フロンテイア物理講演会, 山形大学理学部, 1月30日(2014) 曲面が関わる量子物性 現代幾何学と物性科学との新たな融合領域への展開 尾上 順 E-mail: [email protected] 東京工業大学 理工学研究科原子核工学専攻 原子炉工学研究所 謝辞:JST-PRESTO/CREST, 文科省新学術領域No.21200032 重力場による曲がった3次元空間を動く光子 1916 一般相対論 重力場により時空間が歪む 1920 重力レンズ効果の観測 Albert Einstein (14 March 1879 – 18 April 1955) 曲がった空間を動く光子 光 which where X is a "squeezing from zero only for different will be significantly 19, 20) of the sures the potential: strength Using (4) and 膜を考えます。次に,この膜の形に沿った量子井戸型ポ とその電子透過 電子状態 —the a very small range of values of q3 around q3 0. superscrip = limV„(qs) take q3-0 in all coeffi- is=giy+ In this case we can safely oo g0 テンシャルで膜を上下から挟むことにより,膜の内部に qています。今後はこうした理 cients of E(I. (11) [except of course in the term G»3 ——G3» = is tofrom (If a specific exampleThe required guide our and is result containing V„(qz}]. 証できるか,そのための実験 (5) (9) 粒子を閉じ込めます。その後,ポテンシャル障壁の高さ intuition, we can imagine the harmonic binding We now parameter" mea- Quantum mechanics in Riemannian space =-,' mX Free electrons on a curved surface with R. C. T. DA COSTA を十分高くし,膜厚ゼロの極限を取ります。すると,曲 a periodic uneven (凹凸) structure qs, with X eventually going to infinity, 52 which gives (qs) ~ )2/mX). Before going to the Schrodinger equation it is worthwhile to briefly review the mathematical element where properties (the k2 Wgdq, dqzcoordinate of the us the system (1). Let of Vs(qs) 1984 can equation. W coordinates ( 即した物性推算が重要である surdS= area we are hoping for the existence of a surface Since 面接線方向に運動する粒子の有効ハミルトニアンが,あ 'Tr n„'-det —, &„X face) and wave function, depending only on the variables q& 6.擬一次元導体の朝永 1 2 led to the introduction of and )を用いて以下の形に求まりま る曲線座標系(q q2, we are naturally ,q where G=det . B By", +V, る曲率効果 a new16) wave function y from which, in the event of =1+ Tr(o. )x= (i2) . +det(&„)q3 '„)q, G, &'s given in (q, f(q(, qz, qz) 2m Bq3 Bt 。 す = into two parts a separation X(q„qz, qz) X,(q(, qz)X„(q, ) we will be Z(q 1, qs, q 2), able to define the surface density probability now gives the desired result: Expression (8) normal 2 2 be easily separated by write part, Equation (12) can now 本節と次節では,前述の曲 ! 1 ! ! transf»T"e ade(luate I'dqz I'J (qz) (q~qz) ij 2 Ix ̂ Ix. * ! g g the j vol+(h −k) setting ( 1 X=X((q„qz, ) t) xX„(qz, t), where the submation H(t =− -x can2 m be readilyi,6 "2m a)(q„q„ !q i from !q ! 表面物性に顕在化する例を g j=1inferred (io) X(q ~(qzut qz) (t((q(r qzo "tangent" andqz)"normal, stand forq3)] and n lf(q(&qzs . scripts ume dV expressed in terms of the curvilinear FIG. 1. Curvilinear coordinateprocedure system based yields on the The usual the respectively. * have coordinates using (4) weij は行列[g Really,tensor q, . metric surface S of parametric equation r ここで mq„qz, は粒子の有効質量,g ]の逆行列 は,擬一次元凹凸電子系の =r(q~, q2). ijIntroducing this substitution into (7) we are left following equations. dV=f(q((qz(qz)dSdqz ~ 。hO'とB kX„p" は 成分,g は行列[g ij]の行列式を表します 11)with 状態です。一般に一次元性の + V(, (qz)X„=ia (13) k~. ! r $ % 2m Bg, それぞれ曲面の平均曲率・ガウス曲率と呼ばれる量であ k' (X I' &'X+ 1" sf 2. sX &f 2 — + zh 2+ 2f +V„(qz)X X z SI~~ 4fz s 2 り,曲面上の各点における曲がり具合を表します。