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基礎科学特別講義XII＠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

http://www.materia.titech.ac.jp/~hirayama/2009hirayamalabHP/

A brief review of electronic states in Bulk Materials

atom

bulk material

nucleus

closed shell

the most outer shell

valence electron(free electron)

in solid

“(Nearly) free electron model !”

ex. Cu (1s)2 (2s)2 (2p)6(3d)10(4s)1

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Free electrons in bulk materials

wide space in the solid ⇒ free electrons never see the boundary!(surface)

Free electrons in bulk materials

2 2

2

2 2

2

2 2

2

( ) ( ) ( ) ( )2

: ( ) 0

( ) ( )2

( ) exp( )

2

(1) : ( ) 1

(2) : ( ) ( )

dx V x x E x

m dx

free electron V x

dx E x

m dx

x A ikx

kE

m

boundary condition x dx

boundary condition x L x

　　 　

　　　　　

　　

　　

　 　

　 　

L

x

X=0 X=L

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Free electrons in bulk materials

2 2

:

( ) exp( )

2; ( int )

2

Solution

x ikxL

kE k l l eger

m L

１

　 　　( )E k

k

two electronsper each state(Pauli’s rule)

Electronic Band Structure in bulk materials

( )E k

k

a

a

Brillouin zone

E

＊one electron per atom⇒L/a atoms in solid

2 2 22 /

L

a L a

The band can accommodate

electrons

Electronic band

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( ) exp( ) sin( )x i x xa aL

１

The state @ Bz boundary: ka

E

k

a

2

a

split

E

E>

Band gap in bulk materials

Band gap in bulk materials

E

k

a

2

a

split

gap

band

band

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4

3

2

1

0

3.02.52.01.51.00.50.0

+ブランチ

－ブランチ

エネルギー（eV)

L点での波数の虚数部q (nm-1)

CBM

VBM

Ban

d g

ap

Ag(111)のreal line

Quantization 1: 2D-square well

5

4

3

2

1

0-6 -4 -2 0 2 4 6

22

( )2

E k km

2 2

22 2 2 2 21 1

( ) ( ) ( ) 42 2 2 2

gE k k k g k k g V

m m

( )E eV

Re k k1( )nm

Im k 2

gk

2

gk

BZ boundary

gap

k k i 120.5 , 12.5 , 1

ga nm g nm V eV

a

5

4

3

2

1

0-6 -4 -2 0 2 4 6

22

( )2

E k km

2 2

22 2 2 2 21 1

( ) ( ) ( ) 42 2 2 2

gE k k k g k k g V

m m

( )E eV

Re k k1( )nm

Im k 2

gk

2

gk

BZ boundary

gap

k k i 120.5 , 12.5 , 1

ga nm g nm V eV

a

0

-3 -2 -1 0 1 1( )k nm

( )x bulk vacuum

surface

0

-3 -2 -1 0 1 1( )k nm

( )x bulk vacuum

surface

* Surface state

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after N.Lang & W.Kohn, Phys.Rev.B1,4555(1970)

表面固体中 真空側

正電荷バックグランド

rs=2の時の電子密度分布

rs=５の時の電子密度分布

表面垂直方向の位置

正/負電荷密度

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xcV

2 2

2

Fk

m

0z

FFermi level

Vacuum level

Work function

(Potential)

energy

VacuumSolidSurface

e-

e-e-

e-e-

e-

e-

紫外線

e-

電子放出

金属

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http://www.ifw-dresden.de/institutes/iff/research/SC/arpes/method/index/?set_language=de

Angle-resolved Photo Electron Spectroscopy (ARPES)

UV light

Sample

（Work function: Φ ）

electron

emission

electron detector

(energy analyzer)

electron

energy: Ekin

Energy

Φ

Ekin

valence

band

Density of

states

Ef

θEvac

E

UV light

Sample

（Work function: Φ ）

electron

emission

electron detector

(energy analyzer)

electron

energy: Ekin

Energy

Φ

Ekin

valence

band

Density of

states

Ef

θEvac

E

// 2

2sin

kin

kin

E h E

mEk

//( )E k

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Ultraviolet Photoelectron Spectrum (UPS)from clean & oxidized Si((111) surfacesEastman & Grobman, PRL 28,1378(1972)

