Experimental Test of the Numerical Renormalization Group Theory for Inelastic Scattering from Magnetic Impurities

0.22 MB·English

by  Christopher Bauerle

Experimental Test of the Numerical Renormalization Group Theory for Inelastic
Scattering from Magnetic Impurities
Christopher B¨auerle1, Franc¸ois Mallet1, Felicien Schopfer1,⋆,
Dominique Mailly2, Georg Eska3, and Laurent Saminadayar1
1Center for Low Temperature Research, CRTBT-CNRS, BP 166 X, 38042 Grenoble Cedex 9 France,
6 2Laboratoire de Photonique et Nanostructures, route de Nozay, 91460 Marcoussis, France,
0
3Physikalisches Institut, Universit¨at Bayreuth, Universit¨atsstrasse 30, 95440 Bayreuth, Germany
0
(Dated: February 6, 2008)
2

n Wepresentmeasurementsofthephasecoherencetimeτφ inquasione-dimensionalAu/FeKondo
a wiresandcomparethetemperaturedependenceofτφwitharecenttheoryofinelasticscatteringfrom
J magnetic impurities (Phys. Rev. Lett. 93, 107204 (2004)). A very good agreement is obtained for

9 temperaturesdownto0.2TK. BelowtheKondotemperatureTK,theinverseofthephasecoherence
time varies linearly with temperature overalmost one decade in temperature.

]
ll PACSnumbers: 73.23.-b,73.63.Nm,73.20.Fz,72.15.Qm
a
h
-
The Kondo problem has been fascinating physicist for ence problem, as it allowsto compare experimental data
s
e more than 30 years. Such a large interest is due to the withtheoreticalresultsforalltemperatures,rangingfrom
m
fact that this model, first introduced to describe trans- wellaboveT downtozerotemperature. Itisthusofcru-
K
. port properties of metals containing magnetic impuri- cialimportancetocheckwhetherthistheorycandescribe
t
a ties, is a generic model for the description of many solid correctlyexperimentaldata,asitcouldgivenewinsights
m
statephysicsphenomena[1]. Concerningmagneticalloys, into the problem of the low temperature decoherence in
- the ”historical” Kondo solution first allowed to describe mesoscopic metallic wires.
d
the transportproperties attemperatures largerthan the In this Letter, we compare this recent theory with ex-
n
o characteristictemperatureofthemodel,the Kondotem- perimentalresultsandshowthattheNRGcalculationde-
c perature T [2]. The major breakthrough has been the scribes very well the experimental data at temperatures
K
[
pioneering work of Wilson who was able to calculate the around and below the Kondo temperature. In the first

1 ground state of the Kondo problem at all temperatures part of this Letter we analyze the phase coherence time
v using his Numerical Renormalisation Group (NRG)[3]. of quasi one-dimensional gold wires containing magnetic
2 AnotherimportantcontributionwasduetoNozi`ereswho ironimpurities ofref.[11], denotedasAuFe1 andref.[12],
6
showed that the zero temperature limit of the Kondo denotedasAuFe2,whereasinthesecondpartweusethe
1
model can be described within Landau’s Fermi liquid NRG calculation to interpret the phase coherence time
1
0 theory[4]whichmadeprecisepredictionsfortransportas as a function of temperature of an extremely pure gold
6 well as thermodynamic properties. Since then, many ex- sample (Au1). In table I, we list the geometrical and
0 perimentshaveconfirmedtherelevanceofthisapproach. electrical parameters of these samples.
/
t
a
m Recently,ithasbeensuggestedthatscatteringbymag- Sample w t l R D cimp
netic impurities[5, 6] is responsible for the experimen- (nm) (nm) (µm) (Ω) (cm2/s) (ppm)
-
d tally observedsaturation of the phase coherence time τφ AuFe1 180 40 155 393 200 3.3
n of electrons in metals at low temperatures[7], renewing
AuFe2 150 45 450 4662 56 45
o the interest in Kondo physics. Though the Altshuler-
c Au1 120 50 450 1218 241 <0.015
Aronov-Khmelnitzky (AAK) theory describes the tem-
:
v peraturedependence ofthe phasecoherencetimeofelec-
TABLEI:Samplecharacteristics: w,t,l,R,correspondtothe
i
X trons in pure metals[8], no exact solution is available for width,thickness,lengthandelectricalresistance,respectively.
r thephasecoherencetimeinthepresenceofKondoimpu- Disthediffusioncoefficientandcimp istheimpurityconcen-
a rities. Onlyahightemperatureexpansion,the Nagaoka- tration extracted from theNRGfits.
