Table Of ContentTransport through side-coupled multilevel double quantum dots in the Kondo regime
J. A. Andrade,1,2 Pablo S. Cornaglia,3,2 and A. A. Aligia3,2
1Centro At´omico Bariloche, CNEA, 8400 Bariloche, Argentina
2Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina
3Centro Ato´mico Bariloche and Instituto Balseiro, CNEA, 8400 Bariloche, Argentina
Weanalyzethetransportpropertiesofadoublequantumdotdeviceintheside-coupledconfigu-
ration. Asmallquantumdot(QD),havingasinglerelevantelectroniclevel,iscoupledtosourceand
drainelectrodes. AlargerQD,whosemultilevelnatureisconsidered,istunnel-coupledtothesmall
QD.AFermiliquidanalysisshowsthatthelowtemperatureconductanceofthedeviceisdetermined
4 by the total electronic occupation of the double QD. When the small dot is in the Kondo regime,
1 an even number of electrons in the large dot leads to a conductance that reaches the unitary limit,
0 while for an odd number of electrons a two stage Kondo effect is observed and the conductance is
2 strongly suppressed. The Kondo temperature of the second stage Kondo effect is strongly affected
by the multilevel structure of the large QD. For increasing level spacing, a crossover from a large
r
a KondotemperatureregimetoasmallKondotemperatureregimeisobtainedwhenthelevelspacing
M becomes of the order of the large Kondo temperature.
1
1 I. INTRODUCTION
]
l Double quantum dots (DQDs) laterally defined in
l
a semiconductor heterostructures are highly tunable elec-
h tronic devices whose energy level spacings and charging
-
energiescanbedeterminedbysettingthesizeandgeome-
s
e tryofeachdot,andtheircouplingbetunedusingmetallic
m gates.1–6 The great tunability of the DQDs parameters,
. as the interdot coupling, open the possibility of applica-
t
a tions in both classical and quantum computing,7–11 and
m allowadetailedanalysisoftheinterplaybetweeninterfer-
- ence and correlation phenomena.12–15 The properties of
d these devices can be probed through transport measure-
n
mentsusingmetallicelectrodes. WhenoneQDhavingan
o
odd number of electrons and a single relevant electronic
c FIG.1. (coloronline)Schematicrepresentationofthedouble
[ levelistunnelcoupledtometallicsource-drainelectrodes, quantum dot device in the side-coupled configuration. QD d
the electric conductance increases below a characteristic is coupled to left (L) and right (R) metallic electrodes. The
2
temperatureT whichsignalsthebuildupofKondocor- orbitals of QD D are coupled to QD d through a tunneling
v K
relations. The Kondo effect is associated to the screen- barrier.
3
6 ingoftheQDsmagneticmomentbytheFermiseaofthe
9 metallic electrodes. In the so-called side-coupled config-
7 uration (See Fig. 1), were only one of the QDs is cou- hybridization between the small QD and the leads is
. pledtotheelectrodes,theinterdotcouplingcancompete smaller than the thermal energy, the Coulomb blockade
1
0 with the Kondo effect leading to a rich variety of corre- diamonds change their structure as a function of the in-
4 lated regimes. Namely, two-stage Kondo physics for a terdot tunneling coupling signaling a change in the un-
1 smallside-coupledQDhavingasinglerelevantelectronic derlying electronic structure associated to the multilevel
v: level,13,15andatwo-channelKondoeffectforalargeside- nature of the QDs.
i coupled QD.12 A device with an intermediate size of the In this article we analyze the transport properties of a
X
side-coupleddothasbeenpredictedtobearealizationof DQD device in the side-coupled configuration. We con-
r the Kondo box problem.16–25 The properties of this type sider the multilevel nature of the side-coupled dot and
a
of devices have received considerable theoretical and ex- the intra- and inter-dot Coulomb interactions are prop-
perimental attention. However, most theoretical studies erly taken into account. We obtain an exact relation be-
have focused on simplified models considering a single tween the zero-temperature conductance and the charge
interacting level on each dot,13,14,26–29 a continuum of in the DQD device (assuming energy independent lead-
levels of the side-coupled dot,30,31 or no interactions on dot couplings) which generalizes Friedel’s sum rule for
the large dot.32 thedouble-dotdevice. Weanalyzethedifferentstrongly-
Recent experimental results for a DQD device in the correlated-electron regimes that occur at finite temper-
side-coupled configuration,33 show a change in the level atures using Wilson’s numerical renormalization group
structure of the DQD as a function of the interdot tun- (NRG)andcharacterizetheelectronictransportthrough
neling coupling. In a spectroscopic regime, where the the device.
2
In the weak coupling regime (for temperatures higher the energy level splitting on QD D we include a single
thanthecouplingtotheleads)weuseaperturbativeap- electron energy term:
proach,startingfromtheexacteigenstatesforthesystem
(cid:88)
oftwodotsisolatedfromtheleads. Inthestrongcoupling He = (cid:15)˜Dαd†DασdDασ. (4)
regime, we also use a slave-boson mean-field approxima- σ,α
tion to help the interpretation of the NRG results. We
Finally,
showthattheusualeven-oddasymmetryintheCoulomb
blockade valleys, due to the Kondo effect, can be com- (cid:88) (cid:88) (cid:104) (cid:105)
H = V c† d +h.c. , (5)
pletely altered in the large interdot hopping regime. V kν νkσ dσ
The effect of increasing the interdot hopping on the ν=L,R k,σ
conductance is very different for different occupations.
