基礎科学特別講義 東大駒場キャンパス

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1 基礎科学特別講義 東大駒場キャンパス 東京工業大学 大学院総合理工学研究科 材料物理科学専攻平山博之 hirayama.h.aa@m.titech.ac.jp 009hirayamalabHP/ A brief review of electronic states in Bulk Materials (Nearly) free electron model! atom closed shell the most outer shell bulk material ex. Cu (1s) (s) (p) 6 (3d) 10 (4s) 1 nucleus valence electron (free electron) in solid 1

2 Free electrons in bulk materials wide space in the solid free electrons never see the boundary! (surface) Free electrons in bulk materials d ( x) V ( x) ( x) E ( x) m dx free electron : V ( x) 0 d ( x) E( x) m dx ( x) Aexp( ikx) k E m boundary condition(1) : ( x) dx 1 boundary condition() : ( x L) ( x) x L X=0 X=L

3 Free electrons in bulk materials Solution : 1 ( x) exp( ikx) L k E ; k l ( l int eger) m L Ek ( ) two electrons per each state (Pauli s rule) k Electronic Band Structure in bulk materials Ek ( ) *one electron per atom L/a atoms in solid E Electronic band a Brillouin zone a k The band can accommodate L / a L a electrons 3

4 > Band gap in bulk materials The Bz boundary: 1 ( x) exp( i x) sin( x ) L a a k a E split E E a a k Band gap in bulk materials E split band a a k band gap 4

5 Band gap Ag(111) のreal line エネルギー (ev) 4 CBM 3 + ブランチ 1 VBM ブランチ L 点での波数の虚数部 q (nm -1 ) Quantization 1: D-square well E 1 1 ( k ) ( ) ( ) 4 k k g k m m k g V g 5 E ( ev ) BZ boundary * Surface state surface ( x) bulk vacuum E ( k ) k m g k a 0.5 nm, g 1.5 nm, V g 1 ev a gap Re k k g 1 k ( nm ) Im k k k i k ( nm ) 5

6 正 / 負電荷密度 正電荷バックグランド rs= の時の電子密度分布 rs=5 の時の電子密度分布 固体中 表面 真空側 表面垂直方向の位置 after N.Lang & W.Kohn, Phys.Rev.B1,4555(1970) 6

7 Work function Solid (Potential) energy Surface Vacuum Vacuum level F Fermi level V xc k F m 0 z 紫外線 電子放出 e- e- e- e- e- e- e- e- 金属 7

8 Angle-resolved Photo Electron Spectroscopy (ARPES) electron detector (energy analyzer) electron energy: E kin UV light θ electron emission Sample (Work function: Φ ) E vac E f Energy E kin Φ Density of E states valence band E h E k me kin Ek ( ) kin // // sin 8

9 Ultraviolet Photoelectron Spectrum (UPS) from clean & oxidized Si((111) surfaces Eastman & Grobman, PRL 8,1378(197) Ultraviolet Photoelectron Spectrum (UPS) from clean Cu((111) surfaces Kevan, PRL 50,56(1983) 9

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11 Electron Density of States (DOS) of bulk materials x y z k E( k) m k k k m here k lx, ly, lz L L L l, l, l : 0, 1,, 3, x y z Electron Density of States (DOS) of bulk materials NE ( ) ne ( ) L me / 3/ ne ( ) m ( E) E E 3 E number of electrons in the sphere 4 k 3 4 k dk k L 3 L ( ) D ( E ) N E k me 3 3 L L 3 dim ension : states /( ev m ) E 3/ 3 D ( E ) E dim ension : ev 11

12 Electron Density of States (DOS) of Nanosheet (D) d x z y n z 3 n z n 1 z n z 1 n z n z 3 k z ( circle ) k, k x y Quantization z k n d // z z E k x k y k z const. m ( sphere ) kx, ky lx, ly L L Electron Density of States (DOS) of Nanosheet (D) d x z y n z 3 n z n 1 z n z 1 n z n z 3 k z ( circle ) k, k x y D x y z z D z z Quantization // z k z n z d D D z z E( k ) k k E ( n ) k E ( n ) m m m k E( k ) E ( n ) ( sphere ) E k m k k const. x y z ND ( E) n D ( E) L n D ( E) L m D ( E) E k kdk 0 L L m ND ( E) k E( k D ) Ez ( nz ) L 1

13 Electron Density of States (DOS) of Nanosheet (D) d x z y n z 3 n z n 1 z n z 1 n z n z 3 k z ( circle ) k, k x y L m D Quantization // z k z n z d ( sphere ) E k m k k const. x y z ( E D ) dim ension : states /( ev m ) 5 m 4 3 E n z E k n z m E dim ension : ev Electron Density of States (DOS) of Nanosheet (D) subband structure Ag(111) nanofilm on Si substrate z// QWSs x x-y// free electron y after Matsuda,Ohta,Yeom; Phys.Rev.B65,08537(00) 13

14 ψ^ ψ^ interface Surface Si sub. Ag film Vacuum z=-d/ z=+d/ z ML 奇数 = Z (ML).0 11ML 偶数 = Z (ML) 14

