Supplemental information: Cotunneling drag effect in Coulomb-coupled quantum dots

Transcription

1 Supplemental information: Cotunneling drag effect in Coulomb-coupled quantum dots A. J. Keller, 1, J. S. Lim, David Sánchez, 3 Rosa López, 3 S. Amasha, 1, J. A. Katine, 4 Hadas Shtrikman, 5 and D. Goldhaber-Gordon 1, 1 Department of Physics, Stanford University, Stanford, California 9435, USA School of Physics, Korea Institute for Advanced Study, Seoul 13-7, Korea 3 IFISC (UIB-CSIC, E-71 Palma de Mallorca, Spain 4 HGST, San Jose, CA 95135, USA 5 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 961, Israel A. CAPACITANCES To extract the capacitances in our device, we begin by assuming the usual electrostatic model for a double quantum dot (DQD [1], which has a capacitor between each dot and its source and drain leads (C Si and C Di, for i {1, }, each dot and its gate (C P i, and a capacitor between the dots (C m. We define C i to be the total capacitance of dot i, e.g. C 1 = C P 1 + C S1 + C m + capacitances to ground. The electrostatic energy of the system Q V / is given by: Ũ(N 1, N = 1 U 1N U N + UN 1 N 1 e i {1,} U i {C P i V P i N i + C Si V Si N i } 1 e U {C P 1V P 1 N + C P V P N 1 + C S1 V S1 N + C S V S N 1 } where we let Q i = e N i. The addition energies of the system are given by: U 1 = e C C 1 C Cm = e X ( C 1 (1 U = e C 1 C 1 C Cm = e X (3 C C m U = e C 1 C Cm = e C m (X 1 (4 where 1 X = ( (5 1 C m C 1C It follows that C m /C = U/U 1 and C m /C 1 = U/U, so we may also write 1 X = ( (6 1 U U 1U The addition energies can may be extracted simply from transport measurements and we use the values reported in our previous work []. We obtain X = 1.6 in terms of the addition energies U 1 = 1. mev, U = 1.5 mev, and U =.1 mev. We then obtain C 1 = 13 af, C = 17 af, C m = 9. af. The effective dot levels are given by: µ 1 (N 1, N = Ũ(N 1, N Ũ(N 1 1, N (7 µ (N 1, N = Ũ(N 1, N Ũ(N 1, N 1 (8

2 The slopes of linear features seen in source-drain bias spectroscopy correspond to conditions on the effective dot levels. The slopes may be related to the capacitances. The slopes m i1 and m j1 in Supp. Fig. 1(a are: m i1 = C P 1 C S1 (9 m j1 = These can be used to solve for the remaining capacitances: C P 1 = C 1 X C P 1 C 1 /X C S1 (1 m i1 m j1 m i1 m j1 (11 C S1 = C 1 X m j1 m i1 m j1 (1 The capacitances C P and C S are similarly related to the slopes m i and m j in Supp. Fig. 1(b. We find C P 1 = 17.5 af, C S1 = 6.9 af, C P = 11.9 af, C S = 53.9 af. At this point we can check for self-consistency using other measurements. Supplementary Figure 1(c and Supp. Fig. 1(d show part of the charge stability diagram (note the axes are swapped between (c and (d. In Supp. Fig. 1(c, the peak in G 1 corresponds to constant µ 1. The following condition holds: U 1 C P 1 V P 1 + UC P V P = (13 Therefore we would expect that the measured slope m = V P 1 / V P =.166 should equal UC P /U 1 C P 1 = 567. The large discrepancy indicates that the typical electrostatic model of a DQD is insufficient to describe our data. To account for the discrepancy we introduce direct capacitances between reservoir S1 and dot (C S1,, and vice versa. We also introduce capacitances between gate P1 and dot (C P 1,, and vice versa, though these will be small. The total capacitances C 1 and C will now include these extra capacitances. To Eqn. 1 we must add corresponding terms: 1 e U 1 {C P,1 N 1 V P + C S,1 N 1 V S } 1 e U {C P 1, N V P 1 + C S1, N V S1 } 1 e U {C P 1,N 1 V P 1 + C P,1 N V P + C S1, N 1 V S1 + C S,1 N V S } (14 While the definition of U 1, U, U are unchanged, we must reinterpret the slopes of the linear features observed in measurements. For Supp. Fig. 1(a,b the slopes m jk, m ik ; k {1, } are given by: m jk = V Sk E Ck C P 1 + UC P k, k = V P k e (E Ck C Sk + UC Sk, k (15 m ik = V Sk V P k = E CkC P 1 + UC P k, k E Ck C Sk + UC Sk, k (16 where k = 1( if k = (1. Additional measurements are needed to solve for all capacitances. The slope of the feature observed in Supp. Fig. 1(c,d is: n k = V P k V P k = E CkC P k,k + UC P k E Ck C P k + UC P k, k (17 where k = 1 for (c and k = for (d. We can now solve for four of the capacitances. It may be shown that:

