Intensity (arb. units) Kohn Anomaly in MgB 2 and Related Compounds by Inelastic X-ray Scattering Γ Γ M FWHM (meV) Γ M Mode Energy (meV) (h 0 0) (c) E 2g Linewidth 30 20 10 0 B 1g E 2g A 2u E 1u 300 K 16 K 44 K 100 80 60 40 20 0 (b) Dispersion (a) Spectra along Γ - M Energy Transfer (meV) 0 5 100 (2.31, 0, 0.01) (2.27, 0, -0.01) (2.23, 0, -0.03) (2.19, 0, 0.03) (2.15, 0, 0.01) (2.11, 0, -0.01) Fig. 1. Phonon dispersion in pure MgB 2 (see [3]). (1a) Selected IXS spectra (room temperature) with fits (solid lines). The softening and broadening of the E 2g mode (indicated by red arrows) are readily apparent. (1b) Phonon dispersion determined by fits to the data (points) and the results of ab initio calculations (lines - Bohnen et al. , Phys. Rev. Lett . 86 (2001) 5771; dashed lines are modes not expected to be observed from phonon polarization considerations). ( 1c) Measured linewidth after correction for resolution, and the theoretical calculation (see text). In general, there is excellent agreement with the theoretical calculation. The solid horizontal bar shows the diameter of the sigma Fermi surfaces projected into the basal plane which is the expected position for the Kohn anomaly. MgB 2 has generated an immense amount of i n t e r e s t s i n c e i t s h i g h T c , ~ 3 9 K , w a s f i r s t demonstrated a few years ago [1]. This is the highest known critical temperature for a simple metallic material and rather outside the range for T c using standard estimates. One of the especially interesting properties of MgB 2 is that the electron-phonon coupling (EPC) that causes this high T c is predominantly to a particular phonon mode (with E 2g symmetry at Γ ), which makes in v estigating the b eha v ior of this mode v ery important. F urthermore, the rele v ant sections of the F ermi surface ( “ sigma ” sheets) are nearly simple cylinders along the c * a x is, thus the effects of the electron-phonon coupling should b e clear and easy to see in phonon spectra. A difficulty with MgB 2 , howe v er, is that a v aila b le single crystals are small, ~ 0 . 0 1 mm 3 , so that the con v entional method of in v estigating phonon dispersion, inelastic neutron scattering, cannot b e applied. H owe v er, the high b rilliance of S Pring- 8 means that, in principle, such in v estigations can b e carried out with X -rays, assuming a suita b le, highly efficient, spectrometer is a v aila b le. W e used the BL35XU spectrometer [2], which pro v ides ~ ~ 3 × 1 0 1 0 photons / s in a 4 me V b andwidth and a small ( 1 00 micron diameter) spot on the sample. G ood q uality inelastic X -ray scattering ( IXS ) spectra could b e measured from samples of si z e ~ 0 . 5 × 0 .2 × 0 . 05 = 0 . 005 mm 3 in a b out 12 h, using a 6 me V resolution setup at 1 5 . 8 ke V (corresponding to the ( 888 ) b ack reflection in silicon). The EPC of the E 2g mode to the sigma surfaces is e x pected to cause strong softening and a huge increase in the linewidth of the phonon mode as one reduces the in-plane component of the pro b ed momentum transfer from v alues larger than the diameter of the sigma F ermi surfaces to smaller v alues. The softening is an e x treme e x ample of a Kohn anomaly, while the b roadening corresponds to the reduction in phonon lifetime that occurs as the additional decay channel (e x citation of the electronic system) turns on at smaller momentum transfers. The momentum transfer dependence is easily understood q ualitati v ely b ased on the shape of the sigma F ermi surfaces : since phonon energies are negligi b le on the 54 References [1] J. Nagamatsu et al. : Natu re 410 (200 1 ) 63 . [ 2 ] A . Q . R . B a ron et al. : J. Phy s. & Che m. Solid s 61 (2000) 46 1. [ 3 ] A . Q . R . B a ron, H . Uchiy ama , Y . T a n a k a , S . T sutsu i, D . I s hik a w a , S . Lee, R . Heid, K . -P . Bohnen, S . T a ji ma a nd T . I s hik a w a : Phy s. Rev . Le tt. 92 (2004) 1 97004 . [ 4 ] S . Lee et al. : Phy s ic a C 397 (2003) 7 . [ 5 ] A . Q . R . B a ron, H . Uchiy ama , S . T sutsu i, S . Lee a nd S . T a ji ma : in prep a r at ion . Alfred Q . R . B a ron a a nd Hiro s hi Uchiy ama b ( a ) SPrin g -8 / J ASRI (b) S u percond u c t ivi t y Re s e a rch L a bor at ory, ISTEC E- ma il: B a ron@ s prin g 8 . or . jp Fig. 2. Phonon dispersion in carbon-doped material, Mg (B 1-x C x ) 2 [5]. Selected IXS spectra (room temperature) with fits (solid lines) from (a) x = 0.05 sample and (b) x = 0.125 sample. (c) shows the phonon dispersion determined from the fits and a guide to the eye (dashed lines). The E 2g mode “pops up” when the carbon content is increased to 12.5%, as expected from filling the sigma surfaces. The presence of the extra mode evident at higher momentum transfers and other details are being analyzed s c a le of m o st elec t ron stat e s , EPC i s only po ss ible for phonon s wi t h m o m en ta a ppropri at e t o m ove elec t ron s wi t hin t he nei g hborhood of a Fer m i su rf a ce . Given t he ne a rly circ u l a r projec t ion of t he s i gma su rf a ce s in t o t he b asa l pl a ne, sma ll phonon m o m en ta c a n m ove elec t ron s fro m one loc at ion t o a no t her a cro ss t he Fer m i su rf a ce . However, for l a r g er m o m en ta (Q > 2 k F ), a ll a ppropri at e stat e s a re filled, t h us t he co u plin g i s red u ced . Fi gu re 1 s how s t he re su l ts fro m a p u re M g B 2 sam ple a lon g t he Γ M line [ 3 ] of t he b asa l pl a ne – t he s of t enin g a nd bro a denin g as one m ove s in s ide t he s i gma su rf a ce s a re i mm edi at ely cle a r . F u r t her m ore, t he ag ree m en t wi t h t heore t ic a l c a lc u l at ion s i s excellen t , bo t h in di s per s ion a nd linewid t h, even for t hi s very ex t re m e c as e of st ron g co u plin g of s in g le m ode . Thi s provide s cr u ci a l confir mat ion t h at M g B 2 i s a n ex t re m e c as e of a conven t ion a l BCS / Eli as hber g - t ype su percond u c t or . H a vin g de m on st r at ed t he effec ts of t he EPC on t he E 2 g m ode a nd t heir g ood ag ree m en t wi t h t heore t ic a l re su l ts , we t ried t o m odify t he sam ple a bi t ; t he recen t g row t h of s in g le cry sta lline c a rbon-doped M g (B 1 -x C x ) 2 [ 4 ] provide s a n excellen t oppor tu ni t y . In p a r t ic u l a r, t he s i gma s hee ts of t he Fer m i su rf a ce s a re hole su rf a ce s , s o t h at elec t ron dopin g wi t h C c a n be expec t ed t o clo s e t he m a nd re m ove t he elec t ron phonon co u plin g. Th us , we inve st i gat ed t he beh a vior of t he E 2 g m ode in sam ple s wi t h 2%, 5% a nd 1 2 . 5% c a rbon dopin g , corre s pondin g t o t r a n s i t ion t e m per atu re s of 35 . 5, 30 a nd 2 . 5 K, re s pec t ively . The s of t enin g re ma in s cle a r in t he 2% a nd 5% sam ple s ; however, as eviden t in Fi g. 2, in t he 1 2 . 5% sam ple, t he s of t enin g g oe s a w a y, a nd t he E 2 g m ode “pop s u p”, s howin g t he re m ov a l of t he EPC fro m t he s e su rf a ce s. Thi s i s in g ood ag ree m en t bo t h wi t h expec tat ion s re ga rdin g t he ch a n g e s in T c a nd wi t h one c a lc u l at ion sugg e st in g t h at a bo ut 9% c a rbon dopin g s ho u ld be su fficien t t o fill t he s i gma b a nd s. x = 0.02 x = 0.05 x = 0.125 E 2g E 1u ? LA Γ M (h 0 0) Mode Energy (meV) 120 100 80 60 0 40 20 (c) Mode Dispersion (2.22, 0, -0.03) (2.37, 0, 0.01) (2.29, 0, 0.03) (2.13, 0, -0.01) 0 5 100 (b) 12.5% Carbon (2.37, 0, 0.01) (2.29, 0, -0.03) (2.21, 0, -0.03) (2.13, 0, 0.01) 0 5 100 (a) 5% Carbon Intensity (arb. units) Energy Transfer (meV) 55
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