15 /research_stm.html /ScanningTunnelingMicroscope_schematic.png /400px-ScanningTunnelingMicroscope_schematic.png 15

16 Field Ion Microscope (FIM) CRC Crit.Rev.Sol.State Mat.Sci.10,391(198) 16

17 Si(111) 表面 17

18 Si(001) 表面 groups.physics.umn.edu/.../surface_image.gif : dangling bond 18

19 Si-Si 結合 ( 原子分子 ) Si-Si 結合 ( 多原子分子 = 固体 ) 反結合軌道 伝導帯 バンドギャップ sp 3 軌道 sp 3 軌道 sp 3 軌道 結合軌道 価電子帯 表面局在電子準位 ( ダングリングボンド準位 ) Si(001) 表面の STM 像のバイアス電圧依存性 19

20 STMによる empty state と filled state のイメージング Si(111)3x1-Ag 表面 Filled state 像 Empty state 像 0

21 エネルギー 真空準位 (Vacuum level) 仕事関数 (Work function) フェルミ準位 (Fermi level) フェルミエネルギー (Fermi energy) バンドの底 (Band bottom) L L R R vacuum level connect A B L L apply bias R voltage R L tunneling current L ev R R reduce space L no current L ev R nano space V wide space V 1

22 ~ I STM image di/dv image I di dv ev s (E) ( E) T( E) de Potential barrier for tunneling Sample surface L L ev STM tip R R 0 V L V E R ev ev space: s V 0 V L R ev ev L R ev V E

23 MgB superconducting gap M.Miyazaki & H.Hirayama 3

24 Si(111) 3x 3-Ag 表面における電子波の STM 観測 * ドメイン境界 ( 一種の岸壁 ) * 正しくは STM をベースとした di/dv 像観察ですが 原子 1~5 個位が抜けた欠陥 原子数個が表面に余分に存在 Hirayama Lab. 室温での測定! 固体表面の電子の波は見える! ホイヘンス (Huygens) の原理 (1678 年 ) ある時刻の波面から出される素元波の包絡面が新しい波面になる ( 波が伝播すること, 反射, 屈折, 回折の現象を説明することができる ) 4

25 Life time broadening repeat bouncing in Quantum Well leakage e leakage normalized R Re i ( reflection ) n 1 n n 3 n 4 kd n ; E k m d nm, 0.9 R (0.1d igit).0 E( ev ) A B C D (c) Φ ^ mapping of an (a) Electron standing (b) n= quantized state in artificially constructed waves at Ag(111) surface one-dimensional wells at two-dimensional quantum well (33nmx33nm) Ge/Si(001) surface at Si(111) 3x 3-Ag surface (0nmx0nm) (0nmx0nm) 5

26 Previous study (PES) 0ML Ag/Si(111) Surface state peak Ag/HOPG Disappearance of the surface state peak surface state depopulation! Ag/Si G. Neuhold & K.Horn Phy.Rev.Lett (1997) Two-step growth on Si(111)7x7 1.6ML 3.ML 6.4ML 1.8ML 80K 300K 1.6ML 3.6ML 6.4ML 1.8ML reference 300K *Magic thickness: θ ~6.4ML 300x300nm 6

27 Pixsel Frequency (%) Height distribution of electronically grown Ag θ =6.4ML 6.4ML 6.4ML? magic thickness B A ML 1ML 0.8 B C C A K 80K 300K Topography (nm) *θ ~6.4ML: magic thickness! 0.15V E vs. k 0.5V 0.35V 0ML Ag 0ML, 31.5x31.5 nm E( k) k E m 0 Sawa, Aoki & Hirayama Phys.Rev.B 80, (009) Bottom: E 0 Effective mass: m 7

28 Results 1: contd. Thickness-dependence E vs. k plot Sawa, Aoki & Hirayama Phys.Rev.B 80, (009) θag (ML) Bulk -63 E0 (mev) As the thickness decreases, keep the parabolic dispersion E 0 : upward shift! STS Results : contd. E 0 shift around the dislocation E 0 shifts downward around the dislocation! The downward shift in E 0 lattice displacement around the dislocation! 8

29 Discussion : Misfit strain effect Lattice const. Ag < Si in-plane tensile strain! Ag Si change in the band gap at L-point Surface State supported by the real line at the band gap causes the shift in the surface state! Discussion : contd. Theoretical calculation by G. Neuhold & K.Horn Phy.Rev.Lett (1997) E 0 as d (strain ) E 0 around dislocations (strain ) 9

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31 Quantization : D-cylindrical well ( x, y) ( r, ) k a j r // n 1 1 x, y ( r, ) Ex, yx, y ( r, ) m r r r r ( r, ) ( r, ) x, y x, y x, y ( r, ) R ( r)expil d 1 d l ( ) R r Ex, yr ( r) m dr r dr mr ( r, ) J ( k r)exp il x, y l // E j n xy, ma Quantization : D-cylindrical well * Bessel function J ( z ) 0 J ( z ) 1 J ( z ) J ( z ) z kr Zero cross points: j n l 0 l 1 l l 3 n n n n l Jl ( z) cosz z 4 k a j // n 31

