Tunable Non-local Coupling between Kondo Impurities
Daniel Tutuc, Bogdan Popescu, Dieter Schuh, Werner Wegscheider, Rolf J. Haug
TTunable Non-local Coupling between Kondo Impurities
D. Tutuc ∗ , B. Popescu † , D. Schuh , W. Wegscheider § , and R. J. Haug Institut f¨ur Festk¨orperphysik, Leibniz Universit¨at Hannover, Appelstr. 2, 30167 Hannover, Germany and Institut f¨ur Experimentelle und Angewandte Physik,Universit¨at Regensburg, Universit¨atsstrasse 31, 93053 Regensburg, Germany (Dated: October 17, 2018)We study the tuning mechanisms of Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interac-tion between two lateral quantum dots in the Kondo regime. At zero magnetic field we observethe expected splitting of the Kondo resonance and estimate the non-local coupling strength as afunction of the asymmetry between the two Kondo temperatures. At finite magnetic fields a chiralcoupling between the quantum dots is observed in the Kondo chessboard and we probe the presenceof the exchange interaction by analyzing the Kondo temperature with magnetic field.
PACS numbers: 73.21.La,73.23.Hk,73.63.Kv
The high versatility and tunability of semiconductorquantum dots (QDs) have attracted increasing interestin the past years and their investigation as magnetic im-purities has allowed even the realisation of the artificialKondo system [1–3]. Kondo effect is a signature of spinentanglement in a many-electron system, where delocal-ized electrons screen a localized spin, leading to the for-mation of a new singlet ground state. The main signatureof Kondo effect is the formation of a peak in the densityof states at the impurity site due to successive spin-flipsat temperatures below T K , the so-called Kondo temper-ature , essentially the energy scale describing the bindingenergy of the spin singlet formed between the localized,unpaired electron and delocalized electrons in the leads.Kondo impurities also interact with one another viacarrier mediated spin-spin interactions that competewith the local interactions that give rise to Kondo ef-fect. The competition between the two effects is aform of quantum entanglement between two or morespins, usually discussed in the framework of the two-impurity Kondo problem [4]. The Ruderman-Kittel-Kasuya-Yosida (RKKY) is an example of such a car-rier mediated interaction [4]. In quantum dot systems,even if the observation of the RKKY-Kondo competi-tion has been previously claimed [11, 12] a conclusiveunderstanding is missing [7, 8], and the observed re-sults have also been explained in terms of a Fano an-tiresonance [9]. While the description in terms of Fanoantiresonance is suitable for high interdot hopping, theexplanation in terms of RKKY is more appropriate inthe limit of strongly localized electrons. In between, thetwo effects can coexist [10, 26]. Due to the fact thatthe RKKY interaction provides a way for spin entangle-ment control beyond the nearest-neighbor restraint, andits strong dependence on the Fermi wavevector, the im-plications to quantum information processing cannot beunderestimated [14].In this letter we study the tuning mechanisms of theRKKY exchange interaction strength in a system withtwo lateral QDs coupled to a central open conductingregion [11]. As mentioned before, T K describes the bind-ing energy of the spin singlet formed between the confined V V G3 V V S I I V G4 V G1
500 nm
FIG. 1. AFM image of our device defined by oxide lines(bright/yellow). Quantum dots, 1 and 2, are connected toa common source S, and each to individual drains. Six in-plane gates, G1 to G6, control the potentials of the dots andcoupling to the leads. The arrows mark the measured trans-port paths. electron spin and the leads, with an analytical form givenby T K ∼ exp( − / (2 D C K )) [4], where T K isthe Kondo temperature of quantum dot 1(2), D C thedensity of states (DOS) in the leads, and K the Kondocoupling to the leads. Assuming a continuous DOS in thecentral region, the RKKY exchange interaction strengthis given by J ∼ K K [7]. Therefore by tuning the tun-nel coupling between the quantum dot and the leads, theKondo coupling K i is also tuned, and with it the strengthof the exchange interaction. Using this method we studyhow the exchange interaction strength J changes by tun-ing the system into an asymmetric state, more explicitlyby tuning the coupling of QD1 to the central region withrespect to QD2. Furthermore, a perpendicular magneticfield induces Landau levels