Table Of ContentThe 2-Channel Kondo Model I: Review of Experimental Evidence for its Realization
in Metal Nanoconstrictions
Jan von Delft1,∗, D. C. Ralph1, R. A. Buhrman2, S. K. Upadhyay1, R. N. Louie1,
A. W. W. Ludwig3, Vinay Ambegaokar1
1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853, USA
2School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA
3 Department of Physics, University of California, Santa Barbara, CA 93106 , USA
(February 4, 1997)
Certain zero-bias anomalies (ZBAs) in the voltage, temperature and magnetic field dependence
of the conductance G(V,T,H) of quenched Cu point contacts have previously been interpreted to
be due to non-magnetic 2-channel Kondo (2CK) scattering from near-degenerate atomic two-level
8 tunneling systems (Ralph and Buhrman, 1992; Ralph et al. 1994), and hence to represent an
9 experimental realization of the non-Fermi-liquid physics of the T = 0 fixed point of the 2-channel
9 Kondo model. In this, the first in a series of three papers (I,II,III) devoted to 2-channel Kondo
1
physics, we present a comprehensive review of the quenched Cu ZBA experiments and their 2CK
n interpretation,includingnewresultsonZBAsinconstrictionsmadefromTiorfrommetallicglasses.
a WefirstreviewtheevidencethattheZBAsareduetoelectronscatteringfromstucturaldefectsthat
J are not static, but possess internal dynamics. In order to distinguish between several mechanisms
4 proposed to explain the experiments, we then analyze the scaling properties of the conductance
1 at low temperature and voltage and extract from the data a universal scaling function Γ(v). The
theoreticalcalculation ofthecorrespondingscalingfunctionwithinthe2CKmodelisthesubjectof
2
papers II and III. The main conclusion of our work is that the properties of the ZBAs, and most
v
notablytheirscalingbehavior,areingoodagreementwiththe2CKmodelandclearlydifferentfrom
8
several otherproposed mechanisms.
4
0
PACS numbers: 72.15.Qm, 72.10.Fk, 63.50.+x, 71.25.Mg
2
0
7
9 Contents B Scaling Analysis of Experimental Data 14
/ 1 First Test of T1/2 and V1/2 Behavior 14
t
a I Introduction 2 2 Scaling Collapse . . . . . . . . . . . 14
m 3 Universality . . . . . . . . . . . . . 15
- II The Nanoconstriction 3 C Upper Bound on the Energy Splitting ∆ 15
d
n
III Ballistic Point Contact Spectroscopy 4 VII Related Experiments 16
o
A Titanium Nanoconstrictions . . . . . . 16
c
: IV Experimental Facts for Quenched Cu B MechanicalBreakJunctionsMadefrom
v
Samples 5 Metallic Glasses . . . . . . . . . . . . . 18
i
X
V The 2-channel Kondo (2CK) Interpreta- VIII Conclusions 18
r
a tion 7 A Summary . . . . . . . . . . . . . . . . . 18
A Two-state systems . . . . . . . . . . . . 7 B Open Questions and Outlook . . . . . . 19
1 Slow Fluctuators . . . . . . . . . . . 7
2 Two-LevelSystems . . . . . . . . . 7 APPENDIXES 20
B Successes of the 2CK Interpretation . . 8
C Open Questions in the 2CK Scenario . 9 A Ruling out Some Alternative Interpre-
1 Conductance Transitions . . . . . . 9 tations 20
2 Strong Magnetic Field Dependence 10 1 Static Disorder. . . . . . . . . . . . . . 20
3 Microscopic Nature of the TLS . . . 10 2 Magnetic Impurities . . . . . . . . . . . 21
D Summary of Assumptions of 2CK Sce- 3 TLS Population Spectroscopy . . . . . 21
nario . . . . . . . . . . . . . . . . . . . 11 4 Properties of External Circuit . . . . . 22
5 Charge Traps and Other Possibilities . 22
VI Scaling Analysis of G(V,T) 12
A The Scaling Ansatz . . . . . . . . . . . 12 B MagneticFieldDependencein2CKSce-
1 General Scaling Argument . . . . . 12 nario 22
2 Back-of-the-envelope calculation of 1 H-Tuning of ∆. . . . . . . . . . . . . . 22
Γ(v) . . . . . . . . . . . . . . . . . . 13 2 Channel Symmetry Breaking by H . . 22
1
I. INTRODUCTION scattering mechanism.
For very small eV/k and T (< 5K), RB observed
B
The study of systems of strongly correlated electrons non-ohmicZBAsintheV-andT-dependenceofthecon-
that display non-Fermi-liquid behavior has attracted ductance signals of unannealed, ballistic nanoconstric-
widespreadinterestinrecentyears,fueledinpartbytheir tions. The qualitative features of these anomalies (such
possible relevance to heavy-fermion compounds [3,4,5] as their behavior in a magnetic field, under annealing
and high-T superconductivity materials [6,7,8]. On the and upon the addition of static impurities), which are
c
theoretical front, one of the consequences was a renewed reviewed in detail in the present paper, lead to the pro-
interest in various multi-channel Kondo models, some of posal [1] that the ZBAs are caused by a special type of
whichwerepredictedbyNozi`eresandBlandin[9]tocon- defectinthenanoconstrictions,namelytwo-levelsystems
tain non-Fermi-liquid physics. Some of the most recent (TLSs). This proposal has recently received strong sup-
advancesweremadebyAffleckandLudwig(AL)(see[10] port from a number of subsequent, related experiments
and references therein), who developed an exact confor- (brieflyreviewedinsectionVII)byUpadhyayetal. onTi
malfieldtheory(CFT)solutionfortheT =0fixedpoint constrictions [19]and by Keijsers et al. on metallic-glass
ofthe multichannelKondomodels. Onthe experimental constrictions [20,21].
front,anexperimentperformedbytwoofus(RB)[1,11], ThereareatleasttwotheoriesforhowTLSscancause
that investigated certain zero-bias anomalies (ZBAs) in ZBAsinnanoconstrictions. Inthefirst,basedonZawad-
the conductance of quenched Copper nanoconstrictions, owski’s non-magnetic Kondo model [23,24], the interac-
has emerged as a potential experimental realization of tionbetweenTLSsandconductionelectronsisdescribed,
the 2-channel Kondo (2CK) model and the correspond- at sufficiently low energies, by the 2CK model (reviewed
ing non-Fermi-liquid physics [2,12,13,14]. Although crit- in Appendix B of paper II), leading to an energy de-
icisms of the 2CK interpretation [15,16] and alternative pendent scattering rate and hence a ZBA. In the sec-
mechanisms for the ZBAs have been offered [17,18], the ond, Kozub and Kulik’s theory of TLS-population spec-
2CK scenario has recently receivedimportant additional troscopy [17,18], the ZBA is attributed to a V-induced
suppportfromexperimentalresultsonZBAsinconstric- non-equilibrium occupation of the upper and lower en-
tions made fromTitanium [19]andfrom metallic glasses ergy states of the TLSs (see Appendix A3).