3 3 m jk m ik n k = X m ik m jk C P k,k + X 1 C k C P k C m (18 m jk m ik = X C P k + X 1 C P k, k (19 m ik m jk C k C m Equations (18 and (19 may be solved for C P k and C P k, k, yielding C P 1 = 17.4 af, C P 1, = 1.37 af, C P = 11.7 af, C P,1 = 1.9 af. In Supp. Fig. 1(e,f, the slopes are given by: p k = V P k V SD k = E CkC S k,k + UC S k E Ck C P k + UC P k, k ( It may be shown that: m jk m ik p k = X m ik m jk C S k,k + X 1 C k C S k C m (1 m jk = X m ik m jk C Sk + X 1 C Sk, k ( C k C m Solving the system of equations (1 and ( yields C S1 = 6.3 af, C S = 53. af, C S1, = 7.19 af, and C S,1 = 13.1 af. We emphasize that capacitances like C S,1 are bigger than or comparable to C m, and cannot be neglected. The capacitances are summarized in Supp. Table I. By accounting for all of these capacitances, one can take measurements in such a way as to apply a source-drain bias without gating the dot (changing the effective dot levels, and have independent effective gates for each dot, as done in the manuscript and described in the supplementary information of previous work []. Value (af C C 17 C m 8.97 C P C P 11.7 C P 1, 1.37 C P,1 1.9 C S1 6.3 C S 53. C S1, 7.19 C S, TABLE I. Experimentally derived capacitances.

4 4 (a m i1 = (b m i = -.1 V SD1 (µv - m j1 = 8499 di 1 /dv SD1 (e /h.4. V SD (µv - m j = 749 di /dv SD (e 1. /h (c n 1 = (d -3-4 V P (mv n = V P (mv -7 (e di 1 /dv SD1 (e /h V P (mv p 1 = di /dv SD (e /h (f.5 p = V P (mv di 1 /dv SD1 (e /h.4. di /dv SD (e /h V SD (µv - V SD1 (µv Supplementary Figure 1. Determining capacitances. (a Source-drain bias spectroscopy for dot 1. V P = 3.4 mv is chosen to be away from the triple points. Slopes m i1, m j1 are indicated. (b Source-drain bias spectroscopy for dot. V P 1 = mv is chosen to be away from the triple points. Slopes m i, m j are indicated. (c,d The capacitance of gate P to dot 1 (c and gate P1 to dot (d may be obtained with these zero source-drain bias measurements. Slopes n 1, n are indicated. (e,f The capacitance of reservoir S to dot 1 (e and S1 to dot (f may be obtained. Slopes p 1, p are indicated. (e V P = 3.4 mv. (f V P 1 = mv. B. LIMITS ON INTERDOT TUNNELING The series conductance G series at a triple point of the DQD places a limit on the interdot tunneling energy scale t. The series conductance is measured by applying an ac+dc bias voltage to both leads of dot 1 and using one current amplifier attached to both leads of dot to measure current. In the case of zero dc bias [, supp. info]: G series = 64 t 3Γ 1 ( Γ1+Γ where Γ i is the total tunnel rate between dot i and its two leads S, i and D, i. In Supp. Fig. (a we present conductance in a region of gate voltage containing the triple points of the DQD. For simplicity, here we are just varying the gate voltages V P 1,P, rather than independently controlling ε 1,. We switch measurement configurations in Supp. Fig. (b and measure G series at several dc biases, including zero bias. In contrast with transport through the individual dots (Supp. Fig. (a, conductance through the two dots is at our noise floor. Perhaps a very faint signal can be observed for certain gate voltages at V SD = µv, but even there, G series < 1 3 e /h for all measured dc biases. Given that Γ 1 15 µev and Γ 47 µev, we find that t <.1 µev at zero dc bias. We do not rule out a dependence of t on the bias voltage, but at least we do not observe appreciable G series at non-zero bias either. To rule out the possibility that the triple points have drifted out of the measurement window owing to charge instability in the device, we immediately repeat the measurement of (a in Supp. Fig. (c, and see that the features have barely moved. e h (3

