Kondo-assisted switching between three conduction states in capacitively coupled quantum dots
Pierre Lombardo, Roland Hayn, Denis Zhuravel, Steffen Schäfer
KKondo-assisted switching between three conduction states in capacitively coupledquantum dots
Pierre Lombardo,
1, 2, ∗ Roland Hayn,
1, 2
Denis Zhuravel, and Steffen Sch¨afer
1, 2 Aix-Marseille University, Faculty of Science, St J´erˆome, Marseille, France Institut Mat´eriaux Micro´electronique Nanosciences de Provence, UMR CNRS 7334, Marseille, France Bogolyubov Institute for Theoretical Physics, Kyiv 03143, Ukraine (Dated: June 30, 2020)We propose a nanoscale device consisting of a double quantum dot with strong intra- and inter-dot Coulomb repulsions. In this design, the current can only flow through the lower dot, but istriggered by the gate-controlled occupancy of the upper dot. At low temperatures, our calculationspredict the double dot to pass through a narrow Kondo regime, resulting in highly sensitive switch-ing characteristics between three well-defined states – insulating, normal conduction and resonantconduction.
Soon after the advent of the first single-electron tran-sistors in the late 1980s,[1, 2] with differential con-ductance plots governed by the hallmark
Coulomb di-amonds due to charging electrons one by one to thecentral quantum island,[3, 4] it became clear that theKondo effect[5] would present a viable route to over-come the Coulomb blockade and to restore optimal con-ductivity. Kondo physics has since been observed ina large variety of nanoscale devices, ranging from theoriginal GaAs/AlGaAs[6] and Si/SiGe[7] heterostruc-tures to more exotic devices involving single-moleculejunctions[8–11] or carbon nanotubes.[12] The sharpnessof the concomitant zero-bias or Abrikosov-Suhl-Kondoresonance in the differential conductance,[6–8, 12–15]offers an attractive way to implement highly sensitiveswitching properties on which e.g. future molecular elec-tronics might depend.[16, 17]Like other quantum phenomena, the resonant spin-flipcorrelations underlying the Kondo effect are limited tovery low temperatures and persist only up to some tens ofmilli-Kelvin in the usual semiconductor-based devices, orat best some tens of Kelvin in molecular electronics,[11,16] although Kondo temperatures as high as 100 K havebeen reported for magnetic impurities on surfaces.[18, 19]The possibility to observe Kondo-assisted electronictransport in an otherwise Coulomb-blocked quantum dot(QD), hinges on the fact that a singly-occupied dotlevel is available for resonant tunneling from the leads,and is thus generally directly controlled via the gatevoltage.[6, 7, 16] For the ⊥ -shaped device proposed inthis Letter by contrast, schematically depicted in Fig. 1,the gate (lead 2) does not directly control the energylevel ε of dot 1 in the conduction channel between theleft (L) and right (R) lead, but addresses a second QD.And it is the occupation of this upper dot, labeled 2,which ultimately triggers the onset of the current viaan interdot Coulomb repulsion. As we will show in thelast part of this Letter, the observed abruptness of thecurrent onset is intimately linked to the presence of aKondo feature in the spectral densities and to its ex-act location with respect to the Fermi level. Experimen- FIG. 1. Nanoscale device where two single-level QDs, ε and ε , with intradot Coulomb repulsion U and U , and in-terdot Coulomb coupling U , are connected to three uncor-related metallic leads. A electron can tunnel from the left(L) lead via the lower dot to the right (R) lead, thus repre-senting the channel of a transistor. The top lead controls thecharge on the upper dot which, although isolated from thechannel, triggers the current through the latter and thus actsas a gate. The tunneling between leads and their respectivedots is summarized by the hybridization functions Γ α where α ∈ { L, R, } . tally, capacitively coupled quantum dots without inter-dot tunneling have been realized e.g. in GaAs/AlGaAsheterostructures,[20, 21] and tunable interdot couplingswith minimal residual interdot tunneling were imple-mented using bilayer graphene on silicon substrate.