Table Of ContentVortex phases in mesoscopic cylinders with
suppressed surface superconductivity
W. V. Pogosov
Moscow Institute of Physics and Technology,
141700 Dolgoprudnyi, Moscow region, Russia
2
0
0
2 sample (multivortex phase). This state usually appears
n Vortex structures in mesoscopic cylinder placed in exter- at lower fields and larger sample sizes as compared to
nal magnetic field are studied under the general de Gennes
a thegiant-vortexphase[7,8,10,11]. Notethatmultivortex
J boundaryconditionfortheorderparametercorrespondingto state corresponds to the Abrikosov flux-line lattice for
the suppression of surface superconductivity. The Ginzburg-
8 the bulk superconductors.
Landau equations are solved based on trial functions for the
2 The phase diagram of mesoscopic superconductor is
order parameter for vortex-free, single-vortex, multivortex,
strongly influenced by the boundary condition for the
] andgiant vortexphases. Theequilibriumvortexdiagrams in
n order parameter. In general case it is given by the de
the plane of external field and cylinder radius and magneti-
o Gennes boundary condition [18,19]:
zation curves are calculated at different values of de Gennes
c
”extrapolationlength”characterizingtheboundarycondition
- i
r fortheorderparameter. Thecomparisonoftheobtainedvari- n(−i∇−A)ψ = ψ, (1)
p b
ationalresultswithsomeavailableexactsolutionsshowsgood
u
s accuracy of our approach. where n is the unit vector normal to the sample sur-
at. PACS numbers: 74.60.Ec, 74.24.Ha face, b is the de Gennes ”extrapolation length”, A is
the vector potential, and ψ is the order parameter. Here
m
andbelowthefollowingdimensionlessvariablesareused:
- distances, magnetic field, and the order parameter are
d
measured in units of coherence length ξ(T), bulk upper
n
o critical field Hc2, and −α/β, respectively, with α, β
c I. INTRODUCTION being the Ginzburg-Landau coefficients. The ”extrapo-
[ p
lation length” b has a physical meaning of a length scale
2 Recent achievements in electronic device miniaturiza- of the order parameter variation at the sample surface.
v tion allow one to study the mesoscopic superconducting Microscopic considerations show that b depends on the
2 samples with sizes of the order of the coherence length properties of interface, it is maximum for an ideal sur-
9 ξ(T). Such structures attract a considerable current in- facewiththemirrorreflectionofquasi-particlesandmin-
4
terestasapossiblebasisforlowtemperatureelectronics. imum for the rough surface with the diffusive reflection
8
The superconducting state was studied experimentally [18–21]. For the superconductor-dielectric (or vacuum)
0
1 for different-shaped samples: discs, loops, double loops, interface we haveb→∞in the former caseand b∼ξ(0)
0 dots etc. [1–4]. It was shown that the sample shape and inthelattercase. Thevalueofbdependsalsoonthesur-
/ sizes affect significantly the phase diagrams of the meso- face orientation provided that a superconducting gap is
t
a scopic superconductors. anisotropic. ItfollowsfromEq. (1)thattheorderparam-
m
The vortex phases in mesoscopic superconductors are eteris suppressedinthe vicinity ofthe sample surfaceat
- commonlystudiedwithintheframeworkoftheGinzburg- b∼ξ(0). For the superconductor-normalmetal interface
d
Landau theory [5–17]. As it is well-known from the mi- b is always small, b ∼ ξ(0), because of diffusion of nor-
n
croscopic theory, the Ginzburg-Landau approach gives malelectronsfromthemetaltothesuperconductor. The
o
c accurate results provided that the order parameter un- ”extrapolationlength”binthiscaseisafunctionoftem-
: dergoes only slight spatial variation on the lenght scale perature and various characteristics of metal and inter-
v
of ξ(0). This means that the Ginzburg-Landau theory face[19–23]. There arealsopossibilities for the enhance-
i
X can be used in the temperature range not far from T . ment of the order parameter at the interface that can
c
r However, it is known from the experience that it is also be described by negative b values. It can be realized by
a able to give reasonable results beyond this limit. The choosing the suitable material as a surrounding medium
Ginzburg-Landau solutions for axially symmetric meso- [24,25], i.e., a superconductor having a higher transition
scopic samples (cylinders, discs) can be subdivided into temperature then the material of the mesoscopic sam-
two different types [6–11]. In the first case the modulus ple. Another possibility is to use a semiconductor as
of the local order parameter is axially symmetric inside a surrounding medium, such that there is a overlap of
the sample. The superconducting vortex-free state, the the band gap of the semiconductor with the supercon-
single-vortex state, and the giant-vortex state belong to ducting gap. For the case of isotropic superconductor-
this type of solutions. In the second case the axial sym- vacuum ideal interface (b → ∞), the magnetic prop-
metry is brokenandavortexcluster is formedinside the erties of mesoscopic cylinders and discs, their equilib-
1
rium and non-equilibrium phase diagrams were studied H(R)=H . (4)
e
in numerous papers, see e.g. Refs. [5–15], using differ-
ent approaches. The vortex structures allowing for the Next, we expand all variables in powers of κ:
enhancedsurfacesuperconductivitywerestudiedinRefs.
