Macroscopic Aharonov--Bohm Effect in Type-I Superconductors

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by  Catrin Ellenberger

Europhysi
s Letters PREPRINT
5
0
Ma
ros
opi
Aharonov(cid:21)Bohm E(cid:27)e
t in Type-I Super
ondu
tors
0
2
Catrin Ellenberger, Florian Gebhard, Wolfgang Bestgen
n
a Fa
hberei
h Physik, Philipps(cid:21)Universität Marburg, D(cid:21)35032 Marburg, Germany
J

6
2
PACS. 73.23.-b (cid:21) Ele
troni
transport in mesos
opi
systems.

PACS. 74.25.Sv (cid:21) Criti
al
urrents.
]
n PACS. 74.25.Op (cid:21) Mixed states,
riti
al (cid:28)elds, and surfa
e sheaths.
o PACS. 74.40.+k (cid:21) Flu
tuations (noise,
haos, nonequilibrium super
ondu
tivity, lo
alization,
c et
.).
-
r
p
u
s
. Abstra
t. (cid:21) In type-I super
ondu
ting
ylinders bulk super
ondu
tivity is destroyed above
t
a the (cid:28)rst
riti
al
urrent. Below the se
ond
riti
al
urrent the `type-I mixed state' displays
m
(cid:29)u
tuation super
ondu
tivity whi
h
ontributes to the total
urrent. A magneti
(cid:29)ux on the
- axis of the
ylinder
an
hange the se
ond
riti
al
urrent by as mu
h as 50 per
ent so that
d half a (cid:29)ux quantum
an swit
h the
ylinder from normal
ondu
tion to super
ondu
tivity:
n
the Aharonov(cid:21)Bohm e(cid:27)e
t manifests itself in ma
ros
opi
ally large resistan
e
hanges of the
o

ylinder.
c
[

1
v
4
Introdu
tion. (cid:21) The super
ondu
ting
ondensate
an be used as a `quantum ampli(cid:28)er'.
2
6 Forexample,mi
ros
opi
e(cid:27)e
tssu
hasquantuminterferen
eofele
tronwavesbe
omema
ro-
1 s
opi
ally measurable as (cid:29)ux quantization. In this work the ma
ros
opi

ondensate is used
0 to make the Aharonov(cid:21)Bohme(cid:27)e
t prominent on ma
ros
opi
s
ales.
R J
5
Considerawireofradius witha
urrent paralleltothewireaxis. Thesuper
ondu
tiv-
0 J
/ ity of a type-I super
ondu
torbreaksdown when the
urrentex
eeds the Silsbee
urrent
1.
t The Silsbee
urrent is determined from the
ondition that the magneti
(cid:28)eld at the surfa
e
a H J = c0RH /2 c0
m of the wire is equal to the upper
riti
al (cid:28)eld
,
1
, where is the speed of
J >J
- light. For
1, the so-
alledintermediate state forms in the wire with alternatindg normal
d and super
ondu
ting domains [1℄. The diameter of the super
ondu
ting domains shrinks
d
n
and (cid:28)nally the intermediate state
ollapses when be
omes of the order of the
oheren
e
o ξ
length . However,super
ondu
tivitystill remainsin theformofsmalltemporal(cid:29)u
tuations.
c
: In this `mixed state' for type-I super
ondu
tors, super
ondu
ting (cid:29)u
tuations and ele
tri
al
v
and magneti
(cid:28)elds
o-exist. The mixed state was (cid:28)rst predi
ted by L.D. Landau in 1938 [2℄
i
X as asurfa
e statefor hollowsuper
ondu
ting
ylinders. Its existen
ewaslater veri(cid:28)edexper-
r imentally by I.L. Landau and Yu.V. Sharvin [3℄. As later shown by Andreev and Bestgen [4℄
a
the (cid:29)u
tuation indu
ed super
ondu
tivity is destroyed when the thi
kness of the (cid:29)u
tuation
ξ
layer be
omes smaller than the
oheren
e length , and super
ondu
tivity is ultimately de-
J
stroyed at the se
ond
riti
al
urrent
2 whi
h was (cid:28)rst
al
ulated by Andreev [5℄. An
H
externally applied magneti
(cid:28)eld a parallel to the wire axis
ontributes to the suppression
J J
of the super
ondu
ting phase and redu
es
1 and
2 a

ordingly [6,7℄.
(cid:13)