and edge states. At high fieldsthe edge states in the leads remain partially spin unpo-larized, so Kondo effect can still be observed in trans-port [18–21].Our device was produced by local anodic oxidationwith an AFM [27–29], on a GaAs/Al x Ga − x As het-erostructure containing a two-dimensional electron gas(2DEG) 37 nm below the surface (electron density n e =3 . × m − , and mobility µ = 65 m /V s mea- a r X i v : . [ c ond - m a t . s t r- e l ] F e b -0.2 0 0.2 (mV)V bias g ( e / h ) (a) 300 500 700 80120160 (mK)T K1 J ( (cid:109) e V ) g (b) (d) -0.2 0 0.2 (mV)V bias (c)0.20.30.4 g ( e / h ) g (mV)V bias -0.3 0 0.3 -180-170-160 V ( m V ) G g FIG. 2. (color)(a) Conductance g through QD2 showing azero-bias anomaly, when QD1 is tuned in a non-Kondo val-ley (inset). (b) Color scale plot of g as a function of gateG5 and bias voltage, showing the splitting of the QD2 ZBA.Dot+arrow mark the splitting. (c) Conductance g showing asplit zero-bias anomaly when QD1 is tuned in a Kondo state(inset) taken along the dashed line in (b). The black arrowsmark the position of the split ZBA. (d) Exchange interactionstrength J as a function of T K , fit to 1 /ln ( x ) (line). sured at 4.2 K). The structure consists of two quantumdots (Fig. 1), QD1 and QD2, connected each to an indi-vidual drain, and to a central common reservoir, furthercoupled to the Source reservoir S via an 1D constriction.The distance between the QDs is about 600 nm. Us-ing standard lock-in technique we measure the differen-tial conductances g = dI /dV S and g = dI /dV S acrosseach dot in parallel by applying an ac excitation of 10 µ Vwith a frequency of 83 . He/ He dilution refrigeratorwith an electron temperature of ∼ .
25 meVfor QD1, 0 . ≈ µ eV. Gates G2 and G3 are used to controlthe coupling of the two dots to the central region, whileG5 and G6 are used as plunger gates.Both quantum dots can be individually tuned in theKondo regime and, by changing the electron number oneby one, we can switch from a Kondo to a non-Kondo val-ley. Figure 2(a) shows a Kondo resonance of QD2, or theso-called zero-bias anomaly (ZBA), as a function of dcbias, while the inset shows the corresponding situationin QD1, i.e. tuned to the middle of a Coulomb valleythat does not exhibit a Kondo resonance. To estimatethe Kondo temperature we use T K = ( wπδ ) / (4 k B ) [4],where w is the Wilson number, k B the Bolzman con-tant and δ the half width at half maximum of the Kondoresonance [15]. A value of 280 mK is obtained for T K . Changing the electron number in QD1 by one ( V G from -37 to -20 mV) brings it into a Kondo state (in-set of Fig. 2(c)), characterized by a Kondo temperature T K (cid:39)
720 mK [16] for V G = −
48 mV. The color plot of the differential conductance g through QD2 as a func-tion of bias voltage and gate G V G = −
54 mV. From V G ∼ = −
175 mV the ZBA starts to deviate to the rightalong with a decrease in conductance, that suggests thebeginning of a splitting. At V G (cid:39) − ∼ µ V bias, a shoulder emerges at ∼ − µ V (marked in Fig. 2(b) by two dots and arrows).The asymmetric peak splitting is interpreted in terms ofcoupling asymmetry, QD2 being stronger coupled to thecentral region. The fact that we see a dependence on theKondo energy scale indicates that the splitting is causedby the RKKY exchange interaction. The splitting of theKondo resonance gives direct access to J using the re-lation eV bias = ± J/ µ eV. However theZBA splitting is observed only in QD2, while in QD1 wesee just a suppression of the Kondo resonance, probablydue to the strong asymmetry between the two Kondoscales. By adjusting V G between -42 mV to -56 mV thecoupling of QD1 to the central region is tuned, and theKondo temperature of QD1 at the same time. There-fore we can measure the magnitude of the splitting asa function of T K (Fig. 2(c)). The analytical relationfor the Kondo coupling K i ∼ − / (2 D C ln ( T Ki )) gives anasymptotic behavior of the RKKY exchange interactionstrength J as a function of the Kondo temperature T K ,and from a fit to J = a + b/ln ( T K ) (line in Fig. 2(d))[17],we obtain a ∼ . b ∼ − . a isextracted as the maximum bandwidth of the central re-gion [7], while b contains information about the densityof states in the leads.Kondo effect can be observed not only at zero mag-netic field, but also at finite perpendicular magnetic field.Since one of its main conditions is the presence of bothspin up and spin down in the leads, the observation