Microwave-assisted transport via localized states in degenerately doped Si single electron transistors
MMicrowave-assisted transport via localized states in degenerately doped Si singleelectron transistors
A. Rossi ∗ and D.G. Hasko † ∗ Microelectronics Research Centre,University of Cambridge, J.J. Thomson Avenue,Cambridge, CB3 0HE, U.K. † Centre for Advanced Photonics and Electronics,University of Cambridge, J.J. Thomson Avenue,Cambridge, CB3 0FA, U.K. (Dated: October 29, 2018)Resonant microwave-assisted and DC transport are investigated in degenerately doped siliconsingle electron transistors. A model based on hopping via localized impurity states is developedand first used to explain both the DC temperature dependence and the AC response. In particular,the non-monotonic power dependence of the resonant current under irradiation is proved to beconsistent with spatial Rabi oscillations between these localized states.
PACS numbers: 03.67.Lx, 71.30.+h, 71.55.Gs, 72.10.Fk, 72.15.Rn, 72.20.Ee, 73.20.Fz, 73.23.Hk
I. INTRODUCTION
The single electron transistor is a widely used de-vice technology for the exploitation of quantum mechani-cal effects (Coulomb blockade) for applications includingmetrology and quantum information processing. Thefirst devices were fabricated using a combination of ametal (aluminium, to form the leads and the island) andan insulator (aluminium oxide, to form the tunnel bar-riers); such devices are known as metallic single elec-tron transistors (metal SETs). Much of the understand-ing of single electron effects in this type of device wasobtained by measuring their electrical characteristics asa function of temperature and magnetic field. Today,metal SETs have a well understood behaviour, exhibit-ing regular Coulomb oscillations (constant gate periodand peak height) and a temperature dependence that isconsistent with the charging energy given by the experi-mental Coulomb gap.The difficulty in fabricating these lithographically de-fined metal island SETs has largely limited their opera-tion to very low temperatures (usually requiring a dilu-tion refrigerator), so that many other material combina-tions have been explored to find an alternative fabricationroute. One of the most successful has been the degener-ately doped silicon single electron device, fabricated us-ing silicon-on-insulator technology. This material systemwas used to demonstrate some of the earliest examplesof single electron classical logic and memory in semi-conductor devices, as well as providing a route towardsqubit implementation. This approach offers significantscalability and operating temperature advantages com-pared to most metal SET systems. ∗ Electronic mail: [email protected]
In contrast with metal SETs, the electrical character-istics of degenerately doped silicon SETs usually exhibita disordered characteristic (varying gate period and peakheight). Furthermore, single electron transistor charac-teristics are observed without the need for a different ma-terial to form the tunnel barriers, as all of the island, theleads and the tunnel barriers are made from degeneratelydoped silicon. In semiconductor materials a very effectivebarrier can be formed by free carrier depletion, howeversuch a depletion process is significantly constrained bythe very high doping density used to fabricate these de-vices. The doping densities used for degenerately dopedsilicon SET fabrication are very much higher than situa-tions where depletion is usually employed. The minimumdoping level is set by the need to be above the Mott-insulator transition ( ≈ × cm − for P doped silicon)to avoid carrier freeze-out at low temperature and themaximum doping level is set by the solid solubility limit( ≈ cm − for P doped silicon). Coulomb blockade de-vices have been successfully demonstrated over the wholeof this doping density range with lateral dimensionsdiffering only by a factor of ∼
3. It is interesting thata change in doping density of more than two orders ofmagnitude makes little difference in the size or geometryof devices needed to show Coulomb blockade character-istics.The earliest silicon SETs were fabricated with con-strictions to define the location and strength of the tun-nel barriers, but it was soon found that these wereun-necessary for the purpose of demonstrating Coulombblockade characteristics. At that time, it was thoughtthat fluctuations in the surface potential on the side wallsof the silicon were the origin of the tunnel barriers. Thisidea was supported by the change in characteristics ob-served each time the device was cycled between the roomambient and the measurement temperatures, on the prin-ciple that the surface traps would be refilled in some ran- a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l dom way each time. Further support for this idea comesfrom the observation that the temporary application ofa large bias voltage to the substrate (particularly duringcool-down) is effective in shifting the overall characteris-tics of the system, say to decrease conduction in a devicewhere the Coulomb peaks are poorly defined. The disorder in the Coulomb peaks makes it difficultto extract an unambiguous charging energy. Devices ofthis type can show a very wide range of charging