Quantum-dot thermometry ∗ E.A. Hoffmann, N. Nakpathomkun, A.I. Persson, and H. Linke Physics Dept. and Materials Science Institute, University of Oregon, Eugene, OR 97403-1274, USA H.A. Nilsson and L. Samuelson Solid State Physics/The Nanometer Structure Consortium, Lund University, Box 118, S-221 00, Lund, Sweden We present a method for the measurement of a temperature differential across a single quantum 8 dot that has transmission resonances that are separated in energy by much more than thethermal 0 energy. Wedeterminenumericallythatthemethodisaccuratetowithinafewpercentacrossawide 0 range of parameters. The proposed method measures the temperature of the electrons that enter 2 thequantumdotandwill beusefulinexperimentsthataim totest theorywhichpredictsquantum n dotsare highly efficient thermoelectrics. a J 1 In the ongoingdevelopmentof effective thermoelectric be written [17, 18] 1 materials and devices, low-dimensional systems are of ∞ particularly great interest, because to optimize the per- I = 2e [f (ε)−f (ε)]τ(ε)dε, (1) ] s d i formance of a thermoelectric, it is crucial to control the h −∞ c energy spectrum of mobile electrons [1, 2, 3, 4, 5]. De- Z -s vicesforhigh-efficiencythermal-to-electricpowerconver- where fs−,d1 = eξs,d +1 are the Fermi-Dirac distributions l sion based on quantum dots defined by double barriers in the nanowire’s source and drain leads, respectively, r mt embedded in nanowires have been proposed [6]. Such andtheirargumentsareξs,d =(ε−µs,d∓eV/2)/kBTs,d. systems have great advantages, because they select the We assume the bias voltage, V, is applied symmetrically t. energies at which electrons are transmitted [7, 8], and across the dot. For the case of a quantum dot or single- a because nanowires can be contacted in highly ordered electron transistor (SET) with well-separated transmis- m arrays[9] with the potentialfor large-scaleparalleloper- sion maxima as a function of gate voltage, Eq. (1) - ation. predicts the characteristic Coulomb blockade diamonds d n which appear in the differential conductance, G, as a o function of bias voltage and gate voltage. In order to measure quantitatively the dependence c of thermopower and energy-conversion efficiency on the [ I transmission spectrum of a quantum dot, it is necessary H 2 to apply and determine accurately a temperature differ- v entialacrossthedot. Traditionallyforthethermoelectric 0 s d characterization of mesoscopic devices such as quantum 5 V 2 point contacts [10], quantum dots in 2DEG’s [11], car- .1 bheoantinnagncouturrbeenst[g1e2n,e1r3a,t1es4]a,atenmdpnearnaotwurireedsi[ff1e5r,e1n6ti]a,latnhaact (cid:39)TH (cid:39)Ts 0 (cid:39)T 1 ismeasuredinseparatecalibrationexperiments. Herewe d T 7 propose a technique that measures the actual electronic 0 0 temperature differential across a quantum dot and does : not require separate calibration. The basic concept is as v FIG. 1: The heating setup and the temperature landscape. Xi froelslpoownss:etthoeacnhaanpgpelieind hcuearrteinngt vaocrltoasgsea, Vqu,ainstummeadsuorteidn. Thesourcecontactiswarmedwithavoltage-balancedheating H current and can be biased for thermocurrent measurements. r This signalcontainsinformationaboutthe electrontem- a Electron transport through the quantum dot is determined peraturesatthesourceanddrain,butitalsodepends on bythelocaltemperaturesofthesourceanddrainsidesofthe the dot’s energy-dependent transmission function, τ(ε). dot, Ts,d =∆Ts,d+T0. (Color online) However,onecanobtainthenecessaryinformationabout τ(ε) from conductance measurements. Together, these In a typical experiment, an ac heating current is used twomeasurementsallowonetodeterminethesourceand to modulate the