[22]While unarguably more complex, we will show in the fol-lowing that, at low enough temperature, this double-dotsetup offers the possibility to construct a quantum de-vice capable of switching in a highly sensitive mannerbetween three states instead of two: insulating, normalconduction and Kondo-assisted resonant conduction.To investigate the setup proposed in Fig. 1, wewill study the corresponding single-impurity Andersonmodel[23] within the framework of the non-crossingapproximation (NCA)[24–26]. The model’s electronic a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Hamiltonian contains contributions from the leads, thedots, and from tunneling between leads and dots, H = H leads + H dots + H tun . The three uncorrelated metallicleads, α ∈ { L, R, } , are described by H leads = (cid:88) αkσ ε αk c † αkσ c αkσ , (1a)with c † αkσ and c αkσ the creation and annihilation opera-tors of σ -electrons in state k on the lead α . The contri-bution from the dots is H dots = (cid:88) i ∈{ , } ( ε i n i + U i n i ↑ n i ↓ ) + U n n , (1b)where n i = n i ↑ + n i ↓ is the total electronic occupation ondot i , with n iσ = d † iσ d iσ the number of σ -electrons onthe dot, and d † iσ ( d iσ ) the corresponding creation (an-nihilation) operators. The sum in the above expressionthus describes two individual Anderson dots, with theupper dot serving as gate, V g ≡ ε , and the lower onedefining the conduction channel, V ch ≡ ε . The last termin eq. (1b) accounts for the capacitive coupling betweenboth dots and can be implemented experimentally as de-scribed in Refs. 20–22.Finally, the lead-dot tunneling is given by H tun = (cid:88) α ∈{ L,R } kσ (cid:16) t αk d † σ c αkσ + t ∗ αk c † αkσ d σ (cid:17) + (cid:88) kσ (cid:16) t k d † σ c kσ + t ∗ k c † kσ d σ (cid:17) , (1c)where the first line describes the coupling of electrons ondot 1 to the left and right lead, while the second line al-lows electrons on dot 2 to tunnel to the upper lead. Formost practical cases, the tunneling amplitudes t αk onlydepend on k via the corresponding level energy, ε αk , suchthat the tunneling to each lead is entirely summarized bythe hybridization strength[27] Γ α ( ε ) = 2 π | t α ( ε ) | N α ( ε )where N α ( ε ) is the spin-summed DOS on lead α ( ∈{ L, R, } ). Here, we further assume that all three leadsare identical metals, with a DOS characterized by a sin-gle wide band, and that the tunneling between any leadand its dot is governed by the same only weakly energydependent physical process such that all three hybridiza-tion strengths are the same, Γ L ( ε ) = Γ R ( ε ) = Γ ( ε ).Specifically, we take broad Gaussian lead DOSes, of half-width D = 70 Γ, where Γ = Γ L (¯ µ ) + Γ R (¯ µ ) is the to-tal hybridization strength of the lower dot at the meanchemical potential ¯ µ = ( µ L + µ R ). Γ will henceforthserve as our energy unit which, with k B set to unity,will also be our unit of temperature. Furthermore, weassume zero voltage and temperature bias such that¯ µ = µ L = µ R = µ and T = T L = T R = T .The conductance G ( T ) = e I ( T ) and Seebeck coef-ficient S ( T ) = − I ( T ) / [ eT I ( T )] are given in terms of -10.5 -10 -9.5 -9V g / Γ G / G FIG. 2. Conductance as a function of gate voltage V g ≡ ε for high and low temperatures. Parameters (in units of Γ) are V ch ≡ ε = − .
5, intradot U = U = 8 and interdot U = 6.At high temperatures, the conductance varies smoothly frominsulating to metallic behavior. At low temperatures, a sharpconductance resonance is observed at the transition from theinsulating to the ordinary metallic state. energy-weighted integrals,[28] I n ( T ) = 2 h (cid:90) ε n (cid:18) − ∂f∂ε (cid:19) τ eq ( ε )d ε . (2)The integrand comprises the Fermi function, f ( ε ), andthe equilibrium transfer function, τ eq ( ε ) = π A ( ε )Γ( ε ).Similarly, the occupancies of the dots j = 1 , A j ( ε ) via[29] n j = (cid:90) f ( ε − µ j ) A j ( ε )d ε .The A j ( ε ) = − π Im (cid:80) σ G r jσ ( ε + iδ ) are readily ob-tained from the corresponding retarded dot Green’s func-tions G r jσ ( ε + iδ ) = (cid:104)(cid:104) d jσ ; d † jσ (cid:105)(cid:105) . In the atomic limit,the dot Green’s functions would show sharp transitionsbetween the 4 local eigenstates of each dot. The hy-bridization with the leads and the interdot Coulomb re-pulsion, however, create correlations and fluctuations be-tween these 16 local states, and we address the latterwithin the framework of the NCA, an approximation suit-able for resonant spin-fluctuations underlying the Kondoeffect which are expected in the parameter regime underinvestigation.In our setup, we adopt a rather large intradot Coulombrepulsion, U = U = 8, and an appreciable but nottoo large capacitive interdot coupling, U = 6. Theenergy level of the lower dot in the channel is fixed at V ch ≡ ε = − .