∞ ∞ ∞
[s2id4e,2re5d]. wInitRhienf.[t2h4e]tlohweecsatseLoafnmdaeusolsecvoeplicapdpisrcosxwimasatcioonn-, ψ = ψ2nκ12n, A= A2nκ12n, H = H2nκ12n.
n=0 n=0 n=0
which first was proposed in Refs. [7,10]. In Ref. [25] the X X X
Ginzburg-Landauequationsweresolvednumericallyand (5)
self-consistently for superconducting state in long cylin-
We substitute expansions (5) to Ginzburg-Landau
ders.
equations (2), (3) and to boundary conditions (1), (4)
Inthispaperwefocusonmagneticpropertiesofmeso-
and equate powers of κ. It is easy to show that the vec-
scopiccylindersunderthegeneralboundaryconditionfor
tor potential and the magnetic field at leading order are
the order parameter corresponding to the opposite case
given by:
of suppressed surface superconductivity. For this pur-
pose, we propose a variational approach and solve the
H r
Ginzburg-Landau equations without straightforward in- A0 =eφ e , H0 =He. (6)
2
tegration using trial functions for the order parameter.
These trial functions involve a set of variational param- Theorderparameteratleadingorderψ0isdeterminedby
eters yielding, as we show, more accurate quantitative thefirstGinzburg-Landauequation(2)andtheboundary
description of the spatial distribution of the order pa- condition (1) at H = H0. This condition is now given
rameterthanfrequentlyusedlowestLandaulevelapprox- by:
imation. The approachis applicable to all vortex phases
(the vortex-free, the single-vortex, the multivortex, and ∂ψ0 ψ0
+ =0. (7)
thegiantvortexstates)andanyvaluesofdeGennes”ex- ∂r b
trapolation length”. The comparison of variational cal-
In the next order one has from Eq. (3):
culations with some availableexactresults demonstrates
goodaccuracyof our approximation. The model enables
r i
ustocalculatetheequilibriumHe-Rdiagramofthecylin- H2 =− dreϕ |ψ0|2A0+ 2(ψ0∗∇ψ0−ψ0∗∇ψ0) . (8)
der, where He is the external field and R is the cylinder Z0 (cid:20) (cid:21)
radius. Themagnetizationcurvesofthecylinderarecal-
Thus, at leading order the magnetic field is uniform in-
culated.
sidethesampleandthemagnetizationequalszero. Phys-
ically, this implies that the additional field generated by
the Meissner current and by the vortices is of the order
II. MODEL
of 1/κ2 in comparison with the external uniform field.
Below, we shall calculate the energy of the sample at
Let us consider a cylindrical type-II superconductor
leading order, whereas the magnetization will be found
placedinthe uniformexternalmagneticfieldH parallel
e atthenextorderusingEq. (8). ItwasshowninRef.[27]
to the cylinder axis. The sample is assumed to be much
that this approximation is accurate not only for high-κ
longer than London penetration depth λ(T). Therefore,
materials but also for moderate-κ superconductors (e.g.
boththeorderparameterandthemagneticfieldarecon-
κ≈4) with sizes comparable to ξ(T).
