SUPPORTING INFORMATION. Gigahertz Single-Electron Pumping Mediated by Parasitic States

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1 SUPPORTING INFORMATION Gigahertz Single-Electron Pumping Mediated by Parasitic States Alessandro Rossi, 1, Jevgeny Klochan, 2 Janis Timoshenko, 2 Fay E. Hudson, 3 Mikko Möttönen, 4 Sven Rogge, 5 Andrew S. Dzurak, 3 Vyacheslavs Kashcheyevs, 2 and Giuseppe C. Tettamanzi 6, 1 Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge, CB3 0HE, U.K. 2 Faculty of Physics and Mathematics, University of Latvia, Riga LV-1002, Latvia 3 School of Electrical Engineering & Telecommunications, The University of New South Wales, Sydney 2052, Australia 4 QCD Labs, QTF Centre of Excellence, Department of Applied Physics, Aalto University, PO Box 13500, FI Aalto, Finland 5 School of Physics, The University of New South Wales, Sydney 2052, Australia 6 Institute of Photonics and Advanced Sensing and School of Physical Sciences, The University of Adelaide, Adelaide SA 5005, Australia Electronic mail: ar446@cam.ac.uk; Phone: +44(0) Electronic mail: giuseppe.tettamanzi@adelaide.edu.au; Phone: +61(0) S1

2 I. PUMP TUNABILITY AT HIGH FREQUENCY As discussed in the main article, it is important to operate the pump at a working point that ensures robust current quantisation. Tunable-barrier semiconductor pumps perform best when tuned at the first plateau with complete charge emission. 1 Figure S1 shows a significantly wider extension of the 1 plateau in these operating conditions, which confirms an enhanced robustness of the pumping protocol. Figure S1 also reveals that both the emission and capture limits can be tuned at gigahertz frequencies. However, at different frequencies, the optimal operation point may slightly shift, as one can observe by comparing the positions of the quantised current plateaus in the panels of Fig. S1. This can be attributed to the frequency-dependent attenuations and rlections of the driving signal in the coaxial line of the cryostat. These non-idealities become usually more prominent for increasing frequency. As a result, in order to operate the pump, one has to increase the overall ac power at the signal generator and, consequently, re-adjust the dc voltages at some gate electrodes. Therore, experiments similar to those of Fig. S1 have proved instrumental for coarse tuning the pump at gigahertz, in preparation for the time-consuming high-averaging measurements. (a) (b) f = 1.1 GHz f = 2.0 GHz I P / (c) (d) f = 2.5 GHz f = 3.55 GHz Figure S1: Pumped current as a function of top gate and C2 gate voltages. Experimental parameters: (a) f = 1.1 GHz, V C1 = 0.3 V, V OB = 0.98 V, V IB = 0.51 V, and P ac = 3.6 dbm. (b) f = 2.0 GHz, V C1 = 0.3 V, V OB = 0 V, V IB = 0.52 V, and P ac = 3.5 dbm. (c) f = 2.5 GHz, V C1 = 5 V, V OB = 7 V, V IB = 0.52 V, and P ac = 3.7 dbm. (d) f = 3.55 GHz, V C1 = 0 V, V OB = 6 V, V IB = 0.57 V, and P ac = 5.0 dbm. S2

