現代物理学への招待 低温物理学 超伝導

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物理学セミナー 050601

微小超伝導体における新しい渦の観測

物理学専攻 神田晶申 „ アウトライン „ メゾスコピック超伝導体とは? „ 渦糸状態の特徴 „ 実験方法 „ 巨大渦糸状態の観測 „ 渦糸状態間転移の温度依存 „ まとめ

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超伝導とはどんな状態か?

„ 金属の抵抗の温度依存性 青: 通常の金属 赤: 磁性不純物を含む金属(近藤効果) 緑: 超伝導金属 抵抗 温度 5 0 ) (T = ρ + BT ρ 臨界温度TC以下で抵抗が 完全にゼロ(完全導電性) TC

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いろいろな超伝導体

元素(51種) 合金(約1000種) 金属化合物(約500種)    窒化物、炭化物など 金属間化合物(約200種) 有機物(約20種) トコトンやさしい超伝導の本 (下山淳一、日刊工業新聞社) 液体窒素(77K) BCS理論の壁

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 磁場中の振る舞い  

超伝導体と完全導体

„ ゼロ磁場で冷却(ZFC: zero-field cooling) 常伝導状態(T>Tc) 超伝導状態(T<Tc) 磁場印加 冷却 Js Js:遮蔽電流 レンツの法則:外部磁場の変化を妨げるような 磁場を作る向きに電流が流れる。(電磁誘導) 完全導体では、抵抗ゼロなのでその電流は減衰せず流れ続け る。従って、磁場は永遠に侵入できない。 ZFCでの超伝導体の振舞は完全導体として理解できる。

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 超伝導体と完全導体の違い

„ 磁場中で冷却(FC: field cooling) 常伝導状態(T>Tc) 超伝導状態(T<Tc) 完全導体の性質: 磁場の時間変化がないの で磁場侵入のまま。 冷却 Js 超伝導体: 磁場を完全にはじき出す。 (完全導電性とは独立の性質) 完全反磁性(マイスナー 効果) Js:遮蔽電流

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超伝導体の2大特徴

„ 完全導電性 „ 完全反磁性(マイスナー効果) „ 超伝導体は単なる「抵抗が無限に小さくなった金属」ではない! „ 超伝導になる ・・・ 新しい状態への『相転移』  浮き磁石 磁力線の歪みに由来する力と重力がつ りあう。

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超伝導はこわれやすい

„ 磁場、電流、温度が大きすぎると超伝導は壊れる。 電流密度 超伝導状態 温度 臨界電流密度:Jc 臨界磁場:Hc 臨界温度:Tc 磁場 高臨界温度、高臨界磁場、高電流密度の実現が実用化の課題

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2種類の超伝導体

HC1 磁場 Tc マイスナー状態 (完全反磁性) Hc 磁場 マイスナー状態 (完全反磁性) Tc 温度 HC2 混合状態 (渦糸状態) 0 0 第1種超伝導体 第2種超伝導体 ξ: コヒーレンス長(クーパー対の拡がり) λ: 磁場侵入長 (遮蔽電流の流れる範囲) 2 / 1 / > =

λ

ξ

κ

κ

=

λ

/

ξ

<1/ 2 Js Js:遮蔽電流 温度 HC2は10Tに達することもある (超伝導電磁石に使える!) Hcは0.01T程度 (小さい!)

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第2種超伝導体の混合状態

遮蔽電流 印加磁場 渦糸(vortex) 中心部直径ξ程度が常伝導で、量子化磁束       が貫く。 その周りλ程度の範囲にΦ0を作るための超伝導電流の渦が流れる。 渦糸の周りでオーダーパラメタ      の位相は2π変化する。 wb 10 2 2 / 15 0 − × = = Φ h e 渦糸は三角格子を組む (アブリコゾフ) ) exp(i

θ

Ψ = Ψ

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混合状態に電流を流すと・・・

電流 電流によって、渦糸はローレンツ力を 受け、動き出す。 電圧、ジュール熱発生 超伝導破壊   だめ!! 解決法 超伝導体中に、意図的に欠陥、不純物を導入する。 渦糸は、そこにピン止めされ、大電流まで動かない。

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ピン止め中心のつくりかたと超電導ナノ工学

„ 従来のピン止め中心 „ 結晶中の不純物、空孔、転位、析出物、結晶粒界 „ 材料の焼きなましで非超伝導層をつくる(Nb−Ti合金) „ 重イオン照射により柱状欠陥を導入する „ 空間分布などの制御が困難 „ これからの方向 超伝導ナノ工学 „ ナノテクを駆使した渦糸配置制御 „ 電子ビームリソグラフィー、収束イオンビーム加工… „ 新しい特性、機能を発現させる Moshchalkov (ベルギー) M: 磁化

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メゾスコピック超伝導体とは?

