超伝導の基礎 II
有本 靖
高エネルギー加速器研究機構
2022-09-06 OHO’22 加速器セミナー

目次• BCSの基礎概念• 磁束の量子化• Josephson効果

BCS理論の基礎概念

金属自由電子ガス
Surface

EnergySurface
OutsideInsideOutsideU
ElectronPositionh2 2r (~r) = E (~r))
2me<latexit sha1_base64="BvkfX2N2KXVTA5ZI22DRud5Vpes=">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</latexit>

シュレディンガー方程式

周期的境界条件 : 一辺の長さLの立方体結晶. 表面で完全に反射金属自由電子ガス
シュレディンガー方程式の解<latexit sha1_base64="kWie2Zbp5AkNAAS7bXH+FRN4qrg=">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</latexit>

1(x, y, z) = p exp[i(kx x + ky y + kz z)]
L3
2⇡lx
kx =
L
2⇡ly
ky =
L
2⇡lz
kz =
L
<latexit sha1_base64="LH4pGZFX5bZJwnu4LEZe6k4AJ48=">AAAHuXicnVRLb9NAEJ6UR0p4NIUjlxUVqEih2UQVIFCliiLEoRKN1BeKq+A4m8SNX7WdNm4UxOPAAU78O8QfQOLEP+DbjQNJTZwKr7w7++3MN7Ozs1v3LDMIOf+Wmbtw8dLl7PyV3NVr128s5Bdv7gZu1zfEjuFarr9f1wNhmY7YCc3QEvueL3S7bom9emdDru8dCz8wXWc7jDxxYOstx2yahh4CquV/sk6tx+6tMa3p60a/rHkms2q9QX9zwLSjo/6ALWPK1hgvMM2zS6pnZQxGww2D+5rjhnpL03LgiRI80SRPlMJj6XVh9cXRE19Yw9g6Nd1p1KzBkPw0QX46SX46K8hcLb/EV7j6WFIoxcISxd+Wuzj3nTRqkEsGdckmQQ6FkC3SKUCrUok4ecAOqA/Mh2SqdUEDysG2Cy0BDR1oB30Ls2qMOphLzkBZG/Bi4fdhyehurNOA3FTocJT+2ZhuDpr/9tJX7DLKCGM9ZrWBhtQGOttypHt+yyodqyy01I5C6v2JOVD5mO5xxO6hjbI3Xdv7L+YQci/lXPrwbIO1AdkHdzqvgTYeg8wRO4My/rUi+Bf+mbd5BnMTeCE+v4C/4RW+zz/yQ/6Kv+YRb3Lnwy/xiW/ShmIwEIUAj+Ad6BhY3+Nv+Qvu8Jb1Q19Ijc5Bf6LyBHvVFzB2VW59haflIWmdnovzMlfHNB8gzyZQU2XqYILlGOeQxrKNeyctRpkWKltpOxrtQ2bexmr67QzB/Fh5kCfmKWS2j8lbU0jcv+mWHvQjVR/GzHpOZmn6bUzfyd9Xpqk4GdZc1UfQFJDkCckaNIBZQIeIrirTjN86Wc1tdet1pScjC1L2WqQdhUutorIz1a2Tvov0/MxbK3UairuJWZHKaJxWUT0uvEoLW0U77gM5wUtfOvuuJ4Xd8krp4cpqZXVp/Vn85s/TbbpDy6ivR7ROL2kL0RqZSuYk8y7zPvs0q2fb2cOh6lwmtrlFE182+A2h9Kyw</latexit>(lx = 0, ±1, ±2, · · · )(ly = 0, ±1, ±2, · · · )(lz = 0, ±1, ±2, · · · )エネルギー固有値

h2 2
E=(kx + k2y + k2x )
2me
h2
2
2
2
=
(l
+
l
+
l
x
y
x)
2
2me L
<latexit sha1_base64="Iy/cBnlp3simDUlgXnlllxJZFvA=">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</latexit>

