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1 018 年 11 月 1-14 日名古屋大学工学研究科 工学部応用物理学特論 応用物理学特別講義 ( 集中講義 ) 東京大学理学系研究科物理学専攻 Lecture Slides (PDF files) 長谷川修司 1.Nanoscience and Surface Physics ナノサイエンスと表面物理 Nanoscience in Nobel Prize.Atomic Arrangements at Surfaces 表面原子配列構造 Scanning Tunneling Microscopy, Electron Diffraction 走査トンネル顕微鏡 電子回折 3.Surface Electronic States 表面電子状態 Surface states 表面状態 Rashba Effect ラシュバ効果 Topological Surface States トポロジカル表面状態 Band Bending バンド湾曲 4.Surface Electronic Transport 表面電気伝導 Space-Charge-Layer Transport and Surface-State Transport 空間電荷層伝導と表面状態伝導 Atomic-Layer Superconductivity 原子層超伝導 エネルギーダイヤグラムと仕事関数 Energy Diagram & Work Function 真空準位 Vacuum Level Bulk Term E V Surface Term 電子エネルギー Electron Energy 物質 Material 真空 Vacuum Work Function E F フェルミ準位 Fermi Level 物質中の電子を外に取り出すのに必要なエネルギーの最小値 The minimum energy necessary for taking an electron out of the material 物質中の最高占有エネルギー準位にある電子を真空準位に上げるのに必要なエネルギー The energy necessary to excite an electron at the highest occupied level to the vacuum level 物質の外 : 無限遠ではなく 物質の表面の直上 ( 表面から鏡像力の影響を無視できる程度の距離 ~1 μm) の真空中 Outside of the Material = a position away from the surface (~1 μm) at which the image force is ignored, not a position at infinite. 表面項 Surface Term 電子の滲みだしと表面電気二重層 Spill out of electrons & Surface Electric Dipole Layer Electron Density normalized by Bulk value 物質 Material 表面 Surface 真空 Vacuum Distance from Surface in unit of Fermi wavelength 電子のエネルギー (=- 電位 ) Electron Energy (= - Potential) 平行平板コンデンサー Parallel-Plate Capacitor 低 Low ー ー ー ー ー 高 High バルク項 ( 交換相関エネルギー V XC )-( 運動エネルギー ) Bulk Term (Exchange-Correlation Energy V XC )-(Kinetic Energy) それぞれの電子の周りには電子密度の低い領域 ( 正電荷を帯びた領域 ) が存在 真空中の電子に比べて安定化 ( エネルギーが下がる V XC ) Low-el-density area (positively charged area) around each electron Each electron is more stabilized than that in vacuum (Energy lowering V XC ) クーロン孔, 相関ホール (Correlation Hole) 電子間のクーロン反発によって他の電子を遠ざけている ( 相関相互作用 ) (Correlation Inter. due to Coulomb repulsion) フェルミ孔, 交換ホール (Exchange Hole) 同じスピンを持つ電子どうしは, パウリの排他原理による交換相互作用による反発がはたらき他の電子を遠ざけている (Exchange Inter. due to Paul s Exclusion Principle) 4 3 ρ: 電子の数密度 ( 個 /cm 3 ) Number density of electrons (1/cm 3 ) 3 R S :1 個の電子が占める体積 (cm 3 ) Volume occupied by an electron (cm 3 ) R S RS 無次元化 Dimensionless rs a B :Bohr Radium (0.5A ) R S 3/ 4 a B 物質 1/ 3
2 仕事関数の電子密度依存性 Dependence of Work Function upon Electron Density Work Function r 3/ 4 a 1/ 3 S B r : 大きい Large S 低電子密度 Low Density バルク項の寄与大 Bulk Term: Large 表面項の寄与が小 Surface Term : Small 金属単結晶の仕事関数 Work Function of Metal Single Crystal Face Orientation Crystal Structure Metal Bulk Term Surface Term r S : 小さい Small 高電子密度 High Density バルク項の寄与小 Bulk Term: Small 表面項の寄与大 Surface Term: Large In unit of ev 表面項が違う : 原子数の表面密度が大きいほど表面二重電気層が強くなり 仕事関数が大きくなる Different Surface Term: As larger the atom density at surface is, stronger the surf. Electric Dipole Layer is. larger Work Function 電子を物質から取り出して無限遠にもっていくのに必要なエネルギーは 取り出す結晶面によらずに同じ fcc 金属 :(111)>(100)>(110) bcc 金属 :(110)>(111)>(100) From Energy Levels to Band Formation Surface States Shockley & Tamm States Electron Energy Bulk Conduction Band Bulk Bands & Surface-State Bands Surface Conduction Band Clean Surfaces Surface States decouple from bulk states Dangling Bonds Adsorbed Surfaces (Surface Alloys) Surface States Adsorbates Anti-Bonding State Dangling-Bond State Bonding State Hybridized Orbital Atomic Orbitals Dangling Bonds Isolated Atoms Bulk Valence Band Surface Valence Band Bulk States Metals Semiconductors Topological surface states -Low-D Electronic Systems -Broken (Space-Inv.) Symmetry -New Periodicity
