Paramagnetic Meissner effect in mesoscopic samples

10 Pages·0.34 MB·

by  V. A. Schweigert

Paramagnetic Meissner effect in mesoscopic samples.
V.A. Schweigert † and F.M. Peeters ‡
† Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia,
‡ Departement Natuurkunde, Universiteit Antwerpen (UIA), B-2610 Antwerpen, Belgium
Electronic mail: [email protected]
0
(February 6, 2008)
0
0
2
We consider a defect–free superconducting disk (and
n
Using the non-linear Ginzburg–Landau (GL) theory, we
cylinder)immersedinaninsulatingmediawithaperpen-
a
studythemagneticresponseofdifferentshapedsamplesinthe
J dicular (along cylinder axis) uniform magnetic field H0.
0 field–cooled regime (FC). For high external magnetic fluxes, The behaviourof the superconductor is characterizedby
the conventional diamagnetic response under cooling down
1 itsradiusR(andthicknessdforthediskcase),magnetic
] cAansecboenfdo-lolorwdeerdtbrayntshiteiopnafrraommaagngeiatinctMvoeritsesxnesrtaetffeetcota(PmMulEti)–. fipeelndetHra0t,iotnheλc(Toh)er=enλc(e0ξ)((1T)−=T/ξT(0))−(11/−2 Tle/nTgct)h−s,1/w2haenrde
n vortex state, with the same vorticity, occurs at the second c
T is the critical temperature. To reduce the number of
o critical field which leads to thesuppression of PME. c
c independent variables we measure the distance in units
- PACS number(s): 74.24.Ha, 74.60.Ec, 73.20.Dx of the sample radius, the vector potential A~ in c¯h/2eR,
r
p and the order parameter Ψ in −α/β with α, β being
u the GL coefficients [12]. Then tphe GL equations become
s The Meissner effect is considered the most important
mat. cphearcroanctdeurcisttoircpisrocpoeorlteydodfoswupnerincotnhdeucptrivesiteyn.ceWohfenanaesux-- (cid:16)−i∇~2D−A~(cid:17)2Ψ= Rξ22Ψ(1−|Ψ|2), (1)
ternal magnetic field the field is expelled and it behaves
-
d as a diamagnet. However, some samples show a param- R2
n agnetic response under cooling. The finding of PME (or −△3DA~ = λ2f(z)~j2D. (2)
o Wohllebeneffect)inhigh-T superconductors[1]initiated
c
c the appearance of several models interpreting PME as Here, the indices 2D, 3D refer to two-dimensional and
[
evidence of non-conventional superconductivity in these three-dimensional operators;

