Saturation of resistivity and Kohler's rule in Ni-doped La$_{1.85}$Sr$_{0.15}$CuO$_{4}$ cuprate

0.99 MB·

by  A. Malinowski

Saturation of resistivity and Kohler’s rule in Ni-doped La Sr CuO cuprate
1.85 0.15 4
A. Malinowski,1 V. L. Bezusyy,1,2 and P. Nowicki1
1Institute of Physics, Polish Academy of Sciences, al. Lotniko´w 32/46, 02-668 Warsaw, Poland
2International Laboratory of High Magnetic Fields and Low Temperatures, ul. Gajowicka 95, 53-421 Wroclaw, Poland
(Dated: January 11, 2017)
WepresenttheresultsofelectricaltransportmeasurementsofLa1.85Sr0.15Cu1−yNiyO4thinsingle-
crystal films at magnetic fields up to 9 T. Adding Ni impurity with strong Coulomb scattering
7
1 potential to slightly underdoped cuprate makes the signs of resistivity saturation at ρsat visible in
themeasurementtemperaturewindowupto350K.Employingtheparallel-resistorformalismreveals
0
2 that ρsat is consistent with classical Ioffe-Regel-Mott limit and changes with carrier concentration
n ndecarseaρssiantg∝th1e/ir√enff.ecTtivheercmoonpceonwterratmioenasausrenm=en0t.s15shoyw. tThahteNcliastseincadlsutnomloocdailfiizeed mKoobhilleer’scarrurileersis,
a fulfilled for magnetoresistance in the nonsupe∼rcondu−cting part of the phase diagram when applied
J
tothe ideal branch in theparallel-resistor model.

0
1 PACSnumbers: 74.72.Gh,74.25.F-,74.25.fg,72.15.Lh

]
n
Increasing evidence for well-defined quasiparticles in cay)rate1/τ,theDrudepeakiscenteredatω=0regard-
o
underdoped cuprates seems to corroborate a view that less of how strong scattering becomes [2, 23]. Electron-
c
- they are normal metals, only with small Fermi surface. electroninteractionsmakeτ frequency-dependentbutfor
r Fermi-Diracstatisticunderlyingthequantumoscillations Fermi-liquid-like ω2 dependence σ(ω) still peaks at ω=0
p
u [1],single-parameter-quadraticinenergyω andtemper- (Ref. [2]). Thus large impurity-induced ρres may fa-
s atureT -scalinginopticalconductivityσ(ω,T)(Ref.[2]), cilitate approach to Ioffe-Regel-Mott limit in dc (ω=0)
t. T2 resistivity behavior extending over substantial T- LSCO transport at lower T before the spectral weight is
a
region in clean systems [3] and fulfillment of typical for transferred to higher-energy excitations at larger T. Ni
m
conventional metals Kohler’s rule in magnetotransport impurityisagoodcandidatebecauseitsstrongCoulomb
- [4] are observations in favor of Fermi-liquid scenario. scatteringpotentialintheCuO planesallowstoachieve
d 2
n On the other hand, in cuprates with significant dis- large ρ at moderately high T [28, 29].
o order manifested by large residual resistivity ρ , pure In this paper we report transport and thermopower
res
c T2 resistivity dependence has not been reported so far measurements on slightly underdoped x=0.15 LSCO
[
as for Bi2Sr2CuO6+δ [3, 5, 6] or observed only at rela- with added Ni impurity. The obtained ρsat corresponds

1 tively narrowdoping- and T-regionas in La2−xSrxCuO4 totheclassicalvalueforsmallFermisurfaceandchanges
v (LSCO) [3, 7]. The clear violation of Kohler’s scaling as 1/√n. The Fermi-liquid quasiparticle picture holds
8 in underdoped LSCO and YBa Cu O [8, 9] (although in the nonsuperconducting part of the phase diagram,
2 3 7
6 not necessarily meaning breakdown of Fermi-liquid de- as revealed by ρ T2 dependence and classical Kohler’s
5 ∝
scription [10]) has served almost as a hallmark of their rule for magnetoresistance, both hidden under the large
2
peculiar normal-state properties for two decades. resistivity of the system.
