Crossover from Thermal Activation to Quantum Interlayer Transport at the Superconducting Transition Temperature of Bi-2212

1 MB·English

by  S. O. Katterwe

Crossover from Thermal Activation to Quantum Interlayer Transport at the
Superconducting Transition Temperature of Bi Sr CaCu O
2 2 2 8+δ
∗
S. O. Katterwe, A. Rydh, and V. M. Krasnov
Department of Physics, Stockholm University, AlbaNova University Center, SE-10691 Stockholm, Sweden
(Dated: February 2, 2008)
Weperform adetailed studyof temperature,bias anddopingdependanceofinterlayertransport
in the layered high temperature superconductor Bi2Sr2CaCu2O8+δ. We observe that the shape of
8 interlayer characteristics in underdoped crystals exhibit a remarkable crossover at the supercon-
0 ductingtransition temperature: from thermalactivation-typeatT >Tc,toalmost T−independent
0 quantum tunneling-type, at T < Tc. Our data indicates that the interlayer transport mechanism
2 may change with doping: from the conventional single quasiparticle tunneling in overdoped, to a
progressively increasing Cooper pair contribution in underdoped crystals.
n
a
PACSnumbers: 74.72.Hs74.45.+c74.50.+r74.25.Jb
J

8
1 How does the high temperature superconductivity constrains on the nature of the c−axis pseudogap. In

(HTSC) emerge with decreasing temperature? This the supplementaryinformation[15]wediscuss singleQP
]
n highly debated question is crucial for understanding the characteristics, self-heating and clarify determination of
o mechanism of HTSC. The superconducting transition in parameters.
c HTSC appears to be unusual, with a seeming lack of We study two batches of crystals: the Y-doped Bi(Y)-
-
r changes in the quasiparticle (QP) density of states at 2212withthemaximumTc ≃94.5KandthepureBi-2212
p Tc and persistence of the normal state pseudogap above withthemaximumTc ≃86K.Detailsofmesafabrication
u
s Tc [1, 2]. However, there is no consensus about the and characterizationare described elsewhere [7, 8, 13].
. T−evolutionof the energy gap. Results obtained by dif- Figs. 1 a) and b) show the T− variation of I−V and
at ferent experimental techniques range from complete T− dI/dV(V) curves for a Bi-2212 mesa, containing N = 7
m independence of the superconducting gap, ∆SG, in the intrinsic Josephson junctions, see Fig. 1 c). The follow-
- whole T− range [3], to strong T−dependence at T =Tc ingcharacteristicfeaturesareseenindI/dV curves: The
d [4, 5, 6, 7, 8, 9]. In some experiments, coexistence of sharp ”coherence” peak appears at T < T and is, in
c
n
o the twoenergy scalesbelow Tc wasreported[7, 8, 9, 10]. analogy with conventional low-Tc junctions, attributed
Theexistingcontroversyisoneofthemajorobstaclesfor to the sum-gap voltage V =2N∆ /e [7, 8, 11, 14]. A
c g SG
[ understanding HTSC andrequires further T−dependent broader hump is seen in the whole T− range and is at-
studies. tributed to the c− axis pseudogap[7, 8, 11]. At elevated
1
Intrinsictunnelingspectroscopyutilizestheweakinter- T thedI/dV curvescrossinnearlyonepoint[8,14]. The
v
1 layer (c−axis) coupling in layered HTSC. Unlike surface T−dependence of these features is plotted in Fig. 1 d)
2 probe techniques, it is perfectly suited for T−dependent Figure 2 a) shows dI/dV(V) curves (in a semi-log
9 studies and provides information about bulk electronic scale) for the most underdoped Bi(Y)-2212 mesa with
2 propertiesofHTSC.Inrecentyearsdifferentwaysofim- N = 34 junctions. The thin and thick lines in Fig. 2 a)
1. proving this technique were employed [7, 8, 11, 12, 13, represent dI/dV(V) curves below and above Tc, respec-
0 14]. Yet, interpretation of intrinsic tunneling character- tively. It is seen that the shape of interlayer characteris-
8 istics may not be free from misconceptions, associated tics exhibits a remarkable qualitative change at Tc:
0 bothwiththeproblemofself-heating[15]andthelackof (i)AboveT theslopeofln[dI/dV](V)curvesdecreases
c
:
v completeunderstandingoftheinterlayertransportmech- withincreasingT,whilethevoltagescaleremainsalmost
i anism. the same. The curves lean towards the horizontal line
X
Herewestudytemperature,biasanddopingdependen- dI/dV =σN, correspondingto the normalstate conduc-
ar ciesofintrinsictunnelinginsmallBi2Sr2CaCu2O8+δ (Bi- tance, and cross almost in one point.
2212) mesa structures. We observe an abrupt crossover (ii) Below Tc the ln[dI/dV](V) curves have the same
in the shape of current-voltage characteristics (IVC’s), T−independent slope. With increasing T the curves re-
from thermal activation (TA) like at T > T , to main parallel and move towards the vertical axis V = 0
c
T−independent quantum tunneling like at T < T . Our (i.e., the voltage scale decreases).
