# A Robust Protocol for Entropy Measurement in Mesoscopic Circuits

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Device Design and Layout

## 3. Measurement Protocol

## 4. Common Problems

## 5. Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QD | Quantum Dot |

CS | Charge Sensor |

2DEG | Two-Dimensional Electron Gas |

QPC | Quantum Point Contact |

DC | Direct Current |

## References

- Hartman, N.; Olsen, C.; Lüscher, S.; Samani, M.; Fallahi, S.; Gardner, G.C.; Manfra, M.; Folk, J. Direct entropy measurement in a mesoscopic quantum system. Nat. Phys.
**2018**, 14, 1083–1086. [Google Scholar] [CrossRef] - Kleeorin, Y.; Thierschmann, H.; Buhmann, H.; Georges, A.; Molenkamp, L.W.; Meir, Y. How to measure the entropy of a mesoscopic system via thermoelectric transport. Nat. Commun.
**2019**, 10, 5801. [Google Scholar] [CrossRef] [PubMed] - Pyurbeeva, E.; Mol, J.A. A Thermodynamic Approach to Measuring Entropy in a Few-Electron Nanodevice. Entropy
**2021**, 23, 640. [Google Scholar] [CrossRef] [PubMed] - Gehring, P.; Sowa, J.K.; Hsu, C.; de Bruijckere, J.; van der Star, M.; Le Roy, J.J.; Bogani, L.; Gauger, E.M.; van der Zant, H.S. Complete mapping of the thermoelectric properties of a single molecule. Nat. Nanotechnol.
**2021**, 16, 426–430. [Google Scholar] [CrossRef] [PubMed] - Rozen, A.; Park, J.M.; Zondiner, U.; Cao, Y.; Rodan-Legrain, D.; Taniguchi, T.; Watanabe, K.; Oreg, Y.; Stern, A.; Berg, E.; et al. Entropic evidence for a Pomeranchuk effect in magic-angle graphene. Nature
**2021**, 592, 214–219. [Google Scholar] [CrossRef] [PubMed] - Smirnov, S. Majorana ensembles with fractional entropy and conductance in nanoscopic systems. Phys. Rev. B
**2021**, 104, 205406. [Google Scholar] [CrossRef] - Sela, E.; Oreg, Y.; Plugge, S.; Hartman, N.; Lüscher, S.; Folk, J. Detecting the universal fractional entropy of Majorana zero modes. Phys. Rev. Lett.
**2019**, 123, 147702. [Google Scholar] [CrossRef] [PubMed][Green Version] - Child, T.; Sheekey, O.; Lüscher, S.; Fallahi, S.; Gardner, G.C.; Manfra, M.; Kleeorin, Y.; Meir, Y.; Folk, J. Entropy measurement of a strongly correlated quantum dot. arXiv
**2021**, arXiv:2110.14158. [Google Scholar] - Elzerman, J.M.; Hanson, R.; Beveren, L.H.W.v.; Witkamp, B.; Vandersypen, L.M.K.; Kouwenhoven, L.P. Single-shot read-out of an individual electron spin in a quantum dot. Nature
**2004**, 430, 431–435. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hanson, R.; Kouwenhoven, L.P.; Petta, J.R.; Tarucha, S.; Vandersypen, L.M. Spins in few-electron quantum dots. Rev. Mod. Phys.
**2007**, 79, 1217. [Google Scholar] [CrossRef][Green Version] - Barthel, C.; Kjærgaard, M.; Medford, J.; Stopa, M.; Marcus, C.M.; Hanson, M.P.; Gossard, A.C. Fast sensing of double-dot charge arrangement and spin state with a radio-frequency sensor quantum dot. Phys. Rev. B
**2010**, 81, 161308. [Google Scholar] [CrossRef][Green Version] - Field, M.; Smith, C.G.; Pepper, M.; Ritchie, D.A.; Frost, J.E.F.; Jones, G.A.C.; Hasko, D.G. Measurements of Coulomb blockade with a noninvasive voltage probe. Phys. Rev. Lett.
**1993**, 70, 1311–1314. [Google Scholar] [CrossRef] [PubMed] - Sprinzak, D.; Ji, Y.; Heiblum, M.; Mahalu, D.; Shtrikman, H. Charge distribution in a Kondo-correlated quantum dot. Phys. Rev. Lett.