Suhl (NS) expression, was able to describe the experi-
mental data at temperatures T T [9]. In the op- ThephasecoherencetimeofsampleAuFe1andAuFe2,
K
≫
posite limit (T T ), Fermi liquid theory predicts a as extracted from standard weak localization measure-
K
≪
T2 dependence of the inelastic scattering rate[4]. This, ments is shown in figure 1. Both samples display a dis-
however, has never been observed experimentally. Only tinct plateau at a temperature above T 0.3K. This
≥
very recently, Zara´nd and coworkers have been able to plateau is caused by Kondo spin flip scattering due to
obtain an exact solution for the inelastic scattering time the presence of iron impurities which leads to dephas-
in Kondo metals using Wilson’s NRG[10]. This calcula- ing. Decreasing the temperature, the magnetic impurity
tion constitutes a major breakthrough for the decoher- spinisscreenedandasaresult,thephasecoherencetime
2
increases again.
The temperature dependence of the measured phase
coherencetime τ inthe presenceofmagneticimpurities
φ
can be described in the following way
1 1 1 2
= + + (1)
τ τ τ τ
φ e e e ph mag
− −
where 1/τ = a T2/3 = [e2R√DkB]2/3T2/3 cor-
e−e theo 23/2h¯2L
responds to the electron-electron interaction term[13],
1/τ = bT3totheelectron-phononinteraction,while
e ph
−
1/τ correspond to the contribution due to magnetic
mag
impurities. Toaccountforthescatteringoffmagneticim-
purities, we use on one hand the common Nagaoka-Suhl
expression[9, 14]
FIG. 1: (color online) Phase coherence time as a function of
1 1 cimp[ppm] π2S(S+1) temperature for two Au/Fe Kondowires: sample AuFe1of
= = (2) ⋄
τ τ 0.6[ns] (π2S(S+1)+ln2(T/T )) ref.[11], sample AuFe2 (sample B) of ref.[12]. The green
mag NS K ◦
dotted line (denoted AAK) corresponds to the assumption
where S is the impurity spin, c the impurity concen- that only electron-electron and electron-phonon interaction
imp
trationexpressedinparts-per-million(ppm)andthepref- contributetodephasing. Theblackdashed-dottedlines(NS1
andNS2)takeinadditionaccountforthemagneticscattering
actor of 0.6[ns] has been calculated taking the electron
usingtheNSexpression,whereasforthered(NRG1)andblue
densityofgold[6]. Ontheotherhandweusetheinelastic
(NRG2) solid lines, the NRG calculation has been employed
scattering rate calculated by NRG[10]
for the contribution of the magnetic impurities. The inset
shows 1/τ versus T on a linear scale for sample AuFe2 in
φ
1 1 σ(w)
= =A inel c (3) order to emphasize thelinear regime.
imp
τ τ σ ∗
mag inel(NRG) 0
where σ(w) is the inelastic scattering cross section at
inel
finite energy w, σ = 4π/k2 the elastic scattering cross athightemperatures (T >10TK). One clearlysees that
0 F
the NS expression deviates already from the NRG data
section at zero temperature and A a numerical constant
in units of [s 1] to express the impurity concentration at relatively high temperatures (T 5K) as already
− ∼
pointed out in ref.[10] and fitting of experimental data
c in ppm.