c
data indicates that the interlayer transport mechanism ExperimentalcurvesfromFig. 2a)haveacharacteris-
inBi-2212changesinunderdopedBi-2212,togetherwith tic V-shape with voltage-independent slope. The overall
development of the pseudogap. The observed simple TA shape of the curves at T >Tc closely resembles thermal
behavior in the whole normal state region puts strong activationcharacteristics[16],whichcanbe describedby
a simple expression [15]:
dI n(T) U eV
(T,V)∝ exp − TA cosh , (1)
∗Electronicaddress: [email protected] dV T (cid:20) k T(cid:21) (cid:20)2k T(cid:21)
B B
2
FIG.1: (Coloronline). a)I−V andb)dI/dV(V)characteris-
ticsforaBi-2212mesaatdifferentT. c)Periodicquasiparticle FIG.2: (coloronline). a)Asemi-logplotofdI/dV(V)curves
branches. d)Temperaturedependenceofthepeak,humpand foranunderdopedBi(Y)-2212mesaatdifferentT. Notethat
crossing voltages (perjunction). below Tc the curves maintain the same slope, while above
Tc the slope changes progressively with T. A characteristic
crossingpointofdI/dV(V)curvesatT >Tcismarkedbythe
where n is the concentration of mobile charge carriers, arrow. b)Theeffectivetemperature,obtainedfromtheslopes
ofthedI/dV(V)curvesatafinitebiasV/N =30mV [15]. A
andU is the TAbarrier. Indeed, the cosh termrepro-
TA
remarkable crossover from the thermal activation, T =T,
duces the rounded V-shape of ln[dI/dV](V) curves with eff
the slope that monotonously increases as 1/T. to quantum,Teff =const,behavior occurs at Tc.
InFig. S10[15]weshowthatthedatafromFig. 2a)in
thewholenormalstate,T >T ,isverywelldescribedby
c
Eq.(1) with constantU ≃24meV andT−linearn(T). (the exp term in Eq.(1)). Experiments on small mesas
TA
It also reproduces the crossing point [16], which accord- provide a possibility to measure the small-bias QP re-
ing to Eq.(1) occurs at eV ≃ 2U . Apparently, the sistance, RQP, in the superconducting state [11, 12, 19],
TA
TAbarriershouldbe associatedwiththec−axispseudo- whichisotherwiseshuntedbythesupercurrent[20]. This
gap. Thus,theappearanceofthecrossingpointissimply isfacilitatedbythe largespecific capacitance(highqual-
the consequence of a fairly T−independent c−axis pseu- ity factor) of intrinsic junctions [18], which causes the
dogap [1, 3, 7, 8], while the crossing voltage represents pronounced hysteresis in the IVC’s and allows junctions
the pseudogapenergy. Indeed, a correlationbetween the to remain in the resistive state even below the critical
hump and the crossing voltages is seen from Fig. 1 d). current,seeFig. 1c). Therefore,biasyieldanadditional
According to Eq.(1) the slope of the curves in Fig. parameter for our studies, which may render crucial for
2 a) should be proportional to the reciprocal temper- correct interpretation of the data.
ature: d/dV(ln[dI/dV]) = (e/2k T)tanh(eV/2k T) ≃ Fig. 3a)showsthedc−resistanceR =V/I atdiffer-
B B dc
e/2k T. Fig. 2 b) shows the effective temperature, ent I for the same Bi-2212 meas as in Fig. 1. Measure-
B
T , obtained from the slopes at V/N =30mV [15]. It ments were done by first applying a large current, suffi-
eff
is seen that in the normal state T ≃ T, confirming cient for switching to the last QP branch, see Fig. 1 c),
eff
the TA nature of interlayer transport at T > T . How- and then ramping it back to the desired value. The R
c dc
ever, at T < T an abrupt saturation of T (T) oc- at the smallest current I = 5µA coincides with the con-
c eff
curs, typical for quantum tunneling transport [17, 18]. ventional ac− resistance, R . It drops at T < T , since
ac c
This is the central observation of this work. Remark- the finite quality factor of the junctions causes retrap-
ably, the crossover is clearly distinguishable at T → T , ping to the superconducting branch at a finite current
c
althoughno other spectroscopicfeatures can be resolved [18]. From Fig. 3a) it is seen how non-linearity develops
in dI/dV(V) curves. with decreasing both bias and T. At small I the curves
So far we discussed dI/dV(V) at finite bias. Another approach an asymptotic, allowing a confident estimate
straightforward way to investigate the TA behavior is of the zero-bias quasiparticle resistance, RQP, without a
0
to consider T−dependence of zero-bias resistance, R0, need for extrapolation.
3
1000
R QP
UD(T=85K),
c 0
OP(93K),
100 OD(92.5K). R
K) TA
/
Ω
( Bi(Y)-2212
T 10
/0 2 OD(92.5)
R A
T
R OP (93)
1 R/01
R
ac UD (85)
0.01 0.02 0 0.03 0.04
50 100 150 200
1/T (K-1) T (K)
FIG.4: (Coloronline). T−normalizedzero-biasquasiparticle
resistance, R0QP,extrapolated from IVC’sat T <Tc andRac
measured with a small ac-current for Bi(Y)-2212 mesas with
FIG. 3: (Color online). a) T− dependence of the dc− resis- different doping. Dashed lines represent TA fits, Eq.(1), at
tance at different bias currents for the same Bi-2212 mesa. T > Tc. Inset shows relative values of the excess QP resis-
Panels b-d) show the asymptotic zero bias resistance: b) in tance with respect to theTA fit of thenormal state. A large
the double-logarithmic scale; c) as a function of 1/T; and d) excess resistance appears below Tc in the overdoped mesa.
normalized by T: a clear linear thermal activation behavior The excess resistance rapidly decreases with decreasing dop-
(dashed line) is observed in the whole normal state region. ing and becomes negative in theunderdoped mesa.