EDPS
ien
es
2 EUROPHYSICSLETTERS
In this work we investigate the stability of the mixed state in type-I super
ondu
tors for
R ≪R H
a hollow
ylinder of inner radius i in a magneti
(cid:28)eld a with an additional magneti
Φ
(cid:29)ux line of strength a on the
ylinder axis. We showJth(Φat)the Aharonov(cid:21)BohmΦ e/(cid:27)Φe0
t leads
to strong os
illations of the se
ond
riti
al
urrent
2 a as a fun
tion of a where
Φ0 =hc0/(2e)
isthe(cid:29)uxquantum. Thee(cid:27)e
tisparti
ularlymarkedforthe`one-dimensional'
R ≪ ξ J (Φ )
mixed state for whi
h i . In this
ase,
2 a
anJvar<y Jby(aΦlm0/o2s)t <50Jpe<r
Jent(.0)If the
applied
urrent through the wire is (cid:28)xed at some value
1
2
2 , the
Φ
modulation of between integer (cid:29)ux quanta will lead to a periodi
onset and breakdown of
J Φ Ω V
the super
ondu
ting
urrent s as a fun
tion of . The resistan
e at applied voltage
be
omes V V Jz
Ω= = ≈Ω 1− .
s
J J +Jz n J (1)
n s (cid:18) (cid:19)
Therefore,theAharonov(cid:21)Bohme(cid:27)e
tleadstoperiodi
(cid:29)u
tuationsin theresistan
eoftype-I
super
ondu
tors in the mixed state.
J ≃ J
Model and super
ondu
ting
urrent. (cid:21) In the mixed state with
2, the dynami
s
of the super
ondu
ting (cid:29)u
tuations at the inner
ylinder surfa
e is des
ribed by the time-
dependent, linear Ginzburg(cid:21)Landau equations with Langevin for
es. In dimensionless
ylin-
ρ,φ,z
dri
al
oordinates they read [7(cid:21)9℄
∂Ψ
−Ψ+LΨ=f(ρ,ϕ,z,t)
∂t , (2)
with the di(cid:27)erential operator
2 2
1 ∂ ∂ 1 ∂ iγ iδ ∂ iβ
2
L=− ρ − − ρ− − + ρ +iǫt
ρ∂ρ ∂ρ ρ∂ρ 2 ρ ∂z 2 (3)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
and the Gaussian (cid:29)u
tuations
1
hf(ρ,ϕ,z,t)f∗(ρ′,ϕ′,z′,t′)i= δ(ρ−ρ′)δ(ϕ−ϕ′)δ(z−z′)δ(t−t′)
ρ (4)
ǫ β γ δ
The dimensionless parameters , , and are given by
2eξ 2eH ξ2 H
ǫ= E γ = a = a
¯hν , ¯hc0 H ,
4eξ3J eΦ Φ
2
β = ¯h 2R2 , δ = ¯hc0aπ = Φa0 . (5)

0
ξ = ξ(T) ν = 8k (T −T)/(π¯h) e
is the
oheren
e length, B
is the relaxation frequen
y, and
m E
and aretheele
tron
hargeandmass,respe
tively. istheele
tri
al(cid:28)eldwhi
hdrivesthe
J = J +Jz z
ǫto≪tal1
urrent n s in -dire
tion. For type-I super
ondu
torsJwza≪s sJhown in [4℄ that
. Using the linearized Ginzburg(cid:21)Landau equations implies that s .
Thegeneralexpressionforthetotallongitudinal(cid:29)u
tuation
urrentis
al
ulatedalongthe
lines of [4,9℄ and gives the result
∞
2eTk ∂λ
nmk
Jzs = πǫ¯hB dk ∂k Fnm(k), (6)
nm Z
X−∞
Catrin Ellenberger et al.: Ma
ros
opi
Aharonov(cid:21)Bohm Effe
t 3
with ∞ k
2
Fnm(k)= dτ exp τ − λnmk′ dk′
ǫ   . (7)
Z0 k−Zτ
  
λ L k = k +ǫt k
nmk z z
is the lowest eigenvalue of as a fun
tion of where is the wavenumber
z λ
nmk
ve
tor in -dire
tion. The spe
trum Rn=0
,mank b∼eρo|bmt|aei−nαedρ2with the help of a variψational
method, whi
h starts from the ansatz for the radial part of . We
n=0 α
may restri
t ourselves to the dominant
ontribution, [10℄; is a variational parameter
whi
hdeterminestheradialextensionofthesuper
ondu
tingregion[4,9℄. Fromthevariation
prin
iple we then (cid:28)nd for λm,km =λo0p,mt,k
β2
λ =−(m−δ)γ+4α(|m−δ|+1)− (|m−δ|+1),
mkm 16α2 (8)
with
γ2+4k β β2(|m−δ|+2)
m
2− − =0
8α2 8α3 , (9)
−β(|m−δ|+1)
k =
m
4α . (10)
The
riti
al
urrent
an be evaluated analyti
ally in the vi
inity of the
riti
al
urve [8℄
Jz
but for our further analysis expli
it expressions for s are not required. We merely state
that the experimental parameters
an be tuned in su
h a way that the (cid:29)u
tuation
urrent is
severalper
entofthe total
urrent,i.e., the resistan
e(cid:29)u
tuationsa

