Electronic Interactions in Semiconductor Quantum Dots and Quantum Point Contacts

A dissertation submitted to the

Graduate School

of the University of Cincinnati

in partial fulﬁllment of the

requirements for the degree of

Doctor of Philosophy

in the Department of Physics

of the College of Arts and Sciences

Tai-Min Liu

M. S. National Chung Cheng University, Chia-Yi, Taiwan

B. S. National Taiwan Normal University, Taipei, Taiwan

July 2011

Committee Chair: Andrei Kogan, Ph.D. Abstract

We report several detailed experiments on electron transport through Quantum Point Contacts (QPCs) and lateral Quantum Dots (QDs), created in a Single-Electron Transistor (SET). In the experiment for QPCs, we present a zero-bias peak (ZBP) in the differential conductance, G, which splits in an external magnetic field. The observed splitting closely matches the Zeeman energy and shows very little dependence on gate voltage, suggesting that the mechanism responsible for the formation of the peak involves electron spin. We also show that the mechanism that leads to the formation of the ZBP is different from the conventional Kondo effect found in QDs. [1] In the second experiment, we present transport measurements of a QD in a spin-flip cotunneling regime and a quantitative comparison of the data to the microscopic theory by Lehman and Loss. The differential conductance is measured in the presence of an in-plane Zeeman field. We focus on the ratio of the nonlinear G at bias voltages exceeding the Zeeman threshold to G for those below the threshold. The data show good quantitative agreement with the theory with no adjustable parameters. We also compare the theoretical results to the predictions of a phenomenological form used for the determination of a heterostructure g-factor and find good agreement between the two.

In the third experiment, we report the magnetic splitting, ∆K , of a Kondo peak in G for a

QD while tuning the Kondo temperature, TK , along two diﬀerent paths in the parameter space: varying the dot-lead coupling at a constant dot energy, and vice versa. At a high magnetic ﬁeld,

B, the changes of ∆K with TK along the two paths have opposite signs, indicating that ∆K is not a universal function of TK . At low B, we observe a decrease in ∆K with TK along both paths, in agreement with theoretical predictions. Furthermore, we find ∆K /∆ < 1 at low B and ∆K /∆ > 1 at high B, where ∆ is the Zeeman energy of the bare spin, in the same system. [2] In the last experiment, we report the zero-bias differential conductance, of an SET in the Kondo regime as a function of temperature, T, and an in-plane magnetic field B. Scaled plots of both the T - and B-dependent data show universal behavior. At moderate and high B, the magnetoconductance data show good agreement with renormalization group calculations in the spin-1/2 Kondo regime. At very low B, we observe a non-monotonic behavior, which may due to the presence of multiple orbital dot levels with similar energies. Further study is required to confirm this assumption.

ii iii Acknowledgments

Almost six years have passed since I began my PhD studies in Cincinnati, OH, and I could not have come this far without the many people present along the journey. I am deeply thankful for all the guidance, support, help, advice, and encouragement that I have received from these wonderful people over the past years. Without them, I would not be able to accomplish this dissertation.

First and foremost, I am grateful for my advisor Andrei Kogan. Andrei’s excellent insight and enthusiasm for Physics inspire me to devote myself to research in mesoscopic Physics. I have learned a lot through his guidance not only in the knowledge of physics but also in laboratory skills. If I were to become a professor in the future, I would look to Andrei as a role model. In addition, I also appreciate his support and understanding in my life outside of physics. Without it, I could not have spent an incredible and beautiful time with my wife I-Chun and my daughter Leah when the little one was born in Jan, 2011.

At the same time, I am thankful for all the help and encouragement from my PhD degree committee: Howard Jackson, Michael Ma, and Philip Argyres. I specially appreciate Howard’s mentoring and advising in both my academic career and everyday life. I will never forget his generosity in taking all of the students out for a fancy dinner at APS March Meetings too. Mike and Philip are two excellent professors. While Mike’s lecture is comprehensive, Philip’s is very accurate; I have learned and enjoyed a lot in their classes. I would also like to thank Mike’s generosity in taking us out for lunch when his class ends in a quarter and to hang out with graduate students. Many professors, Paul Esposito, Rohana Wijewardhana, Rostislav Serota, Frank Pinski,

Alex Kagan, Brian Meadows, Leigh Smith, Hans-Peter Wagner, Joseph Scanio, Young Kim, Mike

Sokoloﬀ, Kay Kinoshita, who are not in my committee are sincerely acknowledged for their lecturing and supporting. I am grateful to M. Jarrell, R. Serota and M. Ma for helpful discussions.

People down in the 1st ﬂoor of Geo/Phys building are the most wonderful ones in the world. I cannot imagine what my life would have become if Robert Schrott, Mark Ankenbauer, and John

Markus were not there. Bob and Mark gave us a lot of help in building the lab. They are also who introduced real American life to me. I have enjoyed a lot of wonderful time eating lunch with them either on the loading dock or oﬀ campus. John is another mentor I have had at UC; without his advice I could have made a lot of mistakes during my study. The work he has done to our

iv lab is tremendous; his enthusiasm to Physics will be always the example that I will push myself to pursue. There is one more thing that should and must be mentioned, which is spinning TOPS!

Thank you John for all the fantastic experiences related to tops.

I would also like to thank many people in Physics department: Richard Gass, Larry Bortner and W. Henry Leach gave me a lot of help when I was teaching the undergraduate recitations and labs. Donna Deutenberg and Elle Mengone help me to deal with all the paper work and purchases.

It is my honor to have the opportunity to collaborate with many excellent people and to meet professionals during my PhD research. I thank Steven Herbert from Xavier University for his help in our device fabrication and his encouragement; I thank Michael Melloch from Purdue University for his generosity in giving us the 2DEG material. Without this high quality material, we could not have reported any results presented in this thesis; I thank Jeﬀ Simkins, Robert Jones, and

Ronald Flenniken for their assistance in helping me become an expert with the facilities in the cleanroom. I thank Theo Costi from Institut f ¨urFestk¨orperforschung, Germany for his helpful discussion on magnetoconductance and his generosity in sharing his NRG calculation with me.

The most important and wonderful collaboration I have experienced was to work with Sergio Ulloa and his student Anh Ngo from Ohio University. I have learned a lot from them and have received more than what I expected through this collaboration. I hope we can continue to work together on more projects in the future.

A number of people in Andrei’s lab have contributed to the work done in my thesis, and I would like to acknowledge them. Amir Maharjan, Maryam Torabi established the lab before I joined the group. They set up most of the dilution refrigerator measurement circuitry that I used to perform the electron transport experiments together with two undergraduate students, Adam Simpson and

Kristen Herrmann. Bryan Hemingway is the fantastic partner and colleague in our lab; we have worked together on several successful projects. Besides the academic relationship, Bryan, Aaron

Wade, a post-doc fellow in our department, and Patrick Malsom, a graduate student, are also good friends of mine who help me to lower my “antisocial barrier”. I currently hold my record of drinking on the night they took me out to Arlins in Clifton.

A group of friends who overlap with me at UC are highly appreciated. I have spent the wonderful

ﬁrst year with Hyundoek Song, Herbert Fotso and the entire class of 2005. I specially thank Melodie

Fickenscher for her humongous help in proofreading my dissertation; she is so sweet, and I hope

v she can ﬁnd a good job in Cincinnati area. Aaron, also one of the spinning tops “cluster” at UC, gave me a lot of help and useful advice during my PhD studying and job searching. Without his assistance, I probably would not be writing the Acknowledgments in Berlin, Germany now; so far

I enjoy my Europe job interviewing trip.

I would also like to thank my former advisors in Taiwan, Chih-Ta Chia at National Taiwan

Normal University, Chia-Lian Cheng at National Dong Hwa University, Tai-Huei Wei at National

Chung Cheng University, and Yu-Ming Chiang at National Taiwan University for initiating me into the research world and for their teaching and training. I specially thank Chih-Ta who personally funded me US$2,000 when I left for the US in 2005.

A round of thanks has to be extended to the members of the Cincinnati Taiwanese Presbyterian

Ministry and Church by the Woods in Cincinnati, the First Baptist Church in Sioux Falls, my former teachers, my relatives, and my friends for their fantastic support. Last but most certainly not least,

I would like to call attention to my family: my mom Su-Hsia Chien and my dad Chi-Kuang Liu, my parents-in-low Chwan-Yi Chiang and Mei-Lien Chuang, my brothers Tai-Tsung Liu and Tai-Yu

Liu, and my lovely wife I-Chun Chiang. Thank you for your unconditional love and support. You are the world to me.

I acknowledge ﬁnancial support from these University of Cincinnati Awards: the URC Grant of the University Research Council, the Mary J Hanna Fellowship of the Department of Physics, and the ERC Cleanroom Awards of the Institute for Nanoscale Science and Technology .

vi Contents

1 Introduction 1

2 Transport 6

2.1 GaAs/AlGaAs heterostructure ...... 7

2.1.1 2 dimensional electron gas ...... 7

2.1.2 Hall measurement ...... 8

2.2 Measurement setup and electrical characterization ...... 10

2.3 Transport in a quantum point contact ...... 13

2.4 Transport in a lateral quantum dot ...... 18

3 Magnetic Splitting in a Quantum Point Contact 22

3.1 Introduction ...... 22

3.2 “0.7 structure” and zero-bias anomaly ...... 23

3.3 Magnetic splitting of zero-bias peak ...... 25

3.3.1 QPC in three diﬀerent conﬁgurations ...... 25

3.3.2 Zero-bias peak splitting in Coulomb blockade free regime ...... 26

4 Spin-Flip Cotunneling in a Quantum Dot 34

4.1 Introduction ...... 34

4.2 Cotunneling transport regime of quantum dot ...... 36

4.2.1 Microscopic Model ...... 38

4.3 Tunneling rate of a quantum dot ...... 40

4.4 Γ and energy dependence of cotunneling through the Zeeman splitting ...... 42

vii 5 Magnetic-Field-Induced Crossover to a Nonuniversal Regime in a Kondo Dot 48

5.1 Introduction ...... 48

5.2 Transport measurements of a Kondo dot ...... 50

5.3 Kondo temperature and scaling ...... 53

5.4 Kondo peak splitting in magnetic ﬁeld ...... 55

5.5 Crossover to a nonuniversal regime ...... 57

5.5.1 Magnetic splitting: tune TK via dot energy ...... 59

5.5.2 Magnetic splitting: tune TK via tunneling rate ...... 59 5.5.3 Orbital eﬀects due to B-ﬁeld ...... 60

6 Magnetoconductance of a Single-Electron Transistor in the Kondo Regime 65

6.1 Introduction ...... 65

6.2 Universal scaling of conductance as a function of temperature and magnetic Field . . 66

7 Conclusion 75

A Device Fabrication 78

A.1 GaAs substrate preparation ...... 78

A.2 Photolithography: Karl Suss MJB 3 ...... 79

A.3 E-beam lithography: Raith 150 ...... 80

A.4 Metal deposition and lift-oﬀ ...... 81

A.5 Ohmic contact ...... 83

A.6 Packaging and bonding ...... 85

B Peak splitting extraction 86

viii List of Figures

2.1 Lattice constant of semiconductors and schematics of 2DEG ...... 7

2.2 A schematic of the heterostructure and the energy band structure ...... 8

2.3 SdH oscillations and the Integer Quantum Hall eﬀect ...... 10

2.4 Transport measurement circuit ...... 11

2.5 Noise characterization ...... 12

2.6 Schematic of the split-gate QPC ...... 14

2.7 Quantized conductance plateau of QPC ...... 14

2.8 Non-linear diﬀerential conductance of QPC ...... 17

2.9 SEM image of SET device ...... 18

2.10 Coulomb blockade oscillation ...... 19

2.11 Energy diagrams and Coulomb blockade diamonds ...... 21

3.1 Devices micrograph and cotunneling spectroscopy ...... 24

3.2 Nonlinear conductance of a QPC ...... 25

3.3 Conductance map of three diﬀerent QPC conﬁgurations ...... 27

3.4 Kondo peak splits in a magnetic ﬁeld ...... 28

3.5 Gate dependent of ZBP splitting ...... 29

3.6 Zero-bias peak for two QPC conﬁgurations ...... 31

3.7 Comparison of ZBP ∆ and Zeeman ∆Z splitting ...... 33

4.1 Cotunneling process through a Zeeman split orbital ...... 37

4.2 Cotunneling transport spectroscopy ...... 38

4.3 Tunneling rate as a function of VS ...... 41

ix 4.4 Cotunneling transport conductance ...... 43

4.5 Comparison of microscopic calculations to measurements ...... 45

4.6 Comparison of the conductance ratio for tunneling rate and energy dependence . . . 46

5.1 Single Electron Transistor and transport measurements ...... 51

5.2 Conductance scaling and Kondo temperature ...... 54

5.3 Electron and mixing chamber thermometer temperatures ...... 55

5.4 Method of extracting the Kondo splitting ...... 56

5.5 Kondo splitting ∆K as a function of gate voltage VG for several B-ﬁelds ...... 58

5.6 Deviation of ∆K,0 from ∆ as a function of B-ﬁled ...... 60

5.7 Deviation ∆K,0 − ∆ as a function of VS along a constant ϵ0 path ...... 61 5.8 Magnetic shifting of dot energy level ...... 63

6.1 Temperature dependent Kondo eﬀect ...... 67

6.2 Kondo peak in magnetic ﬁeld ...... 69

6.3 Magnetic ﬁeld dependent Kondo eﬀect ...... 70

6.4 Magnetic ﬁeld dependent Kondo eﬀect ...... 71

6.5 TK vs BK ...... 72 6.6 Magnetoconductance for two SETs ...... 74

A.1 Schematics of mask aligner ...... 80

A.2 Clear dose for PMMA 950 A3 ...... 82

A.3 Characteristic curve of ozone cleaning ...... 83

A.4 Au-Ge Ohmic contact system ...... 84

A.5 wire bonding and epoxy bonding ...... 85

B.1 Peak splitting extraction at low B-ﬁelds ...... 87

B.2 Representative traces of peak splitting at low B-ﬁelds ...... 88

x List of Tables

4.1 Capacitance ratio αG for diﬀerent VS ...... 41

B.1 Splitting with/without leveling ...... 87

xi Chapter 1

Introduction

Rapid miniaturization of components in electronics has generated interest in the behavior of con-

fined electrons in a solid-state environment; it also has drawn attention to possibly discovering and eventually exploiting related quantum phenomena. Systems that produce a sufficient degree of electronic confinement to make quantum effects visible are called mesoscopic because their dimensions fall between the two well-understood limits: infinite conductors and nature-made atoms.

Research in mesoscopic systems has rapidly accelerated during the past two decades due to advances in fabrication techniques on the sub-micron scale, and has, in recent years, increasingly focused on many-body phenomena because of interactions between the electrons [3, 4, 5, 6, 7, 8].

Even the simplest possible mesoscopic arrangements are strongly influenced by many-body effects, but many of their properties remain unexplained [9]. This motivates us to research the low dimensional systems: lateral Quantum Dots (QDs), created in a Single-Electron Transistors (SET), and Quantum Point Contacts (QPCs). In this thesis, we use a combination of photolithography and e-beam lithography techniques to fabricate nanostructures on a GaAs/AlGaAs heterostructure, grown by molecular-beam epitaxy (MBE) with modulation doping, which contains a 2 dimensional electron gas (2DEG) 85 nm underneath the surface. In order to observe quantum phenomena of confined electrons, we first need to construct structures with dimensions on the order of a few

Fermi wavelengths. The magnetotransport measurements give the 2DEG material electron sheet density 4.8×1011 cm−2 and mobility 5×105 cm2V−1s−1, which corresponds to a Fermi energy of

17 meV and Fermi wavelength of ∼ 36 nm; therefore, the characteristic dimensions of our devices are on the order of 100 nanometers. Metallic electrodes are patterned on the surface of the 2DEG

1 material to form QPC or SET devices depending on the design geometry. By applying negative voltages on these metallic electrodes, electrons of the 2DEG are depleted underneath them and eventually are shaped into a quasi-1D electron transport channel for a QPC device or an isolated quasi-0D electron droplet for an SET device. Electron transport through a device is then measured via the ohmic contacts (source and drain leads), made by using a rapid thermal annealing technique

(RTA), which provides the lab circuit an access to the 2DEG of the devices. A pair of electrodes with a gap (a so-called “split gate”) produces a QPC, a quasi-1D wire. The convenience of such

QPCs is that one can continuously vary the conﬁnement potential by changing the gate voltages.

Two QPCs separated by a small region, a quantum dot, form a single-electron transistor (SET).

The term SET reflects that the on-site Coulomb repulsion between electrons trapped in the dot region can be made larger than the temperature; as a result, an electron can be loaded (unloaded) onto (off) the dot individually, which makes it possible to study few-electron states of a QD. Since the signal generated by the many-body quantum phenomena is very weak compared to the thermal energy of the environment, our experiments must be performed at sub-Kelvin temperatures in a dilution refrigerator; therefore, a very quiet circuit is essential for a successful measurement. Given the fact that the width of the Fermi step is approximately 3.5 kT, a measurement performed at 10 millikelvin, the base temperature of our refrigerator, will be affected by the source-drain noise of

3.0 microvolts or higher, which sets the upper bound of the noise tolerance. For our lab circuitry setup, the measured noise level is around 2 microvolts using a 1M ohm resistor as a test sample, which satisﬁes the requirement to perform a good measurement.

