Mechanism of thermally activated c-axis dissipation in layered High-T$_c$ superconductors at high fields

6 Pages·0.16 MB·English

by  A. E. Koshelev

Mechanism of thermally activated c-axis dissipation in layered High-T
c
superconductors at high fields
A.E. Koshelev
Argonne National Laboratory,Argonne, IL 60439, and
Institute of Solid State Physics, Chernogolovka, Moscow District, 142432, Russia
We propose a simple model which explains experimental behavior of c-axis resistivity in layered
High-Tc superconductorsathighfieldsinalimitedtemperaturerange. Itisgenerallyacceptedthat
the in-plane dissipation at low temperatures is caused by small concentration of mobile pancake
vortices whose diffusive motion is thermally activated. We demonstrate that in such situation a
finite conductivity appears also in c-direction due to the phase slips between the planes caused by
the mobile pancakes. The model gives universal relation between the components of conductivity
6 which is in good agreement with experimental data.
9
9
1

n
a A well known property of layered High-Tc supercon- pancake vortex renormalized by the intervortex interac-
J ductors (e.g., Bi SrCa Cu O (BiSCCO)) is an exis- tions [12]. In this case a very natural assumption is that
2 2 2 x
2 tence of extended low-temperature tails in the tempera- thelatticeisstronglypinnedandthein-planedissipation
1 ture dependencies of the both in-plane [1,2] and out-of- is caused by the motion of mobile pancake vortices with

plane [2–5] components of resistivity for magnetic field the small concentration n and diffusion constant D. In-
3
appliedalongthec-axis. Animportantexperimentalfact planeconductivitywithinthismodelcanbeestimatedas
v
7 isthatthethermallyactivatedbehaviorforbothρab and follows. In-plane transport current j drives mobile pan-
9 ρ ischaracterizedbythesamevalueofactivationenergy cakeswithvelocityv =µs(Φ /c)j. Hereµisthe average
c 0
0 (see Ref. [4]and Fig. 1). This indicates that dissipations pancake mobility, which is connected with the diffusion
2
in both directions have essentially the same origin. constant D by the Einstein relation, D = µT. Pancake
1
The origin of the tail for the in-plane resistivity com- motion produces electric field E = (Φ /c)vn. Therefore
5 0
9 ponent is qualitatively understood. It is theoretically theconductivityinab-planeσab inthismodelisgivenby
/ expected that at fields larger than the crossover field
t 2
a B ≈ Φ /(γs)2 and temperatures larger than the de- c T
cr 0 σ = (1)
m ab
coupling temperature Tdec the system can be treated as (cid:18)Φo(cid:19) snD
- a set of almost independent two-dimensional (2D) vor-
d Both n and D presumably have Arrhenius-type temper-
tex lattices in the layers [6] (here γ is the anisotropy
n aturedependenciesbuttheir temperaturedependence as
o of the London penetration depth and s is the interlayer
well as detailed mechanism of in-plane motion have no
c spacing). The in-plane dissipation can therefore be at-
: tributed to the thermally activated motion of point de- influence for our further consideration.
v
Inthe regionwherethe conductivity inthe c-direction
i fects inthe 2Dlattices (weak disorder)or weaklypinned
X is much larger than the normal conductivity it has the
mobile pancake vortices (strong disorder) [7]. For a 2D
r superconductor with weak pinning a crossover between Josephson origin and is caused by phase slips between
a neighboringlayers. Itwasassumed[3,8]thatthesephase
freefluxflowandthermallyactivatedmotionshouldtake
slips are similar to the phase slips in a single Josephson
place at the melting temperature of the vortex lattice
junction with the typical area Φ /B and therefore the
T ≈ 0.014sΦ2/(4πλ)2 as observed in the thin films of 0
m 0 Ambegaokar-Halperinformulaforthe resistivityofsmall
a−Nb Ge [9] and in numerical simulations [10]. This
3 Josephson junction can be used to describe the temper-
seems not to be the case for available samples of the
ature and field dependencies of ρ . Unfortunately this
BiSCCOcompoundwherecrossoverto thethermallyac- c
physically transparent assumption does not have micro-
tivated behavior at high fields (> 1 T) takes place at
scopic support.
