Application Note

Electron Thermometry
QFilter-II
The performance and accuracy of quantum electronics is substantially degraded when the temperature of the electrons in thedevices is too high. The electron temperature can be reduced with appropriate thermal anchoring and by filtering both the
low frequency and radio frequency noise. Ultimately, for high performance filters the electron temperature can approach thephonon temperature (as measured by resistive thermometers) in a dilution refrigerator. In this application note, the method formeasuring the electron temperature in a typical quantum electronics device using Coulomb blockade thermometry is described.
This technique is applied to find the readily achievable electron temperature in the device when using the QFilter provided by
QDevil. With our thermometry measurements, using a single GaAs/AlGaAs quantum dot in an optimized experimental setup, we
determined an electron temperature of 28 ± 2 milli-Kelvin for a dilution refrigerator base temperature of 18 milli-Kelvin.

1Application Note
INTRODUCTION
Quantum electronics experiments play a central role in thedevelopment of new quantum technologies. Quantum effects
occurring at ultra low temperatures have shown tremendous
potential. Investigating these physical phenomena typicallyrequires the use of dilution refrigerators which allow to cool
experimental setups down to phonon temperatures of theorder of 10 mK. The performance of the quantum electronics
is, however, also dictated by the temperature of the electrons in
the device.This is typically found to be higher than the phonon
temperature normally measured with resistive thermometerssuch as ruthenium-oxide devices. To determine the electron
temperature, different types of devices and measurement
techniques are available [1], such as tunnel junctions between
a normal metal and a superconductor (NIS junction) [2],superconducting tunnel junctions (SIS junction) [3], shot noise
thermometry [4], and thin-film thermometry near the metalinsulator transition [5]. In this application note we give a detailed
description of the Coulomb blockade thermometry method[6-10], as pioneered by J.P. Pekola [6,7] and based on wellestablished theory [11,12]. For this purpose we use a multi-gate
GaAs/AlGaAs single quantum dot.

METHOD AND INSTRUMENTATION
DEVICE FABRICATION
For the Coulomb blockade thermometry, we make use of asemiconductor single quantum dot device, through which
the transport signatures of single electron tunneling can bemeasured. This requires at least a source (S) and a drain (D)electrode and a plunger gate (PG) to control the potential of thequantum dot.
Many different material systems and gate electrode layouts canbe used for this purpose [13], including commercially available
transistors. In this application note, we make use of a multi-gate
quantum dot device defined in a two dimensional electron gas(2DEG), formed in a GaAs/AlGaAs heterostructure.
Carefully shaped metallic gates are used to apply anelectrostatic potential and confine the quantum dots laterally.
Although the full device consists of many gates, for thismeasurement we will only use the gates controlling a single
quantum dot, shown in figure 1c. Next to the source, drain, andplunger gate electrodes, this also includes several barrier gates(TB, BB, LB, RB), which give more control over the electrostaticsof the quantum dot. More details about the fabrication of thisdevice can be found in [14,15].
EXPERIMENTAL SETUP
The quantum dot sample is placed on a sample holder (QBoard
by QDevil), which accommodates the connection of the on-chipelectrodes to the transmission lines in the cryostat, and ensures

