Aug 26, 2010 - We study the interference between the Fano and Kondo effects in a side-coupled double quantum dot system where one of the quantum dots ...
PHYSICAL REVIEW B 82, 085325 共2010兲
Tunable Fano-Kondo resonance in side-coupled double quantum dot systems Chung-Hou Chung1,2 and Tsung-Han Lee1,3 1Electrophysics
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 renormalizationgroup 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 competes 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 states c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 between a localized state with finite width and a conduction band.1 The hallmark of the Fano resonance is the asymmetric line shape in tunneling density of states 共TDOS兲 of the conduction band. One example of Fano resonance is the transport through low-dimensional electronic 共Fermi兲 system with local impurities. The Kondo effect2 plays 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.10,11 The 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 electrons in such systems can be served as an alternative approach 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 interference 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 1098-0121/2010/82共8兲/085325共7兲
dot 1.16,17 In the Kondo limit where charging energy on each dot is large, an antiferromagnetic 共AF兲 spin-exchange coupling is generated via the second-order hopping between two dots competes with the Kondo effect, leading to local spinsinglet ground state for arbitrary finite values of J, so-called “two-stage Kondo effect.”16–18 Previous studies on the sidecoupled 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 generalize 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 numerical renormalization-group 共NRG兲 approaches are applied 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 between dot 1 and the isolated dot 2 which side coupled to it.16 H=
⑀kck†␣ck␣ + 兺 兺 共t␣ck†␣d1, + H.c.兲 兺 k,␣=L,R ␣=L,R k, + 兺 ⑀didi†di + i
Uin↑i n↓i + JS1S2 , 兺 i=1,2
共1兲
where tL and tR denote the tunneling amplitudes to the left and right leads, respectively, and ck†␣ creates an electron in
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©2010 The American Physical Society
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CHUNG-HOU CHUNG AND TSUNG-HAN LEE
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tL2 L + tR2 R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 U particle-hole symmetry, we have ⑀di = − 2i . Note that in the Kondo limit where the charging energy EC is 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 = −1 Im 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 leads L = R兲: 0 = −1 Im G0c 共 = 0兲 with G0c 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
␦ G c共 − i 兲 =
⌫ 0 G 共 − i兲Gd1共 − i兲G0c 共 − i兲. 0 c 共2兲
Using Eq. 共2兲, we have4
␦c共兲 = − ⌫0 ⫻ 关共q2c − 1兲Im Gd1共 − i兲 − 2qc Re Gd1共 − i兲兴
Re G0c 共 − i兲 Im G0c 共 − i兲
共4兲
and it can be treated approximately as a frequencyindependent constant.3,4 Following 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 spinexchange coupling J via both RG and NRG has been computed 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 for c共兲 by analyzing both the real and the imaginary parts of Gd1共兲 at finite J via NRG and compare them with those via perturbative 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 Ⰶ TK with TK ⬇ D0e−U1/⌫ being the Kondo temperature for dot 1, Gd1共兲 is well approximated by the single Lorentzian,16
z ˜ + i + iT K
共5兲
TK
with z = c ⌫ being the quasiparticle weight at the Fermi energy, and ˜TK = z⌫ = cTK being an energy on the order of the Kondo temperature, TK. The precise value of the universal constant c relating TK and ˜TK depends on the definition of TK. Here, we define TK as 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 ˜ → Jz .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.16 However, for ⱖ TK, the above simple Lorentzian approximation fails to account for the large logarithmic tail in Im Gd1共兲. Therefore, 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 interference between the Kondo resonance and the broadened impurity level, we obtain a more accurate description for the Green’s function of the dot 1 共Ref. 3兲 0 0 ˜ 共兲G0 共兲, 共兲 + Gd1 共兲T Gd1共兲 = Gd1 d1 d1