h と Lo & & Old Bt クーロン相互作用のために ず,フェルミ準位 E F の近く electrons bound in a curved surface ' 成されます。その結果,系の k はConducting g ij の関数ですので,結局曲面の形さえ決まれば, — etd( ()(I in [-, Tr(n„)]' X, = can be written a iamore We are now ready to take into account the effect Using (3) this term in effective mass form ofその形状を記述する計量テンソル the ・local potentialchange (i4) V(, (q&). Since in the limit when Ĥ が一意に g ij により useful のようなべき乗則に従います "sees" two X-~ ・generation potential the wave function of attractive andsteep repulsive potentials (M'-tf}=a v, (q„q, ) =one-dimensional is just then(w){*!w−E Expression (13)2m on both sides of the surface, its value barriers 決まります。 F* ,a= Koshino and H.different Aoki, Phys.from Rev. Bzero 71 (2005) Schrodinger- equation for a particle bounded by for only 073405. will beM.significantly N. Fujita and O. Terasaki, Phys. Rev. B 72 (2005) 085459. the transverse ここで現れる指数 a は TLL 式( 1 )かっこ内の第一項は,通常の Schrödinger 方 potential V„(qz), and can be ignored small a very range of values of q3 around q3 —0. R.C.T. da Costa, Phys. Rev. A 23, 1982 (1981). howExpression in allk&future (14), of where and k2calculations. curvatures are the principal In this case we can safely take q3-0 in all coeffi- 2 pres- パラ 程式で目にするラプラシアン演算子 interesting, due to the TLL is much the ever, surface of E(I. (11) [except of course in the term ! の曲線座標表示 cients S, andmore作用の強さを表す のとき m = −1, 0, 1 の 3 つの値が,それぞれ許されます.これらをまとめ ますと, (n, l, m) = (2, 0, 0) 量子力学 (n, l, m) = (2, 1, 0), (2, 1, 1), (2, 1, −1) となります. 上記で既に記載していますが,軌道を区別しやすいように,方位量子数 の値によって軌道に名前が付けられています. ユークリッド空間 vs リーマン空間 l = 0 → s 軌道 l = 1 → p 軌道(3 重縮退) l = 2 → d 軌道(5 重縮退) l = 3 → f 軌道(7 重縮退) l = 4 → g 軌道(9 重縮退) ⎧ ⎛ h ⎞⎡ ∂ 2 ∂ Λ ⎤ e 2 ⎫ + + 2 ⎥ − ⎨−⎜ ⎬ R(r)Y (θ, φ ) = ER(r)Y (θ, φ ) 2 ⎟⎢ 2 h, i, · · · と続きます(実際に安定 す.その後は,アルファベット順に,g, 8m π ∂ r r ∂ r r 4 πε r ⎣ ⎦ ⎝ ⎠ ⎩ e 0 ⎭ に存在する元素はウラン(原子番号 Z = 92)原子の f 軌道までです.そ 2 2 は principle,d は diffusion,f は という具合です.ここで,s は sharp,p 1次元ユークリッド空間箱の自由電子 fundamental の頭文字が由来で,これらは分光学から名付けられたもので の上の g 軌道に電子が詰まるのは Z = 121 以上の超重元素(super heavy .したがって,波動関数 ψ100 は 1s 軌道,ψ200 は 2s elements)1) からです) 軌道,ψ21-1 , ψ210 , ψ211 は 2p 軌道と名付けられます. 1) 現在,Z = 113 までの元 素が確認されています. 水素原子の n = 3 までのエネルギー準位について,ボーアモデルの結果 とシュレーディンガー方程式の結果を図 5.3 に示します.ボーアモデルで € 2 ∂2ψ − = Eψ 2 2m ∂x E3 E3 E2 水素原子 (x, y, z) → (r, θ, φ) Ψ(x, y, z) → R(r)Y(θ, φ) 3d m=+1 m=0 m=−1 2s 2p E1 1s ボーアモデル シュレーディンガー方程式の結果 図 5.3 水素原子のエネルギー準位 水素原子——前期量子論との比較(ボーアモデルとシュレーディンガー方程式の結果の比較) 2.0 1s R10 4.0 R210 1.0 0 0.8 0.6 R 0.4 2s 20 0.2 0 −0.2 0 0.40 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r/aB 3s 0 0.5 0.3 0.1 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r/aB 4.0 6.0 r/aB 8.0 0 2.0 0.15 4.0 6.0 r/aB 0 0.764 8.0 0.08 6.0 10.0 14.0 18.0 r/aB 図 5.5 0 2.0 8.0 0.04 0.05 0 4.0 5.236 r/aB r2R210 0.06 R210 0.10 0.10 0.10 0.05 0.1 2.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r/aB 0.15 r2R210 0.2 R30 0.20 −0.10 0 2.0 0.7 2 r2R10 0.5 2.0 0.4 R210 0.3 0.30 1 #$ #' 1 " = 2 &r2 ) + 2 * r #r % # r ( r m=+2 m=+1 m=0 m=−1 m=−2 3p エ ネ ル ギ ー E1 第5章 m=+1 m=0 m=−1 3s E2 エ ネ ル ギ ー 82 € 動径関数 は,各エネルギー準位に対して 1 つの軌道しかありませんが,シュレーディ 0.02 6.0 10.0 14.0 18.0 r/aB 0 2.0 6.0 10.0 14.0 18.0 r/aB 1s∼3s 軌道における動径関数と関連する値 pz = Y10 = ! 