stant along cylinder axis. We use the cylindrical coordi-
natesystemwithcoordinatesr, ϕ,z andunitvectorse , We present ψ0 as a Fourier series:
r
e , e . ∞
ϕ z
The system of Ginzburg-Landauequations is given by ψ0(r,ϕ)= fLj(r)exp(−iLjϕ). (9)
[26]: LXj=0
|ψ|2ψ−ψ+(i∇+A)2ψ =0, (2) For the axial symmetric distribution of the modulus of
the orderparameterinside the sample the only one term
inEq. (9)isnonzero. Thevortex-free,single-vortex,and
1 i giantvortexphaseswith angularquantummomentumL
rot H= A|ψ|2+ (ψ∗∇ψ−ψ∗∇ψ) , (3)
κ2 2 correspond to the harmonics with Lj = 0, 1, and L, re-
(cid:20) (cid:21)
spectively. The modulus of the order parameter in the
whereHandψ arethedimensionlesslocalmagneticfield multivortex phase is not axially symmetric. Note that
and the order parameter (H = rot A, H = He ); κ = in this case the symmetry of the vortex configuration
z
λ(T)/ξ(T) is the Ginzburg-Landau parameter. Equa- imposes a restriction on functions f : some of these
Lj
tions(2)and(3)mustbesupplementedbytheboundary functions equal zero. It was shown in Refs. [7,10] that
conditions for the order parameter (1) and the magnetic taking into account only two main terms in right-hand
field: side of Eq. (9) is enough for an accurate calculation of
2
theenergyofthemesoscopicsuperconductorinmultivor- contributions,G andG . The formeris the bulk energy
b s
tex state. The vortex cluster with L vortices on a ring of the sample and the latter is the surface energy. These
andnovortexatthe axis(ring-likeconfiguration)canbe contributions are given by [20,23,26]:
described as a mixture of two components with L1 = 0
and L2 = L ((0 : L) state). The vortex cluster with one G = −|ψ|2+ 1|ψ|4+|(−i∇−A)ψ|2+
vortexat the cylinder axis and (L−1) vorticesona ring b
2
"
corresponds to L1 = 1 and L2 = L ((1 : L) state). The Z
contribution from other harmonics is small and can be
κ2H2−2κ2HH dV, (13)
neglected, when we consider few-fluxoid cylinders that e
#
can accommodate just few vortices [7,10].
UsingEqs. (2)and(9)itcanbeeasilyshownthateach
function fLj(r) has the following asymptotic at r →0: G = 1 |ψ|2dS. (14)
s
b
fLj(r)∼rLj. (10) Z
The integration in Eqs. (13) and (14) is performed over
Besides, each function f (r) meets boundary condition
Lj the sample bulk and surface, respectively. Note that the
(7). These conditions for f (r) are valid both for the
Lj general boundary condition for the order parameter (1)
giant vortex and the multivortex phases. It is a rather
can be obtained phenomenologically by minimization of
complicated task to find f (r) from the first Ginzburg-
Lj thefreeenergyfunctionalG=G +G withrespecttothe
Landau equation (2) due to its non-linearity. Instead of b s
orderparameterψ andthe vectorpotentialA[20,23,33].
thestraightforwardintegrationofthisequation,itispos-
Substituting expansion (9) to Eqs. (13) and (14) and
sibleto usetrialfunctionsforthe coordinatedependence
taking into account Eq. (6) we obtain the expression for
f (r). Notice that different variational procedures al-
Lj the energy of the multivortex state (per unit length of
lowing one to solve approximately the Ginzburg-Landau
the cylinder):
equations were used in numerous papers for mesoscopic
[7,10,15], bulk [28–31], and different-shaped [32] super-
R 1
conductors. OnecaneasilyshowthatifEq. (9)isasolu- G =2π rdr f4 +f4 +4f2 f2 −
tionofthefirstGinzburg-Landauequation,eachfunction b Z0 "2 L1 L2 L1 L2
fLj(r) can be represented as the following series expan- (cid:0)df 2 df 2 (cid:1)
sion in powers of r/R without loss of generality: f2 −f2 + L1 + L2 −κ2H2+
L1 L1 dr dr e
fLj(r)=exp(cid:18)−qLjRr22(cid:19)uX∞=0pLuj(cid:16)Rr(cid:17)Lj+2u, (11) fL21(cid:18)H2er −(cid:18)Lr1(cid:19)2(cid:19)+fL22(cid:18)(cid:18)H2er(cid:19)− Lr2(cid:19)2#, (15)
where pLj are constants. The value of q can be found
u Lj
from boundary condition (7): 2πR
G = f2 (R)+f2 (R) . (16)
R 1 ∞ pLj(L +2u) s b L1 L2
q = + u=0 u j . (12) (cid:0) (cid:1)
Lj 2b 2 ∞ pLj The energy of any state having axially symmetric mod-
P u=0 u ulus of the order parameter with angular momentum L
Our approach is to considerPthe coefficients pLj as vari- reduces to Eqs. (15) and (16). In this case we must put
u
ationalparameters and to minimize the free energy with L1 = L, fL2 = 0. Using Eqs. (11) and (12) we find
respect to pLj. The exponential prefactor in Eq. (11) the energy G from Eqs. (15) and (16) by a straight-
u
forward integration as an explicit function of variational
takes into account the suppression of the order param-
parameters pL1, pL1, pL1 and pL2, pL2, pL2. The result-
eter at the contact with a surrounding material. First 0 1 2 0 1 2
ing expression, however, is rather cumbersome and we
term in the expansion describes the behavior of the or-
do not present it here. Finally, values of the variational
der parameter in the central part of the sample. Other
parametersat eachR are found numerically by the min-
terms specify the order parameter in the sample as a
imization of the free energy. This procedure yields the
whole. As it will be shown below, taking into account
local order parameter and the energy of the cylinder.
