Transport properties of InGaAs based devices

DOTTORATO DI RICERCA IN FISICA – XIX CICLO

sede amministrativa ` DEGLI STUDI DI MODENA E REGGIO EMILIA UNIVERSITA

TESI PER IL CONSEGUIMENTO DEL TITOLO DI DOTTORE DI RICERCA

Transport properties of InGaAs based devices

CANDIDATO: Daniele Ercolani ` Universita di Modena e Reggio Emilia

RELATORE: Prof. Lucia Sorba ` di Modena e Reggio Emilia Universita

Foreword Even though only my name is printed on the cover of this booklet, it is the result of the joint effort of a multitude of people. First of all I must thank my tutor, Lucia Sorba, for the constant support given to me in all these years. Her help has definately not been limited to scientific advice and tutoring, but has ranged from waking me up in the morning when I slept too late to taking care of annoying paperwork at the University of Modena. She has bought without a blink countless liters of liquid helium even when results to be obtained were uncertain, has stayed up late in the laboratory to discuss and correct my work, answered stupid questions, fought continously with her enormous energy against all bureocratic and technical difficulties encountered, and has withstood my attitude, even in these last, frenetic months. Thank you very much, Lucia. Next, the other present and past group members: Giorgio Biasiol, the growth-master, for the exceptional samples grown at will, and the many suggestions and droplets of scientific wisdom on all aspects of the growth; Flavio Capotondi, who’s done the hard work on the InGaAs samples back when they were still a total mystery, with his stakanovism, long nights and weekends in the laboratory and in the cleanroom, and his curious mind about all aspect of physics. And about beer. My lab-brother Giorgio Mori, whom I cannot praise enough, for the endless discussions on the meaning of life, the universe and all the rest (including physics). Marco Lazzarino, the wizard who has introduced all of us to the black magic of processing and fabrication, with enthusiasm and dedication, and for all the talks about politics, mountains, travels, food, life, etc. Silvano De Franceschi, with his jolly attitude and religious passion about magnetotransport on nanothings, for coping with my ignorance and disbelief and shading light in this “Coulomb what?” thing. My fellow Ph.D. student Tomaˇz Mlakar, and the Ph.D.-student-to-be Massimo Mongillo for their help and support, especially in this last period of frenetic craziness. Last but not least, Stefan Heun, for many things among which the proofreading of all the stuff that follws, back when it was still in beta version... I must not forget the patience with which Luca Businaro has stuck in my head the basic and advanced notions of EBL and SEM. Did I write ‘patience’ ? Well... anyhow Luca has spent an enormous amount of time with me, and by now I have learned to read the subtitles, so, yes, patience

Foreword

and attention. Thanks Luca. Actually I owe a lot also to many other Lilit group members, starting hierarchically from Stefano Cabrini down to the Ph.D. students, Mauro Prasciolu (always available for advice and discussion, and always smiling), Radu, Arrigo, Matteo, and yes, also “la vita è dura” Alessandro. And then come the people at the NEST in Pisa: “papà” Pasqualantonio Pingue whith his sardonic humor and all the help, suggestions, consulting, praying and swearing at the EBL (sorry again, Pask, for hitting the column with the sample holder!), and Franco Carillo, who also has always found five minutes for me when I needed, and more if I needed more. Special thanks to Aleˇs, Federico and Paolino of the mechanical workshop: it is a pleasure to see them working with their hands and huge machinery to fabricate tiny, perfect, pieces of art. Always in good mood, rarely mistaken when giving suggestions on the design of things to be made, and really enjoying a work well done. Moreover, they have promptly saved me in several situations in which I urgently needed their help. Same goes for Stefano Bigaran, the “everything” technician, the man who solves problems, who never sits, who brings back to life computers (and also kills them sometimes!). And how to forget the countless times that the electronics technicians Stefano, Fabio or Andrea have fixed some board, some controller, or resoldered the horrible lump of wires of one of the low-temperature inserts! I also have to mention the support I have recived in the other half of time, from my parents and their spouses, patient and understanding, to whom I owe most, and from my friends, who are still my friends even though I’ve been increasingly spending my time in the laboratory and less and less with them. Thanks guys. At last Cecilia, the women I want to spend my life with, and who happens also to be my wife, deserves a very special aknowledgment. First, because she’s not asked to divorce yet, even though she has been more like a widow for the past months, while I was measuring ans writing, calling at 10 P.M. saying “Sorry, sorry, I’ll be home in 15 minutes” and instead arriving at after midnight; when I was writing or studying or even measuring also in the weekends, not giving any help to solve problems at home or do housework or preparing dinner, and so on. Second, because I would not even have graduated withour her support, pressure and deadlines. And last, for all the rest. In synthesys, all these people (and others that I am certainly forgetting now) deserve credit for the good things that may appear in the pages that follow. Mistakes, errors, and misinterpretations are, instead, my sole responsability.

