量子化磁束動力学シミュレーション研究グループ研究会
九州工業大学サテライト福岡天神
2017年1月31日火曜日

イムズ11階

14:00

超伝導のシミュレーション
ミクロからマクロへ

大阪府立大学工学研究科電子・数物系専攻数理工学分野

加藤勝

Institute for nanofabrication research

内容

分子動力学シミュレーション
熱伝導を取り入れたシミュレーション
準粒子緩和を取り入れたシミュレーション
現象論的シミュレーション
超伝導ネットワーク
カイラル磁性体と超伝導体の２層構造における磁束構造
微視的シミュレーション
ナノ構造超伝導体の超伝導転移温度
磁場中のナノサイズ超伝導体における奇周波超伝導ワークショップ
「ナノ構造超伝導体
中の渦糸物理」を
１月２９日〜３０日に開催しました。

Institute for nanofabrication researchhttps://sites.google.com/site/nanoscvortexopu/home

ICCP10のミニシンポジウム（柳澤先生企画）で講演しました。

10th International Conference
on Computational Physics
16-20 Jan 2017, Macao

Institute for nanofabrication research

Many thanks to
Toyota RIKEN

N. Nishida

NIMS

K. Hirata

NIMS

S. Ooi

Osaka Prefecture Univ.

T. Ishida

大阪府立大学高専

佐藤修

Univ. of Tokyo

T. Tamegai

Tokyo Inst. Tech.

S. Okuma

Univ. of Elctro-Comm.

N. Kokubo

Akita Univ.

M . Hayashi

Tohoku Univ.

H. Ebisawa4Institute for nanofabrication research

分子動力学シミュレーション

Institute for nanofabrication research

分子動力学シミュレーション法

渦糸を２次元空間で一つの粒子として扱う。
渦糸のサイズが小さく、相互作用の範囲が広い場合渦糸が動く→エネルギーが消費される
渦糸の運動には抵抗力が発生する
渦糸の運動に伴う、質量×加速度の項（慣性項）は相対的に小さい
渦糸を記述する方程式
2
d ri
dri
imp
f
M 2 +η
= fdi + fpi + fvi + fidtdt

超伝導体中の渦糸のInstitute for nanofabrication research

分子動力学シミュレーション
渦糸の運動方程式

η

dri
dt

C. Reichhardt, C. J. Olson, J. Groth, S. Field, and F. Nori PRB 53 R8898 (1996)
Misko and Peeters PRB 74, 174507(2006)= fdi + f

imppi

+ fvi + fi

η
⎛
⎞⎟
2
⎜⎜ r̂
rj ri − rj ⎟⎟ij2⎟
fvi = f0 ∑ ⎜⎜ − rj2⎟⎟⎟2j ⎜⎜⎜ rij
r
r
−
r
j i
j ⎟
⎝
⎠

f

ri :渦糸の位置

粘性抵抗率
渦糸間の反発力1
fd = j ×Φ0
外部電流による駆動力c⎛ r − r − r p ⎞⎟
f
⎜⎜ p
i
j ⎟
p
imp
p
⎟⎟r̂ij 不純物によるピン留め力
fpi = ∑ ri − r j Θ⎜
⎜⎜
⎟⎟
λ
j rp
⎝
⎠
fi f (t1 ) ⋅ fj f (t2 ) = 2ηkBT δij δ (t1 − t2 )

熱揺らぎ力

f0 =

Φ20
8π 2λ 3

Institute for nanofabrication research

渦糸格子の融解

730 Ketterson
G. Blatter,
V.B. Geshkenbein
The Physics of Superconductors(K. H.Bennernann) Chap.10
G. Blatter and V.B. Geshkenbein
10 VortexJ.B.
Matter
729

H

H

渦糸液体
渦糸液体
渦糸固体

o
o
Fig.II10.3.
InDisordereda defected crystal the vortex
lines are pinned by the disorder poten. 10.2. Schematic Purephase diagram
for a strongly fluctuating typesuperconsuperconductor
superconductor
The vortex solid is transformed into a vortex-glass, while the vortex-liquid rem

tor. The vortex lattice undergoes a first-order melting transition transforminga (viscous) liquid. At low fields the solid is only weakly disordered - the Br
vortex solid into a vortex liquid. Transverse screening produces
an exponenglass is free of dislocations and exhibits a superconducting response in all directi
leading
to disordered vortex-glass (possibly a frozen vortex-liquid) ca
y weak interaction between the vortex lines at low magnetic fields,
The morestrongly
entrance of the melting line. The flux lines in the vortex-liquid
are entangled
a finitedensity of dislocations
lting in an ohmic longitudinal response, hence the vortex-liquid- and normal
allic phases are separated by a mere crossover at HC2