フェルミ球電子は1つの状態に最大2つしか入れない<パウリの排他律>kz

エネルギー準位の低い順に電子を詰めていった時の系の電子の最大エネルギー
2Fermi sphere<latexit sha1_base64="T04K6Spg1pzZCyBCwMDRk5dm5KU=">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</latexit>

h
k2F
2me
h2
2/3
2
=
3⇡ n
2me

E0F =

<latexit sha1_base64="KWBQQyoCLz9LbSwEXHzw0fNw5nA=">AAAG9nicnVS7bhNBFL0JYEJ4JIGSZkWElCLE11EENEgRQREFEomUF9hRWK/HzpB9sTsOtixTQEEBouAD6BAtBS38B+IHaPkDztyswYnxJmJHO3v3zL3n3jnzqMa+Tg3z95HRU6fPFM6OnRs/f+HipYnJqcsbadRMPLXuRX6UbFXdVPk6VOtGG19txYlyg6qvNqt7S3Z8c18lqY7CNdOO1XbgNkJd155rAO1MzjjxTsWoluksd+9U0meJ6cwHGaK6nYpaVkmgH3e7O5PTPMfyOINGKTOmKXtWoqnRH1ShGkXkUZMCUhSSge2TSylamUrEFAPbpg6wBJaWcUVdGkdsE14KHi7QPfQN/JUzNMS/5Uwl2kMWH2+CSIeuZz412HVBD742v9PnOw7Pf2fpCLutso1vNWMNgBraBXp8ZM/35JFl2hcVGjIjQ60/Naeix/CMPfYYrafecO/4v5gN7FbOunSQOQBrDXYC7nxeD62/BquRcwR1+P2q4rf8hnd5BP8a+Gy2fik/4VXe4lf8lB/yI25zncOXv9RrfkBLwuChCgUexXvw8TC+yS94mUNu+D/didzqQvTPRSfESz+Lb1O0TQTP02EwOl+LkzKX+zxvQGcNVItS24dY9rEOeSxrOHc2oqe0ErXyZtSbh1U+wGj+6TRgvi0Z7IrFghyf4/CpmR04f8MjY/i3ZX94x+7nQZWGn8b8mfy9ZerC6WAskr4NTwXLrpDdgx4wH+gB4srO1NldZ3fzrpx6V/xsZWnOXIu0Lrj1KkqcllNncxfp3pG71vrUhLuOvyLNozEtYPdEyGojAqm2Pwc0wU1fOnqvDxob83Olm3MLqwvTi3ezO3+MrtI1msH+ukWLdJ9WUK1H7+gLfaVvhVbhQ+Fj4dOB6+hIFnOFDj2Fz78BcQ1wpg==</latexit>

Fermi運動量pF =q

2me E0F

銅の場合のFermi エネルギーEF0 = 1.1x10-18 J,
TF = EF0/kB=80,000 K

k
2 π/L

kx 2 π/L

2 π/L

ky

電気抵抗の起源• 結晶格子が完全な周期性を持つ場合電気抵抗は零. (量子力学)• 周期性を乱す要因

銅の抵抗率の温度依存性- 格子振動 → 温度を下げると減少

格子振動* 原理的にはT=0Kで抵抗は0に- 格子欠陥, 不純物等 → 温度を下げ
抵抗率ても変化しない (常伝導の限界)

Resistivity (Ω m)

格子欠陥

OFHC

99.999%Temperature (K)温度

Cooper対• Cooperのモデル (1956年)- T=0K- Fermi球の内側はすべて電子が占有

pz

Fermi sphere- Fermi球の少し外側に2つの電子をおく* p1 = –p2s2- Fermi面の真上にある2つの電子を考えてそれらの間に相互作用が働くとしたとき, この電- (答) Yess1

p1

• クーパー問題

子対が束縛状態をつくるか?

py

p2

px

Cooper対• 電子が固体中を進むと結晶格子のイオンが影響を受ける• 電子通過後 イオン同士が近づき正の電荷密度が増える.
• 2つ目の電子が引力を受ける
格子面

(b)

格子の歪み

Lattice planese–~1000 Å

Deformation amplitude(a)1000 Å

2π~ VF ωDZ

ZDistance from electron
電子からの距離H. Ibach and H. Lüth, Solid-state physics, 4th ed. (Springer Berlin, Heidelberg, 2009).

結晶内のCooper対
Cooper対は長い距離で結合しており,
Cooper対同士は重なり合っている.

Cooper対の結合エネルギーは10-4 ~ 10-3 eVと非常に低いので熱運動で分解しない
ように低温を維持する
Metal ion

Cooper pairSingle electron必要がある.