共6兲
0 共兲, describing where the bare Green’s function on dot 1, Gd1 a local impurity level with a level broadening ⌫ and LDOS 0 d0 ⬅ −1 Im Gd1 共 = 0兲, is given by
共3兲
with qc being defined as qc = −
0 Gd1共兲 ⬇ Td1 共 − i兲 =
0 Gd1 =
n/2 1 − n/2 + − ⑀d1 + i⌫ − ⑀d1 − U1 + i⌫
共7兲
↑ ↓ with n = 具nd1 + nd1 典 being the average occupation number on ˜ dot 1 and Td1共兲 is the scattering T—matrix corresponding to the Kondo resonance, given approximately by3
˜T 共 − i兲 ⬇ d1
bei2␦ ˜ + i − ⑀K + iT K
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 symmetry, we have n = 1 and ⑀K = 0. Here, ␦ in Eq. 共8兲 corresponds to the phase shift associated with the Kondo resonance scattering, and it gives ␦ = / 2 in the case of particle-hole symmetry. 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 Ⰶ TK and Ec Ⰷ ⌫兲 of our system and for J = 0, the bare Green’s function on dot 1, 0 0 共兲, are approximately given by Re Gd1 共兲 ⬇ 0 and Gd1 0 共兲 −Im Gd1 ⬇ 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 broadened impurity 共dot 1兲 level via Eq. 共6兲:
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Re Gd1共兲 ⬇ 共d0兲2
b 2 + ˜T2
K
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Im Gd1共兲 ⬇ − d0 − 共d0兲2
˜ bT K 2 ˜2 +T
K
1
0 Gd1 共 = 0兲
being the LDOS of dot 1 at with d0 = − Im = 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 expressed in terms of the well-known Fano line shape,3,4
␦c共,J = 0兲 ⬇ 0
冉
冊
q2c + 2qc⑀ − 1 + , ⑀2 + 1
˜ 兲. Tⴱ = ˜TK exp共− ˜TK/J
共9兲
共10兲
The second-order self-energy correction to the retarded 0 simply gives the expression16 Green’s function Gd1 ⌺共兲 = S共S + 1兲
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 ⬍ Tⴱ Ⰶ TK 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 effect. We will discuss how the Fano line shape for c共兲 is affected in the presence of the antiferromagnetic spinexchange coupling. Summing up all leading logarithmic vertex diagrams leads to the following scaling equation for the dimensionless vertex function:16 ˜ 兲兴 d关ˆ 共兲J ˜兴 d关␥共,T K ˜ 兴2 ⬅ = 关ˆ 共兲J dl dl
共11兲
˜ / ˜T⬘ 兲. Here, with the scaling variable defined as l ⬅ log共T K K −1 ˆ 共兲 ⬅ 共兲 / z = z Im Gd1共兲 is the rescaled effective density of states of dot 1. Integrating this differential equation up ˜ / 兲, one obtains the dimensionless vertex functo l ⬅ log共T K tion in the leading logarithmic approximation,16 ˜ 兲= ␥共,T K
1 ˜T 兩兩 K log ⴱ + log ⴱ ˜T2 T T 2
K
ⴱ
with the second scale T defined as
共12兲
˜J2 Gd1共兲 4z2
共14兲
where S = 1 / 2. The Green’s function of dot 1 after including self-energy and vertex correction is given by16 J Gd1 共兲 =
−⑀
where ⑀ = ˜ K and  = d0⌫共q2c − 1兲. Note that in general the TK Dyson equation approach in Eq. 共6兲 is also valid for both Ⰶ TK 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
共13兲
z , ˜J2共兲S共S + 1兲 −1 zGd1 共兲 − Gd1共兲 4z
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 ⬃ Tⴱ Ⰶ TK 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 couJ 共兲 obtained pling J ⬎ 0, the real and imaginary parts of Gd1 in Eq. 共15兲 via perturbative RG approach lead to an analytical expression for the correction to the LDOS on dot 1, ␦Jc共兲, J J ␦Jc共兲 = − ⌫0关共q2c − 1兲Im Gd1 共兲 − 2qc Re Gd1 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 parameters we used are U1 = U2 = D0 = 1, ⑀d1 = ⑀d2 = −0.5, ⌳ = 2, and ⌫L = ⌫R = 0.1 with D0 being the bandwidth of the conductionelectron baths. 共Here, we set D0 = 1 as the unit of all parameters.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 spinexchange 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 Tⴱ Ⰶ Ⰶ TK. 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 ⱕ Tⴱ: 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 completely 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
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CHUNG-HOU CHUNG AND TSUNG-HAN LEE
J=0.001 RG fit via Eqs. (5), (15) J=0.0015 RG fit via Eqs. (5), (15)
1
(b)
(a)
0.4
J=0.0025 J=0.002 J=0.0015 J=0.001 J=0
1
0.2
0.8 ReGd1(ω)
ImGd1(ω)
0.8 0.6
J=0.0025 J=0.002 J=0.0015 J=0.001 J=0
0
0.6 2
*
3(ω/Τ )
-0.4
0.2
10
-4
-2
10 ω/ΤK
10
-2
0 -2 10
0
10
0
10
ω/Τ*
2
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 different 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 for Ⰶ Tⴱ, see Eq. 共17兲.