energies,within the same transistor or within a batch of similardevices, even when fabricated together. Furthermore, incontrast to metal SETs, their temperature dependenceis not well predicted from the charging energy and stan-dard theory. Indeed, devices made in this laboratory haveexperimentally demonstrated maximum operating tem-peratures from a few tens of Kelvin up to nearly roomtemperature, despite being fabricated to the same speci-fications and from the same material.Finally, the more recent observation of high quality fac-tor resonances during broadband microwave spectroscopyis another dramatic difference between the behaviour ofdegenerately doped silicon SETs and metal SETs. Previ-ous measurements on metal SETs showed only low qual-ity factor resonances (quality factor (cid:62) Thesestanding waves cause the amplitude of the microwavesignal at the device to vary with frequency and thedevice current is influenced by this amplitude throughheating effects. The current passed by a device underfixed source-drain and gate bias increases with temper-ature in a characteristic way due to the co-tunnellingeffect. This is confirmed by the experimental obser-vation in the case of metal SETs that the device cur-rent due to the microwaves always increases and thatthe frequency dependence is unaffected by the bias con-ditions of the device under test. This is in contrastto the experimental observations on the degeneratelydoped silicon SETs, where both increases and decreasesin device current due to the microwaves were seen andthe frequency dependence was influenced by the devicebias conditions. The quality factors of these resonancescan be very high (up to around 400,000) and the peakshape varies from positive/negative gaussian to differen-tial. The very large number of high quality factor res-onances (more than 1000 seen in the frequency range1-20GHz) and the Rabi-like oscillations under pulsed ex-citation conditions, strongly indicates a quantum me-chanical origin for these effects.In this paper, we attempt to explain the behaviour ofdegenerately doped silicon SETs on the basis of local-ized transport resulting from significant charge trappingon the sidewalls during cool-down. This charge trappingis influenced by the details of the device fabrication andcan explain the relative independence of the required de-vice geometry on the doping density. We will firstly de-scribe the observed temperature dependence and disor-dered Coulomb blockade response at DC in terms of hop- FIG. 1: Schematic diagram of the radiation coupling methodusing separate DC and AC lines. The DC lines are low passfiltered (LPF) by MiniCircuits BLP1.9 units. Top inset: SEMimage of a typical SET. Bottom inset: schematic cross sectionof the silicon-on-insulator structure. ping transport via impurity states. Next, we will discusshow trapped charges can give rise to high quality fac-tor resonances during microwave spectroscopy and whatcharacteristics these resonances might be expected to ex-hibit.
II. EXPERIMENTAL DETAILS
Devices investigated in this work were fabricated us-ing degenerately P-doped silicon-on-insulator wafers (seebottom inset of Fig. 1). Devices were fabricated usingelectron beam lithography and a sacrificial metal maskto transfer the pattern by reactive ion etching. Patternedsubstrates were thermally oxidized at 1000 ℃ to reducethe surface trap state density, which in turn reduces theincidence of random telegraph signals. A micrograph ofa typical measured SET is depicted in the top inset ofFig. 1.Measurements were carried out in a specially con-structed probe that could be directly immersed in a liquidcryogen, so that the device under test and a substantialpart of the connecting leads, were all efficiently cooled.All measurements discussed here were performed in liquidhelium, at a temperature of 4.2K. DC measurements weremade using Keithley 236 source-measure units (SMU)controlled by a LabView programme via the GPIB bus.The DC measurement connecting leads were directly cou-pled to the source, drain and gate terminals of the deviceand individually low pass filtered by MiniCircuits BLP1.9inline units at room temperature with cut-off frequencyof 1.8 MHz. This filtering is essential to prevent elec-tromagnetic noise (due to external sources) from propa-gating to the device under test, which would otherwisebe subject to strong electron heating. Microwave signalswere provided by an Agilent E8257D source capable ofdelivering up to +15dBm via a flexible sma-terminatedcable to the probe.AC signals can be applied to the source-drain leadsvia a biasing circuit; a schematic of the coupling circuit isshown in Fig. 1. A 50 Ω metal film resistor is provided toimpedance match the waveguide and microwave source,so that standing waves are inhibited. 100 nF ceramiccapacitors are used to decouple any DC voltage on thewaveguide from reaching the device. At 100 MHz (thelowest frequency used here) the impedance of each ca-pacitor is < µ H inductor (impedance > Thisindirect coupling method resulted in a global time depen-dent potential for the whole device, whereas the more ef-ficient coupling technique presented here results in a timedependent potential difference between device electrodes(mainly source and drain).