Nanoelectronics has been proposed as the next generation electronics. However, for it to become realistic would require functional nanodevices that can provide a way for the purpose to utilize the quantum eﬀects. Almost all nanodevices currently studied involve QPCs, the simplest nanostructure, in their design. Ballistic transport of QPC is now understood by using the Landauer formalism [10, 11, 12], which describes the ballistic conduction in 1D system as transmission problem . Yet, the non-linear I-V characteristics of many QPCs show a Zero-Bias Peak (ZBP), which is challenging to explain by using single-particle eﬀects alone. Recently, Rejec and Meir [6] report in

Nature a possible impurity formation in the QPC to explain the origin of the ZBP. Nevertheless, it is not clear what role the electronic interactions play in point contacts and how that depends on their geometry, which motivates us to investigate the electron transport in a QPC with a tunable

2 potential.

The primary advantage of SETs (or lateral QDs) in Kondo physics is that a lot important microscopic dot parameters can be tuned in-situ: the dot energy as well as the dot occupancy are tuned by adjusting the plunge gate voltage; the dot-lead coupling is manipulated by the voltages on the electrodes that create the potential barriers between the dot and leads; the Fermi energy of the leads can be modulated by the bias across the source-drain leads. This makes an SET a very convenient system to study Kondo physics. A QD which is loaded with an odd number of electrons is considered as a single site magnetic impurity since it contains a net spin, and the unconﬁned electrons in the leads screen the spin to form a many-body singlet; this is much like the bulk material system that Kondo studied in the 60’s: conduction electrons screen out the impurity atoms. A QD provides a perfect system to model Kondo physics and hence improve the understanding of Kondo physics. For instance, we are able to study the non-equilibrium Kondo eﬀect by applying a drain source bias, which is something that can not be reached in bulk material systems. In addition, the capabilities of manipulating Kondo temperature, dot energy, dot-lead coupling, etc., allows us to investigate direct comparisons between the theoretical calculations and the experimental measurements under various perturbations.

Possible applications of SETs (or lateral QDs), besides the fundamental and conceptual research interests, are proposed because the quantum dots exhibits lots of spin eﬀects. To date, most electronics are based on controlling the electron charges in the devices; whereas electronics based on the spins of electrons, namely spintronics [13, 14, 15, 16], is expected to improve the electrical world by reducing the energy cost and speeding up the information processing. For instance, QDs have been used as spin ﬁlters [17, 18, 16] and spin pumps [19] in various applications of spintronics.

Quantum dots are also “famous” for the potential application in quantum computing. For example, under a magnetic ﬁeld, the dot state splits into two states: spin-up and spin-down by the Zeeman energy, thus providing a two level system that can be regarded as a quantum bit, also known as a qubit [20, 21, 22]; such that a quantum computer could be made. However, to make all these nice ideas become realistic still requires further studies to solve unknown questions.

In this dissertation, we study the fundamental phenomenon and report several detailed experiments on electron transport of our nanodevices. In Chapter 3, we present a zero-bias peak in the diﬀerential conductance of a QPC, which splits in an external magnetic ﬁeld. The peak is observed

3 over a range of device conductance values starting signiﬁcantly below 2e2/h. The observed splitting closely matches the Zeeman energy and shows very little dependence on gate voltage, suggesting that the mechanism responsible for the formation of the peak involves electron spin. Precision

Zeeman energy data for the experiment are obtained from a separately patterned SET located a short distance away from the QPC. The QPC device has four gates arranged in a way that permits tuning of the longitudinal potential. We show that the agreement between the peak splitting and the Zeeman energy is robust with respect to moderate distortions of the QPC potential. We also show that the mechanism that leads to the formation of the ZBP is diﬀerent from the conventional

Kondo eﬀect found in quantum dots. [1] This work is done in collaboration with Bryan Hemingway,

Dr. Steven Herbert at Xavier University, and Dr. Michael Melloch at Purdue University.

In Chapter 4, we present detailed transport measurements in a quantum dot in a spin-flip cotunneling regime and a quantitative comparison of the data to microscopic theory. [23] The differential conductance G is measured in the presence of an in-plane Zeeman field. We are interested in the ratio of the nonlinear conductance values at bias voltages exceeding the Zeeman threshold, a regime that permits a spin-flip on the dot, to those below the Zeeman threshold, when the spin-flip on the dot is energetically prohibited. The data obtained in three different oddly-occupied dot states show a good quantitative agreement with the theory with no adjustable parameters. We also compare the theoretical results to the predictions of a phenomenological form used previously for the analysis of non-linear cotunneling conductance and specifically the determination of a heterostructure g-factor, and find a good agreement between the two. This experimental work is done with Bryan

Hemingway, Dr. Steven Herbert, and Dr. Michael Melloch; the theoretical work is done by Dr.

Anh Ago and Dr. Sergio Ulloa at Ohio University.

Kondo peak is suppressed at zero bias with a magnetic ﬁeld, and eventually the split peaks are recovered at the bias voltages approximately equal to the Zeeman energy of the dot. Theory

[24] predicts the splitting should be equal to the Zeeman energy and a universal function of Kondo temperature, TK ,. In Chapter 5, we study the magnetic splitting, ∆K , of a Kondo peak in the differential conductance of a quantum dot while tuning the Kondo temperature along two different paths in the parameter space: varying the dot-lead coupling at a constant dot energy, and vice versa to examine the proposed predictions. At a high magnetic field, B, the changes of ∆K with

TK along the two paths have opposite signs, indicating that ∆K is not a universal function of TK .

4 At low B, we observe a decrease in ∆K with TK along both paths, in agreement with theoretical predictions. Furthermore, we ﬁnd ∆K /∆ < 1 at low B and ∆K /∆ > 1 at high B, where ∆ is the Zeeman energy of the bare spin, in the same system. [2] This project is done in collaboration with

Bryan Hemingway, Dr. Steven Herbert, and Dr. Michael Melloch.

In Chapter 6, we report the zero-bias conductance of an SET in the Kondo regime as a function of temperature, T, and magnetic field, B, oriented parallel to the plane of the device. Scaled plots of both the T - and B-dependent data show universal behavior. At moderate and high B, the magnetoconductance data show good agreement with renormalization group (NRG) calculations in the spin-1/2 Kondo regime. At very low B, we observe a non-monotonic behavior: as B increases, the conductance initially increases and only starts to decrease at a finite B. A possible explanation of this effect is due to the presence of multiple orbital dot levels with similar energies. This project is progressing with collaboration with Bryan Hemingway, Dr. Steven Herbert, Dr. Michael Melloch,

Dr. Anh Ngo, Dr. Sergio Ulloa, and Dr. Theo Costi at IFF, Germany.

5 Chapter 2

Transport

The purpose of this chapter is to provide a brief review of basic electron transport in 0D and 1D nanostructures. 0D and 1D structures are also known as nanodevices or mesoscopic devices, which indicate structures that are larger than the atomic (microscopic) scale but smaller than the bulk

(macroscopic) scale. Electron transport in macroscopic systems has been well described via the

Boltzmann or similar formalism; in contrast, transport in mesoscopic systems shows interesting quantum eﬀects, yet many unexplained phenomena remain.

In the first section, I will introduce a semiconductor heterostructure that contains a 2 dimensional electron gas (2DEG) with high electron mobility and density. On this material, we fabricate single electron transistor (SET) and quantum point contact (QPC) devices in order to study electron transport for confined mesoscopic systems, which are beyond the Boltzmann regime. In the second section, I will describe the experimental setup which is designed to detect the very weak electron transport signal generated by these nanodevices. I will discuss the basic physics of electron transport for the 1D QPC system in the third section and for the 0D Quantum Dot (QD) system in the fourth section. Readers who are interested in learning more about electron transport in nanostructures can find comprehensive details in references [25, 26, 27, 9].

6 a) b)

e- e- e- e- e- + + + + +

DEC undoped GaAs n-AlGaAs

z c) 2DEG e- e- e- e- e- + + + + + EF

Figure 2.1: (a)Band gap as a function of lattice constant for several III-V semiconductors[28]. Schematics of 2DEG formation: (b) before doping; (c) after doping [26].

2.1 GaAs/AlGaAs heterostructure

2.1.1 2 dimensional electron gas

Various mesoscopic devices have been fabricated on semiconductor heterostrutures for the purposes of both fundamental studies and applications. In our research, we use GaAs/AlGaAs heterostructure to make the devices since this material provides a 2DEG which has high electron mobility and density. A typical heterostructure which consists of an AlxGa1−xAs alloy layer on top of a thick GaAs substrate is grown by molecular beam epitaxy (MBE) since these two semiconductors have closely-matched lattices [Fig. 2.1(a)].

To introduce carriers (electrons in our case), the AlGaAs layer in the heterostructure is n- doped with Si while the GaAs layer remains undoped. After doping, some electrons can migrate into the GaAs layer, where they are trapped at the interface of GaAs/AlGaAs. This is due to the discontinuity, ∆EC , in the energy band which sets up a barrier to prevent those electrons from being driven back to the donors in the AlGaAs layer by electrostatic repulsion [Fig. 2.1(b)(c)]. In the z direction, the electrons are trapped in a roughly triangular potential well which is about 10 nm wide at the energy of the electrons [26, 29]. It is often seen that all electrons occupy the same

7 a) b)

i-GaAs 5nm doped

n-Al0.3Ga0.7As 60nm undoped

i-Al0.3Ga0.7As 20nm

i-GaAs 500nm

Al0.5Ga0.5As(5nm)/GaAs(5nm) x20

i-GaAs 800nm 2DEG

Semi-insulating substrate

Figure 2.2: (a) A schematic of the heterostructure on which our nanodevices are fabricated for this research. (b) Energy band structure of a modulation-doped heterostructure [30].

lowest quantized energy state for the motion in the z direction; nevertheless, electrons can still move freely in the x and y directions. This is the so-called two dimensional electron gas (2DEG).

A schematic and a band structure diagram of the GaAs/AlGaAs heterostructure used to fabricate our nanodevices are shown in Fig. 2.2. The material is grown via modulation-doped MBE technique by our collaborator, Prof. Michael Melloch at Purdue University. We note that an undoped AlGaAs spacer layer is grown to further reduce the coulombic interaction between the conducting electrons and the donor impurities such that the low-temperature mobility can be sig- niﬁcantly enhanced [30, 31].

2.1.2 Hall measurement

To characterize the 2DEG of our heterostructure, we perform a magnetotransport measurement through a Hall bar with a magnetic ﬁeld perpendicular to the heterostructure at 4.2K. The mea-

11 −2 5 2 surement gives the sheet density n2d = 4.8×10 cm and mobility µ ≥ 5×10 cm /V sec. We are then able to further deduce the mean free path of the electron in the 2DEG: lMFP = 5.8µm. The Hall bar, shown in Fig. 2.3(a), is cooled in a dilution refrigerator at liquid helium temperature in

8 non-circulating mode. We simultaneously measure the longitudinal current, ICD, through connecting terminals C and D, and the dc voltage, VAB, across A and B with a vertical magnetic field from 0 to 8.52 T; then we flip the polarity of the magnetic field and redo exactly same measurement.

The longitudinal resistance is given by Rxx = VCD/ICD, and the transverse resistance is given by 1 | − ˜ | ˜ Rxy = Vxy/ICD, where the Hall voltage is Vxy = 2 VAB VAB , and VAB is the voltage across A and

B when the magnet polarity is flipped. Since the electron density n2d stays constant as a function of B-field for a 2D electron system, the number of occupied Landau levels must change as the level separation grows due to the increasing B-field. This results in two characteristics of the 2DEG: the longitudinal resistance showing strong oscillations with zeros at high B, known as the Shubnikov- de Hass (SdH) effect [26] and the transverse resistance showing plateaus, known as the Integer

Quantum Hall (IQH) eﬀect. Fig. 2.3(b) shows the SdH oscillations and the quantized plateaus in

11 −2 conductance for the 2DEG of our heterostructure. One can extract n2d = 4.8 × 10 cm from Eq. (2.1)

2e 1 n2d = (2.1) h P1/Bm

− where P1/Bm is the average of (1/Bm 1/Bm−1), Bm is the magnetic ﬁeld as Rxx reaches a local minimum, and m is the index of the IQH plateaus.

An alternative method to obtain n2d is based on the Hall eﬀect. The transverse resistance

Rxy = B/(en2d) depends linearly on B at low ﬁelds. From the slope of the Rxy as a function of B,

11 −2 we extract n2d = 4.7 × 10 cm , which is consistent to the result from using the SdH oscillations.

5 2 −1 −1 Once we know the n2d, we can then deduce the electron mobility µe = 5 × 10 cm V s from Eq. (2.2)

1 µe = (2.2) en2dRsquare

where Rsquare = 25.7Ω is the measured square resistance of the Hall bar mesa. Using Eq. (2.3)

v m∗µ l = F e (2.3) MFP e

∗ where vF is the Fermi velocity, and m is the eﬀective mass of the electron in GaAs, we ﬁnd the

9 a) b) 15 4 L = 110 mm W= 50 mm Rxy 3 ) A L 10 3 Rxx (k V w W C B D 4 2

I W ~ Rxy (k y 5 6 ) Rxx x 8 1

0 0 0 2 4 6 8 B (T)

Figure 2.3: (a) A schematic of the Hall bar on which we perform magnetotransport measurements. (b) The longitudinal resistance (Rxx) shows Shubnikov-de Hass (SdH) oscillations, and the transverse resistance (Rxy) shows the Integer Quantum Hall (IQH) eﬀect, which are the characteristics of 2D electrons.

mean free path of the electron in our 2DEG is lMFP = 5.8µm. The mean free path is much longer than the dimensions of our devices (couple hundred nanometers), which guarantees the transport measurements we performed are in the ballistic transport regime.

2.2 Measurement setup and electrical characterization

In order to perform electron transport measurements, we used a rapid thermal annealing technique to make Ni/Ge/Au ohmic contacts to the 2DEG of the heterostructure on which we fabricated the devices. The ohmics and gates were wirebonded to a chip carrier, which allows us to connect the laboratory circuitry to the devices. We applied a small AC excitation voltage across the source and drain leads of the devices and measured the corresponding current by using lock-in techniques.

Lock-in techniques are commonly used in lab experiments to improve the signal-to-noise ratio of measurements. Fig. 2.4 shows the sketch of our measurement circuitry. The dc voltages on gates and the bias voltage on ohmic contacts (source drain leads) are supplied by a National Instrument

6703 voltage card. All outputs are ﬁltered and the voltage is divided to deliver |Vgate| ≤ 2V for gates and |Vds| ≤ 5mV for drain and source leads. The bias voltage is also inductively coupled to

10 DI Lock-in CA

Vac

S DVds

D Vds

Figure 2.4: Transport measurement circuit.

an AC voltage Vac at frequency fac=17Hz from a lock-in ampliﬁer. The total root-mean-square

(RMS) voltage across the source drain leads is Vtotal = Vds +∆Vds, where ∆Vds is a small excitation voltage from Vac; this excitation voltage is reduced by the transformer and the voltage divider before going through the devices. We measure the responding AC current ∆I at frequency fac by using DL instruments Model 1211 current pre-ampliﬁer (CA) together with the lock-in ampliﬁer.

In this way, we can measure the diﬀerential conductance, G = ∂I/∂Vds ≈ ∆I/∆Vds, at any given

2 Vds through our devices [29]. The conductance quantum e /h ≈ 1/(25.8kΩ) is often used as the unit of the diﬀerential conductance in our data [26, 25].

To see the phenomenon in which we are interested, our devices must be cooled to millikelvin temperatures and we must have a very quiet circuit. Since the width of the Fermi step is approximately 3.5kT [32, 33], a measurement at 10 mK temperature will be aﬀected by ∼ 3µV or higher noise between the source and drain leads; this sets the upper bound of the noise level. The noise across the source and drain leads of our circuit is characterized at room temperature by using a

1MΩ resistor as a test sample in the dilution refrigerator. The root mean square (rms) noise is described in Eq. 2.4 [34]

√ 2 − 2 VNoise = VDC VGND, (2.4)

11 4

2 VDC VGND V) 1 V

m Noise 6 ( 4 Noise

,V 2

GND 0.1 ,V

DC 6 V 4

2 0.001 0.01 0.1 1 10 100 F (kHz)

Figure 2.5: Noise characterization using a DL Instruments 1201 voltage amplifier. A 1M ohm resistor is connected to the source drain leads as a test sample while the rms noise level is measured. VGND is the rms noise of the amplifier itself, which is not part of the measurement circuit; VDC is the total output from 1201 while DC-couple mode is used to measure the rms noise across the test sample. The rms noise of the measurement circuit VNoise (Eq. 2.4) is characterized at several cut-off frequency F; the bandwidth for each cut-off frequency goes from ∼10 Hz to F.

12 where VDC is the total noise voltage while we use the DC-couple mode of DL Instrument 1201 voltage amplifier to measure the circuit noise; VGND is the rms noise of the voltage amplifier itself. Fig. 2.5 shows the noise level as a function of the cut-off frequency F for our circuit; the bandwidth for each cut-off F goes from ∼ 10 HZ to F. VNoise ≤ 3µV satisfies the requirement to perform successful measurements.

2.3 Transport in a quantum point contact

A quantum point contact is a short and narrow constriction; it can be considered as a quasi-1D wire connected to two electron reservoirs one on each side. The most studied QPC is formed by two depletion split gates on top of the heterostructure containing the 2DEG [Fig. 2.6(a)]. The width of the constricted channel formed between the two reservoirs on each side of the QPC is controlled by the gate voltage. Fig. 2.6(b) shows a typical saddle-like potential proﬁle of this kind of QPC once the constriction channel is formed. We note one can make QPC devices not only by this split-gate technique, but also by semiconductor etching [35, 36, 37, 38].