**2002**, 88, 176805. [Google Scholar] [CrossRef][Green Version] - Elzerman, J.; Hanson, R.; Greidanus, J.; Van Beveren, L.W.; De Franceschi, S.; Vandersypen, L.; Tarucha, S.; Kouwenhoven, L. Few-electron quantum dot circuit with integrated charge read out. Phys. Rev. B
**2003**, 67, 161308. [Google Scholar] [CrossRef][Green Version] - Aleiner, I.L.; Wingreen, N.S.; Meir, Y. Dephasing and the Orthogonality Catastrophe in Tunneling through a Quantum Dot: The “Which Path?” Interferometer. Phys. Rev. Lett.
**1997**, 79, 3740–3743. [Google Scholar] [CrossRef][Green Version] - Silva, A.; Levit, S. Peculiarities of the controlled dephasing of a quantum dot in the Kondo regime. EPL (Europhys. Lett.)
**2003**, 62, 103. [Google Scholar] [CrossRef] - Kang, K.; Khym, G.L. Entanglement, measurement, and conditional evolution of the Kondo singlet interacting with a mesoscopic detector. New J. Phys.
**2007**, 9, 121. [Google Scholar] [CrossRef] - Buizert, C.; Koppens, F.H.L.; Pioro-Ladrière, M.; Tranitz, H.P.; Vink, I.T.; Tarucha, S.; Wegscheider, W.; Vandersypen, L.M.K. In Situ Reduction of Charge Noise in GaAs/Al
_{x}Ga_{1−x}As Schottky-Gated Devices. Phys. Rev. Lett.**2008**, 101, 226603. [Google Scholar] [CrossRef] [PubMed][Green Version] - Liang, S.; Nakamura, J.; Gardner, G.; Manfra, M. Reduction of charge noise in shallow GaAs/AlGaAs heterostructures with insulated gates. Appl. Phys. Lett.
**2020**, 117, 133504. [Google Scholar] [CrossRef] - Mittal, A.; Wheeler, R.; Keller, M.; Prober, D.; Sacks, R. Electron-phonon scattering rates in GaAs/AlGaAs 2DEG samples below 0.5 K. Surf. Sci.
**1996**, 361, 537–541. [Google Scholar] [CrossRef] - Beenakker, C.; van Houten, H. Quantum transport in semiconductor nanostructures. In Solid State Physics; Academic Press: Cambridge, MA, USA, 1991; Volume 44, pp. 1–228. [Google Scholar]
- Maradan, D.; Casparis, L.; Liu, T.M.; Biesinger, D.E.F.; Scheller, C.P.; Zumbühl, D.M.; Zimmerman, J.D.; Gossard, A.C. GaAs Quantum Dot Thermometry Using Direct Transport and Charge Sensing. J. Low Temp. Phys.
**2014**, 175, 784–798. [Google Scholar] [CrossRef][Green Version] - Topinka, M.; LeRoy, B.; Westervelt, R.; Shaw, S.; Fleischmann, R.; Heller, E.; Maranowski, K.; Gossard, A. Coherent branched flow in a two-dimensional electron gas. Nature
**2001**, 410, 183–186. [Google Scholar] [CrossRef] [PubMed] - Rosen, I.T.; Andersen, M.P.; Rodenbach, L.K.; Tai, L.; Zhang, P.; Wang, K.L.; Kastner, M.; Goldhaber-Gordon, D. Measured potential profile in a quantum anomalous Hall system suggests bulk-dominated current flow. arXiv
**2021**, arXiv:2112.13123. [Google Scholar] - Mukhopadhyay, U.; Dehollain, J.P.; Reichl, C.; Wegscheider, W.; Vandersypen, L.M.K. A 2 × 2 quantum dot array with controllable inter-dot tunnel couplings. Appl. Phys. Lett.
**2018**, 112, 183505. [Google Scholar] [CrossRef] - Dehollain, J.P.; Mukhopadhyay, U.; Michal, V.P.; Wang, Y.; Wunsch, B.; Reichl, C.; Wegscheider, W.; Rudner, M.S.; Demler, E.; Vandersypen, L.M.K. Nagaoka ferromagnetism observed in a quantum dot plaquette. Nature
**2020**, 579, 528–533. [Google Scholar] [CrossRef] [PubMed][Green Version]

