Tunable Fano-Kondo resonance in side-coupled double quantum dot systems
Chung-Hou Chung1,2and Tsung-Han Lee1,31Electrophysics Department, National Chiao-Tung University, HsinChu, Taiwan 300, Republic of China 2Department of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
3Department of Physics, National Tsing-Hua University, HsinChu, Taiwan 300, Republic of China 共Received 17 May 2010; published 26 August 2010兲
We study the interference between the Fano and Kondo effects in a side-coupled double quantum dot system where one of the quantum dots couples to conduction-electron bath while the other dot only side couples to the first dot via antiferromagnetic共AF兲 spin-exchange coupling. We apply both the perturbative renormalization-group共RG兲 and numerical renormalization-group 共NRG兲 approaches to study the effect of AF coupling on the Fano line shape in the conduction leads. With particle-hole symmetry, the AF spin-exchange coupling com-petes with the Kondo effect and leads to a local spin-singlet ground state for arbitrary small coupling, so-called “two-stage Kondo effect.” As a result, via NRG we find the spectral properties of the Fano line shape in the tunneling density of statesc共兲 of conduction-electron leads shows double dip-peak features at the energy
scale around the Kondo temperature and the one much below it, corresponding to the two-stage Kondo effect; it also shows an universal scaling behavior at very low energies. We find the qualitative agreement between the NRG and the perturbative RG approaches. Relevance of our work to the experiments is discussed.
DOI:10.1103/PhysRevB.82.085325 PACS number共s兲: 72.15.Qm I. INTRODUCTION
Fano resonance is the quantum interference effect be-tween a localized state with finite width and a conduction band.1The hallmark of the Fano resonance is the asymmetric line shape in tunneling density of states共TDOS兲 of the con-duction band. One example of Fano resonance is the trans-port through low-dimensional electronic共Fermi兲 system with local impurities. The Kondo effect2plays an important role if these impurities carry unpaired spins. Recently, there has been growing interest both theoretically and experimentally in the Fano resonance associated with the Kondo effect via the scanning tunnel microscope 共STM兲 measurements of noble-metal surfaces3–9 as well as in quantum dot devices.10,11The Fano resonance in these systems in general arises from two quantum interference effects: 共1兲 between the broadened local level and the continuum conduction band and共2兲 between the Kondo resonance in the local level and the conduction band. The combined two effects give rise to rather complicated line shape in STM measurement of the TDOS. The Fano resonance in TDOS of conduction elec-trons in such systems can be served as an alternative ap-proach to study the Kondo effect in addition to the local density of states共LDOS兲 of the quantum dot. The Fano line shape in TDOS of conduction electrons in the leads of a single Kondo dot system has been extensively studied, and it is sensitive to both the spatial phase of the free-conduction electrons and the scattering phase shift associated with the Kondo effect.
Very recently, the Fano resonance has been extended experimentally12 and theoretically13–15 to the side-coupled double quantum dot system where the competition between Kondo and Fano effects gives rise to change in conductance profile. In this paper, we investigate the Fano-Kondo inter-ference in the side-coupled double quantum dot systems where only one of the two dots共dot 1兲 connects to the leads while the other isolated dot 共dot 2兲 is side coupled to
dot 1.16,17In the Kondo limit where charging energy on each dot is large, an antiferromagnetic 共AF兲 spin-exchange cou-pling is generated via the second-order hopping between two dots competes with the Kondo effect, leading to local spin-singlet ground state for arbitrary finite values of J, so-called “two-stage Kondo effect.”16–18 Previous studies on the side-coupled double dot systems have been mostly focused on the dip of LDOS on dot 1 upon applying the AF spin-exchange coupling. However, little is known about the feedback effect of the two-stage Kondo effect mentioned above on the TDOS of conduction electrons in the leads. In this paper, we gener-alize the Fano line shape in TDOS of electrons in the leads as a result of the two-stage Kondo effect in side-coupled double quantum dot system. The systematic perturbative and nu-merical renormalization-group 共NRG兲 approaches are ap-plied here in the cases both with and without particle-hole symmetries. We find as a consequence of the two-stage Kondo effect, the spectral property of the Fano line shape in TDOS of the leads develops an asymmetrical double dip/ peak structure; it also shows an universal scaling behavior at very low energies. We compare our NRG results with the perturbative RG analysis.