ofKondo effect in magnetic field requires largely unpolar-ized leads. In our sample we can observe Kondo effect upto 5.5 T in both QDs. This corresponds to a filling factorof the leads ν leads (cid:39) ν = n e h/eB , where n e isthe electron density, h is the Planck constant, e the ele-mentary charge and B the magnetic field), showing thatthe leads are not fully polarized. In the same magneticfield range, the quantum dot filling factors are lower, i.e. ν QD < ν leads , and basically their properties are governedby the lowest Landau level (LL0) formed at the edge andLandau level 1 (LL1) formed at the core of the dots. Be-cause the tunnel coupling of the core to the leads is nothigh enough, Kondo effect in magnetic field involves onlytransport through the edge of the dot, which is closer tothe leads. By changing the total spin of the edge by one can change Kondo transport through the dot. Thetotal spin number in the edge is given by the number ofelectrons and it can be changed in two ways: either weadd an electron from the leads or from the core of the dot.The first mechanism is the result of increased voltage ona nearby plunger gate, increasing also the total electron G6-30-20-100 V ( m V ) G g (a) V ( m V ) G g (c)4.7 5.0 5.3 g B (T)B (T) g (b)-4.7 -5.0 -5.3-30-20-100 -30-20-100-30-20-100(e) (f)G6 FIG. 3. Differential conductance g through QD1 as a func-tion of V G and magnetic field, for negative (a) and posi-tive (d) magnetic field polarity. Differential conductance g through QD2 as a function of V G and B measured for nega-tive (b), and positive (c) magnetic field polarity, measured si-multaneously with (a), and (d) respectively. (e)-(f) Schematicdiagram of the sample in magnetic field: with black - AFMoxide lines, orange - edge states. Landau Level 1 (dark grey)and Landau level 0 (light grey) are marked in the QDs. number by one, while the second mechanism is the resultof adding one flux quantum to the dot by increasing themagnetic field, and redistributing an electron from thecore to the edge [19].The result of these two mechanisms is a Kondo ef-fect modulation as a function of gate voltage and mag-netic field in a regular pattern of high-low differentialconductance, generally referred to as the Kondo chess-board [18–22]. In Fig. 3(a) there is shown the expectedKondo chessboard pattern exhibited by QD1, with alter-nating regions of high (bright) and low (dark) differentialconductance as a function of the voltage applied on G6and magnetic field. In contrast, QD2 exhibits a morecomplicated pattern (Fig. 3(b)). By changing the direc-tion of the magnetic field the situation is reversed. Nowwe observe a regular chessboard pattern in the trans-port through QD2 (Fig. 3(c)) - here the total electronnumber remains unchanged, because gate G6 has a verysmall lever arm on QD2, and one sees only a modulationof the conductance by the magnetic field. At the sametime QD1 (Fig. 3(d)) exhibits a pattern similar to theone in Fig. 3(b). The data presented in Fig. 3 (a) and(b), respectively (c) and (d), are acquired in parallel,in the same range of V G and B-field, so the differencesarise only from changing the polarity of the perpendicu-lar magnetic field, which in fact changes the direction ofelectron transport in the edge states formed in the cen-tral region between the dots. In the first case (Fig. 3(a) and (b)), the edge state picture corresponds to the onedepicted in Fig. 3(e), that is the edge state transport di-rection is clockwise in the central region, therefore trans-port through QD2 will be sensitive to potential changes inthe edge states generated by electron transport throughQD1, i.e. the sample will behave as a current divider. Asa consequence QD2 will exhibit a combination of its ownchessboard pattern and a negative of the QD1 pattern(Fig. 3(b)). In the second situation (Fig. 3(c) and (d))the edge state transport direction is reversed - as depictedin Fig. 3(f), i.e. the electrons move counter-clockwise inthe edge states in the central region. Hence, now QD1will exhibit a combination of both chessboard patterns(Fig. 3(d)). The measurements in Fig. 3(a)-(d) demon-strate a chiral coupling between the quantum dots viathe edge states formed in the central region. Using theseconductance plots one can identify the regions were onlyone or both quantum dots exhibit transport through aKondo state. Along the dashed lines in Fig. 3(a) and (d)two such situations are marked, that is between ± .