To study electron transport in a 1D channel, we connect the QPC device to the measurement circuitry (Fig. 2.4) via ohmic contacts on both sides of the channel. The most important feature observed for a QPC is the conductance quantization in the ballistic regime. The diﬀerential conductance shows quantized steps in units of 2e2/h as the width of the channel (voltage on gates) increases [Fig. 2.7(a)]. This conductance quantization was ﬁrst discovered by van Wees et.al. [39] and Wharam et.al. [40] and now is understood as a consequence of full transmission of 1D subbands

[Fig. 2.7(b)] with spin degeneracy (2 for the electron).

The data shown in Fig. 2.7(a) are the results of the numerical calculation using the adiabatic transport model in single particle regime described in Ref. [25], and the simulation program is coded with exact parameters of our heterostructure, i.e., sheet electron density, Fermi energy, eﬀective electron mass, etc.

Given a smooth saddle-like potential, the spatial variation of the potential in the transport (x) direction is much slower than in the transverse (y) direction using an adiabatic approximation, as shown in Fig. 2.7(a) inset, the corresponding two-dimensional Schr¨odingerequation can be written

13 a) b) Electrode

Source Y Drain X

X Y

Electrode

Figure 2.6: (a) A schematic of a split-gate quantum point contact illustrating how a conﬁned channel is formed between the source and drain leads via the electrostatic depletion (blue: electrons of 2DEG; white: depleted area). (b) The approximated potential of a QPC is a smooth function in the x direction, but the potential energy changes rapidly along the y direction. Electrons in the channel see very little barrier in the x direction; however, electrons are conﬁned in the y direction.

a) b) 30 12 50 25 10 0 d(x)

y(x) (nm) 20 8 -50 E /h) -500 0 500 F 2 6 x (nm) 15 G (e

4 E (meV) 10 12 a=10 2 14 5 a=10 0 0 0 50 100 -0.2 0.0 0.2 d0 (nm) kx (1/nm)

Figure 2.7: (a) Calculated conductance as a function of channel width [d0 → d(0)] using the adiabatic transport model and Launder formula shows the quantization and its dependence on the confined potential profile. Inset: Shorter (blue solid line) and longer (red dotted line) confined potential profiles. (b) Energy subbands of a QPC. The differential conductance shows a plateau whenever a subband comes below the Fermi energy.

14 as

{ ( ) } ¯h2 ∂2 ∂2 − + + V (x, y) ψ(x, y) = Eψ(x, y), (2.5) 2m∗ ∂x2 ∂y2 where m∗ is the effective mass of electron in 2DEG, and V (x, y) is the potential. Using both the adiabatic approximation and the hard wall confinement (see ref. [25] for details), the 2D Schrödinger equation can be simplified as a 1D equation

{ } ¯h2 ∂2 − + E (x) ϕ (x) = Eϕ (x), (2.6) 2m∗ ∂x2 n n n

th where En(x) is the eﬀective potential determined by the spatial variation of the n energy eigenvalue of the solution in the transverse (y) direction. For simplicity, we assume the constriction potential is symmetric (quadratic in our calculation), and then the eﬀective potential can be expressed as

¯h2n2π2 1 E (x) = ≈ V − m∗ω2x2, (2.7) n 2m∗d(x)2 0 2 x

th where d(x) is the width of the channel [Fig. 2.7(a) inset]. The transmission coeﬃcient for the n mode thus has a simple form as described in ref. [41]

1 Tn(E) = − − . (2.8) 1 + e 2[E V0]/¯hωx

2 Further expanding eq. 2.7 into a quadratic form, En(x) ≈ En(1 − αx ), where α is the curvature

2 2 2 ∗ 2 parameter of the potential proﬁle, En = ¯h n π /(2m d0) with d0 the minimum separation of the channel (Fig. 2.7 inset), we can rewrite the transmission coeﬃcient as

1 Tn(E) = − − , (2.9) 1 + e βn[E En] where √ 2m∗ βn = 2 . (2.10) α¯h En

Plugging in the transmission coeﬃcient into the two terminal Landauer formula, [10, 11, 12] the

15 corresponding conductance of the QPC is given as

2e2 ∑ 2e2 ∑ 1 G = Tn(EF ) = , (2.11) −βn[EF −En] h h n 1 + e where EF is the Fermi energy of the electron in the QPC channel. Fig. 2.7 shows the quantized conductance plateaus calculated for two diﬀerent α in the zero source-drain bias regime. The linear conductance steps an amount of 2e2/h each time an additional subband enters below the Fermi energy.

Differently from the linear regime, one can apply source drain bias to drive the QPC into a nonlinear regime where the Landaurer-Büttiker model [10, 11] needs modification. Due to the high bias, the number of subbands which permit successful transport from the right lead is different than the number for the left lead; this results in very interesting features in conductance as a function of bias voltage shown in Fig. 2.8. The equation used to generate the nonlinear conductance map is derived from

{ } − − − 2e2 ∑ 1 1 + eβn[µL e(ϕ0 Vds/a) εn] I = IL + IR = ln (2.12) βn[µR−e(ϕ0−Vds/a)−εn] h n βn 1 + e where µL(R) is the chemical potential of the left (right) lead, ϕ0 is the barrier height at zero bias, and a is the barrier shape parameter. The channel barrier changes as ϕ = ϕ0 − Vds/a at ﬁnite

Vds with µL − µR = eVds and En = eϕ + εn, where eϕ is the electrostatic potential and εn is the conﬁnement energy. Here we assume εn is independent of Vds. Thus the diﬀerential conductance,

G = ∂I/∂Vds, is given by ( ) 2e2 1 ∑ 1 1 ∑ 1 G = − ( − 1) . (2.13) −βn[µL−e(ϕ0−Vds/a)−εn] −βn[µR−e(ϕ0−Vds/a)−εn] h a n 1 + e a n 1 + e

Using the Hard Wall Conﬁnement approximation, where a = 2, the nonlinear G is simpliﬁed to

2 ∑ [ ] e L R G = Tn(EF ) + Tn(EF ) , (2.14) h n

L(R) where EF stands for the Fermi energy of the left (right) lead. As expected, the conductance accumulates at N × 2e2/h at zero bias as shown in Fig. 2.8, which is consistent with the quantized

16 A E

Vds=0 4 C L R

B E V >0 ds D 3

L R /h) 2 C E

G (e 2

Vds=0 L R

1 B E D A V =0 L ds 0 R -2 0 2

Vds (mV)

Figure 2.8: Calculated non-linear diﬀerential conductance as a function of bias. Each trace represents a speciﬁc channel width (d0 in our code); the width is increasing from bottom to top. A: Small (G< 2e2/h) conductance at zero bias is expected as a subband is approaching the Fermi level. B: The conductance can increase to half 2e2/h with the bias as part of the subband comes below the Fermi level of the right lead (R). C: The second conductance step is predicted with a value of 2 × 2e2/h as two subbands are below the Fermi level. The conductance decreases to 3/2 × 2e2/h while the negative bias is applied; This is due to the fact that part of the subband goes above the Fermi level of the right lead (R).

17 a) b) Vg 200 nm gate Source Cg

Source dot Drain

CS CD Drain Vds

Figure 2.9: (a) Scanning Electron Microscope image of our SET device. A QD, indicated by the dashed circle, is formed between gates. Source and Drain are connected to the heterostructure 2DEG via ohmic contact. (b) The capacitor model of a quantum dot.

plateaus in linear conductance. Importantly, at ﬁnite bias, the conductance reaches half of 2Ne2/h, which can be understood as the bias eﬀect described above and is illustrated by the diagrams from

(A)-(D).

2.4 Transport in a lateral quantum dot

Quantum dots, quasi-0D systems, refer to electrons being confined into a very small volume in all spatial dimensions. The quantum dot used in our research is called a lateral quantum dot, which is formed at the 2DEG of a semiconductor heterostructure by using an electrostatic depletion technique. Fig. 2.9 shows SEM images of the smallest electrodes of our QD devices. The lithography diameter of our dot is about 0.3 µm, which is much less than the electron mean free path in the 2DEG of our heterostructure (lMFP = 5.8µm). By applying negative voltages on these gate electrodes, electrons will be depleted underneath the gates and finally confined in a tiny area which has a local minimum potential. We can change the voltages on these gates to manipulate the shape and the size of the dot. To perform the electron transport through the dots, we connect the source and drain ohmics to the measurement circuit as illustrated in Fig. 2.4

An isolated dot can be modeled as a classical capacitor. The total charge on the the dot is given ∑ as Q = CdotVdot, where Cdot = Ci is the eﬀective dot capacitance considering the contribution

18 0.4

U /h) 2 0.2 G (e

N N+1 N+2 N+3 N+4

0.0 -900 -800 -700

V g (mV)

Figure 2.10: Coulomb blockade peaks observed at zero bias shows the charging eﬀect of the QD: Peaks are separated by the charging energy U; electrons are added onto the dot one by one as the voltage becomes less nagtive.

from all the gates, and source and drain electrodes. Since the number of electrons on the dot at a speciﬁc voltage setting is ﬁxed except at charge degeneracy cases, adding one electron to the dot changes the dot potential by ∆Vdot = e/Cdot. The dot energy thus changes accordingly

2 by ∆Edot = e /Cdot, which is also known as the charging energy, U. In a zero-bias transport experiment, we only scan the voltage, Vg of one speciﬁc electrode—the plunge gate; the voltage needed to add one electron on to the dot is ∆Vg = e/Cg. In terms of charging energy, it gives

∆Vg = (1/αg)U/e, where αg = Cg/Cdot. This charging effect is easily observed in transport measurements at low temperature since the thermal fluctuation is reduced (T < ∆ϵ, where ∆ϵ is the energy level spacing of a QD). The differential conductance exhibits a peak whenever an electron is loaded (or unloaded) onto (or from) the dot (Fig. 2.10).

Another interesting and important transport feature of a QD is the appearance of Coulomb blockade diamonds. This feature can be observed when both the source-drain and plunge gate ∑ voltages are scanned. Based on the capacitor model, the dot energy is changed by ∆E = (−e)αiVi.

19 Here we introduce the dot chemical potential deﬁned as µ(N) ≡ E(N) − E(N − 1) [34]. Since we keep the source lead grounded while we scan the gate Vg and the source drain bias Vds in our measurements, the dot chemical potential is then expressed as

µdot(N) = µ0(N) − eαgVg − eαdVds. (2.15)

In the zero-bias case, a spike in the diﬀerential conductance (Coulomb charging peak) is seen whenever the dot chemical potential lines up with the source-drain lead potential µdot = 0. In the non-zero bias case, a peak happens not only at µdot = 0 (where the dot energy level lines up with the chemical potential of the source lead) but also at µdot = (−e)Vds (where the dot energy level lines up with the chemical potential of the drain lead). In conductance as a function of Vds and

Vg plots, two branches of resonant tunneling, know as Coulomb blockade diamonds, are seen. The diamond edges can be described by two linear equations:

dV α ds = − g (2.16) dVg αd dV α ds = d (2.17) dVg 1 − αd

Fig. 2.11 shows a calculated result using our dot parameters. The diamond edges indicate peaks in conductance. Inside the diamonds, electron transport is blockaded. The Coulomb blockade diamonds are very important since we rely on them to extract the capacitance ratios (used as energy lever arms) from the two slopes of the edges and the charging energy U from the diamond size.

20 Vg (mV) -750 -700 -650

1.0 2

D G (e

0.5 2 /h) 0 (mV) A

ds B V

C 0.0 -2

A B C D

Vds V U+De ds S D S D S D S D

Figure 2.11: Conductance as a function of Vds and Vg shows a typical characteristic of QD– the Coulomb blockade diamonds. Current is blockaded within diamonds because of the Coulomb repulsion between electrons. A-D: Several cases in the diamond plot: A sequential tunneling (peak in G) is observed whenever the dot chemical potential enters the transport window, which is illustrated in the energy diagrams.

21 Chapter 3

Magnetic Splitting in a Quantum Point Contact

We report a zero-bias peak in the differential conductance of a Quantum Point Contact (QPC), which splits in an external magnetic field. The peak is observed over a range of device conductance values starting significantly below 2e2/h. The observed splitting closely matches the Zeeman energy and shows very little dependence on gate voltage, suggesting that the mechanism responsible for the formation of the peak involves electron spin. Precision Zeeman energy data for the experiment are obtained from a separately patterned single-electron transistor located a short distance away from the QPC. The QPC device is fabricated in a GaAs/AlGaAs heterostructure containing 2-dimenional electron gas and has four gates arranged in a way that permits tuning of the longitudinal potential.

We show that the agreement between the peak splitting and the Zeeman energy is robust with respect to moderate distortions of the QPC potential. We also show that the mechanism which leads to the formation of the ZBP is diﬀerent from the conventional Kondo eﬀect found in quantum dots.

3.1 Introduction

Current-voltage characteristics of ballistic Quantum Point Contacts (QPCs) [40] – narrow channels contacted by macroscopic conductors – have proven diﬃcult to understand despite the geometric simplicity of the QPC devices. Ballistic ﬂow of electrons in QPCs produces plateaus in the linear

22 conductance G separated by 2e2/h, found in many experiments and well understood [39]. Yet, the non-linear I-V characteristics of many QPCs show a zero-bias peak (ZBP), which is challenging to explain by single-particle effects alone. In this report, we focus on the properties of the ZBP at relatively low conductance values, < 0.5e2/h. This regime differs from the near-opening regime, in which the so-called “0.7 anomaly” in the linear conductance is often observed [3, 4, 5, 42] and where the effect of spin correlations on nonlinear transport has been of significant interest [5, 6, 7, 43].

Recent experiments [7] and theory [6] suggested that, near the opening, localization of unpaired spins in QPCs may occur and produce a ZBP due to an analog of the Kondo eﬀect [5, 44]. At the same time, an interpretation of the ZBP that does not involve electron spin was recently proposed

[8]. What we report here is that the ZBP related to electron spin can occur at conductance values significantly lower than 0.7 × 2e2/h even though the Kondo effect is controllably suppressed. This experimental work thus adds to the puzzle and shows that spin-dependent effects in point contacts can be important even when the tunneling is weak, in addition to the “near opening” regime.

3.2 “0.7 structure” and zero-bias anomaly

The ballistic transport of electrons in a QPC revealing quantized conductance has been fully explained within the non-interacting Fermi-liquid formalism. However, the deviations from the simple

QPC quantizations [45, 46, 3, 5] have also been observed, yet not fully understood [47]. Among the deviations, the extra shoulder-like plateau, typically occurring at a conductance of ∼ 0.7(2e2/h) at the zero source drain bias, is the most extensively studied feature [3, 4, 5, 48, 42, 43, 44, 38, 6].

Thomas et al. ﬁrst studied the 0.7 structure and relate it to the possible spin polarization of the

1D electron gas. In addition to the linear conductance approach, Cronenwett et al. [5] observe the zero-bias anomaly (ZBA) peak, associated to the 0.7 plateau seen in the linear conductance, when the source drain bias is scanned. They ﬁnd that the temperature dependence of the ZBA can be

fitted into a universal Kondo-like scaling function, which suggests the origin of the 0.7 structure might be a possible magnetic impurity forming in the QPC [44, 6]. What we are interested in is to understand the ZBA peak by means of magnetic field. The ZBP that we study occurs at conductance values significantly lower than 0.7 × 2e2/h even though the Kondo effect is avoided.

This regime is diﬀerent from the “near opening” regime and is unlikely to be related to the 0.7

23 a) b) c) -3 VT 8x10 200 nm 200 nm B = 8 T Source /h) V 2 4 V G G G (e Drain V 0 B -200 0 200 Vds (mV)

Figure 3.1: (a) Micrograph of a four-gate QPC nominally identical to that used in measurements with the gate voltage labeling convention shown. (b) Micrograph of an SET placed ∼ 150 µm away from the QPC device on the same chip for Zeeman energy measurements. (c) The plot of the nonlinear conductance of the SET device in the spin-ﬂip cotunneling regime showing characteristic steps at Vds = ∆Z /e. The step-to-step width is two times the Zeeman splitting ∆Z /e.

structure.

Our ZBP measurements are obtained with a semiconductor QPC sample that has 4 independent gates, which we use to manipulate the device potential profile along the direction of the flow of the QPC current. When no significant distortion of the potential is present, we find a clear ZBP at conductance values substantially below the first plateau. The ZBP splits with the application of an in-plane magnetic field B, applied perpendicular to the current flow direction. Further, we show that the result is robust against moderate distortions of the longitudinal potential. Distorting the potential by a large amount, however, produces a real bound state, likely localized between the device gates, as evidenced by the characteristic Coulomb blockade (CB) diamond and a zero-bias peak that we attribute to the conventional Kondo physics found in quantum dots. Importantly, when the QPC potential is “smooth” and the CB is not observable, the ZBP is still present and shows clear splitting with the magnetic field applied. The splitting closely matches precision Zeeman energy data, defined as ∆Z = |g|µB B , where g is the effective electron g-factor, µB is the Bohr magneton and B is the magnetic field, which we obtain independently from a Single-Electron

Transistor (SET) on the same chip. This shows that the ZBP in this regime is still related to the electronic spin, and rules out the conventional Kondo physics due to an accidental trapping of an unpaired electron in the device, which would produce CB features in addition to the ZBP.

24 2 /h) 2 G (e

0 -4 0 4

Vds (mV)

Figure 3.2: Nonlinear conductance of our QPC device shows the characteristic features: fully quantized at zero bias and half quantized at higher bias. The zero bias peak, which is not predicted using single particle model, is clearly seen.