5, while the upper dot level, V g ≡ ε ,varies. The resulting overall conductance is illustrated inFig. 2. At high temperature, it shows a smooth crossoverbetween a badly insulating and a badly conducting be-havior. At low temperatures, the insulating and the n σ , n σ G / G -9.5 -9.45 -9.4 -9.35 -9.3V g / Γ -0.8-0.6-0.4-0.20 S / ( k B / e ) FIG. 3. Dot occupancy per spin (top panel), conduc-tance (middle), and Seebeck coefficient or thermopower (lowerpanel) for the same parameters as in Fig. 2 as a function of V g ≡ ε close to the transition and for various temperatures.Note that the total occupancy is twice the n jσ displayed inthe upper panel, both for the lower dot (1), and the upperdot (2). metallic state are much closer to their defining charac-teristics, and the transition is very sharp. Moreover, anadditional narrow regime of enhanced conductance existsright above the transition voltage, V cg ≈ − .
4, in which G is found to approach its theoretical maximum given bythe conductance quantum G = e h . Note that this sce-nario is generic in the sense that it does not hinge on ourspecific choice of parameters: different Coulomb repul-sions and hybridizations naturally shift the value of V cg ,but the characteristic switching behavior between threeconduction states remains robust. Conversely, a com-plete removal of the upper dot produces a quite differentscenario in which the channel is insulating by default,and conductance is observed only for specific values ofthe gate voltage yielding the hallmark Coulomb diamond plots.These findings are examined in greater detail in Fig. 3,displaying the dot occupancies, the conductance and thethermopower for the same parameters as in Fig. 2, buta much narrower range of gate voltages near the transi-tion between insulating and metallic behavior. As is wellknown, the Kondo effect requires a (spin-summed) dotoccupancy not too far from unity. We therefore chooseparameters such that the total occupancy of both dotsis close to two. As obvious from the symmetry of theoccupancies in the upper panel of Fig. 3, our setup relieson an effect analogous to the famous communicating ves-sels to ensure that n passes through the relevant regimeupon variation of V g . The upper dot is close to doublyoccupied for V g (cid:28) V cg , which forces the occupancy ofthe lower dot to be small, thus impeding charge trans-port through the channel. For V g (cid:29) V cg , both dots areclose to simply occupied and an ordinary metallic statearises which, at low temperature, has a conductance ofapproximately G . For low temperatures, we further-more observe a strongly enhanced conductance on themetallic side of the transition, V g (cid:38) − .
4. For our low-est temperature, T = 0 . G . The figure also showsthat the maximum is strongly eroded with rising temper-ature. This, and the fact that the enhanced conductanceis only observed in a very narrow gate voltage regime,hints to resonant spin fluctuation underlying the Kondoeffect as a plausible cause. For our system, we indeedfind a Kondo temperature of T K ≈ · − , in perfectagreement with the observed erosion of the conductancemaximum. This scenario is further corroborated by thethermopower displayed in the last panel of Fig. 3, sincea negative or sign-changing thermopower are other hall-marks of Kondo-mediated transport. [30]The low-temperature spectral densities displayed inFig. 4 provide further support for a Kondo-enhancedconductance on the metallic side of the transition, andunveil the details behind the sudden switching behaviorfrom virtually no to almost perfect conductance. Part(a) shows the spectral densities of the lower dot (in red)and upper dot (in blue) on the metallic side of the tran-sition, V g ≈ V cg + 0 .
04. Both densities are dominated bytheir respective upper and lower Hubbard bands, sepa-rated by the intradot Coulomb U = U = 8, as expectedfor roughly half-filled Anderson dots. More importantly,a sharp Kondo resonance is present in both spectral den-sities exactly at the Fermi level (as is most obvious fromthe insets in the lower panel of the figure). It is thus theKondo contribution to the spectral density at the Fermilevel which allows to overcome the Coulomb blockade andto restore a perfectly metallic behavior – an effect thathas already been observed in other Kondo systems at lowtemperature.[31] On the insulating side of the transition, V g ≈ V cg − .
04, shown as part (b) of Fig. 4, the spec-tral weight of the lower dot is predominantly transferred A ( ε ) A ( ε ) A ( ε ) -10 10000.20.40.6 A ( ε ) (a)(b)(i ) FIG. 4. Spectral densities of the lower and upper dot, A ( ε )and A ( ε ), with part (a) representing the metallic side, V g ≈ V cg + 0 .