0
. In contrast to overdoped cuprates where large cylin- The 4-point transport measurements were carried
1
drical Fermi surface yields a carrier density n=p+1 (p out on the c-axis aligned single-crystal films grown on
0
being doping level) [11–13], the total volume of Fermi isostructural LaSrAlO substrate by the laser ablation
7 4
1 surface in underdoped systems is a small fraction of the method from the polycrystalline targets [30, 31]. The
: firstBrillouinzoneandcorrespondston=pthroughLut- thermopower, which is not sensitive to the grain bound-
v
tinger’stheorem[14–17]. Thissmallnshouldbereflected aries and porosity effects in cuprates [32], was measured
i
X in zero-field transport. In normal metals, resistivity ρ on the samples cut from the targets.
r saturates in the vicinity of Ioffe-Regel-Mott limit ρIRM Figure 1 shows the systematic change in ρ(T) with
a
whereelasticmeanfreepathlmin becomescomparableto Ni, from superconducting y=0 specimen with midpoint
interatomicdistance[18,19]. Incuprates,however,signs T =34.6K to y=0.08one exhibiting insulating behavior
C
of saturation are seen at ρsat much larger than ρIRM atlowT. InhighT,a changeofslope in portionof ρ(T)
calculated from the semiclassical Boltzmann theory [20– curves that increases with T foreruns the approaching
23]. Moreover, ρ(1000 K) ( ρsat) in LSCO changes saturation. Variation of dρ/dT, visible even in the y=0
∼
as 1/x, while for n x (Ref. [24]) the theory predicts data, authenticates the slope decreasing that becomes
∝
ρsat 1/√x. more pronounced with increasing y, as can be seen in
∝
Theabovecanbeexplainedbybreakdownofthequasi- Fig.2(a). Resistivityinthisregionisdescribedextremely
particle picture due to strong inelastic scatteringat high well by the parallel-resistor formula
T manifested by disappearing of a Drude peak in σ(ω)
(Refs. [23, 25–27]). In systems where impurity scatter- 1 1 1 1 1
= + = + . (1)
ing dominates the carrier relaxation (quasiparticle de- ρ(T) ρ (T) ρ a +a T +a T2 ρ
id sat 0 1 2 sat
2
1.0 y=0 1.0y=0.02 1.0y=0.03 5 1.0 y=0. 04 1.0 y=0. 06 1.0 y=0 .08 m/K) y=0.0 4 y=0.035 m)
2 /350 K00..68 00 ..68 0 .8 0.8 0 .9 0 .9 m c2 y(=a0.)02 m c (b)
K350 00..24 0.4 0.6 0.6 00..78 -3 (10 y=0.06y=0 (res1
/ 0.00 1503000.20 150300 0 150300 0 150300 0 1503000.80 150300 dT
T (K) T (K) T (K) T (K) T (K) T (K) / y=0.08
1 y=0.08 d1 0
m) 0.00 0.03 0y.06 0.09 100 200 300 0 0.04 0.08
c2 T (K) ) y
(m1350K cm)1 (c) -2 m K 6 (d)
y=0
small c
0 m FS
0 100 200 300 2 ( m 4
T (K) 1/ 5
n - 0
FizIeGd.r1e.sis(tCivoitloyrfoonrlianes)erTieesmopfeLraat1u.8r5eSdr0e.p15eCndue1−ncyeNioyfOn4orsmpeacl-- sat l a FrgSe a (122
imens. Their resistivities at T=350K are depicted in lower
0 0
inset. Upper inset shows the fits of Eq. (1) to 150K-350K 0 0.04 0.08 5 10 15
data for selected specimens. y 1/n
FIG. 2. (Color online) (a) Temperature derivatives of
La1.85Sr0.15Cu1−yNiyO4 resistivity. The dashed horizontal
The ρ term is the ideal resistivity in the absence of line is a guide to the eye. (b) The increase in residual re-
id
saturation [19] and the additive-in-conductivity formal- sistivity of the samples with the smallest ρres for a given y.