Ultraviolet Photoelectron Spectrum (UPS) from clean Cu((111) surfacesKevan, PRL 50,526(1983)

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Electron Density of States (DOS) of bulk materials

2 2

2 2 2 2( )2 2

2 2 2, ,

, , : 0, 1, 2, 3,

x y z

x y z

x y z

E k k km m

here l l lL L L

l l l

k k

k　 　

　

Electron Density of States (DOS) of bulk materials

number of electronsin the sphere

323 3

3/ 230

3 3 2 2 3

422 4

3( ) 23 32 2

kkk dk L L

N E k mE

L L

3/ 2

3 2 3

3/ 2

2 3

2( )( )

3

2( )( )

2

mEN En E

L

mn EE E E

E

5

4

3

2

1

0151050 E

3 ( )D E

3 ( )D E E

3dim : /( )ension states eV m

dim :ension eV

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Electron Density of States (DOS) of Nanosheet (2D)

dx

y

z

,x yk k

zk

1zn

2zn

3zn ・・

1zn

2z

n 3zn ・ ・

// z zQuantization z k nd

2

2 2 2 .2

x y zE k k k constm

( )sphere

( )circle

dx

y

z

dx

y

z

,x yk k

zk

1zn

2zn

3zn ・・

1zn

2z

n 3zn ・ ・

// z zQuantization z k nd

2

2 2 2 .2

x y zE k k k constm

( )sphere

( )circle

2 2

, ,x y x yk k l lL L

Electron Density of States (DOS) of Nanosheet (2D)

2 22 2 2

2 2

2 2

2 20 2

2 22 2

( ) ( ) ( )2 2

2( ) ( )

2 22

( ) ( ) ( )2 22

D x y z z D z z

D D z z

k

D D z z

E k k E n k E nm m

mk E E n

kdkL L m

N E k E E n

L

k

k

k

dx

y

z

,x yk k

zk

1zn

2zn

3zn ・・

1zn

2z

n 3zn ・ ・

// z zQuantization z k nd

2

2 2 2 .2

x y zE k k k constm

( )sphere

( )circle

dx

y

z

dx

y

z

,x yk k

zk

1zn

2zn

3zn ・・

1zn

2z

n 3zn ・ ・

// z zQuantization z k nd

2

2 2 2 .2

x y zE k k k constm

( )sphere

( )circle

22 2

2

22 2

( )( )

( ) 2( )

2

DD

DD

N En E

L

n E L mE

E

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Electron Density of States (DOS) of Nanosheet (2D)

5

4

3

2

1

0151050

2 ( )D E2dim : /( )ension states eV m

Edim :ension eV

E

1zn 2 3 42

2 2

2z z zE k n

m

2

m

5

4

3

2

1

0151050

5

4

3

2

1

0151050

5

4

3

2

1

0151050

2 ( )D E2dim : /( )ension states eV m

Edim :ension eV

E

1zn 2 3 42

2 2

2z z zE k n

m

2

m

2

2 2

2

2D

L m

d

xy

z

,x yk k

zk

1zn

2zn

3zn ・・

1zn

2z

n 3zn ・ ・

// z zQuantization z k nd

2

2 2 2 .2

x y zE k k k constm

( )sphere

( )circle

dx

y

z

dx

y

z

,x yk k

zk

1zn

2zn

3zn ・・

1zn

2z

n 3zn ・ ・

// z zQuantization z k nd

2

2 2 2 .2

x y zE k k k constm

( )sphere

( )circle

Electron Density of States (DOS) of Nanosheet (2D)

Ag(111) nanofilm on Si substrate

after Matsuda,Ohta,Yeom;

Phys.Rev.B65,085327(2002)・・・

z// QWSs

xy

x-y// free electron

Ag(111) nanofilm on Si substrate

after Matsuda,Ohta,Yeom;

Phys.Rev.B65,085327(2002)・・・

z// QWSs

xy

x-y// free electron

“subband structure”

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zz=+d/2z=-d/2

Surfaceinterface

Ag film VacuumSi sub.

