Quantum Noise Measurement of a Carbon Nanotube Quantum Dot in the Kondo Regime

2.1 MB·English

by  Julien Basset

Quantum Noise Measurement of a Carbon Nanotube Quantum Dot in the Kondo
Regime
J. Basset,1 A.Yu. Kasumov,1 C.P. Moca,2,3 G. Zar´and,2,4 P. Simon,1 H. Bouchiat,1 and R. Deblock1
1Laboratoire de Physique des Solides, Univ. Paris-Sud,
CNRS, UMR 8502, F-91405 Orsay Cedex, France.
2BME-MTA Exotic Quantum Phase Group, Institute of Physics,
Budapest University of Technology and Economics, H-1521 Budapest, Hungary.
3Department of Physics, University of Oradea, Oradea, 410087, Romania
4Freie Universita¨t Berlin, Fachbereich Physik, Arnimallee 14, D-14195 Berlin, Germany.
The current emission noise of a carbon nanotube quantum dot in the Kondo regime is measured
2
atfrequenciesν oftheorderorhigherthanthefrequencyassociatedwiththeKondoeffectk T /h,
1 B K
0 with TK the Kondo temperature. The carbon nanotube is coupled via an on-chip resonant circuit
2 to a quantum noise detector, a superconductor-insulator-superconductor junction. We find for
hν ≈ kBTK a Kondo effect related singularity at a voltage bias eV ≈ hν, and a strong reduction
n
of this singularity for hν ≈ 3k T , in good agreement with theory. Our experiment constitutes a
a B K
new original tool for the investigation of the non-equilibrium dynamics of many-body phenomena
J
in nanoscale devices.
6
2 PACSnumbers: 73.23.-b,72.15.Qm,73.63.Fg,05.40.Ca

]
l
l How does a correlated quantum system react when via a superconducting tunnel junction of size 240×150
a
probed at frequencies comparable to its intrinsic energy nm2 (figure 1). Each transmission line consists of two
h
- scales ? Thanks to progress in on-chip detection of sectionsofsamelengthlbutdifferentwidths,thusdiffer-
s high frequency electronic properties, exploring the non- ent characteristic impedances Z ≈ 110Ω and Z ≈ 25Ω
e 1 2
m equilibrium fast dynamics of correlated nanosystems is (figure 1a). Due to the impedance mismatch, the trans-
now accessible though delicate. In this respect, the missionlineactsasaquarterwavelengthresonator, with
.
t Kondoeffectinquantumdotsisamodelmany-bodysys- resonances at frequencies ν = nv/4l = nν , with v the
a n 1
tem,wherethespinofthedotisscreenedbythecontacts’ propagation velocity and n an odd integer [13]. The
m
conduction electrons below the Kondo temperature T two transmission lines are close to one another to pro-
K
-
d [1–3]. The Kondo effect can then be probed at a single vide a good coupling at resonance and are terminated
n spin level and in out-of-equilibrium situations. It leads by on-chip Pd resistors. The junction has a SQUID ge-
o to a strong increase of the conductance of the quantum ometry to tune its critical current with a magnetic flux.
c dot at zero bias due to the opening of a spin degenerate The carbon nanotube (CNT) is first grown by chemical
[
conducting channel, the transmission of which can reach vapor deposition on an oxidized undoped silicon wafer

2 unity. This effect has been extensively studied by trans- [29]. An individual CNT is located relative to prede-
v port and noise experiments in the low frequency limit fined markers and contacted to palladium leads using
0 [4–11]. Howeverthenoiseinthehighfrequencylimithas electron-beam lithography. The junction and the res-
7
notbeenexploredexperimentallydespitethefactthatit onatorarethenfabricatedinaluminum(superconducting
5
allows to probe the system at frequencies of the order or gap ∆ = 182µeV). A nearby side-gate allows to change
1
smaller than k T /h characteristic of the Kondo effect the electrostatic state of the nanotube. The system is
. B K
0
[12]. In this letter we present the first high frequency thermallyanchoredtothecoldfingerofadilutionrefrig-
1
noise measurements of a carbon nanotube quantum dot eratorofbasetemperature20mKandmeasuredthrough
1
in the Kondo regime. We find for hν ≈ k T a Kondo low-passfilteredlineswithastandardlowfrequencylock-
1 B K
: effect related singularity at a voltage bias eV ≈hν, and in amplifier technique.