liquid兲, the very low-energy crossover of Im Gd1共兲 for Ⰶ TK vanishes as 共 / Tⴱ兲2, following the Fermi-liquid behavior: − ⌫ Im Gd1共兲 ⬇ a1
冉 冊 Tⴱ
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 decreases, indicating the Kondo effect is suppressed. At a much lower energy scale, Tⴱ ⬇ Ⰶ TK, the Kondo dip-peak structure 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. 2 and 3兲. We can get an analytical understanding of this behavior as follows: in the Kondo regime Ⰶ TK, the real J 共兲 is approximately given by 关see Eq. 共15兲兴 part of Gd1
Re
J 共兲 Gd1
⬇
冉
2 1 −
冉
˜ 2共 兲 3J z 1 − ˜2 16T ˜ 2共 兲 3J ˜2 16T K
冊 冉
K
冊
2
+ ˜TK2 1 −
˜ 2共 兲 3J ˜2 16T K
冊
2.
共18兲
-1
0 ω/ΤK
1
2
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.
From the perturbative RG results, as → Tⴱ, ˜J共兲 diverges, leading to the vanishing LDOS. As decreases to Tⴱ ˜2 from above, the factor 1 − 3J ˜共2 兲 in Eq. 共18兲 first becomes 16TK negative then it approaches 0 as further approaches 0. This explains the additional dip-peak structure seen for 兩兩 → Tⴱ in 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
0.4
ReGd1(ω)
0.2
0 -6 10
-0.2
0.4
0.4
J=0.001 RG fit via Eqs. (5), (15) J=0.0015 RG fit via Eqs. (5), (15)
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4 -100
-10-2 - 10-4 ω/Τ K
-10-6
10
J=0.0025 J=0.002 J=0.0015 J=0.001 J=0
-6
10
-4
-2
10 ω/Τ K
10
0
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 / TK for 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. Dotted lines are RG fits via Eqs. 共5兲 and 共15兲. The other parameters are the same as in Fig. 1.
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0.4
1.3
0.2
1.2 ρc(ω)
ReGd1(ω)
0.2
J=0.0025 J=0.002 J=0.0015 J=0.001
0
J=0.0025 J=0.002 J=0.0015 J=0.001 J=0 fit via Eq. (10)
1.1
0 1
-1.5(ω/T*)
0.9
-0.2
-0.2 -4
-10
2
ω/Τ *
-10
0
-10
-2
-0.4 -2 10
10
0
ω/Τ*
10
2
10
4
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 / Tⴱ for 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 共 / Tⴱ兲 fits to the universal crossover function of Re Gd1共兲 for Ⰶ Tⴱ, see Eq. 共19兲.
˜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. 2 and 3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 for ⱖ TK, as expected. Similar to the KT scaling behavior for Im Gd1共兲, the NRG results for Re Gd1共兲 also show a scaling behavior for ⱕ Tⴱ: Re Gd1共兲 ⬇ g0⬘g⬘共 / Tⴱ兲 共see Fig. 4兲. Here, the scaling function g⬘共 / Tⴱ兲 is again completely universal in the case of particle-hole symmetry. Based on the Fermi-liquid theory, the very low-energy 共 Ⰶ Tⴱ兲 crossover function for Re Gd1共兲 is linear in / Tⴱ 关see, for example, Eq. 共9兲兴, ⌫ Re Gd1共兲 ⬇ − a2
冉 冊
, Tⴱ
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 qc in 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. 3 and 7. 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. 6 and 7, c共兲 shows a dips at ⬇ −TK and ⬇ 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
0 ω/Τ K
2
4
FIG. 5. 共Color online兲 The Fano line shape for c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.
dip 共peak兲 at ⬇ ⫾ Tⴱ correspond 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 limit Ⰶ Tⴱ where the system approaches to the Fermi liquid of local spin singlet, we have the following approximated power-law scaling behavior for ␦c共兲:
1.3
ρc(ω)
-0.4 4 -10
-2
J=0.001 RG fit via Eqs. (5), (15) J=0.0015 RG fit via Eqs. (5), (15)
1.3
1.2
1.2
1.1
1.1
1
1
0.9
0.9
0.8
-100
-10-2 -10-4 ω/ΤK
-10-6
0.8 -6 10
J=0.0025 J=0.002 J=0.0015 J=0.001 J=0 fit via Eq. (10)
10
-4
-2
10 ω/Τ K
10
0
FIG. 6. 共Color online兲 The Fano line shape for c共兲 共in unit of 0兲 on a logarithmic scale of / TK for 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.
