Andreev tunneling in charge pumping with SINIS turnstiles
T. Aref, V. F. Maisi, M. V. Gustafsson, P. Delsing, J. P. Pekola
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Andreev tunneling in charge pumping with SINIS turnstiles
T. Aref , , V. F. Maisi , M. Gustafsson , P. Delsing and J. P. Pekola Low Temperature Laboratory, Aalto University - P.O. Box 13500, 00076 Aalto, Finland Centre for Metrology and Accreditation (MIKES) - P.O. Box 9, 02151 Espoo, Finland Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, SE-412 96 G¨oteborg,Sweden
PACS – Proximity effects; Andreev reflection; SN and SNS junctions
PACS – Single electron devices
PACS – Units and standards
Abstract –We present measurements on hybrid single-electron turnstiles with superconductingleads contacting a normal island (SINIS). We observe Andreev tunneling of electrons influencingthe current plateau characteristics of the turnstiles under radio-frequency pumping. The data iswell accounted for by numerical simulations. We verify the dependence of the Andreev tunnelingrate on the turnstile’s charging energy. Increasing the charging energy effectively suppresses theAndreev current.
Introduction. –
At the present time, there does notexist a quantum current standard though such standardsexist for both voltage and resistance. A strong candidatefor such a standard is a turnstile with a small normal metalisland connected by tunnel junctions to superconductingleads (SINIS) [1]. These hybrid turnstiles pump electronsone at a time producing a well-defined current of I = ef where e is the electron charge and f is the frequency ofpumping. Understanding and eliminating error processesin these turnstiles is vital for realizing a quantum metro-logical triangle (QMT), a key goal in metrology [2]. Inclosing the QMT, the three standards of voltage, resis-tance and current would be compared against each othervia Ohm’s law. Closing the QMT will allow the most ac-curate comparison of the Josephson constant K J = 2 e/h and the von Klitzing constant R K = h/e (i.e. the chargeof the electron and Planck’s constant) to date.There exist several potential quantum current standardcandidates. The NIST seven junction pump has demon-strated a current accuracy of 1 . [3, 4] butis limited to maximum currents of approximately 1 pA.Other candidates with the potential for metrological ac-curacy at metrologically relevant currents include semi-conductor tunable barrier pumps, charge-coupled devicepumps [5], tunnel junction pumps [6–8], quantum dotpumps [9], surface acoustic wave pumps [10] and quantumphase slip nanowire pumps [11]. In general, the accuracyof these quantum currents standards is not yet compara- ble to the accuracy of the quantum Hall and Josephsoneffect standards. A semiconducting quantum dot parallelpump recently demonstrated an accuracy of 1.5 in 10 at54 pA [12]. The minimum required current magnitude forclosing the metrological triangle is about 100 pA with anaccuracy of about 1 part in 10 , which has not yet beenachieved.The SINIS turnstile operates by using energy barriersto control the flow of electrons. The normal metal islandis capacitively coupled to a gate electrode. By applyingan appropriate voltage to the gate, single electrons canbe added and removed from the normal metal island. Inessence, the SINIS turnstile operates as a single-electrontransistor, with the superconducting gap providing extraprotection against unwanted tunneling of the electrons.There are multiple transport processes that can occur ina SINIS turnstile. The dominant one employed in chargepumping is sequential single electron tunneling throughthe insulating barrier. The dominant two-electron processcausing errors is Andreev tunneling. In Andreev tunnel-ing, an electron in the normal metal is reflected as a hole(or a hole as an electron) at the interface of the NIS junc-tion. This forms (or removes) a Cooper pair in the super-conducting electrode and can be alternatively viewed astwo electrons tunneling simultaneously across the insulat-ing barrier [13]. It has been shown that errors arising fromsequential 1 e -tunneling in turnstiles such as environmentalactivation can be effectively suppressed by proper filteringp-1 a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec . Aref et al. including an on-chip capacitively coupled ground plane[14]. Higher order, multiple-electron error processes noteliminated by this filtering may then be observed. Manyexperimental observations of Andreev currents have beenreported previously [15–17]. Andreev processes in deviceswith Coulomb blockade have been studied previously, boththeoretically [18] and experimentally [13, 19]. Only re-cently were individual Andreev reflection events detectedin NIS junctions [20, 21].Here we demonstrate that the Andreev tunneling pro-cess is detectable in single-electron turnstile current pump-ing plateaus and that the resulting error effect can be min-imized by increasing the charging energy of the turnstile.The data is well accounted for by numerical simulationswith reasonable parameter values and the observed behav-ior is physically intuitive.