v
a strong reduction of it for hν ≈ 3k T . These results
i B K To characterize the CNT-quantum dot, we first mea-
X are compared to recent theoretical predictions.
sure its differential conductance dI/dV as a function
S
r The high frequency current fluctuations are mea- of dc bias voltage V and gate voltage V (figure 1c).
a S G
sured by coupling the carbon nanotube (CNT) to a For a gate voltage between 3.05 and 3.2V the CNT’s
quantum noise detector, a Superconductor-Insulator- conductance at zero bias strongly increases, a signature
Superconductor (SIS) junction, via a superconducting of the Kondo effect. The half width at half maximum
resonant circuit (see figure 1a). This allows us to probe (HWHM) of the Kondo ridge yields the Kondo temper-
the emission noise of the CNT at the resonance frequen- ature T = 1.4K in the center of the ridge [1]. This
K
cies of the coupling circuit (29.5 GHz and 78 GHz) by valueisalsoconsistentwiththetemperaturedependence
measuring the photo-assisted tunneling current through of the zero bias conductance. The Kondo temperature is
thedetector[13]. Theprobedsampleconsistsoftwocou- related to the charging energy U of the CNT quantum
pled coplanar transmission lines. One line is connected dot,thecouplingΓtotheelectrodesandtheposition(cid:15)of
tothegroundplaneviaacarbonnanotubeandtheother the energy level measured from the center of the Kondo
2
a) Z2 Z1 1µm b) ν (GHz) the Josephson junction [17]. Figure 1b shows the I(V)
0 25 50 75 100 125
VDARR aa b a junctionVsG I(nA)D00..24 ν 1 = 42690. 5ΩGHz ν 3 3=3708 GΩHz 240000Re[Z()] (νΩ) owvfiiatthheeqmujaaugtninocetntioic2nflRiunex[t.Zh(eTνs)h]ueb(sgFuaibgp.garep1gbiro)enswofhnoiarcnhIcCeissmtphaeuxaskimeydiizeealddt
l l
VSA source 5N00Tnm 0.0 0 frequencies ν1 = 29.5 and ν3 = 78GHz. Using the
super2conductor gate drain 0.0 0.1VD(mV)0.2 0.3 hweeigihntfearndthewidcothupolifngthebertewsoeennantcheepejuankcstioofnRaen[Zd(νth)]e,
c)
CNT [13]. We then translate a photo-assisted tunneling
1 2
d (PAT) quasi-particles current measurement into a cur-
V(mV)S -10 1SVd/)eI(h /2 rareennndttνteh3m.roisuTsgihohnetrhnaeotiidsoeetrmencetbaosreutawrneemdenethnttehcefourmrrteehanestuefrrmeedqisusPeiAonncTienscouiνsr1e-
0
of the CNT at a given resonance frequency is estimated
-2
as follows. r is given at the resonant frequency ν by
3.0 3.1 3.2 n n
VG (V) e2|Zt(νn)|2δνn/(hνn)2IQP,0(VD+ hνen) with Zt the tran-
simpedance of the coupling circuit, defined as the ra-
tio between the voltage fluctuations across the detector
FIG.1: (a)Sketchofthesample: acarbonnanotube(bottom
and the current fluctuations through the source, δν the
n
electron microscope picture) is coupled to a superconductor-
widthoftheresonancepeakandI (V )theI-V char-
insulator-superconductor (SIS) junction (top electron micro- QP,0 D
acteristic of the detector [13]. This value has been cali-
scope picture), used as a quantum detector, by a supercon-
brated in a previous experiment with the same design as
ducting resonant circuit. This circuit is constituted by two
the one used in the present work [13]. r is then calcu-
transmission lines, placed close to one another, made of alu- n
minumandterminatedbyon-chipPdresistors. TheSISjunc- lated using the ratio of the calibrated sample corrected
tion is fabricated by shadow angle evaporation. The carbon according to the square area under the corresponding
nanotube is CVD grown, connected with Palladium contacts peak of Re[Z(ν)] (figure 1b), the value of ν , the super-
n
andside-gated. (b)Lowercurve: I(V)ofthedetectorinthe conducting gap and the tunnel resistance of the detector
subgap region. Upper curve: The real part of the impedance junction.