ordingto (1) shouldbe
measurable.
J (H ,Φ )
Se
ond
riti
al
urrent. (cid:21) Inourtheory, the
riti
al
urrent
2 a a abovewhi
hthe
super
ondu
ting (cid:29)u
tuations vanish is determined by the
ondition
[λ (β,γ,δ)]=1.
Mmin m,km (11)
λ
If the minimum over all m,km was noti
eably smaller than unity the
orresponding super-

urrentwouldbe
omesolargethat theuseofalinearizedGinzburg(cid:21)Landauequationswould
λ
not have been justi(cid:28)ed. If the minimum over all m,km was substantially larger than unity
Jz
s wouldbe tiny. Our theory appliesto the transition regionwhere the super
urrentis small
but measurable.
J
Solving (11) for
2 leads to the se
ond
riti
al
urrent as a fun
tion of the external
magneti
(cid:28)eld and applied (cid:29)ux in the interior of the
ylinder. In SI units we (cid:28)nd:
H Φ c2R2¯h
0
J
2 Ha Φ0 = β
2 4eξ3
(cid:18)
2 (cid:19) 3
c2R2¯h H 2 1
0 3
= eξ3 "A 1+s1−(cid:18)H
a2(cid:19) 48A2 (12)
2 2  1/2
H A H 1
−(cid:18)H
a2(cid:19) 161+s1−(cid:18)H
a2(cid:19) 48A2# ,
 
where 1+(m−δ)γ
A=
6(|m−δ|+1) . (13)
4 EUROPHYSICSLETTERS
δ m J
For a given (cid:29)ux , in (13) is the integer whi
h gives the minimal
2 [8℄. Figure 1 gives a
J γ δ
three-dimensional plot of
2 as a fun
tion of (external (cid:28)eld) and (
entral (cid:29)ux).
β (γ,δ) R ≪ξ
Figure 1 (cid:21) The normalizedse
ond
riti
al
urrent
2 for i .
The se
ond
riti
al
urrent
an be almost halved by the appli
ation of a suitable ve
-
tor potential even if no magneti
(cid:28)eld is present. This is a ma
ros
opi
realization of the
Aharonov-Bohm e(cid:27)e
t.
Con
lusions. (cid:21) In this work we have studied the super
urrent in a hollow
ylinder of
type-I super
ondu
ting material in the presen
e of an external magneti
(cid:28)eld and a (cid:29)ux line
onthe
ylinderaxis. We haveshownthat the se
ond
riti
al
urrent, abovewhi
h(cid:29)u
tuation
super
ondu
tivity breaks down, periodi
ally
hanges as a fun
tion of the applied (cid:29)ux, by as
mu
h as 50 per
ent in the absen
e of an external (cid:28)eld. These Aharonov(cid:21)Bohm os
illations
lead to a periodi

hange of the resistan
e of the
ylinder, see (1). The latter are measur-
able be
ause the
urrents due to the super
ondu
ting (cid:29)u
tuations are sizable, of the order
of a few per
ent of the total
urrent. Therefore, a suitably tuned experiment with type-I
super
ondu
ting
ylindersshouldbeabletoverifytheAharonov(cid:21)Bohme(cid:27)e
tbyma
ros
opi
resistan
e measurement.
∗∗∗
We thank F. Rothen and M. Pae
h for useful dis
ussions.
Referen
es
[1℄ London F., Super(cid:29)uids I.(Wiley, New York)1950.
[2℄ LandauL.D.,private
ommuni
ationwithD.Shoenberg;see: D.ShoenbergSuper
ondu
tivity.
Cambridge: Cambridge UniversityPress, 1938; page 59.
Catrin Ellenberger et al.: Ma
ros
opi
Aharonov(cid:21)Bohm Effe
t 5
[3℄ Landau I.L.andSharvin Yu.V.,Pis'maZh.Eksp.Teor.Fiz.,10(1969)192andJETPLett.,
10 (1969) 121.
[4℄ Andreev A.F. and Bestgen W., Zh. Eksp. Teor. Fiz., 10 (1973) 453 and Sov. Phys. JETP,
37 (1973) 942.
[5℄ Andreev A.F., Pis'ma Zh. Eksp. Teor. Fiz., 10 (1969) 453 and JETP Lett., 10 (1969) 291.
[6℄ Bestgen W., Zh. Eksp. Teor. Fiz.,76 (1979) 566 and Sov. Phys. JETP, 49 (1979) 285.
[7℄ Bestgen W., Sol. State Commun.,36 (1980) 441.
[8℄ Ellenberger C., Ph.D. Thesis, Marburg 2001.
[9℄ Bestgen W., Z. Phys., 269 (1974) 73.
[10℄ Bestgen W. and Rothen F., Low. T. Phys., 77 (1989) 257.