The RQP grows with decreasing T at T < T , and (OP)mesaandturns negative alreadyinmoderatelyun-
0 c
is usually described either in terms of power-law depen- derdoped (UD) mesas.
dence [20], inherent for single QP tunneling in the pres- Inset in Fig. 4 shows the magnitude of the excess QP
ence of a d−wave gap (see Fig. S7 [15]), or in terms of resistance with respect to the TA fit, R , in the nor-
TA
TA [21]. The double-logarithmic plot, Fig. 3 b), demon- mal state (dashed lines). The quality of the fit could
strates that R0(T) is not described by the power law in be judged from the flatness of the normal state region.
any extended T−range for our mesas. Neither is it per- A progressive decrease of the excess QP resistance at
fectly described by the Arrhenius law exp(UTA/kBT), T <Tc with decreasing doping is obvious.
as demonstrated in Fig. 3 c). On the other hand, Fig. Westartdiscussionoftheobservedphenomenabyrec-
3 d) demonstrates that the ratio R0/T follows very ac- ollecting the expected T−variationfor conventionalsin-
curately the Arrhenius law (dashed line) in the whole gle QP tunneling characteristics. As discussed in sec. I
normal region, consistent with the TA expression Eq.(1) of the supplementary information, provided the physi-
and with the finite bias behavior,Fig. 2 b). However,at cal requirement that ∆ and the quasiparticle lifetime
SG
T <T , RQP/T deviates downwardsfrom the Arrhenius increasewith decreasingT,allsingleQPtunneling char-
c 0
law. This is consistent with saturation of the effective acteristics (including d−wave) exhibit the following uni-
T (T < T ), as shown in Fig. 2 b). Therefore, it is a versal T−dependence:
eff c
consequence of the same crossover. A) Opening of the superconducting gap at T < T
c
In Fig. 4 we analyze doping dependence of R0(T)/T. leadstoappearanceofthelargeexcessQPresistancedue
In the normal state, R0(T) varies in the TA manner for torapidfreezingoutofQP’supontheircondensationinto
alldopinglevels. Inthesuperconductingstate,RQP also Cooper pairs (see inset in Fig. S7 [15]).
0
continues to grow with decreasing T, but the rate of the B)RQP continuestogrowwithdecreasingT bothdue
0
growth with respect to the TA behavior, RTA, (dashed tofreezing-outofthermalQP’sanddue togrowthofthe
lines) strongly depends on doping: QPlifetime,whichdecreasesthenumberofavailablesub-
For the overdoped (OD) mesa, RQP(T < T ) grows gap QP states. All this leads to a progressive growth of
0 c
muchfasterthaninthenormalstate,i.e.,thereisasharp the slopeofdI/dV(V)characteristicswithdecreasingT.
onset of the excess resistance at T = T . However, the The sudden appearance of excess resistance at T <T
c c
excess resistance at T < T decreases rapidly with un- in OD mesas indicates that the superconducting transi-
c
derdoping. Itbecomessmallinthenearoptimallydoped tionhereisconventional,inasensethatitisaccompanied
4
by opening of the superconducting gap, and is also con- shots in underdoped Bi-2212 could arise from doping-
sistentwithsingleQPtunneling mechanismofinterlayer dependent nano-scale inhomogeneity, observed at the
transport. surfaceofBi-2212[26,27], whichinthis caseshouldper-
However, the superconducting transition in under- sists also in the bulk of Bi-2212 crystals.
doped mesas is clearly abnormal. We emphasize again Such interpretation implies that not only the elec-
thatappearanceofthenegativeexcessresistanceandthe tronicstructure,butalsothec−axistransportmechanism
crossover to T−independent slope reflect the same phe- changeswithunderdoping: fromcoherentanddirectional
nomenon at zero and finite bias, respectively. Therefore, [15] single QP tunneling in overdoped, to progressively
any explanation of the observed unusual T−dependence increasingpaircontributioninunderdopedBi-2212. The
must account for both zero and finite bias behavior. latter is apparent only in the phase coherent state at
At the first glance, the lack of excess resistance at T <Tc,isalmostT−independent,andisconsistentwith
T < T in UD mesas could be consistent with the pre- multiple Andreev reflection mechanism of the interlayer
c
cursor superconductivity scenario of the pseudogap [3], transport.
according to which the gap does not open and QP’s do Finally we note that the reported amazingly trivial
not start to pair at T , but at much higher T∗. Then thermalactivationbehaviorinthewholenormalstatere-
c
onecouldarguethatthe lackofexcessresistancereflects gion, puts strong constrains on the nature of the c−axis
simply the lack of dramatic changes in the QP spectrum pseudogap. Even though, the similar TA-behavior can
upon establishing of the global phase coherence at T . be ascribed to several processes (e.g., inelastic tunneling
c
However, the same argument would make it difficult to via the impurity [27, 28], or elastic tunneling via a reso-
explain the abrupt crossover to T−independent slope. nant[29]stateinthetunnelbarrier,orCoulombblocking
Furthermore, as we noted above, the slope of dI/dV(V) of tunneling [7]), it does not involve any gap, nor angu-
should continue to increase with decreasing T for single lardependenceintheQPspectrum,butassumesinstead
QP tunneling. Therefore, the observed crossover to con- thatthereissomeconstantblockingbarrierforinterlayer
stantslopecanhardlybeexplainedintermsofsingleQP hoping [16]. This may indicate that the ”large” c−axis
tunneling [15]. pseudogapisnotapairinggap,whichwouldnaturallyex-
plain it’s indifference to such classical depairing factors
This conclusion brings us to one possible interpreta-
tion of the observedphenomena. If the c−axistransport as temperature and magnetic field [7].