III. DC TRANSPORT
A typical stability plot, measured at a temperature of4K, is shown in the inset of Fig. 2(b) showing aperiodicCoulomb oscillations and unequal peak heights. Clearlythis device must meet the two requirements for the ob-servation of Coulomb blockade; the thermal energy mustbe smaller than the charging energy and the tunnel bar-rier impedance must be large compared to the resistancequantum. The second condition requires that the tun-nel barrier impedance should be at least 100kΩ and thefirst condition requires that the tunnel barrier does notextend by more than one or two squares along the de-vice axis (otherwise the capacitance of the island is toolarge). However, a doping density of ≈ cm − givesa sheet resistance of about 1600Ω / (cid:3) at room tempera-ture, dropping by a factor of about 1.6 at 4K, givinga “tunnel barrier”resistance of only a few kΩ from geo-metrical considerations alone. Clearly, the carrier trans-port in the tunnel barrier region must be influenced bysome other mechanism that greatly increased the resis-tance. To explain the magnitude of the current observedat a Coulomb peak, the tunnel barrier resistance mustbe nearly two orders of magnitude larger than the valueexpected from the geometry and from the doping density. It was previously suggested that the high tunnel barrierresistance is a result of the depletion of free carriers bycharge trapped at the sidewalls of the silicon. Of course,a high density of traps at the sidewalls is expected due tothe non-(100) orientation of the surface. In general, thedensity of traps due to unsatisfied dangling bonds at thesilicon surface is dependent on the crystalline orientationof the sidewalls. Oxidation is routinely used to passi-
FIG. 2: Depletion effect due to sidewalls trapping. a) Cartoonshowing a top-view of the nano-structure. Depleted regionsare mostly formed in the vicinity of the side edges and wherethe channel is narrower. b) Cross sectional electrostatic po-tential at the generic position marked by the dotted line in(a). Fermi energy is bent because of charge accumulation atthe sidewalls. Ionised donors and neutral sites with boundelectrons are represented by plusses and crosses, respectively.Inset: a device stability plot: drain-source current as a func-tion of drain-source voltage, V DS , and side gate voltage, V G .Blue-to-red scale corresponds to current varying from -60 pAto +60 pA. vate such dangling bonds, but little difference in devicebehavior was seen for the earliest devices made withoutoxidation and those with oxidation. Oxidation is effec-tive in reducing random telegraph noise, but has littleeffect on device geometry necessary for the observationof Coulomb Blockade. Moreover, the depletion depth dueto these traps would be fixed (and estimated to be muchtoo small to explain the tunnel barrier size) for a givendoping density. So that an increase in the doping densitywould require a corresponding decrease in the physicalsize of the tunnel barrier, contrary to experimental ob-servation. Another mechanism for trap formation arisesfrom the fate of dopant atoms in the silicon etched awayto form the sidewalls. Reactive ion etching is used to formthe island, tunnel barriers and leads by removing the un-wanted silicon. This process proceeds by transportinggas reactants to the silicon surface, where a chemical re-action takes place to form volatile products, which areremoved by the vacuum pumping system. Typical sil-icon dopants, such as phosphorus, do not form volatilecompounds with the CF and SiCl reactive etch gasesat the usual etch temperatures. The phosphorus atomsremain on the surface after etching and become incor-porated into the oxide during passivation. These atomsare no longer able to act as dopants, but do act as traps.The density of these traps directly depends on the dop-ing density of the silicon so that a change in this dopingdensity has no direct effect on the depletion depth in thesilicon. This mechanism explains why all tunnel barriersappear similar for a wide range of doping density.Transport in highly doped silicon SETs is strongly lo-calized in and around the SET island and