Thelines show theunitarity limit assuming n=0.15 y (thick
ism stems from existence of the minimal scattering time
−
line) and - forcomparison -n=0.15 (thinline), n=0.15 0.7y
τmin, equivalent to Ioffe-Regel-Mott limit, which causes (dashed line) and hypothetical n=0.15 2y (dotted line−). (c)
the shunt ρsat to always influence ρ in normal metals The product ρsat√n with the arithm−etic mean (ρsat√n)av
[33, 34]. The formula was used for overdopedLSCO [35]
(red dot) for the 32 measured samples. Solid lines show the
butwiththelarge-Fermi-surfaceρ value[34]asafixed
sat expectedy-dependenceforsmallandlargeFermisurface. (d)
parameter. The excellent fits of Eq. (1) with all free pa- The parameter a2 as a function of inverse carrier concentra-
rameters including ρsat to ρ(T) in 150 K-350 K interval tion (open circles). Solid circles mark the normalized values
aredepictedinupperinsettoFig.1. Extendingthefitin- an2 =a2(ρsat√n)av/(ρsat√n)andsolidlineisthelinearfitto
tervaldownwardsto lowerT for small y does not change them.
significantly the obtained fitting parameters, which are
presented in Fig. 2(b-d).
The Ni-induced residual resistivity, calculated as the Fermi level EF and the carriers from the energy in-
1/ρres=1/ρsat+1/a0, accelerates with y such that at terval Wσ are responsible for conduction. In addition, a
large y substantially exceeds s-wave scattering unitarity shift bWD between the centers of WD and Wσ bands is
limit ∆ρ =(~/e2)(y/n)d for n=0.15=const, depicted assumed. The best fits of the formula determining S in
res
as thin solid line in Fig. 2(b). Here, d=c/2= 6.6˚A the model (Eq. (1) in Ref. [40]) are shown as thick lines
∼
is the average separation of the CuO planes in the in Fig. 3. The discrepancies at low T come from the
2
La1.85Sr0.15Cu1−yNiyO4 films [31]. To find the actual limitations of the model derivated under the assump-
n(y) dependence we carried out the thermopower mea- tion WD ∼= kBT. Above y=0.15, the model also fails
surements. for larger T, well above Tmax (WD > 380 meV, while
Ni doping increases the positive Seebeck coefficient S, kBT ∼= 26 meV at 300 K). For y=0.17 and y=0.19, S
as canbe seenin Fig.3. Taking into accountthe univer- changes as 1/T above 250 K, consistent with the
∝ ≈
sal correlation between S(290K) and hole concentration formulaS(T)=(kB/e)(Ea/(kBT+const)indicativeofpo-
fulfilled in most of cuprate families [36–38], this strongly larons. ThethermalactivationenergyEa estimatedfrom
suggests decreasing of carrier density with y. The S(T) the best fit for y=0.19, Ea=32.4 0.2 meV, is in good
±
curvesinLa1.85Sr0.15Cu1−yNiyO4 retainthe specific fea- agreement with Ref. [41].
tures of thermopower in underdoped cuprates: the ini- At region of interest corresponding to high T, the
tial strong growth of S with increasing T is followed asymmetrical narrow-band model works very well. The
by a broad maximum and subsequent slight decrease in obtained asymmetry factor b = (0.06-0.08), although
−
S [39]. We find that the phenomenological asymmet- very small, is essential for good fits. The ratio F of n to
rical narrow-band model [40] describes the experimen- the total number od states n is slightly above half-
DOS
tal S(T) curves very well at high T, above their maxi- filling (0.51-0.53) and thus consistent with the sign of S
mum at T . In this model, a sharp density-of-states and in good agrement with literature data for x=0.15
max
peak with the effective bandwidth W is located near LSCO [40, 42]. The Fermi level, E =(F–0.5)W –bW ,
D F D D
3

surfacewiththeheight2π/dandtakingIoffe-Regel-Mott
S kB Ea const condition as lmin≈a (i.e. kFlmin≈π, see Ref. [46, 47]),
e kBT where a is the lattice parameterin CuO2 plane,one gets
ρsmall=(√2π~/e2)d/√n=0.68/√n mΩcm [26]. This is
100 IRM
clearlydistinguishablefromthe largeFermisurfacecase,
)
K ρlarge=0.68/√1+n, inapplicable to the system. A sim-
/ IRM
V ple formal statistics for all 32 measured samples shows
S ( 0.60.0 yNi 0.1 0.2 y=0 tthioantwthitehpmroedanuc=tmρseadti√annafnordnze=ro0.1sk5e−wynehsass[4a8]d.istribu-
∼
0.01 0.09
WD0.4 0.02 0.11 Thepreciselocationofthelarge-to-smallFermisurface
W/ 0.03 0.13 transitiononthe phasediagramofcupratesisstillunder
10 0.2 0.04 0.15
0.05 0.17 debate. Inthe bismuth-basedfamily,the expectedlinear
0.0 0.06 0.19 relationshipbetweennestimatedfromT andfromLut-
C
0 100 200 300 400 tingercountisobtainedonlyassuminglargesurfacefrom
overdoped specimens down to p=0.145 inclusive [49]. In
T (K) ∼
La1.85Sr0.15Cu1−yNiyO4,Nidopingeffectivelymovesthe
FIG. 3. Temperature dependence of Seebeck coefficient for systemtowardssmallerpbutthe smoothevolutionofall
La1.85Sr0.15Cu1−yNiyO4 fromy=0(bottom)toy=0.19(top). the fitted ρ(T) parameters down to y=0 points toward
The solid lines for y 0.15 and the dashed ones for y>0.15
≤ smallFermi surfaceat p=0.15. This finding is consistent
are the best fits to the model from Ref.[40]. The thin solid
withrecentHallmeasurementsinYBa Cu O indicating
lineisthebestfitofthermally-activatedtransportformulafor 2 3 y
thatFermi-surfacereconstructionwithdecreasingdoping
y=0.19. Inset: TheWσ/WD ratioasafunctionofNidoping.
