Magnetic flux tuning of Fano-Kondo interplay in a parallel double quantum dot system

3.9 MB·English

by  R. R. Agundez

Magnetic flux tuning of Fano-Kondo interplay in a parallel double quantum dot
system
R. R. Agundez,1 J. Verduijn,1,2 S. Rogge,1,2 and M. Blaauboer1
1Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
2Centre for Quantum Computation and Communication Technology,
University of New South Wales, Sydney NSW 2052, Australia
We investigate the Fano-Kondo interplay in an Aharonov-Bohm ring with an embedded non-
interacting quantum dot and a Coulomb interacting quantum dot. Using a slave-boson mean-
field approximation we diagonalize the Hamiltonian via scattering matrix theory, and derive the
conductance in the form of a Fano expression, which depends on the mean-field parameters. We
3
predictthatintheKondoregimethemagneticfieldleadstoagappedenergylevelspectrumdueto
1
0 hybridisationofthenon-interactingQDstateandtheKondostate,andcanquantum-mechanically
2 alter the electron’s path preference. We demonstrate that an abrupt symmetry change in the Fano
resonance, as seen experimentally, could be a consequence of an underlying Kondo channel.
n
a
PACSnumbers: 72.10.Fk,73.21.La,73.23.-b,73.63.Kv
J

7
1 I. INTRODUCTION So far, only a small number of dedicated Fano-Kondo
studies have been reported.17–20 Sasaki et al.20 per-
]
l formed transport measurements through a side-coupled
l One of the most fascinating manifestations of many-
a double QD configuration and found a strong modulation
body physics in mesoscopic systems is the Kondo effect.
h of the Kondo conductance by the Fano effect. Their re-
- It was originally observed in the 1930s in gold wires and sults are in good agreement with a tight-binding model
s formallyexplainedin1964byJunKondo1 inmetalsthat
e presentedinaparalleltheoreticalpublication.21 Inathe-
m contain magnetic impurities. The Kondo effect became oretical work, Fang and Luo22 have calculated the con-
an intense topic of research in the context of mesoscopic
. ductance in an Aharonov-Bohm interferometer contain-
t systems2 after its observation in quantum dots (QDs) in
a ing a Kondo QD coupled to a non-interacting QD. They
m the late nineties.3 The Kondo effect in QDs is revealed predictaFano-Kondoeffectthatcoulddecoupledthetwo
as a zero-bias conductance enhancement at low temper-
- dots through the manipulation of the Kondo resonance
d atures when the QD is spin polarized and strongly cou- by varying the inter-dot coupling.
n pled to the conduction electrons (such as a QD that is
o tunnel-coupled to leads). Since its discovery in QDs, the
In this publication we investigate the Fano-Kondo in-
c Kondo effect has been observed and studied in a variety
[ terplay in an Aharonov-Bohm ring, containing a non-
of exciting mesoscopic systems: individual atoms,4 car- interacting QD and a Coulomb interacting QD. Using
1 bonnanotubes,5 molecules,6 buckyballs7 andgraphene.8
a scattering matrix approach combined with the slave-
v
Since the Kondo transport can strongly influence the boson mean-field approximation (SBMF), we calculate
5
3 Fano phenomenon, there has recently been a lot of in- the conductance and the Fano parameter. The deriva-
2 terest in the interplay between these two effects in QDs. tion of the Fano form for the conductance allows us to
4 This interest comes from the use of Fano resonances as analyzein-depththeFano-Kondophenomenonand, spe-
. a tool to study coherence and to provide valuable trans- cially, the origin of the transport features. In addition,
1
0 portinformation,9,10 thatmayresultinfuturenano-scale wepresentaqualitativecomparisonwiththeexperiment
3 deviceapplications.11–13 TheFanoeffectresultsfromthe performed by Verduijn et al.9, and suggest an explana-
1 interference between a scattering event within a contin- tion of their observations in terms of a Fano-Kondo in-
: uum of states (background process) and another one by terference phenomenon in an Aharonov-Bohm geometry.