3.3 Magnetic splitting of zero-bias peak

3.3.1 QPC in three diﬀerent conﬁgurations

The diﬀerential conductance G = dI/dVds of our QPC is measured via standard lock-in techniques with the excitation voltage of approximately 3.9 µV RMS at 17 Hz. The four gates of our QPC

11 are arranged on top of a GaAs/AlGaAs heterostructure (electron sheet density n2D = 4.8 × 10 cm−2 and mobility µ = 5 × 105 cm2/V sec at 4.2K) as shown in Figure 3.1(a). The same voltage

VG is applied to the two opposing gates, and the voltages VT and VB can be tuned to adjust the longitudinal potential proﬁle. A nearby SET device [Figure 3.1(b)], patterned approximately 150

µm away from the QPC, is used to measure the Zeeman energy and the electron temperature via spin-ﬂip cotunneling spectroscopy [49, 50, 23]. In this regime, the conductance of the SET shows steps at Vds = ∆Z /e [Figure 3.1(c)] and the steps slopes can be used to obtain the electron temperature, about 55 mK in our devices, as described in Refs. [12] and [15]. Fitting the Zeeman energy linearly with the magnetic ﬁeld gives the exact heterostructure g-factor |g| = 0.20730.0013

25 [2], which is much smaller than the bulk GaAs value |g| =0.44.

Figure 3.3 shows three representative gate voltage settings (a)—(c) and the corresponding nonlinear conductance maps (d)—(f) and the zero bias conductance curves (g)—(i). In each presented measurement, the voltage VG, applied to the opposing center contacts, is scanned. The voltages

VT and VB applied to the top and the bottom gates, respectively, control the longitudinal potential proﬁle: by setting both VT and VB to zero, as shown in Figure 3.3(a), we produce a “short” constriction, formed by the center gate alone. Applying a moderate negative voltage to VT and VB [Figure 3.3(b)] increases the constriction length. We note that, in both regimes, a zero-bias peak in the nonlinear conductance is observed over a range of values of VG, and the linear conductance rises

2 to a value close to 2e /h monotonically. As expected, the pinch-oﬀ voltage for the VT = VB = 0 data is more negative than for the VT = VB = −662 mV data. An example of a strong distortion of the potential is shown in Figure 3.3(c). A dramatic change in the conductance properties is observed in this regime: a characteristic “diamond” appears in the nonlinear conductance plot [Figure

3.3(f)], and the linear conductance displays sharp peaks before the 2e2/h plateau is reached, both typical features of a quantum dot in a CB regime [51]. We attribute this behavior to a quantum dot forming between the electric ﬁeld fringes created by the middle and the top gates, as shown in

Figure 3.3(c). Importantly, a zero-bias peak is present across the diamond [Figure 3.3(f)], which splits when an in-plane magnetic ﬁeld is applied [Figure 3.4(b)]. We interpret this feature as the conventional Kondo eﬀect often observed in quantum dots. With the ability to control the device regime, we thus avoid this Kondo feature to investigate the ZBP in QPCs.

3.3.2 Zero-bias peak splitting in Coulomb blockade free regime

The “smooth”, CB-free regime persists over a range of voltages VT , making it possible to compare the ZBPs observed at diﬀerent aspect ratios of the device potential. As VG is scanned, both conﬁgurations show bunching of the non-linear traces at Vds ∼ 2 mV that occurs as the lowest transport band enters the transport window, as commonly observed in split-gate point contacts

[4, 5, 7]. The ZBP, clearly seen in the data, splits when the magnetic field is applied. We define the peak splitting ∆/e as half the separation between the two peaks. We note that the separation between the two peaks in bias voltage for a conventional spin-1/2 Kondo effect is expected to be approximately twice the ratio of the Zeeman energy to the electron charge (∆Z /e) , and the same is

26 a) b) c) VT = -622 mV VT = -900 mV VT V V V G G G VG VG VG VB VB VB = -622 mV 2 G (e /h) d) 0 1 2 e) f)

4 4 1

0 0 0 (mV) (mV) (mV) ds ds ds V V V -1 -4 -4 -1720 -1670 -1620 -1510 -1460 -1410 -1500 -1450 -1400

VG(mV) VG(mV) VG(mV) g) h) i)

2 2 2 /h) /h) 2 2 /h)

1 1 2 1 G(e G(e G(e 0 0 0 -1720 -1670 -1620 -1510 -1460 -1410 -1500 -1450 -1400

VG(mV) VG(mV) VG(mV)

Figure 3.3: (a-c) Three representative device settings discussed in the text. Areas accessible to electrons are schematically shown in blue. (a) Short constriction, with VT and VB both set to zero. (b) Long constriction, with VT and VB both at a negative bias. (c) A strongly distorted potential, resulting in a formation of a quantum dot between the gates. (d-f) Nonlinear conductance maps for the three regimes shown above. A Coulomb diamond, clearly seen in (f), is not present in either (d) or (e). The ZBP is seen in each data set. (g-i) Linear conductance data corresponding to the plots shown in (d-e). (i) shows an onset of Coulomb oscillations signaling a formation of a quantum dot in the constriction. No such oscillations are present in (g) or (h).

27 a) b) B = 0 T B = 9 T 0.5 0.5 2

0.4 G (e G (e 2 0.0 1 2 0.0 /h) (mV) /h) (mV)

ds 0.2 ds V V

-0.5 0 -0.5 0.0 -1460 -1440 -1420 -1400

VG(mV) VG(mV)

Figure 3.4: Non-linear conductance data for the constriction in a Coulomb blockade regime with the gate voltages set as shown in Figure 3.3(c). (a) The portion of the data shown on Fig 3.3(i) corresponding to the CB diamond region at zero magnetic ﬁeld. (b) The same gate scan as in (a) with a 9 T in-plane magnetic ﬁeld present. The horizontal lines marked by the arrows show the Zeeman bias voltage threshold.

true for the voltage difference between the spin-flip cotunneling steps which we use to measure the g-factor. It is thus interesting to compare ∆/e to ∆Z /e. First, we note that ∆/e shows no strong dependence on the gate voltage. Apart from relatively small departures [16, 2], a similar behavior is expected for the conventional Kondo effect. Representative data at B= 9 T for the short and

B= 7 T for the long constrictions are shown in Figure 3.5. The behavior of the ZBP in both regimes is similar: as the device becomes more open, the two peaks become less defined, however, the splitting stays close to the values expected from the Zeeman energy data, marked on the plots with the dashed lines. Next, we fix the gate voltage and focus on the dependence of the splitting on the magnetic field. Figure 3.7 shows the comparison of the peak splitting (∆/e) to the Zeeman splitting (∆Z /e). We find that the splitting of the ZBP increases approximately linearly with the field, and follows closely the Zeeman splitting data obtained from cotunneling measurements. For comparison, we also show the peak splitting obtained from the data shown in Figure 3.4(b), when

Coulomb blockade is present. As expected for the conventional Kondo effect, we find a value which is close to ∆Z /e. The response of the ZBP in QPCs to an in-plane magnetic field varies, and no common picture currently exist. Several groups reported a ZBP that splits at channel conductances comparable to

28 a) B = 9T b) B = 7T 2 2

1 1 8 8 6 6 4 4

2 2 /h) /h) 2 2 0.1 0.1 G(e G(e 8 8 6 6 4 4

2 2

0.01 0.01 -200 0 200 -200 0 200 V (mV) Vds (mV) ds

Figure 3.5: Evolution of the split peak with the gate voltage for the short constriction (a): VG from -1634 (bottom) to -1593 mV (top) at B=9T and the long constriction (b): VG from -1507 (bottom) to -1470 mV (top) at B=7T. Dashed lines: Zeeman bias voltage threshold.

29 2e2/h but not at lower conductances [52, 5]. Chen et al. [8] reported zero-bias peaks in QPCs that did not split with the magnetic ﬁeld at all, and concluded that the phenomenon did not involve spin. The magnetic splitting of the ZBP signiﬁcantly larger than the bulk GaAs Zeeman energy was reported earlier [5], and attributed to the enhancement of the g-factor in 1-dimensional conductors

[53, 42, 54, 55]. Such enhancement of the g-factor in open channels has been reported in several experiments: Thomas et al. [53] found the eﬀective g-factor enhanced from 0.4 to ∼ 1.2. Koop et al. [42] found a g-factor enhanced by as much as a factor of ∼ 3 as compared to the bulk material, and a very recent work [55] also reported the enhanced g-factor in an open channel as well as its dependence on carrier density. Compared to these observations, our measurements are performed at relatively low (less than e2/h ) conductance values, i.e. in the tunneling regime when no actual

1-dimensional channel is formed. This may explain the absence of a similar g-factor enhancement in our data. Importantly, the peak splitting we report is in a regime where no signatures of Coulomb blockade ( no conductance “diamond”) for a conventional Kondo eﬀect are present, and also a direct comparison between the ZBP splitting and the Zeeman energy measured on the same chip is possible.

A geometry that favors a formation of a bound state inside the channel was used previously by Sfigakis et. al. [43], who performed extensive measurements of the temperature dependence of the zero-bias anomaly and concluded that the 0.7 structure and the singlet Kondo effect in a wire are two distinct effects. In that work, a two-gate geometry with small extensions near the ends of the contact was used, and, therefore, localization of an electron and the overall transmission of the channel were controlled by the same gate voltage. Thus, in the low conductance regime that we focus on in this work, it was not possible to independently control the degree of the confinement of the electron and the channel conductance. By contrast, the four-gate geometry used in our device allows one to controllably create or eliminate the bound state even at the very low conductance values. Our findings suggest that spin-dependent phenomena influence QPC transport even when the tunneling is relatively weak, in addition to the near-opening regime studied extensively by other groups recently [5, 7, 43].

In summary, we have observed a good quantitative agreement between the electron Zeeman energy and the magnetic splitting of a ZBP in a quantum point contact at conductance values signiﬁcantly below the ﬁrst plateau. This result is robust with respect to moderate distortions of

30 a) b)

1 1 /h) /h) 2 2 G(e G(e

0 0 -4 0 4 -4 0 4 V (mV) V (mV) c) ds d) ds -3 60x10 0.8 6T 3 T 0.6 /h) /h) 2 40 2 G(e G (e 0.4

9T 9 T 20 0.2 -200 0 200 -200 0 200

Vds (mV) Vds(mV)

Figure 3.6: (a), (b) Observed ZBPs for the short constriction: VG from -1735 (bottom) to -1697 mV (top) and long constriction: VG from -1512 (bottom) to -1484 mV (top) at zero magnetic field. Data shown in the figure are obtained by scanning VG only. (c), (d) Magnetic field dependence of the peak shape for the two configurations, showing that the splitting increases with the field in both cases. The traces are taken at VG = −1630mV (short) and VG = −1490mV (long).

31 the longitudinal potential of the QPC achieved via additional gates in the device design, and shows that even a relatively weak tunneling current in a QPC may be inﬂuenced by spin-dependent eﬀects.

Signiﬁcant distortions of the potential produce a conventional bound charge state accompanied by the Coulomb blockade and Kondo transport features similar to those found in quantum dots.

Coulomb blockade behavior is not present when the QPC potential is smooth. This suggests that an accidental trapping of charge in the channel and the ensuing singlet Kondo eﬀect as observed in quantum dots is not the origin of the ZBP observed in our sample, even though the behavior of our ZBP in magnetic ﬁeld and that of a Kondo peak is very similar: both exhibit splitting in bias voltage which is close to twice the Zeeman energy divided by the electron charge and is approximately independent of the gate voltage.

32 150 Kondo splitting

100 V) m /e ( D 50

0 0 2 4 6 8 10 B (T)

Figure 3.7: Comparison between the Zeeman energy, ZBP splitting in diﬀerent regimes, and Kondo splitting. Filled circles: Zeeman splitting obtained from SET cotunneling transport measurements. Triangles: The splittings of the ZBP in the short constriction (VB = VT = 0). Squares: the splittings of the ZBP in the long constriction (VB = VT = −622 mV). Diamond: Kondo splitting at the mid-point of CB valley extracted from ﬁgure 3.4(b).

33 Chapter 4

Spin-Flip Cotunneling in a Quantum Dot

We report detailed transport measurements in a quantum dot in a spin-flip cotunneling regime, and a quantitative comparison of the data to microscopic theory. The quantum dot is fabricated by lateral gating of a GaAs/AlGaAs heterostructure, and the conductance is measured in presence of an in-plane Zeeman field. We focus on the ratio of the nonlinear conductance values at bias voltages exceeding the Zeeman threshold, a regime that permits a spin flip on the dot, to those below the Zeeman threshold, when the spin flip on the dot is energetically forbidden. The data obtained in three different odd-occupation dot states show good quantitative agreement with the theory with no adjustable parameters. We also compare the theoretical results to the predictions of a phenomenological form used previously for the analysis of non-linear cotunneling conductance, specifically the determination of the heterostructure g-factor, and find good agreement between the two.

4.1 Introduction

Electronic transport in nanoscale devices, such as nano wires, semiconductor quantum dots, molecules, single atoms, and carbon nanotubes, [56, 57, 58, 59, 60, 61, 62] has been of signiﬁcant current interest. This is due in part to its use as a spectroscopic tool for precision studies of fundamental phenomena such as the Coulomb blockade and the Kondo eﬀect, and because of the relevance of

34 these devices in spintronics and quantum computation. [20, 21, 22] For spintronics application purposes, it is important to understand how the spin state of a nanosystem couples to its host surroundings. Spin-dependent transport can be conveniently studied in tunable quantum dots

(QD)s [63, 64, 49, 50, 16]. Using a dot weakly coupled to the “leads” with an applied in-plane magnetic field, Kogan et al. [49] showed that the differential conductance G = dI/dVds exhibits steps at Vds values given by the ratio of the Zeeman energy and the electron charge, e, and used a phenomenological fit to the transport data to measure the heterostructure g-factor.[49, 2] Later,

Lehmann and Loss [23] developed a microscopic theory to calculate the conductance through a QD in this regime, which included a phonon-assisted spin-flip mechanism. In this chapter, we present extensive transport data of a quantum dot dot in the spin-flip cotunneling regime and compare the results to a microscopic description.[23] Importantly, we measure all dot parameters needed for the calculation of the conductance, which enables a direct comparison between the data and the microscopic theory without any adjustable parameters, and find excellent quantitative agreement between the data and theory.

We present data obtained for three different choices of the dot potential defined by the voltages on the confining gates, which correspond to three different occupancies of the dot. We focus on the ratio of the device conductance above and below the Zeeman threshold as function of the tunneling rate and the dot energy. Since the orbital part of the wave function of the two Zeeman spin states is the same, the tunneling probabilities for each electron crossing the dot depend only on its spin and the spin of the dot. Therefore, a useful insight can be obtained from the ratio of the device conductance above and below the Zeeman threshold, i.e., when the bias across the dot matches the ratio of the Zeeman energy and the electron charge. If the coupling to the leads is extremely weak (i.e., the tunneling rates between the dot and the leads are much smaller than the spin relaxation rate on the dot) one might expect this ratio to be approximately 2: at large biases, there are two possible dot states (the ground spin state and the excited spin state) available upon the completion of each tunneling event, whereas at low biases, the dot has to remain in the ground spin state. In practice, however, the spin relaxation rate due to intra dot processes is usually very slow, compared to the tunneling rates in transport experiments between the dot and the leads.[65, 66] In that regime, therefore, exchanging spin with the leads is the dominant mechanism of the dot spin relaxation. Predicting the device conductance in this regime requires a formalism that includes a

35 complete set of rate equations, as we use in this work for a single-orbital two spin dot.[23] Our calculations and measurements both do reveal a nontrivial value for the conductance ratio ≈ 2.4, indicating the role of the current leads in providing spin relaxation in the dot.

Further, we show that this ratio is independent of the dot-lead tunneling rate Γ over approximately one decade, 0.02 < Γ < 0.2 meV, but varies slightly with the dot energy, exhibiting a slight minimum in the middle of the CB valley. Finally, we compare the shape of the nonlinear conductance as function of the dot bias as obtained from our calculations to the predictions of a phenomenological form used in the earlier work. [49] For a given Zeeman energy, we ﬁnd excellent agreement between the two, which means that either method provides a valid choice for using cotunneling transport for g-factor measurements. For the device used in this work, we ﬁnd, using both methods, the g-factor to be 0.2073 0.0013.

4.2 Cotunneling transport regime of quantum dot

The QD we have studied is created by gating a GaAs/AlGaAs heterostructure. Ti/Au electrodes of our SET are patterned via e-beam and photolithography followed by lift-oﬀ. The 2DEG under the electrodes is statically-depleted to form an electron droplet, i.e., a QD, connected on both sides to the electron reservoirs, source and drain [Fig.4.1(c)]. We estimate the diameter of the

QD to be ∼ 0.13 µm, which contains tens of electrons. From magnetotransport data we ﬁnd that the 2DEG has a mobility of 5 × 105 cm2/(Vs) and an electron density of 4.8×1011cm−2 at 4.2K.

The device is oriented parallel to the magnetic ﬁeld within 1 degree, and is cooled in a Leiden

Cryogenics dilution refrigerator to a base electron temperature Tele ∼ 55mK. We use standard lock-in techniques to measure the diﬀerential conductance through the QD.