1.5

1.0

0.5

0.0

ψ^2

-4 -2 0 2 4

Z (ML)

11ML 奇数=9

2.0

1.5

1.0

0.5

0.0

ψ^2

-4 -2 0 2 4

Z (ML)

11ML 偶数=10

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http://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/ScanningTunnelingMicroscope_schematic.png/400px-ScanningTunnelingMicroscope_schematic.png

http://physics.berkeley.edu/research/crommie/research_stm.html

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CRC Crit.Rev.Sol.State Mat.Sci.10,391(1982)

Field Ion Microscope (FIM)

http://www.nims.go.jp/apfim/fim.html

http://www.ornl.gov/info/ornlreview/rev28-4/text/atoms.htm

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Si(111)表面

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groups.physics.umn.edu/.../surface_image.gif

www.sljus.lu.se/stm/NonTech/big/Si001.jpg

Si(001)表面

： dangling bond

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sp3軌道 sp3軌道

結合軌道

反結合軌道

Si-Si結合（２原子分子）

sp3軌道

Si-Si結合（多原子分子＝固体）

伝導帯

価電子帯

バンドギャップ

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

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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

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真空準位(Vacuum level)

フェルミ準位 (Fermi level)

バンドの底 (Band bottom)

エネルギー

仕事関数 (Work function)

フェルミエネルギー (Fermi energy)

space space

L R

vacuumlevel

LR

L R

L R

connectA B

applybias voltage

reduce

space

L RL

R

V

eV

nano

tunnelingcurrent

L

LR

V

eV

current

wide

no

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eV

s

dEETEI

)()(STM image

)(EdV

dIdI/dV image

I

eV

L

R eV

0

V

V E

eV

L

R eV

0

V

V E

2

L R eV

～ ～

Potential barrier

for tunneling

L RL

R

V

eV

s:space

Samplesurface

STM tip

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MgB2 superconducting gap

http://www.insp.upmc.fr/axe1/Dispositifs%20quantiques/AxeI2_more/SPECTRO/SpecHD.HTM

.

M.Miyazaki & H.Hirayama

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Si(111)√3x√3-Ag表面における電子波のSTM観測＊

＊正しくはSTMをベースとしたdI/dV像観察ですが・・・

taken @ Hirayama Lab.室温での測定！

ドメイン境界（一種の岸壁）

・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

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。）

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Life time broadening

e

repeat bouncing

in Quantum Well

Rei ( )reflection

leakageleakage

1.0

0.8

0.6

0.4

0.2

0.02.01.51.00.50.0

( )E eV

normalized2

R

0.9 0.5R (0.1d )igit

1n 2n 3n 4n

kd n2

2;2

E km

2 , 0.9d nm

A B

C

D

(a) Electron standingwaves at Ag(111) surface(33nmx33nm)

(b) n=2 quantized state inone-dimensional wells atGe/Si(001) surface(20nmx20nm)

(c) |Φ|^2 mapping of an artificially constructedtwo-dimensional quantum wellat Si(111)√3x√3-Ag surface(20nmx20nm)

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Previous study (PES)

G. Neuhold & K.HornPhy.Rev.Lett 78 1327 (1997)

20ML Ag/Si(111)

Disappearance of the surface state peak

↓surface state

depopulation !

Surface state peak

Ag/HOPG

Ag/Si

1.6ML 3.2ML 6.4ML 12.8ML

1.6ML 3.6ML 6.4ML 12.8ML

80K⇒300K

reference300K

Two-step growth on Si(111)7x7

＊Magic thickness: θ ～6.4ML 300x300nm2

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Height distribution of electronically grown Ag films@ θ =6.4ML

6.4ML? magic thickness

A

B

C

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Pix

sel F

req

uen

cy (

%)

0.60.40.20.0-0.2Topography (nm)

A

B

C

1ML 1ML

6.4ML

300K 80K⇒300K

＊θ ～6.4ML: magic thickness!