„ サイズ:超伝導コヒーレンス長 ξ や磁場侵入深 さ λと同程度. „ 渦糸配置? „ アブリコゾフの三角格子 „ 試料端との相互作用 の競合によって決まる

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Multi- and giant vortex states

(d) (b) (c) (a) Vorticity L = 5 Cooper-pair density Phase of the order parameter

Giant vortex state (巨大渦糸状態) Multivortex state (多重渦糸状態)

(渦度)

他の量子系では、巨大渦糸は見つかっていない。 ) 1 ( 2

π

n n >

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超伝導の理論

„ 現象論− 仮定の下に理論を構築。実験結果をよく説明 „ London理論 1935年 „ ギンツブツグ−ランダウ理論  1950年 „ 後に微視的理論から導かれた。 „ さまざまな複雑な状況に適用可能な強力な理論 „ 微視的理論− 完全に現象を説明する

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Theoretical formalism

„ Dimensionless Ginzburg-Landau equations:

(

)

(

)

(

)

2 2 2 * * 2 1 1 2 i A A A i

ψ ψ ψ ψ

ψ

ψ ψ ψ

ψ

κ

− ∇ − = − ⎡ ⎤ ∇ × ∇ × = ∇ − ∇ − ⎣ ⎦ r r r r r r r r

(

)

0 boundary 0 A A n i A ψ ∞ = ⋅ − ∇ − = r r r r r

• Boundary conditions:

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試料サイズによって、渦糸状態はどう

変わるか?

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Disks with radius R << ξ

„ No vortices can enter the sample. Only the Meissner

state (

L =

0) is stable.

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Disks with R

ξ

„ Several vortices can enter the sample, but the

boundary imposes its symmetry on the vortex configuration.

„ Only axially symmetric states or Giant vortex states

can nucleate.

Cooper-pair density L = 2 R = 2ξ

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Disks with R

„ Stabilization of the multivortex state: in some

magnetic field regions, several single vortices nucleate on one shell in the disk.

Cooper-pair density L = 5 R = 4ξ

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Disks with R

4ξ (free energy)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

H

0

/H

c2 6 5 4 3 2 1 L = 0 R = 4.0ξ

F/F

0 Multivortex states

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Disks with R

4ξ (L = 3)

„ The magnetic field distribution for different values of

the externally applied magnetic field.

-4 -2 0 2 4 (a) H=0.525Hc2 y/ ξ (b) H=0.65Hc2 -4 -2 0 2 4 -4 -2 0 2 4 (c) H=0.75H c2 y/ ξ x/ξ -4 -2 0 2 4 (d) H=0.8H c2 x/ξ 2004/12

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Disks with R

„ In some magnetic field regions more shells of vortices

can become stable.

„ Different vortex configurations with the same total

number of vortices can nucleate.

Cooper-pair density L = 13

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Disks with R

„ The combination of the giant vortex state and the

multivortex state becomes possible.

Cooper-pair density L = 14

Phase of the orderparameter

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L = 8 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 x/ξ y/ ξ

Disks with R

6ξ (Free energy)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 L vortices on a ring L - 1 vortices on a ring + 1 vortex in the center

Giant vortex state with vorticity L

L - 2 vortices on a outer ring + 2 vortices on a inner ring

L - 3 vortices on a outer ring + 3 vortices on a inner ring

R = 6.0ξ F/F 0 H 0/Hc2 L = 9 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 x/ξ y/ ξ L = 15 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 x/ξ y/ ξ L = 11 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 x/ξ y/ ξ -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 x/ξ y/ ξ L = 13 0.7 0.8 0.9 1.0 1.1 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 14 13 12 R = 6.0ξ F/F 0 H 0/Hc2

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Disks with R

20ξ

„ Different configurations for L = 16

-20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 x/ξ y/ ξ -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 x/ξ y/ ξ -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 x/ξ y/ ξ B.J. Baelus, PRB 2004

