Measuring Temperature Gradients overNanometer Length ScalesEric A. Hoffmann,*,† Henrik A. Nilsson,‡ Jason E. Matthews,†Natthapon Nakpathomkun,† Ann I. Persson,† Lars Samuelson,‡and Heiner Linke*,†

Physics Department and Materials Science Institute, UniVersity of Oregon, Eugene,Oregon 97403-1274, and Solid State Physics/The Nanometer Structure Consortium,Lund UniVersity, Box 118, S-221 00, Lund, Sweden

Received November 11, 2008

ABSTRACT

When a quantum dot is subjected to a thermal gradient, the temperature of electrons entering the dot can be determined from the dot’sthermocurrent if the conductance spectrum and background temperature are known. We demonstrate this technique by measuring the temperaturedifference across a 15 nm quantum dot embedded in a nanowire. This technique can be used when the dot’s energy states are separated bymany kT and will enable future quantitative investigations of electron-phonon interaction, nonlinear thermoelectric effects, and the efficiencyof thermoelectric energy conversion in quantum dots.

Quantum and finite-size effects modify the thermal propertiesof electronic mesoscopic devices. Examples of novel thermaland thermoelectric phenomena demonstrated by mesoscopicdevices include the quantization of the electronic thermalconductance,1 energy-modulated thermovoltage2 in quantumpoint contacts (QPCs), and nonlinear phenomena, such asthermal diode behavior in quantum dots.3 Applications ofmesoscopic thermal effects include the use of low-dimen-sional systems with sharp features in the electronic densityof states for efficient thermoelectric energy conversion.4 Forexample, quantum dots5,6 and nanowires7,8 are being exploredfor use as efficient thermoelectric devices.

A challenge when performing a low-temperature, quantita-tive, mesoscopic thermal experiment is the measurement ofthe temperature difference across the small device underconditions where electronic and lattice temperatures can bevery different. In the field of mesoscopic thermoelectrics,such thermometry is often accomplished using the thermo-voltage of a QPC to measure temperature differences withina two-dimensional electron gas.1,2,9,10 Thermal experimentsusing micrometer-length nanotubes and nanowires oftenexploit the temperature dependence of local resistors todetermine the lattice temperature difference across thedevice.8,11-15

Recently, we proposed a novel thermometry technique,16

which uses a two-terminal quantum dot to measure separatelythe electron gas temperature on the source and drain sides

of a quantum dot that has been placed in a temperaturegradient. Crucially, this technique measures the temperatureof precisely those electrons which enter the dot, rather thanthe temperature of separate electrons (or phonons) in thevicinity. This can be a significant advantage, for example,in the study of thermal phenomena in quantum dots thatdepend on electronic temperature, such as inelastic processesor thermoelectric phenomena. In such experiments, the useof quantum-dot thermometry significantly simplifies devicelayout and fabrication, because the same quantum dot is usedfor both experiment and thermometry. An example applica-tion of this local thermometry technique is to embed aquantum dot into a nanowire in order to answer fundamentalquestions about the nature of electron-phonon (e-ph)interaction and its role in heat flow in nanowire-baseddevices.

Here we present the experimental demonstration ofquantum-dot thermometry. Specifically, we use a quantumdot defined by a double barrier within a nanowire that isplaced on an electrically insulating substrate and to whichmetal Ohmic contacts are attached (see Figure 1a). Duringthermometry measurements, a temperature difference andbias voltage, V, are applied along the nanowire and acrossthe quantum dot. A heating current, IH, heats the electrongas in the metallic source contact (sc) while the electron gasin the metallic drain contact (dc) is expected to stay at ornear the background cryostat temperature, T0, creating atemperature difference over the length of the nanowire(Figure 2). In the following, we discuss how we measurethe temperature rises, ∆Ts,d, above T0 experienced by the

* Corresponding authors, ehoffma1@uoregon.edu and linke@uoregon.edu.† University of Oregon.‡ Lund University.

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10.1021/nl8034042 CCC: $40.75 2009 American Chemical SocietyPublished on Web 01/21/2009

electron gas inside the nanowire near the source and drainsides of the quantum dot, respectively (see Figure 2).