77 Tand ± .
97 T (black+red part on the dashed line) QD1exhibits transport through a Kondo state. In the ± . ± .
04 T interval QD2 also exhibits Kondo transport(Fig. 3(c)), therefore from ± . ± .
97 T (red parton the dashed lines) both quantum dots are in the Kondoregime. It is worth mentioning that the sum of the differ-ential conductances through the two dots does not changewith the magnetic field direction, hence a departure fromthe Onsager symmetry relations [24] is unlikely.In order to probe the exchange interaction between thetwo quantum dots in perpendicular magnetic field in de-tail we investigate the temperature dependence of QD1taken along the dashed lines in Fig. 3(a) and 3(d). InFig. 4(a) the sharp steps in the differential conductanceat − .
77 T and − .
97 T mark the onset of Kondo ef-fect in QD1, and it can be observed that around − . g ( e / h ) B (T) -4.8 -4.9 -5.0 T ( K ) K B (T) s T ( m K ) T ( m K ) (a) (b) FIG. 4. Temperature dependence of QD1 for negative (a)and positive (b) magnetic field polarity, taken along the whitedashed lines in Fig. 3(a), and 3(d) respectively. T K ( ⊗ ) ver-sus magnetic field for negative (c), and positive (d) polarity;with green( ◦ ) the s fitting parameter. ior, suggesting the presence of a second energy scale. Forthe opposite polarity (Fig. 4(b)), due to the change inthe edge state chirality, an extra step in the conduc-tance of QD1 marks the onset of Kondo effect in QD2 at ∼ . g ( T ) = g ( T (cid:48) K / ( T + T (cid:48) K )) s with T (cid:48) K = T K / ( √ /s −
1) [3] to fit the measured tem-perature dependence within the ± .
77 T to ± .
97 T in-terval, we obtain a change in T K with magnetic field(Fig. 4(c) and 4(d)). From ± . ± . T K shows a monotonic decrease, which is fol-lowed by an increase up to ∼ ± .
95 T, then it decreasesrapidly. The position where T K starts to increase coin-cides with the onset of Kondo effect in QD2. The fittingprocedure is done with the s parameter left free and theobtained values are shown along with T K in Fig. 4(c)and (d).As mentioned before, at high perpendicular magneticfields Kondo effect involves transport only through Lan-dau Level 0 (LL0) energy states. Since the measurementis done at fixed gate voltage, i.e. at fixed energy, as themagnetic field is increased, the ground state moves lowerin energy and the electron wavefunction is compressed,thus the Kondo coupling is reduced, which explains theinitial decrease of T K . In this region s shows a sim-ilar decrease with magnetic field, with values between2.5 and 0.5 which indicate a clear departure from the0 .
22 value for spin [3]. However similar values havebeen previously reported in the Kondo chessboard [19].On the other hand, the increase and decrease of T K around ∼ ± .
95 T, which is correlated with the onset ofKondo effect in the other quantum dot, cannot be ex-plained by single dot physics, although in this region s isremarkably close to the 0 .