II. MODEL HAMILTONIAN
Our starting Hamiltonian for the side-coupled double dot system is the single-impurity Anderson model for dot 1 with additional antiferromagnetic spin-exchange coupling be-tween dot 1 and the isolated dot 2 which side coupled to it.16
H =
兺
k,␣=L,R ⑀kck†␣ck␣+兺
␣=L,R兺
k, 共t␣ck†␣d1,+ H.c.兲 +兺
i ⑀didi † di+兺
i=1,2 Uini↑ni↓+ JS1S2, 共1兲where tL and tR denote the tunneling amplitudes to the left
lead ␣= L , R with spin . This tunnel coupling leads to a broadening of the level on dot 1, the width of which is given by ⌫=⌫L+⌫R= 2共tL2L+ tR2R兲, with L/R the density of
states in the leads. Here, i = 1 , 2 labels the two dots and Si
=共1/2兲兺⬘di
†
⬘di⬘is their spin. Each dot is subject to a
charging energy, U1⬇U2= U = EC. In the presence of
particle-hole symmetry, we have ⑀di= − Ui
2. Note that in the
Kondo limit where the charging energy ECis large, the direct
hopping between the two dots are strongly suppressed and an antiferromagnetic spin-exchange coupling J⬎0 is generated via the second-order hopping processes.
The physical observables of our interest are 共i兲 LDOS of impurity on dot 1: d1共兲=−1Im Gd1共兲 and 共ii兲 the TDOS of the conduction electron c共兲: c共兲=0+␦c共兲, where
0=L/R is the constant density of states of the bare
conduction-electron leads 共here we assume symmetrical leadsL=R兲:0= −1 Im Gc 0共= 0兲 with G c 0共− i兲 being the
bare conduction-electron Green’s function, and␦c共兲 is the
correction to the LDOS of the conduction electron due to the coupling between leads and the quantum dot system:
␦c共兲=
−1
Im␦Gc共− i兲. Here, the correction to the
conduction-electron Green’s function␦Gc共− i兲 is given by
␦Gc共− i兲 =
⌫
0
Gc0共− i兲Gd1共− i兲Gc0共− i兲.
共2兲 Using Eq.共2兲, we have4
␦c共兲 = − ⌫0⫻ 关共qc
2− 1兲Im G
d1共− i兲
− 2qcRe Gd1共− i兲兴 共3兲
with qcbeing defined as
qc= −
Re Gc0共− i兲 Im Gc0共− i兲
共4兲 and it can be treated approximately as a frequency-independent constant.3,4Following Ref.16, below we apply both perturbative RG and NRG approaches to calculate these quantities in the presence of particle-hole symmetry. Though the LDOS on dot 1关or equivalently the imaginary part of the Green’s function on dot 1, Im Gd1共兲兴 at finite AF
spin-exchange coupling J via both RG and NRG has been com-puted in Ref.16, the real part of Gd1共兲, Re Gd1共兲, which is
also needed to analyze the spectral property of the Fano line shape in the TDOS of the conduction-electron leads关c共兲兴,
has not yet been calculated by either perturbative RG or NRG approach. In the following, we provide a numerical and analytical analysis on the Fano line shape forc共兲 by
ana-lyzing both the real and the imaginary parts of Gd1共兲 at finite J via NRG and compare them with those via perturba-tive RG approach.
First, we discuss the case for J = 0. For J = 0 and in the presence of particle-hole symmetry共⑀di= −Ui/2兲, it has been
known that in the Kondo regime ⰆTK with TK
⬇D0e−U1/⌫being the Kondo temperature for dot 1, Gd1共兲
is well approximated by the single Lorentzian,16
Gd1共兲 ⬇ Td10 共− i兲 = z
+ iT˜K+ i
共5兲 with z = cTK
⌫ being the quasiparticle weight at the Fermi
en-ergy, and T˜K= z⌫=cTK being an energy on the order of the
Kondo temperature, TK. The precise value of the universal
constant c relating TK and T˜K depends on the definition of
TK. Here, we define TKas the half width of the transmission
T共兲⬅−⌫ Im Gd1共兲. From fitting Gd1共兲 with the NRG
data, we get c⬇0.5. Note that by Fermi-liquid theory and principles of renormalization group, the AF spin-exchange interaction also gets renormalized by the same z factor: J
→˜J z.