Our method makes use of the fact that, when the nanowiredrain is electrically grounded, the applied temperaturedifference, ∆Ts - ∆Td, causes a net electron charge current,Ith, known as the thermocurrent. Under appropriate gate andbias conditions (see Figure 1b), Ith depends only on thetemperature of the electrons entering the dot, and on the dot’senergy-dependent transmission function, τ(ε). Transmissioninformation can be obtained from conductance spectroscopy;specifically, the second differential conductance, ∂2I/∂V2,provides the necessary information about the dot’s transportproperties, and a mathematical comparison of Ith to ∂2I/∂V2

determines ∆Ts,d. The original theoretical proposal16 for thismethod considered the regime in which the width of the dot’stransmission resonances, Γ, is larger than the thermal energykT and where, for a given Γ, a numerically determinedintegration factor is needed to determine ∆Ts,d.

Here we report an experiment in the regime where Γ ,kT. In this limit, ∆Ts,d can be determined directly from theratio Ith/(∂2I/∂V2), without numerical calibration. To demon-strate this, we begin with the Landauer equation for the two-terminal current through a one-dimensional constriction17

I) 2eh ∫-∞

∞[fs(ε, V)- fd(ε, V)]τ(ε) dε (1)

where fs,d-1 ) es,d + 1 are the Fermi-Dirac distributions in

the source and drain, respectively. We assume that the biasvoltage, V, drops symmetrically by V/2 across each barrierso that s,d ) (ε - µ ( eV/2)/kTs,d, where µ is controlled bythe gate voltage. In the following, we assume that transmis-sion resonances are separated by many kT and that Γ , kT.We consider a single resonance, which is centered at energyε0 and is so sharp that τ(ε) limits to a Dirac delta function,τ(ε) ) Aδ(ε - ε0), where A is an unknown constant. Insertingthis expression for τ(ε) into eq 1 and integrating gives

I)A2eh

[fs(ε0, V)- fd(ε0, V)] (2)

In general, current through the nanowire depends on bothTs and Td. However, using the gate and bias voltages, onecan align either the source or drain electrochemical potentialwith the resonance at ε0, while keeping the oppositeelectrochemical potential several kT from any resonance (seeFigure 1b and insets of Figure 3a). In such a configuration,only one Fermi-Dirac distribution in eq 2 is finite near ε0

and current through the dot depends on only one temperature:Ts or Td. Therefore, under these single-temperature biasconditions, the thermocurrent, Ith, can be written

Ith|s,d )∆Ts,d∂I

∂Ts,d

where the subscripts denote which respective electrochemicalpotential, source or drain, is aligned with the resonance. Afterdifferentiating eq 2 with respect to Ts,d, Ith becomes

Ith|s,d )∆Ts,dA2eh [- ∂fs,d

∂s,d

s,d

Ts,d] (3)

This alone is not sufficient to determine ∆Ts,d because Ais unknown. However, the second differential conductance,∂2I/∂V2, provides additional information and resolves thisissue. Under the same single-temperature bias conditions,and again using eq 2, the second differential conductance is

∂2I

∂V2|s,d )e2

4k2

1Ts,d

A2eh [( ∂fs,d

∂s,d

2fs,d - 1

Ts,d] (4)

The bracketed parts of eqs 3 and 4 have terms in common.This is the primary reason for comparing Ith to ∂2I/∂V2 ratherthan the (conventional) differential conductance, ∂I/∂V. Inparticular, we consider the ratio

R ≡ Ith ⁄∂

2I

∂V2(5)

To approximate R under the single-temperature bias condi-tions, we define Rs,d as the ratio of eq 3 over eq 4

R ≈ Rs,d ≡ Ith ⁄∂

2I

∂V2|s,d)-∆Ts,d

4k2

e2Ts,d

s,d

2fs,d - 1

and A conveniently drops out. By expressing this ratio in

Figure 1. (a) An SEM image of the InAs nanowire with sourceand drain contacts. The indicated InP/InAs/InP quantum dot (QD)embedded in the nanowire is not resolved by the SEM. The voltage,V, biases the nanowire electrically, and the heating current, IH, biasesthe nanowire thermally. The voltage probe (p) assists in tuning theheating voltages, V( (see text for details). (b) Differential conduc-tance as a function of bias voltage and gate energy with brightercolor indicating larger differential conductance. The dot is in thefew-electron Coulomb-blockade regime and, in the center Coulombdiamond, is filled with an odd number of electrons. The averagecharging energy of the dot is 12.9 meV. The (blue) horizontal linesindicate regions where the assumptions leading to eq 6 are satisfiedand at which gate and bias values the thermometry data was taken(see Figure 4 and Figure 5).