[20,21]. Thoughthetwotheoriesmakequitesimilarpredictions
In a series of three papers (I, II, III) we shall present fortheshapeoftheZBA,theymakedifferentpredictions
a detailed analysis of these ZBA experiments and their fortheV/T-scalingbehaviorofG(V,T). WhereasKozub
2CK interpretation. The present paper (I) is a compre- and Kulik’s theory predicts that G(V,T) does not obey
hensive review of the ZBA experiments that attempts anyV/T-scalingrelationatall,the2CKscenariopredicts
tointegrateallexperimentalresultsonthequenchedCu, [2] that in the regime T ≪ TK and eV ≪ kBTK (where
Tiandmetallicglassconstrictionsintoacoherentpicture TK is the Kondo temperature), the conductance G(V,T)
(while postponing allformaltheoreticaldevelopments to should obey a scaling relation of the form
papers II and III). Paper II contains a calculation of the
G(V,T)−G(0,T)
non-equilibriumconductancethroughananoconstriction = F(eV/k T), (1)
Tα B
containing 2CK impurities, which is compared with the
Cu experiments. Paper III, which is the only paper of
where F(x) is a sample-dependent scaling function.
thethreethatrequiresknowledgeofAL’sconformalfield
Moreover, AL’s CFT solution of the 2CK problem sug-
theorysolutionofthe2CKmodel,describesabosonicre-
gested that by scaling out non-universal constants, it
formulation[22] of their theory that is considerably sim-
should be possible to extract from F(x) a universal (i.e.
pler than those used previously and is needed to derive
sample-independent) scaling curve Γ(x), and that the
certain key technical results used in paper II.
conductance exponent α should have the universal non-
Let us begin by briefly summarizing the quenched Cu Fermi-liquidvalueα= 1,instrikingcontrasttotheusual
ZBA experiments and how they inspired the theoretical 2
Fermi-liquidvalue[25]ofα=2. Sincenocalculationhad
work presented in papers II and III.
been provided in Ref. [2] to support the statement that
RB used lithographic techniques to manufacture α= 1,itsstatus uptonowhasbeenthatofaninformed
quenchedCu constrictionsofdiameters assmallas 3 nm 2
guess rather than a definite prediction, a situation that
(see Fig. 1), and studied the conductance G(V,T,H)
is remedied in papers II and III.
throughtheso-callednanoconstriction(orpointcontact)
A detailed scaling analysis [2] showed that the data
as a function of voltage (V), temperature (T) and mag-
of RB indeed do obey the above scaling relation, with
netic field (H). Their constrictions were so small that
α=0.5±0.05. Itshouldbeemphasizedthattheverifica-
theywereabletodetectelectronscatteringatthelevelof
tionofscalingwasaverysignificantexperimentalresult:
individualimpuritiesordefectsintheconstriction. Since
firstly, the scaling relation (1), by combining the V- and
the energy dependence of the scattering rate can be ex-
T-dependenceofG(V,T)forarbitraryratiosofV/T,con-
tractedfromthe voltagedependence ofthe conductance,
tains much more information than statements about the
such an experiment probes the actual electron-impurity
separate V- or T-dependence would; and secondly, an
2
accurate experimental determination of the scaling ex- 5 and 20 mV), implying that some new, large energy
ponent α is possibly only by a scaling analysis of all the scale is involved. These two phenomena are not generic
data (for a detailed review of this central ingredient of to TLS-induced ZBAs, however, since they are observed
thedataanalysis,seesectionVI). Accurateknowledgeof neither in metallic-glass constrictions nor in Ti constric-
α is very important, since α succinctly characterizes the tions, which in fact conform in all respects to what is
low-energycriticalpropertiesofthephysics,enablingone expected for 2CK physics. We shall suggest that the
toeliminatemanyotherwiseplausiblecandidatetheories twophenomenainvolve(asyetpoorlyunderstood)“high-
for the ZBA (such as that by Kozub and Kulik). energy” physics associated with the strongly-interacting
The experimental value for α agrees remarkably well system of electrons and atomic tunneling centers. Such
withtheCFTpredictionofα= 1;furthermore,thescal- physics is beyond the scope of the existing 2CK model
2
ing curve Γ(x) is indeed the same for all three samples and its CFT treatment, which deals only with the “low-
studied in detail by RB, in accord with the CFT ex- energy” aspects of the problem.
pectation that it should be universal and hence sample- Paper I is organized as follows: In section II we de-
independent. Thus,thisresultconsiderablystrenghtened scribe the fabrication and characterization of nanocon-
the case for the 2CK interpretation of the RB experi- strictions,andsummarizesomeelementsofballisticpoint
ment,withinwhichtheexperimentaldemonstrationthat contactspectroscopyinsectionIII. InsectionIVwesum-
α= 1 is,remarkably,equivalenttothedirectobservation marize the main experimental facts associated with the
2
of non-Fermi-liquid physics. ZBA in the Cu samples, which we state in the form of
Nevertheless, this scaling behavior can conceivably nine properties, (Cu.1) to (Cu.9). The 2CK interpre-
also be accounted for by some other theory. Indeed, tation is presented in section V, where its assumptions
Wingreen, Altshuler and Meir [15](a) have pointed out aresummarizedandcriticallydiscussed. SectionVI con-
that an exponent of α= 1 also arises within an alterna- tains a scaling analysis of the G(V,T) data at H = 0.
2
tive interpretation of the experiment, based not on 2CK The related ZBA experiments on Ti and metallic-glass
physicsbutthephysicsofdisorder. (Webelievethatthis nanoconstrictions are discussed in section VII. Finally,
interpretation is in conflict with other important experi- wesummarizetheresultsandconclusionsofthispaperin
mental facts, see section A1). section VIII. Appendix A describes experimental argu-
It is therefore desirable to develop additional quanti- mentsforrulingoutanumberofconceivableexplanations
tative criteria for comparing experiment to the various for the ZBA that could come to mind as possible alter-
theories. One possible criterion is the scaling function natives to the 2CK scenario. In Appendix B we discuss
Γ(x). Averystringentquantitativetestofanytheoryfor possiblesourcesofmagneticfielddependenceinthe2CK
the RB experiment would therefore be to calculate the scenario, concluding it is essentially H-independent.