D. E. Trieste, January 22, 2007

Contents Table of contents

i

Introduction

1

1 Instruments and Techniques 1.1 Molecular beam epitaxy . . . . . . . . . . . . . . 1.1.1 Growth apparatus . . . . . . . . . . . . . 1.1.2 MBE growth process . . . . . . . . . . . . 1.1.3 Growth rate calibration . . . . . . . . . . 1.1.4 The MBE system at Laboratorio TASC . 1.2 High resolution X-ray diffraction . . . . . . . . . 1.2.1 Bragg’s law . . . . . . . . . . . . . . . . . 1.2.2 Diffractometer . . . . . . . . . . . . . . . 1.2.3 Strain and alloy composition in epilayers . 1.3 Atomic force microscopy . . . . . . . . . . . . . . 1.3.1 AFM operation . . . . . . . . . . . . . . . 1.4 Two dimensional electron gases . . . . . . . . . . 1.5 Device fabrication . . . . . . . . . . . . . . . . . 1.5.1 Optical lithography . . . . . . . . . . . . 1.5.2 Electron beam lithography . . . . . . . . 1.6 Transport measurements . . . . . . . . . . . . . .

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5 5 6 7 9 11 12 12 13 14 17 18 21 23 23 24 26

2 InAlAs step-graded buffers 2.1 Lattice-mismatched growth . . . . . . . 2.1.1 Heteroepitaxy . . . . . . . . . . . 2.1.2 Previous results for the growth of 2.2 Structural properties . . . . . . . . . . . 2.2.1 The buffer . . . . . . . . . . . . . 2.2.2 Strain relief model . . . . . . . . 2.2.3 XRD measurements . . . . . . . 2.2.4 Surface and interfaces . . . . . . 2.3 Transport measurements . . . . . . . . .

. . . . . . QWs . . . . . . . . . . . . . . . . . .

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29 29 30 34 34 36 37 39 42 44

. . . . . . . . . . InGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ii

CONTENTS

3 Scattering mechanisms in In0.75 Ga0.25 As quantum wells 3.1 Low temperature scattering mechanisms . . . . . . . . . . 3.1.1 Ionized impurity scattering . . . . . . . . . . . . . 3.1.2 Alloy disorder scattering . . . . . . . . . . . . . . . 3.1.3 Interface roughness scattering . . . . . . . . . . . . 3.2 Scattering in 30 nm thick In0.75 Ga0.25 As QW . . . . . . . 3.2.1 Schrödinger-Poisson simulations . . . . . . . . . . 3.2.2 Mobility versus carrier density measurements . . . 3.3 InAs QWs . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Mobility versus density . . . . . . . . . . . . . . . 3.4 High mobility samples . . . . . . . . . . . . . . . . . . . . 3.4.1 Mobility anisotropy . . . . . . . . . . . . . . . . . 3.4.2 InAs inserted samples . . . . . . . . . . . . . . . . 3.5 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . 4 InGaAs few-electron QDs 4.1 Quantum dots . . . . . . . . . . . . . . . . . . . . 4.1.1 Transport through quantum dots . . . . . . 4.1.2 Spin configurations in few-electron quantum 4.1.3 Kondo effect in quantum dots . . . . . . . . 4.2 In0.11 Ga0.89 As/GaAs structures . . . . . . . . . . . 4.3 E-beam lithography of the gates . . . . . . . . . . 4.4 Measurement issues . . . . . . . . . . . . . . . . . . 4.4.1 Effective electronic temperature . . . . . . . 4.4.2 Few-electron regime . . . . . . . . . . . . . 4.4.3 QPC curve interpretation . . . . . . . . . . 4.5 A few-electron QD in the single QW sample . . . . 4.5.1 Stability diagram . . . . . . . . . . . . . . . 4.5.2 Landé g-factor . . . . . . . . . . . . . . . . 4.6 A few-electron QD in the double QW sample . . . 4.6.1 Few electron regime . . . . . . . . . . . . . 4.6.2 g-factor . . . . . . . . . . . . . . . . . . . . 4.6.3 Kondo effect . . . . . . . . . . . . . . . . . 4.7 Summary of quantum dots results . . . . . . . . . 4.8 Schottky gates on In0.75 Ga0.25 As samples . . . . . 4.8.1 Suspended bridges . . . . . . . . . . . . . .

. . . . . . dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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49 50 53 54 55 56 57 59 61 63 65 66 70 71

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75 78 78 82 83 85 89 90 90 92 93 95 95 97 101 101 106 109 114 115 116

Conclusions

121

Appendices

123

CONTENTS

iii

A Optical lithography recipes 123 A.1 Mesa etching . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.2 Ohmic contacts . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A.3 Top gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 B E-beam lithography recipes 127 B.1 Few-electron quantum dots lithography . . . . . . . . . . . . 127 B.2 Suspended metal bridges lithography . . . . . . . . . . . . . . 128 C Other publications

131

Bibliography

132

iv

CONTENTS

Introduction Semiconductor technology has revolutionized the second half of the twentieth century. Although the majority of the revolution has been based on silicon, increasing demands in terms of speed and functionality have generated interest in alternative semiconductor materials. Most noteworthy among these are the compound semiconductors made of group III and group V elements, of which GaAs and related alloys are the most developed. At present, III-V compound semiconductors provide the basis material for a number of well-established commercial technologies, as well as new cutting-edge classes of electronic and optoelectronic devices. Just a few examples include highelectron-mobility transistors and heterostructure bipolar transistors, diode lasers, light-emitting diodes, photodetectors, electro-optic modulators, and frequency-mixing components. The operating characteristics of these devices depend critically on the physical properties of the constituent materials, which are often combined in heterostructures containing carriers confined into regions of the order of a few nanometers. One of the major advantages of ternary alloys with respect to binary compounds is the possibility to tune, in the range defined by the constituent binaries, their physical properties, such as band gap, effective mass, and the Landé g-factor, by changing the alloy composition. One of the most studied alloy systems, both for optical and electronic applications, is Inx Ga1−x As. At an indium concentration (x) of 0.53, this material is lattice matched to InP and today largely used in light-emitters for optical fiber communications since its emission wavelength of 1.55 µm is within the optimum transmission window of silica fibers. For electronic applications, the possibility to suppress the Schottky barrier at the metal-semiconductor interface, by increasing the indium concentration over 0.75, makes high In concentration Inx Ga1−x As alloys the ideal material to create highly transmissive junctions. This peculiar feature makes this system appealing for studying the transport properties at semiconductor/superconductor or semiconductor/ferromagnet junctions at low temperatures. Furthermore, at high indium concentrations the Landé g-factor of the system increases towards the value of InAs (around -15); thus the resulting large Zeeman spin splitting under the application of weak magnetic fields makes this alloy a promising candidate for spin-valve mesoscopic