In the strong-coupling regime, we show that there is a describes the coupling between QD d and the left (L)
subtlecompetitionbetweenthelevelspacinginthelarge andright(R)electrodes, whicharemodeledbytwonon-
quantum dot and the effective Kondo coupling between interacting Fermi gases:
both dots, leading to a crossover at a very small energy (cid:88)
scale related with a second-stage Kondo effect. Hel = (cid:15)kc†νkσcνkσ. (6)
The rest of this article is organized as follows. In Sec. ν,k,σ
IIwedescribethemodelandbasicformulasforthetrans-
The conductance through the system is given by3,35,36
portcalculations. InSec. IIIwepresenttheconductance
abnetdwteheenDthQeDeloecctcruopdaetsioanndintthheeDreQgDimfeoorfthweeaekx-pceoruimpleinng- G= e(cid:126)2ΓΓR+ΓΓL (cid:88)(cid:90) d(cid:15)(cid:20)−∂f∂((cid:15)(cid:15))(cid:21)Adσ((cid:15)). (7)
tal parameters of Ref. [33]. In Sec. IV we present exact R L σ
results for the zero temperature conductance. In Sec. V
Here A ((cid:15)) is the spectral density of the small QD and
we present numerical results for the conductance in the d
we have assumed proportional (Γ ∝ Γ ) and energy
Kondo regime. In Sec.VI we calculate the conductance L R
independent dot-lead hybridization functions:
andthemagneticsusceptibilityforamodelwithtwoand
three quasidegenerate levels in the side-coupled QD. We Γ =2πρ (E )V∗ (E )V (E ). (8)
L(R) L(R) F L(R) F L(R) F
also analyze an effective model in the slave-boson mean-
field approximation. where E = 0 is the Fermi energy of the electrodes,
F
ρ ((cid:15)) is the electronic density of states of the left
L(R)
(right) electrode, and V ((cid:15)) equals V for (cid:15)=(cid:15) .
L(R) kL(R) k
II. MODEL In the zero temperature limit: −∂f((cid:15)) →δ((cid:15)), and the
∂(cid:15)
conductanceisproportionaltothespectraldensityatthe
The double quantum dot device is described by the Fermi level (cid:80) A (0). As we show in Sec. IV assuming
σ dσ
following Hamiltonian a Fermi liquid ground state, A (0) can be written as a
dσ
function of the total electronic occupation of the DQD.
H =HC +Ht+He+HV +Hel . (1) In the regime of weak lead-QD couplings or high tem-
peraturesΓ=Γ +Γ (cid:28)k T,wecancalculatethecon-
L R B
Here H describes the electrostatic interaction in the
C ductance through the system, to lowest order in Γ/k T,
constant interaction approximation34 B
replacing in Eq. (7) the exact spectral density A ((cid:15)) of
dσ
the isolated DQD:
H = (cid:88) U(cid:96)(Nˆ −N )2
C (cid:96) (cid:96)
2 1 (cid:88)
(cid:96)=d,D Adσ((cid:15))= Z (e−βEi +e−βEj)|(cid:104)Ψj|d†dσ|Ψi(cid:105)|2
+U (Nˆ −N )(Nˆ −N ), (2) i,j
dD D D d d
×δ[(cid:15)−(E −E )], (9)
j i
where Nˆ = (cid:80) d† d is the occupation of the small
d σ dσ dσ
dot (QD d), Nˆ = (cid:80) d† d is the occupation of
D σ,α Dασ Dασ
the large dot (QD D), N = C V /U , C is the ca-
(cid:96) g(cid:96) g(cid:96) (cid:96) g(cid:96) where |Ψ (cid:105) and E are the exact eigenfunctions and
i i
pacitance of dot (cid:96) with its corresponding gate electrode, eigenenergies of the DQD, and Z =(cid:80) e−βEi is the par-
U is the charging energy and U is given by the QDs i
(cid:96) dD tition function. Replacing this in Eq. (7)
mutual capacitance.2,4–6
Ht =(cid:88)tαdD(cid:16)d†dσdDασ+h.c(cid:17), (3) G= e(cid:126)2kBΓT (cid:88)(Pi+Pj)f(Ei−Ej)f(Ej −Ei)
i,j
σ,α
(cid:88)
× |(cid:104)Ψ |d† |Ψ (cid:105)|2 (10)
describes the tunneling coupling between the different j dσ i
n
orbitals on the side-coupled QD D and the smaller QD
d having a single relevant electronic level. To describe where Pi =e−βEi/Z.
3
The spectral function of the DQD can be calculated
non-perturbativelyasafunctionofthetemperatureusing
Wilson’sNumericalRenormalizationGroup(NRG).37–39
This allows a calculation of the conductance in the full
range of temperatures with a high precision.
III. PERTURBATIVE RESULTS IN THE
WEAK-COUPLING REGIME
In this section we analyze the conductance through
the DQD in a regime of weak coupling to the electrodes
in which the thermal energy k T is larger than the hy-
B
bridization energy Γ. In this regime, Γ can be treated
perturbativelyandtheelectrodesserveasaspectroscopic
probe of the DQD in transport measurements. Conduc-
tance maps are generated sweeping the gate voltages of
FIG.2. (coloronline)Leftpanel: Conductancemapforadou-
eachQD(parametrizedherebyN andN )whichmod-
d D blequantumdotdeviceintheside-coupledconfiguration. The
ify the total charge on the DQD and the distribution of
interdot coupling is t = 0.01meV. Other parameters are
dD
charge between the QDs. For a DQD in the spectro- U =0.25meV, U =0.7meV, U =0.1meV, δ=0.02meV,
D d dD
scopic regime, a hexagonal structure is expected in the andk T =0.01meV.Rightpanel: Totalchargeofanisolated
B
conductance maps with conductance peaks occurring at DQD for the same parameters and gate voltages as the con-
gate voltages such that there is a charge degeneracy on ductance map on the left panel. The lines indicate a change
the DQD which allows charge fluctuations between the by one electron in the charge of the ground state of isolated
DQDandtheleads.34 Fortheside-coupledconfiguration, DQD.TheDQDisemptyatthelowerleftcornerofthefigure
andisoccupiedwitheightelectronsattheupperrightcorner.
the conductance is only sensitive to charge fluctuations
TheapproximateoccupationofeachQDisindicatedbetween
on QD d which is connected to the electrodes. As it can
parentheses as ((cid:104)Nˆ (cid:105),(cid:104)Nˆ (cid:105)).
