Supplementary Information for Pseudospin Resolved Transport Spectroscopy of the Kondo Effect in a Double Quantum Dot. D2 V exc I

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1 Supplementary Information for Pseudospin Resolved Transport Spectroscopy of the Kondo Effect in a Double Quantum Dot S. Amasha, 1 A. J. Keller, 1 I. G. Rau, 2, A. Carmi, 3 J. A. Katine, 4 H. Shtrikman, 3 Y. Oreg, 3 1, 3, and D. Goldhaber-Gordon 1 Department of Physics, Stanford University, Stanford, California 9435, USA 2 Department of Applied Physics, Stanford University, Stanford, California 9435, USA 3 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 961, Israel 4 HGST, San Jose, CA 95135, USA I. QUANTITATIVE LIMITS ON THE INTER-DOT TUNNELING D1 V exc I S1 FIG. S1. Circuit for measuring the series conductance G series of the double quantum dot system as discussed in the text. For all the measurements reported in the main text of this paper, we have made the voltages on the gates labeled CU and CL sufficiently negative so as to suppress inter-dot tunneling. We have confirmed that the inter-dot tunneling is negligibly small over a range of gate voltage settings. To check the inter-dot tunneling, we have measured the conductance of the dots in series, as shown in Fig. S1. When the gate voltages are set to a triple point then three dot states are degenerate, e.g. (,), (1,), and (,1). At these triple points a finite inter-dot tunneling gives a finite series conductance 1. Using the above circuit we have found the conductance at the triple points to be below our measurement threshold, indicating the inter-dot tunneling is small, as detailed quantitatively below. We can quantitatively limit the inter-dot tunneling energy scale t as follows. At zero-bias, the series conductance G series can be calculated 2 and is given by: G series = 64 t 2 3Γ 1 ( Γ1+Γ 2 2 ) e2 h (S1) In this equation, Γ 1 / h is the total tunnel rate between electrons on dot 1 and the leads S1 and D1 (with Γ 2 defined similarly). Using this equation, we can put quantitative limits on t. For gate voltage settings close to those used for the data in Figure 2 of the main text we had Γ 1 = 44 µev and Γ 2 = 31 µev and we found G series < e 2 /h which gives t <.6 µev. For the data in Figure 4 of the main text, we have Γ 1 = 2 µev and Γ 2 = 22 µev and G series < e 2 /h, which gives a limit t <.25 µev. Both limits on t are below the thermal energy scale of 2 µev set by our electron temperature of 22 mk. II. RELATING CHANGES IN DOT ENERGIES TO GATE VOLTAGES To perform the bias spectroscopy measurements shown in Figures 2 and 4 of the main text, for given values of V S1 and V we need to adjust the voltages on gates P1 and P2 so as to set the energy E of the orbital states, as well as the energy difference δ between the states. Achieving this control first requires us to determine the capacitance factors that describe how the voltages V P1,V P2, V S1 and V affect the orbital states. We can then use these factors to determine how to vary V P1 and V P2 so as to control E and δ. This process is described below. The electrochemical potential energy of the dot states can be related to the voltages. For dot 1, this relationship can be written as: = e(α P1 V P1 + ξ 1,P2 V P2 + α S1 V S1 + ξ 1, V ) ()

