2018年11月12-14日 名古屋大学工学研究科・工学部 エネルギーダイヤグラムと仕事関数 Energy Diagram & Work Function 応用物理学特論・応用物理学特別講義（集中講義） 電子エネルギー Electron Energy 物質 東京大学理学系研究科物理学専攻 真空準位 Material 真空 Vacuum Lecture Slides (PDF files) EV 長谷川 修司 Vacuum Level http://www-surface.phys.s.u-tokyo.ac.jp/KougiOHP/ Surface Term １．Nanoscience and Surface Physics ナノサイエンスと表面物理 Work Function Nanoscience in Nobel Prize ２．Atomic Arrangements at Surfaces 表面原子配列構造 EF Bulk Term Scanning Tunneling Microscopy, Electron Diffraction フェルミ準位 Fermi Level 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、 • 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 Band Bending バンド湾曲 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 ４．Surface Electronic Transport 表面電気伝導 The energy necessary to excite an electron at the highest occupied level to the Space-Charge-Layer Transport and Surface-State Transport vacuum level 空間電荷層伝導と表面状態伝導 「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 Atomic-Layer Superconductivity 原子層超伝導 Outside of the Material = a position away from the surface (～1 μm) at which the image force is ignored, not a position at infinite.
バルク項 (交換相関エネルギーV )-(運動エネルギー) 表面項 XC Surface Term Bulk Term (Exchange-Correlation Energy VXC)-(Kinetic Energy) 電子の滲みだしと表面電気二重層 Spill out of electrons & Surface Electric Dipole Layer 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 ） 表面 Surface XC 平行平板コンデンサー Low-el-density area (positively charged area) around each electron ⇒ Each electron is
Parallel-Plate Capacitor more stabilized than that in vacuum (Energy lowering VXC) 物質 Material 真空 Vacuum 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 ＋ ー （相関相互作用） (Correlation Inter. due to Coulomb repulsion) ＋ ー ＋ ー 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 ＋ ー 物質 Potential) 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 ＋ ー RS － (Exchange Inter. due to Paul’s Exclusion Principle)
ρ：電子の数密度（個/cm3） Number density of electrons (1/cm3) 4 低 高 R3 ：1個の電子が占める体積（cm3） Volume occupied by an electron (cm3) High S Low 3 無次元化 1/3 電子のエネルギー（＝－電位） (= Energy Electron 1 Dimensionless 3/ 4 Electron Density normalized by Bulk value Electron Density 1 3 ⇒ 3 ⇒ rS 4 3 RS R 4 a S a B Distance from Surface in unit of Fermi wavelength 3 B：Bohr Radium (0.52Å) 仕事関数の電子密度依存性 金属単結晶の仕事関数 Work Function of Metal Single Crystal Dependence of Work Function upon Electron Density Face Orientation 1/3 3/ 4 Crystal Structure Metal rS aB ：大きい Large Work Function rS ⇒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
Surface Term rS ：小さい Small ⇒高電子密度High Density In unit of eV ⇒バルク項の寄与小 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。 Bulk Term: Small Different Surface Term: As larger the atom density at surface is, stronger the surf. Electric Bulk Term 表面項の寄与大 Dipole Layer is. ⇒ larger Work Function fcc金属：(111)>(100)>(110) Surface Term: Large bcc金属：(110)>(111)>(100) 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。
From Energy Levels to Band Formation Surface States ― Shockley & Tamm States ―
Bulk Bands & decouple from Clean Surfaces Adsorbed Surfaces Electron Surface-State Bands bulk states (Surface Alloys) Energy Surface Conduction Band Surface States Dangling Bonds Adsorbates Bulk Conduction Band Surface States Anti-Bonding State Dangling-Bond State