BCSの初歩• Bardeen, Cooper, Schrie er- Fermi球内の電子も対になる• このときの引力は格子振動(フォノン)によりPy

Fermi sphere, p2/2me = EF

-p''

p'

与えられる.

• Fermiエネルギー近傍 ±2 ωd• クーパー対は同じ運動量P=0を持つ- 同じ量子状態に入っている.
- クーパー対はボーズ粒子として振る舞う• エネルギーギャップΔによって特徴付けられ

ff

る

2ωD

-p
p

-p'
p''

Px

BCSの初歩• BCS基底状態: エネルギーギャップ(Δ)によって不対電子とエネルギー準位は分離されている.
• クーパー対を分解するには2Δが必要.

(a)
Energy(b)Single electrons

Energy gap 2Δ
BCS groundSingle electrons

δEpairBCS groundエネルギーギャップの温度依存性• エネルギーギャップは温度依存性を示す: Δ(T)
• BCS理論から算出された関数Δ(T)と測定値は非常によく合っている.

H. Ibach and H. Lüth, Solid-state physics, 4th ed. (Springer Berlin, Heidelberg, 2009).

普遍定数<latexit sha1_base64="TaDQXCooRcmz1ThQawQ25w3Y/E8=">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</latexit>

T=0でのギャップ関数:<latexit sha1_base64="8Snhw33AiqlEfK73Dv+zF6UQCAE=">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</latexit>✓(0) = 2h!D exp -1
V0 N(EF )!D : Debye周波数◆,

V0 : 電子-格子相互作用ポテンシャル
N(EF ) : E=EFにおける単一電子の状態密度✓kB Tc = 1.14h!D exp <latexit sha1_base64="77zq/T4UFvDcrI8TsjLiUha1Q3c=">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</latexit>

臨界温度:

1
V0 N(EF )◆<latexit sha1_base64="ynX/oPn+6SsMTCHYq/bCnZNSvPI=">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</latexit>(0) = 1.76kB Tc

P Schmüser, Superconductivity, CAS - CERN Accelerator School :Superconductivity and Cryogenics for Accelerators and Detectors (2004).

同位体による違い• 前のスライドの一番目の式によると, 臨界温度はDebye周波数に比例.
• また, Debye周波数は原子量Mの平方根の逆数に比例.
<latexit sha1_base64="KXX1CQGmLpak/h3iHASYUVQwWi0=">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</latexit>• 臨界温度:p

Tc / !D / 1/ M

H. Ibach and H. Lüth, Solid-state physics, 4th ed.
(Springer Berlin, Heidelberg, 2009).

超伝導電流と臨界電流 I• δtの間E0を印加~ = hK~ = -2e~E0 t,
P
n
e
h
s
~js = ~
K,
クーパー対の超伝導電流密度:2me<latexit sha1_base64="4SgYhc8PFtUa+G088zQ+cA2mSBQ=">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</latexit>

クーパー対の運動量:<latexit sha1_base64="Buyl3I5w7lsG3G7Qh6qoaoVLH1U=">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</latexit>

ns: 電子の数密度

⇣
⌘
~ ·R
~ を掛けたもの
電流が流れた時の波動関数は位相項 exp iK<latexit sha1_base64="Q0S8fUKYbahYccx3SeRKXhYA69g=">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</latexit>

BCS理論の方程式はそのまま適用可能

運動エネルギーの増加分δEpair が2Δ未満であれば超伝導状態は保たれる超伝導電流と臨界電流 II
Ef• 散乱中心には2つのタイプがある

電子
Ei- 不純物に起因するもの- 格子振動によるもの

Ei=Ef• 不純物: 固定された重い標的と考えられるので, 弾性散

金属イオン

乱となり, Cooper対の電子のエネルギーは増えない• 格子振動による散乱: 平均熱エネルギーがエネルギー(a)
ギャップよりも小さい場合は無視できる.
(T<Tc)(b)

Energy• 抵抗ゼロで電流輸送できる条件：
Single electrons<latexit sha1_base64="Pl5KzgphFjK+lXU3BlMhmOTPs40=">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</latexit>