2

Introduction

devices working at relatively high temperatures. A major problem in the development of devices based on such alloys is the lack of substrates with suitable lattice parameter. Consequently, in order to realize structures with good electrical properties, a careful control of structural defects, related to the strain relief, is necessary. In this thesis, we have investigated the structural and transport properties of two-dimensional electron gases formed in In0.75 Ga0.25 As/In0.75 Al0.25 As quantum wells grown on GaAs (001) substrates. Using a graded buffer to accommodate the lattice mismatch between GaAs and Inx Ga1−x As, and inserting thin InAs layers in the quantum well to reduce alloy disorder scattering, we are able to obtain low temperature electron mobilities well above 30 m2 /Vs in unintentionally doped structures with a carrier density of the order of 3 × 1015 m−2 . By increasing the carrier density to ∼5 × 1015 m−2 with the aid of a top metal gate, the mobility can exceed 50 m2 /Vs. In order to characterize such structures, several issues have been explored: 1. We have optimized the step-graded buffer layer structure employed to accommodate the lattice mismatch between the GaAs substrate and the InGaAs layer in order to minimize the residual strain inside the conductive channel. We found that a careful control of the residual strain in the quantum well region is necessary to obtain a high electron mobility in such systems. 2. The scattering processes limiting the low temperature mobility in these systems are deduced from the dependence of the electron mobility on the two-dimensional electron gas density. From such analysis we have inferred that the mobility is essentially limited by background ionized impurities and by alloy scattering. 3. By partially suppressing these main scattering mechanisms the electron mobility of the sample is increased to the extremely high values indicated above. The electron mobilities achieved are high enough to envisage the fabrication of mesoscopic devices on these samples, such as quantum point contacts or quantum dots. In particular, few-electron quantum dots are thought to be the basis for the practical realization of Qbits, the building blocks of quantum computers, using the spin degree of freedom as carrier of information. The high Landè g-factor of Inx Ga1−x As would make spin manipulation in Inx Ga1−x As based quantum dots much more efficient, with respect to GaAs based quantum dots, in terms of the relaxation of both low temperature and high magnetic field requirements. We have fabricated few-electron quantum dots with integrated charge readout in lower indium concentration Inx Ga1−x As quantum wells, measuring the electron g-factor and studying other spin-related issues, like the

Introduction

3

Kondo effect. We have also begun addressing the problems related with nanometer-sized Schottky gate fabrication on high indium concentration Inx Ga1−x As quantum well structures. Chapter 1 of this thesis desctibes the growth procedure and characterization techniques employed to study the Inx Ga1−x As quantum well samples and devices. A thorough analysis of the structural properties of the InAlAs/InGaAs quantum wells and the relation between their structural and transport properties are presented in chapter 2. The dependence of the electron mobility on the carrier density is studied in chapter 3 in order to determine the low temperature scattering processes limiting the electron mobility. Several modifications to the sample design resulting from this analysis are shown and discussed, including the reduction of the quantum well width and the insertion of thin InAs layers in the quantum well. Chapter 4 is dedicated to the study of mesoscopic devices fabricated from two-dimensional electron gases formed in Inx Ga1−x As quantum wells. The concept of quantum computing is introduced, showing the need of electrically tuned few-electron semiconductor quantum dots. Low temperature transport experiments in low indium content, few electron quantum dots with integrated charge readout are described, focusing on the spin effects. The Landé g-factor of the electrons in the quantum dots is measured, and the Kondo effect in the weak coupling regime studied. To conclude with, the issues related to the fabrication of nanostructures on high indium content samples are addressed and discussed.

4

Introduction

Chapter 1

Instruments and Techniques The aim of this chapter is to briefly describe the instruments and techniques employed in this thesis. The first section considers the basic principles of molecular beam epitaxy (MBE) and, in particular, focuses the attention on the MBE machine installed at the Laboratorio TASC INFMCNR. Sections 1.2 and 1.3 describe the main morphological characterization techniques: first, high resolution X-Ray diffraction (XRD) which gives precise quantitative information on the crystal structure and is of fundamental importance for MBE growth calibration of indium alloys, and then Atomic Force Microscopy, used to probe microscopically the topography of sample surfaces. In section 1.4 the growth of Inx Ga1−x As/Inx Al1−x As heterostructures containing a two-dimensional electron gas is discussed, while section 1.5 is focused on the fabrication techniques that are used to pattern samples with Hall bar devices and nanometer sized top metal gates needed to perform transport measurements. Finally, Section 1.6 is dedicated to the description of the experimental setup used to perform transport measurements at cryogenic temperatures.