seenbythedetectorexhibitsseveralresonances. (c)Differen-
To measure the quantum noise of the CNT, we modu-
tialconductancedI/dV ofthecarbonnanotubeasafunction
S
late its bias voltage V and monitor the modulated part
of voltage bias VS and gate voltage VG. It exhibits a Kondo S
ridge for gate voltage between 3.05 and 3.20 V with an in- of the PAT current through the detector for a given de-
crease of conductance at zero bias. tector bias voltage VD. VD selects the frequency range
of the measurement [13]. We have thus access to the
derivative of the PAT current versus CNT bias voltage
ridge, according to Bethe-Ansatz [14, 15]:
dI /dV at a given frequency. Using the previously
PAT S
(cid:112) (cid:104) π (cid:105) estimated coupling coefficient, we translate this quan-
TK = UΓ/2exp −8UΓ|4(cid:15)2−U2| . (1) tity into the derivative of the current noise SI at one
of the resonance frequencies versus V , dS /dV . This
S I S
FromU =2.5meV,deducedfromthesizeoftheCoulomb quantity is plotted in the center of the Kondo ridge, i.e.
diamond, and TK = 1.4K, we obtain Γ = 0.51meV. The (cid:15)=0 at two frequencies (Figure 2a and b). For each fre-
asymmetry A = (ΓL − ΓR)/(ΓL + ΓR) = 0.67 of the quency, the data exhibit a region close to VS = 0 where
contacts is deduced from the zero bias conductance. dS /dV = 0. This corresponds to |eV | < hν, where
I S S
To characterize the superconducting resonant circuit the system does not have enough energy to emit noise at
whichcouplesthedetectorjunctiontotheCNT,wemea- a frequency ν. The observation of this zero noise region
surethesubgapI(V)characteristicofthejunctionwhich is a strong evidence that we are indeed only measuring
dependsontheimpedanceofitselectromagneticenviron- the emission noise of the CNT. For |eV | > hν the sys-
S
ment [17]. In the case of a superconducting transmission tem emits noise at ν. For the first resonance frequency
line resonator [13, 16], resonances appear in the subgap ν = 29.5GHz, with hν ≈ k T , the measured deriva-
1 1 B K
region VD <2∆/e due to the excitation of the resonator tive of the noise shows a singularity for bias voltages
modes by the ac Josephson effect [18]. These resonances close to the measured frequency. At higher bias volt-
are related to the real part of the impedance Z(ν) seen agesdS /dV ismuchsmoother. Forhν ≈2.7k T the
I S 3 B K
by the junction: previoussingularityisnearlyabsentanddS /dV versus
I S
V is practically flat.