200 nm 200 nm DC GateSiO AlAlO CuAlPatterned Ground PlaneRF gateAl BiasLeadBiasLead GPCCuAlGPCRF andDC gate GroundPlaned)b) c)a)
Fig. 1: Experimental set-up (a) Layout for sample of type one.The leads are coupled to the ground plane by large groundplane couplers (GPC) yet the turnstile itself is off the groundplane. (b) A SEM micrograph of the first sample type showingthe turnstile and combined RF gate. The GPC and groundplane are not shown. This is sample S1 with junction area 50nm by 125 nm. (c) The second sample set-up with the samplesin the patterned part of the ground plane. The device designis modified to include a separate RF and DC gate. (d) A SEMmicrograph of the second sample type showing the turnstileand the DC gate on top of the RF gate. This is sample S4with junction area 60 nm by 70 nm.
Experiment. –
To form the SINIS turnstile, a nor-mal copper island is connected to two superconductingaluminum electrodes using double angle evaporation withan intermediate oxidation step. Two types of devices werefabricated using electron beam lithography. The first typeof sample used one step of lithography and aligned the de- I ( μ A ) V (mV)a) −1−0.500.51−4 −3 −2 −1 0 1 2 3 4160μV320 μV×10 -3 I / ( e f )- A g S3 10 MHz sined) 010200 0.2 0.4 0.6 0.8 1 1.2 1.4 160 μV320 μV×10 -3 I / ( e f )- A g S1 10 Mhz sinec) −5051015200 0.2 0.4 0.6 0.8 1 1.2 1.4 I / ( e f ) A g
320 μV 160 μVS1 10 MHz sineb) 00.20.40.60.811.21.40 0.2 0.4 0.6 0.8 1 1.2 1.4
Fig. 2: 10 MHz sine wave current pumping with simulations.(a) DC current vs voltage envelope measurement for sample S1(green points). The red and blue lines are simulations of themeasurement. (b) Current pumping plateaus for low chargingenergy sample S1 with a 10 MHz sine wave. (c) Close up of (b)showing the Andreev tunneling. (d) Close up of current pump-ing plateaus of sample S3 (high charging energy, low resistance)with a 10 MHz sine wave showing the absence of Andreev tun-neling due to higher charging energy. Bias voltages are: 160 µ V (blue), 240 µ V (red) and 320 µ V (green) with a magentadotted line at I = ef and simulations shown as black dashedlines in figures b-d. vice with a ground plane as shown in fig. 1(a). Large padsensured capacitive coupling between the device and theground plane, strongly suppressing environmental noise[14, 20]. These large pads are marked ground plane cou-plers (GPC) and are electrically isolated from the groundplane by atomic layer deposition (ALD) grown aluminumoxide. This device design allows simple single sample fab-rication but is difficult to parallelize. A single gate lineis used as both a DC gate and an RF gate. A scanningelectron microscope (SEM) micrograph of the first type ofsample is shown in fig. 1(b).The second type of sample involves multiple steps oflithography as shown in fig. 1(c). In the first step, aground plane is patterned to allow device fabrication ontop of it. The turnstile sits in the patterned gap in theground plane and only the leads are directly above it. Pat-terning the ground plane in this manner prevents forma-tion of short circuits when wire bonding. After lithogra-phy, an insulating layer of silicon oxide is deposited byplasma enhanced chemical vapor deposition (PECVD).A second step of lithography is performed to form theturnstiles and individual DC gates for each turnstile.Thep-2ndreev tunneling in charge pumping with SINIS turnstiles
180 μV260 μV×10 -3 I / ( e f )- A g S2 10 MHz sineb)−50510150 0.2 0.4 0.6 0.8 1 1.2 ×10 -3 I / ( e f ) A g S1 50 MHz sine180 μV260 μVd)−101234560.2 0.4 0.6 0.8 1 1.2260 μV180 μV×10 -3 I / ( e f )- A g S1 10 MHz sinea)−50510150.2 0.4 0.6 0.8 1 1.2 1.4 ×10 -3 I / ( e f )- A g
260 μV180 μVS1 10 MHz squarec)−50510150.2 0.4 0.6 0.8 1 1.2
Fig. 3: Close ups of frequency and waveform dependence ofcurrent pumping plateaus (a) Pumping sample S1 with a 10MHz sine wave showing the enhanced tunneling caused by theAndreev process. (b) Pumping of sample S2 (high charging en-ergy, high resistance) with a 10 MHz sine wave showing the sup-pression of the enhanced Andreev tunneling. At high pumpingamplitude, there is increased back tunneling compared to sam-ple S3 due to the higher resistance of the junctions. (c) Pump-ing sample S1 with a 10 MHz square wave showing the differentcharacter of enhanced Andreev tunneling with a square wavedrive. (d) Close up of current pumping plateaus of sample S1with a 50 MHz sine wave showing the smaller relative