1 materials (e.g. see Ref. [2]). However, numerous obser-
1
0v vmaetsioonscsopoficP[8M]Esupinerccoonnvdeuncttioornsailnmdiaccartoestchoepiecx[i3st–e7n]caenodf ~j2D = 2i(cid:16)Ψ∗∇~2DΨ−Ψ∇~2DΨ∗(cid:17)−|Ψ|2A~, (3)
1
another mechanism, which may be explained with GL is the density of superconducting current in the plane
1
theory. (x,y), and the external magnetic field is directed along
1
0 Basedonresultsfromaxial-symmetricsolutionsofthe the z-axis. The boundary conditions to Eqs. (1-2) cor-
0 GL equations by Fink and Presson [9], Cruz et al. [3] respond to zero superconducting current at the sam-
0 proposed that PME in their experiments on Pb99Tl01 ple boundary and uniform magnetic field far from the
/ cylinders is caused by a temperature variationof the su- sample. The order parameter and superconducting cur-
t
a perconducting density in a giant vortex state with fixed rentare assumed to be independent of the z−coordinate
m
angular momentum [3]. In such a state, the supercon- which is valid for cylinders as well as for thin d ≪ ξ,λ
- ducting current, which shields the magnetic field in the disks [13–15]. Then f(z) = 1 and f(z) = dδ(z) for the
d
vicinity of the sample boundary (essentially the Meiss- cylindrical and disk geometry, respectively.
n
ner effect), changes its direction expelling magnetic field Superconducting disks and cylinders placed in a mag-
o
c into the sample that can lead to the PME. Thereafter netic field and cooled down transit from a normal to
: this idea, often called flux compression, was exploited in a superconducting state at the critical temperature T ,
v ⋆
i [8,10,11], but a quantitative analysis of PME within GL whichdepends both onH0 andR. However,the unitless
X theory is still missing and several principal questions re- parameters H0/Hc2(T⋆), R/ξ(T⋆), and the angular mo-
r main to be answered: 1) is the vorticity ofthe giantvor- mentumofthe giantvortexsuperconductingstate inthe
a
tex state fixed during cooling down as was assumed in nucleation point T depend only on the magnetic flux
⋆
Refs. [3,10]?,2) ifyes,canthis leadto the appearanceof Φ = πR2H0 piercing through the sample [14]. Here,
PME?, and, finally, 3) can the proposed mechanism ex- Hc2 = Φ0/2πξ2 and Φ0 = hc/2e are the second critical
plain the PME in recent experiments with conventional field and the flux quantum, respectively. With further
macroscopic [3–7] and mesoscopic [8] samples? In this cooling down, the magnetic response is characterized by
Letter, we follow the GL approach and address these only two independent variables H0/Hc2 and Φ.
questionsbystudyingthemagneticresponseofdifferent– Our numerical approach for solving Eqs. (1-2) is de-
shaped samples in the FC regime. scribed in [13–15]. It turns out that an accurate simu-
lation of a multi-vortex state is a hard task in the case
1
of large vorticity L. The latter correponds to the total givenL,themagneticmomentreachesitsmaximumand
angular momentum and the number of vortices in the minimum value at ΦL−1 and ΦL, respectively. Due to
giant-vortexandthe multi–vortexstate,respectively. To angular momentum quantization the magnetic moment
improve the accuracy we apply a non-uniform rectangu- exhibits a strong oscillating behaviour as function of the
lar space grid condensing in the vicinity of the sample magnetic field (see inset of Fig. 3) which agrees with
boundary. However, due to the tremendous computa- Geim’s observations [8]. With decreasing temperature,
tional expenses (e.g. the total number of grid points in thedipolemagneticmomentbecomeszeroforcertainra-
the plane (x,y) was about 160000 for L = 30) we could tio H/Hc2 ∼ 1/(1−T/Tc) and the original diamagnetic
only treat the vortex state with large L in the cylindri- response becomes paramagnetic. This magnetic field is
cal case, when the vector potential is uniform in the z- showninFig.3forΦ=ΦL−1andΦ=ΦL. Notethat: 1)
direction. Inthediskcase,werestrictourconsiderations forthecylindergeometryalargerGLparameterκ=λ/ξ
totheaxialsymmetricsolutionsΨ=ψ(ρ)exp(iLφ)(ρ,φ favours PME, 2) while an increase of the effective pene-
are the cylindrical coordinates), which are shown to be tration length λ2/d suppresses PME in disks, and 3) an
stable in the region H0 >Hc2. increase of Φ favours PME both in cylinders and disks.
When the sample is cooled down below the critical The reason is that the point ρ , where the supervelocity
⋆
temperature T , a giant vortex state appears with angu- changes its direction, shifts towards the sample bound-
⋆
lar momentum L, which is determined by the magnetic ary with increasing angular momentum which is related
flux Φ [10,14,16]. Starting fromthis state wemimick the tothetotalmagneticfluxΦ(L≫1)≈Φ0(L+L1/2)[16].
FC regime by decreasing (increasing) slowly the value of Incylinders,themagnetizationisdirectlyproportional
H0/Hc2 ∼1/(1−T/Tc) (R2/ξ2 ∼(1−T/Tc)) such that to the magnetic moment. In disks: 1) the diamagnetic
thesystemevolvesalongapathwithfixedexternalmag- currents flowing near the sample boundary give larger
netic flux (e.g. see Fig. 1). Using the superconducting contributions to the magnetization than inner paramag-