only first three terms in Eq. (11) is enough for the ac-
Knowing the local order parameter we can calculate
curate calculation of the order parameter distribution,
the magnetization. It is given by:
and we will use this approximation in all calculations of
magnetizationandH -Rdiagrams. Note that trialfunc-
e −4πM =<H >−H , (17)
tion (11) was used in Ref. [32] for the study of surface e
superconductivityinsamplesofdifferentcomplexshapes
where < H > is the averaged magnetic field over the
placed in vacuum (b→∞).
superconductor volume. Taking into account Eq. (8)
TheGinzburg-LandaufunctionalfortheGibbsfreeen-
and expansion (9) we obtain:
ergy G of the cylinder can be written as a sum of two
3
−4πM = 2 Rrdr rdx f2 (x) Hex − L1 + anydesiredaccuracy(forthestateswithaxiallysymmet-
κ2R2 L1 2 x ric modulus of the order parameter)allowing for enough
Z0 Z0 " (cid:18) (cid:19) number of variational parameters pLj. For multivortex
u
f2 (x) Hex − L2 . (18) statestheaccuracyislimitedbythefactthatwetakeinto
L2 2 x account only two main harmonics in the Fourier expan-
(cid:18) (cid:19)#
sion(9). InFig. 2wealsoplottedthespatialvariationof
Functionsf andf arefoundbythemethoddescribed theorderparametercalculatedwithinthelowestLandau
L1 L2
above. Hence,onecancalculatethe magnetizationusing level approximation (dashed lines). In this approxima-
Eq. (18). Inthefollowingsectionweapplythedeveloped tion the order parameter is proportional to the eigen-
approach for the analysis of the behavior of the cylinder function of the kinetic energy operator corresponding to
in the external field. the lowest eigenvalue. We found that this approach re-
mains very accurate not far from Hc2 and Hc3 (see Fig.
2(a)). At lower fields the results of the lowest Landau
III. RESULTS AND DISCUSSION level approximation are not so accurate (Fig. 2(b)).
AsfollowsfromFig. 1,thesuperconductingstatedoes
Comparingthe energiesofdifferent states onecancal- not nucleate at very small cylinder radii, smaller than
culatetheequilibriumH -Rdiagramofthecylinder. The some critical radius, and the sample is in normal state
e
results of our calculations are shown in Fig. 1 for differ- at any applied field. The critical radius tends to zero at
ent b values: b = 1 (a), b = 2.5 (b), b = 5 (c), and b → ∞. There is also the interval of R for each b, when
b→∞ (d). The latter case corresponds to the isotropic thevortexphasedoesnotnucleate,andthetransitionoc-
superconductor-vacuum ideal interface, and was studied curs from the normalto the superconducting vortex-free
in Refs. [5,6,10,11,13–15]. Curve 1 shows the transition state. Every vortex phase with vorticity L>1 can exist
from the normal to the superconducting state (the sur- inthe formofthe giantvortexorthe multivortexconfig-
face critical field Hc3). The oscillatory behavior of the uration. ThedashedlinesonFig. 1showthe boundaries
function Hc3(R) is caused by the fact that the transi- between these states. Below these lines for given L > 1
tion occurs from the normal to the giant vortex states the multivortex state has the lowest energy, and above
with different angular quantum moments L depending these curves the giant vortex state becomes more ener-
on the cylinder radius. Besides, the function Hc3(R) de- getically favorable. In equilibrium state the multivortex
pends appreciablyonthe value ofb: the value ofHc3(R) phasecanexistifthe appliedfieldis smallerthan1 (Hc2
decreases with decrease of b. At R→∞ the dependence in dimensional units) and if the radius of the cylinder is
Hc3(R)tendstothesurfacecriticalfieldforthehalf-space largeenough. For eachb there exists anintervalof small
sample, which was calculated in Ref. [22] as a function cylinder radii, when the multivortex phase is energeti-
of b. Note that Hc3 (∞)=1.695for b→∞ and Hc3(∞) cally unfavorable as compared to the states with axial
=1 for b→0 [18,22]. symmetric distribution of the modulus of the order pa-
Below Hc3 the transitions between different giant vor- rameter. Curve 2 in Fig. 1 shows the lower critical field
tex states take place. Solid lines show the phase bound- that corresponds to the equilibrium boundary between
aries between the states with different vorticities that the vortex-free and the single-vortex states.