error athigher pumping frequency. The vertical scale has been changedcompared to previous close-ups for clarity. The vertical scalein figures a-c are identical. The horizontal scale is not identicalsince the exact amplitude of RF drive applied to an individualdevice is not important. Bias voltages are: 180 µ V (blue), 220 µ V (red) and 260 µ V (green) with a magenta dotted line at I = ef and simulations shown as black dashed lines in all fourfigures. lithography used a three layer mask with a hard germa-nium layer which allowed smaller features than the twolayer PMMA patterning used for the first type. The ad-vantage of this fabrication method is that the separateDC gates enables parallelization. Parallelization is essen-tial for metrological purposes as it is difficult to get largeenough currents from a single device [22]. A SEM micro-graph of the second type of sample is shown in fig. 1(d).Measurements are done at a base temperature of ap-proximately 70 mK. A typical current vs voltage envelopemeasurement reveals the superconducting gap and singleelectron transistor behavior of the turnstile as shown infig. 2(a). To measure the pumped current of the turnstile,a fixed bias voltage is applied to one of the turnstile’s bias leads while current is measured on the other lead. A sinewave of amplitude V AC (resulting in a normalized ampli-tude A g = C g V AC /e ) with a frequency f is applied to theRF gate. As A g increases past a threshold value, the likeli-hood of transporting one electron through the turnstile ina single cycle approaches unity, giving rise to a quantizedcurrent plateau.We observe current quantization as shown in fig. 2(b).Zooming in on the current plateau reveals characteristicexcess current above the expected I = ef behavior due toAndreev tunneling. For samples with small charging en-ergy ( E C < ∆), as shown in fig. 2(c), the current plateauis enhanced above the expected value particularly for low A g values in the plateau. These deviations are suppressedfor samples with large charging energies ( E C > ∆), asshown in fig. 2(d). Amplifier gain is corrected on the 10 − level to match the simulated plateaus. This does not influ-ence the interpretation of excess current on the 10 − level.Qualitatively, as A g is increased for the situation E C < ∆,we encounter an energy threshold that permits Andreevtunneling before the single electron tunneling threshold isencountered, allowing enhanced current flow for low charg-ing energy samples. This charging energy dependence isindicative of the Andreev effect [20, 23].Fig. 3 shows close ups of the current pumping plateausfor a different conditions. In fig. 3(a), pumping of sampleS1 with a 10 MHz sine wave at a different range of biasesthan fig. 2 is shown. In fig. 3(b) the pumping curvesfor the high charging energy, high resistance sample S2shows a reduction below the expected I = ef due to singleelectron back tunneling. This increased back tunneling athigh pumping amplitude is a first order effect, which isdue to the larger resistance of the high charging energysample S2 compared to S3 [24]. Figs. 3(c) and (d) showsample S1 pumped with a 10 MHz square wave and a 50MHz sine wave.The Andreev effect is visible in a wide range of biasesas shown in figs. 2(b) and (c) and 3(a) with no visible de-pendence on bias voltage. Although the exact position ofthe plateau depends on bias, there is no separation of theplateaus due to bias dependence indicating this is not anenvironmental activation effect [14]. By fabricating highcharging energy samples with low resistance, we can limitboth the Andreev and back tunneling effects in the turn-stiles as shown in figs. 2(d). This requires small junctionswith highly transparent tunnel barriers i.e. junctions withlow RC product as have been fabricated previously [24,25].It should be noted that for device S3 with optimizedtunnel barriers for suppressing Andreev and single electronback tunneling there is still significant structure visible atthe 10 − level. The exact origin of this structure is notyet clear. It is possibly due to quasi-particle relaxationin the aluminum leads or noise still penetrating the fil-tering system. Preliminary simulations indicate that thisstructure can be accounted for if we consider overheatedsuperconducting leads but future research is needed in thisarea. The focus here was to demonstrate control over thep-3. Aref et al. Table 1: Sample Parameters.