temperatures 50-70K which is significantly largerthan
Diffusive motion of pancake vortices at high enough
the estimate for the melting temperature T ≈13K ob-
m temperaturesleadstothetimevariationsofthephasedif-
tained using experimental values of the London pene-
ference between the layers. Such pancake-induced phase
tration depth λ ≈ 1800˚A and s = 15˚A. This indicates
ab slipsprovidesacandidatemechanismforthec-axisdissi-
thatinavailableBiSCCOsinglecrystalspinningisstrong
pationwhichwasnotconsideredsofar. InthisLetterwe
enoughtodestroythemeltingtransitioninthefieldrange
obtain c-axis conductivity due to such mechanism. The
1-10 T and the crossover temperature between free flux
key assumption of our model is that the phase slips are
flow and thermally activated regime ( 50-70 K) has to
causedthediffusivemotionofasmallamountofpancake
be interpreted as the depinning temperature of a single
vortices independently in different layers. The model is
1
applicable above the decoupling temperature where the with n = B/Φ . We assume that the phase at given
v 0
globalsuperconducting coherence is broken. This means point is not sensitive to pancake displacements at dis-
that the model can be used to describe dissipation in a tances larger than some phase coherence distance r ,
max
limited temperature range (∼ 40-70 K). which provides upper cutoff to the logarithmically di-
The starting point of our calculation is the Kubo for- verging integrals. The static phase correlation function
mulawhichrelatesσ withthecorrelationfunctionofthe for randomly placed pancakes Eq. (4) actually has the
c
Josephson current: fastest possible descend. Correlationsin the pancake ar-
rangementsalwaysincreasethespatialrangeofS(r). De-
sj2
σ = J drdthsinδφ(0,0)sinδφ(r,t)i (2) cayofthe dynamicpartisdeterminedbythe diffusionof
c
T Z mobile pancakes and is given by
HerejJ istheJosephsoncurrent,δφ=φ2−φ1−2Φπ0sAz is S(t)=exp −2πnDtln(rm2ax/Dt) (5)
the Gauge invariant phase difference between neighbor-
(cid:0) (cid:1)
ing layers; the phases φ (r,t) are mainly determined Substituting Eqs. (4) and (5) into Eq.(3) we obtain [13]
1,2
by the vortex positions R in the layers, φ (r,t) =
1,2i 1,2
Pdisitφrivb(urt−ioRn1a,r2oi(utn))dwthheerceoφrev(orf)a=sinagtalen(vyo/rxte)xi.sWtheecpahlacuse- σc ≈ 2πln(rmax/a10)ln(nrm2ax)nvsTjJn2D (6)
late σ in the lowestorder with respect to the interplane
c Up to logarithmic factors and numerical constant this
Josephson coupling. Within this approximation one can
result immediately follows from Eq. (2) assuming that
neglectcorrelationsbetweenJosephsoncurrentsindiffer-
the interlayer phase difference is correlated at distances
entlayers,whicharealreadyexcludedinEq.(2). Wecan
of the order of the average vortex spacing and at times
alsoneglectinterplanephasecorrelationsinthe sine-sine
of the orderof the typical“phase slip” time t =1/nD.