beinferredfromEq. (10),toobtainalargeconductance, d D
atleasttwolevelsoftheDQDdifferingintheirchargeby
oneelectronneedtobequasidegenerate(|E −E |(cid:46)k T
i j B
so that the product of Fermi functions is not exponen- oftheDQDwhichcoincidewiththoseofQDd. Asitcan
tially suppressed). Furthermore, the electron must be be seen in the right panel of Fig. 2, these segments of
fluctuating thermally in and out of QD d (in order for highconductanceareassociatedtoachangeinthecharge
the matrix element |(cid:104)Ψ |d† |Ψ (cid:105)|2 to be sizable). of QD d. In this figure, the approximate occupation of
j dσ i
each QD at the different regions of the map is indicated
InwhatfollowswefocusouranalysisonaDQDmodel
between parenthesis ((cid:104)Nˆ (cid:105),(cid:104)Nˆ (cid:105)). The charging of the
having three levels on the side-coupled QD with level d D
large dot is accompanied by much weaker peaks in the
energies(cid:15)˜ =−δ,(cid:15)˜ =0,and(cid:15)˜ =δ. Thissimplified
D1 D2 D3
conductance due to a small mixing between the states of
modelpresents,intheweak-couplingregime,manyofthe
the two QDs.
features observed in systems having a larger number of
levels on the side-coupled QD (see Ref. [33]). To further For an intermediate interdot coupling (tdD ∼ δ, see
simplify the discussion of the results, we consider tα to Fig. 3), the DQD states are in a molecular regime where
dD
be level independent and drop the level index α. the charge on each dot is not well defined as all states
We calculate conductance maps of the DQD in the of the DQD involve a large orbital mixing between the
spectroscopic regime using Ec. (10). The Hamiltonian QDs. This is reflected on the conductance map that
of the isolated DQD is diagonalized to obtain its eigen- presents maxima at the charge degeneracy points of the
vectors and eigenvalues for each value of N and N . In DQDwherethetotalchargeintheDQDchanges(asitis
d D
Figs. 2, 3, and 4 we present the results for a set of pa- indicated in the right panel of the figure). The QDs lose
rametersobtainedfromtheexperimentofBainesetal.,33 theiridentityandbehaveasasingleQDwithaneffective
and different values of the interdot tunnel coupling t . charging energy and coupling to the leads.
dD
In the regime of weak interdot coupling (t <δ), there For larger values of the interdot coupling t > δ the
dD dD
is little mixing of the states between QDs (except when system gradually enters a regime in which some of the
thereisadegeneracyoftheenergylevelsofthetwoQDs) wavefunctionspresentastrongmixingbetweentheQDs
andthechargeoneachQDisgenerallywelldefined. Due and other are mostly localized on one of the QD.33 For
to the topology of the device, a peak in the conductance our simplified model with a single level on QD d and
is expected at the lines of charge degeneracy in QD d, three levels on the side-coupled QD, one energy level is
which is the one coupled to the electrodes. This is ob- associated to a bonding state between the level on QD d
served on the left panel of Fig. 2 where segments of high andasymmetriccombinationoflevelsonQDD,andan-
conductance are obtained at the charge degeneracy lines otherlevelisassociatedtothecorrespondingantibonding
4
FIG. 3. (color online) Left panel: Conductance map for FIG. 4. (color online) Left panel: Conductance map for
t = 0.04meV. Right panel: the lines indicate a charge de- t =0.08meV.Rightpanel: thelinesindicatethechargede-
dD dD
generacyontheisolatedDQD,thechargeintheDQDchanges generacypointsoftheDQD,thechargeintheDQDchanges
by one at each line and increases from zero at the lower left by one at each line and increases from zero at the lower left
corner to eight at the upper right corner. Other parameters corner to eight at the upper right corner. The approximate
are as in Fig. 2. occupation of each QD is indicated between parentheses on
different regions of the map. Other parameters are as in Fig.
2.
state. Thesestateshaveapproximately1/2oftheweight
on each QD. The two remaining states of the DQD have
measurestheasymmetryofthecouplingofthesmallQD
most of their weight on the side-coupled QD. This par-
to the left and right electrodes. This allows a direct es-
ticularelectronicstructureoftheDQDisreflectedonthe
timation of the zero-temperature conductance from the
conductance of the device as it can be observed in Fig.
charging diagrams presented on the right panels of Figs.
4. The highest conductance peaks are obtained when
2, 3, and 4.
the bonding (ground state) and anti-bonding (highest
energy) states change their occupation and much lower
peaks are obtained when the two states mainly localized
IV. EXACT RESULTS AT ZERO
on QD D get charged.
TEMPERATURE
This peculiar structure of the wave functions in the
regime of strong interdot tunnel-coupling can be easily
Assuming that the ground state of the system is a
understood in the non-interacting limit or the high ca-
Fermi liquid, we show in Appendix A that the zero tem-
pacitance limit where U = U = U , see Ref. [33].
D d dD perature spectral density of QD d is given at the Fermi
It appears in the general case considering more levels on
energy by
each QD. In the latter case, some states of the DQD
are strongly hybridized between the two QDs, being a 2
combination of several orbitals from each QD. The re- Adσ(0)= sin2(πNσ), (11)
πΓ
mainingstatesoftheDQDaremostlylocalizedonQDd
or in QD D. The value of the hopping t at which the where
dD
crossover takes place depends on the level spacing δ and
1 (cid:90) 0
on the relative value of the intradot (Ud, UD) and inter- Nσ =− ImTr Gσ(ω)dω, (12)
dot (UdD) Coulomb interactions. For UD = Ud = UdD π −∞
the interaction energy given by Eq. (2) depends only on
isthetotalchargeperspinontheDQDandG (ω)isthe
the total number of electrons on the DQD and is inde- σ
localzerotemperatureGreen’sfunctionoftheDQD(see
pendentofthestructureoftheelectronicwave-functions.