Sawa, Aoki & Hirayama Phys.Rev.B 80, 035428 (2009)

20ML

Ag 20ML, 31.5x31.5 nm2

0.15V

0.25V 0.35V

E vs. k

2 2

0( )2

kE k E

m

☑ Bottom: E0

☑ Effective mass: m

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Results 1: contd.

Sawa, Aoki & Hirayama Phys.Rev.B 80, 035428 (2009)

Thickness-dependence

“E vs. k2 ” plot

θAg (ML) E0 (meV)

7 26

10 -6

20 0

30 -11

40 -51

Bulk -63

As the thickness decreases,☑ keep the parabolic

dispersion☑ E0: upward shift !

Results 2: contd.

STS E0 shift around the dislocation

☑ E0 shifts downwardaround the dislocation !

☑ The downward shift in E0

∝ lattice displacementaround the dislocation !

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Discussion : Misfit strain effect

Lattice const.Ag < Si

Ag

Si

in-plane tensile strain ! ・change in the band gap at L-point

・Surface State supported by the real lineat the band gap

↓

causes the shift in the surface state!

Discussion : contd.

Theoretical calculationby G. Neuhold & K.HornPhy.Rev.Lett 78 1327 (1997)

☑ E0 ↑ as d ↓(strain ↑)

☑ E0 ↓ around dislocations(strain ↓)

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http://www.chembio.uoguelph.ca/educmat/chm729/STMpage/stm16.jpghttp://www.almaden.ibm.com/vis/stm/images/stm.gif

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Quantization 2: 2D-cylindrical well

( , ) ( , )x y r

r

2 2 2

, , ,2 2 2

1 1( , ) ( , )

2x y x y x yr E r

m r r r r

, ,( , ) ( , 2 )x y x yr r

, ( , ) ( )expx y Rr r il

2 2 2 2

,2 2

1( ) ( )

2 2R x y R

d d lr E r

m dr r dr mr

, //( , ) ( )expx y lr J k r il

2 2

, 22

n

x y

jE

ma

// nk a j

5 10 15 2000

0 ( )J z

1( )J z

2 ( )J z

3( )J z

1.0

z kr5 10 15 2000

0 ( )J z

1( )J z

2 ( )J z

3( )J z

1.0

z kr

Quantization 2: 2D-cylindrical well

* Bessel function

16.214.813.311.8

13.011.610.28.7

9.88.47.15.6

6.45.23.82.4

0l 1l 2l 3l

1n

2n

3n

4n 16.214.813.311.8

13.011.610.28.7

9.88.47.15.6

6.45.23.82.4

0l 1l 2l 3l

1n

2n

3n

4n

Zero cross points: nj

2( ) cos

2 4l

lJ z z

z

// nk a j

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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)

V

electron

RL

L

:velocity v( , )Lf E

eV

( , )Rf E eV

R eV

V

electron

RL

L

:velocity v( , )Lf E

eV

( , )Rf E eV

R eV

21 2eG T

R h

for a (l,m)

Tip-sample

distance

increase

Au nanowire

radius

small

After Rubio,Agrait,Vieira;

Phys.Rev.Lett.76,2302(1996)

Tip-sample

distance

increase

Au nanowire

radius

small

After Rubio,Agrait,Vieira;

Phys.Rev.Lett.76,2302(1996)

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An example

(111)Au

alkanethiolmonolayer( )insulating

Au nanoparticle:radius r

I V measurement

STM tip

after Wang,Xiao,Huang,Sheng;Appl.Phys.Lett.77,1179(2000)

Increase the number of electrons (charge) on the spherefrom N to (N+1) ・・・・

2

0

(2 1)( 1) ( )

8c c c

e NE E N E N

r

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Graphene: structure

graphite graphene

2004; exfoliation (A.Geim et. al., Univ. Manchester)

Exfoliated graphene(optical microscope image)

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1a

2a

1t

2t

3t

x

y

1a

2a

1t

2t

3t

x

y

xk

yk

1b2b

xk

yk

1b2b

xk

yk

1b2b

1 2 b b

1b

2b

1 2b b

xk

yk

1b2b

1 2 b b

1b

2b

1 2b b

K

M

(0,0)

1(1, )

3

2 2( ,0)3

Ma

Ka

K

M

(0,0)

1(1, )

3

2 2( ,0)3

Ma

Ka

グラフェン： 実格子

逆格子

＊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。

＊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。

グラフェン： バンド分散

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Graphene: electronic states

( ) Fk v k

cf. 22 4E m c c p

6 810 / ; 3.0 10 /Fv m s c m s 　

Angle-resolved photoelectron spectrum(ARPES)

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Fig. 2 Direct measurement of Landau quantization in epitaxial graphene.

