Quantum-dot thermometry

0.65 MB·English

by  E. A. Hoffmann

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.
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