tunnel barri-ers, due to depletion resulting from trapped charge onthe sidewalls. At high temperatures, these traps will bemostly empty due to thermal excitation; so that the re-sistance is little changed from that expected on the basisof the bulk material properties. But, as the temperatureis reduced these empty, neutral traps are able to captureelectrons that tunnel in from the nearby silicon, whichresults in a significant negative charge accumulation atthe sidewall. To maintain overall charge neutrality, thisnegative charge causes depletion of free carriers from thenearby silicon; these being the most likely to be capturedin the traps anyway. This depletion increases the sizeof the tunnel barrier; but in practice, would only affectthe resistance of narrow sections, such as the SET islandand tunnel barriers (Fig. 2). Such depletion moves theFermi energy down into the impurity band, despite thehigh doping density, so that the conduction mechanismchanges from metallic to a hopping behaviour.Such conduction behaviour normally arises only inlightly doped semiconductors at low temperatures whenthe previously free carriers are recaptured by the ioniseddopants and the overlap between carrier wavefunctionson adjacent impurity sites becomes weak. But such char-acteristics can also be observed in systems where the car-rier density is usually high but is subject to depletiondue to a gate voltage or by trapped charge. In that case, the conductivity decreases exponentially as the energy orspatial separation between the localized sites increases.We can apply the analysis of Efros and Shklovskii tothese situations.Fig. 3 shows the electrical differential conductancemeasured as a function of temperature and source-drainvoltage for a highly doped silicon nano-wire device, un-der conditions where the Coulomb Blockade effects willbe mostly suppressed by the applied potential differenceat drain-source and/or gate. Measurements have beenperformed at six different temperatures for each appliedelectric field by means of standard lock-in techniques.The data points shown are the average values for thedifferential conductance from a number of equivalent ex-perimental conditions obtained by sweeping V DS in bothforward and reverse directions. The observed behaviouris typical of a wide range of device types, including thosewith lithographically defined tunnel barriers. It is clearthat both temperature and electric field have a stronginfluence on the transport properties. This current de-pendence is often attributed to co-tunnelling in CoulombBlockade devices, where the tunnel barrier is formed bya material with a large energy gap. An analytic formfor the inelastic co-tunnelling contribution to the sampleconductivity of an SET measured at low temperature Tand applied source-drain voltage V DS has been given byAverin and Nazarov as σ cot = A ( BT + CV DS ) (1)where A, B and C are constants that depend on the na-ture of the tunnel barriers and fundamental constants.However, it is clear that such a relation does not satisfac-torily explain the dependencies shown, where the largestchanges occur at the lower voltages and temperatures.As discussed earlier, the electron density close to thesidewalls is expected to be significantly reduced by deple-tion resulting from electrons trapped in the grown oxide,so that it is appropriate to consider the temperature andelectric field dependence of the hopping conductivity inlow-doped semiconductors to explain the characteristicsin Fig. 3. The conductivity of a disordered system inthe small polaron hopping regime has been discussed byTriberis. This work concludes that Mott’s law gives themain temperature dependence and that the electric fielddependence is given by a function of the square of thesource-drain voltage. Using these relations, the followingfunction has been fitted to the data in Fig. 3 ln [ σ ( T, V DS )] = A + BT − / (2)where A and B are treated as different fitting constantsfor each value of V DS . The functions that model the fielddependence are then fitted as follows A = C + D ( E + V DS ) − (3) B = F ( E + V DS ) − (4)being C = − . ± . D = 0 . ± . F = − . ± .