2 - Epair > 0,
T < Tc (js )Energy gap 2Δ
BCS groundSingle electrons

δEpairBCS ground超伝導電流と臨界電流 III• 輸送電流が流れているときのエネルギー増加量

h2 kF K
Epair =me<latexit sha1_base64="iJD0JXqOPjnURbTys7z1Q66N6xY=">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</latexit>• したがって,<latexit sha1_base64="uClMgW88xY29T+km88Hu78WTIsU=">AAAHTXicnVRLb9NAEJ4USEsKtOVx4rKiQuJQmk1UARekihaEBIhW6gvFpTjOJtnGL9lOaRQFJDhwgBN/jDPijMSVAzeE+HbqlLQhLsIrr8fffvPN7OyjGro6TqT8nBs7dfpMfnzibGHy3PkLU9MzFzfioB05at0J3CDaqtqxcrWv1hOduGorjJTtVV21WW0tmfHNPRXFOvDXkk6otj274eu6duwE0M70a2HVlJvY4v6Olaj9pBvaOuoJK9aesOqR7XStZtWOnpeF1aqryNPiUa9rKfeJHce9gribkkSZaYekXlco4aeScU/0dg/tgrBcJeCwbOLuTM/KecmPGDZKqTFL6bMSzIx9IYtqFJBDbfJIkU8JbJdsitEqVCJJIbBt6gKLYGkeV9SjAnzbYCkwbKAt9A38VVLUx7/RjNnbQRQXbwRPQddTTg12ndGDr4kvBrgFMP8epcvqJssOvtVU1QOaUBPoyZ597r97VmiPq9DgGSW0f5hzzPUYHbGvHqL1qzeaHf6XcgJ7P2NduojsQbUGO4J2tq6DNpiDqZE4hgr5cVXJD/K9bMoc/jXwuXT9YvlCrsot+VbuyqfymezIuvTffFfv5GNaYgUHWSjoKNkCx8H4pnwlH0hfNtxv9lRmdj76l1wn+HM/h2+baxsxnlWHYe/sWvyrcmWAeRN11kA1V2r7iMoe1iFLZQ3nznj0K624Wlkz6s/DVN7DaPbpTKB8hyOYFQsZOTnG0VMzN3T+RnuG4Hd4fzgn7ufhKo0+jdkz+XPL1FlTYCzgvgOmgmVWyOxBB5gL9ACxeWfq9K4zu7nJp95mnskszphrkdYZN6wi+2k+dSZ2kZaP3bWGU2PtOv6KVEaTtIDdEyCq8fA428EYqAlu+tLxe33Y2CjPl27NL6wuzC7eS+/8CbpK1+gG9tdtWqSHtIJsHfqam8xdzl3Jf8r/yP/M/zqgjuVSn0t05Bkf/w0UJY3G</latexit>

h2 kF K
2hkF
Epair ⇠=
js 6 B2meensj<latexit sha1_base64="kMvztylGT68ZUWLZQ8EBdVoGgYI=">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</latexit>• 臨界電流:

ens
jc ⇠
hkF

<latexit sha1_base64="MlM0R7PxPowsDmk0xLa8+I/LwL8=">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</latexit>• 臨界磁場:

Hc =

j j⇠

L c

ens
.
L
hkF

j

磁束の量子化量子論におけるベクトルポテンシャル
電磁場中での運動量<latexit sha1_base64="KQuqGiau+n6xyjZsBpIb1FsQ2z4=">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</latexit>~~ = m~v + qA
p
ドブロイの関係

波長<latexit sha1_base64="6Dvzq5twC0+s4D3TVShCxD5PF3Y=">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</latexit>2⇡h=.
mv + qA<latexit sha1_base64="bPOMW/vvBtyVLfDMEsVca+temb4=">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</latexit>

2⇡h
=
.
p

自由空間内(磁場なし)で Δx移動した時の位相の進み<latexit sha1_base64="RW53jKZ9ZnSwtkzuDJPvL8a1cho=">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</latexit>

'=

2⇡

1
x = me~v · ~xh

電磁場内ではさらに以下の位相だけ進む<latexit sha1_base64="LQ7yAbofYGaPQkkoqhrH6MgOkys=">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</latexit>e~ ~' = - A · x,h0

Aharonov-Bohrm効果

Aharonov-Bohrm効果
外村らによるAB効果の観測

A

Solenoid coil

Φ

B

A

SourceDetector<latexit sha1_base64="fHqJMOdHOAbqNatuWi1c1wtLPyM=">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</latexit>

I

e
~
~
' = '0 + A · dl = '0 +
h

mag .