ends sharply at p=0.16 (Ref. [16]).
Dotted line is thebest linear fit between y=0.02 and 0.15.
The T-linear coefficient in Eq. (1) for three y=0
samples α =0.93-1.0 µΩcm/K is identical as that at
1
crossesthe conduction-bandupper edge at y 0.15. The ncr=0.185 0.005where a changein LSCO transportco-
≈ ±
band-filling F does not show any obvious y-dependence. efficients was found when tracked from the overdoped
This means that Ni does not change n in the system side[35]. Evidently,α1 isnotsensitivetodisappearingof
(provided n remains constant). The primary effect the antinodal regions during degradation/reconstruction
DOS
ofNidopingappearstobelocalization ofexistingmobile of large Fermi surface into arcs/pockets. The linear-in-
carriers,as revealed by decreasing of Wσ/WD ratio with T scattering is anisotropic in CuO2 plane [50] and its
increasing y (inset to Fig. 3). The ratio extrapolates to maximal level at (π,0) for α1=1 µΩcm/K is compara-
zero at y=0.22 0.02, resulting in the average ”localiza- ble [35] with Planckian dissipation limit [51, 52]. De-
±
tion rate” ∆n/∆y=0.7 0.1 hole/Ni ion. Employing the coherence of quasiparticlestates beyond this limit seems
±
simple two-band model with T-linear term [43], where tobeplausibleexplanation[35]ofnegligibleroleofantin-
half-width of the resonance peak Γ corresponds to the odalstatesin the conductivity. While α1 vanishes forall
range of delocalized states, results in the similar phys- specimens with y>0.035, the T2-coefficient a2 changes
ical picture. We found that Γ starts to decrease with linearlywith1/n(compareRef.[3])inthe wholestudied
increasing y above 0.07 and approaches zero for y=0.17 y range (Fig. 2d). When extrapolated outside accessi-
(Ref.[31]). Theresultsareconsistentwithmeasurements ble n, a2 approaches zero at ncr=0.167 0.009. With
±
of local distortion around Ni ions suggesting trapping such estimated error, the result means that strictly lin-
hole by each Ni2+ [44] to create a well-localized Zhang- ear ρ(T) dependence (albeit masked by ρsat) in LSCO
Rice doublet state [45], albeit indicate a slightly lower is observed from the underdoped side only at optimum
∆n/∆y. doping. Interpreting this as an indication of (antiferro-
Havingestablishedthe effective mobilecarrierconcen- magnetic) Quantum Critical Point remains speculative
tration n∼=0.15−y, we can revert to Fig. 2b. As indi- since a2 diverges at such a point [53–56].