Figure 4.1(d) shows the diﬀerential conductance steps at source-drain voltages equal to the

Zeeman energy of the dot. The tunneling between the dot and the leads is relatively weak, so that the Kondo eﬀect in this regime is suppressed by thermal ﬂuctuations. In the Coulomb blockade

(CB) regime, when the QD has an unpaired electron in the dot energy level, the spin degeneracy is removed by the Zeeman ﬁeld, and the level splits into spin up and spin down states. We label the conductances below and above the Zeeman threshold as G0 and G+, respectively. Figure 4.1(a–b) illustrates the possible tunneling processes: In an elastic event (a), the dot is left in the ground

36 a)

D S D b)

c) d) B = 9 T V 6 T G G (e 200 nm +

S 2 /h) ×10 VG VS 2D

D -3 G V 0 B 1.6 -200 200 Vds (mV)

Figure 4.1: (a–b) Cotunneling process through a Zeeman split orbital occurs when the dot is occupied by an odd number of electrons in Coulomb blockade regime. Taking spin-up to be the lower energy state, a spin-down electron from the lead can tunnel onto and off the dot resulting in non spin-flip cotunneling as shown in (a). When the bias voltage exceeds the Zeeman threshold, |eVds| ≥ ∆ = |g|µBB, the spin-up electron can also tunnel off the dot resulting in spin-flip cotunneling as shown in (b).(c) Micrograph of a device nominally identical to the one used in this work. A quantum dot is created after applying negative voltages on electrodes VT , VS, VB, and VG. The electrode VS is used primarily to vary the dot-lead tunneling rate Γ while the plunger gate VG is used to tune the dot energy. Differential conductance is measured through source(S) and drain(D) via standard lock-in technique with 2 µVRMS excitation at 17 Hz. (d) Differential conductance as a function of drain-source voltage Vds in the cotunneling regime shows lower conductance (G0) at |eVds| < ∆ and higher conductance (G+) at |eVds| ≥ ∆. Dashed lines are guides for the average conductance values of G0 and G+.

37 a) b) -3 -3 8x10 20x10 4T 5T 6 15 6T

7T /h) /h) 2|gm B| 2 2 4 B 10 8T G (e G (e 9T 2 5 0 -200 -100 0 100 200 -200 -100 0 100 200

Vds(mV) Vds(mV)

Figure 4.2: (a) Cotunneling conductance as a function of source-drain voltage for several ﬁelds. The curves are vertically shifted for clarity. (b) Fitting the trace to a phenomenological function, one can extract the corresponding electron temperature and g-factor.

state and the electron does not change its energy as it crosses the dot. If the dot is left in an excited state (b), the electron energy is lowered by ∆. The conductance steps at |Vds|=∆/e, can be described by a phenomenological form [67, 49]:

[ ( ) ( )] dI eVds + ∆ eVds − ∆ = Ae + Ai F + F − . (4.1) dVds kBT kBT

In the equation, Ae(Ai) represents the conductance contribution from elastic(inelastic) cotunneling. F is a function deﬁned as F (x) = [1 + (x − 1) exp(x)]/[exp(x) − 1]2. We ﬁt our cotunneling conductance traces to this form to extract the the corresponding electron temperature T and the electron g-factor for our dot.

4.2.1 Microscopic Model

The microscopic theory adopted in this work is proposed by Lehmann and Loss [23]. They consider a dot system in a spin-1/2 ground state connected to two leads [68, 23]. Their model can be

38 described by a Hamiltonian containing three terms

H = H0 + HT + HS. (4.2)

H0 contains three terms: the isolated dot, the ideal leads, and phonons

∑ ∑ ∑

H0 = Eσnσ + Un↑n↓ + ϵlknlkσ + ¯hωqnq, (4.3) σ=↑, ↓ lkσ q

† † where nσ is the number operator deﬁned as nσ = dσdσ, where dσ (dσ) is the creation (annihilation) operator with spin σ =↑, ↓. Eσ is the single particle energy level. U is the Coulomb charging energy. The leads l = L, R are modeled as Fermi liquids with quasiparticle energy ϵlk, where † † k is the wavevector. The fermion operator is nlkσ = clkσclkσ, where clkσ (clkσ) is the creation

(annihilation) operator of fermion. ωq is the phone frequency, and the phonon occupation number † † nq is deﬁned as nq = aqaq, where a (a ) are phonon creator (destroyer).

HT is the Hamiltonian of dot-lead coupling:

∑ † HT = Tlkclkσdσ + H.c.. (4.4) lkσ

, where Tlk is spin-independent tunnel matrix element associated with a tunneling rate:

2π ∑ Γ (ϵ) = |T |2δ(ϵ − ϵ ). (4.5) l ¯h lk lk k

HS is the Hamiltonian of spin-phonon coupling:

∑ ( ) † HS = (Mq, xσx + Mq, yσy) aq + a−q . (4.6) q

∗ The coefficient Mq,i=M−q,i where i=x,y for all q have to be fulfilled to guarantee the hermiticity † † † † of HS. σx = d↓d↑ + d↑d↓ and σy = i(d↓d↑ − d↑d↓) are spin operators. In order to calculate the differential conductance G = dI/dV , we derive the current which crosses the quantum dot from the left (L) to the right (R) lead. The current through the dot can

39 be expressed by [69, 70, 71]

∑ ILR = e WLσ′,RσPσ (4.7) σσ′ where e is electron charge, WLσ′,Rσ is the transition rate for an electron tunneling from L lead (spin σ′) into R lead (spin σ), and can in principle take into account elastic, inelastic, as well as phonon-assisted elastic cotunneling processes; Pσ is the occupation number of electrons in the dot which is governed by a master equation[23]

dP σ = −γ P + γ P , (4.8) dt σσ¯ σ σσ¯ σ¯

cot flip cot with total rate γσσ¯ = γσσ¯ + γσσ¯ . γσσ¯ is the spin-flip rate due to inelastic cotunneling and includes flip processes with a single lead, while γσσ¯ characterizes spin-flip processes due to the spin-phonon coupling. Detailed expressions for the different rates are found in Ref. [23] and [70, 71].

4.3 Tunneling rate of a quantum dot

To examine the cotunneling conductance and the ratio of G+ to G0 quantitatively, we arrange three diﬀerent dot conﬁgurations: COT I (VS = −800 → −872, VT = −816, VB = −1151, VG = −938 →

−792 mV); COT II (VS = −960 → −1025, VT = −750, VB = −1090, VG = −795 → −671 mV); and COT III (VS = −800 → −917, VT = −750, VB = −1090, VG = −1246 → −1008 mV). For each conﬁguration, the dot contains diﬀerent number of electrons. To tune the dot-lead coupling, Γ, we use a previously developed computer control of the dot gate voltages [2] and adjust the voltages

VS and VG so as to maintain the occupancy of the dot and keep the dot energy in the middle of the Coulomb valley. To tune the dot energy |E1 − µ|, we vary the plunger gate voltage VG while keeping voltages on other electrodes unchanged. We focus on the changes of the conductance as well as the ratio G+/G0 as either the tunneling rate or the dot energy is varied. The experiment is performed for all three device conﬁgurations described above.

We measure the tunneling rate Γ = ΓL + ΓR, where ΓL(R) is the tunneling rate from the left(right) lead, by examining the shape of the charging peak as we vary the voltages on the gates.

Figure 4.3(a) shows clearly the evolution of the Coulomb charging peak width as VS is varied. To

40 a) b) 0.4 0.35 1.0 G 2 (e /h)

max 0.00 (meV) 0.2 -785 -770 G VG (mV) G/G 0.0 -10 0 10 -850 -800

VG (mV) VS (mV)

Figure 4.3: (a) Normalized Coulomb blockade peaks taken at VS= -800, -830, and -872 mV (black, red, and blue lines respectively) for dot conﬁguration I (COT I) clearly shows the variation in peak width. (b) Tunneling rate Γ as a function of VS. Tunneling rate increases as the side gate voltage becomes less negative. Inset: Fitting a Coulomb blockade peak (dot) to a thermally broadened lorentzian (red solid line) gives the corresponding Γ. Conﬁguration II and III have similar results which are not shown.

Table 4.1: Capacitance ratio αG, used as the energy lever arm, extracted from Coulomb blockade diamonds of diﬀerent VS has been considered in order to precisely obtain tunneling rate Γ and charging energy U. Shown in the table are parameters of COT III confuguration.

VS (mV) αG Γ (meV) U (meV) -800 0.027 0.19 2.77 -850 0.03 0.06 2.89 -900 0.036 0.03 3.11

41 determine Γ, we ﬁt the CB conductance line shape to a thermally broadened lorentzian (TBL)

[32, 33]

∫ ( ) 2 +∞ e A −2 E G(VG) = cosh h 4kT −∞ 2kT × (Γ/2)π 2 2 dE. (4.9) (Γ/2) + [eαG(VG − V0) − E]

In this equation, V0 is the gate voltage that corresponds to the CB peak maximum, Γ is the associated tunneling rate, αG is the energy lever arm of the dot, and A is a ﬁtting parameter which is related to the dot asymmetry, [72] S = ΓL/ΓR. The dot asymmetry for each VS voltage setting is obtained from the height of the CB peak; [73, 74] S varies from 4 to 51 in our measurements. A slight deviation of αG due to a possible shifting of the position of the dot has been observed and taken into consideration. Table I lists αG and other dot parameters for three diﬀerent choices of

VS. To assign the corresponding Γ for each VS, we use the average of the tunneling rates extracted from the two adjacent CB peaks in the same valley Γ = (ΓLP +ΓRP )/2, where ΓLP (RP ) corresponds to the left(right) CB peak. An approximately linear dependence of Γ on VS, and the TBL ﬁtting to a CB peak are shown in Fig. 4.3(b). The overall conductance decreases with more negative VS values, as expected, because of the reduction of the barriers transmission.

4.4 Γ and energy dependence of cotunneling through the Zeeman

splitting

Fig. 4.4(a) shows the typical features of the cotunneling conductance in presence of the magnetic

ﬁeld, for a valley with an odd-number electron occupation. At each gate voltage, a threshold step is observed; the separation between the steps at positive and negative bias is controlled by the

Zeeman energy, and it is thus independent of the gate voltage. Figure 4.4(b) shows representative traces at several diﬀerent Γs while the dot energy is kept in the mid-point of the Coulomb valley as described above.

Direct comparison of the calculated and experimental conductance traces, such as that in Fig.

4.4(c), shows their excellent agreement. 1 Above the threshold, the measured conductance exceeds

1We have also included averaging of the theoretical conductance curves to mimic the lock-in process used in

42 2 a) G (e /h) b) -3 -3 10 0 20 40x10

/h) -4 2 10

0 G (e (mV) ds

V -5 -1 10 -850 -800 -750 -200 0 200 VG (mV) Vds (mV) c) d) -3 -3 0.50x10 0.50x10

2|gmB| /h) /h) 2 0.25 2 0.25 G (e G (e phenomenological microscopic 0.00 0.00 -200 0 200 -200 0 200

Vds (mV) Vds(mV)

Figure 4.4: (a) Differential conductance as function of Vds and VG at B = 9 T for COT III. The cotunneling trace of any given Γ is taken in the middle of the Coulomb valley. Dashed lines with arrows indicate the conductance threshold across the odd-occupied valley. Dot-dashed lines mark the Coulomb blockade diamond edges. (b) Representative differential conductance traces taken in the middle of Coulomb valley for different Γs of COT II: from 0.072 meV (top) to 0.032 meV (bottom). (c) Microscopic calculation (dotted line) of differential conductance as a function of Vds shows agreement with the experimental measurement (solid line) for the conductance near zero bias, but is slightly off at high biases. (d) Comparison of the microscopic calculation with no adjustable parameters to predictions of a phenomenological form (Eq. 1 in Ref. [49]) used for the data analysis shows good quantitative agreement. The gap width is twice the Zeeman energy.

43 slightly the calculated conductance, arising perhaps from a slight bias dependence in the barrier transmission coeﬃcients and/or the increasing important role of other dot levels ignored in the model. We point out that spin-phonon interaction is expected to reduce the conductance at high bias; moreover, the overshoot seen in the data near threshold is not expected for the strongly asymmetric quantum dots studied here. [23, 75] Its nature is still unresolved.

We have also compared the microscopic theory to the phenomenological form used by Kogan et al. for the analysis of non-linear cotunneling conductance.[49] We speciﬁcally use both approaches to determine the g-factor of the heterostructure, and ﬁnd excellent agreement [Fig. 4.4(d)] between both approaches.

Having obtained the dot energy, Γ, g-factor, and the dot asymmetry, we now focus on the conductances (G0) and (G+) for the three dot conﬁgurations. Figure 4.5 (left panels) shows quantitative agreement between the predictions and the data, for over two orders of magnitude in conductance, as Γ changes. Notice that the theoretical curves are not smooth functions of Γ since the asymmetry factor is not the same for each choice of Γ. The ratio of conductances G+/G0 ≈ 2.4, however, is nearly independent of Γ for all three conﬁgurations [Fig. 4.6(a)].

The calculated conductance trace quantitatively agrees with the data at bias voltages around zero but slightly disagrees at high biases. The spin-phonon interaction always reduces the conductance at high bias regions[23, 75], therefore this disagreement in conductance at bias voltages higher than the Zeeman threshold is not due to the spin-phonon intradot effect and is not understood. We also compare the microscopic theory to a phenomenological form used by Kogan et al. for the analysis of non-linear cotunneling conductance and specifically the determination of a heterostructure g-factor [49], and find a very good agreement [Fig. 4.4(d)]. With the extracted dot parameters, i.e., dot energy, Γ, g-factor, and dot asymmetry, we then apply this theory to calculate the conductance around zero bias (G0) and at the biases above Zeeman energy (G+) for three dot configurations without any adjustable fitting parameters. Fig. 4.5 (Left panel) shows a quantitative agreement between the predictions and the actual measurements. We notice that the theoretical curves are not smooth functions of Γ since the asymmetry factor is not the same for each choice of Γ. The ratio, G+/G0 ≈ 2.4, is independent of Γ for all three configurations [Fig. 4.6(a)]. ∫ collecting the data. Accordingly, the calculated conductance is averaged by G¯ = (1/T ) T G(V + A sin(2πt/T ))dt, 0 0 where A0 is the amplitude of the ac-drive of period T .

44 -2 10 1.0 G0 G+ COT I

COT I G0 G+ -5 0 10-2 10 2.5

G (e COT II 2

/h) /h) 2 x10 G (e -3

COT II -5 10 0 -2 10 2.5 COT III

COT III -5 10 0 0 0.10 0.20 0 1 2 3 G (meV) |E1-m| (meV)

Figure 4.5: Comparison of microscopic calculations (dotted lines) and measurements for G0 (red squares) and G+ (blue circles) of three dot conﬁgurations. Left panel: tunneling rate dependence. Right panel: dot energy dependence.

45 a) COT I COT II 3 COT III 0 / G + G 2

2 3 4 5 6 7 8 9 2 3 0.1 G (meV) b)

3 0 / G + G 2 III I II

-1 0 1 DE (meV)

Figure 4.6: (a) The G+/G0 ratio as function of Γ. Both calculations (lines) and measurements (symbols) show that G+/G0 ≈ 2.4 is nearly independent of the tunneling rate Γ. (b) The conductance ratio as function of dot energy–using as reference the mid-point of the valley. Vertical dashed lines indicate where charging peaks appear (at half of the charging energies) for all three dot con- ﬁgurations. Charging energy values for COT I, II, and III are 2.9, 2.0, and 3.0 meV, respectively. The ratio reveals a minimum at the mid-point of the valley ∆E = 0, but it slightly increases as the dot energy approaches the charging peaks.

46 For the energy dependent experiment, each dot conﬁguration has an approximate ﬁxed tunneling rate and asymmetry. The dot energy is tuned by varying the plunger gate VG while |E1 − µ|/Γ or

(U − |E1 − µ|)/Γ ≥ 4 is maintained to avoid the dot entering the mix-valence regime. [76] We ﬁnd that the conductance increases symmetrically as the dot energy is tuned away from the mid-point of the valley [Fig. 4.5 (Right pannel)]. We examine the ratio of G+/G0 and ﬁnd that it exhibits a minimum at the mid-point of the valley; the ratio increases slightly as the dot energy approach the adjacent CB peaks. Fig. 4.6(b) again shows a good agreement between our calculated and measured results.

Next, we address the variations of G0 and G+ with dot energy. The dot energy is tuned by varying the plunger gate voltage VG, while maintaining |E1 −µ|/Γ or (U −|E1 −µ|)/Γ ≥ 4, to avoid the dot entering the mix-valence regime. We ﬁnd that the conductance increases symmetrically as the dot energy is tuned away from the mid-point of the valley [Fig. 4.5 (right panels)]. We examine the ratio G+/G0 and ﬁnd that although nearly constant at ≈ 2.4, it exhibits a slight minimum at the mid-point of the valley; the ratio increases slightly as the dot energy approaches the adjacent

CB peaks. Figure 4.6(b) again shows good agreement between the calculated and measured results.

In summary, we have presented a systematic study of the differential conductance of a quantum dot in the cotunneling regime for three different dot occupancy configurations. This allowed us to investigate the dependence of tunneling rate and dot energy on conductance and compare the experimental data to microscopic calculations. Independent experiments to determine the parameters of the dot state were performed so that comparisons could be made without use of adjustable parameters. We find overall excellent agreement between the calculations of a simple two-spin quantum dot model and the measurements. We find that the ratio of the device conductance above the Zeeman threshold to that below the threshold is nearly independent of the dot-lead tunneling rate and it is only slightly dependent on the dot energy, with a value ≈ 2.4, in near agreement with the theoretical ratio. The agreement is best in the middle of the Coulomb valley and becomes worse closer to the charging peaks, possibly due to the role of higher excited states not included in our calculations.