D L Miller et al. Science 2009;324:924-927

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)

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Epitaxial grapheneon SiC(0001) substrate

3x3anneal @1120 Kunder Si flux

√3x√3anneal @1370 K

6√3x6√3anneal @1520 K

graphite 1x1anneal @ 1820 K

http://www.chembio.uoguelph.ca/thomas/oldthom/LEEDexpl.htm

http://www.techsc.co.jp/products/leed/bdl.htm

低速電子線回折装置： Low Energy Electron Diffraction (LEED)

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InGaAs/InAlAs k0=0.028(A-1) Ag(111) k0=0.004 (A-1 ) Au(111) k0=0.012 (A-1 ) Bi(111) k0=0.5 (A-1 ) Bi/Ag(111)√3 k0=0.13 (A-1 )

Giant Spin Splitting through Surface AlloyingC. R. Ast et. al., PRL 98, 186807 (2007)

*Ag(Z=47), Au(Z=79),Bi(Z=83)

Giant spin-splitting at alloy surfaces

Bi-induced Ag(111) √3 surfaces

xk

E

Au(111) surfaces

Expected Results ?

PHYSICAL REVIEW B 75, 195414 2007

G. Bihlmayer,1,* S. Blügel,1 and E. V. Chulkov2,3

Bi/Ag(111) √3x√3

Bi/Cu(111) (2012)

S. Mathias et. al., PRL 104, 066802 (2010)

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Spin-splitting at non-magnetic surfaces

Current）

0

t

r

I

0

3( )

4

I

t rB r

r

+Ze

nucleus

electron

+Zeelectron

nucleus

Spin-orbit interaction

( )V r

p

( )V r p

σ

σ

2DEG

The giant spin-splitting:

why only for the Bi/Ag(111)√3x√3 surface ?

2 2( )

4U V

m c σ r p

Rashba-Vicikov splitting

“outward displacement of Bi atoms”

→ distortion of the wave function(G.Bilmayer et. al., PRB75,195414(2007))

→ breaking the in-plane inversion symmetry(J.Premper et. al, PRB76,073310(2007))

theoretically !!

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A new play ground !?

Quantum-Well-Induced Giant Spin-Orbit SplittingS. Mathias et. al., PRL 104, 066802 (2010)

Bi/Cu(111) –(2012) reconstructed surfaces

*Θ～0.5ML*Dealloyed surface layer !* incommensurate/ commensurate

Two-photon photoemission (2PPE)→ surface dispersion in the empty state → giant spin-splitting ～Bi/Ag(111)√3

( cf. very small splitting @ Bi/Cu(111)√3)

The largest spin-splitting @ a new high-Θ phase at Bi/Ag(111) surfaces ?

Results & Discussion

V=+0.6V, I=0.1nA 7.5x7.5nm2 V=+0.2V, I=0.1nA 7.5x7.5nm2

0 < Θ < 1/3 ML Θ > 1/3 ML

*a = 0.048 nm ～√3a (0.050 nm)* lattice // {211}

→ √3x√3-R30°

* local √3 ordering @ small Θ→ weak attractive interaction

* a new stripe structure // {211}* alternating contrast // {011} * a regular arrangement

of protrusions

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High-resolution STM images of the stripe structure

V=+0.2V, I=0.1nA 5.2x5.2nm2

• Centered (unit?) cell0.41nm//{011}0.49nm//{211}

• Overlaid contrastalternation // {011}

Θ = 0.45 ML • fluctuation in the width

of the dark stripes(one- and two-protrusions)

Θ = 0.75 ML• periodic alternation• one-protrusion width dark

stripes• out-of-phase boundary

with the two-protrusionwidth dark stripe