Appendix A). Replacing Eq. (11) in Eq. (7) we obtain
For U ,U > U , states strongly hybridized between
D d dD
the two QDs have a larger interaction energy than those
(cid:88)e2
having a well defined number of electrons on each QD. G(T =0)=α sin2(πN ), (13)
(cid:101) σ
h
In the next sections, we analyze the low temperature σ
conductanceinthedifferenttunnelcouplingregimes. As
we will show in the next section, the conductance maps where α(cid:101)= (Γ4LΓ+LΓΓRR)2.
inthezerotemperaturelimitareonlydeterminedbythe For a symmetric coupling of the small QD to the left
total occupation N of the DQD and a parameter that and right electrodes (Γ = Γ ) we have α = 1 and the
L R (cid:101)
5
zero temperature conductance reaches the quantum of symmetric situation where Γ = Γ = Γ/2, (cid:15)˜ = 0,
L R Dα
conductancewhenthenumberofelectronsN intheDQD N = N = 1, and U = 0, so that the average
d D dD
isodd(N =N =N/2). Conversely,foranevennumber charge on each QD is 1. When both QDs are in the
↑ ↓
of electrons in the DQD the conductance vanishes. This Kondo regime (U ,U (cid:29) πΓ,t ) charge fluctuations
d D dD
important result indicates that the maximum amplitude on each QD can be eliminated using a Schrieffer-Wolff
oftheconductanceisgivenonlybytheasymmetryofthe transformation53 and the low energy properties of the
couplingtotheleftandrightleads,andtheremainingde- system can be described using a Kondo Hamiltonian:
pendenceontheparameters,includingchargingenergies,
level energies and tunnel couplings enters only through H =J S ·s +J S ·S +H (14)
K K d 0 dD d D el
the value of the occupation in the DQD. For fixed α, all
(cid:101)
sets of parameters that lead to a given occupation in the Here, S with (cid:96) = d,D are spin operators associated to
(cid:96)
DQD will lead to the same value for the conductance at theQDs,ands0 = 21(cid:80)s,s(cid:48)c†0sσs,s(cid:48)c0s(cid:48) istheelectronspin
zero temperature. density on the orbital coupled to QD d. The coupling
The conductance maps analyzed in the previous sec- constants, to leading order in H and H , are J =
t V dD
tion are expected to be completely modified in the zero 8t2 /(U +U ) and J =2Γ/U .
dD d D K d
temperature limit. Irrespectively of the charge distribu- For a sufficiently weak interdot coupling J the sys-
dD
tion inside the DQD, the conductance will be ∼ α(cid:101)2e2/h tem presents a two-stage Kondo effect. As the tem-
in the valleys where the total number of electrons is an perature is reduced, the spin 1/2 of the QD coupled
oddintegerandverysmallinthosewherethetotalnum- to the leads is Kondo screened at a temperature T ∼
K
ber of electrons is even. In the Kondo regime, where the We−1/ρJK,whereρisthelead’slocaldensityofstatesat
hybridization Γ is smaller than the local interactions Ud theFermilevelandW isahighenergycutoff. TheKondo
andUD,theoccupationasafunctionofthegatevoltages effect on QD d generates a peak on its spectral density
is well approximated by the occupation of the isolated of width ∼ k T , at the Fermi energy. This Kondo
B K
DQD away from the charge degeneracy points. At the peak can be associated to a renormalized Fermi liquid
charge degeneracy points, the total charge in the DQD of quasiparticles having a density of states ∼1/k T .54
B K
is N = k+1/2, where k is an integer and the conduc- Its emergence results in an increase of the conductance
tance at T = 0 is e2/h. Based on the calculation of the through the device as the temperature is lowered.
total occupation of the DQD as presented in the right ForJ <T thespinatQDD isscreenedatalower
dD K
panels of Figs. 2, 3, and 4, we expect the conductance temperature13
maps at T = 0 to consist of diagonal stripes of large
caoltnedrnuacttainngcew(itdhelsimtriitpeeds obfylothweccohnadrugcetadnecgee.neracy lines) TK(cid:63) ∼TKe−πJTdDK. (15)
In the next section we show that the temperatures at
We can interpret this expression as the Kondo screening
which the Fermi liquid behavior sets in and the above
of the spin-1/2 in the side-coupled QD by the renormal-
equationissatisfiedcan,insomecases,beextremelylow.
ized quasiparticles associated to the Kondo effect in QD
At intermediate temperatures, the conductance can be
d. WhilethefirststageKondoeffectleadstoanincrease
very different to the value expected in the high temper-
intheconductanceofthedevice,thesecondstageKondo
ature and the low temperature regimes. Depending on
effect suppresses the conductance in agreement with Eq.
the value of the gate voltages (i.e. at different regions of
(13). This suppression of the conductance can be un-
theconductancemaps),thesystemcanbeinthespectro-
derstood as a Fano antiresonance, or interpreted as a
scopic regime or the low temperature regime for a given
blocking effect in the conductance due to the formation
set of parameters.
of a strong singlet between the spins on the two QDs at
Equation (13) is valid for the interactions considered
low energies. For J > T the spins in the QDs are
dD K
in the Hamiltonian of Eq.(1). Its important to point out
alsolockedinasingletnotonlyatlowtemperatures,but
that different interactions to the ones considered (e.g. a
already for temperatures T > T . The conductance is
K
Hund rule coupling in one of the QD) may change the
small for T < J and the Kondo screening does not
dD
nature of the ground state and the value of the zero-
takes place. Shifting the gate voltage of QD d to mod-
temperature conductance.40–51
ify its occupation to ∼ 0 or ∼ 2 results in a single stage
KondoeffectwithaneffectiveKondocouplingthatleads
to a high conductance regime at low temperatures.55
V. NUMERICAL RESULTS
Whentheside-coupledQDineitheremptyordoubleoc-
cupied and QD d is in the Kondo regime, the transport
We now discuss the temperature dependence of the properties are dominated by the Kondo effect on QD d.