3 di P /dv C2 (a.u.) (a) (b) (c) di P /dv C2 (a.u.) di P /dv C2 (a.u.) I P = 0 I P = I P = 0 I P = 21 MHz 105 MHz 908 MHz Figure S2: Derivative of the pumped current with respect to gate C2 voltage as a function of the driving frequency and gate C2 voltage. Experimental parameters: (a) V TG = 2.0 V, V C1 = 0.1 V, V OB = 1.2 V, V IB = 0.52 V, and P ac = 0.3 dbm. (b) V TG = 1.9 V, V C1 = V, V OB = 0.98 V, V IB = 0.51 V, and P ac = 3.6 dbm. (c) Traces denoted by dashed lines in panels (a) and (b) as functions of incremental voltage. The traces are shifted vertically for clarity by +3 for f = 21 MHz, -2 for f = 105 MHz and -7 for f = 908 MHz. II. FREQUENCY DEPENDENCE OF THE CAPTURE CHARACTERISTICS As we discuss in the main article, the electron capture mechanisms are rlected in the rising edge of the current plateau, which shows a distinctive frequency dependence in the experiments discussed. As shown in Fig. S2(a), at low frequency the current rise is monotonic, which results in a single peak in the first derivative of the current with respect to V C2. However, a second peak emerges for increasing frequency, and a third one appears at furtherly high frequency, as illustrated in Fig. S2(b). Note that the previously discussed fects arising from rlection patterns in the coaxial cables are clearly visible by comparing panels (a) and (b). Indeed, the capture boundary line in Fig. S2(a) appears essentially at the same gate voltage for all frequencies, but Fig. S2(b) reveals significant horizontal dependence for a comparable frequency range. In the previous literature, 2,3 peaks in the derivative of the pumped current were associated to excited QD states through which the electron escape was enhanced. This interpretation does not satisfactorily explain the findings presented here. Figure S2(c) shows the first derivatives of the traces reported in Fig. 3(a) of the main text. As discussed in the main text, we argue that the extra peaks that appear for increasing frequency are the signature of states that compete for capturing the electron, as opposed to merely contributing to an enhancement of the escape rates. III. MODEL A. Dinition of the capture and loading probabilities Here we consider a general model for the pumped current which includes the results reported in the main text as special cases. The full initialization phase consists of loading (L), followed by capture (C). We enumerate states in the order with which they emerge from the Fermi sea during capture, starting with the main dot k = 0, following by k = 1... N parasitic states. The total current in the capture-limited regime is I P /() = Pk C, (1) where Pk C is the probability for the electron to be captured and eventually delivered to the drain in state k. We compute Pk C = P k(t ) as the final value of the time-dependent occupation probability at time t when the entrance barrier is so high that all charge exchange with the source fectively ceases. The initial condition P k (t 0 ) = Pj L is the probability for the state k to be loaded bore any back-tunnelling starts at t 0. We first solve the capture problem for arbitrary Pk L subject only to complete initial loading condition N k=0 P k L = 1. Next, we solve separately the problem of initial loading, assuming that the time-dependence of the only parameter k=0 S3

4 controlling both energies and rates is symmetric with respect to t = 0. This condition holds for harmonic modulation of the entrance barrier voltage V IB (t) with t = 0 corresponding to the most positive value during the pumping cycle, which separates the loading phase at t < 0 from the capture phase at t > 0. We characterise the tunnel coupling between the source and a state k by the in-tunnelling (γk in ) and the backtunnelling (γk back ) rates, dined by the Fermi golden rule without the Pauli blocking factors as γk in = g kγk back = g k 2π ρ M 2 k, where g k is the quantum degeneracy of the discrete state k, ρ is the density of continuous states in the lead, and M is the tunnelling matrix element; the averaging is done over the lead conduction modes at energy ɛ k. B. Solution for the capture problem We denote t k the time moment when the energy of a state k crosses the Fermi level on the way up. Consider a particular time interval t [t j... t j+1 for j = 0... N 1. The rate equation for the probability P k (t) of the state k to be occupied is dominated by back-tunnelling of electrons for k j and by in-tunnelling (loading) for k > j, 0 k j : j < k N : d dt P k(t) = γk back (t) P k (t), (2) d dt P k(t) = +γ in k (t) [ 1 P k (t). (3) The initial condition is P k (t 0 ) = Pk L. Equations (2) and (3) disregard the fect of thermal fluctuations in the lead, and make Markovian assumptions in the use of time-dependent rates. Time evolution from t = t j to t j+1 dined by Eqs. (2) and (3) can be expressed explicitly as [ tj+1 0 k j : P k (t j+1 ) = P k (t k ) exp γk back (t ) dt (4) t k [ tj+1 { t j < k N : P k (t j+1 ) = P k (t j ) + γk in (t) [1 P l (t j ) exp W j (t )dt, (5) t j t j where W j (t) = S j (t) = t + S j (t )W j (t ) exp t j k=j+1 l=j+1 [ t W j (t )dt t k=0 } dt S j (t) dt, γ in k (t), (6) j P k (t k ) exp k=0 [ t γk back t k For t > t N, only back-tunnelling remains possible for any of the states, [ P C k = P k (t ) = P k (t N ) exp t t N (t ) dt γk back (t ) dt. (7). (8) For rates that depend exponentially on time, as dined in the main text, Eq. (5) can be integrated leading to finite but cumbersome algebraic expressions in terms of Pk L, δ, t/τ and η. For N = 1, the explicit form is P0 C = exp ( e v+δ )P0 L, (9) ( P1 C = exp ( e v ) 1 exp [ η e v P0 L + η e δ+ t/τ exp{ e v+δ (1 e t/τ )} exp [ η e v ) P L e t/τ 1 η e δ+ t/τ 0. (10) In the large δ limit, the above equations simplify to P C 0 = 0 and P C 1 = exp ( e v ) exp [ (1 + η) e v P L 0. Taking into account that Z 1 = ηx 1 = η e v, and P L 0 + P L 1 = 1, and using Eq. (1), we see that in this limit the result agrees with Eq. (1) of the main text. In Fig. S3, we show characteristic behaviour of the total current for N = 2, g = 4 and a range of values for δ and η. In general, the parameter δ controls the distance between features corresponding to the different states, and η regulates the height of the local maximum. S4