We are grateful to E.Silva for stimulating discussions
inUDmesasisnotsolelyduetosingleQP’s,itmustalso
and to A.Yurgens, T.Benseman and G.Balakrishnan for
involve pairs. The corresponding multi-particle tunnel-
providingBi(Y)-2212andBi-2212singlecrystals. Finan-
ing [22, 23] and the multiple Andreev reflection [24, 25]
processes are well studied for conventionallow−T junc- cial support from the K.&A. Wallenberg foundation and
c
tions. Bothprocessesaresimilar[25] andarealmostT− theSwedishResearchCouncilisgratefullyacknowledged.
independent, becausetheydonotrelyonthepresenceof
thermallyexcitedQP’s,butdirectlyinvolveCooperpairs
I. SUPPLEMENTARY INFORMATION
from the Fermi level. The multi-particle current occurs
via elastic conversion (dissociation or recombination) of
Cooper pairs into QP’s; while the multiple Andreev re- A. Nomenclature
flectionprocessisduetotransmutationofaquasielectron
into a quasihole with creation of a Cooper pair and can ARPES:angularresolvedphotoemissionspectroscopy,
beuniversallydescribedintermsofinelastictunnelingin Bi-2212: Bi2Sr2CaCu2O8+δ,
the presence of time-dependent phase difference due to Bi(Y)-2212: Y-doped Bi-2212,
the ac-Josephsoneffect [25]. DoS: density of states,
Importantly, the multi-particle processes result in al- ∆: energy gap,
mostT−independentandV−exponentialsubgapcurrent ∆T: self-heating,
[22, 23], which essentially coincides with the description Γ: depairing factor, reciprocal quasiparticle lifetime;
of the observed crossover, see Fig. 2. Furthermore, the HTSC: high temperature superconductor,
abruptness of the crossover at T , points towards the I: current,
c
coherent multiple Andreev reflection mechanism of the IJJ: intrinsic Josephson junction.
interlayer current, because it requires the time-periodic IVC: Current-Voltage characteristic,
ac-Josephson effect [25] and, consequently, abruptly dis- N: number of IJJ’s in the mesa,
appears simultaneously with phase coherence at T . n: concentration of mobile charge carriers,
c
We may further speculate why the multiparticle pro- OD, OP, UD: overdoped, optimally doped, under-
cesses become progressivelymore important with under- doped, respectively;
doping. The multiparticle current decrease much faster QP: quasiparticle,
than the single QP current with decreasing the interface R0: zero bias resistance (in the normal state),
transparency [22, 25]. Therefore, the single QP domi- RQP: zero bias quasiparticle resistance in the super-
0
nates over multiparticle current, unless there are micro- conductingstate,isdifferentfromR0 becauseofthepro-
shots in the tunnel barrier [23]. The required micro- nounced hysteresis in the IVC’s at T <T ;
c
5
R : tunnel (normal) resistance, In Fig. S5 a) and b) we show numerically simulated
n
R : thermal resistance of a mesa, SIS characteristics for different single QP tunneling sce-
th
SIS: Superconductor-Insulator-Superconductor, narios. We assumed, t(ϕ1,ϕ2) =const for incoherent-
SIN: Superconductor-Insulator-Normalmetal, nondirectional, t(ϕ1,ϕ2) ∝ δ(ϕ1 − ϕ2) for coherent-
STM: scanning tunneling microscope, nondirectionalandt(ϕ1,ϕ2)∝[cos(kx)−cos(ky)]2δ(ϕ1−
TA: thermal activation, ϕ2) for coherent-directional tunneling [31]. The simu-
lations were made for ∆(ϕ = 0) = 35meV, at low T
T : superconducting critical temperature,
c and small depairing Γ ≪ ∆(0). Detailed discussion of
U : thermal activation barrier,
TA dI/dV characteristics can be found in Ref. [30]. Irre-
V: voltage,
spective of the scenario,the zero-biasresistancediverges
Vg: the sum-gap voltage Vg =2N∆/e (the conductance tends to zero) at T → 0 and Γ = 0.
ThecapitalM/Sinfrontofthefigureorequationnum- The divergence is removed by the finite Γ, see the inset
berindicatesthatthereferenceismadetotheManuscript in Fig. S6 b). Therefore, dI/dV(V = 0,T → 0) pro-
or the Supplementary information, respectively. videsthe informationaboutthe value ofΓ. We alsonote
that dI/dV(V) characteristics exhibit a sharp peak at
the sum-gap voltage V = 2∆/e ( except for the curve
g
B. Influence of the quasiparticle life time on dI/dV A).
characteristics It is expected that interlayer QP tunneling in Bi-2212
single crystals should be predominantly coherent (pro-
We have argued in the manuscript that the observed videdthatthe singlecrystalis pureenoughsothatthere
crossovertoT−independentslopeandthecorrelatedap- isnomomentumscatteringupontunneling)andstrongly
pearanceofthenegativeexcessresistanceareinconsistent directional,withdominatingantinodaltunneling[31,32].
with the single QP tunneling mechanism of the c−axis Indeed, the curve C in Fig. S5 resembles most closely
transport. However, the interlayer tunneling in Bi-2212 the experimental characteristics (cf. with the curves at
may depend on a number of parameters [30, 31, 32]: T = 13.4K from Fig. M1a and b), see also a discussion
in Refs. [30, 34].