v
i means of a discrete energy level (resonant process). A
X
Fanoresonancemanifestsitselfasanasymmetricconduc- The paper is organized as follows: In section II we in-
r tance line-shape.14 Fano resonances have been observed troduce the Anderson Hamiltonian for a non-interacting
a
in many different physical systems and extensively stud- QD and a Coulomb interacting QD connected in paral-
ied in nanoscale systems.15 Scientific awareness of the lel. We diagonalize this Hamiltonian resulting in a set
interaction between the Fano and the Kondo effect be- of self-consistent equations, which we solve numerically
gan with STM measurements on metal surfaces,16 how- to obtain the conductance. In section III we present our
ever, it found its greatest impact in mesoscopic systems. calculations. TheseleadustopredictaFanofootprintof
The Fano-Kondo effect is observed when Kondo trans- anunderlyingKondochannelandtoshowthatthemag-
port takes over the background scattering process in the netic flux can open a gap between two resonances and
Fano effect. The trademark of such interaction is a con- alter the electron’s path preference for transport. We
ductance valley when tuning the energy level of the res- conclude in IV with a discussion of applications of our
onant process. results.
2
II. MODEL & METHODOLOGY state in the lower QD. We then have to create a boson
whenremovinganelectronfromtheQDtoleaveitinthe
WeconsidertwoQDsembeddedinanAharonov-Bohm empty state, fd,σ → fd,σb†d. Since Ud → ∞ double oc-
ring which is attached to two 1D conducting leads. Fig. cupancy is forbidden which is imposed by the constraint
1showsaschematicrepresentationofthenanostructure. b†b +(cid:80) f† f =1. Withinthisexactrepresentation,
d d σ d,σ d,σ
A magnetic field is used to create a magnetic flux (Φ) the Hamiltonian H of the system [Eq. (1)] becomes:
(cid:88) (cid:88)
H =(cid:15) c† c −t (c† c +c.c.)
SB F j,σ j,σ 0 j,σ j+1,σ
j,σ j,σ
(cid:88) (cid:88)
+ (cid:15) f† f +λ (b†b + f† f −1)
- - - n n,σ n,σ d d d d,σ d,σ
n,σ σ
(cid:88)
− (t c† f +t c† f +c.c)
u,L −1,σ u,σ u,R 1,σ u,σ
σ
(cid:88)
− (t c† f b† +t c† f b† +c.c).(2)
d,L −1,σ d,σ d d,R 1,σ d,σ d
FIG. 1. Tight-binding model for two QDs embedded in an
σ
Aharonov-Bohm ring nanostructure. The QDs are labeled
Here the constraint has been added by means of a La-
withindexesuanddfortheupperandlowerQD,respectively.
grange multiplier, λ . Since the Hamiltonian H [Eq.
d SB
(2)] remains difficult to solve, we use the mean-field ap-
through the ring, which gives a phase contribution (φ)
proximation were fluctuations around the average value
to the electron’s wavefunction. The Hamiltonian of the
of the boson operators in Eq. (2) are neglected. By sub-
system is given by
stituting the boson operators with their real expectation
(cid:88) (cid:88)
H =(cid:15) c† c −t (c† c +c.c.) values, b →< b >, the slave-boson Hamiltonian [Eq.
F j,σ j,σ 0 j,σ j+1,σ d d
(2)] is replaced by an effective mean-field Hamiltonian,
j,σ j,σ
(cid:88)
+ (cid:15)nfn†,σfn,σ+Udfd†↑fd↑fd†↓fd↓ HSB,MF =(cid:15)F(cid:88)c†j,σcj,σ−t0(cid:88)(c†j,σcj+1,σ+c.c.)
n,σ
j,σ j,σ
(cid:88)
− (tn,Lc†−1,σfn,σ+tn,Rc†1,σfn,σ+c.c). (1) +2((cid:15)d+λd)fd†fd+λd(<bd >2 −1)
n,σ
+2(cid:15) f†f −2(t c† f +t c†f +c.c)
We assume that each QD has one single energy level la- u u u u,L −1 u u,R 1 u
beled (cid:15)n (with n = u for the upper QD and n = d for −2<bd >(td,Lc†−1fd+td,Rc†1fd+c.c). (3)
the QD in the lower path). In the leads each site has a
The factor of 2 takes into account the contribution from
single-particle energy at the Fermi energy (cid:15) . The hop-
F both spin directions. We now minimize the ground state
ping energy between neighboring sites in the leads is t .