1.1

Molecular beam epitaxy

Molecular beam epitaxy (MBE) is an Ultra-High-Vacuum (UHV) based technique for producing high quality epitaxial structures with monolayer (ML) control. Since its introduction in the 1970s as a tool for growing highpurity semiconductor films, MBE has evolved into one of the most widely used techniques for producing epitaxial layers of metals, insulators, semiconductors, and superconductors, both at the research and the industrial production level. The principle underlying MBE growth is relatively simple: it consists essentially in the production of atoms or clusters of atoms by heating of a solid source. They then migrate in an UHV environment and impinge on a hot substrate surface, where they can diffuse and eventually are incorporated into the growing film. Despite the conceptual simplicity, a

6


Figure 1.1: Schematic drawing of a generic MBE system (top view).

great technological effort is required to produce systems that yield the desired quality in terms of material purity, uniformity, and interface control. An exhaustive discussion on the principles and applications of the MBE technique can be found in Ref. [1]. The choice of MBE with respect to other growth techniques depends on the desired structures and needs. In particular, MBE is the proper technique when abruptness and control of interfaces and doping profiles are needed, thanks to the low growth temperature and rate. Besides, the control on the vacuum environment and on the quality of the source materials allows a much higher material purity, as compared to non UHV based techniques, especially in Al containing semiconductors for applications in high mobility and high speed devices.

1.1.1

Growth apparatus

A schematic drawing of a standard MBE system is shown in Fig. 1.1. Some basic components are: • The vacuum system consists in a stainless-steel growth chamber, UHVconnected to a preparation chamber, where substrates are degassed prior to growth. All the components of the growth chamber must be able to resist bake-out temperatures of up to 200◦ C for extended periods of time, which are necessary to minimize outgassing from the

Instruments and Techniques

7

internal walls. • The pumping system must be able to efficiently reduce residual impurities to a minimum. The pumping system usually consists of ion pumps, with auxiliary Ti-sublimation and cryogenic pumps, for the pumping of specific gas species. Typically the base pressure of an MBE chamber is from 10−11 to 10−12 mbar, which determines an impurity concentration below 1015 cm−3 in grown structures. • Liquid N2 cryopanels surround internally both the main chamber walls and the source flanges. Cryopanels prevent re-evaporation from parts other than the hot cells and provide thermal isolation among the different cells, as well as additional pumping of the residual gas. • Effusion cells are the key components of a MBE system, because they must provide excellent flux stability and uniformity, and material purity. The cells (usually from six to ten) are placed on a source flange, and are co-focused on the substrate to optimize flux uniformity. The flux stability must be better than 1% during a work day, with day-today variations of less than 5% [2]. This means that the temperature control must be of the order of ±1◦ C at 1000◦ C [1]. The material to be evaporated is placed in the effusion cells. A mechanical or pneumatic shutter, usually made of tantalum or molybdenum, is placed in front of the cell, and it is used to trigger the flux coming from the cell (see Fig. 1.1). • The substrate manipulator holds the wafer on which the growth takes place. It is capable of a continuous azimuthal rotation around its axis to improve the uniformity across the wafer. The heater behind the sample is designed to maximize the temperature uniformity and minimize power consumption and impurity outgassing. Opposite to the substrate holder, an ionization gauge is placed which can be moved into the molecular beam and is used as a beam flux monitor (BFM). • The Reflected High Energy Electron Diffraction (RHEED) gun and detection screen are used to calibrate precisely the material fluxes evaporated by the effusion cells; with RHEED it is possible to monitor the epitaxial growth monolayer by monolayer. A thorough description of this tool is given in section 1.1.3.