imp
with this expression at temperatures T < 10T should
Thegreendottedlineinfigure1,denotedasAAK,cor- K
therefore be avoided. From the NRG fitting procedure
responds to the assumption that only electron-electron
we obtain an impurity concentration of approximately
and electron-phonon interaction contribute to the elec-
3.3ppm and T =0.4K 0.05K for sample AuFe1 and
tron dephasing and that there is no other mechanism K
±
45ppm and T = 0.9K 0.05K for sample AuFe2[15],
for decoherence at low temperatures. For the simula- K
±
tion we have used a = 0.41ns 1K 2/3 and b = in good agreement with TK values for Au/Fe found in
theo − −
0.8ns 1K 3. Keeping these values fixed, and adding the literature[16]. It is worth mentioning that the mea-
− −
surement of the phase coherence time is a very precise
the contribution due to magnetic impurities using the
method to determine the Kondo temperature as well as
NS expression for S=1/2, we obtain the black dashed-
the magnetic impurity concentration. In resistance mea-
dotted lines for sample AuFe1 and AuFe2, denoted as
surements, the determination of the Kondo temperature
NS1 and NS2. The NS expression describes relatively
isnotstraightforwardsincetheentiretemperaturerange
well the observed temperature dependence of τ at tem-
φ
down to the unitary limit (T T ) is necessary to ex-
peraturesaboveT butfailstodescribethedesaturation K
K ≪
tract the Kondo temperature with a satisfactory preci-
of τ due to the screening of the magnetic impurities at
φ
sion. The NRG fitting procedure of τ on the contrary
temperatures T . This is not at all surprising since φ
K
≤ allows to extract the Kondo temperature with high pre-
the NS expression results from a perturbative expansion
cision if temperatures only slightly lower than T are
in(T/T )andbreaksdownforT T . Instead,taking K
K K
→ attained.
the inelastic scattering rate obtained by NRG we obtain
averygoodagreementwiththeexperimentaldatainba- Atthelowesttemperaturesweobservedeviationsfrom
sically the entire temperature rangeas shownby the red the NRG theory. A probable explanation for the devia-
(NRG1) and blue (NRG2) solid lines for sample AuFe1 tion is fact that the spin of Fe in Au is not exactly 1/2
and AuFe2, respectively. For the fitting procedure, we andthe impurity spinmightnotbe completely screened.
have adjusted the magnetic impurity concentration such Thiseventuallyleadstointeractionsofmagneticimpuri-
thattheNSexpressionandtheNRGcalculationcoincide tiesandasaturationofτ [12]. Anotherpossibilityisthe
φ
3
presenceofanothermagneticimpuritywithamuchlower sentially the same as for the Au/Fe wires with the only
Kondo temperature. This latter issue will be discussed difference that the wire has been evaporated in an evap-
in the last part of this article. orator which is exclusively used for the evaporation of
The temperature evolution of τ below T deserves extremely pure gold. The gold of purity 5N5 has been
φ K
several comments: after a slow increase of τ the tem- evaporated directly on a silicon wafer without a stick-
φ
perature dependence is almost linear in temperature ing layer. In addition, special care has been taken for
over almost one decade in temperature, as emphasized the sample design such that there is no influence on the
in the inset of figure 1. This relatively weak temper- phasecoherenceduetothetwodimensionalcontactpads
ature dependence explains the fact why the pioneering (see inset of figure 2). For this wire the phase coherence
experiments[14, 17] have not succeeded to observe the lengthatthelowesttemperaturesismorethan20µm. To
Fermi liquid regime. Comparing the experimental re- our knowledge, this is the largest coherence length ever
sults with the NRG calculation, we clearly see that the obtained in a metallic wire and confirms the high purity
Fermi liquid regime can only be reached for tempera- of the sample.
tures typically below 0.01TK. Moreover, the AAK be- Todeterminetheeffectiveelectrontemperatureofthis
haviour is only recoveredat extremely low temperatures sample, we have measured the Altshuler-Aronov correc-
(T <0.001K). tion to the resistivity at very low temperatures. A mag-
It is noteworthy,that the calculated quantity by NRG neticfieldof40mThasbeenappliedinordertosuppress
isσinel andnotthephasecoherencetimeτφ measuredin weaklocalizationcorrectionto the resistivity. Infigure2
a weak localization experiment. In fact, σinel has been we plot the resistance correction as a function of 1/√T.
calculatedin the limit of zero temperature and finite en- For measuring currents below 0.7nA, the sample is in
ergy σinel(w,T = 0), whereas in a transport experiment thermal equilibrium (eV < kT) in the entire tempera-
one measures σinel(w =0,T). The fact that the numeri- ture range and the resistance correction follows the ex-
calresultsdescribethiswellthe experimentaldataletus pected 1/√T temperature dependence down to 10mK.
concludethatthese twoquantitiesarenotverydifferent, This clearly shows that the electrons of a mesoscopic
at least for kBT ǫF, ǫF being the Fermi energy. sample can be cooled to such low temperatures. Fit-
≪
ting the temperature dependence of the resistance cor-
rection to ∆R(T) = α /√T (dotted line in figure2),
exp
wedetermineα andcompareittothe predictedvalue
exp
[13] of ∆R(T) = 2R2/R L /L = α /√T, where
K T theo
L =p¯hD/k T is the thermal length and R = h/e2.