5 5 (b (a S1 D1 (c x1-3 (e /h Γ D1 Γ S1 1 t Γ D Γ S S D V SD = - V V SD = -1 V V SD = V V SD = 1 V V SD = V (d Supplementary Figure. Limits on interdot tunneling. (a Interdot tunneling refers to direct hopping from dot 1 to dot or vice versa. (b Measured di1/dvs1 (left and di/dvs (right at zero dc bias near triple points of the DQD. (c Measured Gseries = di/dvsd. The bias voltage is applied to both leads of dot 1 and current is measured by one current amplifier connected to both leads of dot. Five different dc biases were applied as indicated. No features are observed, except perhaps a faint signal when VSD = µv near the upper left of the plot. (d Immediately after the measurements of (c, we repeat the measurements of (b at lower resolution just to ensure that the features have not drifted. di1/dvs1 is again at left.

6 6 (a.1 (b.1 G (e /h G 1 (e /h (c -ε.1 4 I 1 (pa (d - -ε.1 I (pa (e -ε. (f -ε. (g -ε. (h -ε. -ε. -ε. Supplementary Figure 3. Measurements with zero applied magnetic field. (a Measured di 1/dV SD1 and (b di /dv SD at zero dc bias near triple points of the DQD. (c,e,g Measured I 1 and (d,f,h I for: (c,d V SD1 = 1 µv; (e,f V SD1 = 3 µv; (g,h V SD1 = 5 µv. Note that in (c h the axis scaling differs from (a,b. No bias is applied across dot. In all cases, a drag current can be observed through dot despite no explicit bias being applied to dot. C. DRAG AT ZERO FIELD Figures 4 show measurements taken in a large Zeeman field and small out-of-plane field. This is a regime where our (spinless theory applies, and is where we took most drag measurements. For reference, Supplementary Figure 3 shows measurements taken in zero applied field. Qualitatively similar features are observed, including a finite drag current I many linewidths away from resonance. A notable difference compared to the finite Zeeman field case is that the sign of I changes as a function of ε. This should not be surprising: even in the spinless theory we find that depending on tunnel couplings the direction of drag current can reverse in this way. This is true for both our model and the model in [3] (see their Fig. 3b.

