曲面が関わる量子物性 - 山形大学理学部物理学科

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