5 δ=1, η=0.3 δ=1, η=1 δ=1, η=5 δ=1, η=20 δ=2, η=0.3 δ=2, η=1 δ=2, η=5 δ=2, η=20 δ=3, η=0.3 δ=3, η=1 δ=3, η=5 δ=3, η=20 δ=5, η=0.3 δ=5, η=1 δ=5, η=5 δ=5, η=20 Figure S3: Modelled pumped current I P / as a function of dimensionless gate voltage v for N = 2, g = 4 and different combinations of δ and η. The continuous black line shows the total current, computed from Eq. (1) using the solution (4)-(8) to the capture problem, and assuming P0 L = 1, Pk>0 L = 0. Red, green and blue lines show components corresponding to P2 C, P1 C and P0 C, respectively. The dashed black line shows the total current, computed using the same expressions for Pk C, but for a different distribution of the initial loading probabilities, Pk L, as given by Eqs. (18). C. Solution for the loading problem The loading starts at t = t L N < 0 when all the states are empty and the state with the lowest time-dependent energy ɛ N (t) crosses the source Fermi level, ɛ N (t L N ) = 0, on the way down, ɛ N (t L N ) < 0. This is followed by other states at t L N 1 < tl N 2 <... tl 0 such that ɛ k (t L k ) = 0 and tl 0 < 0. The rate equation for the probability Pk L (t) of the state k to be loaded by the time t in the low-temperature limit can be written as [ d dt P k L (t) = Θ(t t L k )γk in (t) 1 Pl L (t) Θ(t L k t)γk back (t) Pk L (t), (11) l=0 where Θ(t t L j ) is the Heaviside step function. The initial condition P k(t) = 0 for t < t L k rlects the assumption of complete emission in the previous pumping cycle. Summing up Eqs. (11) for all k gives a straightforward solution for the total instantaneous loading probability, γ in j (t )dt. Consequently, the individual proba- N l=0 P l L(t) = 1 exp [ Y (t), in terms of Y (t) = N j=0 Θ(t tl j ) t t L j bilities Pk L (t) are given by t Pk L (t) = Θ(t t L k ) γk in (t ) exp [ Y (t ) dt. (12) t L k A common time dependence for individual rates, γk in(t)/γin N (t) = const = K k K k+1, allows us to compute Y (t) and Pk L(t) by piecewise integration over time intervals t [tl j... tl j 1 in which Y (t) = Y (tl j ) + K t j γn in(t ) dt. The result can be expressed in terms of Y j = Y (t L j ) = N l=j+1 K lz l and Z l = t L l 1 γ t L N in (t) dt, l P L k =(K k K k+1 ) k j=0 K 1 ( j e Y j e Yj 1), (13) t L j S5