Bonding State
Hybridized Orbital Bulk Valence Band Atomic Orbitals Dangling Bonds Surface Valence Band Isolated Atoms Bulk States -Low-D Electronic Systems -Broken (Space-Inv.) Symmetry Metals Semiconductors -New Periodicity ⇔ Topological surface states Various Surface States Various Surface States Mono-Layer Graphene Au(111) Image charge Metal on Si(111) （Monolayer Graphite） Bi(111) Bi alloys 1. Shockley states (extended) Tamm states (localized) Cond. Band Cond. Band Cond. Band Cond. Band Chemical bonding, E
Surface Potential Surface Spin Spin Spin Spin 2. Image states Conduction band Energy Valence Band Valence Band Valence Band Valence Band Image charge EF 表面空間電荷層 Valence band 3. Surface space-charge layer Wavenumber k Rashba effect Topological SS Bending of bulk bands Spin-degenerated Spin-split 2 2 2 2 2 2 E ( mc ) ( pc) 4. Topological surface states Quantum Hall Effect p k E (Relativistic) Spin-orbit coupling ← Edge states of Q(S)H phase 2m* 2m* m 0
HgTe (QW), Bi1-xSbx, Bi2Te 3, Bi2Se3, Free-Electron-like E pc ck (Non-relativistic) Massless Dirac Electrons
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 結晶のバンド構造 拡張ゾーン形式 還元ゾーン形式 周期的ゾーン形式 状態密度 Si ３次元ブリルアン領域と表面ブリルアン領域 Bands of Three-Dim. Si Crystal Extended Zone Scheme Reduced Zone Scheme Periodic Zone Scheme Density of States Conduction Band
2D Brillouin Zone (k space) Projection along <111> direction
Valence Valence Band Reciprocal Lattice Points
3D Brillouin Zone (k space)
Wavenumber k ARPES (Angle-Resolved Photoemission Spectroscopy) ARPES Apparatus & Spectra
Electron energy analyzer Photoelectrons UV X-Ray Band dispersion Eb(k//) Electron Detector Energy Eb vs. Wavenumber k// Sample Energy Conservation Momentum (Wavenumber) Conservation ex Ekin=hν-Eb-φ k// = k//
Kinetic Energy
Work Function
Eb Binding Energy 東北大学・理・物理 高橋隆研究室
Spectra from Si(111) Surfaces Mono-Layer Ag on Silicon : Si (111)-√3×√3-Ag Surface Si(111)-7×7 Clean Surf. Si(111)-√3×√3-Ag Surf. 2D Metal X. Tong, et al ., Phys. Rev. STM Image ） 1990
By depositing B57 Mono-layer Ag , 9015 (1998)
Clean Si 手塚 Ｍ論（東京大学 Surface Surface-State Bands & Surface Space-Charge Layer Theory: Surface Bands of Monolayer Ag on Si(111)
Si(111)-7×7 Si(111)- Ag-Ag 結合状態 Surface states are in the bulk band gap. ダングリング・ E √3×√3-Ag (Bonding State) ボンド状態 （ 軌道由来）
E 5p Dangling-Bond State (5p orbital origin) 分散が 大きい分散 Conduction Band ほとんど無い Large dispersion
No Dispersion (eV)
拡がった状態 E Band Gap 局在状態 Extended Sate 2 2 2 Localized Sate p k EF 伝導度が高い E 伝導度が低い k// k// High Conducitivity 2m* 2m* Low Conductivity 空乏層 ホール蓄積層 Depletion Layer Hole-Accumulation Free-electron-like state Layer Valence Band Electron Energy Energy Electron
Wavenumber k H. Aizawa and M. Tsukada, Surf. Sci. 429 (1999) L509
Experiment: Surface Bands of Monolayer Ag on Si(111) DOS Fermi Surface Band S. Hasegawa, et al., Prog. Surf. Sci. 60 (1999) 89. Dispersion Angle-Resolved Photoemission T. Hirahara, et al., e-J. Surf. Sci. Nanotech. 2 (2004) 141. T. Hirahara, et al., Surf. Sci. 563 (2004) 191. 3D 2 2 2 2 Fermi-Surface Mapping Band Dispersion k kx k y kz
Fermi Level Fermi Level ) -1 (Å - [112] 2D 2 2 2 k kx k y Wave Vector k Vector Wave