Name E C / ∆ R T (kΩ) ∆ ( µ eV) A CH (nm )S1 0.63 160 220 30S2 2.2 1400 210 N/AS3 1.4 430 210 N/AS4 0.75 110 210 30Andreev error process. SINIS pumps are theoretically pre-dicted to achieve an accuracy of 1 part in 10 with a cur-rent of 30 pA for a single pump [26]. The experimentallymeasured accuracy level to date is on the 10 − level i.e. 1part in 10 . Theory. –
For the simulations shown in figs. 2 and3, we calculate the average current flowing through theturnstile by numerically solving a master equation. Thecurrent through the left barrier (which is equal to the cur-rent through the right barrier in the steady state) is givenby: I = e (cid:88) n [Γ LI ( n ) − Γ IL ( n )] P ( n, t ) (1)where P ( n, t ) is the probability for finding n electrons onthe central island at time t . The tunneling rate fromthe left lead to the island is Γ LI ( n ) = Γ SN ( E + L ( n )) +2Γ AR ( E ++ L ( n )) and the tunneling rate from the island tothe left lead is Γ IL ( n ) = Γ NS ( E − L ( n )) + 2Γ AR ( E −− L ( n )).The factor of two in front of the Andreev rate Γ AR ,compared to the single electron rates Γ NS,SN , is to ac-count for the Andreev process transporting two electrons.The energy required to add or remove a single electronto/from the island (denoted by + or − ) from the leftor right lead (denoted by L or R ) is given by E ± L,R = ± E C ( n − n g ± / ± eV L,R where E C = e / C is thecharging energy with C being the total capacitance ofthe island, n is the number of excess electrons on theisland and n g = C g V g /e is the normalized offset charge(the effective charge induced by applying a voltage V g tothe gate with capacitance C g ) [27]. The energy requiredto add or remove two electrons to/from the island (de-noted by ++ or −− ) in the Andreev process is given by E ±± L,R = ± E C ( n − n g ± ± eV L,R .The time dependence of the probability is given by themaster equation [27, 28]: dP ( n, t ) dt = − Γ n,n P ( n, t )+Γ n − ,n P ( n − , t ) + Γ n +1 ,n P ( n + 1 , t )+Γ n − ,n P ( n − , t ) + Γ n +2 ,n P ( n + 2 , t ) (2) b) e R T Γ P / Δ −6 −5 −4 −3 Δ I(e R T )t0 200 400 600 800 d) e ( V L - V R ) / Δ −3−2−10123 n g −1 −0.5 0 0.5 1 1.5 2a) P −3 −4 −5 c) e ( V L - V R ) / Δ −2−1012 n g −1 −0.5 0 0.5 1 1.5 2 Fig. 4: (a) Probability evolution during charge pumping. Theprobability P ( n, t ) is shown for different values of n : n = 0(green), n = 1 (blue), n = 2 (dotted black) and n = − n = 0 to n = 1 is shown in solid green. TheAndreev tunneling rate from n = 0 to n = 2 is shown in dottedgreen and is on top of the relaxation rate from n = 2 to n = 1via single electron tunneling through the right electrode (thickblack). Likewise, the single electron tunneling rate through theright junction from n = 1 to n = 0 is shown in solid blue, theAndreev rate n = 1 to n = − n = − n = 0 throughthe left junction (thick red). (c) Stability diagram of sampleS1. The blue diamonds are the thresholds for single electrontunneling, the red diamonds are the thresholds for Andreevtunneling and the thick black line is the pumping cycle used in(a) and (b). (d) Stability diagram of sample S2. The diamondsare reversed in order because of the higher charging energy ofsample S2. whereΓ n − ,n = Γ SN ( E + L ( n − SN ( E + R ( n − n +1 ,n = Γ NS ( E − L ( n + 1)) + Γ NS ( E − R ( n + 1))Γ n − ,n = Γ AR ( E ++ L ( n − AR ( E ++ R ( n − n +2 ,n = Γ AR ( E −− L ( n + 2)) + Γ AR ( E −− R ( n + 2))Γ n,n = Γ n,n − + Γ n,n +1 + Γ n,n − + Γ n,n +2 . Equation 2 tracks the flow of the probability. The higherorder Andreev tunneling effects are included by consid-ering state changes from n to n ± SN , or the rate at which single electrons tun-nel from the normal metal to the superconductor Γ NS , areobtained using first order perturbation theory [29, 30].p-4ndreev tunneling in charge pumping with SINIS turnstilesThe Andreev tunneling rates Γ AR , are given in equa-tion 3 in reference [26] using second order perturbativecalculations. These rates depend on the charging energy E C , the superconducting gap ∆, and the tunneling resis-tance R T . Values for these parameters are obtained fromDC current vs voltage envelope measurements (see insetof fig. 1) which depend only on Γ NS,SN . Γ AR has an ad-ditional fitting parameter since its magnitude is controlledby the quantity (¯ h/R T e ) /N where N is the