Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Alphenaar, B.W.

Beenakker, C.W.J.

Houten, H. van

Molenkamp, L.W.

Staring, A.A.M.

Leiden University Scholary Publications

Semicond Sei Technol 9 (1994) 903 906 Prmted in the UKINVITED PAPERSawtooth-like thermopower oscillationsof a quantum dot in the Coulombblockade regimeL W Molenkamp [, A A M Staringt, B W Alphenaar! i,H van Houten l and C W J Beenakker§1 Philips Research Labs 5600 JA Eindhoven The NetherlandsijInstituut-Lorentz Umversity of Leiden 2300 RA Leiden The NetherlandsAbstract Current heatmg is used to measure the thermopower of a quantum dotm the Coulomb blockade regime We observe sawtooth-like oscillations äs afunction of gate voltage in the thermovoltage across the dot These observationsare compared with measured Coulomb blockade oscillations m the conductanceand with theoryIn the past few years our understandmg of transportphenomena in semiconductor nanostiuctuies has m-creased considerably Most research has focused onpurely electncal pioperties, and we refer the reader to[1] (quantum balhsüc and quantum diffusive transport)and [2] (transport in the Coulomb blockade regime) foran overview of the transport properties encountered mthese devices In addition, expenments have started ontheimal and thermoelectiic quantum transport phenom-ena in semiconductor nanostructures, äs documented msome recent reviews [3, 4] In this paper we present someof our recent results [5, 6] on electnc and thermoelectnctransport phenomena of a quantum dot m the Coulombblockade, or smgle-electron tunnelling, regimeSingle-electron tunnelling is the dominant mechamsmgoverning the transport properties of a quantum dot thatis weakly coupled to reseivoirs by tunnel barners Attemperatures T such that kB T« e2/C, with C thecapacitance of the dot, it leads to novel transportphenomena, such äs the 'Coulomb blockade oscillations'm the conductance m the linear, and the 'Coulombstancase' m the nonlinear transport regime In thermo-electnc transport, the Coulomb blockade should lead to[7] sawtooth-like oscillations in the thermopower S(S = Al^htrmo/Ar, where ^hcrmo is the thermovoltagemduced by a temperature difference ΔΓ across the dot)äs a function of the Feimi energy m the reservoirsExamples of all of these phenomena follow belowThe samples used for the expenments are definedelectrostatically in the two-dimensional electron gas| Piesent address Hitachi Cambndgc Labs Cavendish LaboratoryCambridge CB3 OHE UKFigure 1 Schematic top view of the 0 7 x 0 8 pm2 quantumdot adjacent to a 2 μιη wide 20 μηι long channel Gates AD and F (hatched) defme mdividually adjustable tunnelbarners and gate E controls the electrostatic potential ofthe dot the gaps between gates D and E and betweengates E and F are pmched off in the expenment For thethermovoltage expenment an AC heatmg current / ispassed through the channel and the thermovoltageVth = νλ — \/2 is measured across the dot and the oppositereference pomt contact defined by gates B and C(2DEG) of (Al,Ga)As heterostructures We use electron-beam lithography to fabncate Ti-Au gates of dimensionsdown to 0 l μηι The (AI, Ga)As wafer used here has anelectron density ns « 3 7 χ 101 1 cm~ 2 and a mobilityμ κ l O6 cm2 V " 1 s"1 The layout of the patterned Ti Augates is shown m figure l Gates A, D and F define twoadjustable tunnel barners, and two additional gates, Band C, define a narrow channel A pomt contact m theboundary of this channel, opposite to the dot, is used äsa reference voltage probe m the thermopower measure-0268 1242/94/050903 + 04 $19 50 © 1994 IOP Publishing Ltd 903L W Molenkamp et al-05 -04 -03 -02 -01Ο Ο-1-05-1 -09 -08 -07 -06Gate voltage (V)Figure 2. Electrical conductance of the quantum dot m thelinear regime, äs a function of the voltage applied to gateE The excitation voltage was 9 μΥ.ment (see below). The sample is immersed in liquidhelium in the mixing chamber of a dilution refrigeratorat a temperature of 45 mK and at zero magnetic field.The signals are measured using low-frequency lock-intechniques.Perhaps the most striking manifestation of theCoulomb blockade on the transport properties of aquantum dot is the occurrence of a periodic series ofpeaks in the conductance of the dot ('Coulomb blockadeoscillations') äs a function of the electrochemical potentialof the dot, which in our case can be varied by changingthe voltage on gate E. The peaks in the conductanceoccur for those gate voltages where the free energy of adot containing N electrons equals that of a dot containingN — l electrons. As an example, we show in figure 2 theCoulomb blockade oscillations observed for the dot usedin the thermopower experiments. One observes a longseries of peaks in the conductance, with consecutive peaks(for less negative gate voltages) corresponding to theaddition of a single electron to the dot. At gatevoltages > —0.3 V, the electron