(i)TheQPlife time,1/Γ,which,accordingtoARPES
data,hasasubstantialT−dependencecloseto T ,where In what follows, we will analyze the incoherent-
c
it increases roughly linearly with T; [9, 33] nondirectional single QP tunneling characteristics, rep-
resenting an ”extreme” d−wave case with the weakest
(ii)Momentumconservationupontunneling(coherent
sum-gappeak. The conclusionswill, however,be univer-
vs. incoherent tunneling);
(iii) The directionality of c−axis tunneling, i.e., the sal for any single QP tunneling scenario.
angular dependence of the tunneling matrix element NowweconsidertheeffectofT (pointiv)andΓ(point
t(ϕ1,ϕ2), caused by a non-spherical Fermi surface [31, i) on dI/dV characteristics. It is clear that both param-
32]. eters will smoothen dI/dV(V) characteristics and fill-in
the dip in conductance at V =0. However, they do this
(iv) Temperature and (v) angular dependence of the
in a slightly different manner. This is illustrated in Figs.
gap in the DoS and the points (i-iii) above. Deviations
from pure d−wave symmetry were reported in recent S6 and S7.
ARPES experiments [9]. TheincreaseofΓ leadstosmearingofthe QPgapsin-
gularity and simultaneous filling-in of the sub-gap states
It may not be obvious that experimental data can not
in DoS, Eq.(S2). This naturally leads to smearing of
bedescribedbythesingleQPtunnelingwithafortunate
the sum-gappeak and to filling-in of the zero-biasdip in
combination of all those parameters. Therefore, below
dI/dV, as shown in the inset in Fig. S6 b).
we want to dwell upon this statement.
TheincreaseofT leadstothedecreaseof∆. Bothfac-
We start with considering the consequences of points
torsresultintheincreasednumberofexcitedQP’sabove
(ii, iii) and (v), i.e., coherence and directionality of tun-
the gap, which leads to rapid filling-in of the zero bias
neling and the symmetry of the order parameter. The
dip in conductance. However, the increase of T (unlike
single QP tunneling current is given by:
Γ) keeps the gap singularity in DoS unchanged. From
2π 2π +∞ Fig. S7 it is seen that this leads not only to filling-in
I(V)= dϕ1dϕ2dE (2) of the dip, but also to development of the maximum at
Z0 Z0 Z−∞ V = 0, representing the so called zero-bias logarithmic
t(ϕ1,ϕ2)ρ(E,ϕ1)ρ(E+eV,ϕ2)f(E)[1−f(E+eV)], singularity [35].
Theoriginofthesingularityisverysimple: atelevated
where E is the energy of the QP with respect to the T thereisasubstantialamountofthermallyexcitedQP’s
Fermisurface,ϕ1,2aretheanglesinthemomentumspace just above the gap. At V = 0 the partly filled gap sin-
of the initial and the final state of the QP, and ρ(E,ϕ1) gularities in the two electrodes are co-aligned, causing a
ρ(E+eV,ϕ2) are the corresponding QP DoS: largecurrentflowfromoneelectrodetoanother,whichis
exactlycompensatedbythe counterflowfromthe second
ρ(E,ϕ)=ℜ[(E−iΓ)/ (E−iΓ)2−∆(ϕ)2], (3) electrode. However,exactcancelationisliftedatanarbi-
p
6

Incoherent,
a)
non-directional
d-wave tunneling.
V) 70 T = 5K, 0.5,
m ∆(T) = 35 meV. 3.2,
(
R n 5.8,
I 35 Γ = 11.1
(meV) 21.6,
40.0,
63.7.
0
0 35V (m V) 70
n b)
R
V 1
d
I/
d 1 T = 5K
n
R
0,1 (0) 0,1 Γ 1.36
V
d
I/0,01 Γ (meV)
d
1 10
0,01
0 35 V (mV) 70
FIG. 5: (Color online). Simulated single QP characteris- FIG.6: (coloronline). Effectofthedepairingfactoronsingle
tics for different tunneling scenarios. A: incoherent, non- QP characteristics. Shown are simulated characteristics at
directional, d−wave; B: coherent, non-directional, d−wave; constantT and∆forvaryingΓ. IncreaseofΓtendstoreduce
C: coherent, directional, d−wave; D: s−wave. Note that the theslopeofdI/dV characteristics. Insetinpanelb)indicates
sharpsum-gappeakandthehalf-gapsingularityindI/dV are that filling-in of the zero bias dip with Γ occurs in a power
inherent to most scenarios. law manner.
trarysmallvoltageacrossthejunction,leadingtoasharp
maximum in dI/dV. From Fig. S7 it is seen that the
zero-bias maximum in d−wave junctions is pronounced
even in the extreme case of weakest sum-gap peak. We
havecheckedthatthezero-biaslogarithmicsingularityis
inherent for single QP characteristics,irrespective of the
It is impossible to maintain the slope under the physical
tunneling scenario.
requirements that Γ increases and ∆ decrease simulta-
Interestingly, the experimental characteristics do not neously with increasing T because all of the three pa-
exhibit the zero-bias logarithmic singularity, see Fig. rametersworkinthe samedirection-decreasetheslope.