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Disks with R

50ξ

„ Triangular lattice in the center

Cooper-pair density

L = 232

Cooper-pair density

L = 44

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Theoretical prediction for size dependence of

vortex states

sample size

R<<ξ Bulk

MVS +GVS

GVS

Meissner (No vortex) (Baelus et al., 2004) triangle lattice several shells single shell radius R=2ξ R=4ξ R=6ξ R=50ξ

„   small samples →GVSs are preferred „   large samples →MVSs

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Stability of the vortex states

L = 2

L = 3

L = 4

Theoretically, vortex configuration corresponding to the sample shape is stable.

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Anti-vortex?

(Moshchalkov)

In type II, v-av patterns are unstable.

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Experimental probes for mesoscopic

superconductors - direct method

・ Scanning SQUID microscopy (Kadowaki)

Nb disk (50 µm)

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Experimental probes for mesoscopic

superconductors - indirect methods

・ Resistance measurement; cusps in Tc(B) is obtained. (Moshchalkov)

・ Magnetization measurement by ballistic Hall Magnetometry (Geim, Moshchalkov)

–Numerical study (minimization of the free energy) is essential in order to identify the vortex states

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Multiple-small-tunnel-junction (MSTJ) method

superconductor normal metal lead SIN junction -20 -15 -10 -5 0 5 10 15 20 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 I( n A ) V(m V) I I (0.1 nA) e Js B Vg = ∆( , )/

- By using small tunnel junction ( ), one can detect change in local energy gap, which is related to the supercurrent, Js, flowing underneath the junction.

- By using multiple small tunnel junctions, one can study

supercurrent distribution.

ξ

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Example: Magnetic response of mesoscopic rings

Cu leads

Al ring

V I

This voltage change has two origins:

(1) smearing of the energy gap due to pair-breaking by the magnetic field.

--- monotonic decrease of V as a function of B

(2) decrease of the energy gap by the supercurrent underneath the junction.

Current was fixed at 100 pA. B ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ∆ = ∆ 2 27 2 1 ) 0 ( ) ( C S S J J J (Bardeen, 1962)

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Disks for Multiple-small-tunnel-junction (MSTJ)

measurement

Fabricated using e-beam lithography and angle evaporation technique.

Cu Al disk radius: 0.75µm, thickness: 33nm tunnel junction resistance > 20 kΩ (no proximity effect)

- Four tunnel junctions are attached to the periphery of the disk.

- Voltages, VA, VB, VC, and VD, at I = 0.1 nA were measured simultaneously as a function of perpendicular magnetic field and temperature.

bias current I

VA VB VC VD

120 o

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How to distinguish GVS from MVS

Contour plots of the current density

By comparing voltages of junctions at the disk periphery one can distinguish between MVS and GVS. Axial symmetry: VA = VB = VC = VD Non-Axial symmetry: VA ≠ VB ≠ VC ≠ VD A B C D 120 o only symmetric with respect to central axis

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Magnetic field dependence of voltage in decreasing B

sweep

T=0.03K

MVS at L = 2, 4 – 11

¾ Each voltage jump corresponds to a transition of vortex states with ∆L = -1.

¾ The sample is symmetric with respect to the central axis, so VA and VD (VB and VC) can be compared. ¾To remove the effect of small

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Magnetic field dependence of voltage in increasing B

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Theoretical study

Ginzburg-Landau theory, taking into account the demagnetization effect. (R = 5 ξ, d = 0.1 ξ, κ = 0.23) (V. A. Schweigert et al.(1998))

MVS:L = 3 - 6 (theory) L = 4 - 6 (exp)

MVS:L = 2 - 10 (theory) L = 2, 4 - 11(exp)

increasing B decreasing B

Theoretical calculations confirm the identification of GVS and MVS by MSTJ method, except for L = 3 and 11.

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L = 3 ?

Cooper-pair density

L = 3 L = 6 L = 9

„ The

L

= 3 state has trigonal symmetry, corresponding to

the angle .

„ For the

L

= 6 and 9 states, the difference in d

V

A/

dB

and

d

V

D/d

B

is large, presumably due to the effect of defects.

AOD

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Effect of defects

- At L = 0 state, all curves are parallel to each other, indicating no defect near the junctions.