I(V )=Re[Z(2eV /h)]I2/2V , (2) S
D D C D
The high frequency noise of quantum dots in the
with I = π∆/(2eR ) the critical current [18], R = Kondo regime has been studied theoretically at equilib-
C N N
28.6kΩ the normal state resistance of the junction and rium using the numerical renormalization group (NRG)
∆ the superconducting gap of the electrodes. Equa- technique [19]. Non-equilibrium results for the finite-
tion 2 accounts for the effect of the electromagnetic frequency noise are theoretically much more demand-
environment on the tunneling of Cooper pairs through ing. They were obtained only for peculiar values of
3
fortheleft(L)andright(R)leads,σ standsforthethree
a) c) LowBiasregime Paulimatrices,andtheoperatorψ destroysanelectron
-1V) 4 ddtheaectaoorhye wreitnhc e E ofspinσ inleadα∈{L,R}. Wepαaσrametrizethedimen-
-1z. 2 theory 2 hνννν1 sionless exchange couplings j as j = j v v , with
pA².H 0 e²/h) eV the factors {vL,vR} = {cos(φαβ/2),sinα(βφ/2)} aαccoβunting
dS/dV(bI)S--424 0dI/dV (S Large Biasregime g(E) fφ(o2rer2et/lhahet)eadssiyntm2o(mφthe).teryTof=th0e qcuonandtuucmtandcoet,acsosG(φ(T)==A,0)an=d
-1-1z.V) 2 E atiTonhse Kinonthdeo HCaNmTiltqounaianntuamssudmotesartehaftroczheanr.ge Tfluhcetrue--
H
dS/dV(pA².IS--042 0.2 VS0(m. 5V) 000.8..36dI/dV(e²/h)S eV hνννν3 fbaogereecs,oVmthSpea(cid:28)trhedUeo/wreei.ttihcAaeslxrapeesfirurimlsttsesnbtteaapsle,odtnooesndeo(tn3el)rymcfaionnrebatinhades rsvahotalitlo-l
-2 -1 VS0 (mV) 1 2 g(E) TKRG/TK, we computed the equilibrium conductance by
usingNRG[30]andcomparedittoexperimentaldatain
the center of the Kondo ridge ((cid:15) = 0). This enabled us
to establish T ≈ 3.7 TRG (see the appendix). There-
K K
FIG. 2: (a) Derivative of the current noise and differential fore, the condition hν (cid:29) k TRG for our FRG approach
conductanceoftheCNTversusV inthecenteroftheKondo B K
S to apply is certainly met for the frequency ν , and still
ridge. Left axis : measured dS /dV (in red) at 29.5GHz as 3
I S reasonably satisfied for ν .
a function of bias voltage V . The black solid line corre- 1
S Within the Kondo model, we can express the Fourier
sponds to the calculated dS /dV with a fitted decoherence
I S
transform of the emission noise S as :
rate incorporating both intrinsic and extrinsic decoherence I
(see text), while the dashed line is the theoretical prediction e2 (cid:18) eV hν T (cid:19)
with the intrinsic decoherence rate. Right axis : Differential SI(VS,ν)= h kBTKRG s k TSRG,k TRG,TRG,A ,
conductance dI/dV of the CNT in units of e2/h. (b) Same B K B K K
S (4)
data at 78GHz. Inset : measured conductance, symmetrized
with respect to the bias voltage, in units of e2/h. The black where s is a dimensionless function, which we calculate
solid line correspond the the FRG calculation of the conduc- bysolvingnumericallytheFRGequation(seeappendix).
tance with the same fitted decoherence rate as the one used Since the measurement temperature satisfies T (cid:28) TK,
forthenoisemeasurements. (c)Schematicpictureofaquan- we have taken T = 0 in the calculations and checked
tumdotintheout-of-equilibriumKondoregimetogetherwith that a finite but small temperature does not affect our
thedensityofstatesfortwodistinctbiasvoltages. Whenthe results. Note that no fitting parameter has been in-
quantumdotisvoltagebiased,theKondoresonancesplitsin cluded at this level, since the asymmetry parameter A
two with a splitting given by the applied bias voltage. This and TRG ≈ 0.38 K were extracted from the experimen-
leads to an increase of the emission at frequency eV = hν. K
S tal data. The dashed lines in figure 2a and b show the
Theamplitudeoftheresonancepeaks,andthustheemission
calculated dS /dV curves for frequencies ν =29.5GHz
noise at eV = hν, can be reduced by decoherence effects I S 1
S
and ν = 78GHz, respectively. The computed curves
induced by the applied bias voltage. 3
are only shown in the bias range |V | < 1 mV, where
S
the Kondo Hamiltonian in equation (3) is appropriate to
parameters (strongly anisotropic exchange couplings) of describe the physics of the CNT quantum dot. For both
the Kondo problem using bosonization methods [20], frequencies,thetheoreticalcurvesexhibitsharpsingular-
and by using non-equilibrium real time renormalization ities at eVS = hν, much more pronounced than the ex-
group approaches [21, 22]. The latter approaches as- perimental ones. This especially holds for the resonance
sume hν,eVS (cid:29) kBTKRG, with TKRG the Kondo temper- frequency, ν3 = 78GHz, where the resonance is almost
ature defined from the renormalization group. Impor- completelyabsentexperimentally. Thesingularityatthe
tantly, TKRG differs from TK (defined experimentally as threshold eVS ≈ hν is related to the existence of two
the HWHM of the differential conductance) by a nu- KondoresonancesassociatedwiththeFermilevelsofthe
merical factor, which has to be determined (see below). two contacts. Inelastic transitions between them lead to
Here we employ the real time functional renormalization an increase of dSI/dVS for frequencies corresponding to
group (FRG) approach developed in Ref. [22] to com- the energy separation, hν ≈eVS (figure 2c).