arethesumsofangularquantummomentsofallvortices With increasingthe externalfield the cylinder can fol-
and giant vortices. In the giant vortex state the order low rather complex set of phase transitions. It can come
parameteris stronglysuppressedin the inner part ofthe fromthe giantvortex to the multivortex states and then
cylinder, and this state can be referred to a surface su- backtothe giantvortexphase. Forexample,atR=4.4,
perconductivity. Forillustration,the spatialdistribution b = 1 the following set of transitions occurs: 0 → 1 →
of the order parameter in the giant vortex phase with (0:2)→2→(0:3)→3→(0:4)→4. At R=4.1, b=∞
L = 2 is plotted in Fig. 2 at b = 1, R = 3.9, H = 0.9 the set of transitions is: 0 → 1 → 2 → (0:3) → 3 →
e
(a) and b →∞, R = 4.56, H = 0.5 (b) (solid lines). In (0:4)→4→5→6→7→8→9→10→11.
e
the former case the order parameter is also suppressed The transitions between the phases with different vor-
at the sample surface because of the small b value (e.g., ticities are always discontinuous, they occur when the
superconductor-normal metal interface). To check the energies of different states become equal. The transi-
accuracyof our approachwe took into accountnext sev- tions between the states with the same vorticity may be
eralterms inexpansion(11), thus increasingthe number continuous as well as discontinuous. Continuous phase
of variational parameters. As we found, this practically transitions occur between the multivortex and the gi-
didnotchangethecalculatedorderparameterforalmost ant vortex states. In this case, with increasing of the
all points of H -R diagram shown in Fig. 1. This result applied field the intervortexdistances decrease, and vor-
e
implies that our variational calculations are close to the tices merge into the giant vortex located at the cylinder
exact solutions of the Ginzburg-Landau equations since axis. The similar results were obtained in Refs. [7–11]
expansion (11) is written without any loss of generality. for the case b = ∞ and in Ref. [24] at b < 0 (enhanced
Therefore, one can find Ginzburg-Landau solution with surface superconductivity). In our calculations we also
4
took into account a possibility of existence of the mul- earized. Aresultingequationhasthefollowinganalytical
tivortex clusters with central vortex. It turned out that solution [5]:
such configurations can be more favorable energetically
thangiantvortexstatesinsomeregionsofHe-Rdiagram f (R)=rLexp −Her2 Φ He−1,L+1,Her2 ,
shownin Fig. 1. However,in allthese cases the ring-like L 4 2H 2
(cid:18) (cid:19) (cid:18) e (cid:19)
clustershavethelowerenergy. Forcylindersthickerthan
(19)
shown in Fig. 1 we found that at any b > 0 the ground
state can be represented by configurations with the cen-
where Φ is Kummer function. Function (19) must meet
tral vortex. The transitions between multivortex states
theboundarycondition(7). Thisyieldsthetranscenden-
ofthesamevorticitywithandwithoutcentralvortexare
tal equation for the surface critical field allowing one to
discontinuous.
findHc3(R)exactly. Thecomparisonofthevariationally
With increasing of the cylinder radius the number of
calculated Hc3 with this exact dependence shows good
vortices, which it is able to accomodate, increases, and
agreementwithanaccuracybetterthan1%forallvalues
finallyvortexarraytransformsto the classicaltriangular
of b and R under study. The lower critical field of the
Abrikosovflux-line lattice far fromthe surface. The first
cylinderHc1 versusRwascalculatednumericallyinRef.
step on this way is the appearance of the clusters with
[12] at b → ∞. The comparison of this result with our
centralvortex. However,wedonotanalyzeherethetran-
dependenceHc1(R)revealsthesameaccuracy. Thus,our
sition from the mesoscopic to the macroscopic behavior
results appear to be a good approximation to the exact
andrestrictourselvesonfew-fluxoidcylinders,whichcan
Ginzburg-Landausolutions for the mesoscopiccylinders.
accomodateonlyfewvorticesbeforethethansitiontothe
Insummary,weanalyzedthesuperconductingstatein
normal state and whose magnetic properties can be de-
long mesoscopic cylinder with suppressed surface super-
scribedintermsofmixture ofonlytwoharmonicsinEq.