1.1.2

MBE growth process

In general, three different phases can be identified in the MBE process [1]. The first is the crystalline phase constituted by the growing substrate, where short- and long-range order exists. On the other extreme, there is the disordered gas phase of the molecular beams. Between these two phases, there is

8


Figure 1.2: Different surface elemental processes in MBE.

the near-surface region where the impinging molecular beams interact with the hot substrate. This is the phase where the phenomena most relevant to the MBE process take place. Atomic or molecular species get physisorbed or chemisorbed on the surface where they can undergo different processes (Fig. 1.2). Atoms can diffuse on a flat surface (a), where they can reevaporate (b), meet other atoms to form two-dimensional clusters (c), reach a step where they can be incorporated (d), or further migrate along the step edge (e) to be incorporated at a kink (f). The MBE growth of III-V semiconductors uses the so called three temperatures method [1], in which the substrate is kept at an intermediate temperature between the evaporation temperature of the group III and group V source materials. Group V species have a much higher vapor pressure than group III atoms, therefore typical cell working temperatures are lower for group V evaporation (around 300◦ C for As) than for group III species (around 1100◦ C, 800◦ C and 1000◦ C, for Al, In, and Ga, respectively). At the substrate temperature, the vapor pressure of group III species is nearly zero; this means that every atom of the III species that impinges on the substrate is chemisorbed on its surface; in other words group III atoms have a unit sticking coefficient. The high vapor pressure of the group V species favors, on the contrary, the re-evaporation of these species from the sample surface. Due to the higher group V species volatility with respect to group III, growth is usually performed with an V/III beam flux ratio much higher than one. This flux imbalance does not affect the one-to-one crystal stoichiometry between III-V species. In fact, as shown by Foxon and Joyce [3, 4], in the case of homoepitaxial growth of GaAs, As atoms do not stick if Ga atoms are not available on the surface for bonding. So, in the case of GaAs, the growth rate is driven by the rate of impinging Ga atoms on the substrate.


9

The flux J of atoms evaporated from an effusion cell can be described as [1] aP 22 √ J = 1.11 × 10 × cos θ mol cm−2 s−1 , (1.1) 2 d MT where a is the aperture area of the effusion cell, d is the distance of the aperture to the sample, θ is the angle between the beam and the normal to substrate, M is the molecular weight of the beam species, T the temperature of the source cell, and P is the vapor pressure of the beam; the vapor pressure is itself a function of the source cell temperature as A + B log T + C , (1.2) T where A, B and C are material-dependent constants. For a growth rate of about 1 µm/h the typical fluxes are ∼ 1016 atoms cm−2 s−1 for group V elements and ∼ 1015 atoms cm−2 s−1 for group III. In the case of alloys with mixed group III elements, such as InGaAs and InAlAs, the reactions with the group V elements are identical to those observed in the growth of binary compounds, such as GaAs [3, 4]. The only difference is that the optimum growth temperature range is driven by the less stable of the two group III atoms, i.e. by indium in the case of InGaAs and InAlAs alloys. In fact, Turco et al. [5] observed that the incorporation of In in InAlAs alloys grown on GaAs substrates decreases for samples grown at temperatures higher than 500◦ C, while significant Ga or Al re-evaporation takes place only at higher temperatures (about 650◦ C for Ga, and about 750◦ C for Al). In the case of substrate temperatures below 500◦ C a unit sticking coefficient can be assumed for the growth of In-based alloys; the resulting growth rate and composition are simply derived from the two binary growth rates that form the alloy. For example if RInAs , and RGaAs are the growth rates for InAs and GaAs respectively, then the total growth rate of the alloy is RInGaAs = RInAs + RGaAs while the indium concentration x is the same as in the gas phase and is given by log P =

x=

1.1.3

RInAs . RInAs + RGaAs

(1.3)

Growth rate calibration

To grow ternary alloys as Inx Ga1−x As and Inx Al1−x As with known indium concentration x, it is necessary to measure accurately, prior to growth, the three growth rates RGaAs , RAlAs , and RInAs . GaAs and AlAs The growth rates of GaAs and AlAs are determined by the intensity oscillations of the specular spot of the RHEED signal during the growth of

10


shutters open RHEED intensity (Arb. Units)

GaAs

shutters closed

AlAs

0

10

20

30

40

50

Tim e (s)

Figure 1.3: RHEED oscillations. In the left panel an actual measurement for GaAs and AlAs grown on GaAs (001). In the right panel a schematic view of the relationship between RHEED intensity and monolayer coverage θ.