0 energyofEq. (3)withrespecttotheLagrangemultiplier,
TakingintoaccounttheAharonov-Bohmphasepickedup
λ , and the boson expectation value, <b >,
by the electron (φ), the hopping between the 1D leads d d
and the ring structure is given by tn,L = tne±iφ4 and ∂ <HSB,MF > =2<f†f >+<b >2 −1=0,(4a)
tn,R = tne∓iφ4, where tn,L and tn,R (n = u,d), describe ∂λd d d d
the hopping of an electron from the upper (u) or lower ∂ <H >
SB,MF =λ <b >2 −R =0, (4b)
(d) QD to the left (L) or right (R) lead. The operators ∂ <b > d d d
d
c (c† ) and f (f† ) correspond to the annihilation
j,σ j,σ n,σ n,σ where
(creation) operators in the leads and in the QDs, respec-
tively. Finally a Coulomb interaction is present in the R =<b >(t <c† f >+t <c†f >
d d d,L −1 d d,R 1 d
lower QD with strength U .
d +t∗ <f†c >+t∗ <f†c >). (5)
We calculate the conductance of this nanostructure in d,L d −1 d,R d 1
three regimes, depending on (cid:15) , the energy level of the
d The fermion expectation values in Eq. (4) are obtained
lower QD, 1) the Kondo regime ((cid:15) < (cid:15) , U > (cid:15) −(cid:15)
d F d F d by diagonalizing the mean-field Hamiltonian [Eq. (3)].
and Γ < (cid:15) −(cid:15) , where Γ is the tunnel coupling of
d F d d We take as ansatz the linear combination of atomic or-
the lower QD to the leads), 2) the mixed-valence regime bitals, |Ψ (cid:105) = (cid:80) ak|n(cid:105)+(cid:80) ak|j(cid:105), where ak and ak
((cid:15) < (cid:15) , U > (cid:15) − (cid:15) and Γ ≈ (cid:15) − (cid:15) ) and 3) k n n j j j n
d F d F d d F d are the probability amplitudes to find the electron, with
the empty orbital regime ((cid:15) > (cid:15) ). For simplicity, we
d F momentumk andenergyw =(cid:15) −2t cos(k),atsitej in
F 0
take U → ∞ and use the infinite-U slave boson mean-
d the leads and at quantum dot n=u,d, respectively. We
field approximation to diagonalize the Hamiltonian H
assumethatthetransportofelectronsisdescribedbyan
[Eq. (1)].23,24 Within this scheme we introduce a bo-
incoming plane wave that is reflected and transmitted at
son creation (annihilation) operator b†d(bd) that acts on theAharonov-Bohmnanostructure. Assuminganincom-
the lower QD. Now the b†|0(cid:105) state represents the empty ing electron from the left lead (L), where the chemical
d
3
potential is at (cid:15) +V , yields: ak = eik·j +r e−ik·j ground scattering amplitude. When |q| >> 1 the elec-
F L jL LL
for j < 0 and ak = τ eik·j for j > 0, where 0 ≤ k ≤ tronic transport is dominated by the resonant process
jL RL
kL = arccos(2VtL0). τRL and rLL correspond to the trans- abnadckgthroeucnodnpdruoccteasnscsetatratksetsoacoLnotrreibnutztiea,nthsehFapane.opAasratmhe-
missionamplitudefromthelefttotherightleadandthe
eter increases. When both processes contribute signifi-
reflection amplitude in the left lead, respectively.25,26.
cantly |q| ≈ 1, and the line-shape becomes asymmetric.