銅酸化物高温超伝導体では、渦糸の固体から液体状態
への融解現象がある。

The interplay between quenched disorder and thermal fluctuations
only manifests itself in the phenomenon of creep. Thermal fluctuation
the individual vortex lines lead to a dynamical sampling, and hence ther
it of zero driving force: Does the dissipation and hence theresistivity
averaging,of thepdisorder potential over the spatial extent of the mean t

強い揺らぎ、超伝導電子対の広がりの小ささd(logR+δR)/dT

characteristics,a achamberstep corresponding
to thetranwith a cryocooler.
Themelting
electric resistance
R is 120number of vortices. Note that another upper straight line
sition emerges½VðþIÞ
in almostthe samefield,
as indicatedby I. The
that merges with the melting line near T c is 1.5
not an intrinsic− Vð−IÞ%=2I
with
an applieddc current
Institute for nanofabrication researchphenomenon, but it is related to the critical1.0current of themagnetic
field wasthe
appliedparallelnature
to the cofaxis
of Bi2212.
the arrows, whichsuggests
intrinsic
theThe melting
transition
the vortex
step, i.e., the melting
transitionof the ofvortex
latticelattice(see can be 100IJJs. When the applied current exceeds I c of0.5the 0th branch
0.0
detected byof the c-axisofresistanceRc
above thisin a field during the H scan, Rc rapidly increases
Supplemental Material
formeasurements
the current dependence
thefield because the IJJs transit to a resistive state. Thus, theand/or the critical current I c of the IJJ stacks due to an
melting transition
and the detection by S.
I c Ooi,measurement
T.thatMochiku,
M. Tachiki,
Hirata114, 087001
position ofand
thisK.
extrinsic
linePRL
is dependent
on the appliedabrupt change of the interlayer couplingreflects the80[17]). Moreover,inthemeasurementsofR,
periodicalignment of pancake vortices [13].c Figure 1 shows the

微小超伝導体での渦糸格子の融解(2015)

0

10

-1

R (Ω)

10

-2

10

60

l

-3

10

30 µA

w

0

10

2.5
2.0

120

1.5

40

1.0
0.5

100

0.0

20
80

5×5 µm2
0

1

10

60

40 70

80.0 K

75

80

85

90

T (K)

-4

10

20 10-1 30
R (Ω)

10

140

H (Oe)

H (Oe)110d(logR+δR)/dT

magnetic field dependence of Rc at 80.0 K in 5 × 5 μm2
sample. Although the appearance of Rc changes with theappliedl current due to the strong nonlinearity of the I-V
characteristics, a step corresponding to the melting tranµAw
sitionemerges in almost the 30same
field, as indicated by
the arrows, which suggests the intrinsic nature of thestep, i.e., themelting transition of the vortex lattice (see5 µm
Supplemental Material for the current dependence of themelting transition and the detection by I c measurement[17]). Moreover, in the measurementsof Rc , periodic
10 µA

5 µm
40
50
H (Oe)

60

70

10 µA

-2
FIG. 1 (color online).
Magnetic field dependence of c-axis
10resistance with different applied currents in 5 × 5 μm2 IJJs.
Although there is a -3large current dependence, the melting
10transitions are observed
as steplike enhancements of the resis80.0 K
tance in both cases indicated by arrows. The inset shows a top-4view of 5 × 5 μm2 z-shaped
IJJs by a SIM.
10

融解転移温度の振動
10

20

30

40
50
H (Oe)

60

70

FIG. 2 (color 20
online). d logðRc þ δRÞ=dT map on the H-T
plane of 5 × 5 μm2 IJJs. δR is set to 1 mΩ to improve the
5×5 µm The melting transition line and theappearance of the image.
0 penetrations can be recognized as a white
individual vortex
oscillating line and as fine white slash lines, respectively. The red
dotted lines are used as a guide to count the number of vortices.
70
75
80
85
90
The i2 th lines of the individual
penetrations
are emphasizedbyT (K)the red lines.