3 Various Surface States 1. Shockley states (extended) Tamm states (localized) Chemical bonding, Surface Potential. Image states Image charge 表面空間電荷層 3. Surface space-charge layer Bending of bulk bands Image charge Conduction band Valence band 4. Topological surface states Quantum Hall Effect Spin-orbit coupling Edge states of Q(S)H phase E F HgTe (QW), Bi 1-x Sb x, Bi Te 3, Bi Se 3, Surface Energy E Mono-Layer Metal on Si(111) Cond. Band Valence Band Wavenumber k p k E m* m* Free-Electron-like (Non-relativistic) Various Surface States Graphene (Monolayer Graphite) E Cond. Band Valence Band ( mc ) ( pc m 0 Au(111) Bi(111) Spin Cond. Band E pc ck Massless Dirac Electrons ) Spin Valence Band Spin Bi alloys Cond. Band Spin Valence Band Rashba effect Topological SS Spin-degenerated Spin-split (Relativistic) バンド構造とブリルアン領域 Band Structure and Brillouin Zone Nearly Free Electron Approximation 3 次元 Si 結晶のバンド構造 Bands of Three-Dim. Si Crystal 3 次元ブリルアン領域と表面ブリルアン領域 拡張ゾーン形式 Extended Zone Scheme 還元ゾーン形式 Reduced Zone Scheme 周期的ゾーン形式 Periodic Zone Scheme 状態密度 Density of States Conduction Band D Brillouin Zone (k space) Projection along <111> direction Valence Band Reciprocal Lattice Points 3D Brillouin Zone (k space) Wavenumber k
4 ARPES (Angle-Resolved Photoemission Spectroscopy) ARPES Apparatus & Spectra UV X-Ray Photoelectrons Sample Electron Detector Electron energy analyzer Band dispersion E b (k // ) Energy E b vs. Wavenumber k // Energy Conservation E kin =hν-e b -φ Momentum (Wavenumber) Conservation k ex // = k // Kinetic Energy Work Function E b Binding Energy 東北大学 理 物理高橋隆研究室 Mono-Layer Ag on Silicon : Si (111)- 3 3-Ag Surface Spectra from Si(111) Surfaces D Metal Si(111)-7 7 Clean Surf. Si(111)- 3 3-Ag Surf. STM Image By depositing Mono-layer Ag Clean Si Surface 手塚 M 論 ( 東京大学 1990) X. Tong, et al., Phys. Rev. B57, 9015 (1998)
5 - Surface-State Bands & Surface Space-Charge Layer Theory: Surface Bands of Monolayer Ag on Si(111) ダングリング ボンド状態 Dangling-Bond State 分散がほとんど無い No Dispersion 局在状態 Localized Sate 伝導度が低い Low Conductivity E Si(111)-7 7 k // 空乏層 Depletion Layer E Si(111)- 3 3-Ag Ag-Ag 結合状態 (Bonding State) (5p 軌道由来 ) (5p orbital origin) 大きい分散 Large dispersion 拡がった状態 Extended Sate 伝導度が高い k // High Conducitivity ホール蓄積層 Hole-Accumulation Layer Electron Energy E (ev) Surface states are in the bulk band gap. Conduction Band Band Gap E F E p k m* m* Valence Band Free-electron-like state Wavenumber k H. Aizawa and M. Tsukada, Surf. Sci. 49 (1999) L509 Experiment: Surface Bands of Monolayer Ag on Si(111) Angle-Resolved Photoemission Band Dispersion S. Hasegawa, et al., Prog. Surf. Sci. 60 (1999) 89. T. Hirahara, et al., e-j. Surf. Sci. Nanotech. (004) 141. T. Hirahara, et al., Surf. Sci. 563 (004) 191. Fermi-Surface Mapping 3D DOS Band Dispersion Fermi Surface k k x k y k z Fermi Level Fermi Level Wave Vector k [11] (Å -1 ) D k k x k y Wave Vector k - [110] (Å -1 ) m*=0.13m e Parabolic dispersion E = h (k xk y ) m* Free-electron like Metallic state Circular Fermi Surface 1D k k x Free-Electron System Isotropic and free-electron-like Metallic D electron system 0D E p k m* m*