a good thermalization of the sample. For our experiment, we
fixed the GaAs chip on a QBoard daughterboard and wirebonded all gates and ohmic contacts to the bonding pads. Thisdaughterboard was then placed on a QBoard motherboard,
inside a loading puck for the Triton Cryofree dilution fridgebottom-loading system (see figure 2a). After loading, the QBoard
connects to two 24-wire twisted-pair looms, used for DC signals, inside the dilution fridge. This setup is illustrated in figure 1.
A list of the instrumentation is provided in table 1 in Appendix A.
Hot electrons enter the dilution refrigerator at room temperatureand need to be thermalized at the thermal stages at lower
temperatures. To lower the electron temperature for theexperiment, special care should be taken on two points whenplacing the DC wiring in the cryostat. First, the wiring should bethermally anchored at each temperature stage of the cryostat.
Here we achieve this with the use of copper bobbins. Second,
final thermalization below 1K and filtering of high frequency noiseis achieved with the use of the QFilter-II, provided by QDevil. Thisis a two part filter consisting of a filter bank with RC low passstages, which filters the electronic noise from 65 kHz up to the
GHz regime and thermalizes the electrons, and a filter bank with
LC low pass stages, which filters frequencies from 225 MHz to
the THz regime (see figure 3a). Transmission data and further
technical details can be found in the QFilter-II product brochure.
Additional RC filtering of the DC lines occurs on the QBoard.
On the outside of the cryostat, it is useful to split the two 24-wirecables into separate connectors for each electrode to simplify
the electrostatic control. In this experiment we make use of two
QBox breakout boxes, provided by QDevil. They are connectedto the cryostat with double shielded low-noise 24 channel
twisted pair cables, with Fischer connectors on each end.
Each channel is connected to a BNC connector on the breakout box. To prevent any ground loops (see the section test of
background noise), only a single ground should be used in theexperiment. We choose to use the ground of the magnet power
supply of the cryostat as common ground. To ensure the propagation of this common ground, the extended shielding of the 24channel cable is connected to the conductive flange of the QBox
and the cryostat with metal collars (see figure 2b), while thesebreakout boxes are otherwise isolated from any other groundcarrying equipment or equipment racks.
To apply computer-controlled voltages to the sample electrodes,
we choose to use the 24-channel QDAC, provided by QDevil,featuring 20-bit resolution and output noise under 20 nV/Hz1/2,
which we connect with BNC cables to the breakout box. For
extra filtering, we insert additional 1.9 MHz BNC low pass filters(BNP-1.9+ from Mini-Circuits) for all applied voltages on the
breakout box. Again, to prevent any ground loops, the QDAC isonly grounded through the BNC cables, powered by an external
isolated power supply, and otherwise isolated from any otherground carrying equipment or equipment racks.

2

Application Note

Breakout boxa

BNC cables

D/A Converter

VPG, VDG, VTB, VLB, VRB, VBB
VoltagedividerFischer cable
Twisted pair loom
Cryogenic ﬁlter

VSDC

VSAC

50 K

4K
1K
100 mK
Copperbobbins
Lock-in ampliﬁer

18 mK

ID

VD
Sample holder
Digital multimeter

Current pre-ampliﬁer
VD

Dilution refrigeratorbc

200 nm
VTB
QD

S

VLB
VPG

D

VRB
VBB

VDG

Figure 1. Overview of the experiment. a) Experimental setup, showing the use of a QDevil QDAC D/A converter, a Stanford Research Systems SR830
lock-in amplifier, a QDevil QBox breakout box, an Ithaco 1211 Current preamplifier, a Keysight 34465A digital multimeter, a QDevil QFilter and a QDevil
QBoard sample holder. Furthermore the BNC cables, Fischer cable and cryogenic cabling are shown schematically. b) Image of the QDevil QFilter-II,
as used in this application note. c) Scanning electron microscope image of the GaAs/AlGaAs device used in this application note. The quantum dot(QD), source (S), drain (D) and plunger gate (PG) are indicated, as well as the barrier gates at the top (TB), bottom (BB) and on the left and right of thequantum dot (RB, LB), and a distant gate (DG). Image courtesy F. Fedele (from F. Fedele thesis [14]).