Figure 2. A schematic (not to scale) of the temperature profile alonga nanowire (nw) with an embedded quantum dot (QD), a heatedmetallic source contact (sc), and an unheated metallic drain contact(dc). In the source and drain contacts, the electron gas temperaturerises by ∆Tsc and ∆Tdc above the cryostat temperature, T0,respectively. In the nanowire, the electron gas temperature risesby ∆Ts,d near the source and drain sides of the quantum dot,respectively (∆Tsc g ∆Ts g ∆Td g ∆Tdc).

780 Nano Lett., Vol. 9, No. 2, 2009

bias voltage and defining the voltage values Vs,d0 ≡ -2(ε0

- µ)/e, Rs,d can be written

Rs,d )∆Ts,d2ke

(V-Vs,d0) coth( e

4k

V-Vs,d0

∆Ts,d + T0) (6)

This equation is the main theoretical result of this Letter.Note that it is only valid when either the source or drainelectrochemical potential is near a resonance of the quantumdot, but not both, and when Γ,kT. In the remainder of thepaper we will discuss our experimental device and setup,present experimental data, explain how ∆Ts,d are extractedfrom the data, and conclude with comparison to numericalmodeling.

The experimental device (see Figure 1a) is an InAsnanowire about 55 nm in diameter and 1.2 µm long grownusing chemical beam epitaxy. Two 5 nm InP barriersembedded during growth define an InAs quantum dot about15 nm in length. After growth, the nanowire is depositedonto an n-doped Si wafer capped with a 100 nm electricallyinsulating SiOx layer. The n-doped wafer serves as a back-gate (see Figure 2), and Ni/Au electrical contacts are defined

at both ends of the nanowire using electron beam lithography.For further growth and fabrication details, see ref 18.Electron-electron Coulomb interaction splits degeneratequantum energy levels into single-electron energy states (theso-called Coulomb blockade). The classical charging energyis on average 12.9 meV (see Figure 1b), and the quantumenergy spacing is as large as 40 meV. In all experiments, T0

< 5 K = 0.4 meV guaranteeing that the energy levels arespaced in energy by much more than the thermal energy.Near energy resonances (quantum energy levels split byCoulomb blockade), the transmission function of the double-barrier quantum dot is Lorentzian.19 The InP barriers providelong electron lifetimes and make each Lorentzian very sharp.In fact, theoretical fits of differential conductance peaks thatassume an infinitely sharp Lorentzian (see eq 1 in ref 20) fitour data very well. The inset of Figure 3b is such a fit and

Figure 3. (a) Modeled thermocurrent, Ith (red), and seconddifferential conductance, ∂2I/∂V2 (blue), as a function of bias voltagecalculated using eq 1 and a Lorentzian τ(ε) with Γ ) 10 µeV ,kT ) 190 µeV ) 2.2 K. Near a resonant energy (see insets), bothIth and ∂2I/∂V2 have a zero-crossing, and their ratio (green) behavesaccording to eq 6 (dashed). (Insets) When biased negatively, heatedelectrons flow through the dot, while cold electrons flow whenpositively biased. (b) Experimental data of Ith (red), ∂2I/∂V2 (blue),their ratio (green), and theory eq 6 (dashed) as a function of biasvoltage. For these data, IH ) 150 µA, ∆Ts ) 230 mK, ∆Td ) 160mK, and T0 ) 2.2 K. (Inset) Differential conductance, G, measuredat T0 ) 550 mK in units of 2e2/h × 10-3 as the resonant energy ofthe dot is swept via the back-gate. The (black) dots are experimentaldata, and the solid (blue) line is a theoretical fit (see text for details).

Figure 4. (a) The (green) data points are the ratio R ) Ith/(∂2I/∂V2)measured at four different heating currents, IH, as indicated and atT0 ) 2.94 K. The left data sets determine ∆Ts, and the right sets∆Td. The (violet) dashed lines are eq 6 based on the mean of thehistograms in (b). The data sets and theory have been multipliedand/or offset for clarity. (b) Histograms of the values of ∆Ts,d,obtained by solving eq 6, at IH ) 115 µA shown in (a). The dotindicates the mean of the data, and the error bars encompass 67%of the histogram data.