By substituting the ansatz, |Ψ (cid:105), into the Schr¨odinger
k When the background process takes over as the main
equation using the mean-field Hamiltonian [Eq. (3)], we
mechanism for transport |q| ≈ 0 and the conductance
arrive at the following system of linear equations:
shows a symmetrical dip centered at the resonance.15
−t0rLL+tueiφ4auL+t˜de−iφ4adL =t0 (6a) Below, we show that by tuning the energy level of the
interacting QD we can set the conductance to any of
−t0τRL+tue−iφ4auL+t˜deiφ4adL =0 (6b) the line-shapes mentioned before, and that interference
tue−iφ4rLL+tueiφ4τRL tuned by the magnetic flux leads to interesting behavior
of the transmission. First we use Eq. (7) to write the
+(w−(cid:15)u)e−ikauL =−tue−iφ4e−2ik(6c) conductance, in terms of (cid:15) , in a Fano form:14
u
t˜deiφ4rLL+t˜de−iφ4τRL |((cid:15) +(cid:15) )+qΓ|2
+(w−(cid:15)˜d)e−ikadL =−t˜deiφ4e−2ik,(6d) G=Go ((cid:15)u+(cid:15)o)2+Γ2 , (8)
u o
where we have renormalized the coupling and the en-
ergy level of the lower QD as t˜d =< bd > td and where the amplitude Go, the resonant point (cid:15)o, the peak
(cid:15)˜ = (cid:15) +λ , respectively. By solving Eq. (6) we ob- broadening Γ and the Fano parameter q are given by:
d d d
tain all the probability amplitudes, ak and ak , for an
jL nL 2e2 1
incomingwavefromtheleft. Inthesamewaywesolvefor G = , (9a)
o h η2+1
an incoming wave from the right lead, where the chem-
η φ
ical potential is at (cid:15)F + VR, by using the coefficients (cid:15) =Γ cos2( ), (9b)
ak =e−ik·j+r eik·j forj >0andak =τ e−ik·j for o uη2+1 2
jR RR jR LR
jne<tic0fi,ewldhebrreea0k≤stkhe≤tikmRe=revaerrcscaolss(y2VmtR0m).eStriyn,cceotehffiecmieangts- Γ=Γuη2+η2s+in21(φ2), (9c)
and probability amplitudes for a left and right incoming
η[η2cos(φ)−sin2(φ)−i(η2+1)sin(φ)]
wave are different. With all the coefficients we can com- q = 2 . (9d)
pute the fermion expectation values in Eq. (4).25 We η2+sin2(φ)
2
obtain the boson parameter < b > and Lagrange mul-
tiplier λd from the self-consistendt set of equations [Eqs. The parameter η = Γ˜(cid:15)˜dd is a measure of the Kondo effect
(?? -6)], at equilibrium and zero temperature. From the strength in the lower QD. Eq. (8) reveals a shift by (cid:15)
o
transmissioncoefficientinEq. (6)theconductanceatthe of the center of the resonance away from its bare value
Fermi energy (k = π/2) reads G= 2he2 |τRL(π/2)|2, with (cid:15)u =(cid:15)F =0, as a consequence of interference.
τRL(π/2)=−[(cid:15) −iΓ(cid:15)u]Γ˜[(cid:15)˜de−iφ2i+Γ˜ (cid:15)˜]d+ΓuΓe−Γ˜iφ2cos2(φ). (7) III. CALCULATIONS & RESULTS
u u d d u d 2
In Eq. (7) we have set the Fermi level to zero ((cid:15) = 0) In this section we analyze the result for the conduc-
F
and introduced the coupling strengths Γu = 2tt2u0 and tsaynstceem[Eoqf.e(q8u)a]tfiroonms [tEhqesp.re(v4i)o-u(6s)s]encutimonerbicyalsloyl.viUngnltehses
Γl˜adte=d b2ytt˜2d0L.´opEeqz.et(a7l).2r7edfourceisdetnotitchael QtrDanssamndisssiyomnmcaeltcruic- sstyamtemdetortihcecrawsies:eΓwue=taΓkde=th1et0o0w,iwdheebreantodislimtaiktenanads tthhee
junctions. unit of energy.
Having obtained the ingredients to calculate the con-
ductance in the presence of Coulomb interaction in the
lowerQD,wenowconsidertheregimeinwhichthelower A. Kondo channel as continuum scattering path
arm of the Aharonov-Bohm ring is tuned to the Kondo
regime and scattering occurs within a continuum (back- We begin by showing that our calculations indicate
ground process). In the upper arm transport occurs that the Kondo channel through the lower QD is acting
through the discrete QD energy level (resonant process). as the continuum scattering path within the Fano effect.