conductance. We first review the main results for a For sufficiently large tunnel coupling t the properties
dD
case with a single relevant level on each QD. This prob- ofthesystemcanbebetterunderstoodconsideringbond-
lem has been studied using a variety of techniques in- ing and antibonding states formed between the two QDs
cluding the Numerical Renormalization Group13,14 and and including effective Coulomb interactions for the hy-
FunctionalRenormalizationGroup.52 Wefirstconsidera bridized levels and effective couplings to the leads.13
6
1 energylevelsofQDD inordertoincreasethehybridiza-
h] 0.8 tionwithQDd. Inthisregime,thewave-functionweight
2e/ 0.6 tdD on QD d associated to the spin-1/2 is reduced and the
2 Kondo coupling between QD d and the electrodes is re-
[ 0.4
G ducedaccordingly,asitcanbereadilyshownpreforming
0.2
aSchrieffer-Wolfftransformation. Whent isincreased
dD
0 further, the level structure of the DQD changes (see also
1
Ref. [33]). Two electrons occupy the lowest lying state
/h] 0.8 oftheDQDwhichisabondingstatebetweentheorbital
22e 0.6 tdD on QD d and a linear combination of the orbitals on QD
G[ 0.4 D. ThethirdelectronismainlylocalizedinQDD which
0.2 leads to a strongly suppressed Kondo coupling and an
0 exponentially suppressed Kondo temperature. For the
10−5 10−4 10−3 T/Γ 10−2 10−1 100 highest values of tdD considered, the Kondo tempera-
tures reach values well below the experimentally accessi-
ble range (see top panel in Fig. 5).
FIG. 5. (color online) Conductance for different values of
theinterdotcouplingt =0.01, 0.015, 0.02, 0.03, 0.04, 0.05, On the bottom panel of Fig. 5 we consider a situation
dD
and 0.09 meV. The other parameters are U = 0.25meV, with a single electron of QD d and 3 electrons on QD D.
D
Ud = 0.7meV, UdD = 0.1meV, δ = 0.02meV, and Γ = In this case, the Fermi liquid theory predicts a vanishing
0.2meV. a) Total occupation of tree electrons: Nd = 1 and conductance at zero temperature. The ground state of
ND = 2. b) Total occupation of four electrons: Nd = 1 and the isolated DQD has total spin equal to zero. As in the
N =3.
D case of a single level in QD D, if the singlet-triplet gap
of the DQD is smaller than the Kondo temperature for
QD d, a two stage Kondo effect occurs as the tempera-
As we show below, the main features observed in the ture is decreased. First the spin 1/2 on QD d is Kondo
single level case for the temperature dependence of the screened by the electrons and holes of the electrodes and
conductance, are also observed when multiple levels are thelowenergyexcitationsofthesystemcanbedescribed
consideredintheside-coupledQD.Wecalculatethecon- by a local Fermi liquid of heavy quasiparticles in QD d.
ductance through the system and the magnetic suscepti- A second stage Kondo effect is observed as the spin 1/2
bility using the full density matrix numerical renormal- on QD D is Kondo screened by the Kondo quasiparti-
ization group (FDM-NRG)56,57. In all calculations we cles on QD d. The Kondo screening of the side-coupled
use a logarithmic discretization parameter Λ = 10 and QD leads to a decrease in the conductance below a char-
the z-trick58,59 averaging over four values of z = 0.25, acteristic T(cid:63) which depends strongly on the antiferro-
K
0.5, 0.75, and1. Wekeepupto5000statesandstartthe magnetic coupling between the QDs [see Eq. (15)]. This
truncation after 4 NRG iterations. behavior can be observed in the bottom panel of Fig.
5 where for the lowest values of t the conductance
In Fig. 5 we present the conductance as a function dD
presents a non-monotonous behavior. As the interdot
of the temperature for the multilevel system analyzed in
coupling is increased, the Kondo temperature of the sec-
the previous section. On the top panel of Fig. 5 we
ondstageKondoeffectincreasesexponentially. Forlarge
consider a situation with a single electron on QD d and
∼ 2 electrons on QD D. For small t (cid:46) 0.02meV the enough values of tdD the antiferromagnetic coupling be-
dD
tween the QDs exceeds T , there is no Kondo effect and
results are very similar to what is observed in a decou- K
theconductancedecreasesmonotonously. Whilequalita-
pled QD situation (t = 0): there is an increase in the
dD
tivelythebehavioroftheconductanceisthesameasthe
conductance associated to the Kondo screening of the
one observed for a single level on the side-coupled QD,
spin-1/2 on QD d. The conductance reaches the quan-
the value of the second stage Kondo temperature can be
tum of conductance, as expected from the Fermi liquid
strongly modified in the multilevel case. As we show in
predictions. In this regime, the Kondo temperature is
the next section, it can have a strong dependence on the
only slightly reduced by the presence of the side-coupled
level spacing δ on QD D.
QD. As t is increased there is a decrease of the Kondo
dD
temperature which sets the temperature scale at which The results of Fig. 5 describe qualitatively the behav-
the Fermi liquid behavior is recovered. This reduction of ior of the conductance away from the charge degeneracy
the Kondo temperature can be understood recalling the lines in the valleys with an odd or an even number of
analysis of the nature of the electronic wave functions in electrons in the DQD. The temperature scales for the
the different interdot coupling regimes presented in Sec. different regimes, however, can vary strongly from one
III. For t = 0 two electrons occupy the ground state valley to the other. This is illustrated in Figs. 6 and 7
dD
of QD D forming a singlet and there is a single electron where the behavior of the conductance as a function of
on QD d; a situation that remains essentially unaltered the temperature and N is presented for N = 1 and
D d
for t (cid:28) δ. For t (cid:38) δ, however, it becomes ener- N = 0.2, respectively. For N = 1, and a weak inter-
dD dD d d
getically favorable to have a sizable occupation in higher dot coupling t = 0.015meV, a clear even odd asym-
dD
7
FIG. 6. (color online) Conductance maps for N = 1 and FIG. 7. (color online) Same as Fig. 6 for N =0.2.