3 z 4π r (5.34) はそのまま実数型の球面調和関数として使えますが,Y1-1 , Y11 は複素数を 球面調和関数 the surface, which leads to an effective Hamiltonian for propagation along the curved surface with no ambiguity. It is well known that the effective Hamiltonian involves an effective scalar potential whose magnitude depends on the local surface curvature.2–5 As a result, quantum particles confined to a thin curved layer behave differently from those on a flat plane, even in the absence of any external field !except for the confining force". Such curvature effects have gained renewed attention in the last decade, mainly, because of the technological progress that has enabled the fabrication of low-dimensional nanostructures with complex geometry.6,8–11,7,12–14 From the theoretical perspective, many intriguing phenomena pertinent to electronic states,15–22 electron diffusion,23 and electron transport24–27 have been suggested. In particular, the correlation between surface curvature and spin-orbit interaction28,29 as well as with the external 1次元ユークリッド空間箱の自由電子 magnetic field30–32 has been recently considered as a fascinating subject. Most of the previous works focused on noninteracting electron systems, though few have focused on interacting electrons33 and their collective excitations. However, in a low-dimensional system, Coulombic interactions may drastically change the quantum nature of the system. Particularly noteworthy are one-dimensional systems, where the Fermiliquid theory breaks down so that the system is in a liquid !TLL" state.34 In a TLL state, 2 Tomonaga-Luttinger 2 many physical quantities exhibit a power-law dependence stemming from the absence of single-particle excitations near the2 Fermi energy; this situation naturally raises the question as to how geometric perturbation affects the TLL behaviors of quasi-one-dimensional curved systems. Peanut- increase in the power-law exponent " of the single-particle density of states n!#" near the Fermi energy EF; i.e., n!#" $ #%# − EF#".38,39 The geometric conditions required for the shift in " to be observable are within the realm of laboratory experiments, which implies that our predictions can be verified with existing materials. We first considered noninteracting spinless electrons confined to a general two-dimensional curved surface S embedded in a three-dimensional Euclidean space. A point p on S is represented by p = $x!u1 , u2" , y!u1 , u2" , z!u1 , u2"%, where !u1 , u2" is a curvilinear coordinate spanning the surface and !x , y , z" are the Cartesian coordinates in the embedding space. Using the notation pi & !p / !ui !i = 1 , 2", we introduced the following quantities gij = pi · p j, hij = pij · n, and n = !pi 1次元凹凸周期曲面上の自由電子 & p j"(リーマン幾何学で扱うと便利) / 'pi & p j', where n is the unit vector normal to the surface. Using the confining-potential approach,2,3 we obtained the Schrödinger equation for noninteracting electron systems on curved surfaces as follows: 量子力学 ユークリッド空間 vs リーマン空間 島 弘幸・小野頌 膜を考えます。