The interference between both processes will give rise to Fig. 2(a) shows the different line-shapes for the three
a Fano line-shape in the conductance. The transport is regimes mentioned in section II. When (cid:15) >> Γ the
d d
then characterized by the Fano parameter q, which in probability of an electron at the Fermi level to tunnel to
general is a complex number. The parameter |q| mea- the lower QD is negligible. The conductance is therefore
sures the ratio of the resonant scattering to the back- determinedbyelectronstransversingtheupperQD.This
4
0.2
0.8 0.4 two parallel paths with an impurity in each one. Us-
00..8181 000...81−4604 0−.812 000..0281.0381.81 02.81 40.81 1 iΓndg=pa1r0aΓmuetaenrds (cid:15)sdim=ila−r3tΓod,thweeoonbetsaininqtuhailsiteaxtipveeriamgerenet-,
00..66 00..62 0.6 00.−06.64.6 0.6 00.6 4 ment. Fig. 3(b) shows the calculated conductance for a
0.4
00..44 0.40 0.4 00.34.4 0.4 0.4 0.5 magneticphaseshiftof2π/5,inordertoobtainacompa-
00..22 0.2−4 0−.22 00.02.02.2 02.2 40.2 rabledegreeofsymmetrywiththereportedresonance[in-
setinFig. 3(b)]. Inthiscalculation,avalueof|q|=1.14
−−0044 −−22 00 −0422 −−20444 −−0−042−44 −0−204 402−002−4 40 44420 0 was obtained, which indicates that the measured asym-
metricconductanceprofileoriginatesfromanunderlying
mixed-valence Kondo path to the upper QD bare reso-
FIG.2. (a)Normalizedconductance,G,forthethreedifferent
nance. The degree of symmetry [arg(q)] versus magnetic
regimes: Lorentzian (resonant channel dominates, red dot-
field was calculated in the experiment, and an abrupt
dashed curve), mixed-valence (resonant channel and Kondo
channel contribute, purple dashed curve) and Fano-Kondo jump in symmetry at half a period was found.9 We rec-
(Kondo channel dominates, black continuous curve). (b)-(e) ognize this sudden symmetry change as a trademark of
Normalized conductance, G, versus energy levels, (cid:15) and (cid:15) , an underlying Kondo channel of the transport process.
u d
for different magnetic fluxes through the ring. Fig. 3(a) shows the argument of the calculated Fano
parameter [Eq. (11)] for Γ = 10Γ . A linear change
d u
in symmetry occurs when the lower QD is completely
givesaresonancewithaLorentzianshapecenteredatthe outside the Kondo regime [black continuous line in Fig.
Fermi level (cid:15)u = 0 [red point-dashed curve in Fig. 2(a)] 3(a)]. As soon as the QD enters the Kondo regime (even
for an arbitrary magnetic phase shift φ [Figs. 2(b)-2(e)]. inthemixed-valenceregime)anabruptsymmetrychange
In this regime there is no Kondo effect, η >>1, and Eq. begins to form at half a period [green dashed line in Fig.
(8) reduces to: 3(a)]. This transition becomes more abrupt as the lower
QD enters deeper into the Kondo regime (η <<1).28
(cid:16)2e2(cid:17) 1 (cid:12)(cid:12)(cid:15)u+ηe−iφΓu(cid:12)(cid:12)2
G= . (10)
h η2 (cid:15)2 +Γ2
u u
WtdhoeeWmreaienlswanoteoerfiwdenbdncyoo|nqtlhs|oiewd>ere>errsotp1nha,aethnhetinnictncheeratanhtchneteeinrl.signygsQteaDmndtboterhabanevsepisnoratsthiiesf π0 00.81.91]An[ tnerruC niarDaBrG g=( aq5t) eT > V0oltage [mv]
mixed-valence regime (cid:15) ≈ −Γ . Here, both arms con-
d d
tribute to the transmission through the structure. The 0.7
interference between the resonant upper path and the
−π 00.6.6
mixed-valence lower path, acting as a continuum, gives −1 0 1 −−88 −6 −−44 −2 00 2 44
rise to a Fano shape [dashed curve in Fig. 2(a)] with
its symmetry tuned by the magnetic phase shift φ [Figs.
2(b)-2(e)]. This symmetry is obtain by computing the FIG.3. CalculationsperformedforΓd =10Γu. (a)Degreeof
argument of the Fano parameter9 [Eq. (9d)]: symmetry of the resonance obtained from Eq. (11) for (cid:15)d =
3Γ (black continuous) and (cid:15) = −3Γ (green dashed). (b)
d d d
(cid:34) (cid:35) Calculatedconductanceforanenergyleveloftheinteracting
2
arg(q)=arctan . (11) QDasintheexperiment((cid:15)d =−3Γd). Insetin(b)showsthe
tan(φ)− η2 cot(φ) reported data and fit of the experiment [J. Verduijn et al.,
2 η2+1 2
Appl. Phys. Lett. 96, 072110 (2010)].