conductivity. An asymptotic expansionwasused to sim-
(9).
plifytheGinzburg-Landauequationsathighandmoder-
Now we find the magnetization of the cylinder using
ate values of κ, and the simplified equations were solved
Eq. (18). The results of our calculations at κ = 5 are
by variational method. The equilibrium H -R diagram
e
presented in Fig. 3(a) for b = 1, R = 4.625 and in Fig.
of the cylinder were obtained, where H is the external
e
4(a) for b → ∞, R = 4.05. In these cases the cylinders
field and R is the cylinder radius, at different values of
accommodatesthegiantvorticeswithmaximumangular
”extrapolation length”. We showed that magnetic prop-
quantummomentsequalto5and11,respectively,before
erties of the cylinder depend appreciably on the value of
the transition to the normal state. Jumps in the magne-
”extrapolationlength”andstudiedtheevolutionofH -R
e
tization correspondto the transitions between the states
diagram with changing of ”extrapolationlength”.
with different vorticity. It is interesting that in the state
with suppressed surface superconductivity (Fig. 3(a))
the discontinuity of the field dependence of the magne- ACKNOWLEDGMENTS
tization is less pronounced, especially at high fields. In
thefirstcase(Fig. 4(a))atHe ≈0.61(0.61Hc2indimen-
The author acknowledges useful discussions with K.
sional units) the transition occurs from the multivortex
I. Kugel, A. L. Rakhmanov, and E. A. Shapoval. This
statewith2 vorticestothe giantvortexstatewithangu-
workwassupportedbytheRussianFoundationforBasic
larquantummomentum2(2→(0:2)). Inthesecondcase
Research (RFBR), grants Nos. 00-02-18032 and 01-02-
(Fig. 4(b)) the transition 3→(0:3) occurs at H ≈0.74
e
06526, by the joint INTAS-RFBR program, grant No.
and the transition 4 → (0:4) occurs at H ≈ 0.87. All
e
IR-97-1394, and by the Russian State Program ’Funda-
thesetransitionsarefollowedbyweakjumpsinthe slope
mental Problems in Condensed Matter Physics’.
ofthemagnetization. Thebehaviorofthemagnetization
nearthetransitionsisshowninFig. 3(b)and4(b). Solid
lines denote the equilibrium magnetization, dashed lines
denote the metastable magnetization corresponding to
thegiantvortexphase. Thesimilarbehaviorofthemag-
netization in the vicinity of the multivortex-giantvortex
transitions wasreportedin Ref. [34]for the caseb→∞.
[1] V.V.Moshchalkov,L.Gielen,C.Strunk,R.Jonckheere,
Next we discuss the accuracy of our variational pro-
X. Qiu,C. VanHaesendonck, and Y.Bruynseraede, Na-
cedure. First, we compare the variational results with ture 373, 319 (1995).
known exact solutions for the surface critical field. The
[2] A.K.Geim,I.V.Grigorieva,S.V.Dubonos,J.G.S.Lok,
phases with axialsymmetric distribution of the modulus
J. C. Maan, A. E. Filippov, and F. M. Peeters, Nature
of the order parameter are always energetically more fa- 390, 259 (1997).
vorable with respect to the multivortex state at applied [3] A. K. Geim, S. V. Dubonos, I. V. Grigorieva, K. S.
field higher than the bulk upper critical field (see phase Novoselov, F. M. Peeters, and V. A. Schweigert, Nature
diagramsonFig. 1). Inthevicinityofthesurfacecritical 407, 55 (2000).
field the first Ginzburg-Landau equation (1) can be lin-
5
[4] V.Bruyndoncx,C. Strunk,and V.V.Moshchalkov, Eu-
rophys.Lett. 36, 449 (1996).
[5] V. V. Moshchalkov, X. G. Qiu, and V. Bruyndoncx,
Phys.Rev.B 55, 11793 (1997).
[6] V. A. Schweigert and F. M. Peeters, Phys. Rev. B 57,
13817 (1998).
[7] V. A. Schweigert, F. M. Peeters, and P. S. Deo, Phys.
Rev.Lett. 81, 2783 (1998).
[8] V.A.SchweigertandF.M.Peeters,Phys.Rev.Lett.83,
2409 (1999).
[9] A. K. Geim, S. V. Dubonos, and J. J. Palacios, Phys.
Rev.Lett. 85, 1528 (2000).
[10] J. J. Palacios, Phys.Rev.B 52, R5948 (1998).
