Properties of parallel upper critical field within Continuous Ginzburg-Landau model

12 Pages·0.23 MB·English

by  L. Wang

Properties of parallel upper critical field within Continuous Ginzburg-Landau model
L. Wang, H. S. Lim, and C. K. Ong
Center for Superconducting and Magnetic Materials and Department of Physics, Blk. S12,
Faculty of Science,
National University of Singapore,
2
2 Science Drive 3,
0
Singapore 117542.
0
2
In this paper, we employ a continuous Ginzburg-Landau model to study the behaviors of the

n parallel upper critical field of an intrinsically-layered superconductor. Near T where the order
c
a parameterisnearly homogeneous, theparallel uppercritical fieldisfoundtovaryas(1−T/T )1/2.
c
J
Withawell-localized orderparameter,thesamefieldtemperaturedependenceholdsoverthewhole

8 temperaturerange. The profileof theorderparameter at theparallel uppercritical field may beof
1 aGaussiantype,whichisconsistentwiththeusuallinearGinzburg-Landautheory. Inaddition,the

influences of the unit cell dimension and the average effective masses on the parallel upper critical
]
n field and theassociated order parameter are also addressed.
o
c I. INTRODUCTION different thickness was introduced and efficient comput-
-
ingmethods havebeenadoptedtodetermine the generic
r
p Most high T superconductors (HTSs) have layered properties of the parallel upper critical field Bk of var-
u c c2
structures and the layered superconductivity is closely ious layered superconductors. In the present work, we
s
. related to the behavior of the order parameter. Spatial shall examine various features pertinent to Bk and the
t c2
a variation of the order parameter would aid us to intu- associated order parameter of a typically-layered super-
m itively understand various properties such as the coher- conductor.
- ence lengths of layered superconductors. On the other
d hand, the investigation of the upper critical field B
c2
n
may provide information on the coherence lengths, and II. MODEL
o
in principle, allow the testing of existing theories (for
c
[ example, see Refs. 1–7). Hence, a study involving the
orderparameterandtheuppercriticalfieldshouldprove In the CGL model of Koyama et al.4, layered super-
1 valuable. conductors have been classified into three categories10,
v
2 The phenomenological continuous Ginzburg-Landau one of which the layered HTS Bi2Sr2CaCu2O8 (Bi2212)
2 (CGL) model4 is convenient for a description of the or- may fall into. Following Ref. 8, Bi2212 shall be chosen
3 der parameter and the upper critical field of layered su- as our modeling prototype for layered superconductors
1 perconductors. The coefficients in the CGL free energy as it possesses a large anisotropy11, and thus is suitable
0 are assumed to be spatially dependent. As a result, the for a detailed study that examines the relationship be-
2 amplitude of the order parameter varies, reflecting the tweenthe amplitude of the orderparameterandthe lay-
0 layered nature. The amplitude of the order parame- ered structure. Note, however, that a study involving
/
t ter at weakly superconducting layers may be extremely the phase effect of the order parameter (Josephson cou-
a
m small, corresponding to a weakly linked layer system pling) in Bi221212 may require a LD description1 for an
similar to the Lawrence-Doniach (LD) model. On the appropriate investigation.
