During the process of making physical systems smaller, it can be difficult to tell in which ways physical laws still apply. To illustrate, the phenomenological theory of thermodynamics is dependent on the supposition that the system being examined is made up of a great (approximated by infinitely many) number of particles.
At the point at which this ‘thermodynamic limit’ no longer holds, variations take over the physics and the laws must be adjusted. Nonetheless, it is important to note that in smaller physical systems, the dimensionality of the system can vary alongside the particle quantity. The laws for two-dimensional systems are often distinct from those derived for one or zero dimensional systems.
Figure 1. Device layout with lithographically patterned gates.
Equipment in Use
The equipment in use in this experiment is the Nanonis Tramea Base Package with standard eight inputs and outputs alongside Oxford Instruments’ TritonTM 200 Cryofree dilution refrigerator, as seen in Figure 2.
Figure 2. The experimental set-up used in this application note, showing the integrated Triton and Nanonis Tramea instruments.
This study focuses on both thermodynamics and spin-orbit interactions in zero-dimensional structures and on the level of single electrons. To achieve this, single or double quantum dots (synthetic atoms or molecules) are formed in a semi-conductor heterostructure.
In order to allow for a two-dimensional, strongly conductive layer, a wafer is grown and then patterned with metallic finger gates atop. To reduce the electron gas within this structure, using Coulomb repulsion, negative voltages are applied.
Using this technique, alongside the appropriate gate design seen in Figure 1, it is possible to create zero-dimensional structures (quantum dots), which can seize single electrons. A dilution refrigerator is then utilized to reduce the temperature of electrons within the structures to 40 mK.
Voltages applied to the finger gates can be used to tune both the tunneling of electrons between the reservoirs, or dark structures, and the quantum dots, as well as between a pair of quantum dots forming a double quantum dot. As well as this, voltage applied to the gates can also be used to tune the depth of prospective wells describing the quantum dots.
In order to achieve effective and timely tuning of all tunnel rates and potentials to the specified values, quick scanning of a sizeable parameter space (5-7 gate voltages) is needed. This is made more difficult to achieve by the cross capacitance between the metallic gates.
In most cases, a number of the gates are assigned a fixed voltage and a measurement is then executed to sweep one signal, then, between each sweep step a second gate voltage and seek a variation in current passing through the quantum dot and quantum dot contact, indicating occupation of the dot (ICD, Figure 1).
Where undesired responses are registered, adjustments to a number of the fixed gate voltages or the range of values in the sweep are made, and the process begins again. The charge state of both dots is then examined at the same time, using real-time charge-sensing methods.
Due to their reactivity to outside noise, it is necessary to ensure low noise levels and use low drift control electronics to ensure accurate measurements during these single electron transfer events.
Due to low bandwidth and time-consuming communication protocols, conventional measurement electronics generally take around an hour to perform a single scan of a pair of parameters within the sample space. Device tuning can subsequently be a tiresome job requiring a great deal of protracted repetition.
On the other hand, the use of fast electronics specifically intended for multi-output control measurements, namely those by Nanonis Tramea, it is possible to achieve a much quicker speed and greatly lessen the time taken for device tuning.
Low-noise voltage sources are also needed in this instance, as a high measurement bandwidth is only achievable with a minimal noise background. In addition to this, to ensure exact measurements which can last several days or longer, it is vital to make certain of the long-term stability of the voltage sources.
Figure 3 demonstrates the speed reached when Nanonis Tramea is used while the parameter space of gate voltages is navigated. It is feasible to undertake a vast exploration of the parameter space and to determine the final working point in just a few hours rather than a full day, due to the great speed of this system, with each graph generated in only around three minutes.
Following appropriate tuning of the dot, the measurement speed can be lowered to enhance the signal to noise ratio and sensitivity to minor effects.
To summarize, the above study of spin-orbit interaction at the level of a single electron has shown that spin-orbit coupling is probed via the tunneling currents between two coupled quantum dots. Constant readjustment of the device working point is necessary to determine the correct coupling between dots, which in turn is required to determine spin-flip rates with high fidelity.
It is possible to greatly increase the speed at which tuning the parameter space of viable devices is achieved by making use of the low noise, low drift outputs, and the instantaneous control of the experimental process found in the newly developed Nanonis Tramea.
Such developments in control electronics have allowed for the measurement of the Jarzynski equality – a proposal which links the variabilities of work to the alteration in free energy in a system driven into imbalance, on the level of single electrons.
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.
References and 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
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This information has been sourced, reviewed and adapted from materials provided by Oxford Instruments Nanoscience.
For more information on this source, please visit Oxford Instruments Nanoscience.