直径数μのトロイダル永久磁石を超伝導ニオブで閉じ込めてAB効果を観測
A. Tonomura, et. al., Phys. Rev. Lett. 56, 792–795 (1986).

磁束の量子化 I
Cooper対の波動関数は0 exp(i') =<latexit sha1_base64="VbNeIKg7IhjWvkxlwonaDvmf48I=">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</latexit>=pnc exp(i'),

B

nc は Cooper 対の密度C

リングを一周した時に, 位相Φ=Φ(s) の進みはA

Superconductor

n 2π でなければならない.
<latexit sha1_base64="FvJ9m4lV0jiib01RJzZbgDTsUa4=">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</latexit>Id'ds = n · 2⇡.
ds

バルク内では v = 0<latexit sha1_base64="m+yOGh1qPv3ebHE2PKKJASoDrik=">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</latexit>d'q~ ~ds = A · ds.
ds
h

0

π/2

π

3π/2

2π

磁束の量子化 II<latexit sha1_base64="UcoD9778MZmvS+Tvdm1O80pV7Qc=">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</latexit>

I

h
~ ~
· n · 2⇡ = n ·mag = A · ds =q<latexit sha1_base64="KJs+Paom09cajmOGVbgN9mZCr/s=">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</latexit>

2⇡h
0 =
q
BCS理論では q=-2eなので,<latexit sha1_base64="YXNEBFTlWbht/uYuG5lXOUjUpIc=">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</latexit>

h
0 =
2e
Φ0: 磁束量子

0Josephson効果

Josephson効果
1962年にJosephsonはクーパー対がトンネル効果を示すことを理論的に導いた.
薄い絶縁体の両面を2つ超伝導体で接合すると, トンネル効果によって電流が流れる現象.

Insurator
S1
Superconductor

S2
SuperconductorAJosephson効果におけるシュレディンガー方程式<latexit sha1_base64="OXe4RhljIZxopJsMOWRQNuQ4rfg=">AAAHPnicnVRLb9NAEJ4U6pbwaAtHLhYVCIm2mUQVcEGqKEJIgGilvlBSFcfZJKZ+yXZCoygc4MABTvwifgUS4o64IMSNAwe+nTolbRq3wpbXs9/O983s7KMauk6cMH/JjZ05O25MTJ7Ln79w8dLU9MzljThoRbZatwM3iLaqVqxcx1friZO4aiuMlOVVXbVZ3V3W45ttFcVO4K8lnVBte1bDd+qObSWAdqZD06k0q1ZkVsJau1t5uhLGTrfY63WTnnnj3rzaOIDMW+bjtFPqVSr544ilPtE8IJYGicVefmd6lhdYHnPYKKbGLKXPSjAz9pUqVKOAbGqRR4p8SmC7ZFGMt0xFYgqBbVMXWATLkXFFPcqD24KXgocFdBdtA71yivroa81Y2DaiuPgiME26nvrUYNcF3f/r+OaAbx6ex0fpirrOsoN/NVX1gCbUBHoys+97emaZ2lKFhswoob2DnGOpx+iIffUQb796o73D/1JOYO9lrEsXkT2o1mBH0M7WtfEO5qBrZB5BTf64qvgDv+cm59B3gM+l6xfzC17lLX7LL/kZP+cO19l/80u94ye0LAo2slDQUbwLHxvjm/yaH7LPDfe7NZWZnY/2ldQJfGnn8G9JbSPBs+owzM6uxWmVywOe86izA9SRSm0fUmljHbJU1nDuNKNfaSXVyppRfx668h5Gs09nAuW7EkGvWCjIyTEOn5q5ofM3mhnCvyP7wz5xPw9XafRpzJ7Jv1umLpomxgJpO/BUsPQK6T1oA3OB7iOW7Ewnvev0bm7KqbfET2cWZ8y1QOuCa6+C8Bw5dTp2gR4cuWu1T0206+gVqISXaRG7J0BUzfAk28EYqAlu+uLRe33Y2CgtFG8vLK4uzi7dT+/8SbpK1+gm9tcdWqJHtIJsbfpMf3LjOcP4ZHwzfhg/913HcinnCh16jN9/AW4GiD0=</latexit>