-
d other hand, when the spatial dependence is neglected, TheunitcellofBi2212composesoftwoCuO2bilayers,
n the CGL model is reduced to the anisotropic Ginzburg- separatedbytheBiO-SrOinterlayer,whichisreferredto
o Landau(GL) model. Hence, the CGL model approaches as insulating (I) layerfor convenience. The two adjacent
c the limiting cases of the LD model and the anisotropic CuO2 planes of the bilayer (interplane distance ∼ 3 ˚A)
:
v GL model. are strongly coupled so that they can be treated as a
i In a previous work8, a set of spatial coefficients for single superconducting (S) layer; therefore the distance
X
the CGL model was proposed for a layered supercon- betweentwosuperconductinglayersishalfthec-axislat-
ar ducting system in which the unit cell was assumed to tice constant ∼ 15 ˚A13. Denoting the thickness of the I
compose ofequivalently thick superconducting andinsu- andSlayersasdI anddS,respectively,wemaywritethe
lating layers and no applied magnetic field was present. size of the unit cell D as D = dI +dS. The CGL free
Recently9, a magnetic field parallel to the layers with energy for the system is8,
1 ¯h2 ∂ 2ie 2
F = dV α(T,z)|Ψ(~r,z)|2+ β|Ψ(~r,z)|4+ − A (~r,z) Ψ(~r,z)
z
2 2M(z) ∂z ¯h
Z " (cid:12)(cid:18) (cid:19) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
1
¯h2 2ie 2 1
+ ∇(2)− A~(2)(~r,z) Ψ(~r,z) + B2(~r,z) , (1)
2m(z) ¯h 2µ
(cid:12)(cid:18) (cid:19) (cid:12) 0 #
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
with planar vector ~r = (x,y) and vector potential where α , α , G , G , g and g are the model parame-
0 1 0 1 0 1
A~(~r,z) = (A~(2)(~r,z),A (~r,z)). M(z) denotes the effec- ters. These parameters are found to be relatedto exper-
z
tive masses along the z-direction (c-axis), and m(z) is imental data, to each other, and to the intrinsic param-
the corresponding planar parameter. The GL coefficient eters, dI and dS (see discussions in section IV).
α(T,z)and the effective masses are takenas before (β is LetanexternalmagneticfieldB be appliedparallelto
assumed as a constant)8: the a- or b-axis which is in the y-direction. Thus, the
vector potential can be taken as A~ = (Bz,0,0). Assum-
α(T,z)=[α +α cos(2πz/D)](1−T/T ), (2a)
1 0 1 c ing Ψ(~r,z) = eik~k·~rΨ(z), it follows from Eq. 1 that, by
=G +G cos(2πz/D), (2b) minimizing the free energy,
0 1
M(z)
1
=g +g cos(2πz/D), (2c)
0 1
m(z)
¯h2 ∂2 ¯h2 ∂ 1 ∂ 1 ¯h2k2
− Ψ(z)− Ψ(z)+ (2eB)2(z−z )2+ y Ψ(z)
2M(z)∂z2 2 ∂zM(z) ∂z 2m(z) 0 2m(z)
(cid:20) (cid:21) " #
+α(T,z)Ψ(z)+β|Ψ(z)|2Ψ(z)=0, (3)
with z =h¯k /(2eB). At B =B , the superconducting ically small. To explore the features of the order param-
0 x c2
orderdevelopsintheSlayerfirstsothatonemaychoose eteralongthe z direction,weassumek =0. Finally, we
y
z = D/2. The high order term in Eq. 3, β|Ψ(z)|2Ψ(z), obtain
0
maybeomittedsincetheorderparameteratB isphys-
c2
¯h2 ∂2 ¯h2 ∂ 1 ∂ 1 D
− Ψ(z)− Ψ(z)+ α(T,z)+ (2eB)2(z− )2 Ψ(z)=0. (4)
2M(z)∂z2 2 ∂zM(z) ∂z 2m(z) 2
(cid:20) (cid:21) (cid:20) (cid:21)
For a giventemperature T,the maximum magnetic field spacing or the size of the unit cell, but the order param-
B which satisfies the above equation, together with the eter is still assumed to be discontinuous. Thus, there
boundaryconditionsΨ(0)=Ψ(D)and ∂ Ψ(z) =0, is the possibility of a true 3-D superconductivity with a
∂z z=D
gives a point on the B -T curve. The largest B can be nearly uniformly distributed order parameter even in a
c2
(cid:12)
readily achieved by treating B2 in Eq. 4 as e(cid:12)igenvalue highlyanisotropicsuperconductor. Thissituationcanbe
problems9. obtainedbyjustvaryingthe temperature(seeFig.1(a)).