次に,この膜の形に沿った量子井戸型ポ Δ normal Δlateral テンシャルで膜を上下から挟むことにより,膜の内部に δ − € ∂ψ = Eψ 2m ∂x 水素原子1098-0121/2009/79!20"/201401!4" (x, y, z) → (r, θ, φ) Ψ(x, y, z) → R(r)Y(θ, φ) "= 1 #$ 2#' 1 &r )+ * r 2 #r % # r ( r 2 r r0 粒子を閉じ込めます。その後,ポテンシャル障壁の高さ を十分高くし,膜厚ゼロの極限を取ります。すると,曲 λ FIG. 1. !Color online" Schematic illustration of a quantum hol面接線方向に運動する粒子の有効ハミルトニアンが,あ low cylinder with periodic radius modulation. 1 201401-1 ©2009 The American Physical Society ,q 2)を用いて以下の形に求まりま る曲線座標系(q す 16)。 2 2 ! 1 ! ij ! 2 ! 6 g g (1) Ĥ =− * i j +(h −k) 2 m ! g i, j=1 !q !q ! r $ % ij 幾何曲率項 ここで m * は粒子の有効質量,g は行列[g ij]の逆行列 軽量テンソル 11) R.C.T. da Costa, Phys. Rev. A 23, 1982 (1981). 。h とkは 成分,g は行列[g ij]の行列式を表します それぞれ曲面の平均曲率・ガウス曲率と呼ばれる量であ ベクトルポテンシャル(アハラノフ・ボーム効果) Vector potential A was thought to be introduced mathematically, but its physical meaning was given by Aharonov and Bohm. Y. Aharonov and D. Bohm: Phys. Rev. 115, 485 (1959) Aharonov-Bohm effect phase difference = 0 superconductive coil phase difference = 1/2 λ A. Tonomura et al.: Phys. Rev. Lett. 56, 792 (1986) 1D metallic peanut–shaped C60 polymers In situ FT-IR apparatus FT-IR E-gun K-cell ρ = 1–10 Ωcm Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with an uneven peanut-shaped structure similar to the cross-linked structure of the P08 C120 Electron beam (3 kV, 0.3–0.5 mA) 1017 cm-2s-1 stable isomer predicted using the general Stone-Wales rearrangement. The area colored in sky blue represents a Riemannian curved space in which !-electrons move one-dimensionally. The !r denotes the degree of uneven deformation. 100 nm CsI substrate C60 single crystal ρ = 108 – 1014 Ωcm JJAP. 39, 1872 (2000), JAP. 92, 7302 (2002), EPJD 24, 389 (2003) APL 82, 595 (2003), APL 85, 2741 (2004), PRB 72, 155416 (2005) PRB 74, 195426 (2006), JPCB 110, 22374 (2006), PRB 75, 233410 (2007), APL 92, 094102 (2008), JAP 104, 103706 (2008), PRB 79, 201401 (R) (2009), JAP 108, 033514 (2010), APL 97, 241911 (2010), EPL 98, 27001 (2012), JPCM 24, 175405 (2012), Diamond & Relat. Mater. 33, 12 (2013). ガウス曲率からみたナノカーボンの分類 Gaussian curvature k = κ1κ2 Average curvature h = (κ1+κ2)/2 Maximum curvature: κ1 Minimum curvature: κ2 nanocarbon family 0 + 0, + – (5) peanut-shaped (exotic) nanocarbons (2) P c S Gaussian curvature (1) graphene (2) fullerenes (3) nanotubes (4) Mackay Crystal (Hypothetical) (1) n X (3) +,ー (4) (5) up 凸 κ < 0 down 凸 κ > 0 Physical behaviors arising from 1D metal FIG. 2. Schematic illustration of the structural region considered for obtaining theoretical IR spectra in the case of the P04 isomer. The region marked by the dashed rectangle shows the common capped structure for all the isomers, thus omitting the IR modes originating from this region for comparison with the experimental IR spectra. Tomoonaga-Luttinger liquid states Peierls transition FIG. 1. !Color online" Schematic illustration of C120 stable isomers derived from the GSW rearrangement. The nth C120 isomer is denoted as Pn. Int(EF)/Int(0.5eV) Intensity (a.u.) Intensity (arb. units) Transmittance [arb. unit]! 