[11] J. J. Palacios, Phys.Rev.Lett. 84, 1796 (2000).
[12] E. A. Shapoval, Pis’ma Zh. Eksp. Teor. Fiz. 69, 532
(1999) [JETP Lett. 69, 577 (1999)].
[13] G.F.Zharkov,V.G.Zharkov,andA.Yu.Zvetkov,Phys.
Rev.B 61, 12293 (2000).
[14] G. F. Zharkov,Phys. Rev.B 63, 224513 (2001).
[15] W. V. Pogosov, A. L. Rakhmanov, and E. A. Shapoval,
Physica C 356, 225 (2001).
[16] V. R. Misko, V. M. Fomin, J. T. Devreese, and V. V.
Moshchalkov, Solid StateCommun. 114, 499 (2000).
[17] V.M.Fomin,J.T.Devreese,V.Bruyndoncx,andV.V.
Moshchalkov, Phys.Rev. B 62, 9186 (2000).
[18] P.G.deGennes, Superconductivity of Metals and Alloys
(Addison-Wesley,New York,1994).
[19] P. G. de Gennes and J. Matricon, Rev. Mod. Phys. 36,
45 (1964).
[20] E. A. Andrushin, V. L. Ginzburg, and A. P. Silin, Usp.
Fiz. Nauk 163, 105 (1997) [Phys. Usp. 36, 854 (1993)].
[21] R.O.Zaitsev,Zh.Eksp.Teor.Fiz.48,1759(1965)[Sov.
Phys.JETP 21, 1178 (1965)].
[22] E. A. Shapoval, Pis’ma Zh. Eksp. Teor. Fiz. 64, 320
(1996) [JETP Letters 64, 350 (1996)].
[23] V. P. Mineev and K. V. Samokhin, The Introduction to
the Theory of Unconventional Superconductors, Moscow,
1996 [in Russian].
[24] S. V. Yampolskii and F. M. Peeters, Phys. Rev. B 62,
9663 (2000).
[25] B. J. Baelus, S. V. Yampolskii, F. M. Peeters, E. Mon-
tevecchi,and J. O. Indekeu,cond-mat/0106403 (2001).
[26] D.Saint-JamesandG.Sarma,TypeIIsuperconductivity,
Pergamon Press, 1969.
[27] S. J. Chapman, Q. Du, M. D. Gunzburger, and J. S.
Peterson, Adv.Math. Sciences Appl. 5, 193 (1995).
[28] H.KoppeandJ.Willebrand,J.LowTemp.Phys.2,499
(1970).
[29] J. R. Clem, J. Low Temp. Phys. 18, 427 (1975).
[30] W. V. Pogosov, A. L. Rakhmanov, and K. I. Kugel, Zh.
Eksp.Teor.Fiz.118, 676(2000) [JETP 90,588(2000)].
[31] W. V. Pogosov, K. I. Kugel, A. L. Rakhmanov, and E.
H.Brandt, Phys. Rev.B 64, 064517 (2001).
[32] R. C. Jones, M. K. Keene, and P. Nurwantoro, Physica
C 298, 140 (1998).
[33] K.I.Kugel,W.V.Pogosov,andA.L.Rakhmanov,Phys-
ica C 339, 10 (2000).
[34] F.M.PeetersandV.A.Schweigert,PhysicaC332,266
(2000).
6
FIGURE CAPTIONS
Fig.1. Equilibrium H -R diagram of the cylinder in the external magnetic field at b=1 (a), b=2.5 (b), b=5 (c),
e
b → ∞ (d). Solid lines show the boundaries between the states with different vorticity. Dashed lines correspond to
the boundaries between the multivortex and the giant vortex phases. Curves 1 and 2 show the surface and the first
critical fields, respectively. The dot line denotes the bulk upper critical field.
Fig 2. The spatial distribution of the dimensionless modulus of the order parameterinside the mesoscopic cylinder
in giant vortex state with angular quantum momentum L=2 at b=1, R=3.9, H =0.9 (a) and b→∞, R=4.56,
e
H = 0.5 (b) Solid lines correspond to our variational result, dashed lines to the results of the lowest Landau level
e
approximation. The distance from the cylinder axis r is measured in units of the coherence length ξ(T).
Fig. 3. The equilibrium magnetization of the cylinder with radius R = 4.625 versus applied field at b = 1, κ = 5.
Jumpsinthe magnetizationinFig. 3(a)correspondtothetransitionsbetweenthe stateswithdifferentvorticity. Fig.