3

Application Note
To measure the single electron tunneling current in the quantum
dot, the channel connected to the source or drain of thedevice should be connected to a current pre-amplifier andcan then be measured in a variety of ways, for instance using
a digital multimeter. In our experiment we use an Ithaco 1211
preamplifier with a conversion of 107 V/A, and a time constantof 30 ms. We measure the resulting voltage with a Keysight
34465A 61/2 digit multimeter. Furthermore, to measure thedifferential conductance in the quantum dot, a lock-in amplifier
can be used to apply a small AC voltage on the source or drainelectrode. The resulting oscillation in current will then allow fora sensitive measurement of the differential conductance withthe lock-in amplifier. Here we use a Stanford Research Systems
SR830 lock-in amplifier with a 224 Hz, 3 μV peak-peak AC
excitation voltage (created with an output of 300 mV and the useof a 1/105 voltage divider, as shown in figure 3b). We use thedirect output of the Ithaco preamplifier (bypassing its low passfilter) as the signal input of the lock-in amplifier to sense the AC
current signal. We use a sensitivity of 500 μV, a time constant of
300 ms and a filter roll-off of 24 dB. We convert the measured
signal to a differential conductance by[1]

Where gdiff is the differential conductance in units of e2/h, SLI is
the signal from the lock-in amplifier in V, RQ is the quantum of
resistance h/2e2 (around 12 kΩ), Gpa is the gain of the Ithaco
preamplifier in V/A and Vex is the excitation voltage in V.

a

All experiments shown in this application note are taken at zeromagnetic field. Furthermore, to execute the experiments, we use
the python-based data-acquisition framework of QCoDeS [16].
TEST OF BACKGROUND NOISE
Prior to the experiments, as a good practice, we investigated
the noise background with a current measurement in deep
Coulomb blockade (as outlined in [17]). For this purpose, we set
the Ithaco gain to 108 V/A and measured its direct output witha spectrum analyzer. Here we found the 50 Hz noise peak to
be 180 μVrms, lower than the 300 μVrms limit for low electron
temperature measurements as mentioned in [17]. Furthermore,by sequentially switching equipment on and off, we found thatthe 50 Hz noise peak could be further reduced to 80 μVrms by
disconnecting the dilution fridge resistive thermometry readout
circuit and down to 25 μVrms when the Ithaco pre-amplifier was
switched from line power to battery power.
MEASUREMENT OF QUANTUM DOT SIGNATURES
In order to operate the specific device in the experiment as asingle quantum dot electron thermometer, schematically shownin figure 4a, we first apply negative voltages of -1.2 V (TB) and-0.6 V (BB) to the gates forming the outer barrier of the quantum
dot as shown in figure 1c. This creates a potential barrier andthus isolates the single quantum dot from the rest of the device.
Next, a small bias voltage is applied to the source contact, inorder to create a current between the source and drain contact
which is grounded through the Ithaco current pre-amplifier.

aRC

225 MHz

65 kHzLCb

225 MHz1.45 GHz

b

5 GHz

5 MΩ

AC input

DC+AC
output50 kΩ
50 Ω

DC input

Figure 2. Photographs of the experiment. a) GaAs sample mounted
and wire-bonded in the QBoard, placed in an Oxford puck loading
system. b) Grounding of the breakout box through the metal collar
system on the Fischer cable connector.

Figure 3. Electronics schematics. a) Schematic overview of the
QFilter RC and LC filter banks. b) Schematic of the voltage divider
used to apply DC and AC voltages to the source electrode.

4Application Note
Then the barrier gates of the quantum dot are tuned to the point
where they almost pinch off the current, about 10-50 mV fromthe pinch-off voltage. This is the regime where single electrontransport occurs, which is best visualized in a scan of theplunger gate voltage (see figure 4b). The peaks of current
indicate single electron tunneling through the quantum dot,
while in between these peaks regions of Coulomb blockade areobserved.

Coulomb diamond. The height from zero bias to either the top or
bottom of the diamond is given by the charging energy:

By extending this measurement to a 2D scan of both the sourceand plunger gate voltage, the Coulomb blockade region forms
a diamond shape, as shown in figure 4c. This is commonlyreferred to as a Coulomb diamond and is a typical signature of
single electron tunneling in a quantum dot. Using the lock-in
amplifier we measure the differential conductance at the sametime, as can be seen in figure 4d. These measurements aretaken as line scans of the source voltage, while stepping theplunger gate voltage. We use these measurements to extract
information about the quantum dot, such as the plunger
gate lever arm and the relation between the relevant energy
scales, which we require to calibrate the electron temperature
measurements later.