state found at the previous step as input, we find the netic currents [11] and consequently, the paramagnetic
next steady–state solution to Eqs. (1-2). Doing so we contribution to the magnetization will be strongly sup-
consider only stable solutions and neglect thermal fluc- pressed (inset (b) in Fig. 1), 2) but the smaller trapped
tuations,whichcouldleadtopossibletransitionsbetween magnetic flux, in disks as compared to cylinders, de-
metastable states. This assumption is valid for nor- creasesthediamagneticresponse. Thisisthereasonwhy
mal superconductors where the barriers separating the thickerdisksshowonsetofPMEatlargerH0/Hc2(Fig.3,
metastable states exceed by far the sample temperature, open symbols).
except near points in which the state becomes unstable WhenthesecondcriticalfieldHc2(T)becomessmaller
[17], e.g. near the saddle points. than the applied magnetic field, the giant–vortex state
When calculating the dipole magnetic moment, we transits to the multi–vortex state with the same vortic-
can neglect non-linear effects in the vicinity of the nu- ity (Figs. 4,5). This second-order transition is not fol-
cleation point. The quantum angular momentum L in- lowed by any jumps in the magnetization or the mag-
creases almost proportional to the magnetic flux Φ but netic moment. Just after the transition, all vortices are
remains always smaller than Φ/Φ0. The supervelocity arrangedinaring(0:L).Note,thatthemagneticmoment
vφ = ρ−1(L−Φρ2/Φ0R2), which is oriented along the of the state (0:L) continues to increase with decreasing
azimuthaldirection,changesitssignatρ⋆ =RpLΦ0/Φ. H0/Hc2(T) but with a smaller slope (Fig. 5). With fur-
Therefore, both diamagnetic (ρ>ρ⋆) and paramagnetic ther decreasing H0/Hc2(T) a pair of vortices moves to
(ρ < ρ ) currents exist in any giant vortex state. How- theinnerregionandthe state(2:L-2)appearsforL=20
⋆
ever, the magnetic moment, which can be estimated in (see Fig. 5). This first-order transition is followed by
the lowest Landau level (LLL) approximation as D ∼ a weak jump in the magnetic moment. The derivative
dρρ2v |ψ |2 with ψ being the lowest eigenfunction of dD/dT changes sign and further cooling down results
φ L L
Rthe linearized first GL equation, turns out to be always in the disappearance of PME. The magnetic moment of
diamagnetic both for disks andcylinders. As long as the the diamagnetic state with smaller angular momentum
superconducting density |Ψ|2 remains small, the mag- L = 19 and Φ = ΦL is also affected by the transition to
netic moment almost linearly increases in absolute value the multi–vortex state, which increases the diamagentic
with decreasing H0/Hc2 (inset (a) in Fig. 1). With fur- response (inset (b) in Fig. 4). As the temperature de-
ther cooling down,the LLL approximationbreaksdown. creases, vortices continue to move from the outer to the
Duetonon-lineareffects(mainlyfromthesecondtermin inner shell. Note, that the corresponding weak jumps in
the RHS of Eq. (1)) the order parameter increases more the magnetic moment are practically not visible on the
rapidly in the inner region which leads to an increase of scale used in Fig. 4. Although the state with L = 19 is
the paramagnetic component (Figs. 1,2). energetically more favourable than the one with L=20,
The resulting magnetic moment crucially depends on no vortex exits the system which is in agreement with
theratiobetweenLandΦ/Φ0. Withincreasingmagnetic observations [8]. Since the vorticity remains unchanged
field, the switchingL→L+1ofthe angularmomentum under cooling down, the magnetic response will be al-
of the nucleated state occurs at a certain Φ [14]. For most reversible. The hysteresis causedby the first–order
L
transitionsbetweendifferentmulti–vortexstateswiththe
2
same vorticity is weak (see inset (a) of Fig. 4). Starting study the FC regime and failed to find any paramag-
fromthe pointH/Hc2 =0.9andwarmingup the system netic response. As shown above, one has to go beyond
toH/Hc2 =1.1wefindthemagneticmoment,whichco- the LLL approximation in order to find PME which is
incidesinthescaleofFig.4withthatobtainedbycooling caused by non-linear effects.
down. WethankA.K.Geimforusefuldiscussions. Thiswork
Although the magnetization curves, shown in Fig. 4, is supported by the Flemish Science Foundation (FWO-
agree qualitatively with those from experiments [4–6], a Vl)andthe“InteruniversityPolesofAttractionProgram
number of issues remain unclear: 1) in experimental ob- - Belgian State, Prime Minister’s Office - Federal Office
servations of macroscopic samples a weak hysteresis is for Scientific, Technicaland CulturalAffairs”. One ofus
found in cooling down and subsequent warming up, and (VAS)wassupportedbyaDWTCfellowshipand(FMP)
2) the observed maximum in the magnetic momentum is a research director with the FWO-Vl.
are weaker than those from our calculations which may
be due to the presence of vortex pinning centra. Note
that within GL theory we found a weak PME which is
causedbythecompetitionoflargediamagneticandpara-
magnetic responses of the outer and inner part of the
sample, respectively. Any mechanism which slightly in-