T B K
We obtain a value α = 0.11Ω/K1/2 which is in very
exp
good agreement with the theoretical value of α =
theo
0.109Ω/K1/2.
FIG. 2: (color online) Resistance variation of sample Au1
plottedasafunctionof1/√T fordifferentbiascurrents. The
dottedlinecorrespondstothetheoreticalexpectation for the
resistance correction. The inset shows a SEM photograph of
thegold wire.
Havingnowatheorywhichsatisfactorilydescribesthe
temperaturedependenceofτ inthepresenceofmagnetic
φ FIG. 3: (color online) Phase coherence time as a function of
impurities,letusreexaminethetemperaturedependence
temperature for sample Au1 ( ). The solid green line cor-
of τ in extremely pure goldwires. In a recentarticle[6], ◦
φ responds to the AAK prediction, the black (a), red (b) and
the deviationof τφ fromthe AAK predictionat verylow blue(c)solidlinescorrespondtotheNRGcalculationassum-
temperatureshasbeenassignedtothe presenceofanex- ingTK =40mK,TK =10mK,and TK =5mK,respectively.
tremely small amount of magnetic impurities (typically Theinsetshowstypicalmagneto-resistancecurvesatdifferent
on the order of 0.01ppm). For this purpose we have fab- temperatures.
ricated an extremely pure gold wire (Au1) as shown in
the inset of figure 2. The fabrication procedure is es- Thephasecoherencetimeτ isthenmeasuredviastan-
φ
4
dard weak localization (see inset of figure 3) and the experimentally observed temperature dependence of τ
φ
phase coherence length l is extracted via the Hikami- caused by the presence of magnetic impurities. Below
φ
Larkin-Nagaoka formula[19]. From the relation τ = T the inverse of the phase coherence time varies basi-
φ K
l2/D we then calculate the phase coherence time as dis- callylinearlywithtemperatureoveralmostonedecadein
φ
played in figure 3. We fit the experimental data with temperature. The T2 temperaturedependence predicted
the AAK expression such that an almost perfect agree- by the Fermi liquid theory, on the other hand, can only
ment is obtained at high temperatures (T > 100mK), bereachedfortemperaturessmallerthan0.01T andre-
K
as shown by the green solid line. The prefactor we ex- mains an experimental challenge.
tract from this fit a =0.42ns 1K 2/3 is in very good note added in proof: an exact calculation of τ in pres-
fit − − φ
agreement with the theoretical prediction of a = ence of disorder supports the above findings [24].
theo
0.41ns 1K 2/3. At temperatures below 100mK our
− −
data deviate substantially from the AAK prediction. To
see whether these deviations can be explained by the ACKNOWLEDGELENTS
presence of a very small amount of magnetic impurities,
we simulate the temperature dependence of τ for the
φ We acknowledge helpful discussions with P. Simon, S.
presence of a small amount of magnetic impurities us-
Kettemann, G. Zara´nd, A. Rosch, G. Montambaux, C.
ing the NRG calculations. The black (a), red (b) and
Texier,H.BouchiatandL.P.L´evy. Inaddition,wewould
blue (c) solid lines correspond to a simulation assuming
liketothankG.Zara´ndforprovidinguswiththenumer-
T =40mK,T =10mK,andT =5mK,withanimpu-
K K K icaldataofthe NRGcalculationofσ . We arealsoin-
inel
rityconcentrationofc =0.008ppm,c =0.013ppm,
imp imp debted to R.A. Webb for the gold evaporationof sample
and c =0.015ppm, respectively. It is clear from
imp Au1. This work has been supported by the French Min-
our simulations that only magnetic impurities with a
istryofScience,grants#0220222and#NN/0220112,
Kondo temperature T 10mK and with a concentra-
K the European Comission FP6 NMP-3 project 505457-1
≤
tion smaller than 0.015ppm describe satisfactorily the
“Ultra-1D”, and by the IPMC Grenoble. C.B. and G.E.
experimental data. A possible magnetic impurity with
acknowledge financial support from PROCOPE.
a Kondo temperature in this temperature range is Mn
(T 3mK)[18].