087001-2
FIG. 1 (color online). Magnetic field dependenceof c-axis
resistance with different applied currents in 5 × 5 μm2 IJJs.

2

FIG. 2 (color online). d logðRc þ δRÞ=dT map on the H-T
plane of 5 × 5 μm2 IJJs. δR is set to 1 mΩ to improve theappearance of the image. The melting transition line and theindividual vortex penetrations can be recognized as a whiteMolecular Dynamics Simulation of
Melting Transition of Vortex Lattice

Institute for nanofabrication research

Equation of motion
dri
imp
η
= fpi + fvi + ffi
dt
f (t ) ⋅ f (t ) = 2ηk T δ δ (t − t )
f

i

f

1

j

2

⎛ r ⎞⎟⎜
fijvv = f0 ∑ K1 ⎜⎜ ij ⎟⎟r̂ij⎜⎝ λ ⎟⎟⎠
j

B

ij

1

2

f0 =

Φ20
2 3

8π λ

λ = λ (T )Mesoscopic Square plate (3λ X ３λ）
Vortex lattice structures in Bulk superconductors
with isotropic and anisotropic interactions.

Institute for nanofabrication research

微小超伝導正方形板中での渦糸の運動

N=9 T/Tc=0.01

N=9 T/Tc=0.5

3λ X ３λ, 周期境界条件

N=9 T/Tc=0.9

Institute for nanofabrication research

微小超伝導正方形板中での渦糸の運動

渦糸の軌跡

N=9 T/Tc=0.01

N=9 T/Tc=0.5

N=9 T/Tc=0.9

Institute for nanofabrication researcha chamber with a cryocooler. The electric resistance R is½VðþIÞ − Vð−IÞ%=2I with an applied dc current I. The
magnetic field was applied parallel to the c axis of Bi2212.
The melting transition of the vortex lattice can bedetected by measurements of the c-axis resistance Rc
and/or the critical current I c of the IJJ stacks due to an
abrupt change of the interlayer coupling that reflects thealignment of pancake vortices [13]. Figure 1 shows themagnetic field dependence of Rc at 80.0 K in 5 × 5 μm2
sample. Although the appearance of Rc changes with theapplied current due to the strong nonlinearity of the I-V
characteristics, a step corresponding to the melting transition emerges in almost the Tsame field, as indicated by
the arrows, which suggests the intrinsic nature of thestep, i.e., the melting
transition of ithe vortex lattice (seeiSupplemental Material for the current dependence of the0
melting transition and the detection
by I c measurement[17]). Moreover, in the measurements of Rc , periodic

微小超伝導正方形板中での渦糸の運動
渦糸の位置の分散T1
σ = ⌠ ri (t ) − ri
T⌡
2
i

(

0

)

2

dt

r =

num
tha
phe
IJJ
in a
fie
pos

1
r (t )dt
∫
T

1

10

l

2.0

5 µm

-1

10

R (Ω)

σ

1.5

30 µA

w

0

10

10 µA

-2

1.0

10

-3

10

0.5

80.0 K
-4

10

N=9

0.2

0.4

0.6

0.8

10

20

30

40
50
H (Oe)

60

70

FIG. 1 (color online). Magnetic field dependence of c-axis
resistance with different applied currents in 5 × 5 μm2 IJJs.
Although there is a large current dependence, the melting
transitions are observed as steplike enhancements of the resistance in both cases indicated by arrows. The inset shows a top
view of 5 × 5 μm2 z-shaped IJJs by a SIM.