2 2-167 = G (e 2 /h).15 V P2 (mv) E = δ =.5 di 1 /dv S1 (1-3 e 2 /h) G 1 (e 2 /h) V S1 (μv) = = = ev S V (μv) FIG.. Sum of the zero-bias conductance through dots 1 and 2 in the vicinity of a pair of triple points. Bias spectroscopy of dot 1 at V P2 = 166 mv. The Coulomb diamond associated with the state in dot 1 is clearly visible. The slopes of the diamond edges allow the extraction of capacitance factors as described in the text. Measurement of the zerobias conductance of dot 1 at V P2 = 166 mv. The slope of the line allows the extraction of a capacitance factor as described in the text. A corresponding equation holds for dot 2. In this equation, the coefficients of the voltages depend on the capacitance of the gates to the dots 1 and these factors are what we need to determine. We can directly extract the value of these capacitance factors from the data. Figure shows an example of the total conductance for a pair of triple points. To determine the capacitance factors for dot 1, we set V P2 = 166 mv, where we are away from the charge transition in dot 2 (this position is approximated by the horizontal blue line in Fig. ). At this fixed value of V P2 we perform standard bias spectroscopy of dot 1, as shown in Fig.. These data show the edges of a Coulomb diamond. Along the edge with negative slope, the dot state is aligned with the Fermi energy of the drain lead, which we assign to be. Then we have = along this line. Since V P2 = and V =, Eqn. gives α P1 V P1 + α S1 V S1 =. So the slope of this diamond edge m i is related to the factors by m i = V S1 / V P1 = α P1 /α S1. Similarly, the slope of the other diamond edge m j is related to the capacitance factors by m j = α P1 /(1 α S1 ). Thus with the measured slopes of the Coulomb diamond, one can extract α S1 and α P1. The remaining capacitance factors can be extracted from other measurements. Along the dot 1 Coulomb blockade line in Figure the dot states are aligned with the dot 1 source and drain leads, so =. Thus from Eqn. we have α P1 V P1 + ξ 1,P2 V P2 =. This then relates the slope of this line m 1 to the factors by m 1 = V P2 / V P1 = α P1 /ξ 1,P2. In the data shown in Fig., we plot the zero-bias conductance of dot 1 as a function of V, which gates dot 1. The slope of this line is related to the capacitance factors by m = ξ 1, /α P1. In this way, we can extract all the necessary capacitance factors for dot 1. A similar procedure can be used to extract the capacitance factors associated with dot 2. The energy of the dot states E and δ can be related to and µ 2 by E = 1 δ 2 µ1 + µ 2 µ 2

3 3 Combining this with Eqn. and a corresponding equation for µ 2, we have: 2 e E = δ (αp1 + ξ 2,P1 ) (α P2 + ξ 1,P2 ) VP1 (αs1 + ξ + 2,S1 ) (α + ξ 1, ) VS1 (α P1 ξ 2,P1 ) (α P2 ξ 1,P2 ) V P2 (α S1 ξ 2,S1 ) (α ξ 1, ) V (S3) For V S1 = V =, we define E = at the triple point indicated in Fig., where the dot states are degenerate (δ = ) and they are at the same energy as the source and drain leads. For given values of V S1 and V we can find the necessary gate voltage changes to establish the desired values of E and δ by substituting into Eqn. S3 and solving for V P1 and V P2. III. DOUBLE DOT COULOMB DIAMONDS di 2 /dv (1-3 e 2 /h) (1,1) µ 2 (1,1) (1,) µ 2 (,1) V S1, V (µv) (,) c d f b (1,1) e - (d) 1 (e) (f) FIG. S3. DQD Coulomb blockade diamond from Fig 2 of the main text. The labeled points correspond to the DQD diagrams in through (f). The dot occupations labeled in and are relative to a background occupation denoted (, ). Figure 2 of the main text shows a Coulomb blockade diamond obtained by performing standard bias spectroscopy at a pair of triple points of the DQD. We can understand the processes that give rise to transport along each edge of the Coulomb diamond. The data from Fig. 2 are shown in Fig. S3, while Fig. S3 -(f) show the DQD energy diagrams corresponding to the different points marked in Fig. S3. In these diagrams, the solid lines represent the electrochemical potential energy of the charge states of the DQD (the charge states are labeled relative to some background occupation denoted (, )); see Fig. S3. For example, (1, ) denotes the energy for adding an electron to dot 1 when dot 2 contains electrons. Similarly, µ 2 (, 1) is the energy to add an electron to dot 2 when dot 1 contains electrons. The dashed lines represent the energy to add a second electron to the double dot: for example (1, 1) is the energy to add a second electron to dot 1 when dot 2 contains the first electron. Figure S3 shows the position of the levels at one of the triple points, when (1, ) and µ 2 (, 1) are degenerate with the Fermi energy of the leads. Along the transport line marked by the positive bias voltages on S1 and lower the electrochemical potential of these leads; however, the dot levels are still aligned with the Fermi energy of the drain leads allowing transport through the dots. Conversely, along the line marked by (d) the dot states (1, ) and µ 2 (, 1) are below the Fermi energy of the drain leads and transport occurs when they are aligned with the electrochemical potentials of S1 and. The diamond edge marked (e) corresponds to applying a negative voltage to S1 and to align the electrochemical potential of these leads with (1, 1) and µ 2 (1, 1). Finally, the vertical transport line labeled (f) corresponds to (1, 1) and µ 2 (1, 1) aligning with the Fermi energy of the drain leads.