pute the non-equilibrium frequency-dependent noise and To compute the dashed curves in figure 2, an in-
compare it to the experimental results. We perform the trinsic spin decoherence time τS induced by the large
non-equilibrium calculations using the Kondo Hamilto- bias was included and calculated self-consistently in the
nian, given by : FRG approach [23] (see appendix). The decoherence of
the Kondo effect induced by a large dc voltage bias is
H = 1 (cid:88) (cid:88)j ψ† S·σ ψ . (3) a well-known feature which has indeed been measured
K 2 αβ ασ σσ(cid:48) βσ(cid:48) [24, 25], and has been predicted to lead to a strong
α,β=L,Rσ,σ(cid:48)
reduction of the Kondo resonance due to inelastic pro-
Herethej denotetheKondocouplings,α,βareindices cesses [23, 26, 27]. Since the singularity in the noise at
αβ
4
a) h/τ ≈ α k TRGatan(βeV /k TRG) (similar in shape
2 S B K S B K
to the calculated intrinsic spin relaxation rates), with
α = 14 and β = 0.15 satisfied all criteria above. The
continuous lines in figure 2 show the dS /dV curves
1 I S
computed with this form of h/τ , and fit fairly well the
S
experimental data for both resonator frequencies. As a
V) final consistency check, we also computed the differen-
m 0
(S tial conductance through the dot (taking into account
V theaboveformofτ−1)andcomparedittothemeasured
S
dI/dV curves. A very good agreement is found with-
S
-1 out any other adjustable parameter in the voltage-range
V > 0.1mV, where the FRG approach is appropriate
2.0
(Inset of figure 2b).
1.5
-2 F From the theoretical fits we infer that the experimen-
b) 2 1.0 SV( tally observed noise spectra and differential conductance
0.5 ) can be understood in terms of a decoherence rate, which
0.0 is about a factor of 2 larger than the theoretically com-
1 puted intrinsic rate (see appendix). One possibility for
this discrepancy is that the experimentally observed de-
V) coherence is intrinsic, and FRG - which is a perturba-
m 0 tive approach - underestimates the spin relaxation rate
V(S in this regime (which is indeed almost out of the range
of perturbation theory). Another possibility is that the
-1 experimental set-up leads to additional decoherence.
The experiment also allows to draw a complete map
of the noise in the region of the Kondo ridge. We define
-2 F(VS) = [dSI/dVS(VS)]/[edI/dVS(VS −hν/e)], i.e. the
3.00 3.10 3.20 ratioofthederivativeofthenoisetothedifferentialcon-
V (V)
G ductance shifted in voltage by an amount corresponding
to the measured frequency. For both linear and non lin-
earsystemswithenergyindependenttransmissionatlow
temperaturethisquantityisequaltotheFanofactor[28].
FIG. 3: F(V ) = [dS /dV ]/[edI/dV (V − hν/e)] as a
S I S S S
function of the bias voltage VS and the gate voltage VG at WehaveplottedF(VS)forν1 =29.5GHz(figure3a)and
ν1 = 29.5GHz (a) and ν3 = 78GHz (b). F(VS) is arbitrarily ν3 =78GHz(figure3b). For|eVS|<hν,wheretheemis-
fixed to zero for e|V | < hν. The gray curves on top of the sion noise is zero, F(V ) is arbitrarily fixed to zero. For
S S
colorplotcorrespondtothebiasvoltagedependenceofF(V ) both frequencies the noise is found to be sub-poissonian,
S
at VG = 3.12V and VG = 3.21V. The black arrow indicates withF(VS)closetooneinthepoorlyconductingregions
a value of the F(VS) equal to 1. When the conductance is and a strong decrease of F(VS) along the conducting re-
low, F(V ) is close to one while it is reduced in the highly
S gions. This is qualitatively consistent with the reduction
conducting regions.