001 and E = 0 .
054 fitting parameters ( E isused as a fixed parameter, therefore it does not come withan error). The fit for low temperatures and small source-drain voltages is very satisfactory; at higher voltages andtemperatures the hopping model becomes progressivelyless applicable. This agreement between the experimen-tal data and the hopping model suggests that the ob-served temperature dependence is mostly determined bychanges in the transparency of the tunnel barriers ratherthan by co-tunnelling.Very large charging energies have been estimated in thepast from electrical measurements at low temperaturessuggesting that operation at high temperature should bepossible. However, these estimates often disagree withthe observed maximum operating temperature. In thesecases, it is the degradation of the tunnel barrier resis-tance that is the cause of the reduction in maximum op-erating temperature, rather than the onset of significantco-tunnelling. Longer constrictions seem to give highermaximum operating temperatures than shorter or nar-rower ones. This effect may be explained by the increasein the number of hops taken to pass through the tunnelbarrier. An increase in the number of hops will reducethe energy barrier per hop and so reduce the effect of anincrease in temperature or electric field.In a metallic SET, the electrical characteristics are de-termined by the island capacitance and the tunnel barriertransmission. The charging energy for an extra electronto occupy the island has only one contribution, that dueto the island capacitance. In a degenerately doped siliconSET, hopping transport can modify this behaviour, sincethe energy for an electron to occupy a localized site inthe island now has three contributions. The first is theconventional Coulomb charging energy due to an amountof charge Q on an island capacitance C (we will assumea fixed capacitance), where the localized site is situated.The second contribution is due to the binding energy ofthe site, which will depend on the nature of the localisa-tion. This contribution decreases the system energy byan amount E bi for each electron. The binding energy foran isolated phosphorus atom in a silicon lattice is givenby the Bohr model as ≈ r , that varies with r ; so that while most donors are separated by a spac-ing that is close to the average value, a small fractionhave significantly smaller spacings. At these locationsthe binding energy for one electron is increased and forthe second is decreased. The effects of the disorder inthe donor location and the local influence of the trappedcharge causes the binding energy to be different at eachlocalized site (Fig. 2). The third contribution is due tothe electron interactions between the occupied localizedsites. If the spacing between an occupied site pair is r ,then this increases the energy by an amount E i which FIG. 3: Natural log of a device differential conductance atdifferent temperature and electric field. Measurements weretaken at applied source-drain voltages of 0, 0.02, 0.04, 0.06,0.08, 0.1, 0.12 and 0.14V from bottom to top. Solid lines arethe results from a fit using eq. (2). Inset: total energy on theisland as a function of the stored charge for a classical metalCoulomb Blockade device (dotted line) and a highly doped sil-icon device (solid line). The red dots indicate positions corre-sponding to integer charge. Of note is that the solid parabolais segmented and displaced due to the additional contribu-tions from the binding and interaction energies. Some of theinteger charge positions have multiple solutions correspondingto different charge configurations. depends on the electron charge e , on the permittivity ofthe channel material (cid:15) and on r in the following way, E i = e π(cid:15)r (5)The total energy of the system E tot , then depends onthese three contributions E tot = N !2!( N − (cid:88) i =1 E i ( r ) − N (cid:88) i =1 E bi ( r ) + N e C (6)where N is the number of electrons localized at donorsites within the island. For the purposes of assessing theCoulomb blockade charging energy, we are only interestedin the change in the total energy E tot for the addition orsubtraction of one electron.∆ E tot = N E i ( r ) − E bi + e C (7)So that the classical Coulomb Blockade energy is modi-fied by two additional terms. The magnitude of the firstof these additional terms (the interaction energy) will be N times about 3 meV (assuming a separation of abouthalf the typical dot diameter of ≈