6 Au Adsorption on Si(111)- 3 3-Ag 0.01 ML Au RT 0.03 ML 65 K 電子の海のさざ波 Ripples in electronic sea (Electron Standing Waves) 135 K 0.0 ML Au RT C. Liu, I. Matsuda, R. Hobara, and S. Hasegawa, Phys. Rev. Lett. 96, ( 006). 1. Carrier doping in the surface-state band Increase in band occupation. Hybridization of the localized state and surface-state band Band splitting 0.0 ML Au Parameters of Si(111)- 3 3-Ag Surface Fermi Wavenumberフェルミ波数 Effective Mass有効質量 Electron Density 電子濃度 Density of States状態密度 Fermi Velocity フェルミ速度 Mean Free Path平均自由行程 Relaxation Time 緩和時間 Momentum (Angular) Distribution Curve at EF Δθ Electron Standing Wave on Si(111)-Ag at 65K T. Hirahara, et al., Surface Science 563 (004) STM Image di/dv Images -0.9 V -0.9 V -0.8 V I t exp d d0 電子の波動関数 の絶対値の２乗 が直接見える!!! Electron wavefunction (its square of the absolute value) is directly observed!!! -0.7 V
![Standing Wave 反射波 Ψ r =R exp[i(-k x xk y y)] u(x, y)](/docs-images/90/103006355/images/7-3.jpg "Reflected Wave R= R exp[iη] η: 反射位相シフト Reflection")
7 電子定在波 Electron Standing Wave ポテンシャル障壁 Potential Barrier ステップ Step ドメイン境界 Domain Boundary ρ(e, x)( 局所状態密度 Local DOS) Ψ i Ψ r {1 R R cos(k x x-η)} u(x,y) 次元ブロッホ波 D Bloch Wave 入射波 Ψi=exp[i(k x xk y y)] u(x, y) Incident Wave E=E 0 h (k x k y ) m * u(x,y)= cell function 定在波 Standing Wave 定在波の波長 λ= kx Wavelength of Standing Wave 反射波 Ψ r =R exp[i(-k x xk y y)] u(x, y) Reflected Wave R= R exp[iη] η: 反射位相シフト Reflection Phase Shift 原子像 Atomic Arrangement π 絶縁体 3 3 Plan View インジウム吸着 Si(111) 表面 擬 1 次元金属 4 1 RT(metallic) 60 K ( 電荷密度波 ) A. A. Sarranin, et. al. H. Y. Yeom, S. et Takeda, al., PRL et 8, al (1999) 次元金属 7 3 物質の表面では 独特な原子の並び方をしている 3 次元の物質とは異なる独特な物性を示す S. L. Surnev, et. al. Surface-State Bands of Si(111)-4 1-In Surface Peierls Transition (Quasi-) 1D Metal -Quasi-1D Metallic Surface Anisotropy in Conductivity -Peierls Instability Charge-Density Wave (CDW) Metal-Insulator Transition Temperature Dependence of Conductivity H. Morikawa, et al. 表面科学 5 (004) 407. H.-W. Yeom, et al. PRL 8 (1999) 4898 Band Dispersion T. Abukawa, S. Kono, et al. Surf. Sci. 35 (1995) 33 High Temp Phase Low-Temp Phase Fermi Wavenumber Constant Charge Density CDW RT Si(111)-4 1-In Surface STM Image Energy Gap (Peierls Gap) Linear Fermi Surfaces Bisecting the Brillouin Zone Fermi Surface Mapping 70K
8 Metal-Insulator Transition at Si(111)-4 1-In Surface Atom Displacements (Lattice Distortion) with CDW 反射高速電子回折 (RHEED) Y.J.Sun et al., PRB 77, (008) High-Temp Phase RT MI Transition at~110 K With CDW Peierls Transition // H.W.Yeom et al., PRL 8, 4898 (1999) W. G. Schmidt, et al., Phys. Status Solidi B 49, No., (01) Low Temp Phase 100 K 1E E Resistance at I=0 (kω) 1E E04 10 Electrical Resistance of Si(111)-4 1-In Surface T. Tanikawa, et al., Phys. Rev. Lett. 93 (004) Tc 4 10μm-spacing probe 4 1 RT Graphene on SiC crystal surface Relativistic(Dirac electron E E ck mc pc m 0 Zero Mass High mobility (M. Nagoya U.) Non-relativistic p k E m m 1E Temperature (K) 65 K RT CDW Phase (8 ) Metal Phase (4 1) H.-W. Yeom, et al., PRL 8 (1999) 4898 A. Bostwick, et al., Nature Physics 3, 36 (007).