Secondly, these measurements can be used to estimate therelations between the different energy scales at play: thethermal energy of the electrons kBTe, the level spacing ΔE, andthe tunnel broadening hΓ. Here kB is the Boltzmann constant,
Te the electron temperature, h the Planck constant, and Γ thetotal tunnel rate. In the regime in which hΓ ≪ kBTe ≪ ΔE, the best
estimate of the electron temperature can be made from thesingle electron tunneling measurements.

where e is the elementary charge and ΔE is the level spacingof the quantum dot. Combined with the width of the Coulomb
diamond given by EC/αPG, the lever arm is found.

For the condition ΔE, kBTe ≫ hΓ to be true, the resulting
conductance in the quantum dot must be much smaller than the
conductance quantum 2e2/h, which the differential conductancemeasurement can verify. In the regime of hΓ ≪ kBTe ≪ ΔE, thezero bias conductance peak is given by [12]:[3]aTMC = 25 mK

S

N-1

kBTe

430

d

VS [mV]

10

0
-460

N+1

N

N+1 N+2

VPG[mV]

-340

-80

0.00

VPG[mV]

0.04

0.00

0
0.00

-1
430

VPG[mV]-425 -424
VPG [mV]
TMC = 100 mK

0.03
dI/dVS[e2/h]-

Gmax [e2/h]

dI/dVS [e2/h]

0

-390

1

15

5

N

b
-

b

I [pA]

N-1

-1

D

QD

0

80

-390

Figure 4. Coulomb blockade measurements. a) Schematic overview
of the Coulomb blockade thermometry measurement. b) Plunger gate
voltage trace at a 100 μV source voltage, showing peaks in current
where the single electron tunneling occurs. c) Current measured
through the quantum dot as a function of the voltages applied to theplunger gate and source contacts. b) Differential conductance mapmeasured simultaneously with c).

0.00
0
d

-425 -424
VPG [mV]

TMC sensor [mK]140

Temperature [mK]VS

N

VS [mV]

N+1

I [pA]

1

dI/dVS [e2/h]c

VLB VPG VRB

0.8

0.03

0.03
a

c

FWHM [mV]PARAMETER EXTRACTION FROM QUANTUM DOT
MEASUREMENTSFirst, we extract the lever arm of the plunger gate αPG, which
gives the relation between the energy level of the quantum dot
and the voltage on this gate, given by the ratio of the plunger
gate capacitance to the total capacitance, CPG/CΣ. This ratio isdirectly given by the ratio between the height and width of the

[2]

150

0

TMC
Te
𝜏𝜏e = 9.8 min

0 𝜏𝜏MC = 7.8 min
02Time [hours]

4

Figure 5. Temperature dependence of the zero bias conductancepeak. a) and b) Data and fit with eq. 3 of the zero-bias conductancepeak, at a mixing chamber temperature sensor measurement of a)
25 mK and b) 100 mK. c) Amplitude and full width half maximum(FWHM) of these as a function of the measured mixing chamber
temperature. A linear guiding line through the origin shows the
expected FWHM with the measured mixing chamber temperatureand αPG. d) Time dependence of the resistive thermometer (black) andelectron thermometer (blue) after switching off the mixing chamber
heater at 150 mK. Both datasets are fit with an exponential decay.

5Application Note
where Δε is the energy difference from the resonant tunnelingcondition, and Gmax is given by [12]:conductance peak as well as the Ruthenium-oxide thermometerfor 4 hours after switching off the mixing chamber heater, as
shown in figure 5d.
[4]Both thermalization curves are fitted with an exponential decay,
for which we find a decay time τe of 9.8 minutes for the electron
temperature measurement, which is close to the decay time τMCof 7.8 minutes for the resistive temperature sensor at the mixing
chamber.