In this regime the Fano parameter gives |q| ≈ 1, which
agreeswiththeinterpretationof|q|[forthegreendashed
curve in Fig. 2(a) |q|=1.2]. IntheFano-KondoregimethelowerQDistunedtothe
Kondo state by lowering its energy level to (cid:15) << −Γ
d d
[black continuous curve in Fig. 2(a)]. The shape of the
B. Symmetry change due to Fano-Kondo interplay conductance then changes from a peak to a dip line-
shape. As was observed experimentally,16 this conduc-
In a transport experiment performed by Verduijn tance behavior indicates that the Kondo channel has
et al.9 using a three-dimensional silicon field effect tran- taken over the role of scattering within a continuum in a
sistor (FinFET), where a few arsenic dopants diffuse to Fano effect, interfering with a discrete scattering process
the transistor channel, an asymmetric Fano resonance mediated by the upper QD. For a magnetic flux away
for one of the arsenic atoms was observed at zero-bias from φ = 2πn, transport is dominated by the Kondo
voltage, and speculated that the Fano resonance results channel regardless of the degree of interference [Figs.
fromtheinterferencebetweenaKondotransportchannel 2(b)-2(e)].15,29 WithinthisregimethelowerQDisinthe
and a direct transport process. The geometry consists of Kondo state, then λ → −(cid:15) such that η << 1. In this
d
5
limit |q|=η <<1 and the interference with the Kondo channel, giving the al-
ready mentioned dip line-shape [Figs. 4(a)-4(c)]. More-
(cid:16)2e2(cid:17) |(cid:15) |2 over, two sharp resonances appear when φ ≈ 2nπ [Figs.
G= u . (12)
h (cid:15)2 +[Γ sin2(φ)]2 4(a)-4(c)]. Since these two resonances originate from the
u u 2 contribution of the discrete energy level path and the
Kondo path [|q|≈1 in Fig. 4(f) as indicated by the con-
In contrast with the constant broadening of the peak
tourdashedlinemappingbothresonances],wespeculate
shape [red point-dashed curve in Fig. 2 (a)] the broad-
eningofthedipismodulatedbysin2(φ)asseeninFigs. that they appear because of a hybridization of the non-
2 interacting QD state with the Kondo state, which leads
2(b)-2(e). Furthermore,theexpressionforq [Eq. (9d)]is
toagapopeningbetweentwohybridstates. Thisgapped
in accordance with the appropriate interpretation of the
systemcouldleadtointerestingphysicsaswecontrolthe
phenomenon [for the black continuous curve in Fig. 2(a)
state of the system (under or above the gap) as well as
|q|=0.3].
the size of the gap, by tuning the non-interacting QD
energy level and the magnetic flux, respectively.
We can also use the magnetic field to alter the elec-
C. Tuning the electron’s path preference
tron’spathpreference,eveniftheinteractingQDisdeep
in the Kondo regime and this path becomes the pre-
So far, we have seen that our system can be in three
ferredchannelforthescatteringprocess. Fig. 4(e)shows
differentregimesdependingontherelativepositionofthe
how the conductance behavior changes near φ = 0 and
energyleveloftheinteractingQD.Ourresultsshowthat
the corresponding absolute value of the Fano parame-
the magnetic flux can be used as a tool to tune the con-
ter [Fig. 4(f)] indicates that the transport mechanism
ductance within each of the regimes with characteristic
changes from dominance of the Kondo channel (|q| ≈ 0)
landmarks: no effect in the resonant regime, symmetry
toamechanisminwhichbothscatteringpathscontribute
change of Fano resonance in the mixed-valence regime
(|q| ≈ 1). The effect is a consequence of the decrease
and change of the dip broadening in the Fano-Kondo
of the Kondo temperature via the manipulation of the
regime. It is in this last regime where the magnetic flux
Kondo-QD tunnel coupling by the magnetic flux. The
canhavetheextraroleoftuningthegapbetweentwores-
tunnel coupling decreases until it becomes equally prob-
onances and influences not only the interference at the
able to tunnel to either QD, such that both scattering
endofthering,butaffectstheprobabilityoftheelectron
paths contribute significantly to the conductance. The
taking the lower or upper path.