effective num-ber of conduction channels, N = A/A CH [26]. A is thecross-sectional area of the junction estimated by scanningelectron microscope (SEM) imaging (see fig. 1 for relevantimages and area estimates) and A CH is the effective areaof a conduction channel used as a fitting parameter. The-oretically, A CH ≈ though fitted values are typicallymuch larger than this value and are interpreted as result-ing from inhomogeneities in the thickness of the oxide inthe junctions [20]. In table 1, the fitting parameters E C (in units of ∆), R T , ∆ and A CH are listed for each ofthe samples simulated. Note that Andreev rate fitting pa-rameters can not be determined for high charging energysamples S2 and S3 since the Andreev effect is suppressedbelow the measurement noise.The simulated probability evolution during chargepumping is shown in fig. 4(a) for various charge states.The corresponding rates of tunneling weighted by the oc-cupation probabilities are shown in fig. 4(b). These plotsare for simulations of sample S1 with V L − V R = 200 µ V, A g = 0 .
74 and f = 10 MHz. Only forward tunneling isrelevant here. In figs. 4(c) and (d), we show the calculatedstability diamonds for samples S1 and S2 respectively. Theminimal pair breaking energy for 1 e -tunneling is ∆ so thethreshold for tunneling is E ± L,R = ∆ which is shown as theblue line. For the Andreev tunneling, this pair breaking isavoided so the threshold for tunneling is E ±± L,R = 0 and isshown by the red dashed line.
Results and Discussion. –
In figs. 4(a) and (b),the probabilities and tunneling rates during one pumpingcycle are shown for sample S1. As the transition rate from n = 0 to n = 1 grows, the probability for being in state n = 0 quickly drops in the beginning of the cycle and isreplaced with a probability for being in state n = 1. Atthe same time, there is a detectable Andreev tunnelingrate for going from state n = 0 to n = 2 after which the n = 2 state quickly relaxes to n = 1 by single-electrontunneling. The low level occupation of the state n = 2is also detectable. Likewise, the opposite process wherean electron leaves the turnstile is observable in the secondpart of the pumping cycle. As before, when the rate from n = 1 to n = 0 grows, the most likely state becomes n = 0.The transition for n = 1 to n = − n = − n = 0 state by single-electron tunneling.In figs. 2(b), 2(c) and 3(a), the pumping plateaus forsix different biases and the corresponding fits are shownfor low charging energy ( E C < ∆) sample S1. There is excellent agreement between the fits and the data. Bylooking at the stability diagram shown in fig. 4(c), we canqualitatively understand the observed effect. The stabil-ity diagram shows tunneling thresholds versus normalizedbias voltage, V , and momentary gate charge, n g . Startingin the diamond on the left, as the gate amplitude, A g , isgradually increased, the Andreev threshold (shown as ared dotted line) is encountered first. Thus the Andreevprocess is most noticeable at low A g . As A g is increasedfurther, we encounter the single electron tunneling thresh-old and this process quickly dominates, obscuring the An-dreev effect. This results in the enhanced current pumpingplateau in figs. 2(c) and 3(a). For comparison, as can beseen in fig. 4(d), the high charging energy ( E C > ∆) sam-ple, S2, encounters the single electron tunneling thresholdbefore the Andreev threshold. The Andreev effect is notobserved as the electron has already tunneled before it en-ters the Andreev regime. Thus the plateau in figs. 3(d)and 2(d) is flat with no evidence of the enhanced tunnelingeffect.The lack of dependence on bias seen in figs. 2(c) and3(a) can similarly be explained by looking at the stabilitydiagram. Changing the bias corresponds to changing thehorizontal level of the black line in fig. 4(c). However,the Andreev tunneling thresholds run parallel to the singleelectron tunneling thresholds so this has a very small effecton the observed current.In fig. 3(c), the results from pumping with a 10 MHzsquare wave for sample S1 are shown. The square waveis modeled with an exponential rise to the applied voltagewith risetime 2.5 ns. Compared to the sine wave pumpingof the same frequency, the enhanced current for the squarewave is more peaked and dies away more quickly. At anamplitude of A g = 0 .