temperature T of an ohmic contact H drain temperatures separately. at one end of a nanowire (taken here to be the source contact, Fig. 1)[19] with amplitude ∆T with respect H The two-terminal current through a quantum dot can to the unperturbed device temperature, T0. We are in- terested in the associated electronic temperature rises, ∆T =T −T ,inthe immediate vicinity ofthe quan- s,d s,d 0 tum dot (see Fig. 1). In the case of strong electron- ∗Electronicaddress: [email protected] phononinteraction(forexamplenearroomtemperature) 2 the electronic temperature will drop linearly along the 15 300 ncwpoaehmcnetorpewedaireretleeod,ctbatroneodnwt-h∆peeahTkows,ni∼=o∆renT.∆iHnATt>edtrail∆focwtTtihosetne>mqinup∆atenhTrteadutnmu>arned0soo,,wtahirinosewdisesh∆veoeTxrrts-, -100 7.5 (cid:39)Ts 20% Peak (cid:39)Td 61120284000T (mK)(cid:39)s,d and ∆Td need to be measured. ) 1 0 0 K Assuming an ac heating voltage, VH = V0cos(ωt), the A/ temperaturerisesonthesourceanddrainsidesofthedot ( canbe written ∆Ts,d =βs,dVHγ , where βs,d are unknown -7.5(cid:80)s (cid:80)d (cid:80)s (cid:80)d constantsandγ cantakeonvariousvaluesdepending on the type and strength of electron-phonon interaction in -15 -5 -2.5 0 2.5 5 the heating wire [20]. Here we assume a short heating BiasVoltage (mV) wire and Joule heating, and therefore γ = 2 [20]. In this regime,byanapplicationofthechainrule,therms- amplitude of the ac temperature rises can be written FIG. 2: A plot of ∂I/∂TH (red) and (2kB/e)2∆TH∂2I/∂V2 −1 (blue)asafunctionofbiasvoltageatafixedgatevoltage,as ∂T ∂I ∂I ∆Ts,d ∼=V02∂(Vs,Hd2) =V02(cid:18)∂Ts,d(cid:19) ∂(VH2). (2) aintduirceatreisdeb(ygrteheen)hoisriczaolnctualladteadshveidalEinqe.i(n7F).igI.n3se.tTs:hTehteemppoesri-- tion of theresonant tunnelingenergy of thedot with respect In an experiment, one can measure ∂I/∂ VH2 , the to the electrochemical potentials in the leads. In this model, frequency-doubled response to the ac heating voltage. weusedatransmissionfunctionconsistingofLorentzianswith (cid:0) (cid:1) The differential thermocurrent, a FWHM of Γ = 0.5 meV equally spaced by 5 meV and ∞ ∆TH = 300 mK, ∆Ts = 0.8∆TH = 240 mK, ∆Td = 0.2∆TH ∂I 2e ∂f ξ = ∓ s,d s,d τ(ε)dε, (3) =60 mK, and T0 =230 mK.(Color online) ∂Ts,d h Z−∞(cid:18) ∂ξs,d Ts,d(cid:19) cannot be measured directly. However, we will show 39 300mK thatitcanbe obtainedingoodapproximationfromcon- ductance measurements. 38 (cid:39)T (cid:39)T Under bias conditions, where the source (drain) elec- s d trochemical potential is near a well-defined transmission V) 37 resonance of the quantum dot, while the drain (source) m ( is severalkBT awayfromthe next resonance,the second ge 36 derivative of the current is a lt 35 ∂2I e2 1 2e ∞ ∂f 2f −1 Vo ∼= ± s,d s,d τ(ε)dε. ∂V2 4kB2 Ts,d h Z−∞(cid:18) ∂ξs,d Ts,d (cid:19) (4) Gate 34 (cid:39)Td (cid:39)Ts 33 A key observation is that the integrands in Eq. (3) and Eq. ( 4) are qualitatively very similar: 0mK 32 ∂f 2f −1 1∂f ξ -7 -3.5 0 3.5 7 s,d s,d ∼=− s,d s,d. (5) BiasVoltage (mV) ∂ξ T 2∂ξ T s,d s,d s,d s,d This approximation holds for all ξ , because 2f −1 s,d s,d limits to −ξ /2 when ξ is small, and ∂f /∂ξ goes FIG.3: Thecalculated temperatureriseasafunction ofbias s,d s,d s,d s,d to zero in all other cases. With this approximation, we andgatevoltagesforasingleCoulombblockadediamond. In theregionsindicated,eitherthesourceordrainelectrochemi- can combine Eqs. (3 ) and (4): calpotential,butnotboth,iswithinafewkBT ofaresonant ∂I e2 1 −1 ∂2I energy of the dot. Our assumptions are fulfilled in these re- ∼ = Λ , (6) gions,andthesimulationproducestemperatureplateauspre- ∂T 4k2 T ∂V2 s,d (cid:18) B s,d(cid:19) dicting the correct ∆Ts,d. Fig. 2 shows a slice at 36.75 mV gate voltage, as indicated by thedashed line. (Color online) where Λis aunitless scaling factorintroducedduringin- tegration. In this way, all the information about τ(ε) needed to determine ∂I/∂T is accounted for by mea- s,d suring ∂2I/∂V2. whichshowsthatapproximationsof∆T and∆T canbe s d Substituting Eq.