universalscalingfunctionΓ(x),whichshouldbeafinger-
printofthetheory,andcompareittoexperiment. Papers
II andIII aredevotedtothistask: Γ(x)iscalculatedan- II. THE NANOCONSTRICTION
alyticallywithintheframeworkofthe2CKmodelandits
exactCFTsolutionbyAL,andthe resultsarecompared A schematic cross-sectionalview ofa typicalnanocon-
to the RB experiment. When combined with recent nu- striction (often also called a point contact) is shown in
merical results of Hettler et al. [12], agreement with the Fig. 1. The device is made in a sandwichstructure. The
experimental scaling curve is obtained, thus lending fur- middle layerisaninsulatingSi N membrane. Thiscon-
3 4
ther quantitative support to the 2CK interpretation for tains in one spot a bowl-shaped hole, which just breaks
the Cu constrictions. through the lower edge of the membrane to form a very
The main conclusion of our work is that the 2CK in- narrowopening,assmallas3nmindiameter. Thisopen-
terpretationcanqualitativelyandquantitativelyaccount ing is so small that the resistance signal, measured be-
forallthescalingpropertiesoftheconductancemeasured tweenthe topandbottomofthestructure,iscompletely
in the ZBAs of Cu point contacts. The Ti and metal- dominated by the region within a few constriction radii
lic glass results add further evidence in support of the of the opening. Hence the resistance is sensitive to scat-
2CK interpretation,as opposedto other proposedmech- tering from single defects in the constriction region.
anisms. However,weshallnotethatthe2CKmodeldoes To obtainthe bowl-shapedhole in a Si N membrane,
3 4
notaccountfortwophenomenaobservedinthequenched electron beam lithography and reactive ion etching are
Cu samples. Firstly, the magnetic field dependence of used in a technique developed by Ralls [27] (the details
the low-bias conductance is rather strong (the 2CK ex- relevantto the presentexperiments aredescribedin Ref.
planation for the field dependence that was offered in [11],section2.2). Inultra-highvacuum(<2×10−10torr)
Ref. [2] does not seem to survive closer scrutiny, as dis- and at room temperature the membrane is then rotated
cussed in Appendix B). Secondly, in many (but not all) to expose both sides while evaporating metal to fill the
Cu constrictions the conductance undergoes very sud- hole (thus forming a metallic channel through the con-
den transitions at certain voltages Vc [11,26] (see (Cu.9) striction)andcoatbothsidesofthemembrane. Alayerof
of section IV) if T and H are sufficiently small. These atleast 2000˚A of metal (Cu orTi in the workdescribed
voltagescanbe ratherlarge(V typicallyrangesbetween
c
3
below) is deposited on both sides of the membrane to If, for example, the voltage is large enough to excite
form clean, continuous films, and then the devices are phonons(>5mVforCu),theI-V curveisdominatedby
quenched (see property (Cu.1) in section IV). electron-phononscattering. Inthis case,itcanbe shown
that at T =0, ∆G(V)=− 4e2m2v a3/3π¯h4 τ−1(eV),
F
where τ−1(ε′) ≡ ε′dεα2F (ε) is the relaxation rate
III. BALLISTIC POINT CONTACT 0 (cid:0)p (cid:1)
for an electron at energy ε′ above the Fermi surface.
SPECTROSCOPY R
Thus, due to phonon-backscattering processes, the con-
ductance of any point contact drops markedly at volt-
A constriction is called ballistic if electrons travel bal-
ageslarge enoughto excite phonons [V >5 meV for Cu,
listically through it, along semi-classical, straight-line see Fig. 2(a)]. Furthermore, the function α2F (eV), the
p
paths between collisions with defects or the walls of the
so-called point contact phonon spectrum, can be directly
constriction. This occurs if two conditions are fulfilled:
obtained from ∂ ∆G(V). For any clean, ballistic Cu
V
Firstly, it must be possible to neglect effects due to the
nanoconstriction, ∂ ∆G(V) should give the same func-
V
diffraction of electron waves, i.e. one needs 1/kF ≪ a, tion α2Fp(eV), characteristic of the phonon spectrum,
wherea=constrictionradius. Secondly,theconstriction
andindeednanoconstrictionmeasurementsthereofagree
mustberatherclean(asopposedtodisordered): anelec- with other determinations of α2F . However, the am-
p
tronshouldjustscatteroffimpuritiesonceortwicewhile
plitude ofthe phonon-inducedpeaks is reduceddramati-
traversing the hole. One therefore needs a ≪ l, where l
callyifthereissignificantelasticscatteringduetodefects
is the electron mean free path.
orimpuritiesintheconstrictionregion,ashasbeenmod-
ThequenchedCuZBA-devicesofRBreasonablymeet
eled theoretically [30] and demonstrated experimentally
both conditions: firstly, for Cu 1/k ≃0.1nm,whereasa
F [31]. Therefore, comparing the point contact phonon
is of order 2-8 nm [as determined from the Sharvin for-
spectrum of a given point contact to the reference spec-
mula for the conductance, Eq. (3)]. Secondly, for clean,
trumof acleanpoint contactprovidesanimportantand
annealed devices l ∼ 200 nm (as determined from the
reliabletoolfordeterminingwhetherthepointcontactis
residual bulk resistivity). For devices containing struc-
clean or not.
tural defects, l is reduced to about l >∼ 30 nm [see For voltages below the phonon threshold (V < 5mV
(Cu.4)], which is still about twice the constriction di-
forCu),theV-dependenceof∆G(V)isduetoscattering
ameter. Thus, we shall henceforth regard the quenched off defects. For a set of defects at positions R~ , with an
Cu ZBA-devices as ballistic constrictions. i
isotropic, elastic, but energy-dependent scattering rate
Some aspects of the theory of transport through bal-
τ−1(ε),thebackscatteringconductancehastheform[14]
listic constrictions[28,29]arereviewedin Appendix Aof
paper II. Here we merely summarize the main conclu- ∞
∆G(V)=−(τ(0)e2/h) dω[−∂ f (h¯ω)] (4)
sions. ω o
The differential conductance has the general form Z−∞
× b 1 τ−1(h¯ω− 1eVa+)+τ−1(h¯ω− 1eVa−) .
dI(V) i2 2 i 2 i
G(V)≡ =Go+∆G(V). (2) Xi h i
dV
(cid:12)(cid:12) (cid:12)(cid:12) We factorized out the constant τ(0)e2/h to ensure that
The constant Go,(cid:12)(cid:12) the so(cid:12)(cid:12)-called Sharvin conductance, ∆G has the correct dimensions and order of magnitude.
arises fromelectrons that travelballistically throughthe We assume that the resistance contribution from each
holewithoutscattering. Sharvinshowedthatforaround defect may be calculated independently – that is, that
hole, quantuminterferenceforelectronsscatteringfrommulti-
pledefectsmaybeignored. Thea andb are(unknown)
G =a2e2mε /(2π¯h3), (3) i i
o F constants of order unity that characterize all those de-
tails of scattering by the i-th impurity that are energy-
where a is the radius of the hole, and hence the mea-
independentandofasample-specific,geometricalnature.