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uses T0 (measured with a calibrated Cernox sensor heat sunkin the same way as the device) and the capacitance ratio R) Cg/CΣ (determined from Coulomb Blockade diamonds).The center of the transmission resonance is the only fittingparameter. The fits indicate that temperature smearing is thedominant peak-broadening mechanism over our experimentalbackground temperature range. Therefore, in our analysisbelow, we assume Γ , kT0 and the validity of eq 6.

In mesoscopic experiments, it is standard practice to usean ac heating current IH and take advantage of lock-inamplification techniques in order to improve the signal-to-noise ratio. We use a lock-in amplifier to measure thefrequency-doubled electrical current which flows in responseto IH. This current is the differential thermocurrent, Ith,because it is the change in electrical current which flows asa result of the change in the thermal gradient. In theexperiment, the ac heating current is created by applyingtwo heating voltages, V(, which are out of phase with eachother by one-half cycle. The relative amplitudes of the twoheating voltages are tuned so that their sum is zero at thenanowire’s source contact. This voltage balancing preventsV( from biasing the nanowire. In practice, their sum cannotbe tuned to exactly zero. A concern is that second-ordernonlinear processes create a frequency-doubled current,which appears as a phantom thermocurrent. However,quantitatively this nonlinear signal is 2 orders of magnitudesmaller than the thermocurrent and behaves qualitativelydifferent than the thermocurrent traces. Therefore, the effectof the small residual heating voltage at the nanowire isnegligible. For the data taken at 2.94 K, IH has an rmsamplitude of up to 460 µA and a frequency of 62.5 Hz.Concurrent with measurements of Ith, we apply an ac biaswith an amplitude of 67 µV and a frequency of 40 Hz andmeasure ∂I/∂V using a lock-in amplifier. ∂2I/∂V2 is thencalculated by taking a numerical derivative.

Raw data of Ith and ∂2I/∂V2 as well as their ratios R )Ith/(∂2I/∂V2) are shown in Figure 3b and Figure 4a. In order

to extract ∆Ts and ∆Td from the raw data, each R data pointis assigned a temperature rise by numerically solving eq 6using Mathematica. Histograms of the numerically deter-mined ∆Ts,d data (see Figure 4b) are then used to calculatea mean ∆Ts,d as well as to define error bars, which encompasstwo-thirds of the data points and are not necessarily sym-metric about the mean. The data outside the error bars aretypically systematic errors associated with the divergenceof R when ∂2I/∂V2 approaches zero. Using the mean valuefrom the histogram to plot eq 6 on top of the original dataprovides an additional confirmation that the resulting curveagrees with the data. ∆Ts,d determined from this analysis andplotted versus heating current at a constant backgroundtemperature exhibit a consistent trend and a good signal-to-error ratio (see Figure 5).

To confirm whether the results for ∆Ts and ∆Td in Figure5 are reasonable, we performed finite element modeling. Weused COMSOL’s Multiphysics Electromagnetics and HeatTransfer modes complemented by a custom e-ph couplingterm of the form Σ(Te

n - Tphn), where the coupling constant,

Σ, and the exponent, n, depend on the material and the e-phinteraction mechanism. Distinct electron (Te) and phonon(Tph) temperatures are modeled explicitly. In addition to thenanowire itself, the model includes the metallic leads in theirentirety, the 100 nm SiOx layer, the top 10 µm of the n-dopedSi wafer, and the embedded quantum dot, whose preciseposition along the nanowire was found via SEM imaging.In the metallic leads and nanowire, we use the electricalconductivity, σ, and phonon mean free path, λ, as inputparameters and calculate the electronic and phononic thermalconductivities via the Wiedemann-Franz law and Debyemodel, respectively.

We first calculate the temperature rise in the heated sourcecontact, ∆Tsc, in the absence of a nanowire. For the sourceand drain metal leads we use σAu ) 4.05 × 107 Ω-1 m-1

(measured), λAu ) 100 nm (comparable to the smallestdimension of the Au heating wire), ΣAu ) 1.8 × 109 W m-3

K-5 (averaged value from refs 21 and 22), and nAu ) 5 (fromrefs 21 and 22). The dashed line in Figure 5 shows thepredicted ∆Tsc as a function of heating current. The corre-sponding temperature rise in the drain contact, ∆Tdc, is foundto be negligible and is not shown in the figure. Afterestablishing these temperatures in the metallic source anddrain contacts, we turn our attention to the nanowiretemperatures.