cated by thick solid line, scattering in the samples with After using the parallel-resistorformalism in the zero-
the smallest ρ is in the unitarity limit for nonmag- field transport description, we will employ it to anal-
res
netic impurity. Even assuming ∆n/∆y=0.7 (and uni- ysis of magnetoresistance in the sections below. In
tarity limit), at most only 20% of increase in ρ for thestronglyoverdopedcupratesmagnetoresistanceobeys
res
y=0.08 can be attributed to scattering on magnetic mo- Kohler’s rule [9]. The relative change of resistivity in
ments. Thus, while our previous finding of spin-glass magnetic field B, ∆ρ/ρ , is a unique function of B/ρ ,
0 0
behavior in the system [30] undoubtedly indicates that where ρ ρ(B=0T). In recent re-analysis of x=0.09
0
≡
the magnetic role of Ni ions in the spin-1/2 network of LSCO data from Ref. [9], the modified Kohler’s rule was
the CuO planes cannot be neglected, the scattering on proposed[4]. The isotemperature transverse magnetore-
2
Ni has a predominantly nonmagnetic origin [28]. sistancevs. B/ρ curvesappearedtocollapseontosingle
0
InFig.2cweshowthatthefittedρ agreesunexpect- curve when ρ is replaced by ρ ρ . Here, we propose
sat 0 0 res
−
edlywithρ calculatedfromBoltzmanntheoryforthe an alternative approach for considering the large LSCO
IRM
smallFermisurfacewithnholes. Assumingacylindrical resistivity in the magnetoresistance analysis.
4

-6-20a, a (10 T).TMRLMR133 (a)100T(1K5)0 y2=000.24606-7 -2 a (10T)ORB -4 / (10)0120 5a6BL0M (KT2Ra4)111102T406 00MK0000R -5 1/(0) (10)idid002468 (b) y[B=/0.id0(70,)]2 3y=0.06 TTTTTTTTTTT===========111111111221234567890100000000000KKKKKKKKKKK -4 /(-) (10)orb0res024 TTTTTTTTTTT===========111111111221234567890100000000000KKKKKKKKKKK (c) -5y/ (10)=orb0 024(0B0/.00)26 [1201501 T 1K/0(m K0.5cm)]2
50 100 150 200 250 0 5 10 15 0.0 0.2 0.4 0.6
2 4 2 2 6 2
T (K) [B/ id(0)] [10 T/(m cm)] [B/( 0- res)] [10 T/(m cm)]
FIG. 4. (Color online) Magnetoresistance of y=0.06 specimen. (a) The temperature dependence of the coefficients aTMR and
aLMR. The orbital part aORB (open symbols) and the result of its 5-point adjacent-averaging (solid symbols) are depicted in
left inset. Right inset shows B2-fits to TMR data at selected T. (b) Kohler scaling for the ideal branch, together with that
of y=0.07 specimen for which abscissa scale is enlarged 3 times. (c) Scaling approach from Ref. [4] and direct Kohler’s-rule
scaling attempt in inset.
At the lowest temperatures down to 2 K, all non- trajectories of those scattered against impurities.
superconducting samples exhibit large and negative in- The OMR analysis reveals that Kohler’s rule in non-
plane (I ab) magnetoresistance, both in the transverse superconductingspecimensisfulfilledintheidealbranch
k
(TMR, B ab) and longitudinal (LMR, B ab) configura- of the parallel-resistor model where the influence of the
tion [TMR⊥(B=9T) 5 10−2 at 2 K] [31].kIn the follow- shunt ρ is eliminated [Fig. 4(b)]. Interpreting the
≈ × sat
ing,wefocusonthehigh-T regionwhere,forthewholey modelinthespiritoftheminimalτ leadstotheassump-
0
range studied, magnetoresistance in both configurations tion that ρ is field-independent, at least in the weak-
sat
is positive and two orders of magnitude smaller than in field regime (where actually observed OMR B2 depen-
the low-T region. The typical field dependence of re- dence is expected). Fitting of Eq. (1) to ρ(T∝) measured
sistivity at various temperatures is displayed for y=0.06 at various fields up to 9T does not reveal any system-
specimen in the right inset to Fig. 4(a). Above 35 K, atic change of ρ with B [58]. With ρ (B)=ρ (0),
sat sat sat
ρ increases as B2. Below 60 K, the positive TMR be- OMR of the ideal branch, ∆ρ /ρ (0) can be calculated
id id
gins to decrease with decreasing T and smoothly evolves from the measured quantities employing only one fit-
into negative one at low T. Similar behavior is observed ting parameter ρ (0). The obtained ∆ρ /ρ (0) vs.