correlationfunctioninEq.(2). Thismeansthataverages ps
Using the relation j = 2πcE and Eq. (1) the last
like hcosδφi assumed to be zero and J Φo J
equation can be transformed to
hsinδφ(0,0)sinδφ(r,t)i≈(1/2)Re[S(r,t)],
Φ s2E2
with S(r,t)=hexpi(δφ(r,t)−δφ(0,0))i. σc ≈C B0 T2Jσab (7)
AsfollowsfromEq.(2)thevalueoftheout-of-planecon- with C = 2π . Therefore within our
ln(nrm2ax)ln(rmax/a0)
ductivityisdeterminedbyhowfastthephasecorrelations modelthe ratioofconductivities depends onlyupon ma-
decrease in space and time. The origins of decay of the terial parameters, temperature, and field, and does not
phase correlations in space and time are very different. depend upon the details of the mechanism, responsible
The decay in space is determined by the randomness in for dissipation. Eq. (7) represents the main result of our
the vortex arrangementwhile the decay in time is deter- Letter.
minebythevortexmotion. Inourmodelthestaticphase The model can be verified independently by measur-
difference δφ(0,0)−δφ(r,0) is established by the almost ing the frequency dependence of σ . The frequency de-
c
all rigidly pinned vortices [11] while dynamic behavior pendence can be taken into account by adding the fac-
is caused by the small amount of mobile vortices. In tor cos(ωt) under the integral in Eq.(2). This gives
such a situation the correlation function can be splitted Lorentzian frequency dependence σ (ω):
c
into the separately averaging static and dynamic parts,
S(r,t)≈S(r)S(t) and Eq. (2) becomes, σ (ω) ω2
c ps
= (8)
σ (0) ω2+ω2
sj2 c ps
σ ≈ J drdtS(r)S(t) (3)
c
2T Z The typical “phase slip” frequency
ω = 2πnDln(nr2 ) can be directly related to the in-
with ps max
planecomponentofresistivityρ =1/σ usingEq.(1),
ab ab
S(r)=hexp[i(δφ(0,0)−δφ(r,0))]i, which gives ωps ≈2108[1/s]·T[K]·ρab[µΩ·cm]. The di-
rect correlation between typical frequency in σ (ω) and
S(t)=hexp[i(δφ(0,0)−δφ(0,t))]i. c
ρ providesadditionalexperimentalcheckofthe theory.
ab
Let us discuss the limitations of our model. At low
In absence of correlations between vortex positions in
enoughtemperatureslowingdownofthepancakemotion
neighbor layers the static part is expected to decay at
in the layers leads to increase of the interlayer correla-
distances of the order of the average intervortex spacing
tions so that the diffusion of pancakes in different layers
a . This assumption can be supported by direct calcu-
0
can not be considered independent any more. Quanti-
lation for the case of randomly placed pancakes, which
tatively the influence of the Josephson coupling on the
gives [11]
diffusive motion is characterized by value of the Joseph-
S(r)≈exp −πn r2ln(r /r) (4) sonforce acting onpancake. This force is determinedby
v max
(cid:0) (cid:1)
2
the variation of Josephson energy under small pancake for fields 0.5-2 T a general trend is in agreement with
displacement and can be calculated as Eq. (7), i.e. the ratio decreases with temperature and
field. Takingtheconductivityanisotropyat60Kand1T
f (R,t)=−E d2r∂φv(r−R(t))sinδφ(r,t) (9) (≈2.7·10−6)andusingtherelationbetweentheJoseph-
Jα J
Z ∂rα soncouplingenergyandtheGinzburg-Landauanisotropy
ratio γ, E = Φ2/π(4πλγ)2s, we estimate from Eq. (7)
Neglecting againthe interplane phase correlationwe ob- J 0
the value of γ as 370 which is within the range reported
tain the correlationfunction of this force
in the literature [14]. An abrupt increase of the ratio at
πE2 ln(r /R) lowtemperaturesindicatesprobablytheemergenceofthe
hfJα(0)fJα′(R)i≈δαα′ n J ln(rmax/a)S(t) (10) global coherence in the system and the glass transition.