04, and part (b) the insulating side, V g ≈ V cg − . T = 0 . ) and upper (i ) dotspectral functions around the Fermi level, with darker colorsrepresenting the metallic and lighter the insulating side of thetransition. to the upper Hubbard band (in red), implying a smalloccupancy for this dot, whereas the spectral density ofthe upper dot resides mainly in its lower Hubbard band(in blue), indicating an almost fully occupied gate dot.Some Kondo contributions are still present close to theFermi level, but they do not suffice to preserve the metal-lic character of the system: as inset i shows most clearly,the residual Kondo peak (light red) lies too far above theFermi level to contribute appreciably to the conductancethrough the dot.In summary, we have investigated the electronic trans-port through a double QD with strong intra- and inter-dot Coulomb repulsions in a ⊥ -shaped layout. In thissetup electrons may only flow from source to drain viathe lower dot. The role of the upper dot is to triggerthe flow via an interdot Coulomb repulsion. To achievethis, a gate regulates the occupancy of the upper dot.A low temperatures, a very abrupt switching behavior isobserved upon rising the gate voltage: first insulating, the dot switches suddenly to a resonant regime wherethe conductance is close to the theoretical maximum of G = 2 e /h , before adopting a normal metallic conduc-tance. Our calculations of the Seebeck coefficient andthe spectral densities show that this peculiar switchingbehavior is intimately linked to the Kondo effect andits sharp zero-bias resonance. Although limited to lowtemperatures, our three-state transistor-like device mightfind applications in highly sensitive spectroscopy or openthe road to ternary electronics. ∗ [email protected][1] D. V. Averin and K. K. Likharev, Journal of Low Tem-perature Physics , 345 (1986).[2] T. A. Fulton and G. J. Dolan, Phys. Rev. Lett. , 109(1987).[3] P. Lafarge, H. Pothier, E. R. Williams, D. Esteve,C. Urbina, and M. H. Devoret, Zeitschrift f¨ur Physik BCondensed Matter , 327 (1991).[4] M. A. Kastner, Rev. Mod. Phys. , 849 (1992).[5] A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, England,1993).[6] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu,D. Abusch-Magder, U. Meirav, and M. A. Kastner, Na-ture , 156 (1998).[7] L. J. Klein, D. E. Savage, and M. A. Eriksson, Appl.Phys. Lett. , 033103 (2007).[8] W. Liang, M. Shores, M. Bockrath, J. Long, and H. Park,Nature , 725 (2002).[9] G. D. Scott and D. Natelson, ACS Nano , 3560 (2010).[10] A. K. Mitchell, K. G. L. Pedersen, P. Hedeg˚ard, andJ. Paaske, Nat. Commun. , 15210 (2017).[11] L. H. Yu and D. Natelson, Nano Letters , 79 (2004).[12] J. Nygard, D. H. Cobden, and P. E. Lindelof, Nature , 342 (2000).[13] W. van der Wiel, S. De Franceschi, T. Fujisawa, J. M.Elzerman, S. Tarucha, and L. P. Kouwenhoven, Science , 2105 (2000).[14] O. Klochan, A. P. Micolich, A. R. Hamilton, D. Reuter,A. D. Wieck, F. Reininghaus, M. Pletyukhov, andH. Schoeller, Phys. Rev. B , 201104 (2013).[15] D. P. Daroca, P. Roura-Bas, and A. A. Aligia, Phys. Rev.B , 245406 (2018).[16] P. T. Mathew and F. Fang, Engineering , 760 (2018).[17] B. Limburg, J. O. Thomas, J. K. Sowa, K. Willick,J. Baugh, E. M. Gauger, G. A. D. Briggs, J. A. Mol,and H. L. Anderson, Nanoscale , 14820 (2019).[18] P. Wahl, L. Diekh¨oner, M. A. Schneider, L. Vitali,G. Wittich, and K. Kern, Phys. Rev. Lett. , 176603(2004).[19] M. Tie, S. Gravelsins, M. Niewczas, and A. Dhirani,Nanoscale , 5395 (2019).[20] I. H. Chan, R. M. Westervelt, K. D. Maranowski, andA. C. Gossard, Appl. Phys. Lett. , 1818 (2002).[21] A. H¨ubel, J. Weis, W. Dietsche, and K. v. Klitzing, Appl.Phys. Lett. , 102101 (2007).[22] S. Fringes, C. Volk, B. Terr´es, J. Dauber, S. Engels,S. Trellenkamp, and C. Stampfer, Phys. Status Solidi C , 169–174 (2012).[23] P. W. Anderson, Phys. Rev. , 41 (1961).[24] T. Maier, M. Jarrell, T. Pruschke, and J. Keller, Euro.Phys. J. B , 613 (2000).[25] J. Otsuki and Y. Kuramoto, J. Phys. Soc. Jpn , 064707(2006).[26] F. Zamani, T. Chowdhury, P. Ribeiro, K. Ingersent, andS. Kirchner, Phys. Status Solidi B , 547 (2013).[27] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B , 5528 (1994). [28] T.-S. Kim and S. Hershfield, Phys. Rev. B , 165313(2003).[29] N. S. Wingreen, A. P. Jauho, and Y. Meir, Phys. Rev. B , 8487 (1993).[30] T. A. Costi and V. Zlati´c, Phys. Rev. B , 235127(2010).[31] J. Park, A. N. Pasupathy, J. I. Goldsmith, C. Chang,Y. Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. D.Abruna, P. L. McEuen, and D. C. Ralph, Nature417