Assuming that the total thermal conductance of thenanowire is dominated by the large electronic thermalresistance of the quantum dot, one would expect that ∆Ts ≈∆Tsc and ∆Td ≈ ∆Tdc ≈ 0. However, this expectation is notconsistent with our experimental results, which show that∆Ts is only about half as large as the calculated ∆Tsc, and∆Td is, in fact, on the same order as ∆Ts (see Figure 5).Therefore, in addition to electronic heat flow, the nanowiremust influence ∆Td by some other mechanism; we considerJoule heating and e-ph interaction. Additional modelingresults show that Joule heating inside the nanowire is toosmall to explain the observed ∆Td. Instead, we hypothesizethat e-ph interaction inside the nanowire allows the electron

Figure 5. A plot of the temperature rises, ∆Tsc, ∆Ts, and ∆Td (seeFigure 2) measured at T0 ) 2.94 K as a function of the heatingcurrent, IH. The circular data points and error bars are obtainedfrom histograms (see Figure 4). The dashed line is a numericalcalculation of ∆Tsc. (The change in drain temperature is negligible.)The solid lines are numerical calculations of ∆Ts,d, which use T0 +∆Tsc and T0 as boundary conditions. See text for modeling details.

782 Nano Lett., Vol. 9, No. 2, 2009

gas to couple to phononic heat flow. Conceptually, heatedelectrons inside the nanowire between the source contact andthe quantum dot give thermal energy to the nanowire’sphonons via e-ph collisions, thereby decreasing ∆Ts. Theseheated phonons pass through the quantum dot much easierthan electrons and carry heat along the nanowire. On thedrain side of the dot, warm phonons couple to cool electrons,thereby increasing ∆Td.

To perform a qualitative test of our hypothesis, we usedthe previously calculated electron and phonon temperaturesof the metallic source and drain contacts as boundaryconditions for the nanowire and calculated ∆Ts,d in thepresence of e-ph coupling inside the nanowire. We variedthe parameters σnw, λnw, Σnw, and nnw (we used integer valuesonly) to reproduce the experimental ∆Ts and ∆Td data inFigure 5. The parameters σnw ) 8.7 × 103 Ω-1 m-1, λnw )100 nm (comparable to the nanowire diameter), Σnw ) 8.4× 1010 W m3- K-1, and nnw ) 1 provide a good approxima-tion of the dependence of ∆Ts and ∆Td on IH. Our σnw iscomparable to values found in ref 23 and, for a 2 × 1017

cm-3 carrier concentration, Σnw ) 2.6 × 106 eV s-1 K-1

carrier-1, which is within 2 orders of magnitude of the valuereported for a trench-type quantum wire etched in a 2DEG.24

We conclude that our measured values of ∆Ts and ∆Td areconsistent with a model where e-ph coupling allowssignificant heat flow past the quantum dot into the electrongas on the nanowire’s drain side. We caution that Σnw andnnw are codependent fit parameters and that other combina-tions of the two might also fit our data.

In conclusion, we have developed and demonstrated theuse of a quantum dot with very narrow transmissionresonances for local thermometry of the electrons enteringthe quantum dot. We have measured temperature differencesas large as 0.74 K across a 15 nm quantum dot at 2.94 K.Strictly speaking, the concept of an electron temperature isonly meaningful when the electrons establish a thermaldistribution. The good agreement of our data with theory(Figure 3b and Figure 4)) suggests that in our experimentthe assumption of Fermi-Dirac distributions is resonable andthat any nonequilibrium effects are relatively small. It isimportant to note that the length scale of electron thermal-ization puts an upper bound on the spatial precision to whichthese measured temperatures can be assigned. Our thermom-etry results agree with a finite element model that includese-ph coupling as a pathway for heat flow in nanowires. Thisinsight might be important for thermal managment innanowire-based applications, such as high-speed transistors,LEDs, and thermoelectric devices. Because of the practicalchallenges involved, few experiments have quantified e-ph

interaction in mesoscopic systems.24 As demonstrated here,quantum-dot thermometry could be put to use testingquantitative predictions of the strength and temperaturedependence of e-ph coupling in nanowires.

Acknowledgment. This research was supported by ONR,ONR Global, the Swedish Research Council (VR), theFoundation for Strategic Research (SSF), the Knut and AliceWallenberg Foundation, and an NSF-IGERT Fellowship.

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