- -1 Wave Vector k[110] (Å ) Parabolic dispersion 2 2 2 2 2 k k m*=0.13me h (k +k ) 1D x E = x y 2m* Free-electron like Metallic state Free-Electron System Circular Fermi Surface p2 2k 2 Isotropic and free-electron-like 0D E Metallic 2D electron system 2m* 2m* Au Adsorption on Si(111)-√3×√3-Ag 0.02 ML Au Parameters of Si(111)-√3×√3-Ag Surface 1. Carrier doping in the surface-state band ⇒ Increase in band occupation 2. Hybridization of the localized state and surface-state band Fermi Wavenumberフェルミ波数 ⇒ Band splitting RT 65 K Effective Mass有効質量
0.02 ML Au 0.03 ML 0.01 ML Au Electron Density 電子濃度 Density of States状態密度 Fermi Velocity フェルミ速度 RT Mean Free Path平均自由行程 Relaxation Time 緩和時間
Momentum (Angular) Distribution Curve at EF , 036803 ( 2006). 96
135 K Δθ C. Liu, I. Matsuda, R. Hobara, and S. Hasegawa, Lett. Phys. Rev. ) Electron Standing Wave on Si(111)-Ag at 65K
T. Hirahara, et al., Surface Science 563 (2004) 191–198 STM Image dI/dV Images
-0.9 V -0.9 V
-0.8 V 電子の海のさざ波 sea Ripples in electronic Waves Standing (Electron d I t exp d0 2 -0.7 V
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!!! 電子定在波 Electron Standing Wave インジウム吸着Si(111) 表面
絶縁体 擬１次元金属 ２次元金属 ポテンシャル障壁 Potential Barrier ステップ Step √3×√3 ４×１ √7×√3 ドメイン境界 Domain Boundary ρ(E, x)（局所状態密度 Local DOS） 2 ∝|Ψi+Ψr| 2 2 ∝{1+|R| +2|R|cos(2kx・x-η)}・|u(x,y)| 2次元 ブロッホ波 2D Bloch Wave 原子像 物質の表面では、独特な原子の並び方をしている 入射波 Ψi=exp[i(kx・x+ky・y)]・u(x, y) 定在波 Atomic 2 ⇒３次元の物質とは異なる独特な物性を示す h 2 2 Standing Wave Arrangement Incident Wave E=E0+(kx +ky ) 2m* π u(x,y)= cell function 定在波の波長 λ＝ Wavelength of Standing Wave kx
反射波 Ψr=R・exp[i(-kx・x+ky・y)]・u(x, y) Reflected Wave R=|R|・exp[iη] η：反射位相シフト Reflection Phase Shift 60 K (電荷密度波) Plan View RT(metallic) A. A. Sarranin, et. al. H. Y. Yeom, S. etTakeda, al., PRL et 82al., 4898 (1999) 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 Band Dispersion Si(111)-4×1-In Surface -Peierls Instability T. Abukawa, S. Kono, et al. High Temp Phase Surf. Sci. 325 (1995) 33 Constant Charge-Density Wave (CDW) → Metal-Insulator Transition Charge → Temperature Dependence of Conductivity Density STM Image H. Morikawa, et al. 表面科学 25 (2004) 407. RT Fermi H.-W. Yeom, et al. Wavenumber PRL 82 (1999) 4898 Low-Temp Phase CDW
Energy Gap （Peierls Gap）
Linear Fermi Surfaces 70K Bisecting the Brillouin Zone Fermi Surface Mapping Atom Displacements (Lattice Distortion) with CDW Metal-Insulator Transition at Si(111)-4×1-In Surface
反射高速電子回折（RHEED）
82 @LT Y.J.Sun et al., PRB 77, 124115 (2008)
RT 41 @RT High-Temp Phase MI Transition at～110 K // With CDW ⇒ Peierls Transition
H.W.Yeom et al., PRL 82, 4898 (1999) W. G. Schmidt, et al., 100 K Phys. Status Solidi B 249, No. 2, 343–359 (2012) Low Temp Phase
Electrical Resistance of Si(111)-4×1-In Surface Graphene on SiC crystal surface T. Tanikawa, et al., Phys. Rev. Lett. 93 (2004) 016801. Relativistic（Dirac electron 1E+0710000 10μm-spacing probe 2 E mc2 pc 2 (M. Kusunoki @ Nagoya U.)
1E+061000 m 0 Zero Mass Non-relativistic p2 2k 2
=0 (kΩ) E I RT 8ב2’ E ck High mobility 2m 2m 4ב2’ 1E+05100 Tc 4×1 Resistance at 1E+0410
1E+031 65 K RT 8080 100100 120120 140140 160160 180180 Temperature (K) CDW Phase (8ב2’) Metal Phase (4×1) H.-W. Yeom, et al., PRL 82 (1999) 4898 A. Bostwick, et al., Nature Physics 3, 36 (2007). Rashba Effect The electron energy is determined by its momentum (and spin).