v max
To verify quantatively the validity of Eq. (7) we need
As one can see from this expression the Josephson force to know the temperature dependence of the Josephson
has typical amplitude fJ ∼ aEJ and is correlated at coupling energy. This dependence is determined by the
times ∼ tps and at large distances of the order of rmax. temperaturedependenciesofthetunnelingintegraltand
This means that mobile pancakes within the correlation superconductingorderparameterΨ,E ∝t|Ψ|2. Usually
J
area ∼ r2 will drift under this force with approxi- the first dependence is neglected. However, as recently
max
matelythesamevelocityv ≈µfJ duringthe“phaseslip” was proposed by Bulaevskii et al [15], the semiconduct-
time tps. This drift motion will produce an extra phase ing temperature dependence of the normal-state c-axis
changeδφ≈nrmaxµfJtps ≈rmaxfJ/T. One canneglect conductivity σcn at low temperatures can be ascribed to
the Josephsoncoupling and treatpancake motion in dif- the temperature dependence of t. We therefore assume
ferent layers independently when this extra phase shift that t and σ have the same temperature dependence.
cn
is small, which gives the condition rmaxfJ ≪ T. More The temperature dependence of σcn have been recently
preciseestimaterequiresevaluationofthein-planephase measured by Cho et al [5] for the temperatures down to
coherencedistancermax. Thisisbeyondthescopeofthe ≈ 55Kusing highmagnetic fields (up to 18T).This de-
simple model proposed in this Letter. pendence in the interval 55-90 K can be very well fitted
Severalmechanismslimitapplicabilityofthismodelat by the exponential formula σ ∝exp(T/29[K]). There-
cn
high temperatures. Quasistatic phase fluctuations with foreto verifyEq.(7)wetry to fitthe conductivityratios
the amplitude χ = (δφ)2 /2 lead to additional sup- by the formula
pressionoftheJoseph(cid:10)sonen(cid:11)ergy,whichcanbedescribed
by the Debye-Waller factor EJeff ≈ EJexp(−χ), and σc/σab(B,T)=A(B)exp(2T/29)(1−T/Tc)2T−2 (11)
reduces the ratio σ /σ as compared to Eq.(7). Two
c ab
contributions to χ come from thermal oscillations of the with A(B) being the fitting parameter. The factor
pinned pancakes (χ1) and from the spontaneous inter- 1−T/Tc accounts for temperature dependence of |Ψ|2.
plane phase fluctuations not related to pancake motion Asonecanseethisformulagivesverygoodfittothedata
(χ ), χ=χ +χ . We estimate this contributions as in the temperature range 45-70 K. However the field de-
2 1 2
pendence of the fitting parameter A(B) is weaker than
n T r2 α T J predicted by Eq. (7), A(B) ∝ B−0.52. The most prob-
χ ≈ v ln max L; χ = ln
1 αL T 2 4J Eeffξ2 able origin of this discrepancy is that increasing of the
J field reduces randomness in vortex arrangement within
where α ≈ sΦ j /(cξ) is the Labush parameter, j is the layers and, as a consequence, the static phase corre-
L 0 c c
the critical current, and J = sΦ2/(π(4πλ)2) is the in- lationsdecayinspaceslowerthanispredictedbyEq.(4).
0
plane phase stiffness. Taking j = 106A/cm2, T = 60K, In conclusion, we propose simple model, which pro-
c
B =1T, ln(r2 α /T)=5 we obtain the estimate χ ≈ vides the mechanism for thermally activated dissipation
max L 1
0.15. Using parameters of BiSCCO and taking the loga- along the c-axis in the vortex state of disorderedlayered
rithmic factor in χ as 5 we obtain χ ≈4[K]/(T −T). superconductors. The model gives the universal relation
2 2 c
Therefore the spontaneous interplane phase fluctuations between the conductivity components.
can be ignored at temperatures ∼ 10 K below T . I would like to thank A.I.Larkin for useful discussion.
c
Another important factor, which can not be ignored This work was supported by the National Science Foun-
at high temperatures, is the quasiparticle conductivity. dationOfficeoftheScienceandTechnologyCenterunder
This conductivity has to be added to the phase slip con- contract No. DMR-91-20000. and by the U. S. Depart-
ductivity (2) and therefore it enlarges the conductivity ment of Energy, BES-Materials Sciences, under contract
ratio in comparison with prediction of Eq.(7). No. W-31- 109-ENG-38.
