CRYOFREE

Application note

Physics on the level of single electrons
Andrea Hofmann, Ville F. Maisi, Tobias Krähenmann, Clemens Rössler, Julien Basset, Marc Röösli, Carolin
Gold, Christian Reichl, Werner Wegscheider, Thomas Ihn, Klaus Ensslin – ETH-Zurich

Introduction
When shrinking the size of physical systems, it
often becomes unclear in which way the physicallaws are still valid. For example, thermodynamics
is a phenomenological theory which relies on theassumption that the system under considerationconsists of a large (approximated by infinitely many)
number of particles. When this limit – the so-called
thermodynamic limit – does not hold, fluctuations
dominate the physics, and the thermodynamic lawsneed to be corrected. However, when considering
smaller physical systems, not only the number ofparticles, but also the dimensionality of the system
might change. Many laws have been derived for twodimensional systems, and can be different for the caseof one or zero dimensional systems.

Figure 1. Device layout with lithographicallypatterned gates.

Equipment in use
Nanonis TrameaTM Base Package with standard
8 inputs and outputs and Oxford Instruments’
TritonTM 200 Cryofree® dilution refrigerator (Figure 2).

Experimental set-up
In our experiment, we study thermodynamics andspin-orbit interactions on the level of single electrons
and in zero-dimensional structures. To do this, weform single or double quantum dots (artificial atoms ormolecules) in a semiconductor heterostructure. A wafer
is grown to host a two-dimensional, highly conductive
layer and then patterned with metallic finger-gates on
top. We apply negative voltages in order to deplete
the electron gas in the two-dimensional structure via
Coulomb repulsion. In this way and with a suitable gate
design shown in Figure 1, we are able to form zerodimensional structures (quantum dots) to capture singleelectrons. The structures are then cooled to an electron
temperature of 40 mK in a dilution refrigerator.

Figure 2. The experimental set-up used in thisapplication note, showing the integrated Triton andNanonis Tramea instruments.

CRYOFREE C
Application note

Tuning Procedure
The tunnelling of electrons between the reservoirs (the dark structures) and the quantum dots as wellas the tunnelling between the two quantum dots forming a double quantum dot can be tuned by
the voltages applied to the finger gates. Additionally, the depth of the potential wells describing thequantum dots can be also be tuned by applying suitable voltages on the gates. An efficient tuning of
all tunnelling rates and potentials to the desired values require fast scanning of a large parameter space(5-7 gate voltages), which is further complicated by the cross-capacitance between the metallic gates.
Usually, some of the gates are set to a fixed voltage and then a measurement is performed to sweep
one signal and between each sweep, step a second gate voltage and look for a change in current
passing through the quantum dot and through the quantum point contact (ICD in Figure 1) whichindicates occupation of the dot. If an incorrect response is observed, changes to some of the fixed gate
voltages or the range of values in the sweep are adjusted and the measurement repeated.
We then measure the charge state of both dots simultaneously by employing real-time charge-sensing
techniques. Due to the nature of these single electron transfer events, particularly their sensitivity toexternal noise, low noise and low drift control electronics are required for accurate measurements.
However, traditional measurement electronics typically require about one hour to perform a single scanof any two parameters in the sample space due to low bandwidth and slow communication protocols.
This means tuning of the devices can be a laborious undertaking that requires several slow iterations tocomplete. However, by employing Nanonis Tramea’s fast electronics designed for multi-output control
measurements, we could drastically increase the speed and favourably reduce the time needed to tune
the devices.
A high measurement bandwidth is only possible if the noise background is small, which means that wealso require low-noise voltage sources. Additionally, the long-term stability of the voltage sources is afundamental requirement for precise measurements which in total last several days or more.
Figure 3 illustrates the speed exploited when using Nanonis Tramea as the parameter space of gate
voltages is navigated. Each graph took only about three minutes to complete meaning that a wide
exploration of the parameter space and the determination of the final working point are possible in
only a few hours instead of a full day. Once the dot is properly tuned the measurement speed can be
reduced to optimize the signal to noise ratio and sensitivity to small effects.

CRYOFREE

Application note

Conclusion
In conclusion, we have studied spin-orbit interaction at the level of a single electron and demonstrated that thespin-orbit coupling is probed from the tunnelling currents between two coupled quantum dots. The correct coupling
between the dots is crucial for determining spin-flip rates with high fidelity, and that requires continuous readjustingof the device working point.
By utilising the new Nanonis Tramea’s low noise, low drift outputs combined with real-time based control of theexperiment routine, we have drastically improved the speed at which we tune the parameter space of viable devices.
These advances in control electronics have allowed us to measure the Jarzynski equality - a theorem which relates thefluctuations of work to the change in free energy in a system driven far from the equilibrium - on the level of single
electrons.

Experimental results

Figure 3. Three snapshots of stability diagrams during an initial double quantum dot tuning procedure.
Each plot was acquired in 3 minutes. A series of plots were acquired between each one which are not shown.
The entire procedure from start to finish only requred about two hours.

CRYOFREE

Application note

Further reading“Measuring the Degeneracy of Discrete Energy Levels Using a GaAs/AlGaAs Quantum Dot“ by A. Hofmann, V. F. Maisi,C. Gold, T. Krähenmann, C. Rössler, J. Basset, P. Märki, C. Reichl, W. Wegscheider, K. Ensslin, and T. Ihn –Phys. Rev. Lett. 117, 206803 (2016)“Spin-Orbit Coupling at the Level of a Single Electron“ V. F. Maisi, A. Hofmann, M. Röösli, J. Basset, C. Reichl,W. Wegscheider, T. Ihn, and K. Ensslin – Phys. Rev. Lett. 116, 136803 (2016)“Nonequilibrium Equality for Free Energy Differences“ by C. Jarzynski – Phys. Rev. Lett. 78, 2690
“Integration of superconducting magnets with cryogen free dilution refrigerator systems” by G. Batey, M. Buehler,M. Cuthbert, T. Foster, A. J. Matthews , G. Teleberg, A. Twin – Cryogenics 49 (2009) 727–734“A rapid sample-exchange mechanism for cryogen free dilution refrigerators compatible with multiple high-frequency signal
connections” by G. Batey, S. Chappell, M. Cuthbert, M. Erfani, A. J. Matthews , G. Teleberg – Cryogenics 60 (2014) 24–32

About Triton and Nanonis Tramea
The ultra low temperatures and high magnetic fields provided
by the Triton dilution refrigerator make it a key research tool inrevealing the quantum properties of many materials of interest.
The Triton systems already lead the way in experimentreadiness with high-density RF and DC wiring capability,
unique sample exchange mechanisms, and unbeatablesuperconducting magnet integration. SPECS’ Nanonis Tramea
QTMS is a natural complementary partner to the Triton, with
its fast, multi-channel, multi-functional capability. The systemenables quantum measurements to be carried out on a varietyof samples, as shown in this application note.

About SPECS Zurich

The latest Triton dilution refrigerator with increased
experimental space and cooling power.

SPECS Zurich is a daughter of Berlin-based SPECS GmbH. The
company specialises in the control systems for scanning probe
microscopes and low noise electronics for quantum transportmeasurements.
The goal of the company is to continue the development of
the Nanonis control system to provide their customers witha superior, innovative solution for sensitive measurements
required in SPM and quantum transport. They act as
scientific advisors in the field of SPM and other low noise
measurements, where the ultimate performance in electronics
is required. Together with their distributors, they ensure that
their clients receive outstanding service.

Nanonis Tramea multifunctional, low noise,
low drift and high resolution electronics.

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March 2017, Issue 1