Again, it is found that the peaks of the order parame-
III. RESULTS AND DISCUSSION ter can be fitted by a Gaussian function. The exponen-
tialfactoristhemostsignificantpartoftheGaussianfit,
showingthatthegroundstateoftheCGLlinearequation
The order parameter distribution in a unit cell at dif-
issimilartothatoftheusuallinearGLequation14,15. We
ferent temperatures is plotted in Fig. 1(a). At low tem-
emphasize that this similarity, together with many rea-
peratures,the orderparameterismainlyconfinedwithin
sonable results to be presented, reveals the plausibility
the S layer,signifying a two-dimensional(2-D) state. At
of our methods of calculating B . The fitted ξ (0) is
hightemperatures,iteffectivelypenetratesintotheIlay- c2 ⊥
0.96 ˚A, which comparesfavorablywith some experimen-
ers. Near T , it varies smoothly and is nearly a constant
c tal values of ∼1 ˚A16.
throughout the unit cell, indicating a three-dimensional
(3-D) state behavior. The present model thus correctly The calculated parallel B as a function of temper-
c2
accounts for the behavior of a 2-D state at lower tem- ature is shown in Fig. 1(b) and Fig. 2. Near T , the
c
peratures and a 3-D state near T . Note that the weak feature of B -T is square-root like while far away from
c c2
modulationoftheorderparameternearT maygenerate T , it is linear. The linear behavior in Fig. 1(b) can
c c
a genuine3-Dsuperconductor. This is differentfromthe be understood by identifying B ∝ 1/ξ2 while the lat-
c2
so-called 3-D region of the LD model1, where the inter- ter is proportional to 1 − T/T 9. Note that the rela-
c
layer coherence length is much largerthan the interlayer tionship between B and T is also linear within the
c2
2
anisotropy GL theory, in which Bk (T) = Φ0 , pounds of Tl Ba Ca Cu O (n=1,2,4), Mukaida
c2 2πξk(T)ξ⊥(T) et al.30 repor2ted2thatn−t1he unpp2enr+4critical field generally
whereboththeinterlayerandin-planecoherencelengths
ξ⊥(T) and ξk(T) are proportional to (1−T/Tc)−1/2 so decreases as the number of CuO2 layer increases. They
attributed their results to the effects caused by the dif-
that Bk (T)∝(1−T/T ).
c2 c ferentthicknessoftheeffectivesuperconductinglayersin
Since the parallelB inBi2212rapidlyexceeds acces-
c2 Tl Ba CuO , Tl Ba CaCu O and Tl Ba Ca Cu O ,
2 2 6 2 2 2 8 2 2 3 4 12
sible laboratorymagnetic fields when the temperature is
whose respective c-axis lattice constants are 23.2, 29.3
reducedfromTc,onlythe calculateddatanearTc canbe and 41.9 ˚A. Clearly, the theoretical trend presented in
compared with experiments17,18 (see Fig. 2). By consid-
Fig. 3(b) is consistent with their experimental observa-
eringaconstantsolutionoftheorderparametertoEq.4,
tions.
onecanimmediatelyobtainasquare-rootB -T relation
c2 The mass dependences of the parallel upper critical
near T . Note, however,that with openboundary condi-
c field at zero temperature are shownin Fig. 4(a)and (b).
tions(OBC,Ψ(z)| =0)imposedonEq.4,wehave
z→±∞ Inthesecalculations,thevalueofG (g )wasfixedwhile
1 1
found that there is a linear B -T relation near T (see
c2 c that of G (g ) was varied to obtain the varying M
0 0 ⊥
solidcircles in Fig. 2). The deviationfromthe linear be-
(m ). It is clear that large values of both M and m
k ⊥ k
haviorcanbeunderstoodasadimensionalcrossover(see
resultinalargecriticalfield,whichisconsistentwiththe
Ref. 19 and references therein). Moreover, when using anisotropicGLtheory,theLDmodel1andtheg theory5.