dependence of the Kondo temperature on magnetic flux
canbederivedbywritingtheFanoformforconductance
11 1 1 [Eq. (8)] in terms of the interacting energy level (cid:15)˜d, such
1 1 that the expression found will describe the Kondo reso-
0.5 0.8 0.8 0 nance. Then, the derived state broadening (Γ) is sub-
stituted into the expression for the Kondo temperature
00 02.6 4 0.6 -1 0
using SBMF Hamiltonian,30
11 0.4 0.4 1 1
(cid:118)
−−0022.−−5022 0200.−0204 24222−2−−22 4400 2022.−204 -01−24−2 0 2 2 0 4 kBTK =(cid:117)(cid:117)(cid:117)(cid:116)(cid:15)˜2d+Γ˜2dζ2ζ+2 1 + siζn22+(cid:16)φ21(cid:17)2, (13)
FIG.4. Calculationsperfomedfor(cid:15)d =−5Γd. (a-d)Conduc- with ζ = Γ(cid:15)uu, kB the Boltzman constant and TK the
Kondo temperature.
tance, G, versus the noninteracting QD energy level, (cid:15) , for
u
different magnetic phase shifts, φ, close to zero. (e) Conduc-
tance, G, versus (cid:15) and magnetic phase φ. Figs. (a)-(d) are
u
differenthorizontalcutsof(e). (f)AbsolutevalueoftheFano IV. CONCLUSIONS
parameter [Eq. (9d)] for the same parameters as in (e).
In this paper, we have studied coherent transport
When φ ≈ 2nπ the conductance [Eq. (8)] cannot be through an Aharonov-Bohm ring structure with two em-
written as Eq. (12). Even though the conductance for bedded QDs. In particular, we have analyzed the in-
small magnetic shifts φ [Figs. 4(a)-4(c)] has a distinc- fluence of a Coulomb interacting channel to a discrete
tive line-shape, the antiresonance near (cid:15) = 0, of width scattering transport process in an Aharonov-Bohm con-
u
Γ ∝ (cid:15)˜2 + Γ˜2sin2(φ), remains present for any value of figuration. Three regimes were identified, depending on
d d 2
magnetic flux. This magnetic flux independence is due the contribution of the Kondo channel to the conduc-
to the fact that near (cid:15) = 0 the upper discrete path is tance. These regimes yield different conductance line-
u
close to resonance, such that the transport characteris- shapes: Lorentzian,asymmetricFanoandsymmetricdip.
tics through it remain robust under the influence of a The influence of a magnetic phase shift to these regimes
magnetic flux. Therefore, the only effect remaining is is to tune the Fano resonance asymmetry and to tune
6
the broadening of the symmetrical dip. We have shown ing), producing an alteration in the electron’s path pref-
that the Fano resonance line-shape can be explained by erence.
an underlying scattering mechanism through a Kondo
channel, and that the shape can vary from an asymmet-
ricresonancetoasymmetricdipaswedrivetheCoulomb
blockedQDfurtherintotheKondoregime. Wehavealso
ACKNOWLEDGMENTS
shownthatwecanidentifythepresenceofanunderlying
Kondo channel by an abrupt change in symmetry of the
discrete channel resonance. Furthermore, we have found This work is part of the research program of
agreementwithexperimentaldatathatsupportourfind- the Foundation for Fundamental Research on Matter
ings. Finally, we predict a gap opening between two res- (FOM), which is part of the Netherlands Organiza-
onances that we speculate is a consequence of gapped tion for Scientific Research (NWO). This research was
hybrid states. We also predict that, even if the interact- financially supported by the ARC Centre of Excel-
ing channel is fully in the Kondo regime, we can use the lence for Quantum Computation and Communication
magneticfluxtodiminishitscontributionbyloweringthe Technology (CE110001027) and the Future Fellowship
characteristicKondotemperature(Kondostatebroaden- (FT100100589).
1 J. Kondo, Prog. of Theor. Phys. 32, 37 (1964). 17 V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie,
2 L. Kouwenhoven and L. Glazman, Phys. World 14, 33 and N. S. Wingreen, Science 280, 567 (1998).
(2001). 18 M. Sato, H. Aikawa, K. Kobayashi, S. Katsumoto, and
3 D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, Y. Iye, Phys. Rev. Lett. 95, 066801 (2005).
D. Abusch-Magder, U. Meirav, and M. Kastner, Nature 19 A. W. Rushforth, C. G. Smith, I. Farrer, D. A. Ritchie,
391, 156 (1998); S. Cronenwett, T. Oosterkamp, and G.A.C.Jones,D.Anderson, andM.Pepper,Phys.Rev.