Transition
region
T/Tc

FIG
pla
app
ind
osc
dot
The
the

087001-2

Institute for nanofabrication research

Vortex motion
in mesoscopic square plate

Vortex number dependence of melting

N=10
N=8
N=11
N=7

2.5

2.0

σ

N=9

1.5

1.0

0.5

0.2

0.3

0.4

0.5

0.6

T/Tc
14

0.7

0.8

0.9

Institute for nanofabrication research

3λX3λ正方形板

3.5
3.0
2.5

σ

2.0
1.5
1.0

N=9

0.5

0.6

T/Tc

0.7

15

0.8

Institute for nanofabrication research

5λX5λ正方形板
0.035
0.030
0.025

σ

0.020
0.015
0.010
0.005
0.000

0.6

0.7

0.8

T/Tc
渦糸数による変化はあるが、系統的ではない。
渦糸全体の並進運動の影響？
16

Institute for nanofabrication research

固定境界条件の系

鏡像法を用いた分子動力学法

η

dri
dt

= fdi + f

imp
pi

+f

ad
pi

+ fvi + ffi⎛ r̂ ⎞⎟
mirrorr̂
jm ij⎜⎜ ij ⎟
fvi = f0 ∑ ⎜ ⎟ + f0 ∑ (−1)
⎟⎟
r
rij
⎜
jm
j ⎝ ij ⎠⎛ r − r − r p ⎞⎟
fp
⎜⎜ p
i
j ⎟
p
p
⎟⎟r̂ijfij = ri − r j Θ⎜
⎜⎜
⎟⎟
rp
λ
⎝
⎠
fi f (t1 ) ⋅ fj f (t2 ) = 2ηkBT δij δ (t1 − t2 )
⎛
⎞⎟
⎜⎜
1
πH ⎟⎟
H
fi = f0 ⎜⎜
−
⎟⎟ri
2
Φ0 ⎟⎟
⎜⎜ R − r
⎝ B
⎠
i
17

鏡像（ 8個）Institute for nanofabrication research

Mirror images, Bean force
Mirror images

Confined force from bean model

18

Ordering of 9 vortices in Squaresuperconducting plate

Institute for nanofabrication research

周期境界系に比べ、渦糸が正方形に配列した。
19

Institute for nanofabrication research

渦糸運動と熱伝導を組み合わせたシミュレーション

Institute for nanofabrication research

渦糸運動における熱の発生と熱伝導の重要性

外部電流によって渦糸を駆動する
渦糸運動に伴う熱の発生
コルビノディスク形状では、電流分布と渦糸の運動、熱の発生が非一様
温度分布が渦糸の運動を引き起こす
渦糸の運動（分子動力学法）+ 熱伝導

Institute for nanofabrication research

Nernst Effect and Ettingshausen Effect
Nernst Effect

R. P. Huebener and A. Seher Phys. Rev. 181, 701–709 (1969)

In metal under an external magnetic field and temperature
gradient, which are perpendicular with each other, anelectric field is induced.
In superconductors, because vortices
have entropy SΦ and move toward
low temperature low temperature
region, and electric field is induced.
Ettigshausen effectIn metal under applied magnetic field and electric current
which are perpendicular, temperature gradient appears.
In superconductors under magnetic field and current,
vortices move perpendicular to both and temperature
gradient appears because of their entropy SΦ.

Institute for nanofabrication research

熱伝導を取り入れた渦糸の分子動力学法
F. E. David M. Kato, Physica C 484, 94 (2013)

渦糸の運動方程式

η

dri
dt

= fdi + f

fT = −Sϕ gradT (ri )SϕT (ri )

imppi

+ fvi + fTi + ffi

エントロピーによる力
渦糸のエントロピー
K. Maki PTP 41, 902(1969)

温度

Institute for nanofabrication research

有限要素法を用いた熱伝導シミュレーション
２次元の熱伝導方程式
2
2
⎛
⎞
∂T
∂
T
∂
T
⎟
⎜
⎟
ch
= k⎜ 2 +
−
α
T
−T
+
hq
x,y
(
)
(
)
⎟
a
2
⎜⎝ ∂x
∂t
∂y ⎟⎠

c
h
k = hλh
α

比熱
板の厚さ

λh

熱伝導率
表面における熱伝導係数
基板の温度

Ta
in e

1
q (x,y ) = ∑ ∑ fd (x j ,y j ) ⋅v j
渦糸運動に伴う熱の発生
See(i ) j

有限要素法(FEM)

Institute for nanofabrication research

2 (x3, y3)
ψ3A3

超伝導体を３角形の要素に分割する
各々の要素で面積座標と呼ばれる関数を定義する

3 (x2, y2)
ψ2
A2

1 (x1, y1)
ψ1
A1
２

面積座標
⎧ 1
ai + bi x + ci y )(⎪e
N i ( x, y ) = ⎨ 2Se⎪0⎩

inside of element
outside of element

ai = x j yk , bi = y j − yk , ci = x k − x j

N3(x,y)・Se

N1(x,y)・Se

P(x,y)
N2(x,y)・Se

3∑ N ( x, y ) = 1
e

i

i =1

e

(

)