3
spatial-independent coefficients (AGL) in the OBC sim-
ItisworthmentioningthatasM increases,wefindthat
⊥
ulation, we obtain a linear B -T relation in the whole
c2 the order parameter Ψ(D/2) at the S layer grows while
temperature range (as expected). Note that these calcu-
that at the I layer, Ψ(0), decreases, leading to a larger
lation results are in reasonably and qualitatively agree-
difference of Ψ(D/2)−Ψ(0). Since 1/[Ψ(D/2)−Ψ(0)]is
ment with experiments (which are diverse due to factors
approximately proportional to the strength of interlayer
such as crystal quality, measurement methods, etc). couplingbetweenadjacentSandI layers8,thusM sup-
⊥
The square-root behavior near Tc indicates that the presses interlayer coupling. In contrast, mk is found to
upward curvature of the Bc2-T curve is absent in the enhance interlayer coupling. We attribute this to the ef-
present simulated system. Recent studies6,20 show that fectthatM enhancestheorderparameterinthe super-
⊥
the upward curvature is perhaps not intrinsic. Indeed, conductinglayerwhilem suppressesit. Hence,HTSare
k
such curvature is neither found nor obviousin the WHH intrinsicallyfavorablefora largeM ,whichcorresponds
⊥
approximation21, the d-wave theory22 and the mixed d- to a weakly linked layered system.
and s-wave theory23. Note, however, that the feature of The spatial variation of the order parameter at sev-
the Bc2-T curvature is controversial7 and remains to be eral condensation energies is plotted in Fig. 5(a). The
tested24. According to Fig. 2, the curvature of Bc2-T is temperature is set to zero. It is obvious that the largest
boundary-condition dependent and thus indeed difficult energycorrespondsto the largestorderparameterin the
to arrive at an absolutely conclusive conclusion. The S layer and that the smaller the condensed energy, the
curvature may also be affected by physical phenomena25 broader the order parameter. Fig. 5(b) further shows
such as the spin orbit scattering (for example, see the thatthe criticalfieldincreaseswiththe condensationen-
microscopic theory in Ref. 2). ergy.
The calculated Bc2 at zero temperature is about 700 Up till now, we have set the parameters α0, α1, G0,
Teslaandiscomparablewiththoseextrapolatedfromex- G ,g andg (seeEq.2)intheSlayerthesameasthose
1 0 1
perimentsonBi221217,18andotherHTSsuchasYBCO26 inthe I layer(for example,αS =αI). We shallnow con-
0 0
and Bi2Sr2Ca2Cu3O1027. However, the experimentally sider the case where αS0 6= αI0. The ratios of αS0/αI0 in
extrapolateddata might notbe too reliable28 and a pos- Fig. 6 (a) and (b) are 20 and 2000000,respectively. The
sible new way to detect Bc2 with Josephson plasma has temperature is zero. The non-monotonic trend (first as-
been suggested29. cendingandthendescending)inFig.6(a)isqualitatively
Fig. 3(a) shows the spatial distribution of the order consistent with the data extracted from the numerical
parameter for different D of the unit cell at zero tem- work of Refs. 3,4.
perature. For small D, the order parameter covers the WhentheenergycondensedintheSlayerisextremely
entire I layer but as D increases, its penetrations into large, the system would be in an extreme 2-D state, as
the neighboring I layers become restricted and quickly illustrated in Fig. 6(b) for different unit cell sizes. The
fall to zero. The 3-D (2-D) behavior for small (large) orderparametertotallyresidesintheSlayer. TheSlayer
unit cell is in accordance with the features reflected fully decouples with the adjacent I layers and therefore
in the calculated B -T curves of Refs. 3,4. Fig. 3(b) the system is in an extreme 2-D state. This interesting
c2
presents the variation of the upper critical field with the 2-D behavior can be confirmed by the thickness depen-
size of the unit cell at T = 0 K. The obtained criti- denceoftheuppercriticalfield. Thetheoreticaldatacan
cal field decreases with D, which is qualitatively consis- be fitted by an inverse relation typical of a 2-D system
tent with the g model5. Here a power law (dash line) (forexample,seeRefs.5,15). Itisinterestingtofindthat
3
could not fit the trend well but an exponential fit (solid the Ψ(D/2)-D profile is qualitativelyconsistentwith the
line) is acceptable. Experimentally, for the similar com- B -D trend. Such qualitative consistency can also be
c2
3
found in Figs. 6(a) and 3. 9L. Wang,H. S.Lim, and C. K. Ong,accepted for publica-
The extreme 2-D behavior can be further confirmed tion in Supercond.Sci. Technol. (2001).