gas underneath gate Eis not fully depleted, and we tentatively attribute theirregulär structure in the conductance trace to Fabry-Perot-type transmission resonances [8]. In the scanshown here, the entrance and exit tunnel barriers areboth adjusted to a conductance of about 0.5e2/h; suchrelatively low barriers may enable (higher-order) co-tunnelling processes [9]. This explains, at least partly,the presence of a remanent conductance in the Coulombblockade minima [5].In the nonlinear regime, the Coulomb blockade canbe suppressed by applying a bias voltage Vb across thedot so that eVb > e2/C. For a dot with asymmetricallyadjusted tunnel barriers at entrance and exit this leadsto a second transport peculiarity, namely the Coulombstaircase. An example of this type of l-V curve is givenin figure 3, which was obtained for entrance and exittunnel barrier conductance of Q.15e2/h and 0.02e2//j,respectively. At each step in this curve, an extra electronis added to the dot.From the theory of Coulomb blockade [10] it followsthat the electrochemical potential of the dot in the050025c ooo-ω(3-025-050<s>-̂c25-5 -25 0Voltage (mV)Figure 3. Coulomb staircase obtained at 45 mK for astrongly asymmetrically adjusted quantum dot The füllcurve gives the measured current, the dotted curve thedifferential conductance, äs a function of the applied biasvoltagesingle-electron tunnelling regime varies in a sawtoothfashion with the voltage on gate E. It has proved difficultto directly observe this sawtooth behaviour (however,see [11] for a recent experiment). In a recent paper,Beenakker and Staring [7] showed theoretically that oneexpects this sawtooth behaviour to be directly observablein the thermopower of the quantum dot. Therefore wedesigned an experiment to measure the thermopower ofa quantum dot. It is at this point that the narrow channel,defined by gates B and D in our structure, becomesrelevant: we use it to create a hot-electron reservoir. Thisis possible by virtue of the fact that in the 2DEG in an(AI, Ga)As heterojunction structure at low temperatures,the coupling between hot electrons and the lattice is muchsmaller (typical relaxation time < l ns) than the couplingwithin the electron System (~ps). Thus, by passing acurrent through a suitably dimensioned channel areservoir of hot electrons is created. Using this technique,we have previously been able to observe the quantumsize effects in the thermopower [12], Peltier coefficientand thermal conductance [13] of a quantum pointcontact.In order to observe the thermopower of our quantumdot [6], we use the sample of figure l, with the tunnelbarriers defined by gates B, C, A and D, adjusted toconductances of about OAe2/h each, and current heatingprovided by a small AC current passing through thechannel defined by gates B, C, A and D. The currentheating leads to a small difference in electron temperature(ΔΓ<χ/ 2) across the dot and across the oppositereference point contact (defined by gates B and C).Lock-in detection at twice the AC frequency is then usedto measure a transverse thermovoltage Vlh = Υγ — V2,which equals the difference in thermovoltages across thedot and the reference point contact, äsKh = (Sdol - (1)Here Sdol is the thermopower of the dot and Srcf is thethermopower of the reference point contact. The contri-bution of Srcf to Vth is independent of VE and leads to a904Thermopower oscillations of a quantum dot010-5715 735 775755EF / (e2/2C)Figure 4 (a) Thermovoltage l/th at a heatmg current of58 nA (füll curve) and conductance (broken curve) äs afunction of gate voltage l/E at a lattice temperature of7" = 45 mK (b) Calculated thermopower (füll curve) andconductance (broken curve) of a quantum dot äs a functionof Fermi energy using the theory of [7] The parametersused in the calculations are discussed m the textconstant offsct voltage, which is mimmized m ourexpenment by suitably adjusting the reference pomtcontact [12] Thus, vanations m Vlh äs a function of VEdirectly reflect changes m the thermopower of the dotIn figure 4 we compare measurements of the Coulombblockade oscillations äs a function of VE in the thermo-voltage (füll curve) and conductance (broken curve,obtamed from a separate medsurement) of the dot, at alattice temperature of T = 45 mK The heatmg currentused in the thermovoltage expenment was 58 nA Clearly,the thermovoltage V2 — V\ (and therefore the thermo-power of the dot) oscillates penodically The penod isequal to that of the conductance oscillations, and thuscorresponds to depopulation of the dot by a smgleelectron As expected, the thermovoltage oscillations havea distmct sci\\ tooth lineshape In addition, the conductancepeaks are appioximately centred on the positive slope ofthe thermovoltage oscillations, with the steeper negativeslope occurnng in between two conductance peaks Thesedata compnse a clear expenmental demonstration of thekey charactenstics