M2a). Within the single QP tunneling scenario this is The inset in Fig. S7 shows the calculated zero bias re-
only possible if the depairing factor increases with T at sistance vs. 1/T for incoherent, non-directional,d−wave
sucharate thatit smearsoutthe gapsingularityinDoS single QP tunneling and different Γ. It is seen that a
and thus suppresses development of the singularity. We prominentexcessresistanceappearseveninthisextreme
estimate that the corresponding Γ at T = Tc should be d−wave case and even for very large Γ. The dashed line
in the range 2-5 meV, somewhat smaller than deduced shows that the excess resistance grows approximately as
from ARPES [9, 33]. T−2 in this case. The abrupt appearance of the excess
Now we can substantiate our statement. In Fig. S8 QP resistance is a mere reflection of the abrupt opening
we show the attempt of maintaining the same slope of of the superconducting gap at T . Therefore, the ob-
c
lndI/dV(V)curvesforprogressivelydecreasing∆. Since served T−independent slope and the absence of excess
increaseofbothΓandT leadstofilling-inofthezerobias resistance, observed for UD mesas, are inconsistent with
dip, i.e., to decrease of the slope, the slope can be main- the single QP tunneling and points towards the doping-
tained only if Γ and T move in the opposite directions. dependentchangeintheinterlayertransportmechanism.
7
where P = IV is the dissipated power and R is the
th
effective thermal resistance of the junctions, which is
T−dependent and, therefore, bias dependent [36]. The
1 influence of self-heating on IVC’s of our mesa structures
n
IVRd/d 0,1 2.0 K,RR / 0n10 Γ25(0,T ,0c )1.1(0m, eV)= waFiadgsi.stc3huofsrrsooiomungRholfeyfs.cs[7atu]l)idn.igedofinIVRCe’fss.i[n34th,e36in,s3e7t,o3f8F]i(gs.ee1aalnsdo
Despite relative simplicity of the phenomenon, discus-
24.8 K, T -2
0,01 47.6 K, sionofself-heatinginintrinsictunnelingspectroscopyhas
62.8 K, 1 causedaconsiderableconfusion,alargepartofwhichhas
78.0 K, 0,00 10/,T02 (K-10),04 been caused by a series of publications by V.Zavaritsky
1E-3 93.2 K. [39], in which he ”explained” the non-linearity of intrin-
0 35 70
V (mV) sictunneling characteristicsbyassumingthatthereisno
intrinsic tunneling. The irrelevance of this model to our
subject was discussed in Ref.[38].
A certain confusion might be also caused by a large
spread in R , reported by different groups [14, 36, 40,
th
FIG. 7: (color online). Effect of T and ∆(T) variation on
41, 42]. For the sake of clarity it should be empha-
single QP characteristics. Calculations were made for inco-
sized that those measurements were made on samples
herent, non-directional, d−wave tunneling, the experimental
of different geometries. It is clear that R depends
∆(T) dependence and linear Γ(T). Note that increase of T th
strongly on the geometry [36, 37], e.g., R can be
anddecreaseof∆leadsnotonlytosmearingofdI/dV charac- th
teristics, butalsotheappearanceofthelogarithmiczero-bias much larger in suspended junctions with poor thermal
singularity. Inset shows the zero bias resistance vs. 1/T for link to the substrate [42] than in the case when both
different values of Γ(Tc). It is seen that the excess QP resis- top and bottom surfaces of the junctions are well ther-
tance rapidly develops at T <Tc even in thed−wavecase. mally anchored to the heat bath [14]. For mesa struc-
tures similar to those used in this study (a few µm in-
plane size, containing N ≃ 10 IJJ), there is a consensus
that R (4.2K) ∼ 30−70K/mW (depending on bias)
th
[36, 41] and R (90K)∼5−10K/mW [36]. Larger val-
th
ues R > 100K/mW claimed by some authors [40] are
th
1 unrealisticforourmesasbecausetheycanwithstanddis-
Incoherent, sipatedpowersinexcessof10mW withoutbeingmelted.
n
R non-directional Yet,wenotethattalkingaboutatypicalvalueofR is
Vd d-wave tunneling. equallysenselessastalkingaboutatypicalvalueofacthon-
I/ Γ =1, ∆=35 (meV), T= 59K; tact (Maxwell) electrical resistance: both depend on the
d
Γ =2, ∆=35 (meV), T=55.2K; geometry. Therefore, reduction of mesa sizes provides
Γ =5, ∆=35 (meV), T= 36.2K;
a simple way for reduction of self-heating [37]. Conse-
0,1 Γ =5, ∆=17.5 (meV), T= 32.4K;
quently,variationofdI/dV characteristicswiththejunc-
Γ =7, ∆=10 (meV), T= 0.1K.
tion size and geometry provides an unambiguous way of
0 35V (mV) 70 discriminating artifacts of self-heating from the spectro-
scopic features [36].
How self-heating can distort the IVC’s of Josephson
junctions is obvious: since self-heating rises the effective
T it may affect the IVC only via T−dependent parame-
FIG. 8: (Color online). An attempt to maintain the slope of ters. There are three such parameters: the quasiparticle
single QP characteristics at different T. It is seen that the
resistance,thesuperconductingswitching[18,43]current
slope can be maintained only under the un-physical require-
andthe superconducting gap. They will affect the IVC’s
ment that Γ increases and ∆ decreases with decreasing T.
of Bi-2212 mesas, containing several stacked Josephson
junctions, in the following manner:
The consecutive increase of T upon sequential switch-
C. Analysis of self-heating and non-equilibrium
ing of IJJ’s from the superconducting to the resis-
phenomena
tive state will distort the periodicity of quasiparticle
branches. Each consecutive QP branch will have a
Self-heating can distort IVC’s of Josephson junctions smallerQPresistance(smallerV atgivenI)andsmaller
atlargebias. The temperature rise due to self-heating is switchingcurrent,seethediscussioninRefs. [7,38]. This
given by a simple expression type of distortion of the QP branches becomes clearly
visible (at base T =4.2K) when ∆T &20K [37, 38].