K
≃
For the sake of objectiveness let us point out, how-
ever, the following: assuming magnetic impurities with
a Kondo temperature below the measuring temperature [*] Present address: Laboratoire National de M´etrologie et
leads to an almost temperature independent scattering d’Essais, Trappes, France.
[1] A.C. Hewson, The Kondo Problem to Heavy Fermions,
rate for T T . Any experimentally observed satura-
≥ K Cambridge Studiesin Magnetism (1997).
tion of τφ can therefore always be assigned to magnetic [2] J. Kondo, Prog. Theor. Phys. 32 37 (1964).
impurities with an unmeasurably low Kondo tempera- [3] K.G. Wilson, Rev.Mod. Phys. 47 773 (1975).
ture. One could also argue, that it is curious that for [4] P. Nozi`eres, J. Low Temp. Phys. 17 31 (1974).
the case of gold wires, the observed temperature depen- [5] F. Pierre and N.O. Birge, Phys. Rev. Lett. 89 206804
denceofτ canonlybedescribedsatisfactorilybyassum- (2002).
φ
ing the presence of one specific magnetic impurity with [6] F. Pierre et al. Phys. Rev. B 68 085413 (2003).
[7] P. Mohanty et al. Phys. Rev. Lett. 78 3366 (1997).
a Kondo temperature below the measuring temperature
[8] B.L. Altshuleret al. J. Phys. C 15 7367 (1982);
T 10mK,whereas it is known that the dominant mag-
≤ [9] for a review see G. Gru¨ner and A. Zawadowski, Rep.
neticimpurityingoldisiron. Ifweassumeanadditional Prog. Phys. 37 1497 (1974).
ironconcentration(0.015ppm;TK=500mK)ofthesame [10] G. Zar´and, et al. Phys. Rev. Lett. 93 107204 (2004).
order as for instance Mn (0.015ppm; T =3mK), the [11] P. Mohanty and R.A. Webb, Phys. Rev. Lett. 84 4481
K
temperature dependence of τ does not satisfactorilyde- (2000).
φ
scribe the experimental data as displayed by the dot- [12] F. Schopferet al., Phys. Rev. Lett 90 056801 (2003).
[13] E. Akkermans and G. Montambaux, in Physique
ted black line (d). A possible explanation of these
m´esoscopique des ´electrons et des photons, ed. EDP Sci-
facts, might be the presence of a distribution of Kondo
ences, (2004).
temperatures, which could be significant for very di- [14] C. Van Haesendonck et al., Phys. Rev. Lett. 58 1968
luted Kondo impurities [20]. Such a distribution of (1987).
Kondotemperatureshasalreadybeseeninpointcontact [15] ThefactthattheextractedKondotemperaturesaredif-
experiments[21]. This,however,canonlybeverifiedwith ferent for the two samples is somewhat surprising. How-
phasecoherentmeasurementsathighmagneticfields[22]. ever, these results are consistent with the maxima ob-
served previously in thespin flip rate[12,17].
Wealsonotethatourpresentresultsdonotallowtorule
[16] C. Rizutto, Rep.Prog. Phys. 37 147-229 (1974).
out the predictions of ref. [23]. [17] R.P. Peters et al., Phys. Rev. Lett. 58 1964 (1987).
In conclusion we have shown that the NRG calcula- [18] G. Eskaet al., J. Low Temp. Phys 28 551 (1977).
tionofthe inelasticscatteringratedescribeverywellthe [19] S. Hikami et al., Prog. Theor. Phys 63 707 (1980).
5
[20] S.Kettemann and E.R. Mucciolo, cond-mat/0509251. [23] D.S.GolubevandA.D.Zaikin,Phys.Rev.Lett.81,1074
[21] I.K.Yanson et al., Phys. Rev. Lett. 74 302 (1995). (1998).
[22] P.Mohantyand R.A.Webb,Phys. Rev. Lett. 91066604 [24] T. Micklitz et al., cond-mat/0509583.
(2003).