**Figure 1.**(main panel) False colored scanning electron micrograph (SEM) of the key parts of the entropy sensor. Electrostatic gates (gold) define the circuit in a 2D electron gas (2DEG). The thermal electron reservoir (red) can be rapidly heated by driving current through quantum point contacts (QPCs) far away. (inset) Current through the charge sensor, ${I}_{CS}$, for a wide sweep of the coarse plunger gate, ${V}_{P}$, demonstrating the alignment of the $0\to 1$ transition at the steepest part of the trace to maximize sensitivity.

**Figure 2.**(

**a**) SEM micrograph of the full measurement device showing the large (10 µm square) chambers used for electron thermalization, QPCs 1 and 2 through which Joule heating current ${I}_{H}$ flowed, and QPCs 3 and 4 through which heat diffused but no net current flowed. The dashed rectangle in the upper left is the region shown in Figure 1, including QD and charge sensor. (

**b**) Crosses: broadening of the charge transition ($\mathrm{\Theta}$, left axis), converted to electron temperature (${T}_{e}$, right axis), increases above the sample temperature, ${T}_{s}$ = 100 mK, due to ${I}_{bias}$ driven through QPCs 1 and 2. Solid line: quadratic fit to $|{I}_{H}|<1$ nA data, with deviations seen at higher $|{I}_{H}|$. (

**c**) Extension of panel (

**b**) to higher ${I}_{H}$ and for a range of different sample temperatures. Sub-linear behavior at very large ${I}_{H}$ reflects electron–phonon cooling at higher temperatures.

**Figure 3.**A step-by-step inspection of the analysis procedure that goes into an eventual calculation of $\Delta S$. The fine-tuning plunger gate, ${V}_{D}$, is used to lower the energy of the QD level such that one electron enters from the thermal reservoir. (top) Schematic illustration of ${I}_{H}\left(t\right)$ through one complete 80 ms cycle. (

**a**–

**d**) Charge sensor current through the 80 ms cycle, calculated with respect to the unheated sections, at four locations on the $0\to 1$ transition: ${V}_{D}=-0.5,-0.1,0.1,0.5$ mV. Data shown here are averaged over 1200 square wave cycles. Blue (red) indicates times at which the thermal reservoir is unheated (heated). The relaxation time of the measurement (∼$3\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$) is visible in panels (

**b**,

**c**). (

**e**) Charge sensor current separated into averages over the 4 parts of the square heating wave, where heating is applied with an alternating current direction (${I}_{H}=3,-3$ nA), with zero bias applied in between (${I}_{H}=0$). Fits to the average “cold” and “hot” data are shown in grey. (

**f**) The difference in charge sensor current between the “cold” and “hot” traces. (

**g**) $\Delta S\left({V}_{D}\right)$ obtained by integration of $\Delta {I}_{CS}$ using Equation (1). $\Delta T$ is 28.1 mK, equivalent to 0.011 mV when converted to effective gate voltage, determined from the difference in thermal broadening of heated and unheated ${I}_{CS}$.

**Figure 4.**(

**a**) False-color scanning electron micrograph similar to the entropy measuring circuit from Ref. [1] where the thermal electron reservoir was heated by ${I}_{H}$ through a single QPC (top), with no additional confinement of the heated channel. (

**b**) Using the circuit in panel (

**a**), $\Delta {I}_{CS}$ measurements over the $0\to 1$ transition for 0, 100 and 200 mT of magnetic field applied perpendicular to the plane of the 2DEG. Then, 100 and 200 mT data are offset by 0.05 and 0.1 nA respectively. Illustrates the effect of unthermalized electrons from the heater QPC reaching the QD, for 0 and 100 mT data. Fits to theory for weakly coupled transitions (solid grey) emphasize the deviation of data from theory on the $N=0$ side of the transition. (

**c**) Four segments of ${I}_{H}$ square wave averaged separately, analogous to Figure 3e and made using the circuit in Figure 2a, but without proper balancing to keep the chemical potential of the reservoir at ground. The result is a shift of ${I}_{H}=+5\phantom{\rule{3.33333pt}{0ex}}\mathrm{nA}$ with respect to −5 nA data. Inset: zoom-in to the ${V}_{D}=-0.4\to -0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{mV}$ range of the main panel, showing both lateral and vertical offsets $\pm 5\phantom{\rule{3.33333pt}{0ex}}\mathrm{nA}$ data.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Child, T.; Sheekey, O.; Lüscher, S.; Fallahi, S.; Gardner, G.C.; Manfra, M.; Folk, J. A Robust Protocol for Entropy Measurement in Mesoscopic Circuits. *Entropy* **2022**, *24*, 417.
https://doi.org/10.3390/e24030417

**AMA Style**

Child T, Sheekey O, Lüscher S, Fallahi S, Gardner GC, Manfra M, Folk J. A Robust Protocol for Entropy Measurement in Mesoscopic Circuits. *Entropy*. 2022; 24(3):417.
https://doi.org/10.3390/e24030417

**Chicago/Turabian Style**

Child, Timothy, Owen Sheekey, Silvia Lüscher, Saeed Fallahi, Geoffrey C. Gardner, Michael Manfra, and Joshua Folk. 2022. "A Robust Protocol for Entropy Measurement in Mesoscopic Circuits" *Entropy* 24, no. 3: 417.
https://doi.org/10.3390/e24030417