16 Here, J˜ is slightly different from J due to the large logarithmic tail in Im Gd1共兲. The value of J˜ is obtained
from the fit of Im Gd1共兲 to NRG data: J˜⬇1.1J.16However,
for ⱖTK, the above simple Lorentzian approximation fails
to account for the large logarithmic tail in Im Gd1共兲. There-fore, corrections to the single Lorentzian approximation are needed in this case to more accurately describe Gd1共兲. Via
the Dyson equation approach, taking into account the inter-ference between the Kondo resonance and the broadened im-purity level, we obtain a more accurate description for the Green’s function of the dot 1共Ref.3兲
Gd1共兲 = Gd1 0 共兲 + G d1 0 共兲T˜ d1共兲Gd1 0 共兲, 共6兲
where the bare Green’s function on dot 1, Gd10 共兲, describing a local impurity level with a level broadening ⌫ and LDOS
d0⬅ −1 Im Gd1 0 共= 0兲, is given by Gd10 = 1 − n/2 −⑀d1+ i⌫ + n/2 −⑀d1− U1+ i⌫ 共7兲 with n =具nd1↑ + nd1↓ 典 being the average occupation number on dot 1 and T˜d1共兲 is the scattering T—matrix corresponding
to the Kondo resonance, given approximately by3
T ˜ d1共− i兲 ⬇ bei2␦ −⑀K+ iT˜K+ i 共8兲 with b being a fitting parameter to be fitted with the NRG data for Im Gd1共兲. In the presence of particle-hole
symme-try, we have n = 1 and⑀K= 0. Here,␦ in Eq.共8兲 corresponds
to the phase shift associated with the Kondo resonance scat-tering, and it gives␦=/2 in the case of particle-hole sym-metry. By fitting Eq. 共6兲 with the NRG data, we find b
⬇z/共d0兲2, which is in good agreement with the known
result: −Im Gd1共= 0兲=1/⌫ for a single impurity Anderson
model.2 In the Kondo regime 共ⰆT
K and EcⰇ⌫兲 of our
system and for J = 0, the bare Green’s function on dot 1,
Gd10 共兲, are approximately given by Re Gd10 共兲⬇0 and −Im Gd10共兲
⬇d0. The above approximations lead to the
follow-ing approximated expressions for Gd1共兲 after including the
interference between the Kondo resonance and the broad-ened impurity共dot 1兲 level via Eq. 共6兲:
Re Gd1共兲 ⬇ 共d0兲2
b 2+ T˜
K
Im Gd1共兲 ⬇ −d0−共d0兲2 bT˜K 2+ T˜ K 2 共9兲 with d0= − 1
Im Gd10 共= 0兲 being the LDOS of dot 1 at
= 0. From Eqs.共3兲 and 共9兲, in the Kondo limit the correction
to conduction-electron density of states can therefore be ex-pressed in terms of the well-known Fano line shape,3,4
␦c共,J = 0兲 ⬇0
冉
qc2+ 2qc⑀− 1 ⑀2+ 1 +冊
, 共10兲 where⑀=−⑀K T ˜ Kand=d0⌫共qc2− 1兲. Note that in general the
Dyson equation approach in Eq. 共6兲 is also valid for both ⰆTK and ⬇TK in the presence of large particle-hole
asymmetry:兩⑀d1−⑀F兩ⱕ⌫ 共with⑀F being the Fermi energy of
the leads兲 where the the interference between the Kondo and broadened impurity levels plays an important role in
Gd1共兲.3
III. PERTURBATIVE RENORMALIZATION-GROUP ANALYSIS
Now, we turn on a finite AF spin-exchange coupling J. Following Ref. 16, to gain an analytical understanding we employing the perturbative renormalization-group analysis in the limit of J→0. We restrict ourselves the case with particle-hole symmetry. Though some of the aspects in this case has been studied in Ref. 16, it proves to be useful to summarize its key results for further calculations on the Fano line shape for c共兲 in the presence of AF spin-exchange
coupling J. In the limit J→0, “two-stage Kondo screening” takes place:16–18 the spin of dot 1 first gets Kondo screened below Kondo temperature TK⬇D0e−U1/⌫, the first stage
Kondo effect. Then for energy scale much below TK, the
second stage Kondo effect occurs at ⬍TⴱⰆTK between
dots 1 and 2 via the AF spin-exchange coupling J where the spin on the dot 2 gets Kondo screened. Here, the Kondo resonance peak in electron density of states on dot 1 plays the effective fermionic bath for the second stage Kondo ef-fect. We will discuss how the Fano line shape for c共兲 is
affected in the presence of the antiferromagnetic spin-exchange coupling. Summing up all leading logarithmic ver-tex diagrams leads to the following scaling equation for the dimensionless vertex function:16
d关␥共,T˜K兲兴
dl ⬅
d关ˆ共兲J˜兴
dl =关ˆ共兲J˜兴
2 共11兲
with the scaling variable defined as l⬅log共T˜K/T˜K
⬘
兲. Here,ˆ共兲⬅共兲/z=−1zIm Gd1共兲 is the rescaled effective
den-sity of states of dot 1. Integrating this differential equation up to l⬅log共T˜K/兲, one obtains the dimensionless vertex
func-tion in the leading logarithmic approximafunc-tion,16
␥共,T˜K兲 = 1 2 T ˜ K 2log T ˜ K Tⴱ + log 兩兩 Tⴱ 共12兲
with the second scale Tⴱ defined as
Tⴱ= T˜Kexp共−˜TK/J˜兲. 共13兲
The second-order self-energy correction to the retarded Green’s function Gd10 simply gives the expression16
⌺共兲 = S共S + 1兲˜J2
4z2Gd1共兲 共14兲 where S = 1/2. The Green’s function of dot 1 after including self-energy and vertex correction is given by16
Gd1J 共兲 = z
zGd1−1共兲 −J
˜2共兲S共S + 1兲
4z Gd1共兲
, 共15兲
where J˜共兲 is replaced by ␥共兲/ˆ共兲, and Gd1共兲 is given by either Eq. 共5兲 共the Dyson equation approach兲 or Eq. 共6兲
共the single Lorentzian approximation兲. Note that due to the logarithmic corrections in ␥, the spectral density of dot 1 develops a dip at energies ⬃TⴱⰆTK for any infinitesmall
J, which suppresses the low-energy transmission coefficient
through dot 1. Physically, this comes from as a consequence of the fact that electrons of energy ⬍Tⴱ are not energetic enough to break up the local spin singlet and therefore their transport is suppressed. For a finite AF spin-exchange cou-pling J⬎0, the real and imaginary parts of Gd1J 共兲 obtained in Eq.共15兲 via perturbative RG approach lead to an
analyti-cal expression for the correction to the LDOS on dot 1,
␦c J共 兲, ␦c J共 兲 = − ⌫0关共qc 2− 1兲Im G d1 J 共 兲 − 2qcRe Gd1 J 共 兲兴. 共16兲 Below we present the results via NRG with fits by the perturbative RG calculations.