N i x j , y j = δ ij
e

e

３
１

Institute for nanofabrication research

Heat transport simulation
using the finite element method

Expansion of temperature using the area coordinateseT (x,y,z;t ) = ∑Tk (t ) N k (x,y,z )e,kHeat transport equation!
aT+c T ) =Q
(
∑
j

ij

j

ij

j

j

Coefficients
e
exx
eyy
cij = αI ij + k K ij + K ij()

Qi = αTa ∑ I+ ∑ hq I

aij = chI ij
j

e
ij

j

e
j ij

Integral of product ofarea coordinates⎧⎪I =⎪⎪ ij ∫∫ S N i N j dS⎪
⎨ αβ ⌠⌠ ∂N ∂N ji⎪⎪K = ⎮⎮dS
ij⎪⎪
∂x α ∂x β⌡⌡
S⎪⎩Institute for nanofabrication research

コルビノ円板中の渦糸のシミュレーション

Institute for nanofabrication research

コルビノディスク

有限要素法のメッシュ

不純物の分布コルビノディスク中の渦糸の運動

Institute for nanofabrication research

渦糸のランダムな分布から層状の流れへ変化
中心部での大きな熱の発生により、中心で温度が上昇
渦糸は中心から端の方へ動く
その後、中心部の温度が次第に下がる

Institute for nanofabrication research

コルビノディスク中の渦糸の運動
基板への熱
伝導を良く
する

基板への熱伝導を良くすると、中心部での温度上昇が押さえられる

Institute for nanofabrication research

コルビノディスク中の渦糸の運動

さらに基板への熱伝導を良くすると、層状構造が不安定になり、渦糸分布がスパイラル状になる

Institute for nanofabrication research

コルビノディスク中の渦糸の運動
25

3.0

14

2.5

12

20

10

2.0

15

8

1.5
6

10

1.0
4
5

0.5

2

300.5

301.0

301.5

302.0

100.2

100.4

100.6

100.8

101.0

800.5

Bad thermal contact：voltage is uniform
Good thermal contact：higher around center
Spiral flow：Oscillation around center

801.0

801.5

r=11
r=7
r=3

802.0

Institute for nanofabrication research

Vortex dynamics ：influence of retarded interaction

Institute for nanofabrication research

Vortex dynamics: Relaxation
After vortex motion, there is a trail insuperconducting order parameter.

order parameter
distribution

M. Machida, H. Kaburaki,Phys. Rev. B 50, 1286 (1994)

Molecular dynamics+ retarded interaction

Institute for nanofabrication research

Kato, Fujibayashi, Physics Proceedia in press

In order to incorporate effects of relaxation of orderparameter, or trails of vortices, we consider thecondensation energy field
Equation of motion of vortices

η

dvi
dt

= fdi + f

imp
pi

+f

fRi = −gradVcond (r )

ad
pi

r=ri

+ fvi + fTi + fRi + ffi

Retarded interaction

Institute for nanofabrication research

Retarded vortex-vortex interaction
Local equilibrium condensation energy (from GL
theory)⎛ T (r )⎞⎟⎜⎜⎟⎟E (r ) = E
⎜⎜1 −
T ⎟⎟
2

cond

cond0

⎜⎝

c

⎟⎠

Condensation energy potential
After vortex leaves, condensation energy decreases
V (r )
cond

Relaxation time τ∂V (r ) E (r ) −V (r )
=
cond

cond

∂t

cond

τ

Retarded vortex-vortex interaction
f = −gradV (r )
Ri

cond

r=ri

Institute for nanofabrication research

Effect of retarded interaction
R=10λ Disk
short relaxation time (or small vortex
velocity)

Institute for nanofabrication research

Effect of retarded interaction
R=10λ Disk
long relaxation time(or large vortex
velocity)

Vortex channel flow appears

Institute for nanofabrication research

現象論的シミュレーション

Institute for nanofabrication research

現象論的なシミュレーション

超伝導の巨視的波動関数を扱うΨ (r ) = Ψ (r ) eiθ(r )