(6)intoEq.(2)andsolvingfor∆T obtained from measurement of ∂2I/∂V2 and ∂I/∂ V2 s,d H yields our final result, and knowledge of T . 0 (cid:0) (cid:1) To illustrate the qualitative similarity of Eqs. (3) and ∆T = 1 T2+Λe2V2 ∂2I −1 ∂I − T0, (7) (4), we show numerical calculations of the two in Fig. 2 s,d 2s 0 k2 0 ∂V2 ∂(V2) 2 taken at the gate voltage indicated by the horizontal B (cid:18) (cid:19) H 3 3.0 (cid:39)T dashed line), and a slice through that diamond is shown 1 (cid:39)Ts = 240 mK s asgreensymbolsinFig.2. Inregionsalongthediamond 2.5 ror (cid:39)Td = 60 mK (cid:39)Td ridges(circledareasinFig.3)—whereone,andonlyone, Er0 of the two electrochemical potentials in the source or % ) 2.0 -1 drain is within a few kBT of a transmission resonance— K ( 0.2 0.6 1.0 Eq. (7) yields consistent values in accordance with the T s,d1.5 (cid:42) (meV) (cid:42)T == 02.350 m meKV aresgsuiomnps,tiEoqn.s(t4h)aitsanlolotwvaulisdt,onowtrietveenEqa.p(p4r)o.xiImnaatlellyo,thbeer- (cid:39) 1.0 0 cause it only accounts for one Fermi-Dirac distribution. 0.5 The use of this method requires knowledge of the ap- 0.0 propriate scaling factor Λ, defined in Eq. 6, which needs 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 to be determined numerically. For the particular mod- (cid:39)T (K) eling parameters used here (see caption of Fig. 2), we H found Λ = 0.304 by averaging Eq. (7) over the voltage range from the peak value of ∂I/∂T to 20% of its peak H FIG. 4: The calculated temperature rises as a function of value, where the signal-to-noise ratio in an experiment ∆TH. The calculated values agree well with the expected should be largest. Note that for different parameters, values (solid lines) up to nearly 10 T0. Inset: The percent Λ will differ, but it is insensitive to typical experimen- error as a function of Γ, thefull width at half max (FWHM) tal variations in Γ and ∆T . For example, in Fig. 4, of the transmission function, τ(ε) in Eq. (1). The error is H we show that the use of the same Λ = 0.304 (calculated within 1% over an order of magnitude in Γ. Here ∆Ts is 240 for Γ = 0.5 meV) yields errors in ∆T and ∆T of only mK and ∆Td is 60 mK. (Color online) s d 1% when Γ is varied over nearly an order of magnitude aroundΓ=0.5meV(insetofFig.4)andonlyafew per- dashedlineinFig.3at36.75mV.TheleftinsetofFig.2 centfor∆TH uptoalmost10T0. Toputthis smallerror illustrates that, in this example, when the bias voltage intocontext,notethatthe localtemperaturesTs andTd is negative, the source temperature is the only temper- can be defined only over a distance of about an inelastic ature affecting the current through the dot; therefore, scattering length, such that an accuracy of less than a ∂I/∂T = ∂I/∂T in this bias configuration. In the op- few percent is not necessarily physically meaningful. H s posite configuration,∂I/∂T =∂I/∂T , as shown in the This research was supported by ONR, ONR Global, H d right inset of Fig. 2. Fig. 3 shows Eq. (7) as calculated the Swedish Research Council (VR), the Foundation for from modeled data of Eqs. (3) and (4) across an en- StrategicResearch(SSF),theKnutandAliceWallenberg tire Coulomb blockade diamond (indicated by a white, Foundation, and an NSF-IGERT Fellowship. [1] L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B 47, [11] L. W. Molenkamp, A. A. M. Staring, B. W. 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