2Δ= 3.52k T ∼ 15 meV (T ∼50 K) Appl. Phys. Lett. 92, 094102 (2008). EPL (Europhys. Lett.) 98, 27001 (2012). Phys. Rev. B 79, 201401 (R) (2009). 1125 1115 1105 Charge-Density-Wave (CDW) phonon mode III. THEORETICAL CALCULATIONS 1125 We have carried out first-principles density-functional 33.1 K! 40.0 K! 50.0 K! 60.0 K! 70.0 K! 80.0 K! 90.0 K! 100.0 K! 1115 1105 Peak (g)! Peak (g)! calculations of IR spectra for all C120 stable isomers derived Transmittance [arb. unit]! Transmittance [arb. unit]! Peak (c)! nce [arb. unit]! Fig. 1. for 1D C60 polymers,19 individual vibrational modes are uniquely assigned to their individual local structures. This II. EXPERIMENTS allows us to examine the theoretical IR spectra of the C120 (a) unit of their corresponding (b) isomers regarded as the minimum The present apparatus used for measuring in situ IR 1D polymers by excluding the common structure !similar 104 to ρ(ω)∝¦ω¦ α spectroscopy consists of an ultrahigh vacuum !UHV" chamρ(ω)∝¦T¦ α that of C60" atT=350 their Kedges. Figure 2 shows the schematic ber !a base pressure: 2 ! 10−7 Pa", a Fourier-transform IR Fig. 1. Since the structure of the P04 C120 isomer taken from T =150 K spectroscope !Mattson Research Series", a Knudsen cell !Kregion marked by250 theKdashed rectangle overlapps with a C60 ΔE=12 meV cell", a molecular turbo pump combined with a rotary pump Peak (c)! cage structure, it 150 canK be excluded when the corresponding Res. Limit ω0.66 !Balzers", an 33.1 EB gun gage 103 T 0.59 K!!Omegatron", a partial pressure 0.66 1D P04 polymer is considered. As a result, the region ω !Anelva", and a helium !He" cryostat !Iwatani". Details of 30 K 125 K includes the common 40.0 K!described elsewhere.15,16 C60 films !60 marked by the solid rectangle, which this system have been 0.1P04 C 0.01its corre30 100 and cross-linked structure between the 120(eV) 85 K Binding Energy nm thick" were formed on cesium iodide !CsI" substrates !20 Temperature(K) 50.0 K!(color C60-polymer sponding 1D polymer, is obtained. In this way, we consid(color online). Schematic illustrations of Peierls Fig. 2. online). Schematic illustrations of photomm in diameter and 2 mm thick" by thermal evaporation of K hν=40.8eV (Heered IIα)the IR modes30arising only from the cross-linked strucp p 60.0 K!induced 99.98% pure" in the dynamics K-cell at 673 at K (a) T>T p, (b) T<T p. transition inBquasi-1D metals at (a) T>T p, (b) T<TCp60. powder !Matsubo, carrier isomers, and compared these with the ture of all the C 120 -0.2 0.2 EF 0.4 for 3 min after70.0 residual organic solvents in the powder were K! experimental ones. Binding Energy (eV) removed at 473 k for 2 h in the same UHV chamber. SubseTo obtain the theoretical IR spectra due to the cross80.0 K!the in situ IR spectra of the C60 films quently, we measured linked structure of all the C120 isomers, we replaced each before EB-irradiation and after EB-irradiation !3 kV, 0.5 stick peak with a Lorentzian function with a full width at half 90.0 K! mA" for 1, 3, 5, 10, and 25 h. All the IR spectra were remaximum !FWHM" corresponding to that of the intense and −1 and with 5000 scans corded with a100.0 resolution of 4 cm K! narrow IR peak appearing at 565 cm−1 for the 25 h EB!S / N " 104". The substrate temperature was controlled in the irradiated C60 film. range of 50–300 K within a variation of #0.1 K, using the He cryostat as well as a heating system. IV. RESULTS AND DISCUSSION 1252 from the GSW rearrangement, using the GAUSSIAN03 33.1 K! package.17 In all calculations, we employed a 6–31g !d" basis set and the Perdew–Burke–Ernzerhof exchange-correlation 18 40.0 K! potential, because these conditions have been shown to provide good 50.0 agreement K! in the energy gap and IR spectra between computational and experimental results for C60.12,19 For example, 60.0 the energy K! gap of C60 was estimated to be 1.67 eV, which is in an excellent agreement with the values of 1.6–1.85 eV obtained 70.0 K!experimentally.20 Figure 1 shows a schematic illustration of thermody80.0C120 K!isomers derived from the GSW rearnamically stable rangement, where 90.0the K!nth C120 isomer is denoted as Pn. For example, the first C120 isomer with a dumbbell-shaped cross100.0 K! linkage is named P01, while the last C120 isomer is P24. 1D infinite structure of each isomer should be 1245Ideally, the 1238 considered. However, unlike the estimation of the energy gap A. IRvan spectraHove Singularity 1D Figure 3 shows the time evolution of the IR spectra of an CsI substrate 1340 cm–1 FIG. 3. Irradiation-time evolution of in situ IR spectra of an EB irradiated C60 film. Downloaded 08 Aug 2010 to 131.112.115.47. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp Appl. Phys. Lett. 97, 241911 (2010). 33.1 K! 40.0 K! 50.0 K! 60.0 K! J. Appl. Phys. 108, 033514 (2010) J. Phys.: Condens. Matter 24, 175405 (2012). ポリマー凹凸曲面上に沿って動く電子 (幾何学ポテンシャルを感じながら電子が動く?) 1950年代からの謎 島 弘幸・小野頌 電子 膜を考えます。次に,この膜の形に沿った量子井戸型ポ テンシャルで膜を上下から挟むことにより,膜の内部に 粒子を閉じ込めます。その後,ポテンシャル障壁の高さ を十分高くし,膜厚ゼロの極限を取ります。すると,曲 面接線方向に運動する粒子の有効ハミルトニアンが,あ る曲線座標系(q 1,q 2)を用いて以下の形に求まりま す 16)。 2 2 ! 1 ! ij ! 2 ! 6 g g (1) Ĥ =− * i j +(h −k) 2 m ! g i, j=1 !q !q ! r $ % ここで m * は粒子の有効質量,g ij は行列[g ij]の逆行列 成分,g は行列[g ij]の行列式を表します 11)。h と k は 1次元電子物性に対する幾何曲率効果の予測 in collaboration with Prof. H. Shima (Yamanashi Univ.) Prof. H. Yoshioka (Nara W-Univ.) 2-3 D metal Fermi Liquids (FL) CNT 島 弘幸・小野頌太 膜を考えます。次に,この膜の形に沿った量子井戸型ポ 1D metal テンシャルで膜を上下から挟むことにより,膜の内部に Tomonaga-Luttinger Liquids (TLL) α<1 粒子を閉じ込めます。その後,ポテンシャル障壁の高さ を十分高くし,膜厚ゼロの極限を取ります。すると,曲 面接線方向に運動する粒子の有効ハミルトニアンが,あ る曲線座標系(q 1,q 2)を用いて以下の形に求まりま δr [Å] す 16)。 1Å= 0.1 nm !2 1 2 ! ij ! 2 ̂ 6 (1) H=− * i ! gg j +(h −k) 2 m ! g i, j=1 !q !q ! r $ % ここで m * は粒子の有効質量,g ij は行列[g ij]の逆行列 Phys. Rev. B 79, 201401 (R) (2009). 成分,g は行列[g ij]の行列式を表します 11)。h と k は 1D 凹凸 C60 polymerのin situ高分解能光電子分光 in collaboration with Prof. T. Ito (Nagoya Univ.) Prof. S. Kimura (Osaka Univ./IMS) (b) ρ(ω)∝¦ω¦ α T =150 K ω0.66 103 30 K Intensity (arb. units) 0.1 0.01 Binding Energy (eV) T=350 K 250 K 150 K ω0.66 125 K 85 K C60-polymer hν=40.8eV (He IIα) 0.4 0.2 104 Int(EF)/Int(0.5eV) Intensity (a.u.) (a) ρ(ω)∝¦T¦ α T 0.59 ΔE=12 meV Res. Limit 30 100 Temperature(K) ρ(T) ∝ |T|α ρ(E) ∝ |Ε – EF|α 凹凸periodic: α ∼ 0.6 30 K EF Binding Energy (eV) -0.2 Europhys. Lett. 98, 27001 (2012). 