of the thermopower oscillations of aquantum dotThe theoietical curve of the thermopower in figure4(b) was calculdted using the linear response formahsmof [10] In oidci to obtain the excellent agreement mboth conductance and thermopower behaviour, we had-0955 -0950 -0945VE (V)-0940 -0935Figure 5 Thermovoltage l/th äs a function of VE at latticetemperatures of T = 45 200 and 313 mK obtamed using aheatmg current of 18 nAto perform the calculations for an electron temperaturem the dot of 230 mK, i e higher than the actual latticetemperature m the expenment We encountered thisproblem in several different expenments on transport inquantum dots [5], and tentatively attnbute the effect tod finite amount of hfetime broademng of the energy levelsin the dot However, we cannot rule out a certam amountof electron heatmg due to RF pick-upIn figure 5 we show the behaviour of the thermovoltageoscillations for three different lattice temperatures TlM(Τ]ΛΗ = 45, 200 and 315 mK, respectively), obtamed fora heatmg current of 16 nA One observes that sawtoothlineshape becomes more Symmetrie for higher latticetemperatures, owmg to thermal smearmg For a quanti-tative companson of the magmtude of the observedthermovoltage with the theoretical peak-to-peak valueAVih κ (e/2C7^aU)Ar, one needs to know the self-capacitance C of the dot, and the increase m electrontemperature A7"m our expenment From the temperaturedependence of the conductance oscillations (not shownhere) we find that e2/C χ 0 3 meV From the 200 mKtrace, we mfer that this value for the self-capacitanceimplies that Α Γ α l mK for / = 18 nA A convementmanner of independently determming A J i s by makinguse of the quantized thermopower of the reference pomtcontact BC (cf [12, 13]) However, this techmque provednot to be viable for the present expenment, äs theconductance of pomt contact BC does not exhibit welldefined plateaus (probably due to quantum interferenceeffects) at temperatures below about l K Still anothermanner for estimatmg Δ Γ is by using the crude heatbalance introduced m [12] for the current heatmgprocess W e h a v e c v A T = ( / / Η/ ) 2 ρ τ , 0 5 ί , w i t hcv = (π2/3)(/οΒ r/£F) n/cB the heat capacity per umt areaof the 2DEG, W the channel width, p the channel resistivityand T loss an energy relaxation time [12] SubstitutingΔΓ= l mK, we obtain T,OSS = 2 χ 10~1 0 s This is areasonable number, and consistent with our earlierexpenments on the thermopower of a quantum pomtcontact defined m similar 2DEG material [12]In conclusion, we have presented data on thermaland thermoelectnc transport properties of a quantum905L W Molenkamp et aldot m the Coulomb blockade regime In contrast withthe penodic peak structure ('Coulomb-blockade oscilla-tions') of the conductance with varymg gate voltage, thethermopower of the dot oscillates m a sawtooth mannerFor both effects, the penod corresponds to a penod ofone oscillation per electron added to the dotAcknowledgmentsWe would like to thank O J A Buyk and M A AMabesoone for their expert technical assistance Theheterostructures were grown by C T Foxon at the PhilipsResearch Laboratories m Redhill (Surrey, UK)References[1] Beenakker C W J and van Houten H 1991 Solid StatePhys 44 l[2] van Houten H, Beenakker C W J and Starmg A A M1992 Smgle Chaige Tunnellmg (NATO ASI SenesB vol 294) ed H Grabert and M H Devoret(New York Plenum) pp 167-216[3] Molenkamp L W, van Houten H and BeenakkerC W J 1993 Proc CXVIl-th Cowse Int School ofPhysics 'Ennco Fermi' Semiconductor Supei latticesand Interfaces ed A Stella and L Migho(Amsterdam North-Holland) pp 365 78[4] Gallagher B L and Butcher P N 1992 Handbook anSemiconductots ed T Moss vol l ed P T Landsberg(Amsterdam Eisevier) p 721[5] Starmg A A M 1993 PhD Thesis Eindhoven Umversityof Technology[6] Starmg A A M, Molenkamp L W, Alphenaar B W, vanHouten H, Buyk O J A, Mabesoone M A A,Beenakker C W J and Foxon C T 1993 EurophysLeu 22 57[7] Beenakker C W J and Starmg A A M 1992 Phys RevB 46 9667[8] Smith C G, Pepper M, Frost J E F, Hasko D G,Newbury R, Peacock D C, Ritchie D A and JonesG A C 1991 J Phys Condens Mattet l 9035[9] Averin D V and Nazarov Yu V 1992 Smgle ChmgeTunnellmg (NATO ASI Senes B vol 294) ed HGrabert and M H Devoret (New York Plenum)pp 217-47[10] Beenakker C W J 1991 Phys Rev B 44 1646[11] Field M, Smith C G, Pepper M, Ritchie D A, Frost J EF, Jones G A C and Hasko D G 1993 Phys RevLeu 70 1311[12] Molenkamp L W, van Houten H, Beenakker C W J,Eppenga R and Foxon C T 1990 Phys Rev Leu65 1052[13] Molenkamp L W, Gravier Th, van Houten H, Buyk OJ A, Mabesoone M A A and Foxon C T 1992Phys Rev Leu 68 3765906

Sawtooth-like thermopower oscillations of a quantum dot in the Coulomb blockade regime

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Sawtooth-like thermopower oscillations of a quantum dot in the Coulomb blockade regime

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