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PHYSICAL REVIEW B 82, 085325 共2010兲
CHUNG-HOU CHUNG AND TSUNG-HAN LEE
1.3
1.2
1.2
ρc(ω)
1.3
1.1
1.1
1
1
0.9
0.9
- 10
4
- 10
2
ω/Τ *
-10
0
-10
-2
10
⌫⬘ / ⌫ = ⌫兩Im GJ1共=⑀ 兲兩 关see Eq. 共15兲兴. Since 兩⑀d1兩 = U1 / 2 Ⰷ TK d1 d1 Ⰷ Tⴱ, 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 J 共⑀d1兲 ⬇ Im Gd1共⑀d1兲. Meanwhile, since the therefore Im Gd1 energy of dot 1 is well below the Fermi energy, ⑀d1 = −U1 / 2 Ⰶ 0 and 兩⑀d1兩 Ⰷ TK. Via Eq. 共6兲 we have, therefore, 0 J 共⑀d1兲 ⬇ −Im Gd1共⑀d1兲 ⬇ −Im Gd1 共⑀d1兲 = 1 / ⌫. 1 / ⌫⬘ = −Im Gd1 Hence, the correction to ⌫ is negligible: ⌫⬘ / ⌫ ⬇ 1.
J=0.0025 J=0.002 J=0.0015 J=0.001
V. CONCLUSIONS
-2
10
0
ω/Τ *
10
2
10
4
FIG. 7. 共Color online兲 The Fano line shape for c共兲 共in unit of 0兲 on a logarithmic scale of / Tⴱ for 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 / Tⴱ via Eq. 共20兲.
冋
冉 冊
␦ c共 兲 ⬇ − 共1 − q2c 兲a1 ⴱ 0 T
2
+ 2a2qc
册
. Tⴱ
共20兲
For Ⰶ TK, we find the single Lorentzian approximation Eq. 共5兲 can very well describe Gd1共兲; however, for ⱕ TK, we expect a finite contribution to Gd1共兲 from interference between 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 leads 0. Nevertheless, we find the correction in TDOS in the leads ␦c共兲, which comes as a result of the FanoKondo 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 correction exists in principle, we found that it is negligible as the resonance width of dot 1 at = ⑀d1 almost remains unchanged before and after the electron-electron interaction U1 = 1 and/or the small AF spin-exchange couplings J Ⰶ 1 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 / 2 Ⰶ 0, lies well below the Fermi level 共or the peak/dip associated with the Kondo effect and AF spin-exchange coupling at energy ⱕ TK is well separated from the resonance charge peak of dot 1 with width ⌫, 兩⑀d1兩 Ⰷ TK Ⰷ Tⴱ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 ⌫ 0 共 = ⑀d1兲 = 4t20. The ratio between the cor= −1 / Im Gd1 rected resonance width ⌫⬘ and the bare one is given by
We have studied the Fano resonance in a side-coupled double quantum dot system in the Kondo regime in the presence of particle-hole symmetry. In the range where the energy 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 approaches 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 for ⱕ TK while 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 ⬇ ⫾ Tⴱ becomes 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 Tⴱ Ⰶ ⱕ TK. In particular, compare to the simple Lorentzian approximation for Gd1共兲, we find a better fit to the NRG results for the Fano line shape for c共兲 for Tⴱ Ⰶ ⱕ TK by taking into account the interference between the Kondo resonance 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 consequence 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 expect a smooth crossover 共instead of the KT type transition兲 between the Kondo and local singlet phases due to the potential scattering terms generated in the presence of particlehole asymmetry. Nevertheless, further investigations via NRG are needed to clarify this issue.
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TUNABLE FANO-KONDO RESONANCE IN SIDE-COUPLED… ACKNOWLEDGMENTS
We are grateful for the useful discussions with Tao Xiang and P. Wölfle. We also acknowledge the generous support
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