32 散乱ポテンシャルは位相シフト δ に反映される (a)v=0 の場合 ( 破線 ) (b)v<0 の場合 ( 波動関数は内部に引き込まれる δ>0) (c)v>0 の場合 ( 波動関数は外側に押し出される δ<0) J.J.Sakurai 現代の量子力学 吉岡書店 Quantized conductance in Nanowire (1D) L electron V R Tip-sample distance Au nanowire radius ev L ev R f ( E, L ) velocity : v f ( E, ev ) R increase small for a (l,m) 1 e G T R h After Rubio,Agrait,Vieira; Phys.Rev.Lett.76,30(1996) 3

33 An example STM tip Au(111) Au nanoparticle radius : r alkanethiol monolayer ( insulating ) I V measurement after Wang,Xiao,Huang,Sheng; Appl.Phys.Lett.77,1179(000) Increase the number of electrons (charge) on the sphere from N to (N+1) e E E ( N 1) E ( N) c c c (N 1) 8 r 0 33

34 Graphene: structure graphite graphene 004; exfoliation (A.Geim et. al., Univ. Manchester) Exfoliated graphene (optical microscope image) 34

35 グラフェン : 実格子 逆格子 k y k y b b b 1 b b 1 b 1 k x k x b 1 b 1 b b a t 3 y t t 1 a 1 x M K (0,0) 1 M (1, ) a 3 K (,0) a 3 グラフェン : バンド分散 * これは塚田捷 表面における理論 I 培風館に載っていた図を転載したものです 35

36 Graphene: electronic states ( k ) vf k cf. 4 E m c c p 6 8 vf 10 m/ s; c m/ s Angle-resolved photoelectron spectrum (ARPES) 36

37 Fig. Direct measurement of Landau quantization in epitaxial graphene. D L Miller et al. Science 009;34:94-97 Published by AAAS Graphene: from exfoliation to epitaxial growth Thermal evaporation of Si reconstruction of remained C 3x3 3x 3 6 3x6 3 Graphene (~1x1) 37

38 Epitaxial graphene on SiC(0001) substrate 3x3 K under Si flux 3x 3 K 6 3x6 3 K graphite 1x1 180 K 低速電子線回折装置 : Low Energy Electron Diffraction (LEED) /LEEDexpl.htm 38

39 Giant spin-splitting at alloy surfaces Au(111) surfaces E k x Bi-induced Ag(111) 3 surfaces Giant Spin Splitting through Surface Alloying C. R. Ast et. al., PRL 98, (007) InGaAs/InAlAs k 0 =0.08(A -1 ) Ag(111) k 0 =0.004 (A -1 ) Au(111) k 0 =0.01 (A -1 ) Bi(111) k 0 =0.5 (A -1 ) Bi/Ag(111) 3 k 0 =0.13 (A -1 ) *Ag(Z=47), Au(Z=79),Bi(Z=83) Expected Results? Bi/Ag(111) 3x 3 G. Bihlmayer,1,* S. Blügel,1 and E. V. Chulkov,3 PHYSICAL REVIEW B 75, Bi/Cu(111) (01) S. Mathias et. al., PRL 104, (010) 39

40 Spin-splitting at non-magnetic surfaces +Ze electron nucleus +Ze electron nucleus DEG Spin-orbit interaction V () r V( r) p 0 t Current) I r 0 It r Br () 3 4 r σ p σ The giant spin-splitting: why only for the Bi/Ag(111) 3x 3 surface? Rashba-Vicikov splitting U σ V() r p 4mc outward displacement of Bi atoms distortion of the wave function (G.Bilmayer et. al., PRB75,195414(007)) breaking the in-plane inversion symmetry (J.Premper et. al, PRB76,073310(007)) theoretically!! 40

41 A new play ground!? Quantum-Well-Induced Giant Spin-Orbit Splitting S. Mathias et. al., PRL 104, (010) Bi/Cu(111) (01) reconstructed surfaces *Θ~0.5ML *Dealloyed surface layer! * incommensurate/ commensurate Two-photon photoemission (PPE) surface dispersion in the empty state giant spin-splitting ~Bi/Ag(111) 3 ( cf. very small Bi/Cu(111) 3) The largest a new high-θ phase at Bi/Ag(111) surfaces? Results & Discussion V=+0.6V, I=0.1nA 7.5x7.5nm V=+0.V, I=0.1nA 7.5x7.5nm 0 < Θ < 1/3 ML Θ > 1/3 ML *a = nm ~ 3a (0.050 nm) * lattice // {11} 3x 3-R30 * local 3 small Θ weak attractive interaction * a new stripe structure // {11} * alternating contrast // {011} * a regular arrangement of protrusions 41

42 High-resolution STM images of the stripe structure Centered (unit?) cell 0.41nm//{011} 0.49nm//{11} Overlaid contrast alternation // {011} Θ = 0.45 ML fluctuation in the width of the dark stripes (one- and two-protrusions) V=+0.V, I=0.1nA 5.x5.nm Θ = 0.75 ML periodic alternation one-protrusion width dark stripes out-of-phase boundary with the two-protrusion width dark stripe 4