[1] W. Braunisch, N. Knauf, V. Kataev, S. Neuhausen,
fluences any of the two responses may strongly influence
A. Gru¨tz, A. Kock, D. Khomskii, and D. Wohlleben,
the total magnetic behavior. As an example, in macro-
Phys. Rev.Lett. 68, 1908 (1992).
scopic disks PME disappears after mechanical abrading
[2] M. Sigrist and T.M. Rice, J. Phys. Soc. Jpn. 61, 4283
the top and bottom surfaces [4–6]. A number of sam-
(1992).
plesmadefromthesamematerialasthosedemonstrating
[3] F.delaCruz,H.J.Fink,andJ.Luzuriaga,Phys.Rev.B
PME exhibited only diamagnetic behaviour [4–6]. This
20, 1947 (1979).
indicates the important role played by the sample struc-
[4] D.H. Thompson, M.S.M. Minhaj, L.E. Wenger, and
tural inhomogeneity. For mesoscopic disks, experiments J.T. Chen, Phys.Rev.Lett. 75, 529 (1995).
[8] show PME for rather small angular momenta which
[5] P. Kostic, B. Veal, P. Paulikas, U. Welp, V.R. Todt,
doesnotagreewithoursimulationsforflatcircularmeso-
C. Gu, U. Geiser, J.W. Williams, K.D. Carlson, and
scopic disks. R.A. Klemm, Phys.Rev. B 53, 791 (1996).
To address some of these sample structural issues we [6] L. Pust, L.E. Wenger, and M.R. Koblischka, Phys. Rev.
consider the influence of the superconductor shape on B 58, 14191 (1998).
the magnetic moment by varying radially the thickness [7] A.Terentiev,D.B.Watkins,L.E.DeLong,D.J.Morgan,
of the disk. We limit ourselves to the case of a strong and J.B. Ketterson, Phys. Rev.B 60, R761 (1999)
type-II superconductor (κ ≫ 1) and solve, therefore, [8] A.K. Geim, S.V. Dubonos, J.G.S. Lok, M. Henini, and
only Eq. (1). The dipole magnetic moment of different- J.C. Maan, Nature (London) 396, 144 (1998).
shaped samples is shown in Fig. (6) for H0 = Hc2(T), [9] H.J.FinkandA.G.Presson,Phys.Rev.151,219(1966).
where the scale of the thickness variation is apparent [10] V.V Moshchalkov, X.G. Qiu, and V.Bruyndoncx,Phys.
fromtheinsetsofFig.6. Themagneticmomentbecomes Rev. B 56, 11793 (1997).
morenegative(positive)inmagnifyingglass(crown)like [11] A.E. Koshelev and A.I. Larkin, Phys. Rev. B 52, 13559
samples as compared to flat disks. An increase of the (1995).
[12] P.G. de Gennes, Superconductivity of metals and alloys,
local thickness near the sample boundary (see Fig. 6(c))
(Addison-Wesley,N.Y., 1989).
increasesthePME.Thisstronglysuggestsapossiblenon
[13] P.S.Deo,V.A.Schweigert,F.M.Peeters,andA.K.Geim,
flat geometry of the disks of Ref. [8].
Phys. Rev.Lett. 79, 4653 (1997).
Insummary,the giant–vortexstateremainsstableun-
[14] V.A. Schweigert and F.M. Peeters, Phys. Rev. B 57,
der cooling down and transits to the multi–vortex state
13817 (1998).
with the same vorticity at H0 ≈ Hc2. The paramag- [15] V.A.Schweigert,F.M.Peeters,andP.S.Deo,Phys.Rev.
netic response is caused by a more rapid growth of the Lett. 81, 2783 (1998).
superconducting electron density in the inner region of [16] R.Benoist andW.Zwerger,Z.Phys.B103,377(1997).
the sample, due to non-linear effects, where paramag- [17] V.A. Schweigert and F.M. Peeters, Phys. Rev. Lett. 83,
netic currents flow. The appearance of the multi–vortex
2409 (1999).
state supresses PME and the maximum of the paramag- [18] J.J. Palacios, cond-mat/9908341.
neticresponsecorrespondstoH0 ≈Hc2. Weshowedthat
within the GL theory a paramagnetic response is possi-
ble for large magnetic fluxes, which is in agreementwith
experimentalfindingsondiskswithlargeradia,butdoes
not agree with the experimental results on mesoscopic
disks of Geim et al [8].
After finishing this work,we came awareof a preprint
by Palacios [18] who used the LLL approximation to
3
FIG. 1. The nucleation field as a function of the ra-
dius. Thin curves show the path along which the system
evolves during cooling down. Insets (a) and (b) show the
corresponding dipole magnetic moment and magnetization
(M =Hc−21 d~r(Hz−H0)/d~r,whereintegrationisperformed
R
over the sample volume), respectively. Here, solid and dot-
ted curves correspond to cylinder (κ = 10) and disk (κ = 1,
d=0.1R), respectively.
FIG.2. Thedistributionofthesuperconductingelectrons
density(a),thesuperconductingcurrent(inset)andthemag-
netic field (b) in the cylinder (solid and dashed curves) and
thedisk (inset) for different reduced magnetic fields H/Hc2:
1-1.6, 2-1.3, 3-1.0.
FIG.3. Themagneticfieldcorrespondingtothetransition
fromthediamagnetictoparamagneticresponseasafunction
of the total angular momentum for Φ = ΦL−1 and Φ = ΦL.
Solid and open symbols correspond to cylinders and disks,
respectively. Inset shows the magnetic moment of the cylin-
drical sample at H0=Hc2 and κ=10.
FIG.4. The dipole magnetic moment of cylindrical sam-
ples for different external magnetic fluxes Φ = ΦL−1 and
κ = 10. The hysteretic behaviour is demonstrated in inset
(a) for L=20 and Φ=ΦL−1. Inset (b) shows the magnetic
response for L=19 and ΦL.
FIG.5. Thedipolemagneticmomentandcontourplotsof
thesuperconductingdensity(log-scaleisused)inthevicinity
of the second critical field.
FIG.6. The dipole magnetic moment of different shaped
circular samples at H0/Hc2 =1.
4
L=19 L=20 L=21 0.00
1
1.8
2
-0.05
D
3
1.6
-0.10
(a)
1.0 1.2 1.4 1.6 1.8
1.4 H/H
2
c c2
H
/ 0 1
0
H 2
1.2 ) 3
4 -1
-
0
1
(