sured value of G can be used to estimate the size of the
o
The b account for the fact that the probability that an
constriction. i
electronwillorwillnottraversetheholeafterbeingscat-
Any source of scattering in the constriction that
teredoffthei-thimpuritydependsonthepositionofthe
backscatterselectronsandhencepreventsthemfrombal-
impurityrelativeto the hole. Thea accountforthe fact
listically traversing the hole gives rise to a backscatter- i
ing correction∆G. If the electron scattering rate τ−1(ε) that impurities that are at different positions R~i in the
is energy-dependent, ∆G(V) is voltage dependent. In nanoconstrictionfeeldifferenteffectivevoltages(because
fact, one of the most important characteristics of ballis- theamountbywhichthenon-equilibriumelectrondistri-
tic nanoconstrictions is that the energy dependence of bution function at R~i differs from the equilibrium Fermi
τ−1(ε)canbe directlyextractedfromthevoltagedepen- function f depends on R~ ).
o i
denceof∆G(V),whichimpliesthatballisticnanconstric- In spite of the presence of the many unknown con-
tions can be used to do spectroscopy of electron-defect stants a , b , we shall see that it is nevertheless possible
i i
scattering. to extract general properties of τ−1(ε) from the mea-
4
sured∆G(V,T)data. Forexample,fromEq.(4)onecan in[11]and[32]. Wesummarizethemintheformof9im-
deduce that if portant properties of the ZBA in quenched Cu nanocon-
strictions:
ln[max(T,ε)],
τ−1(ε,T)−τ−1(0,T)∝ (5)
TαΓ˜(ε/T), (Cu.1) Quenching: ZBAs and conductance transitions
(cid:26)
[Fig. 2(a)] are found only in quenched Cu samples,
ln[max(T,V)],
then ∆G(V,T)∝ TαF(V/T), i.e. samples that are cooled to cryogenic tempera-
(cid:26) tures within hours after being formed by evapora-
where Γ˜ and F are scaling functions. tion. Theyarefoundinabout50%ofsuchsamples,
and in a variety of materials, such as Cu, Al, Ag
and Pt. Cu was used in the samples discussed be-
low).
IV. EXPERIMENTAL FACTS FOR QUENCHED
CU SAMPLES
(Cu.2) Amplitude: Typical values for G(V =0) vary from
2000 to 4000 e2/h. The anomaly is only a small
In this section we summarize the experimental facts feature on a very big background conductance: its
relevant to the ZBA in quenched Cu samples. Our in- amplitude [G −G(V =0)] varies from sample
max
terpretation of these facts is postponed to later sections, to sample, from a fraction of e2/h to as large as
where some of them will be elaborated upon more fully, 70e2/h at 100 mK. It’s sign is always the same,
andwheremostofthefiguresquotedbelowcanbefound. with G(V,T) increasing from G(0,T ) as V or T
o
The phenomenon to be studied is illustrated by the are increased. The sample (#1 in Fig. 7) showing
upper differential conductance curve in Fig. 2. Its three best scaling (see (Cu.6) below) had a maximum
essential features are the following: Firstly, the differen- ZBA amplitude of about 20e2/h.
tial conductance shows a drop for |V| > 5 mV, due to
the excitation of phonons,a process whichis well under- (Cu.3) Annealing:
stood(seesectionIII). Secondly,therearesharpvoltage- (a) After annealing at room temperature for sev-
symmetric conductance spikes at somewhat larger volt- eral days, the ZBA and conductance spikes disap-
ages (V ), called conductance transitions in Ref. [1,32], pear, and the conductance curve looks like that of
c
because in the DC conductance they show up as down- acompletelycleanpointcontact[seelowercurvein
wardstepswithincreasingV (seefigure13below). Some Fig. 2(a)].
of their complex properties arelisted in point (Cu.9) be- (b) Nevertheless, such annealing changes the to-
low. tal conductance by not more than 1% or 2% (both
Thirdly, the conductance has a voltage-symmetric dip increases and decreases have been observed), indi-
near V = 0; this is the so-called zero-bias anomaly catingthatthe overallstructureofthe constriction
(ZBA). As a sample is cooled, the temperature at which does not undergo drastic changes.
the zero-bias features become measurable varies from (c) Upon thermal cycling, i.e. brief (several min-
sample to sample, ranging from 10 K to 100 mK. This utes)excursionstoroomtemperatureandback,the
paper is concerned mainly with the regime V < 5 mV amplitude of the ZBA and the position Vc of the
dominated by this ZBA. conductance transitions change dramatically and
TheZBAisaveryrobustphenomenon. Fordecadesit non-monotonically [see Fig. 3(a)]. The complexity
has been observed, though not carefully investigated, in of this behavior suggests that the thermal cycling
mechanical “spear and anvil” point contacts made from is causing changes in the position of defects within
a variety of materials, see e.g. [33]. Even the dramatic theconstriction,andthattheZBAisverysensitive
conductancetransitionshaveprobablybeenseeninearly to the precise configuration of the defects.
ZBA experiments [28], though their presence had not
(Cu.4) Effect of disorder:
been emphasized there.1
(a)Ifstaticdisorderisintentionallyintroducedinto
The advent of the mechanically very stable nanocon-
a nanoconstriction by adding 1% or more of im-
strictions employedby RB alloweda detailed systematic
purity atoms such as Au to the Cu during evapo-
study of the ZBA. Their findings are discussed at length
ration, the zero-bias conductance dip and conduc-
tance spikes disappear completely [see Fig. 3(b)].
Likewise, the signals are absent in samples for
whichwateris adsorbedontothe Si N surfacebe-
3 4
1For example, Fig. 3C of [28] shows a d2I/dV2 spectrum foremetaldeposition(thestandardsamplefabrica-
with sharp signals, more or less symmetric about zero, that
tion procedure therefore involves heating the sam-
are consistent with being derivatives of spikes in the dI/dV
pleto∼100◦Cinvacuum,orexposingitforseveral
conductance curve. Note that these signals are too sharp to
bespectroscopic signals smeared bykT,butareindicativeof hours to ultraviolet light in vacuum, before the fi-
abrupt transitions. nal metal evaporation is done).