1次元電子物性に対する幾何曲率効果の実証 δr = 1.4 Å (a) (b) Int(EF)/Int(0.5eV) Intensity (a.u.) 4 10 α of a one-dimensional αwith an Figure 1. Schematic illustration metallic C polymer ρ(ω)∝¦ω¦ ρ(ω)∝¦T¦ 60 uneven peanut-shaped structure similar to the cross-linked structure of the P08 C120 T =150 K stable isomer predicted using the general Stone-Wales rearrangement. ω0.66 T 0.59 The area meV ΔE=12 Res. Limit colored in sky blue represents a Riemannian3 curved space in which !-electrons move 10 one-dimensionally. 30 K The !r denotes the degree of uneven deformation. 0.1 0.01 Binding Energy (eV) 30 100 Temperature(K) ρ(T) ∝ |T|α ρ(E) ∝ |Ε – EF|α 凹凸periodic: α ∼ 0.6 Europhys. Lett. 98, 27001 (2012). δr [Å] 1Å= 0.1 nm Phys. Rev. B 79, 201401 (R) (2009). The charge polarization of the C120 isomer J. Appl. Phys. 108, 033514 !2010" ma, Onoe, and Nishii −1 ears at 565 cm . Since the FWHM peak is almost equal to that !6 – 8 cm−1" film with just the C60 structure, this sugpolymer formed after saturation of EB es not contain a mixture of cross-linked r is comprised of just one structure. and, the peak at 1182 cm−1 for the prisnot to have changed even after 25 h EB ay be because this mode arises from the an-curved structure22 that remains even in polymer. 29 cm−1 for the pristine film disappeared iation, as shown in Fig. 3, which indicates were completely polymerized. In place of d peaks in the range of 1360– 1400 cm−1 changed gradually with increasing EBup to 10 h !a mixture of cross-linked 5 h irradiation, just one intense narrow at 1340 cm−1 in addition to the peak at ven cross-linked structure". In summary, ization is saturated, the EB-polymerized ts two narrow and highly intense peaks at 1 . The large enhancement in IR intensity s is not clearly understood yet, but is prevon Hove singularity in the vibrational r the 1D C60 polymer.23 We are currently study of this anomaly. ith theoretical spectra ares the IR spectrum !red plot" of the 25 h film with the theoretical spectra !blue plot" cross-linked structures of all the C120 isohown in Fig. 1. Here, we used a scaling of 0.9–1.0 for adjusting the theoretical erimental one in order to compare their es.24 As shown in Fig. 4, the IR spectra cross-linked structures of P04 and P07– mble the experimental spectrum relatively the other isomers. As shown in Fig. 1, the ure of these C120 isomers has a peanuthus supporting the previous results.1,11,12 etical IR spectra were calculated at 0 K, it e these with experimental IR spectra of the easured at a temperature lower than 300 ares the experimental IR spectrum meathe theoretical spectra obtained from the ure of P04 and P07-P11. The IR spectrum cross-linked structure of P08 is in relament with the experimental one than the es, except that the calculated peak around vely intense compared to the experimengly, the comparison between the experiical IR spectra suggests that the 1D med C60 polymer has a cross-linked structure that of P08 as shown in Fig. 6. Gaussian 03 FIG. 5. Comparison of the experimental spectrum !measured at 60 K" with theoretical ones obtained from the cross-linked structure of P04 and P07– P11 isomers. Polarization degree: ca. ± δ = 0.05 Negative Charge !"