To verify the prediction of the model we plot in Fig. 2
temperature dependencies of the conductivity ratio at
differentfields(the data[2]arethe sameasinFig. (1)).
One can see that within the temperature range 50-70 K
3
[1] T.T.M. Palstra et al, Phys.Rev.Lett. 61, 1662 (1988); FIGURE CAPTIONS
N.Kobayashiet al,Physica C159,295 (1989)
[2] R.Buschet al,Phys.Rev.Lett. 69, 522 (1992) FIG. 1. Temperature dependencies of the resistivity com-
[3] G.Bricen˜o, M.F.Crommie, and A.Zettl, Phys.Rev.Lett. ponents ρab and ρc for BiSCCO for fields 0.5,1,2,and 5 T.
66, 2164 (1991) Data are taken from Ref. [2]. Note that in spite of huge dif-
[4] Yu.I.Latyshevand A.F.Volkov, Physica C182,47 (1991) ference in absolute values, ρab and ρc have almost identical
[5] J.H.Cho et al,Phys.Rev. B, 50,6493 (1994) temperature dependencies.
[6] L.I.Glazman and A.E.Koshelev, Phys. Rev. B, 43,2835
(1991)
[7] P.H.Kes et al Supercond.Sci. Technol. 1, 242 (1989) FIG.2. Temperature dependencies of the conductivity ra-
[8] K.E.Gray and D.H.Kim Phys.Rev.Lett. 70, 1693 (1993) tio for the data from Ref. [2]. Solid lines give the fits by
[9] P.Berghuis et al, Phys. Rev. Lett. 65, 2583, (1990); the formula (11), which were made to verify the prediction
P.Berghuis, and P.H.Kes, Phys.Rev.B 47, 262,(1993) of Eq.(7). Table gives the values of parameter A at different
[10] A.E.Koshelev,PhysicaC198,371(1992)andtobepub- fieldsobtained from thefits.
lished
[11] A.E.Koshelev, L.I.Glazman, and A.I.Larkin Phys. Rev.
B, 53,????, (1996)
[12] We should emphasize that this statement is related to
large fields where the melting temperature is close to
the melting point of a single layer. As field decreases
the melting temperature increases due to 3D effects
and at some point becomes larger than the depinning
temperature. This means that the melting transition
is expected to be restored at small fields (< 1 kOe).
Ordered vortex lattice at small fields indeed was ob-
served by decorations (R.N.Kleiman Phys.Rev.Lett. 62,
2331 (1989)), neutron diffraction (R.Cubitt et al Na-
ture(London)365, 407 (1993)), andLorentzMicroscopy
(Harada et al Phys.Rev.Lett. 71, 3371 (1993))
[13] Note that Eqs. (4) and (5) give hexpiδφi = 0, which
is valid only if the interlayer coupling is completely ne-
glected. If small interlayer coupling is included than
the correlation function S(r,t) does not vanish at
large distances and times but saturates at S(∞,∞) =
hcosδφi2 ≪ 1. This large-scale behavior does not influ-
encemuchtheintegralinEq.(2)andresult(6),becausein
higherorderwithrespecttointerlayercouplingonehasto
keep term hexpi[δφ(0,0)+δφ(r,t)]i, which was skipped
in the transformation from Eq.(2) to Eq.(3). This term
term being added in Eq.(3) exactly cancels S(∞,∞) at
large scales and the main contribution still comes from
small distancesandtimeswhere approximations(4) and
(5) are applicable.
[14] K.Okuda et al, J.Phys.Soc.Jpn., 60, 3226, (1991);
J.Martinez et al, Phys.Rev.Lett. 69, 2276 (1992);
F.Steinmeyer et al, Physica C194-196,2401 (1994)
[15] L.N.Bulaevskii et al,unpublished
4
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