L. Kouwenhoven, Science 281, 540 (1998). B 73, 081305 (2006).
4 A. F. Otte, M. Ternes, K. von Bergmann, S. Loth, 20 S.Sasaki,H.Tamura,T.Akazaki, andT.Fujisawa,Phys.
H.Brune,C.P.Lutz,C.F.Hirjibehedin, andA.J.Hein- Rev. Lett. 103, 266806 (2009).
rich, Nature Phys. 4, 847 (2008). 21 H. Tamura and S. Sasaki, Phys. E 42, 864 (2010).
5 J. Nygard, D. Cobden, and P. Lindelof, Nature 408, 22 T.-F.FangandH.-G.Luo,Phys.Rev.B81,113402(2010).
342 (2000); P. Jarillo-Herrero, J. Kong, H. van der Zant, 23 P. Coleman, Phys. Rev. B 29, 3035 (1984).
C. Dekker, L. Kouwenhoven, and S. De Franceschi, ibid. 24 Even though we have limited our theoretical treatment to
434, 484 (2005). an infinite-U approach, the conclusions made also hold
6 W. Liang, M. Shores, M. Bockrath, J. Long, and for finite-U because of the equivalence between both ap-
H. Park, Nature 417, 725 (2002); J. Park, A. Pasupathy, proaches: astrongKondochannelinthepresentedframe-
J. Goldsmith, C. Chang, Y. Yaish, J. Petta, M. Rinkoski, work, (cid:15) <<(cid:15) , is equivalent to having a finite-U system
d F
J. Sethna, H. Abruna, P. McEuen, and D. Ralph, Nature near the particle-hole symmetry point, (cid:15) −(cid:15) = U.
F d 2
417, 722 (2002). 25 P. A. Orellana, G. A. Lara, and E. V. Anda, Phys. Rev.
7 L. Yu and D. Natelson, Nano Lett. 4, 79 (2004). B 74, 193315 (2006).
8 J.-H.Chen,L.Li,W.G.Cullen,E.D.Williams, andM.S. 26 A. Sadreev and I. Rotter, J. of Phys. A 36, 11413 (2003).
Fuhrer, Nature Phys. 7, 535 (2011). 27 R.Lo´pez,D.Sa´nchez,M.Lee,M.-S.Choi,P.Simon, and
9 J. Verduijn, G. C. Tettamanzi, G. P. Lansbergen, N. Col- K. Le Hur, Phys. Rev. B 71, 115312 (2005).
laert, S. Biesemans, and S. Rogge, Appl. Phys. Lett. 96, 28 ThiscanbeseenfromEq.(11).OutsidetheKondoregime
072110 (2010). η >> 1 then η2 ≈ 1, this means that arg(q) = ±π
10 L. E. Calvet, J. P. Snyder, and W. Wernsdorfer, Phys. η2+1 2
at φ = ∓mπ with m odd. As we take the QD to the
Rev. B 83, 205415 (2011). 2
11 D.M.Cardamone,C.A.Stafford, andS.Mazumdar,Nano Kondoregime η2η+21 <1andcot(φ2)needstocompensateto
Letters 6, 2422 (2006). obtain arg(q)=±π. Then as η→0, lim arg(q)=±π
12 D. Darau, G. Begemann, A. Donarini, and M. Grifoni, 2 φ→2zπ∓ 2
with z integer, creating an abrupt symmetry change near
Phys. Rev. B 79, 235404 (2009).
φ=2πz.
13 C. A. Bu¨sser and A. E. Feiguin, Phys. Rev. B 86, 165410
29 When the magnetic phase is φ ≈ 2nπ, with n an integer,
(2012).
there is a battle between the limit (cid:15)˜ → 0 which gives
14 U. Fano, Nuovo Cimento 12, 154 (1935). d
η << 1 and the limit φ → 2nπ which gives sin(φ) << 1.
15 A.E.Miroshnichenko,S.Flach, andY.S.Kivshar,REV. 2
Inthiscasetheconductancecannotbereducetoasimple
of Mod. Phys. 82, 2257 (2010).
Lorentzian expression [Eq.(12)].
16 J. Li, W.-D. Schneider, R. Berndt, and B. Delley, Phys.
30 H.OguchiandN.Taniguchi,JournalofthePhysicalSoci-
Rev. Lett. 80, 2893 (1998).
ety of Japan 78, 083711 (2009).