47 Chapter 5

Magnetic-Field-Induced Crossover to a Nonuniversal Regime in a Kondo Dot

We report the magnetic splitting, ∆K , of a Kondo peak in the differential conductance of a quantum dot while tuning the Kondo temperature, TK , along two different paths in the parameter space: varying the dot-lead coupling at a constant dot energy, and vice versa. At a high magnetic field,

5.1 Introduction

Very often we are surprised by the nature. Different physical phenomena can end up displaying exactly the same behavior and being described by a universal scaling function [77, 78]. Physicists have been using these universality and scaling to describe both equilibrium (e.g. thermodynamics of near critical fluids or ferromagnets) [79] and non-equilibrium (transport, turbulent flow) physics phenomena. A famous and wildly studied quantum system exhibiting universal dependence of properties on a single intrinsic energy scale (called the Kondo temperature, TK ) is the Kondo singlet,

48 a correlated many body ground state of a conﬁned spin interacting coherently with delocalized electrons [80]. While the equilibrium scaling in Kondo systems is now well understood, a lot less is known about the nonequilibrium regime, i.e. when the Fermi sea near the spin conﬁnement site is perturbed. The nonequilibrium Kondo phenomena, governed by the delicate interplay of correlations, spin coherence, and dissipation, can be observed in Single Electron Transistors (SETs)

[81, 82, 83, 84, 85]: electric transport in SETs is strongly affected by the Kondo effect, and the deviation from equilibrium near the spin confinement site (called quantum dot) can be precisely controlled by an externally supplied bias voltage, Vds. One of the open questions is whether the nonequilibrium Kondo physics is universal, and, if so, whether TK , as defined for an equilibrium system, remains the relevant energy scale. Very recently, Grobis et. al. [78] have shown that, at low biases and temperatures, the SET conductance G = ∂I/∂Vds is a universal function of eVds/kBTK , where e is the electronic charge. Their results motivated us to further test the universality via the applied magnetic field in addition to the bias. The splitting of the zero-bias Kondo peak in G(Vds) in an external magnetic field B [24] is another nonequilibrium effect predicted to exhibit universal dependence on TK , with B/TK being the scaling variable [63, 86]. These predictions have not been systematically tested in reported studies [82, 83, 87, 88, 49, 16, 89].

In this chapter, we investigate the dependence of the Kondo peak splitting on TK at different B by employing two independent parameters to tune TK , rather than a single gate voltage [16]. One can expect if the splitting is a universal function of TK , its dependence on the Kondo temperature would then be exactly the same regardless which parameters is used to tune TK . This novel approach tests whether the changes in the splitting with TK have a universal character at each B independent of the observations at other values of B. An attractive feature of this method is that a weak dependence of TK on B due to orbital effects, usually present in experiments with SETs, does not complicate data interpretation. At low B, our data are qualitatively consistent with the predicted universal dependence of the splitting on TK [63, 86] as well as with the earlier experiment [16] but are incompatible with the universality at high B. The tunable parameters in this work are the energy ϵ0 of the local orbital measured from the Fermi energy of the leads, and the effective dot-lead coupling Γ = ΓS + ΓD, where ΓS and ΓD are the tunneling rates from the source and drain leads onto the dot. Increasing Γ or |ϵ0 + U/2|, where U is the quantum dot charging energy, makes TK larger [90]. When B is applied, the peak in G(Vds) splits into two

49 peaks with the maxima at energies ∆K relative to the Fermi energy of the leads. Theory [63, 86] predicts that ∆K /∆ increases with ∆/kBTK , as a universal function, and therefore ∆K decreases with TK . Here ∆ = gµBB is the Zeeman splitting of a spin-degenerate orbital level in the absence of Kondo correlations, g is the g-factor and µB is the Bohr magneton. We ﬁnd a marked change in the ∆K (TK ) dependence as B increases. At low B, ∆K decreases with TK , as predicted[63, 86].

At high B, however, ∆K increases with TK when we increase Γ while keeping ϵ0 constant, but decreases with TK when we vary ϵ0 at constant Γ. We conclude that the predicted universality with respect to TK breaks down at high B [2]. The remainder of this chapter is organized as follows. In section 2, we discuss the transport of the SET in Kondo regime. In section 3, we show the Kondo temperature data of our device and its scaling to an empirical Kondo form deduced from the renormalization group calculation. Finally, we present the details of the magnetic splitting of Kondo peak and its crossover to a nonuniveral regime in the last two sections.

5.2 Transport measurements of a Kondo dot

To measure the diﬀerential conductance G, we use standard lock-in techniques with a 17 Hz, 1.9 µV

RMS excitation voltage added to Vds (Fig. 2.4). Depending on the visibility of the peak splitting which we are intended to investigate, we vary the acquisition time of a single Vds scan between 15 minutes and 4 hours with time constants ranging from 1 s to 30 s and, for the smallest splittings that we record, average several measurements. Reproducible splittings were observed several months apart in measurements separated by multiple gate voltage cycles. Our SETs [Fig. 5.1(b), inset] were fabricated on a modulation-doped GaAs/AlGaAs heterostructure containing a two-dimensional

11 −2 electron gas (2DEG) 85 nm below the surface with the sheet density n2D = 4.8 × 10 cm and mobility µ ≥ 5 × 105 cm2/V sec. The metal gates (5 nm Ti/ 20 nm Au) were patterned via e-beam lithography followed by lift-oﬀ. The tunneling rates ΓS and ΓD are controlled by the pairs (VS,

VT ) and (VS, VB) of gate voltages. We note that VS controls both ΓS and ΓD. A deviation, δϵ, of the dot energy from the middle of the Coulomb valley is a linear combination of the changes in the

50 a) 2 b) G (e /h) 2.0 0 0.4 VT 200 nm -825 Source V VG

(mV) S

G 1.5 Drain V

V -925 B ( K) K T 1.0 -1025 -150 0 150 V (mV) c) ds 0.26 0.8 0.5

II /h) /h) 2 2 I (e (e

I II I II G

0.04 0.3 0.0 -100 0 100 -20 -10 0 10 20 Vds (mV) dVG(mV)

Figure 5.1: (a) Differential conductance G as a function of VG and drain-source bias Vds. (b) TK as a function of δVG for two SET configurations. Inset: An electron micrograph of our SET devices’ gate pattern with the gate voltage labeling convention shown. (c) The Kondo peak in Configuration I(VB = −825 mV, VT = −942 mV, VS = −908 mV, and VG = −914 mV) and in Configuration II (VB = −825 mV, VT = −942 mV, VS = −880 mV, and VG = −984 mV). The dot occupancy is the same in I and in II.

51 gate voltages and Vds:

(δϵ)/e = αS(δVS) + αT (δVT ) + αB(δVB) + αG(δVG) + αds(δVds) (5.1)

where αi = Ci/Ctotal are the mutual capacitances between the dot and the device gates and leads, expressed as fractions of the total capacitance of the dot. Using standard techniques, we ﬁnd

U = 1.4 0.05 meV, αds = 0.4 0.03, αG = 0.024 0.0018, αS/αG = 2.4, αT /αG = 2.5, and

αB/αG = 2.0. We estimate the actual dot diameter to be ∼ 0.13 µm, which gives an average orbital level spacing of ∼ 540 µeV. The external magnetic field B is aligned parallel to the 2DEG plane to 1◦. Using the spin-flip cotunneling spectroscopy method [49], we obtain the base electron temperature for our dilution refrigerator (Leiden Cryogenics) Tele ≤ 55 mK and the heterostructure g−factor |g| = 0.2073 0.0013, which gives ∆ = 12.02 µeV/T. Fig. 5.1(a) shows a Kondo valley between VG = −930 mV and VG = −870 mV flanked by Kondo-free, even-occupied valleys. The Kondo temperature at different gate voltages, obtained from measurements of the temperature dependence of the Kondo conductance, is plotted in Fig. 5.1(b). The zero-bias mid-valley Kondo peak [Fig. 5.1(c)] is shown for two device configurations: I with Γ=0.53 meV, TK = 0.3 K, and

II with Γ = 0.7 meV and TK = 0.63 K. We note that, despite the difference in VG, both I and II correspond to the middle of the Kondo valley, i.e. ϵ0 ∼ −U/2 for both. The choice of the I and II configurations is a compromise between minimizing thermal effects, which favors higher Γ and

TK , and avoiding mixed-valence corrections [76], which limits the largest usable Γ to |ϵ0|/Γ ≥ 0.5

[76]. When sweeping between I and II, the relative electron temperature Tele/TK varies from 0.18 to 0.09, and |ϵ0|/Γ > 1 is maintained. Fig. 5.1 (b) shows the values of TK at different gate voltages in configurations I and II. Using the values of the relative gate capacitance αG and the charging energy U, we evaluate the dot-lead coupling Γ by fitting the TK data at different VG to the Haldane formula [90]

√ [ ] [ ] 2 ΓU πU π(eαG) 2 TK = Exp − Exp δVG . (5.2) 2kB 4Γ ΓU

We ﬁnd a good agreement between the ﬁt and the data, with some asymmetry likely due to a weak dependence of Γ on gate voltage. The curvature of ln[TK (VG)] gives the product ΓU, yielding

52 Γ =0.53 meV and Γ =0.7 meV in I and II. Independently, we estimate Γ from the width of the charging peak with the Kondo effect thermally suppressed, and find good agreement. As expected, the more open configuration II has a stronger coupling between the dot and the leads. Using the height of the Kondo peak to estimate the device asymmetry, ΓS/ΓD, we get 24 (I) and 7 (II).

5.3 Kondo temperature and scaling

To obtain the Kondo temperatures in the I and II conﬁgurations, we ﬁt the temperature dependence of the zero-bias Kondo conductance G(T ) at multiple gate voltages to the scaling form

1/0.22 2 −0.22 G(T ) = G0[1 + (2 − 1)(T/TK ) ] (5.3)

[76]. At each gate voltage, G0 and TK are free parameters of the fit. The measurements are done in zero magnetic field. In our system, the lowest attainable electron temperature is 50-55 mK, measured using cotunneling thermometry.1 Connecting a mixing chamber (MC) thermometer circuit, which is always disconnected during our peak splitting measurements, causes the electron temperature to increase. To account for the difference between the electron temperature and the thermometer readings, during TK measurements, we assume a constant heat load to the sample and model the thermal path from the SET to the mixing chamber as a single link with a thermal conduc- √ 2 2 tance linearly dependent on temperature. This gives the electron temperature Tele = T1 + TMC where T1 is the base electron temperature and TMC is the thermometer reading (Fig. 5.3). Taking

T1 as a floating parameter of the fit, we find that T1 values fall within 10 mK of 95 mK across the

Kondo valley for both I and II configurations. For consistency, in the fits reported here we fix T1=

95 mK and leave G0 and TK free. The uncertainty of 10 mK in T1 causes a 10% change in the Kondo temperature reported by the ﬁt, which we reﬂect in our calculation of the error bars for the

TK . In the analysis of the state I data, we extend the temperature range by including conductance measurements taken at 55 mK (MC thermometer removed) in the ﬁts. In Fig. 5.2, we plot the scaling conductance, 1 − G/G0, as a function of T/TK for several sampled gate voltages. Also plotted are conductance traces vs gate voltage for various temperatures. We note that at higher

1determined from spin-ﬂip cotunneling step width observed with our sample. For the detailed description of the method, see [49]

53 a) 6 b) 6 5 I 5 II 4 4 3 3

0 2 0 2 +15mV +10mV +10mV +8mV 1-G/G +5mV 1-G/G +5mV 0.1 0mV 0.1 0mV -5mV -5mV -10mV -8mV 6 -15mV 6 -10mV 5 5 4 4 2 3 4 5 6 7 2 2 3 4 5 6 7 8 9 0.1 1 0.1 1 T/T T/T c) K d) K

0.40 I 0.90 II

0.80 0.30 /h) /h) 2 2 0.70 G (e G (e 0.20 0.60

0.10 0.50 -930 -920 -910 -900 -890 -1000 -990 -980 -970 VG (mV) VG (mV)

Figure 5.2: Zero-bias conductance scaling. (a), (b): Scaled conductance plots for two device gate voltage configurations, I and II. Different colors correspond to different positions in the Kondo valley relative to the mid-valley point. Black lines: theoretical scaling curve (same in both panels). (c), (d) : Representative conductance measurements show the evolution of the Kondo conductance G(T ) across the Kondo valley. The red line shows G0 values as returned by the fit. The black trace in FIG. (c) shows data recorded at electron temperature ∼ 55 mK, with the MC thermometer circuit disconnected. All other traces (blue) are recorded with the MC thermometer turned on, which increases the base electron temperature to ∼ 95 mK.

54 400

2 2 1/2 Tele=(Tbase +TMC ) T = 98 ± 9 mK 300 base

200 (mK) ele T 100

Tele=TMC 0 0 100 200 300 400

TMC (mK)

Figure 5.3: Electron√ temperature Tele as a function of mixing chamber temperature TMC . Solid 2 2 line: A ﬁt to Tele = T1 + TMC , where T1 = 95 10 mK. Dashed line: Tele = TMC .

temperatures, the conductance data, consistent with prior studies [78], show good scaling but de- viate from the theoretical curve. We restrict the ﬁt range at high temperatures to T < 450 mK, which corresponds to 1.5 TK and 0.7 TK for the conﬁgurations I and II, respectively. Extending the

fit range to higher temperatures would increase TK values by ∼ 30%. Although using the full range brings the overall scaled data closer to the theoretical line, we believe that this approach is not justified because it distorts the scaled plot at low temperatures, giving the temperature dependence an exponent slightly different from the theoretical value of 2, which is not physically correct.

5.4 Kondo peak splitting in magnetic ﬁeld

The equilibrium Kondo effect reveals a peak at zero bias across the Coulomb valley containing of a net spin in the differential conductance. The rise of conductance is due to the dot energy level aligning with the source drain leads chemical potentials, which is equal to the Fermi energy at the zero bias. Kondo effect will be driven to the non-equilibrium regime when the applied DC bias significantly influences the leads chemical potentials. In an applied magnetic field, the Kondo peak suppresses at zero bias but recovers at finite bias voltages [82, 83].

55 a) G

Vds

b) 1.9 T II 0.68 0.19 /h) /h) 2 0.63 2 (e (e

0.16 I I II G G 0.58 0.13 -40 -20 0 20 40

Vds (mV)

c) 7 T 0.42 0.14 II /h) /h) 2 0.09 0.29 2 (e (e I

I II G 0.04 0.15 G -100 -50 0 50 100

Vds (mV)

Figure 5.4: (a) Obtaining ∆K from measured G(Vds). (b), (c) Representative conductance data in conﬁgurations I and II at B= 1.9 T (b) and B=7 T (c). The peak positions as determined by our procedure are marked with arrows next to the traces. The Zeeman voltage scale, Vds = ∆/e, is shown with short dashed lines.

56 In a field, the spin degeneracy Kondo singlet ground state of the dot is broken to favor the spin-polarized ground state. This results in the suppression of the Kondo conductance at zero bias because it requires an energy cost for the spin-flip transition to happen. The applied DC source drain bias thus compensates for this spin-flip energy cost and recovers the Kondo peaks at finite biases. In an experiment, this produces two splitting Kondo peaks separated by the spin-flip energy cost [81]. Whether the energy cost is equal or not to the Zeeman energy becomes an interesting question. Earlier reports from theory [24] and experiment [83] showed that the recovered Kondo peaks should be exactly at Vds = ∆. In contrast, Costi [91] predicted that there is a threshold field to split the Kondo peak, and Moore and Wen [63] predicted the separation of the Kondo splitting should be less than 2∆. Recently, Kogan et al. [49] and Amasha et al. [16] reported that the Kondo splitting agrees with Moore and Wen’s prediction at low fields but disagrees at high

ﬁelds since the splitting exceeds the Zeeman energy. The puzzles of Kondo splitting still remain.

In order to obtain the Kondo splitting, ∆K , from measured G(Vds) [Fig. 5.4(a)], we ﬁrst ﬁt

fit the data near each maximum to an analytical function G (Vds). To account for the slight peak height diﬀerence, we subtract a linear background from the ﬁts to equalize the maxima and then

fit obtain Vds for the left and the right peaks by solving d G /d Vds = 0. ∆K /e is taken as half the difference between these Vds values. We find that the result varies by no more than ∼ 2 µV for different sensible choices of the background slope.2

5.5 Crossover to a nonuniversal regime

We open the discussion of the results by examining how ∆K changes with ϵ0 at constant dot-lead tunneling rate, Γ. Starting with all gate voltages set to I (see Fig. 5.1, caption), we scan VG, and then repeat the experiment with II as the starting point. Fig. 5.5 presents plots of ∆K as functions of the deviation, δVG, of VG from the value that corresponds to I (open squares) and II

(ﬁlled circles). At all B, ∆K decreases as VG is tuned away from the center of the valley, which corresponds to an increase in TK by Haldane formula. This agrees with the earlier observations [16]. Next, we ﬁx the dot energy in the middle of the valley and focus on the changes of the splitting with Γ. 2See appendix B for more details.

57 115 a) 7.5 T

85 -20 0 20 55 b) 3.5 T

V) m ( I /e

K II D

25 -20 0 20 30 c) 1.9 T

0 -20 0 20 dVG (mV)

Figure 5.5: Variation of ∆K with gate voltage VG at different values of B field. δVG = 0 corresponds to configuration I (open squares) and II (filled circles) as defined in the caption of Fig. 5.1. The Zeeman bias voltage scale, ∆/e, for each B is marked with a dashed line. TK in I and II is 0.3 and 0.63 K, respectively

58 5.5.1 Magnetic splitting: tune TK via dot energy

A detailed dependence of the mid-valley splitting ∆K,0 on B for the conﬁgurations I and II is presented in FIG. 5.6. First, we note that the lowest magnetic ﬁeld at which the Kondo peak

onset shows detectable splitting increases with TK . Introducing the corresponding Zeeman scale ∆ ,

onset onset we ﬁnd ∆ = 0.55 kBTK (I) and ∆ = 0.4 kBTK (II). These are in reasonable agreement

onset with the prediction ∆ = 0.5 kBTK [64] and are somewhat lower than the previously reported

∼ 0.86 kBTK [16], and ∼ 0.8 kBTK [88] possibly due to a heavy signal averaging and a lower relative electron temperature (∼1/15 to 1/6 in the present work vs 1/6 in [88], and 1/3 in [16]). Near the onset, the data show a pronounced suppression ∆K,0 < ∆, consistent with, but much stronger, than in the earlier report by Quay et al. [88] who used a carbon nanotube-based SET. We note that in the earlier experiments with heterostructure-based SETs [49, 16], ∆K,0 < ∆ was not observed.