M-2 1
1.0 (b)
1 2 3
-3
1.0 1.2 1.4 1.6 1.8
H/H
c2
0.8
5.0 5.5 6.0 6.5 7.0
x
R/
Fig.1
0
10
3
3
t 0.0
n 2
e
1
r -0.2
r 1
2 u
|
10-1 c -0.4
Y|
0.6 0.8
F F
cylinder ( / =24.76)
0
F F
cylinder ( / =25.88)
-2
10
0 (a)
F F
disk ( / =24.76)
1.001
0
1
1.000
0 1.05
H
2
3
/
H 0.999
1
1.00
3
0.95
0.998
0.0 0.5 1.0 (b)
0.0 0.2 0.4 0.6 0.8 1.0
r
/R
Fig.2
0.0
1.6
D -0.2

k
1.5 =10
-0.4
0 20 40
1.4
F F
/
2 k =21/2
c 0
H
k
/ =2
01.3
H
k
=10
k
d=0.01R, =1
1.2 F =F
k
L-1 d=0.1R, =1
F F
1.1 =
L
1.0
0 20 40 60 80 100 120 140
L
Fig.3
4
0
(2:18)
0.2 L=19
-5
3
0.1
-10
(a) (b)
(4:16)
2
0.0 -15
0.985 0.990 0.995 1.2 1.6
)
2
-0 1
1
(

D
0
L=30
-1
L=20
L=10 k
-2 =10
1.0 1.2 1.4 1.6 1.8
H/H
c2
Fig.4
L=20
32
(0:20)
30
)
4
-
0
1
(

D
28
(2:18)
26
0.998 1.000 1.002
H/H
c2
Fig.5
0.0
-0.1
D
-0.2 (a)

0.0

D
-0.1

(b)

0.0
D
-0.2

(c)

-0.4
10 20 30 40 50
F F
/

0
Fig.6