oftheFanofactorforaconductingchannelsoftransmis-
sion close to one. This result has to be contrasted with
backscatteringnoisemeasurementsintheKondoregime
hν ≈ eV is associated with the transitions between the at low frequency and low bias voltage where the Fano
S
two Kondo resonances pinned at the Fermi levels of the factor was found to be higher than one [10, 11].
contacts, this singularity is also affected by decoherence. In conclusion we have measured the high frequency
However, as shown in figure 2, the computed intrin- current fluctuations of a carbon nanotube quantum dot
sic decoherence time is insufficient to explain the exper- in the Kondo regime by coupling it to a quantum de-
imentally observed suppression of the peak in dS /dV . tector via a superconducting resonant circuit. We find
I S
Therefore, we incorporated a voltage-dependent spin re- that the noise exhibits strong resonances when the volt-
laxationrateinourcalculations,τ−1(V ),whichincludes age bias is of the order of the measurement frequency in
S S
external decoherence. The consistency of this approach good agreement with theory provided that an additional
can be checked against the experiments: a single choice decoherence rate is included which prevents the full for-
of τ−1(V ) must simultaneously reproduce the voltage mation of the out of equilibrium Kondo resonances. Our
S S
dependence of the differential conductance through the experiment constitutes a new original tool for the inves-
dot dI/dV (V ), and those of the ν = 29.5 GHz and tigation of the non-equilibrium dynamics of many-body
S S 1
ν = 78 GHz noise spectra, dS /dV (V ). Further- phenomena in nanodevices.
1 I S S
more, τ−1 should be suppressed for V < TRG. We We thank M. Aprili, S. Gu´eron, M. Ferrier, M. Mon-
S S K
foundthatabias-dependentdecoherencerateoftheform teverdeandJ.Gabelliforfruitfuldiscussions. Thiswork
5
has benefited from financial support of ANR under Con- 20
tract DOCFLUC (ANR-09-BLAN-0199-01) and CNano "extrinsic rate"
Ile de France (project HYNANO), the EU-NKTH GE- Korringa rate (FRG)
OMDISS project, OTKA research Grants No. K73361 15 fitted relaxation rate
and No. CNK80991 and the French-Romanian grant
DYMESYS (ANR 2011-IS04-001-01 and contract PN-II- pert
K
ID-JRP-2011-1). T
B10
k
/
Γ
Appendix : Relation between T and TRG h
K K 5
Toperformthetheoreticalcalculations,onefirstneeds
to find the Kondo temperature. However, the Kondo
0
temperature is defined only up to a prefactor, and its
0 5 10 15 20
value also depends slightly on the physical quantity
pert
from which it is defined. Our experimental Kondo tem- eV / k T
S B K
perature, T , is defined as T = e ∆V /k , with
K K S B
∆V the half-width at half maximum of the measured
S
G(VS) = dI/dVS curves. In the FRG calculations, on FIG. 4: Voltage dependence of the fitted and intrinsic relax-
the other hand, it is defined as a scale (frequency), ation rates, and their difference, termed as extrinsic spin re-
k TRG ≡¯hωRG, where the so-called leading logarithmic laxationrateinthevoltagerange,V ≥0.1mV. Theintrinsic
B K K S
calculationsyieldadivergentinteractionvertexatT =0 ratewascalculatedbyFRG,whiletheformandoverallmag-
temperature. Itcan,however,alsobedefinedasthetem- nitudeoftheextrinsicratewereinferredfromthedifferential
perature, TT−dep, atwhichthelinearconductancedrops conductance data.
K
to half of its T = 0 temperature value. The ratios of
alltheseKondotemperaturesarejustuniversalnumbers
degrees of freedom. This procedure yields a retarded in-
(apartfromapossiblebutpresumablysmalldependence
teraction(j →g (t−t(cid:48),a)),whosecut-offdependence
αβ αβ
of ∆V on the anisotropy, A).