(b) When a strongly disordered region is created
5
near the constriction (by electromigration: a high is possible to extract from F(v) a “universal”scal-
bias (100-500 mV) is applied at low temperatures ingfunctionΓ(v)[showninFig.11(b)below]. Γ(v)
so that Cu atoms are moved around, a method is universalin the sense that it is indistinguishable
controllably demonstrated in [27,34,35]), the con- for all three devices for which a scaling analysis
ductance shows no ZBA either, but instead small- was carried out (they are called sample 1,2 and 3
amplitude, voltage-dependent (but aperiodic) con- below).
ductance fluctuations at low voltage [see Fig. 3(c),
(Cu.7) Logarithms: For V or T beyond the cross-over
(d)]. That these are characteristic of strongly dis-
scales V or T , G(V,T) deviates markedly from
orderedconstrictionsandcanbeinterpretedasuni- K K
the scalingbehaviorof(Cu.6)andbehavesroughly
versalconductancefluctuationsduetoquantumin-
logarithmically: For H =0 and fixed, small T, the
terference, was established in a separate investiga-
conductance goes like lnV for V > V [Fig. 5(a)];
tion [36], [11, chapter 4], [37]. K
similarly, for H =V =0 and T >T , the conduc-
K
tance goes like lnT [Fig. 5(b)].
(Cu.5) Phonon spectrum: For quenched samples, in the
(Cu.8) Magnetic field:
point contact phonon spectrum the longitudinal
(a)Whenamagneticfield(ofupto6T)isapplied,
phonon peak near 28 mV is not well-defined, and
the amplitude of the ZBA in Cu devices decreases
the total amplitude of the spectrum is smaller by
[see Fig. 4(b)]. The change in amplitude can be
about 15% than after annealing. After anneal-
as large as 24 e2/h if H changes from 0 to 6 T.
ing, the longitudinal phonon peak reappears and
For sufficiently small H (< 1T), at fixed T and
the spectrum corresponds to that of clean ballis-
V = 0, the magnetoconductance roughly follows
tic point contacts. Both these differences indicate
G(H,T) ∝ |H| (see Fig. 14 below). However, the
(see p. 4) that the elastic mean free path l in the
available data is insufficient to establish linear be-
annealed samples is somewhat longer than in the
havior beyonddoubt, and, for example, wouldalso
quenched samples. From the phonon spectrum of
be compatible with a |H|1/2-dependence.
the latter, l can be estimated (see section II) to be
l >∼ 30 nm [for the sample shown in Fig. 2(a)], still (b) The ZBA dip undergoes no Zeeman-splitting
in H, in constrast to the Zeeman splitting that is
more than about twice the constriction diameter
foundfordevicesintentionallydopedwithmagnetic
for that device. Note also that the point contact
impurities such as Mn [see Fig. 4(a)].
phononspectrum for a quencheddevice [Fig. 2(b)]
isqualitativelyverydifferentfromthatofastrongly
(Cu.9) Conductance transitions:
disordered constriction [Fig. 3(d)]. These facts,
(a) Voltage-symmetric conductance transitions
viewed in conjunction with (Cu.3b) and (Cu.4c),
(spikes in the differential conductance at certain
imply that the Cu constrictions displaying ZBAs
“transition voltages” V , see Fig. 2) occur only in
c
are still rather clean and ballistic.
quenched point contacts that show ZBAs, but oc-
cur in at least 80% of these. The spikes disappear
(Cu.6) V/T scaling (to be established in detail in sec-
under annealing, just as the ZBA does (Cu.3a).
tion VI):
(b) (i) A single sample can show several such con-
(a)At H =0, the conductance obeys the following
ductance transitions (up to 6 different V s have
scaling relation if both V < V and T < T , but c
K K
been observed in a single sample). (ii) If T and
for arbitrary ratio v =eV/k T:
B H aresmall(sayT <∼1K,H <∼0.5T),Vc istypically
G(V,T)−G(0,T) rather large,with typical values ranging between 5
=F(v). (6)
Tα and 20 mV, well above the typical voltages associ-
atedwiththe ZBA(i.e. V >V ). The spikeshave
c K
a very complex behavior as a function of temper-
Relation (6) allows a large number of data curves
ature (T) and magnetic field (H), including (iii) a
to be collapsed onto a single, sample-dependent
hysteretic V-dependence, (iv) a bifurcation of sin-
scaling curve [e.g. see Figs. 8(a) and 8(b) below].
gle spikes into two separate ones (V ,V ) when
The departure of individual curves from the low-T c1 c2
B 6=0(Fig.12),(v)theH-dependentmotionofthe
scaling curve in Figs. 8(a) and 8(b) indicates that
spike positions V (H) → 0 when H becomes suffi-
V or T has surpassed the crossover scales V or c
K
ciently large (Figs. 12, 13), and (vi) a very rapid
T . From the data, these are related roughly by
K
narrowing of the peaks with decreasing T. They
eV =2k T , with T in the range 3 to 5 K.
K B K K
are described at length, from a phenomenological
(b) F(v) is a sample-dependent scaling function
with the properties F(0) 6= 0 and F(v) ∝ vα as point of view, in Ref. [32].
v →∞, and the scaling exponent is found to have
Any theory that purports to explain the ZBA in Cu
the value α=0.5±0.05.
constrictionsmustbeconsistentwithalloftheaboveex-
(c) By scaling out sample-dependent constants, it
perimental facts. An extention of this list to include the
6
results of the recent related ZBA experiments by Upad- and odd linear combinations of the lowest-lying eigen-
hyayet al. onTi constrictions and by Keijsers et al. [21] states of each separate well), whose eigenenergies differ
onmetallic-glassconstrictionsispresentedinsectionVII. by ∆=(∆2+∆2)1/2. Ultra-fast two-state systems have
z x
In the next section, we shall argue that the 2CK sce- such a large ∆ that ∆ too becomes very large, so that
x
nario provides the most plausible interpretation of the at low temperatures only the lowest level governs the
above experimental facts. A number of alternative ex- physics.
planationsfor the ZBAthatcouldcome to mind aredis-
cussedin Appendix A, but allare found to be in conflict
with some of the above facts. 1. Slow Fluctuators
Thefactthattwo-statesystemsinmetalnanoconstric-
V. THE 2-CHANNEL KONDO (2CK) tions can influence the conductance was demonstrated
INTERPRETATION byRallsandBuhrman[34,35,37],whoobservedso-called
telegraphsignalsinwell-annealeddevices(atratherhigh
In this section, we develop the 2CK interpretation of temperatures of 20-150K).These are slow, time-resolved
the ZBAs in quenched Cu constrictions. It attributes the fluctuations (fluctuation rates of about 103s−1) of the
ZBA to the presence in the constriction region of dy- conductancebetweentwo(orsometimesseveral)discrete
namical structural defects, namely TLSs, that interact values, differing by fractions of e2/h, which can be at-
with conduction electrons according to the non-magnetic tributedtothefluctuationsofaslow two-statefluctuator
Kondo model, which renormalizes at low energies to the in the constriction region.
