldbepartiallycoupledtoatransverse how these renormalization effects feed back on the low one [29]. energy properties of this material. It is possible that re- Within the electron-hole plasmon decay channel sce- ducing the charge density further, perhaps through the nario one might try to model the scattering rate using application of pressure, may enhance such interactions anexpressionfortheplasmondensityofstatesD(ω)and and drive the system into an anomalous metallic state. a simple scheme for coupling of the spectrum to elec- Pressure dependent optical studies may prove to be a tronic excitations [25, 26]. Unfortunately, although as- useful probe in this regard. It would also be interest- pects ofourdataarequalitativelyconsistentwithsucha ing to search for similar effects in other semi-metals like scenario, such a calculation is hard to compare with the graphite or single-layer graphene. data exactly, due to the onset of the interband contribu- TheauthorswouldliketothankM.Dressel,H.D.Drew tion. Thelowenergyprepeakstructureisinterestingand and A.J. 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