9 Rashba Effect The electron energy is determined by its momentum (and spin). spin s k Time-reversal k Time-Reversal Symmetry E(k, )=E(- k, ) But, on Surfaces Momentum (Wavevector) Space-Inversion k s Space-Inversion Symmetry E(k, )=E(-k, ) s Emmanuel I. Rashba Time-Rev. Sym.Space-Inv. Sym. Spin(Kramers) Degenera Spin-Split in Surface States E(k, )=E(k, ) Difference in Energy between Spin and Spin H 1 p m gradv p 1 V( x) 4mc = σ grad V p Spin-Orbit-Coupling Hamiltonian Effective Magnetic Field B Orbital Motion Orbital Motion Eff. Mag. Field B of Electron of Nucleus Electron Nucleus σ Nucleus B σ Interact with spin σ Electron Lorentz Transformation (Rest Frame of Electron) Difference between and =Zeeman Energy by Eff. Mag. Field B Surface States of Au(111) Spin split due to Rashba effect Wavenumber k y (A -1 ) Binding Energy (mev) (b) (a) B C X A (d) (c) 10 nm Wavenumber k x ( M )(A -1 ) (a) (b) G. Nicolay, et al., Phys. Rev. B 65, (001). (c)(d) L. Petersen, et al., Phys. Rev. B 58, 7361 (1998).
10 Band Dispersion of 0 Atomic Layer Bi(111) slab (1st Principles Calculation) Energy Gap Surface States Red: Spin-Up Blue: Spin-Down Conduction Band Bihlmayer (Juhlich, Germany) vacuum Bi vacuum First-principles calculation for free-standing Bi slabs including SOC ARPES of Bi(111) Ultrathin films QWS and SS T. Hirahara, et al., Phys. Rev. Lett. 97, (006). Quantum-Well States Surface States Conductivity of Bi thin film = surface-state conductivity Binding Energy E (ev) Wavenumber k // (nm -1 ) DOS D(E) (ev -1 nm - ) Valence Band Pure Bi Semi-metal in Bulk Conduction band and valence band are connected by the spin-split surface states Binding Energy E (ev) 14 BL thick 30 BL thick 40 BL thick Wavenumber k (A -1 ) Wavenumber (A -1 ) Wavenumber (A -1 ) Topological Surface States Bi 1-x Sb x, Bi Te 3, Bi Se 3, Analogue of Edge States in Quantum Hall States (DEG) Extension to 3D Materials Strong SO Interaction produces effective B. Electronic States of Bi Se 3 (Theory) H. Zhang, et al., Nature Physics (May 009) Cond. Band SS Dirac Cone Val. Band Isolated Atom Atomic Bondings (Atomic Orbitals) Split due to Crystal filed Spin-Orbit Intercation Trivial Ins Non-trivial Ins
11 Chiral Dirac Cone of Topological Insulators and Current-Induced Spin Polarization BiSe3 : Epitaxial Growth & Bands Y. Sakamoto, et al., Phys. Rev. B81, (010). ky RHEED Layer Growth in Quintuple-Layer Unit Momentum p k RHEED Oscillation Bulk: Y. Xia, et al., Nature Physics 009 (May) kx EF M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 8, 3045 (010) k (Å -1) π-rotation of spin Time-Reversal Symmetry E(k, )=E(-k, ) Band Bending of Bulk States Near Surface Energy difference between EF and a core level Measured by Photoemission Spectroscopy Conduction Band EF EF ψ π-rotation ψinitial ψ π-rotation ψscattered1 Valence Band Core Level ψinitial ψ -π-rotation ψscatter Ψscatter = -Ψscatter1 π-rotation Spinor ψ Chiral Fermi Surface Ψscatter1 Ψscatter=0 Destructive interference No backscattering High mobility Core Level
12 Surface States & Surface-Space-Charge Layer(Band bending) Three Channels for Electrical Conduction near Surface - Surface-State Conduction - Surface-Space-Charge-Layer Conduction - Bulk Conduction S. Hasegawa & F. Grey Surface Science 500 (00) S. Hasegawa & F. Grey Surface Science 500 (00) Origins of Band Bending (Origins of SSCL) 外部印加電界 接触電位差 表面状態との電荷のやり取り
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