40 nm) and the second(binding energy) term will be of the order of 44 meV fromthe previous discussion. Clearly, with about 15 electronson the island these two additional terms are expected tocancel and the classical Coulomb Blockade charging en-ergy depends only on the island capacitance. However,as considerable variation from these average approxima-tions are expected, then the actual charging energy willcontain a correction term to the classical result, which isdue to the difference between the binding energy and in-teraction terms. This difference will vary as the electronnumber varies and for a given electron number will alsovary with configuration i.e. if a different set of donors isoccupied with the same number of electrons then the en-ergy difference is also changed. In addition, the estimatefor the interaction term makes no allowance for screen-ing, which will greatly reduce the influence of chargesat the larger separations. Indeed, the screened Coulombinteraction energy will decay rapidly with distance; wedescribe it as follows: E scr = E i e − k s r (8)where 1 /k s is the Thomas-Fermi screening length. Weestimate 1 /k s ≈ IV. TRANSPORT UNDER MICROWAVEIRRADIATION
In a typical experiment, the SET is biased with a fixedsource-drain voltage and held at a fixed gate voltage,while the source-drain current is measured as the mi-crowave frequency is swept. This must be done withthe SET set up as an impedance matched load (or witha matching impedance very close to the SET) or withthe SET in a low quality factor (Q) cavity, which it-self is matched to the waveguide impedance. In the lat-ter case, the coupling efficiency is very greatly reducedand this approach has been used in this work. A typ-ical result is shown in Fig. 4, where the following fea-tures may be identified; large amplitude, low Q fluctua-tions (main plot), medium amplitude, mid Q fluctuations(right inset), and low amplitude, high Q resonances (leftinset). The high Q features have been extensively stud-ied by Cresswell and coworkers, who have demonstrated
FIG. 4: SET response to AC excitation. The large amplitudepeaks have typically low Q value (few tens). Bias voltages:V DS =+0.7 mV, V G =-2.77 V; radiation power: P=+15 dBm.Left inset: a high Q-value peak on an expanded frequencyscale, with amplitude indicated by the double headed arrow.Q ≈ . Right inset: resonances of different line-shape andmedium Q closely spaced in frequency. Different data-set withrespect to the main plot. Bias voltages: V DS =-1 mV, V G =-1 V; P= 0 dBm. that the resonant frequency is affected by both the gateand the source-drain voltages, although the effect can bevery small. Resonant features have previously been ob-served in two-dimensional electron gas systems (siliconMOSFETs and AlGaAs heterostructures), where this be-haviour has been attributed to single particle excitationwithin the Fermi sea. In this previous work, a magneticfield is used to determine the energy difference betweentwo levels, giving rise to a single resonance, resulting inonly small numbers of resonances due to the requirementsfor a separately identifiable spin active system for eachresonance. By contrast, the degenerately doped siliconSET exhibits large numbers of resonances without theneed to apply a magnetic field. The large number of res-onances (many more than the expected number of freeelectrons in the island) suggests that their origin involvessingle particle excitation, with each able to give rise to anumber of resonances. Furthermore, the photon energycorresponds to a small fraction of the thermal energy atthe measurement temperature of 4K, so that the popu-lations in the two levels would be expected to be verynearly equal (of the order 0.49 and 0.51 for the upperand lower energy levels).The AC conduction behaviour of the hopping model,described earlier to explain the DC characteristics of de-generately doped silicon SETs, has been considered pre-viously.