sat id id
forLMR,whichconstitutesasignificantportionofTMR [B/(ρ (0)]2 curves from 110 K up to T =210 K col-
id up
(beingequalto60%ofTMRatT=150Kasanexample) lapseto a single temperature-independentline, spanning
and thus may not be ignored in the analysis. To obtain 50% of T [57]. The similar result was obtained for
up
the orbital part, OMR, at T>35 K, we fitted ρ(B) with y=0.07 specimen. Closer to the superconducting region
the form [∆ρ(B)/ρ(0)]TMR,LMR=aTMR,LMRB2 and next of the phase diagram, for y=0.04, the scaling interval in
calculated aORB =aTMR−aLMR. The extracted coeffi- ρid(B,T) is much smaller (140K-160K) but larger dif-
cients are displayed in Fig. 4(a). A reliable and precise ference between ρ and ρ 0.3ρ emphasizes the
sat res sat
comparisonofthe variouspossible OMRscalingrequires difference between the possible≈OMR scalings and the
moderate numerical smoothing without alternating the scaling approach illustrated in Fig. 4(c) fails completely.
a vs. T dependence [leftinsetto Fig.4(a)]. Employ-
ORB Concluding, the signs of resistivity saturation both at
ing the smoothed a coefficients, a , OMR at any
ORB orb the value ρ and with n-dependence from Boltzmann
field B can be calculated as OMR=a B2. IRM
orb theory reflectmetallic-like characteroftransportdespite
Clearly,Kohler’sruleinLa1.85Sr0.15Cu1−yNiyO4 isvi- smallvolume ofFermisurfaceandstrongdisorderinun-
olated when applied directly to the measured OMR of derdoped LSCO. A fully quantum-mechanical explana-
the specimen [inset to Fig. 4(c)]. Modification of the tion of saturation is still lacking [26, 46, 59–61]. Evi-
rule in the way describedinRef. [4]does not leadto any dentlyhowever,strongelectron-electroninteractions[62]
reasonable scaling range [57]. The ∆ρorb/(ρ0 ρres) vs. do not invalidate the ρIRM limit [27]. The Ni-induced
[B/(ρ0 ρres)]2 lines collapse one onto anothe−r between order-of-magnitude increase of a0/ρsat ratio leaves the
180 K a−nd Tup=200 K, spanning only 10%of Tup. Let’s resulting ρsImRMall√n intact. The revealed omnipresence
notethattheexistenceofsuchascaling-wherethewhole ofτmin eveninbadmetalspoints towardsuniversalityof
absolute resistivitychangeinfield, ∆ρ , is relatedonly theIoffe-Regel-Mottcriterion[46]ratherthanaccidental-
orb
toT-dependentρel-ph ρ0 ρres part-wouldmeaninthe only agreement between ρIRM and saturation [27].
≡ −
classicalpicturethatthefieldactsbetweenthescattering The known huge increase of ρ in LSCO outside our
eventsonlyonthesecarriersthatscatteragainstphonons T-measurement window means the breakdown of quasi-
duringthesubsequentscatteringeventanddoesnotbend particle description. At lower T, the transport in
5
La1.85Sr0.15Cu1−yNiyO4hasacoherentdescriptioninthe In summary, the typical for normal metals
framework of Fermi-liquid theory, where Kohler’s rule is parallel-resistor formalism, employed for analysis of
derivedunder single-relaxation-timeapproximationwith La1.85Sr0.15Cu1−yNiyO4 transport, reveals resistivity
the assumption of small τ-anisotropy over Fermi sur- saturationat the value expected from Boltzmanntheory
face [63, 64]. Fulfillment of the rule when ρ is changed for the small Fermi surface and fulfilment of unmodified
0
bychangingtemperatureindicatesnearlyT-independent Kohler’s rule in the nonsuperconducting part of the
frequency distribution of the phonons involved [65] and phase diagram.
isconsistentwithFermi-arclengthincupratesbeingcon-
stant[66,67]ratherthandecreasingwithdecreasingtem-
perature [68]. Ni doping can restore antiferromagnetic
ACKNOWLEDGMENTS
fluctuations[30,69]thatgiveadditionalT-dependencein
nearly-antiferromagnetic-metals magnetoresistance via
correlationlengthξ (T),OMR ξ4 (T)ρ−2 (Ref.[64]). This researchwas partiallyperformedin the NanoFun
AF ∝ AF 0
ThusfulfillmentofconventionalKohler’srulesuggestsT- laboratory co-financed by the ERDF Project NanoFun
independent ξ within the framework of this theory. POIG.02.02.00-00-025/09.
AF
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