7 7 V 1 C 11 1 C 13 V 3 C 1 C C 3 C 1 C 14 V V 4 C C 4 Supplementary Figure 4. Sketch of the electrostatic model. D. THEORETICAL FORMALISM 1. Electrostatic model We consider two capacitively coupled quantum dots, 1 and. electrostatic equations for the charges Q 1 and Q are given by With the geometry shown in Supp. Fig. 4, the Q 1 = i Q = i C 1i (φ 1 V i + C(φ 1 φ, (4 C i (φ V i + C(φ φ 1, (5 with φ 1 and φ the internal potentials and V i (i = 1,..., 4 the applied voltages. The potential energies for both dots with Q 1 = en 1 and Q = en excess electrons take the form U 1 (N 1, N = U (N 1, N = en1 en φ 1 (Q 1, Q dq 1, (6 φ (Q 1, Q dq, (7 where φ 1 and φ are determined from Eqs. (4 and (5. The electrochemical potential of dots 1 and can thus be written as µ 1 = ε 1 + U 1 (1, U 1 (,, (8 µ = ε + U (, 1 U d (,. (9 We have lumped the gate dependence into the values of the dot levels ε 1 and ε. Let µ li = E F + ev i be the electrochemical potential of lead i and K = i C 1i j C j + C C ij. Since the Fermi

8 8 functions are evaluated at µ S1 µ 1, etc. (see main text, the energies of interest become µ 1 µ l1 = ε 1 E F + e e(c + i C i + C i C 1j V j1 + C C ij V j1 K i j µ 1 µ l3 = ε 1 E F + e e(c + i C i + C i C 1j V j3 + C C ij V j3 K i j µ µ l = ε E F + e e(c + i C 1i + C 1i C j V j + C C ij V j K i j µ µ l4 = ε E F + e e(c + i C 1i + C 1i C j V j4 + C C ij V j4 K i j (3a (3b (3c (3d with V ij = V i V j. It is worth noting that these expressions depend on voltage differences only. Thus, our current expression will be gauge invariant. When one of the dots is occupied, the electrochemical potentials can be evaluated as follows, µ 1 = ε 1 + ε + U 1 (1, 1 + U (1, 1 [ε + U 1 (, 1 + U (, 1], (31 µ = ε 1 + ε + U 1 (1, 1 + U (1, 1 [ε 1 + U 1 (1, + U (1, ]. (3 Hence, Eq. (3 becomes µ 1 µ l1 = ε 1 E F + e 5e(C + i C i + K i µ 1 µ l3 = ε 1 E F + e 5e(C + i C i + K i µ µ l = ε E F + e 5e(C + i C 1i + K i µ µ l4 = ε E F + e 5e(C + i C 1i + K i C i C 1j V j1 + C j C i C 1j V j3 + C j C 1i C j V j + C j C 1i C j V j4 + C j C ij V j1 C ij V j3 C ij V j C ij V j4. (33a (33b (33c (33d Substracting Eq. (3 from Eq. (33, we find that the interdot Coulomb interaction corresponds to U = e C K. (34 In our numerical calculations, we use the experimental parameters reported above: C 11 = C S1, C 1 = C S,1, C 1 = C S1,, C = C S, C = C m and C 13 = C 3 = C 14 = C 4 = together with V 1 = V S1, V = V S, V 3 = V D1 and V 4 = V D.. Hamiltonian and tunnel rates Our model Hamiltonian is H = H + H T, (35 where H = H D + H C. (36