non-Fermi-liquid regime of the 2CK model. We begin by Such telegraph signals were also observed by Zimmer-
briefly recalling in section V.A some properties of two- man et al. [40,41], who studied the conductance of poly-
levelsystems (or slightly more generally,dynamical two- chrystalline Bi films, a highly disordered material with
state systems) in metals. Successes and open questions presumably large numbers of two-state systems. They
ofthe2CKscenarioarediscussedinsubsectionsV.Band were able to measure the parameters of individual slow
V.C, respectively, and its key assumptions are listed, in fluctuatorsdirectly,findingvaluesfortheasymmetryen-
the form of a summary, in subsection V.D. ergy ∆ ranging from as little as 0.08 K to about 1K.
z
They also demonstrated that in a disordered environ-
ment the asymmetry energy of a TLS is a random, non-
A. Two-State Systems monotonic function of the magnetic field, ∆ = ∆ (H)
z z
(as predicted earlier in Ref. [42]), and hence can be
A dynamical two-state system (TSS), is an atom or “tuned” at will by changing H. The reason is, roughly,
group of atoms that can move between two different po- that∆z depends onthe difference δρ=ρL−ρR between
sitions inside a material [38]. In the absence of interac- thelocalelectrondensitiesatthetwominimaoftheTLS
tions,itsbehaviorisgovernedbyadouble-wellpotential, potential. Due to quantum interference effects that are
generically depicted in Fig. 6, with asymmetry energy amplified by the presence of disorder, changes in H can
∆z, tunneling matrix element ∆x. The corresponding induce random changes in δρ and hence also in ∆z.
Hamiltonian is Unfortunatly, experiments on slow fluctuators do not
yield any direct information on the parameters to be ex-
H = 1(∆ τz +∆ τx) , (7) pected for fast ones, since their parameters fall in differ-
TSS 2 z x
ent ranges.
where τx and τz are Pauli matrices acting in the two-
by-two Hilbert space spanned by the states |Li and |Ri,
describing the fluctuator in the left or right well. 2. Two-Level Systems
Depending on the parameters of the potential, the
atom’smotionbetweenthepotentialwellsis classifiedas
Fast fluctuators or TLSs presumably have the same
either slow, fast or ultrafast, with hopping rates τ−1 <
microscopic nature and origin as slow fluctuators, be-
108s−1, 108s−1 < τ−1 < 1012s−1 or τ−1 > 1012s−1, re-
ing composed of atoms or small groups of atoms which
spectively [39]. Slow two-state systems, called two-state
move between two metastable configurations, but with
fluctuators, have large barriers and neglibibly small ∆ ,
x muchlowerbarriers. Therefore,theyannealawayquicker
andthemotionbetweenwellsoccursduetothermallyac-
than slow fluctuators, which is why they were not seen
tivated hopping or incoherent quantum tunneling. Fast
in the above-mentioned Ralls-Buhrman experiments on
two-statesystemshavesufficientlysmallbarriersandsuf-
well-annealed samples [11, p. 265]. Also, whereas slow
ficiently large ∆ that coherent tunneling takes place
x fluctuators “freeze out” as T is lowered (which is why
backandforthbetweenthewells. Suchasystemisknown
they don’t play a role in the ZBA regime of T <5K), at
asatwo-leveltunnelingsystem(TLS),becauseitsphysics
lowT fastfluctuatorscontinueto undergotransitionsby
is usually dominated by its lowesttwoeigentstates (even
tunneling quantum-mechanically between the wells.
7
A fast fluctuator or TLS interacting with conduction complicated non-Fermi-liquid physics characteristic of
electronsisusuallydescribedbythenon-magnetic oror- the 2CK model in the T ≪ T regime. In this respect
K
bitalKondomodel,studiedingreatdetailbyZawadowski thenon-magnetic2CKmodeldiffersinanimportantway
andcoworkers[23,24](itis definedandreviewedinmore from the (1-channel) magnetic Kondo model, for which
detail in Appendices B and C of paper II; for other re- the low-T scaling is of the Fermi liquid form (∝T2).
views, see [39,43,44]):
H =H + ε c† c (8) B. Successes of the 2CK Interpretation
TSS ~k ~kσ ~k′σ
X~k
We now turn to an interpretation of facts (Cu.1) to
+ c† V0 +Vx τx+Vz τz c .
~kσ ~k~k′ ~k~k′ ~k~k′ ~k′σ (Cu.9) in terms of the 2CK scenario [1,2]. Our aim here
X~k~k′ h i is to sketch the physicalpicture underlying the scenario.
Those aspects that require detailed analysis, such as the
Herec† createsanelectronwithmomentum~kandPauli
~kσ scaling behavior (Cu.6) and magnetic field dependence
spin σ. The terms Vo and Vzτz describe diagonal scat- (Cu.8), will be discussed more fully in subsequent sec-
tering events in which the TLS-atoms do not tunnel be- tions.
tweenwells. ThetermVxτx describesso-calledelectron- Qualitative features: The cooling and annealing prop-
assisted tunneling processes. Duringthese,electronscat- erties (Cu.1) and (Cu.3) suggest that the ZBAs are due
tering does lead to tunneling, and hence the associated to structural defects or disorder that can anneal away
barematrixelementsaremuchsmallerthanfordiagonal athightemperatures [althoughthe well-resolvedphonon
scattering: Vx/Vz ≃10−3. spectrumimpliesthatonlyasmallamountofsuchdisor-
Zawadowski and coworkers showed that the electron- der can be present (Cu.5)]. This conclusion is reinforced
assisted term Vxτx renormalizes to substantially larger by the remarkablycomplex andnon-monotonicbehavior
values as the temperature is lowered (as does a similar of the ZBA under thermal cycling (Cu.3c), which indi-
Vyτy term that is generated under renormalization). At cates that the ZBA probes the detailed configuration of
sufficientlylowtemperatures(whereVz ≃Vx ≃Vy),the individual defects, not just the average behavior of the
non-magneticKondomodelwasshown[45]tobeequiva- entire constriction region. Subsequent experiments with
lenttothestandard2-channelKondo(2CK)model,with Ti constrictions have shown that the structural disorder
an effective interaction of the form is located in the “bulk” of the bowl-shaped hole, not on
its surface, and that it is caused by geometry-induced
Hienftf =vK dε dε′ c†εασ 12~σαα′ · 21~τ cεα′σ . stress occuring in the metal in the bowl-shaped part of
Z Z α,α′ σσ′ the constriction [see (Ti.1d), section VIIA].