6 where we set Y N = K N+1 = 0 and Y 1 = + for notational consistency. Note that K l = N Explicitly, for N = 1 and N = 2, j=l γin j /γin N. (N = 1) P0 L =(K 0 K 1 )K0 1, P1 L = 1 P0 L, (14) (N = 2) P0 L =(K 0 K 1 )K0 1, P L 1 =[K 1 K 2 [ K 1 0 e K1Z1 + K 1 1 (1 e K1Z1 ) e K2Z2, P L 2 = 1 P L 0 P L 1. (15) D. Connection between the loading and the capture problem Due to time-reversal symmetry of the sinusoidal single-parametric driving, we expect t L k = t k, ɛ k (t) = ɛ k ( t), and k (t) = g k γback k ( t). This connects the parameters of the loading to those of the capture, γ in Z k = (γ in N /γ in k 1) g k 1 X k 1 (γ in N /γ in k )g k X k. (16) In terms of the parameters δ, /τ and η dined in the main text, Eqs. (14) and (15) can be written explicitly as (N = 1) P0 L = exp( η e v ), (17) 1 + e δ+ t/τ (N = 2) P0 L = exp[ η e v (1 + e δ + e t/τ ), (18) 1 e δ+ t/τ + e 2δ+2 t/τ P1 L = exp( η e v ) 1 + e exp[ η e v (1 + e δ + e t/τ ) δ+ t/τ (1 + e δ+ t/τ )(1 + e δ+ t/τ + e 2δ+2 t/τ ). In general, competition between different states during the loading phase (as described by Pk L ) leads to results, similar to competition at the capture stage only (assumes P0 L = 0), see comparison in Fig. (S3). For the simple case of N = 1 and large δ, large X 0 limit, Eq. (17) gives P0 L = exp( η e v ) = e Z1. Using this value instead of P0 L = 1 in Eq. (1) of the main text leads to Eqs. (2) and (3) with the value η replaced by 2 η, ie. for a single, well-pronounced parasitic state feature the difference between the two loading scenarios is just the doubling of the parameter η. IV. COULOMB BLOCKADE The compact device architecture that we have shown in Fig. 1(a) of the main manuscript can be operated as an error-resilient pump as long as a QD is formed in the region enclosed by the barrier and confinement gates. The desirably large charging energy observed in the Coulomb diamond plot as a function of V TG (see Fig. 1(b) of the Figure S4: Absolute differential conductance of the device as a function of source-drain and confinement gate voltages (V C = V C1 = V C2). Dashed lines are guides for the eye to highlight the boundaries of the Coulomb diamonds. The other gate voltages are: V TG = 3.00 V, V OB = 0.99 V, V IB = 7 V. S6

7 main manuscript) suggests that a very small QD is formed within the transport channel. However, since the top gate extends across the whole two-dimensional electron gas between the ohmics, one needs to perform additional transport measurements to verify that the QD is indeed formed in the region of interest. Figure S4 shows a Coulomb diamond plot as a function of both confinement gates being kept at the same values and swept simultaneously. From this measurement and similar ones as a function of the barrier gate voltages, one can infer that an intended QD is indeed formed in the region highlighted in red in the inset of Fig. 1(a) of the main manuscript. (1) Kaestner, B.; Kashcheyevs, V. Rev. Progr. Phys , (2) Kataoka, M.; Fletcher, J. D.; See, P.; Giblin, S. P.; Janssen, T. J. B. M.; Griffiths, J. P.; Jones, G. A. C.; Farrer, I.; Ritchie, D. A. Phys. Rev. Lett. 2011, 106, (3) Rossi, A.; Tanttu, T.; Tan, K. Y.; Iisakka, I.; Zhao, R.; Chan, K. W.; Tettamanzi, G. C.; Rogge, S.; Dzurak, A. S.; Möttönen, M. Nano Lett. 2014, 14, S7