ih

@

r, t)
1 (~

r, t) + K
1 (~

= -eV

@t
@ 2 (~r, t)
ih
= eV
@t

r, t) + K
2 (~

r, t)
2 (~

r, t)
1 (~
K: 結合パラメーター

ψ1Insurator

S1
Superconductor

ψ2

S2
Superconductor
A
V

Josephson効果におけるシュレディンガー方程式
波動関数<latexit sha1_base64="hjBdEux8bVMGz53L25WsqSsSo2g=">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</latexit>r, t) = |1 (~r, t)| exp(i'1 (~r, t))
1 (~r, t) = |2 (~r, t)| exp(i'2 (~r, t))2 (~波動関数を先の式に代入✓◆

exp(i'1 ) = -eV|@'2@| 2 |-h| 2 |+ ih@t@t

exp(i'2 ) = eV|<latexit sha1_base64="QTGmSTpb+A9s4EfPq+HE4WtGXYA=">AAAHgXicnVTLbtNAFL1pgRTzaAsrxMaiQmrUtJlEFSAhREURKgJEK/WFkqp1nEky1LGN7YRGUVjAggWs+BW+BvEDbPkDztw4JQ/iVtjy+M6Zc8+9c+dR9h0VRkL8SE1NX7h4KT1z2bhy9dr12bn5G7uh1wxsuWN7jhfsl61QOsqVO5GKHLnvB9JqlB25Vz5e1+N7LRmEynO3o7YvDxpWzVVVZVsRoMO576V3UXvRMJdL9bIVlKxy2Cm92vRD1cl3u2bJr7S4X+d+J+oaS6rH5JFhOo+bGaMkT/xF1ffjkYzxCDHk7ohHj3kaIGMaSy8GKQWdwyCn0M1kjcO5BbEi+DHHjXxsLFD8bHrzUz+pRBXyyKYmNUiSSxFshywK8RYpT4J8YAfUARbAUjwuqUsGfJtgSTAsoMdoa+gVY9RFX2uG7G0jioMvgKdJd2NOBXaV0d5fxzcHuAaY/47SYXWdZRv/cqzaABpRHejZnn3u+T2L1OIq1HhGEZ2c5hxyPSZH7Kv7ePvVm8z2/0s5gn2SsC4dRG5AtQI7gHayro13MAddI3MENcW3LSm+ii+iLlLoK+DZeP1CcSS2xL74JN6K1+KNaIuqcD/+lp/FS1pnBRtZSOhIcQyOjfE98UE8E66oOb+s2cTsXLTvuU7w5zaLf5NrGzCeVIdx7+RanFe5OMBcRp0VUMWVOhhSaWEdklS2ce60R7/SkquVNKP+PHTlGxhNPp0RlB9wBL1iPiNnxxg+Ndmx8zfZ0we/zfvDPnM/j1dp8mlMnsnfW6bKmibGPG7bYEpYeoX0HrSBOUB7iMU7U8V3nd7NdT71FvN0ZmHCXHO0w7hm5dhP8anTsXP0dOSu1ZwKa1fRy1EBr6BV7B4PUbVHg7MdjIGa4KbPj97r48ZuYSV/b2V1a3Vh7Ul858/QbbpDi9hf92mNNmgT2dqpW6nHqY3U8/R0OpMW6UKPOpWKfW7S0JN++AfSCZ0i</latexit>@'1@| 1 |-h| 1 |
+ ih
@t
@t

✓
<latexit sha1_base64="kuK8LRuoA4+uJXuL19SSmJdabGU=">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</latexit><latexit sha1_base64="nkLisQwuCtshGKfluM/YyUslkww=">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</latexit>|

1 | exp(-i'1 ) と |<latexit sha1_base64="N/jWagrp6u30KtlgmB3vK0vnz+w=">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</latexit>◆

2 | exp(-i'2 )1 | exp(i'1 ) + K|

2 | exp(i'2 ),2 | exp(i'2 ) + K|

1 | exp(i'1 )を乗じて実数と虚数に分ける.