Time-Reversal Symmetry spin s Time-reversal E(k, ↑)=E(- k, ↓) k k
Momentum （Wavevector） s Space-Inversion Symmetry Space-Inversion E(k, ↑)=E(-k, ↑) But, s on Surfaces k
Time-Rev. Sym.＋Space-Inv. Sym.⇒ Spin(Kramers) Degenera Spin-Split in Surface States E(k, ↑)=E(k, ↓) Emmanuel I. Rashba
Surface States of Au(111)―Spin split due to Rashba effect― Difference in Energy between Spin↑ and Spin↓ ） -1 Å 1 1 （ 2 y X H p V( x) 2 σ grad V p k B 2m 4mc C A ○ Spin-Orbit-Coupling Hamiltonian (d) Wavenumber Effective (b) grad Vp = Magnetic Field B ） Orbital Motion Orbital Motion Eff. Mag. Field B meV of Electron of Nucleus （ Electron Nucleus σ + Nucleus ‐ B σ Interact with spin σ + ‐ (a) (c) 10 nm Lorentz Electron Energy Binding Transformation (Rest Frame of Electron) Wavenumber k ()M （Å-1） Difference between ↑ and ↓ x (a) (b) G. Nicolay, et al., Phys. Rev. B 65, 033407 (2001). ＝Zeeman Energy by Eff. Mag. Field B (c)(d) L. Petersen, et al., Phys. Rev. B 58, 7361 (1998). Band Dispersion of 20 Atomic Layer Bi(111) slab ARPES of Bi(111) Ultrathin films ―QWS and SS― (１st Principles Calculation) Bihlmayer T. Hirahara, et al., (Juhlich, Germany) Phys. Rev. Lett. 97, 146803 (2006).
Conduction (eV) Band vacuum Quantum-Well States E Energy Gap Bi Surface States vacuum Surface States First-principles calculation Red: Spin-Up for free-standing Bi slabs Conductivity of Bi thin film Blue: Spin-Down including SOC Energy Binding DOS D(E) = surface-state conductivity -1 Wavenumber k// (nm ) (eV-1 nm-2) (eV)
Pure Bi E Valence ・Semi-metal in Bulk ・Conduction band and Band valence band are connected by the spin-split surface Binding Energy Binding 14 BL thick 30 BL thick 40 BL thick states Wavenumber k (Å-1) Wavenumber (Å-1) Wavenumber (Å-1)
Topological Surface States Electronic States of Bi2Se3 (Theory) H. Zhang, et al., Nature Physics (May 2009) Bi1-xSbx, Bi2Te 3, Bi2Se3, Analogue of Edge States in Quantum Hall States (2DEG) Cond. Band ⇒ Extension to 3D Materials Strong SO Interaction produces effective B. SS Dirac Cone
Val. Band
Isolated Atom Atomic Bondings Spin-Orbit Intercation （Atomic Orbitals） Split due to Crystal filed ＋＋ ＋ － －
－ ＋ ＋ －－
Trivial Ins Non-trivial Ins Chiral Dirac Cone of Topological Insulators and Bi2Se3 : Epitaxial Growth & Bands Y. Sakamoto, et al., Phys. Rev. B81, 165432 (2010). Current-Induced Spin Polarization
RHEED ky Momentum p k Layer Growth in Quintuple-Layer Unit RHEED Oscillation
Bulk: Y. Xia, et al., Nature Physics 2009 (May) kx E F
0.2
M. Z. Hasan and C. L. Kane, Chiral Fermi Surface 0.4 Rev. Mod. Phys. 82, 3045 (2010) -1 )
-0.1 00.1 k(Å
2π-rotation of spin Time-Reversal Symmetry Band Bending of Bulk States Near Surface E(k,↑)=E(-k,↓) Energy difference between EF and a core level Measured by Photoemission Spectroscopy
Conduction Band E EF F
ψinitial ψscattered1 ψ ψ π-rotation 2π-rotation Valence Band
Core Level
ψinitial ψscatter2 -π-rotation Core Level
Ψscatter2 = -Ψscatter1 ψ ψ 2π-rotation Ψscatter1 +Ψscatter2=0 Spinor Destructive interference ⇒ No backscattering ⇒ High mobility 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 S. Hasegawa & F. Grey Surface Science Surface Science 500 (2002) 84–104 500 (2002) 84–104
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