22 value. Therefore non-localexchange interaction has to be considered. The zero-biastemperature dependence of a split ZBA (either by mag- netic field or by RKKY) follows that of the zero magneticfield Kondo resonance at high temperatures [25, 26] anda change in the ZBA splitting will appear as a change inthe Kondo temperature. Therefore the T K peaks seen inFig. 4(c) and 4(d) are probably not due to an actual in-crease in Kondo coupling, but are attributed to a changein ZBA splitting in the presence of the exchange inter-action between the spins of the dots. Even though byreversing the magnetic field direction the electron trans-port direction is changed, the general behavior of T K isvery similar for both magnetic field directions (Fig. 4(c)-(d)), questioning whether the chirality of the edge statesinfluences also the exchange interaction between the spinsof the dots.In conclusion, we have investigated the exchange in-teraction between the two quantum dots as a function ofKondo temperature asymmetry and perpendicular mag-netic field. At B = 0 we find the expected ∼ /ln ( T K )dependence of the RKKY interaction strength J on theKondo energy scale. At finite magnetic fields we observea chiral coupling between the quantum dots in the Kondoregime and we probe the presence of the RKKY exchangeinteraction by a Kondo temperature analysis.The authors would like to thank N. Ubbelohde, C.Fricke and F. Hohls for their help with the measure-ment setup and data analysis, as well as E. R¨as¨anen,J. Martinek, R. Zitko and R. Lopez for many valuablediscussions. We acknowledge financial support from theGerman Excellence Initiative via QUEST and the NTHSchool for Contacts in Nanosystems. ∗ e-mail: [email protected]; † present ad-dress: School of Engineering and Science, Jacobs Univer-sity Bremen, Campus Ring 1, 28759 Bremen, Germany; § present address: Laboratorium f¨ur Festk¨orperphysik,ETH Z¨urich, Schafmattstr. 16, 8093 Z¨urich [1] D. Goldhaber-Gordon et al., Nature , 156 (1998)[2] S.M. Cronenwett et al., Science , 540 (1998)[3] D. Goldhaber-Gordon et al., PRL , 5225 (1998)[4] A. C. Hewson, The Kondo Problem to Heavy Fermions ,(Cambridge University Press, 1993)[5] R. Lopez et al., Phys. Rev. Lett. , 136802 (2002)[6] R. Aguado and D.C. Langreth, Phys. Rev. Lett. , 1946(2000)[7] P. Simon et al., Phys. Rev. Lett. , 086602 (2005)[8] M.G. Vavilov and L.I. Glazman, Phys. Rev. Lett. ,086805 (2005)[9] G.B. Martins et al., Phys. Rev. Lett. , 066802 (2006)[10] R. Zitko, Phys. Rev. B , 115316 (2010)[11] N.J. Craig et al., Science , 565 (2004)[12] S. Sasaki et al., Phys. Rev. B , 161303 (2006)[13] H.B. Heersche et al., Phys. Rev. Lett. , 017205 (2006)[14] D. Loss and D. P. DiVincenzo, Phys. Rev. A , 120(1998) [15] The HWHM of the ZBA is obtained from a Lorenz fitwith 0 background.[16] The Kondo temperature of each quantum dot was mea-sured while the other dot was not in a Kondo state.[17] Due to limited data range, a linear fit would work as well.[18] M. Keller et al., Phys. Rev. B , 033302 (2001)[19] C. F¨uhner et al., Phys. Rev. B , 161305 (2002)[20] M. Stopa et al., Phys. Rev. Lett. , 046601 (2003)[21] D. Kupidura et al., Phys. Rev. Lett. , 046802 (2006)[22] M.C. Rogge et al., Phys. Rev. Lett. , 176801 (2006)[23] V. Fock, Z. Phys. , 446 (1928); C.G. Darwin, Math.Proc. Cambridge Philos. Soc. , 86 (1930)[24] L. Onsager, Phys. Rev. , 405 (1931)[25] T.A. Costi, Phys. Rev. Lett. , 1504 (2000)[26] J. Martinek and R. Zitko, private communication[27] M. Ishii and K. Matsumoto, Jpn. J. Appl. Phys. , 1329(1995)[28] R. Held et al., Appl. Phys. Lett. , 262 (1998)[29] U. F. Keyser et al., Appl. Phys. Lett.76