d d
different values of t . Other parameters as in Fig. 5. An
dD
even-oddasymmetryisobservedinthelowtemperaturecon-
ductance.
metry can be observed in the low temperature conduc-
the QDs and has therefore a large Kondo temperature.
tanceasthechargeinQDDismodifiedbythegatevolt-
The valleys for N = 2,4, however, have the magnetic
age. A large (small) conductance is obtained for the val- D
momentmainlylocalizedonQDDleadingtoaverysmall
leys with an odd (even) total number of electrons in the
Kondo temperature that for t =0.1meV is well below
DQD. There is, however, an intermediate temperature dD
the temperature range numerically explored (excepting
regime T(cid:63) <T <T where this even-odd asymmetry is
K K valuesofN veryclosetothechargedegeneracypoints).
lost. The behavior of the conductance in this parameter D
regime is qualitatively described by the Hamiltonian of
Eq. (14). The behavior of the conductance for N = 0.2 is pre-
d
Fort =0.035meVtheDQDisinamolecularregime, sented in Fig. 7. In this case a large conductance is
dD
andthetwostageKondoeffectisnotobserved. Thesys- expected at zero temperature for odd values of N . For
D
tem behaves as a single QD that presents the Kondo ef- low interdot coupling t = 0.015meV [see Fig. 7a)]the
dD
fect for an odd number of electrons on the DQD, and Kondo temperature is, however, well below the range of
no Kondo physics otherwise. In the Kondo regime a temperatures numerically explored. As t is increased
dD
Schrieffer-Wolff transformation can be performed to ob- the Kondo temperature increases and becomes higher
tain the Kondo coupling that is determined by the effec- than the minimum temperature considered close to the
tive charging energy of the DQD and the magnetic mo- charge degeneracy points [see Fig. 7 b)]. For larger val-
ment of QD d in the ground-state wave function of the ues of t [see Figs. 7c) and d)] the Kondo temperature
dD
isolated DQD. For larger values of t the Kondo tem- for N ∼1 is strongly enhanced as the interdot bonding
dD D
perature for some of the valleys is strongly suppressed state is formed. In the valley with N (cid:39) 3, the Kondo
D
as the orbital on QD d strongly hybridizes with a com- temperaturereachesamaximuminthemolecularregime
bination of orbitals on QD D and the remaining wave t ∼δandbecomesstronglysuppressedforlarget as
dD dD
functions are mainly localized on QD D. The odd valley the magnetic moment in the DQD becomes increasingly
with N ∼0 is associated to the bonding state between localized on QD D.
D
8
1 1
0.9 0.9
0.8 0.8
0.7 0.7
/h] 0.6 /h] 0.6
2e 0.5 2e 0.5
2 2
G[ 0.4 G[ 0.4 N =1,N =7
0.3 0.3 N =2,N =6
0.2 0.2 N =3,N =5
0.1 0.1 N =4
0 0
1 2 3 4 5 6 7 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102
N T/TK
FIG.8. (coloronline)Conductancevs. totalDQDoccupation
FIG.9. (coloronline)Conductancefordifferentvaluesofthe
for a three level side-coupled QD. NRG (solid dots) and the
occupationintheDQD.Foranevenoccupation,theconduc-
analytic result (lines) of Eq. (13) . The gate voltage in the tanceissuppressedforT (cid:46)T(cid:63) wherethesecondKondostage
QD d is set such that its average occupation is one, while K
sets in. Parameters are as in Fig. 8.
the gate voltage in QD D is varied to change its occupation
from zero to six electrons. Parameters are N = 1, U /Γ =
d D
U /Γ = 7.143, U = 0, t /Γ = 0.0693, δ/Γ = 0.001428,
d dD dD wide range of model parameters.
andΓ=0.07inunitsofthehalf-bandwidthoftheconduction
Figure 9 presents the conductance as a function of the
band of the electrodes.
temperature for different values of the total DQD occu-
pation and the parameters of Fig. 8. For an odd num-
ber of electrons in the DQD, the occupation per spin is
VI. KONDO EFFECT IN MULTILEVEL
N = N/2 and Eq.(13) leads to an unitary conductance
QUANTUM DOTS σ
at zero temperature G(T = 0) = 2e2/h. The observed
increaseintheconductancewhenthetemperatureislow-
In this section we analyze the effect of having several
eredisassociatedtothebuildupofKondocorrelationsas
electronic levels on QD D on the transport properties of
the spin in QD d is screened. The transport properties
thedeviceandontheKondocorrelationsinthetwo-stage
are essentially unaltered by the presence of QD D when
Kondo regime. We consider a few levels on QD D and
it is charged by an even number of electrons. For an
calculate the conductance for the full range of values of
eventotalnumberofelectronsintheDQDthetwostage
the level spacing δ. Since we consider a finite number of
Kondo effect is obtained, as in the single level case, and
levels, the limit of small level spacing δ →0 describes fi-
thezero-temperatureconductancevanishes. Thesecond-
nitesizeQDwithadegenerategroundstate. Weanalyze
stageKondotemperaturegenerallydependsonthenum-
a DQD system in a parameter regime which is different
ber of electrons on QD D, here is the same for total
to the one analyzed in the previous sections. We set
occupation in the DQD of N and 8−N electrons due
U = U = U, U = 0, and t (cid:28) U ,U . Now the
D d dD dD d D to the electron-hole symmetry for the set of parameters
Coulomb interactions determine the charge distribution
considered.
in the DQD, in particular the bonding state obtained in
We now consider the effect of the level spacing δ on
the tdD >δ regime is unfavored by the absence of inter- the second-stage Kondo temperature T(cid:63). When there is
dot Coulomb repulsion (U = 0). This can be easily K
dD a single electron on QD D, there are two limiting cases
seen calculating the expectation value of the interaction
that can be readily solved. For δ → ∞ there is a single
energy term [Eq. 2] for a DQD wave-function with a
relevant level on QD D and the problem reduces to the
single electron on each QD or the two electrons on the
single level case. For δ = 0, we can perform a change of
bonding state.
basisonthedegenerategroundstateofQDD, suchthat
We first analyze the validity of Eq. (13) for a sys- H couples the level on QD d to only one of the states
t
tem having 3 quasidegenerate levels (δ (cid:28) tdD) on the of the new basis. Namely, the symmetric combination of
side-coupled QD calculating the conductance using the the original orbitals on QD D.