IV. COMPARISON TO THE NRG ANALYSIS We have performed the NRG calculations on the system in the presence of particle-hole symmetry. The NRG param-eters we used are U1= U2= D0= 1,⑀d1=⑀d2= −0.5, ⌳=2, and
⌫L=⌫R= 0.1 with D0being the bandwidth of the
conduction-electron baths. 共Here, we set D0= 1 as the unit of all
param-eters.兲 Within each NRG iteration, we keep the lowest 1000 states. For J = 0, we find TK⬇0.005D0. As AF spin-exchange
coupling is increased, both real and imaginary parts of
Gd1共兲 get splitted at = 0. First, as shown in Ref.16, the
imaginary part of Gd1共兲 共proportional to DOS of dot 1兲 at
finite J shows a dip below the characteristic energy scale Tⴱ for any arbitrary J⬎0 关see Fig. 1共a兲兴. For small AF spin-exchange coupling J, the NRG results for Im Gd1共兲 can be
fitted reasonably well by the perturbative RG approach over an intermediate energy range TⴱⰆⰆTK. Furthermore, a
clear universal scaling behavior of the Kosterlitz-Thouless 共KT兲 type is observed from the NRG results of Im Gd1共兲
for ⱕTⴱ: Im Gd1共兲⬇g0g共/Tⴱ兲 关see Fig. 1共b兲兴.16 With
particle-hole symmetry, the scaling function g共/Tⴱ兲 is
com-pletely universal. As pointed out in Ref.16, the ground state of the system at any finite J is a local spin singlet 共a Fermi
liquid兲, the very low-energy crossover of Im Gd1共兲 for
ⰆTKvanishes as共/Tⴱ兲2, following the Fermi-liquid
behav-ior: −⌫ Im Gd1共兲 ⬇ a1
冉
Tⴱ冊
2 , 共17兲where a1⬇3.0 from the fit to the NRG data 关see Fig.1共b兲兴.16
Note that we find the perturbative RG approach via Eq. 共6兲
leads to a better fit to the NRG results for Im Gd1共兲 than
that via Eq.共5兲, as expected.
We now discuss the real part of Gd1共兲. For J=0,
Re Gd1共兲 is antisymmetric with respect to = 0 and it
shows a peak/dip at ⬇ ⫾TK, signature of the first Kondo
effect. As the AF spin-exchange coupling J is increased, the magnitude of the peak/dip in Re Gd1共兲 for ⬇TK
de-creases, indicating the Kondo effect is suppressed. At a much lower energy scale, Tⴱ⬇ⰆTK, the Kondo dip-peak
struc-ture in Re Gd1共兲 gets a further split with a width D⬇2Tⴱ: it develops a negative-valued dip for⬇Tⴱ while it shows a positive-valued peak for ⬇−Tⴱ. In the →0 limit, both positive and negative branches of Re Gd1共兲 vanish 共see
Figs.2and3兲. We can get an analytical understanding of this
behavior as follows: in the Kondo regime ⰆTK, the real
part of Gd1J 共兲 is approximately given by 关see Eq. 共15兲兴
Re Gd1J 共兲 ⬇ z
冉
1 −3J ˜2共兲 16T˜K 2冊
2冉
1 −3J˜ 2共兲 16T˜K 2冊
2 + T˜K2冉
1 − 3J˜2共兲 16T˜K 2冊
2. 共18兲From the perturbative RG results, as →Tⴱ, J˜共兲 di-verges, leading to the vanishing LDOS. Asdecreases to Tⴱ from above, the factor 1 −3J˜2共兲
16T˜K2
in Eq. 共18兲 first becomes
negative then it approaches 0 asfurther approaches 0. This explains the additional dip-peak structure seen for兩兩→Tⴱin the NRG results. This qualitative feature can be captured by the perturbative RG approach. However, the magnitudes of the dip-peak features via perturbative RG are much smaller than those obtained from NRG. We believe the reasons for the deviation are twofolds: first, the overall shape of Re Gd1共兲 predicted via RG is shifted toward the smaller 兩兩 region compared to the NRG results. This leads to a smaller value for 0⬎0 共compared to that via NRG兲 where