9 H D is the Hamiltonian for the dot region, H C corresponds to the reservoir Hamiltonian, H D = ε i d i d i + Un 1 n, (37 H C = i=1, α=s/d,i,k ε αk c αik c αik, (38 and finally H T describes tunneling processes between the dot region and the reservoirs, H T = ( t αik c αik d i + h.c.. (39 α,i,k 9 We regard H T as a perturbation and calculate the probabilities for transitions between initial ψ i and final ψ f states (energies E i and E f, respectively from an expansion of the T -matrix, P if = h ψ f H T + H T G H T + ψ i δ(e i E f, (4 where the resolvent operator is G = (E i H 1. To lowest order in H T (Fermi s golden rule one obtains sequentialtunneling transition rates between the dot states =, 1 = 1, = 1 and d = 11 : Γ αi i = 1 h Γ αif αi (µ i, (41 Γ αi i = 1 h Γ αi[1 f αi (µ i ], (4 Γ αī id = 1 h γ αīf αī (µ ī + U, (43 Γ αī di = 1 h γ αī [1 f αī (µ ī + U], (44 with Γ αi = ρ αi t αi and γ αi = ρ αi t 1 αi. As discussed in the main article, the transmission probabilities depend on energy and t αi thus becomes a function of the charge state. In our numerical simulations of the drag current, we use the experimental values Γ S1 = Γ D1 = Γ 1 = 15 µev and Γ S = Γ D = Γ = 47 µev and choose γ S1 = γ D1 = Γ S1, γ S = Γ S and γ D = Γ D. To second order in H T we obtain cotunneling transition rates involving many intermediate states which must be summed over: γ if = h Tr W i ψ 1 f H T H T ψ i E i H δ(e i E f, (45 where the trace is performed over the lead degrees of freedom and the thermal factor W i obeys ψi W i c αk c αk ψ i = f α (ε αk. (46 i In the following, we focus on the nondiagonal cotunneling rates since the terms γ, γ 11, γ and γ dd do not contribute to the master equations (see below or to the drag current: iī d d = h t αī dε t βi ε µ ī + iη t 1 αī t1 βi ρ ε µ ī U + iη αī ρ βi f αī (ε [1 f βi (ε + µ i µ ī ], = h t αī dε t1 βi ε µ ī + iη t 1 αī t βi ρ ε µ ī U + iη αī ρ βi f αī (εf βi (µ i + µ ī + U ε, = h t αī dε t1 βi ε µ ī + iη t 1 αī t βi ρ ε µ ī U + iη αī ρ βi [1 f αī (ε] [1 f βi (µ i + µ ī + U ε], (47a (47b (47c

10 where a finite broadening η (a small imaginary part has been added to the denominators in order to avoid the divergence associated to the infinite lifetime of the intermediate states. Expansion in powers of 1/η leads to a first term which reproduces the sequential tunneling result. Therefore, to avoid double counting we subtract this term. The next order is independent of η and corresponds to the nondivergent cotunneling expressions. We find iī = β { 4π h Γ αīγ βi n B (µ ī µ i + µ βi µ αī Im Ψ (1 + iβ µī µ αī + β { 4π h γ αīγ βi n B (µ ī µ i + µ βi µ αī Im Ψ (1 + iβ µī + U µ αī Γαī Γ βi γ αī γ βi n B (µ π h ī µ i + µ βi µ αī 1 ( 1 {Ψ U Re + iβ µī µ αī d = β { 4π h Γ αīγ βi n B (µ i + µ ī + U µ αī µ βi Im Ψ + β 4π h γ αīγ βi n B (µ i + µ ī + U µ αī µ βi Im Γαī Γ βi γ αī γ βi n B (µ i + µ π h ī + U µ αī µ βi 1 {Ψ U Re d = β + iβ µ i µ βi Ψ (1 + iβ µī µ αī Ψ (1 + iβ µ i µ βi } Ψ (1 + iβ µ i + U µ βi + Ψ { Ψ (1 + iβ µī + U µ αī +Ψ { 4π h Γ αīγ βi [1 + n B (µ i + µ ī + U µ αī µ βi ] Im + iβ µī µ αī + iβ µ i µ βi Ψ } 1 iβ µī + U µ αī iβ µ } i + U µ βi, (48a + Ψ (1 + iβ µ } i + U µ βi + Ψ (1 + iβ µ } i µ βi Ψ Ψ (1 + iβ µī µ αī + β { 4π h γ αīγ βi [1 + n B (µ i + µ ī + U µ αī µ βi ] Im Ψ (1 + iβ µī + U µ αī Γαī Γ βi γ αī γ βi [1 + n B (µ i + µ π h ī + U µ αī µ βi ] 1 {Ψ U Re + iβ µī µ αī +Ψ + iβ µ i µ βi Ψ Ψ iβ µī + U µ αī }, (48b iβ µ i + U µ βi + Ψ (1 + iβ µ } i + U µ βi + Ψ (1 + iβ µ } i µ βi Ψ iβ µī + U µ αī }. (48c iβ µ i + U µ βi Here, n B (x = 1/(e x 1 is the Bose distribution function, Ψ (Ψ (1 is the digamma (trigamma function and β = 1/k B T is the inverse temperature. The stationary values of the set of probabilities p = (p, p 1, p, p d T follow from the equations = Γp written in matrix form with (Γ 1 + Γ + γ d Γ 1 Γ γ d Γ Γ = 1 (Γ 1 + Γ 1d + γ 1 γ 1 Γ d1 (49 Γ γ 1 (Γ + Γ d + γ 1 Γ d γ d Γ 1d Γ d (Γ d1 + Γ d + γ d, where Γ 1 = α Γ α1 1, Γ 1 = α Γ α1 1, Γ = α Γ α, Γ = α Γ α, Γ 1d = α Γ α 1d, Γ d1 = α Γ α d1, Γ d = α Γ α1 d, Γ d = α Γ α1 d, (5a γ 1 = α,β γ αβ1 1, γ 1 = γ α1β 1, γ d = γ α1β d, γ d = γ α1β d, α,β α,β α,β