∆T =PR (T), The T−dependence of ∆ may lead to appearance of
th
8
of self-heating the ”measured” gap vanishes at T = T ,
c
i.e., simultaneously with the true gap. Thus trivial self-
T(K)=90,80,70,50,30,4.2 T(K)=90,80,70,50,30,4.2 heating simply can not ”hide” the qualitative ∆(T) de-
V) 80 without V)80 0 with pendence. Forexample,thereisnowayinwhichonecan
m 60 self-heating m60 self-heating get the vanishing ”measured” (self-heating affected) gap
R (N40 R (N40 ifthetruegapisT−independent. Therefore,thediscrep-
I I ancy in the measured strong T−dependence of the gap
20 a) 20 b) at T in intrinsic tunneling (and recent ARPES [9]), or
c
0 0 completeT−independenceinSTMmeasurements[3]can
0 20 40 60 80 0 20 40 60 80
V (mV) V (mV) not be attributed to self-heating.
60 To quantitatively analyze the significance of self-
T(K)=90,80,70,50,30,4.2 d)
120 0 V)50 heatinginourmesas,weprovidevaluesofthedissipation
T (K) 4800 e∆2/ (m234000 w withitho ut gBpaoi(pwYep)r-e2Pa2k1=2isImV0e.s8af2omratWthTe. s=tFur4do.i9meKdtmhdeeisssapispr:eavtFiiooornustahtaentanhleeyasrsi-usOmoP-f
c) 10 self-heating the size dependence of V for similar mesas [36] it was
g
0 0 observed that V becomes size-independent, and there-
g
0 20 40 60 80 100 0 20 40 60 80
V (mV) T (K) fore not affected by self-heating, for small mesas with
P(V )<1mW. Allthe mesasstudiedinthismanuscript
g
fall into this category. All of them exhibit perfect pe-
riodicity of quasiparticle branches [7] and none of them
exhibit back-bending at any T, which according to Fig.
FIG.9: (Coloronline). Simulateddistortion ofSIStunneling
S9, implies that self-heating at V is less than T /2 at
characteristics by strong self-heating. a) Undistorted IVC’s g c
T = 4.2K. This is consistent with previous in-situ mea-
at different T. Simulations weremade fortypicalparameters
of ourmesas, T−dependentthermalconductivityof Bi-2212, surements[36],Rth(T =4.2K,V =Vg)≃30−40K/mW.
andforcoherent,directional,d−wavetunneling. b)distorted Therestofthedatapresentedinthemanuscriptisnot
IVC’s at the same base temperatures; c) The mesa tempera- affected by self-heating for the following reasons:
tureasafunctionofbias. d)Temperaturedependenceofthe
The effective T in Fig. M2b) was obtained at
eff
genuinesuperconductinggap(solidline)andthe”measured”
V/N = 30meV at which the dissipation power was only
gapobtainedfromdistortedIVC’s(dashedcurve). Notethat
P ≃ 3.8µW at T = 5.5K (for comparison, P(V ) =
even strong self-heating (T reaches Tc/2 at Vg at 4.2K) does g
0.21mW for the same curve). Therefore, self-heating
not cause considerable distortion of the measured gap. Data
from Ref.[34] here is negligible (sub-Kelvin). Besides, as arguedin the
manuscript, the slope of lndI/dV(V) in Fig. M2 is ap-
parently V and P independent, therefore the same data
back-bending of the IVC at the sum-gap knee. Fig. S9 couldbeobtainedatmuchsmallerorlargerP. Thedata
reproduces the results of numerical simulation of such inFigs. M3b-d)andM4isfreefromself-heatingbecause
the distortion, made specifically for the case of Bi-2212 it was obtained at zero bias.
mesa with the corresponding T−dependent parameters Finally, it is important to emphasize that the concept
(see Ref.[34] for details). Fig. S9 a) shows a set of simu- of heat diffusion is inapplicable for small Bi-2212 mesas
latedundistortedIVC’satdifferentT forcoherent,direc- containing only few atomic layers. The phonon trans-
tional,d−wavetunnelingwitha∆(T),shownbythesolid port in this case is ballistic [36, 45] and the energy flow
line in Fig. S9 d). Panels b) and c) show the distorted from the mesa is determined not by collisions between
IVC’s and the actual junction temperature, respectively. the tunneled non-equilibriumQPwith thermalphonons,
The dashed line in panel d) represents the ”measured” but by spontaneous emission of a phonon upon relax-
gapobtainedfromthepeakindistorteddI/dV character- ationofthe non-equilibriumQP[13]. Thisprocessisnot
istics. Remarkably, the deviation from the true ∆(T) is hinderedatT =0. Therefore,theeffectiveR (andself-
th
marginal,despite largeself-heating,∆T ≃T /2at4.2K! heating)canbemuchsmallerbecauseitisnotlimitedby
c
The robustness of the measured gap with respect to poor thermal conductivity at T = 0, but is determined
self-heating is caused by a flat T−dependence of the su- by the fast, almostT−independent, non-equilibrium QP
perconducting gap at T < T /2. If self-heating reaches relaxationtime. Theconceptofself-heatingbecomesad-
c
T at the sum-gap knee, an acute back-bending appears equate only inthe bulk ofthe Bi-2212crystal,where the
c
in the IVC’s, however even this does not cause princi- dissipation power density and the temperature rise are
ple changes in the ”measured” ∆(T). In experiments on much smaller due to the much larger area of the crystal.
large mesas [40] or suspended structures [42], in which FormoredetailsseethediscussioninRef. [36]. Thenon-
acute self-heating was reported, no clear Ohmic tunnel- equilibriumenergytransferchannelis specific for atomic
ing resistance could be observed at high bias, in stark scaleintrinsicJosephsonjunctionsmadeofperfectsingle
contrast to our mesas, see Fig. M1a, and Refs.[7, 8, 44]. crystals. It can explain a remarkably low self-heating at
SimulationsasinFig. S9clearlyshowthatirrespective very high bias [13].