Re Gd1共兩兩⬍0兲 changes its sign from positive 共negative兲 to
negative共positive兲 for 0⬍⬍0共−0⬍⬍0兲. This makes the magnitudes of these additional dips and peaks smaller as 10-6 10-4 10-2 100 ω/Τ 0 0.2 0.4 0.6 0.8 1 ImG (ω)
J=0.001 RG fit via Eqs. (5), (15) J=0.0015 RG fit via Eqs. (5), (15)
10-2 100 102 ω/Τ 0 0.2 0.4 0.6 0.8 1 J=0.0025 J=0.002 J=0.0015 J=0.001 J=0 K * d1 3(ω/Τ )*2 (a) (b)
FIG. 1. 共Color online兲 共a兲 Im Gd1共兲 关normalized to Im Gd1共 = 0 , J = 0兲⬇−1/⌫兴 of dot 1 with particle-hole symmetry for differ-ent AF spin-exchange couplings J calculated by NRG共solid lines兲 and perturbative RG 共dashed lines兲 via Eqs. 共6兲 and 共15兲 共dashed lines兲. Dotted lines are RG fits via Eqs. 共5兲 and 共15兲. The NRG parameters are U1= U2= D0,⑀d1=⑀d2= −0.5D0, and ⌫=0.2D0 with
D0= 1. For J = 0, we find TK⬇0.005D0. The fitting parameters c ⬇0.5 and J˜⬇1.1. 共b兲 Im Gd1共/Tⴱ兲 关normalized to Im Gd1
共=0,J=0兲⬇−1/⌫兴 shows an universal scaling behavior for ⬍Tⴱ. The dotted-dashed line is the power-law 共/Tⴱ兲2 fit to the crossover function of Im Gd1共/Tⴱ兲 forⰆTⴱ, see Eq.共17兲.
-2 -1 0 1 2 ω/Τ -0.4 -0.2 0 0.2 0.4 Re G( ω ) J=0.0025 J=0.002 J=0.0015 J=0.001 J=0 K d1
FIG. 2. 共Color online兲 Re Gd1共兲 关normalized to −Im Gd1
共=0,J=0兲兴 of dot 1 for different antiferromagnetic spin-exchange couplings J by NRG 共solid lines兲. The dashed line is a fit to the NRG data for J = 0 via Eq.共6兲. The other parameters are the same as in Fig.1. 10-6 10-4 10-2 100 ω/Τ -0.4 -0.2 0 0.2 0.4 Re G( ω)
J=0.001 RG fit via Eqs. (5), (15) J=0.0015 RG fit via Eqs. (5), (15)
10-6 10-4 10-2 100 ω/Τ -0.4 -0.2 0 0.2 0.4 J=0.0025 J=0.002 J=0.0015 J=0.001 J=0 K K -d1
FIG. 3. 共Color online兲 Re Gd1共兲 关normalized to −Im Gd1
共=0,J=0兲兴 on a logarithmic scale of /TKfor different
antiferro-magnetic spin-exchange couplings J by NRG共solid lines兲 and per-turbative RG via Eqs.共6兲 and 共15兲 共dashed lines兲. Dotted lines are RG fits via Eqs.共5兲 and 共15兲. The other parameters are the same as in Fig.1.
J
˜共兲 diverges even further 关see Eq. 共18兲兴. As J is further
increased, the deviations between RG and NRG become more transparent. This is expected as the perturbation theory becomes uncontrolled once the system moves away from the weak-coupling regime. Nevertheless, the perturbative RG approach can still capture the qualitative features of Re Gd1共兲 for Tⴱ⬍兩兩⬍TK共see Figs.2and3兲. Note that the
perturbative RG approach via Eq.共6兲 共the Dyson’s equation兲
can fit the NRG result for Re Gd1共兲 better than that via Eq.
共5兲 forⱖTK, as expected. Similar to the KT scaling
behav-ior for Im Gd1共兲, the NRG results for Re Gd1共兲 also show
a scaling behavior forⱕTⴱ: Re Gd1共兲⬇g0
⬘
g⬘
共/Tⴱ兲 共seeFig. 4兲. Here, the scaling function g
⬘
共/Tⴱ兲 is againcom-pletely universal in the case of particle-hole symmetry. Based on the Fermi-liquid theory, the very low-energy 共ⰆTⴱ兲 crossover function for Re G
d1共兲 is linear in/Tⴱ
关see, for example, Eq. 共9兲兴,
⌫ Re Gd1共兲 ⬇ − a2
冉
Tⴱ
冊
, 共19兲where we find a2⬇1.5 from the fit to the NRG result 共see Fig.4兲.