With Γin and Γout the tunnel rates in and out of the quantum
dot. In contrast, Gmax does not depend on the temperature in
the classical regime with kBTe ≫ ΔE. Therefore, temperaturedependent measurements of the conductance peak will verify, if
indeed only a single quantum level influences the single electron
tunneling (see figure 5).

FURTHER OPTIMIZATION
Finally, improvements were made to the electron thermometry
measurements, by finding the ideal values for several equipment
settings. The ideal lock-in amplifier amplitude was found to be
1 μV (created with an output of 100 mV and the use of a 1/105
voltage divider, as shown in figure 2d), the ideal frequency to
be 117 Hz, the ideal time constant to be 30 ms, and the idealroll-off to be 12 dB. Furthermore we use a 1/103 voltage divider
for the source voltage (see figure 3b), such that we can achievea resolution below 10 nV using the QDevil QDAC. To improve theresolution of the plunger gate, we use one of the distant gates(DG) further away from the dot instead, with a lever arm for the
distant gate of 0.0016 ± 0.0001. With these settings, we scan
the source voltage close to the zero bias point, in order to makesure to find the narrowest point of the conductance peak (seefigure 6a). We fit these conductance peaks around the zero-bias
point with eq. 3 (see inset figure 6b) and find that the electron
temperature stabilizes at 28±2 mK where the Coulomb peak is
the narrowest (see figure 6b).

DATA ANALYSIS
From figures 4c and 4d, the plunger gate lever arm is measured
as 0.074 ± 0.003. The zero bias differential conductance peakat a plunger gate voltage of -424 mV in figure 4d measuresaround 0.02 e2/h, thereby verifying that we are in the regime
ΔE, kBTe ≫ hΓ. Furthermore, the amplitude of this peakdepends on temperature, as shown in figure 5c, thereby also
confirming kBTe ≪ ΔE. This figure shows that there is a good
agreement between the temperatures measured by the resistive
Ruthenium-oxide thermometer on the mixing chamber and theelectron thermometry using the quantum dot at temperatures
above 40 mK. Below 40 mK, the electron temperature isstabilizing, while the resistive thermometer still measuresa cooling of the mixing chamber plate. Next, we show ameasurement of the time constant of the electron thermalization.
Here we continuously measure the width of the zero-bias

a

b

-70

60

0.02
0.00

dI/dVS [e2/h]

0.04
-80

Electron temperature [mK]Distant gate voltage [mV]dI/dVS[e2/h]

28±2 mK

0.04

0.00

-90 -80 -70
DG voltage [mV]

40

-90

28 mK

-10

0
Source voltage [μV]

10

20

-10

0

10

Source voltage [μV]

Figure 6. Electron thermometry measurement with further optimized parameters. a) High resolution scan of the Coulomb peak near zero bias,
using a distant plunger gate and a voltage divider for the source voltage to increase the resolution. b) Measured electron temperature by fitting the
Coulomb peaks with eq. 3 (see inset for the fit resulting in the blue datapoint). A black dashed line shows the measured electron temperature of
28 mK.

6

Application Note
CONCLUSIONREFERENCES

Using single electron transport in a GaAs/AlGaAs quantum
dot, we performed electron thermometry measurements.
The experimental setup was specifically setup for low-noise
measurements, of which a detailed description has been given.

1.

F. Giazotto et al., ‘Opportunities for mesoscopics inthermometry and refrigeration: Physics and applications’,
Rev. Mod. Phys. 78, 217 (2006).

2.

J.M. Rowell and D.C. Tsui, ‘Hot electron temperature
in InAs measured by tunnelling’, Phys. Rev. B 14, 2456
(1976).

3.