4 4 IV. DETERMINING COUPLING ASYMMETRIES..5 (e) V S1 (µv) Data di 2 /dv (e 2 /h) Calculation Γ D1 / Γ S1 =.9 Γ / Γ =.3 Calculation Γ D1 / Γ S1 = 12 Γ / Γ = di 1 /dv S1 (e 2 /h) (d) (f) V (µv) FIG. S4. Comparison of the data shown in and to the results of a simple sequential tunneling model shown in -(f) allow us to identify the correct ratios Γ D1/Γ S1 and Γ /Γ. The color scale for dot 1 shown above also applies to and (e). The color scale for dot 2 shown above also applies to (d) and (f). For understanding the features observed in Fig. 4 of the main text, it is useful to know the ratios of tunnel rates Γ D1 /Γ S1 and Γ /Γ. For the measurements reported in the main text, we are in the regime where the Coulomb blockade peak shapes are dominated by the tunneling rates and can provide information about these ratios. Specifically, the height of the Coulomb blockade peak in dot 1 is given by 4Γ S1 Γ D1 /(Γ S1 + Γ D1 ) 2 e 2 /h, and similarly for the peak in dot 2. For dot 1 we observe a peak height of.29 e 2 /h, and this gives Γ D1 /Γ S1.9 or 12. We have two possible solutions because the peak height measurement alone cannot determine whether the dot is more strongly coupled to lead S1 or D1. Similar measurements for dot 2 give a peak height of.75e 2 /h, corresponding to Γ /Γ.3 or 3. Bias spectroscopy measurements allow us to determine whether a dot is more strongly coupled to its source or its drain. Figure S4 shows spectroscopy measurements of dot 2 as a function of E and V S1. Similarly, Fig. S4 shows measurements of dot 1 as a function of V. We can compare these data with the results of a simple sequential tunneling model that takes Γ D1 /Γ S1 and Γ /Γ as inputs. This model does not include the Kondo physics in the inter-dot region, but should adequately describe the resonant tunneling features. Figure S4 and (d) show the results of this model using Γ D1 /Γ S1.9 and Γ /Γ.3 and the agreement with the data is quite good. In contrast, Fig. S4(e) and (f) show the results of the calculation using Γ D1 /Γ S1 12: the results for the conductance through dot 2 (Fig. S4(e)) are qualitatively different from the data (Fig. S4). A similar qualitative discrepancy in the dot 1 data are observed if we input Γ /Γ 3. This allows us to conclude that Γ D1 /Γ S1.9 and Γ /Γ.3, which will be useful for understanding features in the data as described in the next section. V. E pz = FEATURE IN PSEUDOSPIN SPECTROSCOPY WITH V Figures 4(g) and (h) in the main text show pseudospin-resolved spectroscopy as a function of V. Since lead is the pseudospin down lead, this spectroscopy resolves the pseudospin down peak. As expected, we see an enhancement at only one sign of the bias, in this case V = E pz /e. However, the data in Fig. 4(g) we also observe enhanced conductance along the vertical line at E pz =. This is associated with a pseudospin Kondo process with the dot 2 drain lead as illustrated in Fig. S5. Specifically, an electron on dot 1 can tunnel off the dot into D1, and an electron can tunnel onto dot 2 from (Fig. S5). This maintains energy conservation and results in a pseudospin flip. The

5 5 FIG. S5. Pseudospin-flip process that causes the feature at E pz = in Fig. 4(g) of the main text. This process does not rely on lead, and hence is independent of the voltage V applied to this lead. electron can then tunnel from dot 2 back to, while an electron tunnels back onto dot 1 from S1 (Fig. S5). This type of process (as well as higher order processes) lead to an enhanced conductance through dot 1, but does not give transport through dot 2. We are able to observe this feature at E pz = in Fig. 4(g) because the coupling of lead to dot 2 is not weak (Γ /Γ.3). The corresponding feature in Fig. 4(f) of the main text is very faint because D1 is only weakly coupled to dot 1 (Γ D1 /Γ S1.9). Present address: IBM Almaden Research Center, San Jose, CA 9512 goldhaber-gordon@stanford.edu; Present address: Stanford University, Stanford, California W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev. Mod. Phys., 75, 1 (2). 2 Y. V. Nazarov, Physica B: Condensed Matter, 189, 57 (1993).