NumericalRenormalizationGroup. Wecheckedthatthe
renormalization procedure had converged to the low en- H =√n t (cid:88)(cid:16)d† d˜ +h.c.(cid:17) (16)
t D dD dσ Dσ
ergy fixed point to calculate the conductance and the
σ
occupation. The results are presented in Fig. 8. The
√
gate voltage of QD d was kept fixed (N = 1) and the where d˜ = (cid:80) d / n , and n is the degener-
d Dσ α Dασ D D
occupation on QD D was swept from 0 to 6 by changing acy of the ground state. The problem reduces again
it’s gate voltage. There is an excellent agreement with to the single level case but with a larger hopping am-
√
the Fermi liquid predictions. The small discrepancy be- plitude t˜ = n t . Performing Schrieffer-Wolff
dD D dD
tween the numerical and analytical results is due to a transformation to the single level problem we obtain
small loss of spectral weight in the numerical solutions. J (δ = 0) = n J (δ → ∞), leading to a much
dD D dD
Identical results to those in Fig. 8 were obtained for a largersecond-stageKondotemperatureintheδ =0case
9
1 1 100
0.8 δ
2G[2e/h] 000...0246 eloccupation 000...468 N1 111000−−−321KKT/T(δ=⋆⋆
00..56 Lev 0.2 N2 10−40)
0.4
2µ 00..23 δ 010−3 10−2 10−1 100 101 102 103 104 10510−5
0.1 δ/TK⋆(δ=0)
0
10−8 10−6 10−4 10−2 100 102 FIG. 11. (color online) Slave-boson mean-field theory results
T/T fortheleveloccupationandtheKondotemperatureasafunc-
K
tionoftheenergylevelspacingδ inQDD. Forδ largerthan
FIG.10. (coloronline)Conductance(toppanel)andeffective TK(δ = 0) , it becomes energetically favorable to form the
magneticmomentsquared(bottompanel)asafunctionofthe Kondo correlations with the lowest lying energy level. This
temperature for different values of the energy level spacing reducestheexchangecouplingwiththeelectronbathandthe
δ/T(cid:63)(δ = 0) = 0,0.004,0.4,3.6,7.2, and δ → ∞ on QD D. Kondo temperature.
K
Other parameters as in Fig 8. Only for δ >T(cid:63)(δ =0) there
K
is a sizable change in T(cid:63).
K
odd (3). The system presents a two stage-Kondo effect
as it can be observed in the figure. The conductance in-
[T(cid:63)(δ =0)(cid:29)T(cid:63)(δ →∞)]. creasesattemperaturesbelowthefirst-stageKondotem-
K K
We therefore expect to obtain a crossover from a large perature T which is essentially independent of δ. The
K
KondotemperaturetoasmallKondotemperatureasδis secondstage-KondotemperatureT(cid:63) decreasesasδ isin-
K
increasedfromzero. PerformingaSchrieffer-Wolfftrans- creased but only when δ exceeds T(cid:63)(δ = 0). Note that
K
formation in a system with two levels in QD D we have for δ → ∞, the second stage Kondo temperature falls
fortheHamiltonianoftheisolatedDQDtoleadingorder below the minimum temperature numerically explored.
in Ht and δ/U The behavior of µ2 as a function of the temperature
shows a contribution of the three electrons on QD D
(cid:88)
HDd =J1Sd·SD1+J2Sd·SD2+δ d†D2σdD2σ and suggests that a single electron in an effective or-
σ bitalcouplesantiferromagneticallytoQDdandisKondo
(cid:34) (cid:35) screened60 while the remaining two electrons are doubly
(cid:88)
+J12 d†D2σdD1σSd·SD1+h.c. (17) occupied and effectively decoupled from the rest of the
σ system. For T (cid:29)δ these electrons levels contribute with
µ2 =1/3 to the magnetic moment squared of the DQD
with D2
and with zero for T (cid:28) δ presenting a fast crossover be-
tweenthesetworegimesasafunctionofthetemperature
t2 in contrast to the slow Kondo screening.
J =4 dD
1 U As it was mentioned in the previous section, in the
(cid:18) δ2 (cid:19) second-stage Kondo regime, one can consider that the
J2 =J1 1+ U2 (18) magnetic moment on QD D couples to a renormal-
ized Fermi liquid of quasiparticles formed by the Kondo
J +J
J = 1 2 screening of the magnetic moment of QD d. In the next
12
2 sectionweassumethatthefirst-stageKondoeffectiswell
developed and explore the crossover of T(cid:63) as a function
where, S with(cid:96)=1,2arespinoperatorsassociatedto K
D(cid:96)
of δ using a slave-boson theory in the saddle point ap-
the levels of QD D.
proximation.
As we show below, the crossover from a large Kondo
temperature to a small Kondo temperature does not oc-
cur at δ ∼ J , as it might be naively inferred from an
1
analysis of the isolated DQD, but at a much smaller en- A. Slave-boson mean-field theory
ergy scale δ ∼T(cid:63)(δ =0).