11 11 (a I 1 (pa 1 (c 4 (b -ε I (pa.4 4 I 1 / 3 and I (pa (d 4 -. I 1 / 3 I I (pa di /d(-ε 1 (a.u.. G1 (e /h -ε (mev Supplementary Figure 5. Extended data and analysis from Fig. (V S1 = mv, V S = ; ε =.3 mev where cuts are shown. (a Drive current I 1. (b Drag current I. This is a reproduction of Fig. (e to aid in comparing with Supp. Fig. 5(a. (c Cuts from Supp. Fig. 5(a and Supp. Fig. 5(b. I 1 has been divided by 3 to fit on the same scale. (d Numerically differentiated di /d( ɛ 1 (black, arbitrary units is correlated, positively or negatively, with G 1 (blue, right axis. G 1 has been taken from the cut in Fig. (c and I is the same trace as above in Supp. Fig. 5(c. The correlated features appear generic in region (iii as defined in Fig. (a. E. EXTENDED DATA AND ANALYSIS FROM FIG. In Supp. Fig. 5(a we show the drive current I 1 as a function of ε 1 and ε for the same parameters as used in Fig.. For ease of comparison, Supp. Fig. 5(b reproduces Fig. (e, the drag current I. Supplementary Figure 5(c compares I 1 and I for ε =.3 mev. The drag current is seen to be less than a percent of the drive current, which is typical for region (iii, although the ratio can reach a few percent in region (ii. In Supp. Fig. 5d we compare the vertical cuts in Fig. (c and (e. Peaks and dips in G 1 are correlated (or anticorrelated with peaks and dips in numerically calculated di /d( ε 1. There appears to be correlation near ε 1 = U =.1 mev, and anticorrelation nearer to ε 1 = U + e V S1 =.6 mev. In the middle, G 1 is flat and there appears to be no correlation. As the peaks in G 1 correspond to excited states in dot 1, these features may not be explained satisfactorily by existing theories, and are well-resolved in the low temperature regime. Present address: Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 9115, USA Present address: MIT Lincoln Laboratory, Lexington, Massachusetts 4, USA goldhaber-gordon@stanford.edu [1] W. G. Van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev. Mod. Phys. 75, 1 (. [] S. Amasha, A. J. Keller, I. G. Rau, A. Carmi, J. A. Katine, H. Shtrikman, Y. Oreg, and D. Goldhaber-Gordon, Phys. Rev. Lett. 11, 4664 (13. [3] K. Kaasbjerg and A.-P. Jauho, Phys. Rev. Lett. 116, (16.