Finally, we discuss the behavior for the Fano line shape for c共兲. As indicated in Eq. 共16兲, the Fano line shape for
␦c共兲 is effectively a linear combination of the asymmetric
real part and symmetric imaginary part of the Gd1共兲. The parameter qcin Eq.共16兲 depends on the conduction-electron
reservoir. Following Refs. 3 and 4, qc can be reasonably
treated as a constant. We take a realistic value qc⬇1.4 here,
corresponding to the Co/Au system studied in Refs.3and7. We find c共兲 is asymmetric with respect to = 0 with a
larger magnitude for⬎0 than that for⬍0. As shown in Figs.6and7,c共兲 shows a dips at⬇−TKand⬇Tⴱ as
well as peaks at ⬇TK and ⬇−Tⴱ. The peak 共dip兲 at
⬇ ⫾TK correspond to the first stage Kondo effect while the
dip共peak兲 at⬇ ⫾Tⴱcorrespond to the second stage Kondo effect via spin-exchange coupling. We find a reasonably good agreement between the NRG results and the fit via the perturbative RG approach for TK⬇⬍Tⴱ. 关The fit via Eq.
共6兲 is somewhat better than that via Eq. 共5兲 as the former
gives a better fit to the NRG result for Re Gd1共兲.兴 The above dip-peak structure in the Fano line shape for c共兲 in
the presence of AF spin-exchange coupling can be detected in the STM measurement of the conduction-electron leads as the signature of the two-stage Kondo effect in side-coupled double quantum dot. Note that the/Tⴱ scaling in the NRG results for c共兲 is observed 共see Figs. 5 and 6兲, which
comes naturally from the scaling behaviors for both real and imaginary parts of Gd1共兲 关see Eq. 共3兲 and Fig. 7兴. In the
low-energy limitⰆTⴱwhere the system approaches to the Fermi liquid of local spin singlet, we have the following approximated power-law scaling behavior for␦c共兲:
10-2 100 102 104 ω/Τ -0.4 -0.2 0 0.2 0.4 ReG (ω) 10-2 100 102 104 ω/Τ -0.4 -0.2 0 0.2 0.4 J=0.0025 J=0.002 J=0.0015 J=0.001 *- - * --1.5(ω/T )* d1
FIG. 4. 共Color online兲 Re Gd1共兲 关normalized to −Im Gd1
共=0,J=0兲兴 on a logarithmic scale of /Tⴱfor different antiferro-magnetic spin-exchange couplings J by NRG. The other parameters are the same as in Fig. 1. The dotted-dashed lines are power-law 共/Tⴱ兲 fits to the universal crossover function of Re G
d1共兲 for ⰆTⴱ, see Eq.共19兲. -4 -2 0 2 4 ω/Τ 0.9 1 1.1 1.2 1.3 ρ (ω) J=0.0025 J=0.002 J=0.0015 J=0.001 J=0 fit via Eq. (10)
K
c
FIG. 5. 共Color online兲 The Fano line shape forc共兲 共in unit of
0兲 for different antiferromagnetic spin-exchange couplings J by NRG. The dashed line is a fit to the Fano line shape forms Eq.共10兲 for J = 0. The other parameters are the same as in Fig.1.
10-6 10-4 10-2 100 ω/Τ 0.8 0.9 1 1.1 1.2 1.3 ρ (ω)
J=0.001 RG fit via Eqs. (5), (15) J=0.0015 RG fit via Eqs. (5), (15)
10-6 10-4 10-2 100 ω/Τ 0.8 0.9 1 1.1 1.2 1.3 J=0.0025 J=0.002 J=0.0015 J=0.001 J=0 fit via Eq. (10)
K K -c
FIG. 6. 共Color online兲 The Fano line shape forc共兲 共in unit of
0兲 on a logarithmic scale of/TKfor different antiferromagnetic
spin-exchange couplings J by NRG 共solid lines兲 and perturbative RG via Eqs.共6兲 and 共15兲 共dashed lines兲. The dotted-dashed lines are fits to the Fano line shape form via Eq. 共10兲 for J=0. The dotted lines are the RG fits via Eqs.共5兲 and 共15兲. The other parameters are the same as in Fig.1.
␦c共兲 0 ⬇ −
冋
共1 − qc 2兲a 1冉
Tⴱ冊
2 + 2a2qc Tⴱ册
. 共20兲ForⰆTK, we find the single Lorentzian approximation Eq.