K
In Fig. 10 we present the conductance and the mag- Our starting point is the Hamiltonian of Eq. (17) for
netic moment squared µ2 of the DQD as a function of a system with two levels on QD D and a single electron
the temperature for different values of the energy level oneachQD.WeassumethatthefirststageKondoeffect
spacing in QD D. The parameters are the same as in is well developed an take the local density of states on
Fig. 9 and the number of electrons in the large dot is QD d as that of a non-interacting band with a density of
10
states ρ ∼1/πk T and a bandwidth πk T . The integrals in Eq. (23), Eq. (24), and Eq. (25)
eff B K B K
For simplicity we consider δ (cid:28)U and take J =J = can be solved analytically at T = 0. The resulting
1 2
J =J. The resulting Hamiltonian is: equations were solved numerically for the level occupa-
12
(cid:34) (cid:35) tions in QD D and the Kondo temperature T(cid:63) = ∆(cid:101) =
k
Hml =J SD·s(cid:48)0+(cid:88)d†D2σdD1σSD1·s(cid:48)0 (19) πρeffJ(cid:101)2. The results are presented in Fig. 11. We ob-
serve a slow crossover from a large Kondo temperature
σ
+δd†D2σdD2σ+He(cid:48)l, (20) TK(cid:63)(δ = 0) = T2Ke−π2TJK to a small Kondo temperature
where SD = SD1 + SD2 and s(cid:48)0 = 21(cid:80)s,s(cid:48)f0†sσs,s(cid:48)f0s(cid:48) δTK(cid:63)(cid:38)(δT→K(cid:63)(δ∞=) =0)Tw2Khee−reπTJtKh.eTlehveelsloocwcucpraotsisoonvserNs1etasnidnNfo2r
is the electron spin density of the quasi-particles of the begin to differentiate. For large δ (cid:29)T(cid:63)(δ =0) only the
Fermi liquid where f0†s (f0s) creates (destroys) a quasi- lowest lying level is occupied. K
particle on QD d, and Ifweassociateanenergygain∼T(cid:63) totheformationof
K
(cid:88) the Kondo effect, for small δ it is energetically favorable
H(cid:48) = (cid:15)(cid:48)f† f . (21)
el k νkσ νkσ to occupy partially the highest energy level on QD D
ν,k,σ forming a symmetric combination of the two levels in
QD D in order to increase the hybridization with QD d
The biquadratic interactions between the pseud-
and the Kondo temperature. The energy in this case is
ofermionsgeneratedbythespin-spininteractionsarede-
∼ −T(cid:63)(δ = 0)+δ/2. For δ > 2T(cid:63)(δ = 0) it is however
coupled, introducing two Bose fields B (conjugate to K K
i
the amplitude (cid:80) d† f with i = 1,2) and the con- energetically favorable to empty the highest energy level
σ Diσ 0σ (N → 0) at the expense of having lower energy gain
2
straint on the occupation of QD D is enforced introduc-
due to the Kondo effect ∼ −T(cid:63)(δ → ∞). We therefore
ingtheLagrangemultiplierλ. Thefreeenergyexpressed K
expect a crossover to occur between the two regimes for
in terms of the Bose fields has a saddle point at which
δ ∼T(cid:63)(δ =0).
the latter condense, (cid:104)B (cid:105) = (cid:104)B†(cid:105) = b = (cid:104)f d† (cid:105). At K
i i i 0σ Diσ
the saddle point the effective Hamiltonian is
VII. CONCLUSIONS
(cid:88)(cid:16) (cid:17)
H(cid:101) =J(cid:101) d†Dασf0σ+h.c +δd†D2σdD2σ (22)
σ,α We analyzed the electronic transport through a DQD
+(cid:88)λ(d† d −1)+H(cid:48) , device in the side-coupled configuration. A small quan-
Dασ Dασ el tumdothavingasinglerelevantelectroniclevelistunnel-
σ,α
coupled to source and drain electrodes, a larger QD is
coupled to the small QD but not directly to the elec-
where J(cid:101)=J(b1+b2). Using the equations of motion of trodes (side-coupled QD). We considered multiple levels
the operators d we obtain:
Diσ on the side-coupled QD and Coulomb interactions be-
tween electrons in the DQD. Assuming a Fermi liquid
1 (cid:90) TK (cid:104) (cid:105) ground state, we obtained an exact relation between the
b1 =− dωf(ω)Im G(cid:101)01σ , (23) zero-temperature conductance and the electronic occu-
π
−TK pation of the DQD. Additional interaction terms in the
systemHamiltonian,asaHund’srulecoupling,40–51 may
1 (cid:90) TK (cid:104) (cid:105) leadtonon-Fermiliquidgroundstatesandtoamodified
b2 =− dωf(ω)Im G(cid:101)02σ , (24) zero-temperature conductance formula.42
π
−TK We explored numerically the conductance through the
systemformodelparametersappropriatetodescribetwo
2 (cid:88)(cid:90) ∞ (cid:104) (cid:105) experimentally relevant situations: i) a system with a
Ni =− dωf(ω)Im G(cid:101)iiσ , (25) large level spacing on the side-coupled QD and ii) a sys-
π
σ −∞ tem with up to three quasidegenerate levels in the side-
whereN =(cid:80) (cid:104)d† d (cid:105)isoccupancyofthedifferent coupled QD. In the latter case we considered that the
i σ Dασ Dασ quasidegeneratelevelsweretheonlyrelevantfortheelec-
levels on QD D, G(cid:101)iiσ and G(cid:101)0iσ are the Green’s function tronic transport, as it is expected if the remaining levels
atthei-thlevelofQDD andthecorrelatorofthequasi-
in the side-coupled QD have a much larger energy. In
particleswitheachlevelofQDD respectively. Theseare
theformercase,whichdescribestheparameterregimeof
given by
Ref. [33], only a few levels in the side-coupled QD need
tobeconsideredtoobtainaqualitativedescriptioninthe
−1 weak electrodes-QD coupling regime.
i J(cid:101) J(cid:101)
πρeff We analyzed the low-temperature conductance and
G(cid:101)σ = J(cid:101) ω−λ 0 . (26) the Kondo correlations for the interdot tunnel-coupling
J(cid:101) 0 ω−δ−λ crossover observed in Ref. [33]. Depending on the parity