Josephson効果におけるシュレディンガー方程式<latexit sha1_base64="r+4m/T7nNR70SbH4jEG9JTTHt08=">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</latexit>

-h|

2 @'1

1|

= - eV|

2

1|

+ K|

1 ||

2 | ⇥ cos(@t2
h @| 1 |=K| 1 || 2 | sin( '),
2 @t
2 @'2
2
-h| 2 |
=eV| 2 | + K| 1 || 2 | cos( '),@t2h @| 2 |=K| 1 || 2 | sin( ').
2 @t

'),

⇥2e/h
<latexit sha1_base64="WNMuObDfDLn/IRHWjUq1lMqkTzY=">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</latexit>⇥2e/h<latexit sha1_base64="WNMuObDfDLn/IRHWjUq1lMqkTzY=">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</latexit>

2
2
|
|
@|
1
2
IS = eS= -eS
S: 超伝導体S1, S2の断面積
@t
@t
2e
=
K| 1 || 2 | sin( ') 直流Josephson効果h<latexit sha1_base64="bJNZlEYJ9k1rq12gy3djkSxY0rQ=">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</latexit>@|

位相差を与えるだけで印加電圧にかかわらず電流が流れる超伝導量子干渉
Ia

I = Ia + Ib = I0 (sin 'a + sin 'b )Insurator<latexit sha1_base64="pwzYnPR1UCTttfyT75v9FsXGZd0=">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</latexit><latexit sha1_base64="vKGx6xHx2q+adcSYZh3CvIhO5j8=">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</latexit>

2e'b - 'a =
h

I

C
I

2e
~ =
~ · dsAhΦSuperconductor

SuperconductorIb<latexit sha1_base64="bPBQ95ozdSA4xeIhtk7OxujYWmQ=">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</latexit>'0 = ('b - 'a )/2 とおくと,ee'a = '0 +, ' b = '0 ,hh⇣e ⌘I = I0 sin '0 coshJosephson
current<latexit sha1_base64="tO9u+60qtt320nMZcD/ybmiBYYQ=">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</latexit><latexit sha1_base64="fK98aD3cjRpDQ5Fio37OBg/mEQk=">AAAG+XicnVTJbhNBEK0EcEJY4sCRi0WEFESIy1EEXJAighCRkEikbMiOwrjdtpvMppmxiWWFAxw4wAX+gBviyoErfAbiB7jyB7yujMGJ8SRiRtNT/frVq+rqpRq6Jk6Yv4+Mnjp9Jjc2fnbi3PkLFyfzU5c24qAVKb2uAjeItqpOrF3j6/XEJK7eCiPteFVXb1Z3l+z4ZltHsQn8taQT6m3PafimbpSTANrJ31i+u7zDldj4lbYThU2DjgrimUo9clRX73crzaoT7VdWmub6Tn6a51iewqBRSo1pSp+VYGr0B1WoRgEpapFHmnxKYLvkUIy3TCViCoFtUxdYBMvIuKZ9moBvCywNhgN0F20DvXKK+uhbzVi8FaK4+CJ4FuhayqnBrgt68LfxC33cCTD/HaUr6jbLDv7VVNUDmlAT6PGePe7JPcvUlio0ZEYJ7f3JOZZ6DI/YUw/x9qo3nB3+l3ICey9jXbqI7EG1BjuCdrauwtufg61R4Qha4Permt/yG27yCPoG+Gy6fjE/5VXe4lf8jB/zE+5wnf2Xv/RrfkRLoqCQhYaO5l1wFMY3+QU/YJ8b7k9nMjM7H+1zqRP8pZ3FvyW1jQTPqsOgd3YtTqpc7mPeRJ0NUCOV2j6k0sY6ZKms4dxZj16ltVQra0a9edjKexjNPp0JlO9IBLtioSDHxzh8amYHzt9wzxD8juwPdex+HqzS8NOYPZO/t0xdNAsYC6TtgKlh2RWye1ABc4EeII7sTJPedXY3N+XUO8KzmcUZcy3SuuCWVRQ/I6fOxi7S/SN3reXURLuOXpHm8TItYPcEiGo9PMm2PwZqgpu+dPReHzQ25udKt+YWVhemF++ld/44XaGrNIP9dZsW6SGtIFtF7+gLfaVvuW7uQ+5j7tMBdXQk9blMh57c598b83DH</latexit>

I

Magnetic ﬁeld (mG)

まとめ
• 現象論
- ロンドン方程式: マイスナー効果, 磁場侵入深さ- ピッパード方程式: コヒーレンス長- GL理論: 秩序パラメーターを波動関数にとりいれる. 侵入深さの温度依存,
GLパラメーター• BCS理論- クーパー対が生成される- 臨界電流, 臨界磁場
- 様々なBCS理論による予言は実験で再現されている.
• Josephson効果- ベクトルポテンシャルの位相差によりJosephson電流が流れる