共5兲 can very well describe Gd1共兲; however, forⱕTK, we
expect a finite contribution to Gd1共兲 from interference be-tween the Kondo resonance and the broadened impurity level at dot 1. We find indeed a better agreement between the analytic fits and the NRG results for Gd1共兲 and c共兲 via
perturbative RG approach based on the Dyson’s equation Eq. 共6兲 than those from the single Lorentzian fit via Eq. 共5兲.
We would like to make a side remark here. The “bare” resonance width⌫ of dot 1 共corresponding to the resonance width of an impurity in a noninteracting resonance-level model兲 is proportional to the bare constant density of states in the leads0. Nevertheless, we find the correction in TDOS in the leads ␦c共兲, which comes as a result of the
Fano-Kondo interference and AF spin-exchange coupling, does not lead to a sizable feedback effect on ⌫ within our parameter range. We have checked this numerically via NRG as well as analytically via Eqs. 共6兲 and 共15兲 共combined with the fit to
NRG data兲 where these effects in the leads and on the dots have been properly taken into account. Though this correc-tion exists in principle, we found that it is negligible as the resonance width of dot 1 at =⑀d1 almost remains un-changed before and after the electron-electron interaction
U1= 1 and/or the small AF spin-exchange couplings JⰆ1 共in
unit of D0= 1兲 are introduced. This comes as a result of the
fact that the energy of the dot 1, ⑀d1= −U1/2Ⰶ0, lies well
below the Fermi level 共or the peak/dip associated with the Kondo effect and AF spin-exchange coupling at energy ⱕTKis well separated from the resonance charge peak of dot
1 with width⌫, 兩⑀d1兩ⰇTKⰇTⴱ兲. Some details are as follows:
the bare resonance width of dot 1 is given by 共assuming symmetrical hoppings between leads and dot 1: t = tL= tR兲 ⌫
= −1/Im Gd10 共=⑀d1兲=4t20. The ratio between the
cor-rected resonance width ⌫
⬘
and the bare one is given by⌫
⬘
/⌫= 1⌫兩Im Gd1J共=⑀d1兲兩 关see Eq. 共15兲兴. Since 兩⑀d1兩=U1/2ⰇTK ⰇTⴱ, the frequency-dependent AF spin-exchange coupling is
negligible at =⑀d1, J˜共=⑀d1兲→0 关see Eq. 共12兲兴. We have
therefore Im Gd1J 共⑀d1兲⬇Im Gd1共⑀d1兲. Meanwhile, since the
energy of dot 1 is well below the Fermi energy, ⑀d1
= −U1/2Ⰶ0 and 兩⑀d1兩ⰇTK. Via Eq. 共6兲 we have, therefore,
1/⌫
⬘
= −Im Gd1J 共⑀d1兲⬇−Im Gd1共⑀d1兲⬇−Im Gd10 共⑀
d1兲=1/⌫.
Hence, the correction to⌫ is negligible: ⌫
⬘
/⌫⬇1. V. CONCLUSIONSWe have studied the Fano resonance in a side-coupled double quantum dot system in the Kondo regime in the pres-ence of particle-hole symmetry. In the range where the en-ergy of the dot 1 is on the order of the broadening of its energy level, quantum interference between the Kondo effect and the broadened energy level of the dot 1 gives rise to modification of the Green’s function on dot 1. We apply the perturbative and numerical renormalization-group ap-proaches to describe the Fano line shape in TDOS of the conduction electrons, which depend on both the real and imaginary parts of the Green’s function Gd1共兲 of the dot 1. At J = 0, Im Gd1共兲 shows the Kondo peak forⱕTKwhile
Re Gd1共兲 exhibits a peak 共dip兲 for⬇TK共⬇−TK兲. As a
result of the Kondo effect, the Fano line shape in TDOS of the conduction-electron leads shows a peak 共dip兲 around ⬇TK共⬇−TK兲. At a finite antiferromagnetic spin-exchange
coupling between the two dots, the two-stage Kondo effect leads to the suppression of the density of states on dot 1 as well as an additional dip共peak兲 structure in the real part of
Gd1共兲 at⬇ ⫾Tⴱ from the NRG results. This leads to an additional dip 共peak兲 around ⬇Tⴱ 共⬇−Tⴱ兲 in the conduction-electron LDOS. The splitting between dip and peak in LDOS at⬇ ⫾Tⴱbecomes more pronounced as the AF spin-exchange coupling J is increased. At finite values of
J and for⬍Tⴱ, the NRG results for Re Gd1共兲, Im Gd1共兲, and c共兲 all show distinct universal scaling behaviors in
/Tⴱ. Analytically, we find the perturbative RG approach
can qualitatively capture the above behaviors for TⴱⰆ ⱕTK. In particular, compare to the simple Lorentzian
ap-proximation for Gd1共兲, we find a better fit to the NRG re-sults for the Fano line shape for c共兲 for TⴱⰆⱕTK by
taking into account the interference between the Kondo reso-nance and the broadened impurity level on dot 1 within the Dyson’s equation approach. To make contact of our results in the experiments, the asymmetrical double dip/peak structure and the scaling behaviors in the Fano line shape predicted here in the spectral properties of the TDOS of the conduction-electron leads can be detected by the transport through the STM tips4 as an indication and direct conse-quence of the two-stage Kondo effect in our side-coupled double quantum dot system. Finally, we would like to make a remark on the Fano line shape in TDOS of the leads in our system without particle-hole symmetry. In this case, we ex-pect a smooth crossover 共instead of the KT type transition兲 between the Kondo and local singlet phases due to the po-tential scattering terms generated in the presence of particle-hole asymmetry. Nevertheless, further investigations via NRG are needed to clarify this issue.