XX (cid:0) (cid:1)
By assuming that the ZBA is due to fast TLSs, i.e. a
(9)
specific type of structural defect, the 2CK scenario ac-
countsforallofthepropertiesmentionedintheprevious
The two positions of the fast fluctuator in the L- and R
paragraph. Property (Cu.4a), the disappearance of the
wells correspond to the spin up and down of a magnetic
ZBA upon the addition of 1% Au atoms, can then be
impurity (and L-R transitions to impurity spin flips).
attributed to the TLSs being pinned by the additional
The electrons are labelled by an energy index ε, a so-
static impurities.
calledpseudo-spinindexα=1,2(correspondingtothose
Logarithms and Scaling: Next, we assume that the
two combination of angular momentum states about the
TLS-electron interaction is governed by Zawadowski’s
impurity thatcouple moststronglyto the TLS),andthe
non-magnetic Kondo model, which renormalizes to the
Pauli spin index σ =↑,↓. Evidently, α plays the role of
2CK model at low energy scales. This explains a num-
the electron’s magnetic spin index in the magnetic 2CK
ber of further facts. Firstly, the non-magnetic nature of
model, and since the effective interaction is diagonal in
the interaction explains the absence of a Zeeman split-
σ (which has two values), σ is the channel index.
ting in a magnetic field (Cu.8b). Furthermore, the fact
This(non-magnetic)2CKmodel,withstronganalogies
thatthe 2CKscatteringrate τ−1(ε,T)has alogarithmic
to the magnetic one, yields an electron scattering rate
τ−1(ε,T) with the properties [24,46] form for ε > TK(> T) or T > TK(> ε) [see Eq. (10)]
accounts, via Eq. (5), for the asymptotic logarithmic V-
τ−1(ε,T)−τ−1(0,T) ∝ (10) and T dependence (Cu.7) of G(V,T) for V > VK(> T)
or T > T (> V). Thus, we identify the experimental
ln[max (T,ε)] if T >T , K
K crossover temperature T (≃ 3 to 5K) of (Cu.6a) with
T1/2Γ˜(ε/T) if ∆2/T <T ≪T . K
(cid:26) K K the Kondo temperature of the 2CK model.
Similarly, the scaling form of τ−1(ε,T) for ε,T ≪ T
(Thecondition∆2/T <T isexplainedinsectionVIC.) K
K [seeEq.(10)]accounts,viaEq.(5),fortheobservedscal-
Hence, for T > T or ∆2/T < T ≪ T , it yields [via
K K K ing behavior (Cu.6) of G(V,T) for V <V and T <T .
Eq. (5)] a contribution to the conductance of δσ(T) ∝ K K
To be more particular, the very occurence of scaling be-
lnT or T1/2, respectively. The latter is typical for the
8
havior (Cu.6a), and the fact that the experimental scal- are related and must involve some new “high-energy”
ing curve Γ(v) of (Cu.6c) is universal, can be explained physics,since(Cu.9)occursatalargevoltageV . There-
c
(see section VIA) by assuming that the system is in the fore, our lack of understanding of the latter need not
neighborhood of some fixed point. Assuming this to be affect the 2CK interpretation of the low-energy scaling
the2CKnon-Fermi-liquidfixedpoint,theexperimentally behavior (Cu.6). We conclude with some speculations
observedscalingregime canbe associatedwith the theo- about the microscopic nature of the TLSs, and the like-
retical expected scaling regime of ∆2/T < T <T and lihood that realistic TLSs will have allthe properties re-
K K
V <T . Moreover, the non-Fermi-liquid value of α= 1 quired by the 2CK scenario.
K 2
that is then expected for the scaling exponent (see sec-
tionVIA)agreespreciselywiththevalueobservedforα.
Thus, within the 2CK interpretation, the experimental 1. Conductance Transitions
demonstrationofα= 1 is equivalentto the directobser-
2
vation of non-Fermi-liquid behavior. Finally, it will be The fact that conductance transitions occur only in
shown in paper II that the shape of the universal scal- samplesthathaveaZBA(Cu.9a)suggests[32]thatthese
ing curveΓ(v) is alsoin quantitativeagreementwith the arerelatedtotheZBA:ifthelatterisphenomenologically
2CK model. viewed as the manifestation of some strongly correlated
Number of TLSs: Each 2CK impurity in the constric- state of the system, then conductance transitions corre-
tion can change the conductance by at most 2e2/h.2 spond to the sharp, sudden, “switching off” of the cor-
Therefore, the sample with the largest ZBA of 70e2/h relations as V becomes too large. For example, in the
(sample #2 in Fig.7) wouldrequireup to about40 such 2CK interpretation, interactions of electrons with TLSs
TLSs in the constriction. However, this is still only a in the constrictiongive rise to a stronglycorrelatednon-
relatively small amount of disorder (corresponding to a Fermi-liquidstateatsmallT andV. Onemightspeculate
density of about 10−4 TLSs per atom [11, p.277]3). The that if for some reasona largevoltagecould “freeze”the
samplethatshowedthebestscaling(sample#1inFig.5) TLSs, i.e. prevent them from tunneling, this would dis-
had a significantly smaller amplitude of <∼ 20e2/h, im- rupt the correlationsand give rise to a sudden change in
plying only about 10 active TLSs (that samples with a the DC conductanceandhence aspike inthe differential
smaller amplitude should show better scaling is to be conductance.
expected due to a smaller spread in ∆’s, see (Ti.3) in At present we are not aware of any detailed micro-
section VIIA). scopicexplanationfortheconductancetransitions. Note,
though, that they do not occur in all Cu samples show-
ingZBAs. Moreover,recentexperimentsbyUpadhyayet
C. Open Questions in the 2CK Scenario al. [19] on Titanium constrictions and by Keijsers et al.
[20] on constrictions made from metallic glasses showed
Havingdiscussedthesuccessesofthe2CKscenario,we TLS-induced ZBAs with properties very similar to RB’s
now turn to questions for which the 2CK scenario is un- quenched Cu constrictions, but no conductance transi-
able to offer a detailed explanation, namely the conduc- tions at all (see (Ti.6) and (MG.4) in section VII, where
tancetransitions(Cu.9),thestrongmagneticfielddepen- theseexperimentsarereviewed). Thissuggeststhatcon-
dence (Cu.8a), and the microscopic nature of the TLSs. ductance transitions are not a generic ingredient of the
Weshallpointoutbelowthat(Cu.9)and(Cu.8a)arenot phenomenology of ZBAs induced by TLSs. Moreover, in
generic to TLS-induced ZBAs, and speculate that they thequenchedCusamples,providedthatH andT aresuf-
ficientlysmall,thetransitionvoltageV atwhichthefirst
c
conductancetransitionoccursusuallylieswellaboveT ,
K
thescalecharacterizingtheextentofthelow-energyscal-
2 To see this, we note that in the unitarity limit the scat- ing regime of the ZBA [see Fig. 2(a)]. (In other words,
since they don’t occur near zero bias, the conductance
tering rate of electrons off a k-channel Kondo defect is pro-
portional to ksin2δ (see e.g. [47, Eq.(2.20)]), and the phase transitions need not be viewed as part of the zero-bias
shiftattheintermediate-couplingfixedpointisδ=π/2k[48]. anomaly phenomenon at all, if one restricts this term to
Thus,intheunitaritylimit,thecontributiontotheresistance
refer only to the low-energy regime.)