ギンツブルグーランダウ方程式
2

∗
⎛
1 ⎜!
e ⎞⎟
⎟
αΨ (r ) + β Ψ (r ) Ψ (r ) +
∇
−
A
Ψ
r
=
0
⎜
(
)
⎟
∗⎜
i
c ⎟⎠
2m
⎝
c
roth (r ) = J (r )
4π
∗
∗2
2
e!
e
∗
∗
=
Ψ
r
∇Ψ
r
−
Ψ
r
∇Ψ
r
−
Ψ
r
A
(
)
(
)
(
)
(
)
(
)
∗
∗
2m i
mc
2

{

}

Cf. シュレディンガー方程式21 ⎛⎜ !
e ⎞⎟⎜⎜ ∇ − A⎟⎟ Ψ (r ) = E Ψ (r )2m ⎝ ic ⎟⎠有限要素法を用いて解く。

Institute for nanofabrication research

超伝導ネットワーク
Institute for Nanofabrication Researchvortex

Institute for Nanofabrication Research

Cylinder as the simplest topology
hc
超伝導リング：もっとも単純な超伝導ネットワーク$ #0Cylinder as the simplest topology
Free energy

2evortexhc'(+(2 $ 0 # 2e&Order parameter
垂直な磁場下で転移温度が磁場の増加とともに振動
F #K m%*(-($)( Free0 ,(energy! # eim" ! 0

2
'(
+(
&
F # K *(m % -(
$ 0 ,()(

Order parameterm : integer! # eim" ! 0

Depending on quantum
number m, criticaltemperature may vary
appreciably.

mm : integerm

Depending on quantum
number m, criticaltemperature may vary
appreciably.

リトル-パークス効果←

5/37

磁束の量子化

Joint ESF and JSPS Conference on Vortex Matter in Nanostructured Superconductors (VORTEX V) 8-14 September 2007, Rhodes, Greece Satellite to EUCAS2007 (Brussels)

より複雑な構造ではどうなるのか？

Joint ESF and JSPS Conference on Vortex Matter in Nanostructured Superconductors (VORTEX V) 8-14 September 2007, Rhodes, Greece Satellite to EUCAS2007 (Brussels)

5/37

Institute for nanofabrication research

超伝導ネットワーク
B. Pannetier st al., Phys. Rev. Lett., 53, 1845 (1984)

正方形ネットワーク
磁場依存性

転移温度のピークでは安定な渦糸構造になっている転移温度の減少

転移温度のInstitute for nanofabrication research

Superconducting networks２
Kagome Lattice

M. J. Higgins et al. Phys. Rev. B 1 61, R894 (2007)

Institute for nanofabrication research

超伝導ネットワーク

Magnetic Field Dependence ofSuperconducting
Transition Temperature
転移温度の磁場依存性( 1-Tc / Tc0 )( a/ξ0 )2(Square Lattice)

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Distribution

of

Order

Parameter

andQuantized

Fluxoid

SquareLattice10X10
Φ / Φ0=0.333

PBC
FBC 20X20
FBC 10X10
FBC 5X5
0

0.2

0.4

0.6

0.8

1

Φ / Φ0=0.483( 1-Tc / Tc0 )( a/ξ0 )2

Φ/Φ0
0.52
0.51
0.5
0.49
0.48
0.47

Order

Parameter

0.425 0.475 0.525 0.575
Φ/Φ0

Fluxoid

Institute for nanofabrication research

有限サイズのネットワーク

磁場中の超伝導転移温度が無限系より高くなる。
磁束構造は、形状に依存し、２Φ０磁束、反磁束が現れる。
微小サイズ超伝導板と同様
超伝導状態は、端に局在している。

Institute for nanofabrication research

超伝導ネットワーク

Institute for nanofabrication researchDistributionofQuantized

Fluxoid
SquareLattice20X20
Φ / Φ0=0.0833

Φ / Φ0=0.1666

有限サイズネットワークでは、無限系とは異なり、固有の磁束構造が生じる
巨大磁束、反磁束

部分的に超伝導状態を壊すこともある

Φ / Φ0=0.25

今後の方向Φ / Φ0=0.333

Φ / Φ0=0.4166

３次元、非周期的なネットワー
ク、機能性を持つネットワー
ク、量子ネットワーク

Model for Chiral Helimagnet (CHM)/
Superconductor (SC) bilayer system

Institute for nanofabrication research

We consider a 2D superconductor.
Effects of chiral helimagnet are taken into account as anexternal field. Only z-component of field is considered.
⎛−1 ⎛H z (x,y ) = H 0 cos ⎜⎜2sin ⎜⎜sn
⎝
⎝