9
that for strongly UD mesas it is R0 (circles in Fig. S10
b), rather than R0/T, that is more accurately described
by the Arrhenius law (dashed line).
From comparison of Figs. M2a) and S10a) it is seen
that Eq.(M1) does provide a good fit to experimental
data at T > T , including the crossing point. Simulta-
c
neously a dramatic discrepancy is seen below T . This
c
indicates that the slopes of dI/dV(V) curves in the su-
perconducting state are not determined by the real T,
but by some constant effective temperature, as shown in
Fig. M2 b).
Fig. S10 b) shows the slope of experimental curves
from Fig. M2a) at V/N = 30mV (red squares) and
the logarithm of zero bias resistance (green circles) as
a function of 1/T. Clear linear dependence of both at
T > T indicates that the IVC’s in the normal state
c
have thermal activation nature both at zero and finite
bias, consistent with Eq.(M1). According to Eq.(M1)
theeffectivetemperaturecanbeexplicitlyobtainedfrom
the slope d/dV[ln(dI/dV)] = (e/2k T)tanh(eV/2k T)
B B
≃ e/2k T. Here there are two slight obstacles: First,
B
the tanh term deviated from unity at high T leading
to eventual saturation of the slope at 1/T → 0. The
exact T−depensence according to Eq.(M1) is shown by
the solid (magenta) line in Fig. S10 b). We emphasize
that there are no fitting parameters in this curve. It is
seen that Eq.(M1) reproduces quite well T−dependence
FIG. 10: (Color online). a) Simulated dI/dV characteristics
of the slopes, except for a small offset. This additional
according to Eq.(M1) for the same T as in Fig. M2 a). It is
offset is caused by the fact that the slope of experimen-
seen that at T > Tc they provide a good fit to experimental
tal curves becomes negative at large T [8]. The simple
data, including the crossing point. b) The slope of experi-
mentalcharacteristics from Fig. M2a) atV/N =30mV (red TA expression, Eq.(M1), does not reproduce this nega-
squares, left axis) and the logarithm of zero bias resistance tive slope. However, more advanced TA simulations do
(green circles, right axis) vs. 1/T. It is seen that both fol- reproduce the negative slope and, in fact, all the details
low the TA behavior at T > Tc and simultaneously deviate of experimental dI/dV characteristics at T >Tc [16].
downwards in the superconducting state, indicating that ap-
It should be noted that TA-like behavior is quite uni-
pearanceofthenegativeexcessresistanceandthecrossoverto
versal for many process. Except for pure thermal ac-
quantumc−axistransportarecorrelated. Thedashed-dotted
tivation over the finite barrier without the gap in the
linerepresentstheslope ofresonant tunnelingcharacteristics
electronic DoS [16], it may appear in pure tunneling
(shown for comparison).
characteristics in the presence of the gap in electronic
DoS, due to TA-like behavior of the Fermi factor (even
D. Thermal activation analysis though this would require a very specific correlation be-
tween T−dependent factors mentioned in sec.I); as a re-
sultofinelastictunnelingviatheimpurity[28],orelastic
The TA current is given by a simple expression:
tunneling via a resonant state [29] in the tunnel barrier.
U eV Abrikosov has shown that the later scenario can quanti-
I ∝n(T)exp − TA cosh , (4)
TA tativelyreproducetheinterlayercharacteristicsofHTSC
(cid:20) k T(cid:21) (cid:20)2k T(cid:21)
B B
inthenormalstate[29]. Thebluedash-dottedlineinFig.
from which follows Eq.(M1). S10 b) represents the slope d/dV[ln(dI/dV)] in the case
Fig. S10 a) shows the dI/dV(V) characteristics in a of resonanttunneling with the appropriate energy of the
semi-log scale calculated from Eq.(M1) for the same T resonant state. It follows the simple linear TA behavior
and scale as in Fig. M2 a), in order to facilitate the di- at k T lower than the resonant energy and saturates at
B
rectcomparison. Simulationsweremadeforconstantσ higher T. Comparison with the resonant tunneling cal-
N
and U = 24meV, which were adjusted to obtain sim- culation shows that the saturation of the slope at high
TA
ilar R0(T) values as in the experimental data from Fig. T results in appearance of a negative offset in the linear
M2 a) at T >T ; and for n(T)∝T. Considerable grows d/dV[ln(dI/dV)] vs 1/T dependence at 1/T → 0. Be-
c
of mobile carrier concentrationn(T) with T in UD crys- low the saturation temperature, the actual temperature
tals was reported in Hall effect measurements [46]. The canbeeasilyextractedfromthed/dV[ln(dI/dV)]slopes,
assumption of linear n(T) is supported by observation compensated by the offset at 1/T → 0, as indicated in
10
Fig. S10 b) : and the slope of experimental curves simultaneously de-
viate downwards with respect to the linear TA behav-
e
T = . ior. Therefore, both are consequences of the same phe-
eff 2kB[d/dV[ln(dI/dV)]−Offset(T−1 →0)] nomenon(whichweattributedtodoping-inducedchange
in the c−axis transport) only at zero and finite bias, re-
This expression was used for obtaining the effective T
spectively.
shown in Fig. M2b).
From Fig. S10 b) it is seen that at T <Tc both lnR0
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