As ∆ increases above ∼ kBTK , ∆K,0 < ∆ is replaced with the ∆K,0 > ∆ regime. To our knowledge, such a transition at a ﬁnite B has not yet been reported, although ∆K > ∆ was previously observed experimentally [49, 16, 89] 3, and found theoretically in the very recent calculations by Hong and

Seo [93], and also, for the B >> kBTK regime, predicted by the perturbative method described by Paaske et al. [92]. At yet higher B, our ∆K,0 data for the more open, higher TK conﬁguration

II exceed those for the lower TK configuration I. This is opposite of what we find in the fixed

Γ experiments (Fig. 5.5 and its discussion), in which ∆K,0 decreases with TK regardless of the magnitude of B.

5.5.2 Magnetic splitting: tune TK via tunneling rate

To examine the ∆K (TK ) dependence at ﬁxed energy in more detail, we follow a constant ϵ0, variable Γ path starting in the middle of the Kondo valley (Fig. 5.7). In the beginning of each sweep, the device is set to I. Then, both VS and VG are swept simultaneously so as to keep αS(δVS)+αG(δVG) =

0, and thus maintain ϵ0 ∼ −U/2. The changes in TK during such a sweep come from the changes in Γ only, and the device is being tuned continuously from I to II. At low B, we observe ∆K,0 < ∆ and ∆K,0 decreasing with increasing VS (and also Γ and TK ). At ﬁelds larger then ∼ 4 T, the opposite occurs: ∆K,0 > ∆ and increases as Γ and TK increase. Thus, at high B, scaling with TK

3 The authors of ref. [89] point out that their data are consistent with ∆K /∆ = 1 if one deﬁnes ∆K as the position of the steepest point, rather than the maximum of, G(Vds), as suggested by Paaske et. al. [92]

59 20 I 15 II 10 V)

m 5

)/e ( 0

- D -5 K,0 D

( -10 -15 -20 2 3 4 5 6 7 8 9 1 B (T) 10

Figure 5.6: Deviation of mid-valley splitting ∆K,0 from Zeeman energy ∆ as a function of magnetic ﬁeld. The horizontal dashed line corresponds to ∆K,0 = ∆. The dotted line corresponds to zero peak splitting: ∆K,0=0.

breaks down: changes of ∆K with TK in the constant energy and in the constant Γ experiments have opposite signs. Interestingly, both the high B and the low B trends shown in Fig. 5.6 agree qualitatively with the ∆K → ∆ behavior expected for the limit of small Γ and TK [63, 91, 86].

5.5.3 Orbital eﬀects due to B-ﬁeld

To verify that the presence of the magnetic field does not reverse the expected increase of Γ with VS, as may occur, for example, due to an accidental scattering by impurities near the tunnel barriers, we have compared the charging peak widths in configurations I and II measured at zero bias with a 9 Tesla magnetic field applied. We found the ratio of the widths to be 0.6. This is comparable to the ratio of Γ values in I and II at zero magnetic field (0.78), and indicates that the configuration II remains stronger coupled to the leads than configuration I even at the highest available magnetic

60 15

7.0 T

4.5 T

3.5 T V) m

)/e ( 0 D 2.2 T - K,0 D ( 1.8 T -5

1.9 T

-10 1.7 T

TK increases -15 -910 -890

VS ( mV)

Figure 5.7: Deviation ∆K,0 − ∆ as a function of VS along a constant ϵ0 path. Both VS and VG are swept. The dot energy is set in the middle of the Kondo valley and αS(δVS) + αG(δVG) = 0 is maintained while VS is varied.

61 ﬁeld.

The orbital wavefunction is aﬀected by magnetic ﬁeld [94, 17] and therefore the level energy,

ϵ0, dot-lead coupling, Γ, and TK may depend on B [95]. In our experiments, we find that the level energy shift reaches about 0.3 U at the highest magnetic fields. We take this into account when comparing mid-valley measurements taken at different fields (Fig. 5.7). To find the gate voltage that corresponds to the middle of the valley at each field, we locate the minimum of the zero-bias conductance. When the magnetic field is applied, this point corresponds to the lowest

Kondo temperature in the valley. At several ﬁelds, we also estimated the position of the middle of the valley by taking a mid-point between the charging peaks’ maxima, and found agreement with the data obtained by following the conductance minimum.

Fig. 5.8 presents the shift in gate voltage of the mid-valley point with the ﬁeld for conﬁgurations

I and II. We note that deviations of the individual points from a smooth ﬁt are less than 0.05 U

(a few mV in gate voltage). These give an upper bound for our error in setting the gate voltage at each field. An added benefit of the mid-valley measurements is that the Kondo temperature experiences a minimum, therefore a small deviation of the gate voltage has a negligible effect on

TK . From Fig. 5.4, we can also see that the eﬀect of a few mV change of gate voltage in mid-valley changes the peak splitting by ∼ 1µV or less, and is thus very adequate.

Due to possible changes in Γ with the magnetic field, the Kondo temperature in each configuration somewhat changes with B even if the level energy is controlled well. This does not affect any of our conclusions because at any field, the relationship between TK in configurations I and II is the same as at B = 0: the configuration I is more weakly coupled to the leads than II and thus has lower TK . From experiments on a different, nominally identical device, we find that Γ changes by about 15% between B=0 and B= 9 Tesla, always in the negative direction. We verified this at several occupancies. Assuming that our state behaves similarly, this would change TK at the highest field by roughly 1/3 of its B=0 value. If the change in TK is negative, the prominence of thermal effects will increase at higher field. For configuration II, a 15% decrease in Γ increases

T/TK from 0.08 to 0.12, which is still quite low. If the change in Γ is positive, the parameter ϵ0/Γ in conﬁguration II will change from approximatley 1 to 0.7, still outside the mixed-valence regime

|ϵ0/Γ| < 0.5. Thus, changes in Γ with B are unlikely to inﬂuence our conclusions.

In summary, we have measured the Kondo splitting ∆K in the diﬀerential conductance of an SET

62 Figure 5.8: Shift of the level with the magnetic ﬁeld in conﬁgurations I and II.

63 while tuning the Kondo temperature TK along two diﬀerent paths, ϵ0 and Γ. At low B, we observe a decrease in ∆K with TK along both paths, in agreement with theoretical predictions. Nevertheless, at a high magnetic ﬁeld, a crossover occurs to a regime in which a universal dependence of ∆K on

TK is qualitatively inconsistent with the data. In addition, we observe both ∆K < ∆ (low B) and

∆K > ∆ (high B) regimes in a single SET system, and ﬁnd that the transition between the two regimes occurs at B values comparable to those for the crossover.

64 Chapter 6

Magnetoconductance of a Single-Electron Transistor in the Kondo Regime

We have measured the zero-bias conductance, G, of a single-electron transistor (SET) in the Kondo regime as a function of temperature, T, and magnetic ﬁeld, B, oriented parallel to the plane of the device. Our SETs are fabricated on a GaAs/AlGaAs heterostructure with electron sheet density

4.8 × 1011 cm−2 and mobility 5 × 105 cm2V−1s−1. Scaled plots of both the T - and B-dependent data show universal behavior. At moderate and high B, the magnetoconductance data show good agreement with renormalization group calculations in the spin-1/2 Kondo regime. At very low B, we observe a non-monotonic behavior, which can not be explained by the single impurity Anderson model.

6.1 Introduction

Many theoretical calculations have been performed to study the Kondo effect through a quantum dot for the dependence on temperature, magnetic field, and bias voltage [96, 64, 97, 98]. These theoretical works predict a universal behavior of the Kondo effect; and the Kondo energy, TK , is the only intrinsic energy scale that matters. For instance, the temperature-dependent conductance at different gate voltage, thus different TK (see section 5.3 in the previous chapter), collapses onto

65 a universal function once the conductance is normalized to its zero-temperature value G0; and the temperature is scaled to its corresponding TK . [76, 81] In addition to the scaling eﬀect in the equilibrium Kondo regime, a universal scaling at low-temperature and bias in nonequilibrium

Kondo regime is also reported recently. [78] In the equilibrium regime, a universal behavior of magnetoconductance has been predicted in theory [97, 98] as well; however, neither the scaling of magnetocondutance, nor the Kondo energy determined by the B-dependent data have been reported up to date.

In this chapter, we examine both temperature (T ) and magnetic ﬁeld (B) dependent zero-bias conductance through an SET in the spin-1/2 Kondo regime. We ﬁnd normalized conductance in

T - and B-dependence, G(T )/Gmax and G(B)/Gmax, can be scaled into a universal function of

T/TK and B/BK respectively, with a different exponent (s=0.22 for T -dependent, s=0.55 for B- dependent), which is in good agreement with theory predictions (s=0.22 [76, 96] for T -dependent and s=0.47 for B-dependent). This universal function is derived from a fit to the NRG calculation [76, 96]. We further compare the two Kondo scales, kBTK and gµBBK , and find the ratio kBTK /gµBBK ≈ 1.65. In addition, at low B, we observe a non-monotonic behavior: as B increases, the conductance G initially increases and only starts to decrease after G hits its maximum at B ∼ 0.1BK . This non-monotonic behavior is possibly due to the presence of multiple orbital dot levels with similar energies [99, 100, 101].

6.2 Universal scaling of conductance as a function of temperature

and magnetic Field

The SET devices used in this chapter were made of GaAs/AlGaAs heterosturcture via e-beam and photo lithography followed by lift-oﬀ. The sheet electron density and the mobility of the 2DEG are

11 −2 5 2 n2D = 4.8 × 10 cm and µ ≥ 5 × 10 cm /V sec, which were determined by a magnetotransport measurement at ∼4.2K. We measure the diﬀerential conductance, G = ∂I/∂Vds, through the SETs via standard lock-in techniques at low temperatures with a frequency of 17Hz and an excitation voltage of ∼2µV.

Fig. 6.1(b) shows characteristic Coulomb blockade (CB) diamonds and the signature conductance enhancement of the spin-1/2 Kondo eﬀect at the zero bias for the oddly-occupied valleys in

66 a) b) 2 0 1 2 G (e /h) 200 nm VT 1 S VG VS 0 (mV) ds

D V VB -1 -1150 -950 -750 VG (mV) 2.0 2 c) d) 95 mK 95 mK G (e /h) 2 2 /h) G (e 700 mK 700 mK 1 0.5 -200 0 200 -870 -830 -790 Vds (mV) VG (mV) e) s=0.22 1.0

TH max SET 1

DVG (mV) G/G -30 10 -20 20 -10 30 0.5 0

2 3 4 5 6 7 8 2 3 4 5 6 7 8 0.01 0.1 1 T/TK

Figure 6.1: (a) A Scanning Electron Microscope (SEM) image of our SET device with labeling. (b) Coulomb blockade diamonds with Kondo eﬀect for our SET. Conductance enhancement at zero bias is clear seen for the oddly-occupied valleys. (c) Temperature evolution of the Kondo peak in the middle of the valley, VG = −830 mV. The peak is suppressed with temperature. (from top to bottom) (d) Temperature evolution of the conductance for the whole Kondo valley at Vds = 0. The conductance decreases as temperature increases. (from top to bottom) (e) A plot of scaled conductance G/Gmax vs. T/TK for all measured temperature at selected gate voltages across the Kondo valley. ∆VG = 0 corresponds to the mid-point of the Kondo valley, VG = −830 mV. The red solid line is the theory curve with exponent s=0.22.

67 a transport measurement of our SET [Fig. 6.1 (a)]. The Kondo valley investigated in this work is at the gate voltage range between VG= -880 and -780 mV. The conductance as a function of drain-source bias voltage, Vds, taken in the middle of the valley, shows a typical Kondo peak [Fig. 6.1(c)]; the peak shrinks as the temperature increases. We measure T -dependent conductance at the Vds = 0 across the whole Kondo valley. We note that the Kondo conductance is slightly tilted as shown in Fig. 6.1(d). This is possible due to the gate-dependent dot coupling to the left and right lead. Given the Kondo conductance of a symmetric dot is 2e2/h, [24, 98] the dot coupling asymmetry is estimated as 3 : 1 ∼ 2 : 1. [78, 24] We ﬁt the conductance trace taken at several gate voltages to an empirical form [76]:

1/s 2 −s G(T ) = Gmax[1 + (2 − 1)(T/TK ) ] (6.1)

, where Gmax ≡ G(T = 0), G(T = TK ) ≡ Gmax/2, and the exponent, s, is a fitting parameter; from the fit, we can extract the corresponding TK for each given gate voltage. This empirical form is derived from a fit to the numerical renormalization group (NRG) calculation. [96] Fig. 6.1(e) shows the scaling plot of T -dependent conductance at several representative gate voltages, which is in good agreement with the theoretical calculation. [96] The exponent value s=0.22 of the scaling function evidents that the dot is in spin-1/2 Kondo regime. [76, 81]

To further probe the Kondo eﬀect in the equilibrium regime, we apply a magnetic ﬁeld to our

SETs to investigate the magnetoconductance of a Kondo dot at Vds=0. The B-field is oriented parallel to the device’s 2DEG in order to minimize the dot orbital effect; the mis-alignment of our sample is estimated as 1◦. Since a B-field breaks the spin degeneracy of the Kondo singlet gound state, therefore it results in the suppression of the zero-bias conductance. The Kondo peaks, however, can be recovered by applying a bias voltage to compensate the energy cost that is required for the spin-flip transition to take place (Fig. 6.2). The details of Kondo splitting have been reported in several previous reports [49, 16, 2]. In this chapter, we focus on the B-dependent suppression of the zero-bias conductance through our SETs.

We measure the differential conductance while applying an in-plane B-field to the SETs at the base electron temperature of our dilution refrigerator (Tele ≈55 mK). A Kondo valley shifting with B is observed (Fig. 6.3), possibly due to orbital effects [2]; we were able to track the position of the

68 2 a) G (e /h) b) 1.0 1.5 2.0

0.5 0.5 V ds 0 0 (mV) (mV) ds V

-0.5 -0.5 -870 -840 -810 -870 -840 -810 V (mV) G VG (mV)

Figure 6.2: Kondo diamonds at B=0 T (a) and B=5.5 T (b). The conductance at zero bias is suppressed with B-ﬁeld. Kondo peak is recovered at |eVds| ≈ the Zeeman energy.

two adjacent CB peaks to address the shifting. Two SETs (SET 1 and SET 2) were used in our

B-ﬁeld dependent measurement; We have done a more detailed measurement on SET 1, whereas

SET 2 was used to double conﬁrm the phenomena we observed. SET 1 is also the device that is used in our T -dependent measurement.

Fig. 6.4(a) shows the evolution of B-dependent zero-bias conductance, each conductance trace is centered at the mid-point of the Kondo valley. ∆VG is the voltage difference reference to the mid- point of the valley; ∆VG=0 represents the mid-point at any given B. Same effect is observed when we flip the direction of B; the data is not presented here. We note that the B = 0 conductance trace is not the highest one, which is not what we expected. Overall, the conductance shows a monotonic decrease with B except at low B; this initial increase of conductance is ignored in our analysis. We

ﬁnd that the conductance drops more dramatically at the mid-point [G(B = 9T )/Gmax ∼ 66%] than at the gate voltages approaching the adjacent CB peaks [G(B = 9T )/Gmax ∼ 25%]; this is expected, if the Kondo energy is the only intrinsic universal scale, since the Kondo energy increase as the dot is tuned away from the mid-point of the valley. [90] We do the same analysis as we

69 a) b) 2 -750 right B=1 T V G /h) (mV) 2 mid-valley G (e

B=9T left 0 -900 -900 -850 -800 -750 -10 0 10

VG (mV) B (T)

Figure 6.3: (a) Magnetic ﬁeld evolution of the zero-bias conductance across the whole Kondo valley shows the suppression of Kondo conductance and the shifting of the Kondo valley at B=1, 3, 5, 7, 9 T. (b) CB peaks shifting as a function of B. We deﬁne the mid-valley value as the average shifting of the left and right peak positions.