((

) )

⎞
⎞
β k x | k ⎟⎟ + π⎟⎟⎟ + H appl⎠⎠

Ginzburg-Landau equation is solved using the FEM.
⎡ ⎛
⎤
⎞
h −H
div A) ⎥
⎢1 ⎜
(
α ⎟⎟
1 ⎛⎜
2e ⎞⎟
⎥
F (ψ,A) = ∫ dΩ ⎢ ⎜⎜ β ψ ++⎜⎜−i!∇ + A⎟⎟ ψ +
⎟⎟ +
⎢2⎜
4m ⎝
c ⎟⎠
8π
8π ⎥
β⎟
2

2

⎢⎣ ⎝

2

2

2

ext

⎠

⎥⎦

1. Varying H0, effects of CHM are investigated.
2. Varying Happl, effects of CSL structure are investigated.

Institute for nanofabrication research

CHM/SC Bilayer systemweak field ⎛ ⎛⎞⎞⎟⎜H (x,y ) = H cos ⎜2sin ⎜⎜sn (( β k ) x | k )⎟ + π⎟⎟⎟ + H
⎝
⎠
⎝
⎠
−1

z

0

appl

When H0 is small, there are no vortices but
oscillation of OP occurs.

H0
2πξ

2

= 0.002

Institute for nanofabrication research

現象論的なシミュレーション：
勾配磁場下での渦糸

勾配磁場
引き伸ばし

巨視的波動関数
巨視的波動関数

単一の渦糸は磁場勾配の方向に引き伸ばされる現象論的なシミュレーション：
勾配磁場下での渦糸巨視的波動関数
巨視的波動関数勾配磁場

Institute for nanofabrication research

密度大
勾配磁場

巨視的波動関数

巨視的波動関数密度大

勾配磁場下の渦糸格子は、磁場が大きなところで密度が大きくなる

Institute for nanofabrication researchOrderparameter

CHM/SC Bilayer systemweak field under zero field

Increasing H0 (strength of field from CHM),
vortices appears.

Amplitude
Phase
Magnetic field
Distribution
Magnitude
H0
2πξ

2

= 0.01H0
2πξ

= 0.012
2

H0
2πξ

2

= 0.018

CHM/SC Bilayer system
Upvortex-down-vortex attractive interaction

Institute for nanofabrication research

α = 0.16

1.0

Because attractive interaction between upvortex and down-vortex, pair annihilation of
up-vortex and down-vortex occurs in weak
Vortex-antivortex
field case. Finteraction
H

0.5

Fvv
10

20

30

40

Fvv

-0.5

-1.0

⎛ r −r ⎞
Φ0
⎜⎜ 1
2 ⎟
⎟⎟
Evv = −
K
0⎜
2⎜⎜ λ ⎟⎟
2πλ
⎝
⎠

FH

50

60

Interaction with external field

EH = −

Φ04π

H z (r )Institute for nanofabrication research

CSL/SC Bilayer system
H0
2πξ

2

= 0.012

Under applied field Happl, CHM shows chiral soliton
lattice structures. Down vortices disappear
gradually.
Orderparameter
Amplitude
Phase
Magnetic field
Distribution
Magnitude⎛−1 ⎛⎜H z (x,y ) = H 0 cos ⎜2sin ⎜⎜sn
⎝
⎝

((

) )

⎞
⎞
⎟
β k x | k ⎟ + π⎟⎟⎟ + H appl⎠⎠

Institute for nanofabrication research

CSL/SC Bilayer system
Further increases applied field, lattice
structures of up-vortices becomes modulated
triangular lattice.
AmplitudeOrderparameter
Phase
Distribution
Magnetic field
Magnitude

Such structure may causes pinning of vortices.

Institute for nanofabrication research

まとめ

分子動力学法シミュレーション
渦糸格子融解コルビのディスク：熱発生のため渦糸運動が層流から乱れる
現象論的シミュレーション
有限の超伝導ネットワークにおける磁束曼荼羅
カイラル磁性体における磁束構造
微視的シミュレーション
ナノサイズ超伝導体の転移温度上昇
磁場中のナノサイズ超伝導体における奇周波超伝導の発生