10-2 100 102 104 ω/Τ 0.9 1 1.1 1.2 1.3 ρ (ω) 10-2 100 102 104 ω/Τ 0.9 1 1.1 1.2 1.3 J=0.0025 J=0.002 J=0.0015 J=0.001 *- - * -c
FIG. 7. 共Color online兲 The Fano line shape forc共兲 共in unit of
0兲 on a logarithmic scale of/Tⴱfor different antiferromagnetic spin-exchange couplings J by NRG. The other parameters are the same as in Fig.1. The dotted-dashed lines are power-law fits to the universal scaling function of/Tⴱvia Eq.共20兲.
ACKNOWLEDGMENTS
We are grateful for the useful discussions with Tao Xiang and P. Wölfle. We also acknowledge the generous support
from the NSC under Grants No. 95-2112-M-009-049-MY3, No. 98-2918-I-009-006, and No. 98-2112-M-009-010-MY3, the MOE-ATU program, the NCTS of Taiwan, R.O.C., and National Center for Theoretical Sciences共NCTS兲 of Taiwan.
1U. Fano,Phys. Rev. 124, 1866共1961兲.
2A. C. Hewson, The Kondo Problem to Heavy Fermions 共Cam-bridge University Press, Cam共Cam-bridge, UK, 1997兲.
3H. G. Luo, T. Xiang, X. Q. Wang, Z. B. Su, and L. Yu, Phys. Rev. Lett. 92, 256602共2004兲; 96, 019702共2006兲.
4O. Újsághy, J. Kroha, L. Szunyogh, and A. Zawadowski,Phys. Rev. Lett. 85, 2557共2000兲; Ch. Kolf, J. Kroha, M. Ternes, and W.-D. Schneider,ibid. 96, 019701共2006兲.
5A. Schiller and S. Hershfield,Phys. Rev. B 61, 9036共2000兲. 6M. Plihal and J. W. Gadzuk,Phys. Rev. B 63, 085404共2001兲. 7V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie, and N. S.
Wingreen,Science 280, 567共1998兲;Phys. Rev. B 64, 165412 共2001兲.
8J. Li, W. D. Schneider, R. Berndt, and B. Delley,Phys. Rev. Lett.
80, 2893共1998兲; N. Knorr, M. A. Schneider, L. Diekhoner, P. Wahl, and K. Kern,ibid. 88, 096804 共2002兲; M. A. Schneider, L. Vitali, N. Knorr, and K. Kern, Phys. Rev. B 65, 121406 共2002兲.
9H. C. Manoharan, C. P. Lutz, and D. M. Eigler,Nature共London兲
403, 512共2000兲.
10W. Hofstetter, J. Konig, and H. Schoeller, Phys. Rev. Lett. 87, 156803共2001兲.
11M. Sato, H. Aikawa, K. Kobayashi, S. Katsumoto, and Y. Iye, Phys. Rev. Lett. 95, 066801共2005兲.
12S. Sasaki, H. Tamura, T. Akazaki, and T. Fujisawa, Phys. Rev. Lett. 103, 266806共2009兲.
13W.-R. Lee, J. U. Kim, and H.-S. Sim,Phys. Rev. B 77, 033305 共2008兲.
14T. Tanamoto, Y. Nishi, and S. Fujita,J. Phys.: Condens. Matter
21, 145501共2009兲.
15A. W. Rushforth, C. G. Smith, I. Farrer, D. A. Ritchie, G. A. C. Jones, D. Anderson, and M. Pepper, Phys. Rev. B 73,
081305共R兲 共2006兲.
16C.-H. Chung, G. Zarand, and P. Wölfle,Phys. Rev. B 77, 035120 共2008兲.
17P. S. Cornaglia and D. R. Grempel, Phys. Rev. B 71, 075305 共2005兲.
18E. V. Anda, G. Chiappe, C. A. Busser, M. A. Davidovich, G. B. Martins, F. Heidrich-Meisner, and E. Dagotto,Phys. Rev. B 78, 085308共2008兲.