of a k =2 Kondo impurity is the same as for k = 1, namely
2e2/h (the2 comes from Pauli spin). Thus, there seems to be a clear separation of energy
3Forexample,the6.4Ωconstrictionstudiedin[1]hasadiam- scales governing the ZBA and the conductance transi-
eter of ∼13 nm [estimated via the Sharvin formula Eq. (3)], tions. The latter must therefore be governed by some
and there are 105 Cu atoms within a sphere of this diameter newlargeenergyscaleduetoamechanismnotyetunder-
about the constriction. Assuming on the order of ∼40 active stood. However, due to the separation of energy scales,
TLSs, their density is therefore roughly of order 10−4/atom. the conductance transitions need not affect our descrip-
Although the constriction is believed to be crystalline, not
tion of the low-energy scaling regime of the ZBA below
glassy, it is worth noting that this density of TLSs is about
T (which is ≪eV /k ) in terms of the 2CK model.
the same as estimates for the total density of TLSs in glassy K c B
systems.
9
2. Strong Magnetic Field Dependence of (Cu.6) should not be affected by having H 6=0). The
presently available data is unfortunately insufficient to
Since the electron-TLS interaction is non-magnetic, test this prediction.
i.e. not directly affected by a magnetic field, the 2CK The conclusions of this and the previous subsection
scenario predicts no, or at best a very weak magnetic aresummarizedinassumptions(A3)and(A4)insubsec-
field dependence for the ZBA. This agrees with the ab- tion VD.
sence of a Zeeman splitting of the ZBA for the Cu sam-
ples (Cu.8b) (which was in fact one of the main reasons
for the proposalof the non-magnetic 2CK interpretation 3. Microscopic Nature of the TLS
[1]). However, it leaves the strong magnetic field depen-
dence (Cu.8a) as a puzzle. (Two indirect mechanism for Finally, the 2CK interpretation is of course unable to
H to couple to a 2CK system, namely via H-tuning of answer the question: What is the microscopic nature of
theasymmetryenergy∆ (H)andviachannelsymmetry the presumed TLSs? Now, ignorance of microscopic de-
z
breaking,areinvestigatedinAppendixB;theyarefound tails does not affect our explanation for why the scaling
to be too weak to account for (Cu.8a), contrary to the properties (Cu.6) of the ZBA seem to be universal: be-
interpretation we had previously offered [2].) cause the latter are presumably governed by the fixed
It is therefore very significant that the experiments point of the 2CK model, any system that is somewhere
by Upadhyay et al. on Ti constrictions and by Kei- in the vicinity of this fixed point will flow towards it as
jsers et al. on metallic-glass constrictions show ZBAs the temperature is lowered (provided that relevant per-
withonlyaveryweakorevennoH-dependence[seesec- turbations are sufficiently small) and hence exhibit the
tion VII, (Ti.5), (MG.3)], in complete accord with 2CK same universal behavior, irrespective of its detailed bare
expectations. Thissuggeststhat,justastheconductance parameters.
transitions,thestrongmagnetic field dependence (Cu.8a) However, the quality of the scaling behavior implies
of the quenched Cu constrictions is not a generic fea- some ratherstringentrestrictions onthe allowedproper-
ture of TLS-induced ZBAs. Moreover, Fig. 13 suggests ties of the presumed TLSs, because we need to assume
that in the Cu samples these two properties might be that allactive TLSs (e.g. about10 for sample #1, which
linked, because it shows that the strong H-dependence shows the best scaling) are close enough in parameter
of G(V = 0,H) is related to the fact that the transition space to the non-Fermi-liquid fixed point to show pure
voltage V decreases to 0 as H is increased (Cu.9b,v). scaling.
c
(Inotherwords,ifthestronglycorrelatedstatesetsinat This implies, firstly, that interactions between TLSs
smallerV asV islowered,thevoltage-regime0<V <V (which are known to exist in general [34], mediated by
c c
in which the anomaly can develop is smaller, so that its strain fields and changes in electron density), must be
total amplitude is smaller.) negligible, because they would drive the system away
SincethemaindifferencebetweentheCuconstrictions from the 2CK non-Fermi-liquid fixed point. Secondly,
and the Ti and metallic-glass constrictions seems to be the fact that scaling is only expected in the regime
thattheformercontainTLSswithverysmall∆’s(seethe ∆2/T <T <T canbeusedtoestimatethatT ≃3to
K K K
next subsection), whereas in the latter, being disordered 5K and ∆<∼1K (see section VI for details). Kondo tem-
materials, there will certainly be a broad distribution of peratures in the range of 1-10 K are in good agreement
splittings, we speculate that the conductance spikes and with the most recenttheoreticalestimates for TLSs [49].
strong H-dependence might both be a consequence of However, the condition ∆ <∼ 1K implies that for active
the very small ∆s occuring in the Cu samples, perhaps TLSs the distribution of energy splittings, P(∆), must
duetointeractionsbetweenseveralTLSswithverysmall be peaked below ∆<∼1K. Since ∆=(∆2z+∆x)1/2, both
splittings. the asymmetry energy ∆z and tunneling rate ∆x must
Thus,we concludethatattempts (suchasthosein[2]) be ∼<1K,a value so small that it needs further comment.
toexplaintheH-dependenceoftheZBA(evenatV =0) First note that it is not immediately obvious that val-
purely in terms of the 2CK model, which captures only ues of the bare tunneling rate ∆ exist at all that allow
x
the physics at low energies below T , are misdirected, 2CK physics: For transitions to be able to take place,
K
because the H-dependence would arise, via the conduc- the barrier between the wells must be sufficiently small,
tance transitions, fromthe “high-energy”physics associ- butasmallbarrierisusuallyassociatedwithalargebare
ated with the large scale Vc. ∆x,implyingalargebare∆(and∆setstheenergyscale
This interpretation, according to which a magnetic atwhichthe renormalizationflowtowardthenon-Fermi-
field does not directly affect the low-energy physics of liquid fixed point is cut off). Now, for a TLS in a metal,
the phenomenon (only indirectly via its effect on Vc), the physics of screening can reduce the direct tunnel-
can be checked by doing a V/T scaling analysis at fixed ing rate ∆ by as much as three orders of magnitude
x
but small, non-zero magnetic field. If H is sufficiently under renormalization to T ≪ T [50] (when tunneling
K
smallthattheconductancetransitionsstilloccuratrela- betweenthewells,thetunnelingcenterhastodragalong
tively high voltages(i.e. Vc >TK), the scalingproperties its screening clowd,whichbecomes increasinglydifficult,
10