70 2 2 a) 0T b) G (e /h) 2 2 /h) DVG (mV) G (e -30 9T -20 10 -10 20 0 30 0 0 -50 0 50 0 2 4 6 8 DVG (mV) B(T)

Figure 6.4: (a) Magnetic field evolution of the zero-bias conductance for the whole Kondo valley. Each conductance trace is centered to the mid-point of the valley. (from top B=0 T to bottom B=9 T) We note that the zero B-field conductance trace (red solid line) is not the highest one as we expected. The Kondo valley is exactly the same valley used in temperature dependent measurement. (b) Magnetoconductance at several gate voltages, which are indicated by vertical dashed lines in (a). ∆VG=0 represents the mid-point of the valley. The non-monotonic behavior of magnetoconductance at low B-fields is possibly due to the presence of multiple orbital dot levels with similar energies. [100, 101, 102, 99]

71 s=0.47 1.0

max SET 1 SET 2

0.5 DVG (mV) G/G -30 10 -20 20 TH -10 30 0 0.0 5 6 2 3 4 5 6 2 3 4 5 6 2 0.01 0.1 1 B/BK

Figure 6.5: A plot of scaled magnetoconductance G/Gmax vs B/BK at several representative gate voltages for SET 1 and SET 2 (taken at the mid-point of the valley: black solid line) . The solid red line represents the NRG calculation, which can be expressed as the same universal form [eq:(1)] used for the temperature dependence but with a diﬀerent exponent s=0.47.

do to the T -dependent measurement: we plot G as a function of B at several representative gate voltages [Fig. 6.4(b)], and then we ﬁt the magnetoconductance at B > 1 T for all chosen gate voltages to the same empirical form used for the temperature analysis [eq:(6.1)], simply replacing

1/s 2 −s T by B: G(B) = Gmax[1 + (2 − 1)(B/BK ) ] . From our best fitting, we find the exponent s=0.55 0.02, which is in reasonable agreement with the NRG calculation s=0.47. The scaled conductance G/Gmax as a function of B/BK shows a universal behavior [Fig. 6.5] for all choices of gate voltages for SET 1. Same universal behavior is also applied for SET 2; we only show the conductance at the mid-point of the Kondo valley. Here BK is the corresponding Kondo energy scale defined as G(B = BK ) = Gmax/2. Fig. 6.6 shows a comparison of the two Kondo scales, TK and BK , extracted from the T - and B-dependent measurements respectively. We find both TK and

BK show parabolic behavior, which agrees with the predition of Haldane formula. [90] The ratio of the two Kondo energies is kBTK /gµBBK = 1.65, which is slightly diﬀerent from the prediction

72 of Pustilnik and Glazman’s theory kBTK = gµBBK . [98] In summary, we have reported the T - and B-dependent transport measurements on SETs in the Kondo regime. They both show universal behavior when T and B are scaled to their corresponding Kondo scales TK and BK , which is in agreement with the numerical renormalization group calculations. [96, 64] We further report the ratio of the two Kondo scales kBTK /gµBBK ≈ 1.65. Nevertheless, B-dependent data shows a non-monotonic behavior at low magnetic ﬁeld, which can not be explained by using single level Kondo impurity model. This behavior is possibly due to the existence of multiple orbital dot levels with similar energies. Further studies will be required to understand this phenomena.

73 3 3

2 2 B TK K (T) (K) K T 1 1

BK 0 0 2.0 K B B m /g K T B k 1.0 -20 0 20 D VG (mV)

Figure 6.6: (Upper panel: Comparison of TK to BK as a function of ∆VG for exactly the same Kondo valley of SET 1; Lower panel: The ratio of kBTK to gµBBK .

74 Chapter 7

Conclusion

To conclude, this thesis presents several transport measurements for QDs and QPCs and provides quantitative comparisons between the experimental data and theoretical models, which contribute the existing knowledge of electronic interactions in low dimensional systems. The QDs and QPCs used in the research are fabricated on a GaAs/AlGaAs heterostructure which contains a 2DEG with high electron mobility.

For the QPC research, we have observed a good quantitative agreement between the electron

Zeeman splitting and the magnetic splitting of a ZBP, which exhibits at conductance values significantly below the first plateau. This result is different from the enhancement of g-factor that is often observed for 1D systems and is robust with respect to moderate distortions of the longitudinal potential of the QPC achieved via additional gates in our 4-gate QPC design. This suggests that even a relatively weak tunneling current in a QPC may be influenced by spin-dependent effects.

We ﬁnd when the potential is signiﬁcantly distorted by applying larger voltages on the side gates of our QPC, a conventional bound charge state is produced and is accompanied by the characteristic

Coulomb blockade diamonds and Kondo transport features at zero bias similar to those found in quantum dots. The ZBP observed in this regime is nothing but the Kondo eﬀect, which is often observed for quantum dots. Given CB diamonds and peaks are not present when the QPC potential is smooth, yet the ZBP persists; this suggests that an accidental trapping of charge in the channel which results in a conventional Kondo eﬀect as observed in quantum dots is not the origin of the ZBP observed in our sample. The origin of the peak remains unknown, and further study is required to understand this feature.

75 For the quantum dot research, we have investigated the electron transport through dots in two regimes: 1) a weakly-coupling regime, in which the Kondo eﬀect is heavily suppressed, and the conductance through the dot is mainly from cotunneling. 2) a strongly-coupling regime, in which a correlated many-body state is formed, which results in a Kondo peak in addition to the Coulomb blockade diamonds in transport data.

As the dot is tuned into a deep Coulomb blockade regime, the spin portion of the dot state splits into spin-up and spin-down states with an in-plane field while the orbital wavefunction remains the same. By applying a bias voltage, it provides an energy to flip the electron spin, which results in a transition from non-spin-flip to spin-flip cotunneling regime at the bias equal to the Zeeman energy. We have measured the ratio of the differential conductance of a quantum dot above the

Zeeman threshold to that below the threshold in the co-tunneing regime and compared the data to calculations based on a microscopic theory. We perform independent experiments to determine the parameters of the dot state, so the comparisons could be made without use of adjustable parameters. We ﬁnd a good overall agreement between the theory and our measurements.

As we increase the dot-lead coupling by lowering the tunneling barriers, Kondo effects eventually exhibit and results in the enhanced conductance in the blockaded transport region. We first study non-equilibrium Kondo effect by means of applying a bias voltage across the source and drain leads of a dot; the bias manipulates the chemical potentials on the leads as well as the dot energy. The

Kondo peak is suppressed with an in-plane magnetic ﬁeld and can be recovered at a ﬁnite bias

(symmetric with the zero bias), which compensates the energy cost to flip spin, thus Kondo effect can take place. We have measured the Kondo splitting ∆K in the differential conductance of an

SET while tuning the Kondo temperature TK along two diﬀerent microscopic dot parameter paths,

ϵ0 and Γ. At low magnetic fields, we observe a decrease in ∆K with TK along both paths, in agreement with theoretical predictions. Nevertheless, at a high magnetic field, a crossover occurs to a regime in which a universal dependence of ∆K on TK is qualitatively inconsistent with the data. In addition, we observe both ∆K < ∆ (low B) and ∆K > ∆ (high B) regimes in a single SET system, and find that the transition between the two regimes occurs at B values comparable to those for the crossover.

“Scaling in Kondo” is another interesting topic that attracts not only theorists but also ex- perimentalists. We have reported the measurements on two diﬀerent dependences, the T - and

76 B-dependent zero-bias conductance for the quantum dots in the Kondo regime. Both measurements show universal behavior as the perturbed parameter is scaled to its corresponding crossover energy, TK and BK ; and the universal function agrees with the numerical renormalization group (NRG) calculations using single level Anderson impurity model in the Kondo regime.

77 Appendix A

Device Fabrication

In this appendix, we describe the details of GaAs-based semiconductor fabrication process for our

SET and QPC devices. The recipe, which is originally from Mar Kastner’s group at MIT, is modiﬁed in order to improve the success rate of device making.

A.1 GaAs substrate preparation

• Use a diamond scriber to cut the wafer into the wanted size, in our case, 10x10 mm. Make

sure to scribe the wafer either parallel or perpendicular to the primary ﬂat in oder to address

the anisotropic orientation.

• Make a mark to denote the crystal orientation of the cleaved substrate.

• Clean the substrate:

1. Hot bath cleaning (Apply this when a substrate is ﬁrst brought into a cleanroom):

Acetone(2mins/40◦C) −→ 2-Propanol(2mins/40◦C) −→ 2-Propanol(2mins/rt) water(2mins/rt)

−→ N2 blow dry (rt: room temperature)

2. Cold bath cleaning:

Acetone(2mins/rt) −→ methanol(2mins/rt) −→ 2-Propanol(2mins/rt) −→ water(2mins/rt)

−→ N2 blow dry

78 A.2 Photolithography: Karl Suss MJB 3

• Spin coating (2 µm in thickness of the positive photoresist S1818):

Clean the substrate holder −→ Vacuum the substrate −→ Put on S1818 −→ Spreading: 5

secs @ 0.5K rmp −→ Spinning: 30 secs@6K rmp

• Pre-bake (Soft bake):

Heat the substrate on a hotplate for 60 secs at 115 ◦C

• Mask alignment:

Use the soft contact mode of MJB3 mask aligner −→ Use a test substrate of the same or

similar thickness as the real substrate to setup the proper level hight −→ Use spring loading

stage to make the substrate properly contact the mask (Fig. A.1)

• Exposure:

Exposure dose: 8 mW/cm2 for 13 sec (wavelength: 405 nm) −→ Keep the cooling air on at

least for 30 minutes before turning oﬀ the machine.

• Development and Post-bake:

1. For lift-oﬀ:

1 Prepare developer(351:water = 1:5) −→ Chlorobenzene soaking(90 secs/rt) −→ N2 blow dry the substrate immediately after soaking −→ Oven-bake the substrate (3mins/90◦C)

−→ Develop the pattern on the substrate(1 mins/rt) −→ Stop developing in water(1min/rt)

−→ N2 blow dry −→ Inspect substrate under microscope −→ No post-bake

2. For etching:

Prepare developer(351:water = 1:5) −→ Develop the pattern on the substrate(1mins/rt)

−→ Stop developing in water (1min/rt) −→ N2 blow dry −→ Inspect substrate under microscope −→ 2 mins at 115◦C on a hotplate

• Etching (Mesa isolation):

2 Prepare etchant: NH4OH:H2O2:Water = 1:1:1000 −→ Calibrate the etching rate −→ Etch

1Chlorobenzene soaking can increase the undercut of the resist since it hardens the surface of photoresist S1818. 2Calibrate the rate for several tested GaAs substrate via a proﬁlometer or an AFM. A typical etching rate: ∼5 nm/sec

79 a) b) Mask Holder Mask Substrate Spring Loaded Stage

c) d)

Figure A.1: (a) Load a test substrate that has the same or close thickness as the real substrate. (b) Raise the substrate on the spring loaded stage up to make contact with the mask. (c) The stage is now oriented parallel to the mask. (d) Replace the test substrate by the real one and do the exposure via soft contact mode. (Modiﬁed from the Lab manual of Marvell nanofabrication laboratory, University of California, Berkeley)

the substrate to a desire depth −→ Stop etching in water (1mins/rt) −→ Do cold cleaning

for the substrate −→ N2 blow dry the substrate −→ Inspect substrate under microscope −→ Inspect the features under a proﬁlometer or an AFM if needed

A.3 E-beam lithography: Raith 150

• Spin coating (125 nm thickness of PMMA 950 A3):

Clean the substrate holder −→ Vacuum the substrate −→ Put on PMMA A3 −→ Spreading:

5 secs @ 0.5K rmp −→ Spinning: 45 secs@4K rmp

• Pre-bake (Soft bake):

Place the substrate on a hotplate for 90 secs at 180 ◦C

80 • Alignment and Exposure:

Set EHT at 30 KeV @10µm aperture −→ Measure current −→ Set global coordinate −→

Focus electron beam −→ Do Write Field Alignment −→ Do 3 points alignment (Both global

and local) −→ Expose dose: 400 - 500(Area); 1500 - 3000(Line) µC/cm2 (Fig. A.2)

• Develop and inspection:

Developer(MIBK : 2 Propanol = 1:3) −→ Develop the pattern on the substrate(30 secs/rt)

−→ Stop developing in 2 Propanol(30 secs/rt) −→ N2 blow dry −→ Inspect the developed pattern via e-beam at 10KeV with 30 µm aperture

• Minor etching3:

Prepare etchant: H2SO4:H2O2:Water = 1:2:1000 then dilute 50% with water −→ Calibrate the etching rate(A typical etching rate: ∼0.2 nm/sec) −→ Etch the patterned substrate to a

desire depth −→ Stop etching in water(1mins/rt) −→ Do cold cleaning for the substrate −→

N2 blow dry the substrate −→ Inspect substrate under microscope

A.4 Metal deposition and lift-oﬀ

• Micron-scale pattern deposition:

Remove the oxidation on substrate (HCl:water = 1:1)(1min/rt) −→ Stop oxidation removing

in water (1mins/rt) −→ N2 blow dry the substrate −→ Mount the developed substrate on a clean glass slice with PMMA −→ Bake the slice with the substrate on a hotplate at 90◦C for

1min −→ Put the substrate into the deposition system

• Sub-micron scale pattern deposition:

Minor etching −→ Stop minor etching in water (1mins/rt) −→ N2 blow dry the substrate −→ Mount the developed substrate on a clean glass slice with PMMA −→ Bake the slice with the

substrate on a hotplate at 90◦C for 1min −→ Put the substrate into the deposition system

• Micron-scale pattern lift-oﬀ:

Soak the sample into Acetone for a few minutes −→ Use syringe to help remove loose metals

3Typically used to improve the adhesion of deposited metal and the substrate

81 2 a) Dose: 300 mC/cm

2 b) Dose: 450 mC/cm

2 c) Dose: 540 mC/cm

2 d) Dose:600 mC/cm

Figure A.2: (a)-(d) Developed pattern and its after lift-oﬀ images of PMMA 950 A3 for several e-beam doses.

82 200 pmma 950 A3

Rate: 6 nm/min 150

100 Thickness (nm) 50

0 0 10 20 30 40 50 60 Ozone Exp (minutes)

Figure A.3: Characteristic ozone cleaning curve of PMMA 950 A3

−→ Inspect the features under the microscope with substrate sitting in the Acetone −→ Do

cold cleaning −→ N2 blow dry the substrate −→ Inspect sample under microscope

• Submicron-scale pattern Lift-oﬀ:

Soak the sample into Acetone for a whole night on a stirrer −→ Use syringe to help remove

loose metals −→ Inspect the features under the microscope with substrate sitting in the

Acetone −→ Do cold cleaning −→ N2 blow dry the substrate −→ Inspect sample under microscope −→ Use 1,2-dichloroethane to clean PMMA residue[103] −→ Do ozone cleaning

if needed (Fig. A.3)

A.5 Ohmic contact

• RTA annealing:

Deposit Ni:Ge:Au = 255:525:1050 (A)˚ via metal evaporator −→ check temperature over shoot

of the annealer −→ anneal temperature: 435 ◦C for 30 secs −→ Inspect ohmic contact via

probe station (Fig. A.4)

83 Figure A.4: Gold Germanium alloy phase diagram. (Source: Bulletin of Alloy Phase Diagrams Vol.1 No.2 1980 p51)

84 a) b)

Figure A.5: (a) Wirebonding. (b) Epoxy bonding on ohmics.

A.6 Packaging and bonding

• Packaging:

Use PMMA to mount the sample to the socket (90◦/1min)

• Wire bonding parameters (West Bond 7476D-79):

First bond: 300 mw/ 30 ms; Second bond: 150 mw/ 70 ms; temperature 130◦C

• Epoxy bonding:

EPO-TEK EE129-4 (EPOXY Technology); Cure time: 15mins/100◦C; 24hr/23◦C

85 Appendix B

Peak splitting extraction

In this section, we present several examples of conductance traces recorded at low magnetic ﬁelds.

The data points obtained from the traces presented in this section are labeled with letters A-F

(FIG. B.1). For the two device conﬁgurations I and II, we have chosen a total of six points, three for each, that illustrate three regimes: a Kondo peak with no resolvable splitting, a small but resolvable splitting, and a clearly split peak.

Fig. B.2 shows the traces and illustrates our splitting extraction procedure. First, we fit the region near each peak to an analytical function (red traces). Next, in order to level the peaks, we find the equation of a line tangent to the fit(s) representing the data and subtract this linear function from the fits (blue traces). Here, we report half the separation between these leveled maxima as the Kondo peak splitting. This procedure allows us to consistently assign the splittings to curves in which the overall tilt of the curve, likely due to small change in symmetry with the bias voltage, makes one of the peaks difficult to locate directly. When both maxima in the data are clearly defined, this procedure gives nearly the same result as simply using the positions of the actual conductance maxima without leveling (i.e. using the red traces instead of blue traces). The difference between the two approaches never exceeds 2 µV . For consistency, we use peak leveling for all the data reported in the chapter.

86 0 C F V) m -5 B

)/e ( E -10

- D A K,0

D -15 (

-20 D

9 2 3 1 B (T)

Figure B.1: Peak splitting data taken at low magnetic fields, same as the points shown in Fig. 5.6 in chapter 5. Points A and D correspond to the largest field at which we observe a zero splitting for the given gate voltage configuration. Conductance traces corresponding to the data points labeled with letters A-F are shown in Fig. B.2. The horizontal dash line (black) gives the positions of points with the Kondo splitting ∆K same as the Zeeman splitting ∆. The dotted line (blue) corresponds to ∆K = 0.

Table B.1: The diﬀerence of peak splitting with or without leveling. The data is taken from Fig. B.1

data point with leveling without leveling change C 53.0 µV 52.4 µV 0.6 µV F 47.8 µV 46.3 µV 1.5 µV

87 0.21 0.71 A 0.70 D 0.20 0.69 /h) /h) 2 2 0.19 II I 0.68 G(e G(e 1.76 T 1.2T 0.18 0.67 0.17 0.66

-3 -3 5 5 0 0 x10 x10 -5 -5 -20 -10 0 10 20 -20 -10 0 10 20

Vds (mV) Vds (mV) 0.210 0.70 B 0.69 E 0.200 0.68 /h) /h) I 2 2 0.67 0.190 1.4 T G(e G(e 0.66 0.65 II 0.180 1.78 T 0.64

5 -3

5 -3 0 0 -5 -5 x10 -20 -10 0 10 20 x10 -20 -10 0 10 20 V (mV) ds Vds (mV) 0.64 0.16 C F I 0.62 II 2.2T 2.2T 0.15 0.60 /h) /h) 2 2 G(e G(e 0.14 0.58 0.56 0.13 0.54 -40 -20 0 20 40 -40 -20 0 20 40

Vds (mV) Vds (mV)

Figure B.2: Representative conductance traces in conﬁgurations I and II recorded at low magnetic ﬁelds. Traces A and D show no splitting; C, D : small but detectable splitting; and E, F : well resolved splitting.

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