All of these models consider statistically av-eraged behaviour over a large number of hops in order topredict a value for the conductance. However, in theseSETs individual hops may contribute a significant changein the conductivity due to the small size of the device.The basic underlying mechanism is the change in hop-ping rate between two localized sites, when exposed toelectromagnetic radiation that is resonant with the en-ergy level difference between the sites. This changes theoccupation levels at the two sites away from the valuescorresponding to thermal equilibrium. In an SET withlocalized sites, a change in just two sites may well giverise to a measurable change in the conductance, due tothe very high sensitivity of a tunnel barrier transmissionto wavefunction overlap. Any change in occupancy oflocalized sites close to the edge of a tunnel barrier willhave an exponential effect on the transmission. If thesechanges in occupancy occur at sites further away, thenthe tunnel barrier is affected more indirectly through thepolarisation of intermediate sites, which reduces the over-all change in transmission.Consider two sites separated by a distance that is sig-nificantly larger than the wavefunction overlap, and bya tunnel barrier that reduces the thermally driven tun-nelling rate between the two sites to a very low level, seeleft inset of Fig. 5. Under these circumstances, electro-magnetic radiation that is resonant with the separationbetween these energy levels will cause the electron in theoccupied level to tunnel to and from the unoccupied level,at a rate (the Rabi rate) determined by the intensity ofthe radiation. If the intensity is very high, then the elec-tron will perform these spatial Rabi oscillations, as dis-cussed by Stafford and Wingreen. Such oscillations canproceed coherently until there is an interaction with theenvironment that results in an exchange of energy and/orcauses the electron to tunnel to a different site; such in-teractions determine the lifetime of these oscillations.The changes in site occupancy resulting from these spa-tial Rabi oscillations causes a change in the SET currentby modifying the transmission properties of the tunnelbarriers. This modification is a result of the change inelectrostatic potential resulting from the electron dis-placement in the oscillation. The magnitude of thischange depends on the separation between the sites, thedistances and directions to the tunnel barriers and theRabi rate. The electron displacement due to the Rabioscillation gives rise to a time dependent polarisation,which gives rise to the change in potential at the tunnelbarrier. However, this electric field is screened by the po-larisability of any intermediate structures, such as othertunnelling charges. If the spatial Rabi oscillation is slow,then screening can be very efficient and the affect on thetunnel barrier transmission is correspondingly small. Asthe Rabi rate is increased, the screening is progressivelyovercome and the tunnel barrier transmission is modi-fied.A single electron undergoing a spatial Rabi oscilla-tion will cause a monodirectional resonance (left insetof Fig. 4). However, there is also a natural tendency forsuch systems to couple. The amplitude response of a sys-tem of coupled oscillators is well known and results ina resonance whenever the driving field corresponds to amode frequency. The individual motions of the electronsparticipating in the coupled mode is more complicated than the sum behavior. In particular, the relative ampli-tude for each coupled electron differs on either side of aresonant frequency. In cases where electrons acting inde-pendently would give rise to monodirectional resonancesin opposite directions, the coupled motion would give riseto a differential behavior, as observed in Fig. 4 (right in-set).The width of each resonance directly reflects the life-time of the oscillation, which can vary over a very widerange. Rabi oscillations are usually limited to a max-imum rate that is very much slower than the resonantfrequency of the excitation (say by a factor of 100). Thelowest frequency at which high quality factor resonanceshave been observed in these experiments is of the order1GHz, so that the Rabi rate probably does not exceed afew tens of MHz. In addition, if the Rabi rate is slowerthan the rate at which the electron interacts with theenvironment, then complete oscillations cannot be madeand there is no resulting polarisation to change the SETcurrent.As briefly mentioned above, the energy difference be-tween levels undergoing spatial Rabi oscillations, givenby the resonant frequency, is much smaller than k B T at 4K, so that the Boltzmann function predicts a verysmall difference between the occupancies for these lev-els at thermal equilibrium. The spatial Rabi oscilla-tions, due to continuous wave radiation, can only changethe occupancy levels towards the limiting case of 50%(right inset of Fig. 5), so if these levels were already closeto this figure there would be little change in polarisa-tion between the localized sites and so little change indevice