S is described by the differential equation,
To determine the ratio of TT−dep and k TRG,
K B K
we performed numerical renormalization group calcula- dg(ω,a)
=g(ω,a)q(ω,a)g(ω,a). (5)
tions [31]: We computed the full G(T) curve, extracted dlna
from it the width TT−dep, and for the same parame-
K Here we introduced the matrix notation, g (ω,a) →
ters, we also computed TRG = h¯ωRG/k from the high- αβ
K K B g(ω,a) for the Fourier transform of the retarded interac-
frequency tail of the so-called composite fermions’ spec-
tion, and the matrix q(ω,a) denotes a cut-off function.
tral function, scaling as ≈ C/ln2(ω/ωRG) at large fre-
K In our calculations we have not approximated this lat-
quencies. In this way, we obtained a ratio
ter with Θ-functions, as in Ref. [22], but used a function
corresponding to the real time propagators of Ref. [22],
TRG/TT−dep ≈ 0.3.
K K alsoincorporatingtheeffectofanexponentialdecayrate
Γ=1/τ .
The ratio TTdep/T was then determined from experi- S
K K Thevoltage-dependentdecayrateΓ(VS)hasanintrin-
mental data [32], giving
sic part, Γ (V ), as well as an external contribution,
intr S
Γ (V ). The former contribution can be identified as
ext S
TKT−dep/TK ≈1.2. the Korringa spin relaxation rate, and can be expressed
as
The previous equations yield the ratio, TRG/T ≈0.36,
K K (cid:90)
used in our calculations and quoted in the main text. (cid:88)
Γ = π dωg (ω)g (ω)
intr αβ βα
α,β=L,R
f(ω−µ )(1−f(ω−µ )), (6)
α β
Appendix : Summary of the functional
renormalization group approach with g (ω) the vertex functions in the limit a → ∞,
αβ
µ the electro-chemical potentials of the leads, and f
α
In this work we used the functional renormalization the Fermi function. The rate Γ (V ), as computed
intr S
groupapproachdevelopedinRef.[22],anextensionofthe by FRG is shown in Fig. 4. Rather surprisingly, in the
formalism of Ref. [33]. In this approach, formulated at cross-over regime, eV ∼ k Tpert, it almost saturates,
S B K
thelevelofnon-equilibriumaction,ashorttimecut-offa and only weakly depends on the bias. For a compari-
isintroduced,andincreasedincourseoftherenormaliza- son, figure 4 also shows the total fitted relaxation rate,
tion group (RG) procedure to eliminate the high-energy Γ(V ) = 1/τ (V ) needed to reproduce the differential
S S S
6
conductance and noise data. It is not very far from the double integral of this retarded current vertex taken in
calculated intrinsic contribution, but it is above the lat- the limit a→∞. The emission noise, S (ω), e.g., can be
e
ter, and it apparently includes some extrinsic spin relax- expressed as
ation, too.
In the formalism of Ref. [22], the current operator and
e2 (cid:90) dω˜
the current vertex are also renormalized during the RG S (ω)= S(S+1) Tr{V(ω˜ ,ω˜ )G>(ω˜ )
procedure,andalsobecomenon-local. However,thecur- e 2 2π + − −
rent vertex, V, has a more complicated structure than V(ω˜ ,ω˜ )G<(ω˜ )}, (7)
− + +
theinteractionvertex,andpossessestwonon-trivialtime
arguments, g (ω) ↔ V (ω ,ω ). The evolution of V
αβ αβ 1 2 with ω˜ = ω˜ ± ω and S = 1/2. Here the trace refers to
undertheRGisdescribedbyadifferentialequationsimi- ± 2
the labels α,β =L/R, and the bigger and lesser Green’s
lartoequation(5)(seereference[22]forthedetails). The
noisespectrum,i.e.,theFouriertransformofthecurrent- functions are given as G>αβ/<(ω) = ±i 2π δαβ f(±(ω −
current correlation function can then be expressed as a µ )).
α
[1] D.Goldhaber-Gordonetal.,Nature391,156-159(1998). 1982).