- At L = 1, curves are not parallel

presumably because of a defect

close to (but not at) disk center. increasing B

(43)

The whole L = 8 state

In the

- no hysteresis (2nd order transition)

- additional 1st order transition with hysteresis, possibly due to a transition between different MVSs with the same L.

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Theoretical analysis for L = 8 state

MVS-to-GVS transition appears. No transition with hysteresis

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Effect of defects

- At L = 0 state, all curves are parallel to each other, indicating no defect near the junctions.

- At L = 1, curves are not parallel presumably because of a defect close to (but not) at disk center. increasing B

decreasing B

Defect close to (but not at)

the disk center

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Theoretical analysis for L = 8 state with a defect

Defects lead to additional first order transitions with the same

L.

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まとめ (1)

„ メゾスコピック超伝導体では、巨大渦糸状態(GVS)、多重渦糸 状態(MVS)という新しい渦糸状態が理論的に予言されてきた。 „ MSTJ法によって、はじめて巨大渦糸状態の実験的証拠を得た。 „ 渦糸配置の対称性を考慮 „ 2種類の相転移を観測した(渦度L固定) „ MVS-GVS 転移(2次転移) „ 欠陥に起因するMVS-MVS 転移(1次転移) A. Kanda et al. PRL 93 257002 (2004) 神田他、「固体物理」6月号(2005) 原稿が欲しい人は、神田までメールをください。

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Comparison of current symmetry is less

powerful for squares!

GVS

L = 4 MVS

L = 5 MVS L = 6 MVS L = 3 MVS

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Alternative method to distinguish GVS

from MVS

Temperature dependence of the vortex expulsion fields

0 0.01 0.02 0.03 0.04 0.05 0.01 0.014 0.018 0.022 0 .0 26 V( m V ) B (T ) T=0.1K   0.15K 0.2K 0.25K 0.3K 0.4K 0.5K  0 0.01 0.02 0.03 0.04 0.05 0 .0 1 0 .0 1 4 0.0 1 8 0 .0 2 2 0 .0 2 6 V( m V ) B (T ) L=8 10 12 11 increasing B decreasing B

boundary: L =11

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How to distinguish GVS from MVS

GVS

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Radial dependence of current density

screening current

(distance from the center)

In MVSs, the screening current is almost temperature independent, leading to temperature-independent

transition fields.

B. J. Baelus, A. Kanda, F. M. Peeters, Y. Ootuka and K. Kadowaki, Phys. Rev. B 71 140502(R) (2005).

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Can the criterion be applied to squares?

increasing B 0.75 (µm)2 decreasing B 0.10 K to 1.05 K Two behaviors in decreasing B

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Application of the criterion to squares

(Theory)

Baelus (May, 2005)

MVS

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Experimental results for increasing B

Vortex penetration fields change uniformly as a function of temperature.

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Experimental results for decreasing B

Vortex expulsion fields show two kinds of behavior. The boundary

Lc

increases with square size, showing stabilization of MVS in larger squares.

Lc = 2

Lc = 4

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Stability of the vortex states

L = 2

L = 3

L = 4

Theoretically, vortex configuration corresponding to the sample shape is stable.

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Evaluation of the stability

„ The stability of the multivortex states can be evaluated

by the width of the stability region, ∆H, over which the L state is stable. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 H 0/Hc2 6 5 4 3 2 1 L = 0 R = 4.0ξ F/F 0 ) 2 ( = ∆H L 0 1 2 3 4 5 6 0.4 0.5 0.6 ∆ H/H c2 L

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Shape dependence of the stability

(experiment)

H

triangle

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まとめ

„ メゾスコピック超伝導体の特殊な渦糸状態を、新しい実験 方法で研究 „ メゾスコピック超伝導体の基本的な性質がだんだんと明ら かになってきた。 „ 2種類の渦糸状態がある。(MVSとGVS) „ 渦糸状態間の転移: MVS-GVS(2次転移)、MVS-MVS(1次転移) „ 渦糸状態間転移磁場(磁場下降時)から、MVSかGVSかを判断で きる。 „ 試料サイズが大きくなるほど、MVSが安定化することを確認。 „ 試料形状が、渦糸状態の安定性に影響する。

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Collaborators

„ Natsumi Shimizu, Kumiko Tadano (Tsukuba) „ Ben Baelus, Francois Peeters (Antwerp)

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参照

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