N.E. Booth and D.J. Goldie, ‘Superconducting particle
detectors’, Supercond. Sci. Technol. 9, 493 (1996).

4.

L. Spietz et al., ‘Primary electronic thermometry using theshot noise of a tunnel junction’, Science 300, 1929 (2003).

5.

D.W. Denlinger et al., ‘Thin film microcalorimeter for heat
capacity measurements from 1.5 K to 800 K’, Rev. Sci.
Instrum. 65, 946 (1994).

6.

J.P. Pekola et al., ‘Thermometry by arrays of tunnel
junctions’, Phys. Rev. Lett. 73, 2903 (1994).

7.

Sh. Farhangfar et al., ‘One dimensional arrays and solitarytunnel junctions in the weak Coulomb blockade regime:
CBT thermometry’, J. Low Temp. Phys. 108, 191 (1997).

8.

F. Mueller et al., ‘Printed circuit board metal powder filters
for low electron temperatures’, Rev. Sci. Inst. 84, 044706(2013).

9.

D. Maradan et al., ’GaAs quantum dot thermometry using
direct transport and charge sensing’, J. Low Temp. Phys.
175, 784-798 (2014).

For the electron thermometry analysis, it has been establishedthat the effect of tunnel broadening was negligible and that onlya single quantum level influenced the differential conductancemeasurements. After optimizing all experimental parameters, wefind an electron temperature of 28 ± 2 mK, which we attribute to
the use of a QDevil QFilter-II.

ACKNOWLEDGEMENTS
The measurements shown in this application note were
performed at the Center for Quantum Devices of the Niels Bohr
Institute at the University of Copenhagen in Denmark. A lot ofappreciation goes to J. Kutchinsky, F. Kuemmeth, A. Kühle,M. von Soosten and S. Andresen for their invaluable input andthe many useful discussions. We like to thank F. Fedele for the
fabrication of the here used GaAs quantum dot device, and F.
Ansaloni, B. Brovang, and L. Lakic for setting up the cryogenic
wiring.

Author
Joost van der Heijden
QDevil ApS, Copenhagen, Denmark

10. M. Meschke et al., ‘Accurate Coulomb blockade
thermometry up to 60 Kelvin’, Phil. Trans. R. Soc. A 374:
20150052 (2015).
11. D.V. Averin and K.K. Likharev, in ‘Mesoscopic phenomena
in solids’, edited by B.L. Altshuler, P.A. Lee, and R.A. Webb,
Elsevier, p. 173 (1991).
12. C.W.J. Beenakker, ‘Theory of Coulom-blockade oscillationsin the conductance of a quantum dot’, Phys. Rev. B 44,
1646 (1991).
13. R. Hanson et al., ´Spins in few-electron quantum dots´, Rev.
Mod. Phys. 79, 1217 (2007).
14. F. Fedele, ‘Spin interactions within a two-dimensional
array of GaAs double dots’, PhD thesis, University of
Copenhagen (2020).

QDevil ApS
Fruebjergvej 3
DK-2100 Copenhagen
Denmark
Phone: +45 3699 2145
QDevil Inc
299 California Avenue, Suite 200
Palo Alto
CA 94306, USA
Phone: +1 650 543 3192

15. F. Fedele et al., ‘Simultaneous operation of singlet-triplet
qubits’, 2019 Silicon Nanoelectronics Workship (SNW),
Kyoto, Japan, pp. 1-2 (2019).
16. J.H. Nielsen et al., QCoDeS/Qcodes: v0.23.0.dev1. (2021)
doi: 10.5281/zenodo.596989
17. H.O.H. Churchill, ‘Quantum dots in gated nanowires andnanotubes’, PhD thesis, Harvard University (2012).

© 2021 QDevil ApS. All rights reserved.

7

Application Note
APPENDIX A

Table 1. List of equipment required for the Coulomb blockade electron thermometry measurement, noting the specific models used for theexperiments in this application note.

8