#"$% V. SUMMARY We measured the time evolution of the IR spectra of an EB-irradiated C60 film, using in situ IR spectroscopy, and found that two new highly intense narrow peaks finally appeared around 565 and 1340 cm−1 when the EB-induced C60 polymerization was saturated. Since the FWHM of the former peak was almost equal to that for the pristine C60 film, the EB C60 polymer obtained after saturation of EB polymerization seems to have a single cross-linked structure. When the present experimental IR spectra !300 and 60 K" were compared with the theoretical spectra obtained from the cross-linked structure of all C120 stable isomers derived from the GSW rearrangement !that can be regarded as the minimum unit of their corresponding 1D polymers", the results suggested that the cross-linked structure of the P08 peanutshaped C120 isomer was in relatively better agreement with the experimental IR spectra than the other isomers. This indicates that the 1D metallic peanut-shaped C60 polymer has a cross-linked structure roughly similar to that of the P08 isomer. Although the experimental IR spectra obtained after saturation of EB-induced C60 polymerization indicate that the 1D EB C60 polymer has a single cross-linked structure, "#"&' Positive Charge FIG. 6. Schematic illustration of the cross-linked structure of the P08 isomer. g 2010 to 131.112.115.47. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp 1 Ry : ± δ = 1.0 (r = 0.053 nm) for H atom Europhys. Lett. 98, 27001 (2012). を十分高くし,膜厚ゼロの極限を取ります。すると,曲 まとめ 面接線方向に運動する粒子の有効ハミルトニアンが,あ 1 2 ,q )を用いて以下の形に求まりま る曲線座標系(q A potential driven by a periodic uneven (凹凸) geometry has been curious to affect electronic properties of materials 。 1950s. す 16)since 2 ! 1 2 ! ij ! 2 ̂ ! 6 g g (1) H=− * i j +(h −k) i, j=1 2m ! g !q !q ! r $ % ここで m * は粒子の有効質量,g ij は行列[g ij]の逆行列 First observation of geometric curvature effects on electron behaviors 11)。h と k は 成分,g は行列[g ij]の行列式を表します それぞれ曲面の平均曲率・ガウス曲率と呼ばれる量であ り,曲面上の各点における曲がり具合を表します。h と k は g ij の関数ですので,結局曲面の形さえ決まれば, その形状を記述する計量テンソル g ij により Ĥ が一意に 決まります。 Perspective Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with an 式( 1 )かっこ内の第一項は,通常の Schrödinger 方 uneven peanut-shaped structure similar to the cross-linked structure of the P08 C120 2 の曲線座標表示 程式で目にするラプラシアン演算子 ! Physical quantities Geometric quantities stable isomer predicted using the general Stone-Wales rearrangement. The area colored in sky blue represents a Riemannian curved space in which !-electrons move です 11)。見た目は多少複雑ですが,この表示自体は曲線 quantitative or qualitative correlation ? one-dimensionally. The !r denotes the degree of uneven deformation. 座標系(例えば極座標)を採用すれば平面系でも現れる ものなので,系の曲面性と直接は関係しません。代わり
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