current. However, it is important to point outthat for low dimensional structures at cryogenic tem-perature the electron-phonon interactions are stronglysuppressed. In particular, Tilke and coworkers havefound that for P-doped silicon nano-wires the electron-phonon relaxation time can be as high as few µ s. Single-shot measurements performed on the same device sys-tem investigated here have shown that electron relax-ation time due to microwave excitation has a comparabletimescale. Therefore, we can assume that the thermal-isation processes in the dot occur on a longer timescalethan the photon-induced charge transfer does. This ac-counts for the observation of radiation effects at energymuch lower than k B T .Finally, we turn to explain the effect of a variable ra-diation power on the resonant current. The Rabi rate,Ω R , increases as the square root of the microwave powerunder resonant excitation conditions. The magnitude ofthe device current difference is thus expected to changecorrespondingly. As pointed out earlier, the Rabi ratemust exceed the rate of interaction with environment forany net charge displacement, so that in practice there is athreshold in microwave power that must be exceeded forany resonance to appear. The threshold condition occurswhen T ≈ / Ω R where T is the relaxation time for theresonance. The occupancy of the normally unoccupiedsite increases with microwave power, saturating at 0.5for very high powers. For intermediate power levels, theoccupancy oscillates due to beating effects between theRabi rate and T (see right inset of Fig. 5). The averageoccupation probability of the excited state is given by p = 1 T (cid:90) T p e ( t ) dt (9)where p e ( t ) = sin (Ω R t/
2) is the instantaneous occu-pancy. In order to evaluate the current dependence on ra-diation power (P), we assume that each of the two statescorresponds to currents I g and I e . The instantaneouscurrent is given by the weighted sum of the currents, I ( t ) = I g (1 − p e ( t )) + I e p e ( t ) (10)and the average current is I = 1 T (cid:90) T I ( t ) dt (11)Since we are mainly interested in the current variation,we can set I g =0 which would result in∆ I = I e p = I e T Ω R / − cos ( T Ω R / sin ( T Ω R / T Ω R (12)In order to model the rapid suppression of any excita-tion in case T (cid:28) / Ω R , we introduce an exponentialprefactor as follows ∆ I = 11 + e − α ( T Ω R − k ) (13) · I e T Ω R / − cos ( T Ω R / sin ( T Ω R / T Ω R with α , k fitting constants. As stated earlier, the Rabirate is proportional to the square root of the radiationpower and can be expressed asΩ R = (cid:112) cP + δ (14)being c the constant of proportionality and δ the detun-ing factor. Fig. 5 shows a comparison of the modelledexcess current as a function of the radiation power withexperimental data taken from a resonance at 1.9200 GHzand Q ≈ V. CONCLUSION
The very high doping density does not ensure metallicbehavior in the tunnel barrier and the island of a sili-con SET. Increasing the doping density cannot overcome the sidewalls depletion causing the non-metallic behav-ior due to a consequential increase in sidewall trap sites.Transport through the tunnel barriers and island is domi-nated by hopping through a limited number of sites. Re-arrangement of a fixed number of electrons between alarger number of sites is seen to lead to aperiodic gateoscillations and varying peak heights. Individual trans-fers can be promoted by resonant microwave excitation,resulting in a spatial Rabi oscillation. Such transfersare detected through their effect on the DC transport.The equilibrium population and lifetime of these statesis strongly influenced by the energetics of localized elec-trons. These effects result in each excitation correspond-ing to a different energy, in contrast to Zeeman splitting.These differences could be used to separately address ex-citations/electrons without the need for additional gates,greatly simplifying the problem of realising a multi-qubitquantum computer. Indeed, each radiation-induced res-onance could potentially be considered as a qubit, char-acterised by the resonant feature shape, frequency andwidth. The quantum system initial state would be de-fined by the set(s) of occupied donor sites that minimisesthe free energy in the system. Manipulation and gateoperation could be based on well established NMR pro-tocols.
FIG. 5: Dotted line: experimental current dependenceat resonance (Frequency=1.9200 GHz) for varying radiationpower. ∆ I DS is the excess current with respect to DC con-dition. Bias conditions: V G =-2.0 V, V DS =-1.30 mV. Solidline: calculated current using eq.(13) with I e =9.65 × − A, c =7.33 × rad /s mW , δ =4.96 × rad/s, k =4.98, α =104.04, T =10 − s. Left inset: schematic energy level diagram show-ing two donor locations and the occupied (filled circle) andunoccupied (empty circle) energy levels. Right inset: averageprobability of occupation of an excited state under resonantexcitation against the ratio between the relaxation time andthe Rabi period (eq.(9)). Acknowledgments
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