3(b)shows the behaviorofthe magnetizationinthe vicinityofthe continuousphase transitionatHe ≈0.61Hc2 from
the multivortex state with 2 vortices to the giant vortex phase with vorticity L = 2. Solid lines correspond to the
equilibrium magnetization, dashed line shows the magnetization of the metastable giant vortex phase.
Fig. 4. The equilibrium field dependence of the magnetization of the cylinder with radius 4.05 at b → ∞, κ = 5.
In Fig. 4(b) the magnetization is plotted versus applied field near the transitions from the multivortex states to the
giant vortex phases. Solid lines correspondto the equilibrium magnetization, dashedline shows the magnetizationof
the metastable giant vortex phase.
7
(cid:20)(cid:17)(cid:21)
(cid:20)
(cid:19)(cid:17)(cid:27)
(cid:21)
F
+
(cid:3)
(cid:18)
(cid:3)
H
+
(cid:19)(cid:17)(cid:23)
(cid:21)
(cid:11)D(cid:12)
(cid:19)(cid:17)(cid:19)
(cid:20)(cid:17)(cid:23) (cid:21)(cid:17)(cid:20) (cid:21)(cid:17)(cid:27) (cid:22)(cid:17)(cid:24) (cid:23)(cid:17)(cid:21) (cid:23)(cid:17)(cid:28)
5(cid:18)x 7
(cid:11) (cid:12)
Fig. 1(a)
(cid:20)(cid:17)(cid:25)
(cid:20)
(cid:20)(cid:17)(cid:21)
(cid:21)
F
+
(cid:19)(cid:17)(cid:27)
(cid:3)
(cid:18)
(cid:3)
H
+
(cid:19)(cid:17)(cid:23)
(cid:21)
(cid:11)E(cid:12)
(cid:19)(cid:17)(cid:19)
(cid:19)(cid:17)(cid:26) (cid:20)(cid:17)(cid:23) (cid:21)(cid:17)(cid:20) (cid:21)(cid:17)(cid:27) (cid:22)(cid:17)(cid:24) (cid:23)(cid:17)(cid:21)
5(cid:18)x 7
(cid:11) (cid:12)
Fig. 1(b)
8
(cid:21)(cid:17)(cid:24)
(cid:21)(cid:17)(cid:19)
(cid:20)
(cid:21)
F
+ (cid:20)(cid:17)(cid:24)
(cid:18)
H
+
(cid:20)(cid:17)(cid:19)
(cid:19)(cid:17)(cid:24)
(cid:21)
(cid:11)F(cid:12)
(cid:19)(cid:17)(cid:19)
(cid:19)(cid:17)(cid:19) (cid:19)(cid:17)(cid:27) (cid:20)(cid:17)(cid:25) (cid:21)(cid:17)(cid:23) (cid:22)(cid:17)(cid:21) (cid:23)(cid:17)(cid:19)
5 x 7
(cid:18) (cid:11) (cid:12)
Fig. 1(c)
(cid:21)(cid:17)(cid:24)
(cid:20)
(cid:21)(cid:17)(cid:19)
(cid:21)
+ F(cid:20)(cid:17)(cid:24)
(cid:18)
H
+
(cid:20)(cid:17)(cid:19)
(cid:19)(cid:17)(cid:24)
(cid:21)
(cid:11)G(cid:12)
(cid:19)(cid:17)(cid:19)
(cid:20)(cid:17)(cid:21) (cid:20)(cid:17)(cid:27) (cid:21)(cid:17)(cid:23) (cid:22)(cid:17)(cid:19) (cid:22)(cid:17)(cid:25) (cid:23)(cid:17)(cid:21)
5(cid:18)x 7
(cid:11) (cid:12)
Fig. 1(d)
9
(cid:19)(cid:17)(cid:23)(cid:24)
(cid:19)(cid:17)(cid:22)(cid:19)
I
(cid:19)(cid:17)(cid:20)(cid:24)
(cid:11)D(cid:12)
(cid:19)(cid:17)(cid:19)(cid:19)
(cid:19) (cid:20) U(cid:21) (cid:22)
Fig. 2(a)
(cid:20)(cid:17)(cid:19)(cid:19)
(cid:19)(cid:17)(cid:26)(cid:24)
(cid:19)(cid:17)(cid:24)(cid:19)
I
(cid:19)(cid:17)(cid:21)(cid:24)
(cid:11)E(cid:12)
(cid:19)(cid:17)(cid:19)(cid:19)
(cid:19) (cid:20) (cid:21) U (cid:22) (cid:23)
Fig. 2(b)
10