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⊥
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8L.Wang,H.S.Lim,andC.K.Ong,Supercond.Sci.Tech- different temperatures. (b) Temperature dependence of the
nol. 14, 252 (2001). parallel uppercritical field.
4
FIG.2. Calculated temperature dependences of the par- FIG.4. Upper critical field for (a) perpendicular average
allel uppercritical fieldnear Tc,compared with experiments. mass M⊥ and (b) parallel average mass mk.
Thesolidsquarescorrespondtotheperiodicboundarycondi-
tion with spatial-dependent coefficients (CGL), the solid cir-
cles to the open boundary conditions with spatial-dependent FIG. 5. a) Order parameter distribution and (b) upper
coefficients(CGL)andtheopendiamondstotheopenbound- critical field at different condensation energies.
ary conditions with spatial-independent coefficients (AGL).
The solid line is a fit varying as (1−T/T )0.5. The dotted
c
line signifies thecrossover temperaturefrom 3D to2D in the FIG. 6. Order parameter and upper critical field for
PBC-CGL (solid squares) and OBC-CGL (solid circles) cal- αS0/αI0 =20 in (a) and αS0/αI0 =2000000 in (b). The profiles
culations. of the maximum order parameter vs D seem consistent with
the corresponding Bc2-D trends. The dash line in (b) is ap-
proximatelyaninversefitwhilethesolidlineisanexponential
FIG. 3. (a) Order parameter distribution and (b) upper
decay.
critical field at different sizes of theunit cell.
FIG.7. Typical2Dtemperaturedependenceoftheparal-
lel uppercritical field.
5
Figure1
0.8 900
(b)
(a)
T(K)
0.6
0
600
40
)
a
l
s
e
0.4
T
Ψ
(
74
2
c
B
84
300
84.99
84.9
0.2
0.0 0
0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0
z(a.u.) T/T
c
Figure 2
50
calc. (PBC, CGL)
calc. (IBC, CGL)
calc. (IBC, AGL)
40
expt. (Ref. 17)
expt. (Ref. 18)
fit: ~(1-T/T )0.5
)
a 30 c
l
s
e
T
(

220
c
B
10
0
0.98 0.99 t* 1.00
T/T
c
Figure 3
1.0 1200
calculation
(a) (b)
D (a.u.) exponetional decay
power decay (~D-0.17)
5.8
29.2
0.8 52.5
1000
0.6
)
a
l
s
Ψ e
T
(
2
c
0.4 B
800
0.2
0.0 600
0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 60
z/D D(a.u.)
)LJXUH(cid:3)(cid:23)
(cid:25)(cid:25)(cid:27) (cid:25)(cid:25)(cid:27)
(cid:11)D(cid:12) (cid:11)E(cid:12)
(cid:25)(cid:25)(cid:25) (cid:25)(cid:25)(cid:25)
(cid:12) (cid:12)
D D
O O
V V
H H
7 7
(cid:11) (cid:11)
(cid:21) (cid:21)
F F
% %
(cid:25)(cid:25)(cid:23) (cid:25)(cid:25)(cid:23)
(cid:25)(cid:25)(cid:21) (cid:25)(cid:25)(cid:21)
(cid:19) (cid:21)(cid:19)(cid:19) (cid:23)(cid:19)(cid:19) (cid:25)(cid:19)(cid:19) (cid:19)(cid:17)(cid:19)(cid:19) (cid:19)(cid:17)(cid:19)(cid:24) (cid:19)(cid:17)(cid:20)(cid:19) (cid:19)(cid:17)(cid:20)(cid:24) (cid:19)(cid:17)(cid:21)(cid:19)
0 P (cid:11)D(cid:17)X(cid:17)(cid:12)
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(cid:11)D(cid:17)X(cid:17)(cid:12)
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