75, the square wave pumped plateauis almost flat with a current less than 1 . ef (i.e. closerto the ideal value of I = ef ) while sine wave pumpingwith the same frequency and bias voltage has a plateauwith significant slope at A g = 0 .
75 with a current greaterthan 1 . ef . This behavior can also be qualitatively un-derstood from the stability diagram. The square wavegoes more immediately to the final A g value thus spendingless time in the vulnerable regions of the stability diagram.When we first encounter the Andreev threshold but beforethe single electron threshold is encountered, the Andreevprocess dominates and we see the sharp peak in currentshown in fig. 3(c). As A g is increased, we encounter thesingle-electron tunneling process threshold and the escapeprocess is dominated by it. There is only a small possibil-ity for the Andreev tunneling enhancement to occur sincethe square wave sweeps through that vulnerable regionquickly. Thus the Andreev effect in the pumping plateaudies away more quickly for the square wave than for thesine wave.In figs. 3(d), the pumping plateaus for a 50 MHz sinewave are shown with the Andreev effect visible for lowcharging energy sample, S1. With higher frequency, moreelectrons are pumped per unit time producing a higherp-5. Aref et al. current. Thus the relative accuracy improves comparedto the lower frequency 10 MHz sine wave shown in fig.3(a) since the absolute value of the excess Andreev currentremains roughly the same. The Andreev tunneling contri-bution to the current remains roughly unchanged becausethe same process as described for the lower frequency sinewave takes place. This results in the more flat slope of the50 MHz pumping seen in figure 3(d) but the Andreev pro-cess is still detectable. It should be noted that the samesample parameters listed in table 1 were used for simulat-ing 10 MHz sine wave, 10 MHz square wave and 50 MHzsine wave current pumping demonstrating the robustnessof the simulation.From the simulations, we extract an effective channelarea on the order of 30 nm for the pumping (the effectivechannel area is difficult to determine precisely so the valuesare rounded to the nearest 10’s of nm ). This was consis-tent for both samples S1 and S4 shown in figure 1. Forclarity of presentation, only the data and fits from sampleS1 are shown in figures 2 and 3. Sample S2 and S3 do nothave a discernible Andreev parameter as it is suppressedby their high E C . This is in general agreement with theeffective area of 30 nm found in earlier work [20]. Thesevalues are approximately one order of magnitude largerthan the theoretical channel area of approximately 2 nm ,indicating that roughly only one tenth of the effective areaof the junction is active in agreement with previous results[31, 32].We have shown that the enhanced current is dependenton E C , pumping amplitude and pumping waveform shapebut independent of pumping frequency and bias voltage.These are all characteristic signatures of Andreev tun-neling that can be well accounted for by our theoreticalmodel. Conclusion. –
We have observed the Andreev tun-neling process in current pumping with a single electronturnstile. This error process can be effectively suppressedwith high charging energy leading us one step closer to aquantum current standard and completing the quantummetrological triangle. Andreev reflection as an error pro-cess can be fully eliminated with proper choice of E C asdemonstrated by these experiments. ∗ ∗ ∗ This work was funded in part by the European Commu-nity’s Seventh Framework Programme under Grant Agree-ment No. 218783 (SCOPE), the Aalto University Post-doctoral Researcher Program and the Finnish NationalGraduate School in Nanoscience.
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