[2] S.M. Cronenwett, T.H. Oosterkamp and L.P. Kouwen- [19] M. Sindel, W. Hofstetter, J. von Delft, and M. Kinder-
hoven, Science 281, 540-544 (1998). mann, Phys. Rev. Lett. 94, 196602 (2005).
[3] J.Nygard,D.H.CobdenandP.E.Lindelof,Nature408, [20] A. Schiller and S. Hershfield, Phys. Rev. B 58, 14978-
342346 (2000). 15010 (1998).
[4] Y. Meir and A. Golub, Phys. Rev. Lett. 88, 116802 [21] T. Korb, F. Reininghaus, H. Schoeller, and J. Ko¨nig,
(2002). Phys. Rev. B 76, 165316 (2007).
[5] E.Sela,Y.Oreg,F.vonOppenandJ.Koch,Phys.Rev. [22] C.P.Moca,P.Simon,C.H.Chung,andG.Zarand,Phys.
Lett. 97, 086601(2006). Rev. B 83, 201303(R) (2011).
[6] A. Golub, Phys. Rev. B 73, 233310 (2006). [23] J.Paaske,A.Rosch,J.Kroha,andP.Woelfle,Phys.Rev.
[7] A.O. Gogolin and A. Komnik, Phys. Rev. Lett. 97, B 70, 155301 (2004).
016602 (2006). [24] S. De Franceschi, et al, Phys. Rev. Lett. 89,
[8] C.Mora,X.LeyronasandN.Regnault,Phys.Rev.Lett. 156801(2002).
100, 036604 (2008). [25] R. Leturcq, et al, Phys. Rev. Lett. 95, 126603 (2005).
[9] T. Delattre et al, Nature Phys. 5, 208 (2009). [26] R.C. Monreal, and F. Flores, Phys. Rev. B 72, 195105
[10] O.Zarchin,M.Zaffalon,M.Heiblum,D.Mahalu,andV. (2005).
Umansky, Phys. Rev. B 77, 241303(R) (2008). [27] R.vanRoermund,S.Shiau,andM.Lavagna,Phys.Rev.
[11] Y.Yamauchietal.,Phys.Rev.Lett.106,176601(2011). B 81, 165115 (2010).
[12] P.Nordlander,M.Pustilnik,Y.Meir,N.S.Wingreenand [28] Y.M.Blanter,andM.Bu¨ttiker,Phys.Rep.336,1(2000).
D.C. Langreth, Phys. Rev. Lett. 83, 808-811 (1999). [29] Yu.A.Kasumov,et al,Appl.Phys.A88,687691(2007).
[13] J.Basset,H.BouchiatandR.Deblock,Phys.Rev.Lett. [30] We used the open access Budapest DM-NRG code,
105, 166801 (2010). http://neumann.phy.bme.hu/˜ dmnrg/.
[14] A.M.TsvelickandP.B.Wiegmann,AdvancesinPhysics [31] For our calculations we used the Budapest NRG code
32, 453 (1983). (http://neumann.phy.bme.hu/˜ dmnrg/), developed by
[15] N.E. Bickers, Rev. Mod. Phys. 59, 845-939 (1987). Legeza, O., Moca, C.P., Toth, A. I., Weymann, I., and
[16] T.Holst,D.Esteve,C.Urbina,andM.H.Devoret,Phys. Zarand,G.;FortechnicaldetailsseealsoA.I.Toth,C.P.
Rev. Lett. 73, 3455-3458 (1994). Moca, O. Legeza, and G. Zarand, Phys. Rev. B 78,
[17] G. Ingold, and Y.V. Nazarov, Single-Charge Tunneling, 245109 (2008).
edited by Grabert,H. and Devoret, M.H. (Plenum, New- [32] W.G. van der Wiel et al., Science 289, 2105 (2000).
York, 1992). [33] A.Rosch,J.Kroha,andP.Wo¨lfle,,Phys.Rev.Lett.87,
[18] A. Barone, and G. Paterno,Physics and Applications 156802 (2001)
of the Josephson effect (Wiley-Interscience, New-York,