a GaAs or AlAs film on a GaAs substrate [1]. This technique employs a high energy (up to 20 keV) electron beam, directed on the sample surface at grazing incidence (a few degrees); the diffraction pattern of the electrons is displayed on a fluorescent screen and acquired by a CCD. Thanks to the grazing incidence and the limited mean free path of electrons in solids, the electron beam is scattered only by the very first atomic layers, giving rise to a surface-sensitive diffraction pattern. Besides, the grazing geometry limits the interference of the RHEED electrons with the molecular beams, making the technique suitable for real-time analysis during growth. During crystal growth the intensity of the zero order diffraction spot (the specular spot) is recorded as a function of time. An example of such measurements is shown in the left panel of figure 1.3, where it can be noticed that the intensity of the spot has an oscillatory behavior. This happens because a flat surface, present when a monolayer is complete, reflects optimally the electrons while in a condition in which a half-monolayer has been deposited the electron beam gets partially scattered by the stepped surface. As schematically shown in the right panel of figure 1.3, starting with a flat surface and proceeding with growth, the incident electron beam gets partially scattered by the islands steps of the growing monolayer, thus reducing the reflected intensity. Scattering becomes maximum at half monolayer coverage, while as the new monolayer completes (one Ga or Al plus one As layer) the surface flattens again by coalescence of the islands, and the reflected intensity recovers its value. A progressive dumping of the oscillation intensity is due to an increasing disorder of the growth front as the growth proceeds.


11

Thus, a period of RHEED oscillation corresponds to the growth of a single monolayer. By measuring the time necessary to complete a certain number of oscillations one can calculate the growth rate in monolayer/s for a fixed effusion cell temperature, and easily convert it to units of ˚ A/s knowing the lattice parameter of GaAs or AlAs. This calibration is performed almost daily, prior to sample growth, on a ad hoc substrate. The day-to-day variation of RGaAs and RAlAs , with constant cell temperatures is ∼1%; the long term behavior of these rates, on the other hand is fairly predictable, and is constant (within 1−2%) until the cell is almost empty, unless major changes to the cell environment happen (like refilling, etc.). InAs Unfortunately, InAs growth rates cannot be measured taking advantage of RHEED oscillations. This is because it is very difficult to obtain good quality, monolayer-flat InAs surfaces on any substrate, and it is virtually impossible on GaAs ones. In fact the large lattice mismatch between InAs and GaAs (∼7.2%) favors the formations of 3D islands even after the first one or two monolayers. However the relation ((1.3)) provides an alternative method to evaluate the InAs growth rate, by in-situ measuring the GaAs growth rate by RHEED oscillations, and the In concentration in a thick InGaAs layer by ex-situ X-ray diffraction measurements (see section 1.2.3). The indium flux calibration is a time-consuming operation that involves the growth of several samples of Inx Ga1−x As, and multiple X-ray diffraction measurements on each sample. For this reason, and knowing the relative stability of the fluxes until the cells are almost empty, the In flux calibration is performed only once every few months, after major maintenance operations to the MBE chamber.

1.1.4

The MBE system at Laboratorio TASC

The MBE chamber installed at Laboratorio TASC INFM-CNR in Trieste is mainly dedicated to the growth of GaAs based heterostructures characterized by a very high carrier mobility. Such a system requires some peculiar modifications. Two 3000 l/s cryopumps replace the ion pumps, providing a cleaner, higher-capacity pumping system. All-metal gate valves are mounted to eliminate outgassing from Viton seals. No group-II materials, such as Be, are used for p-doping, since they are known to drastically reduce carrier mobility. High-capacity and duplicate cells are used to avoid cell refilling or repairing for extended periods. Extensive degassing and bake out duration (three months at 200◦ C) were carried out after the installation of the MBE system to increase the purity of the materials.

12

High resolution X-ray diffraction

Figure 1.4: Bragg’s Law. The dots are the direct lattice points; the set of lattice planes where Bragg reflection is taking place is marked by the horizontal lines.

1.2


Since Max von Laue’s first observation of X-ray diffraction by a crystal in 1912 [6], and the Braggs’ quantitative explanation of this effect [7], this technique has proven to be a powerful tool to get accurate quantitative information on crystal structures. The basic idea is that taken a crystal and an electromagnetic radiation with a wavelength smaller than, but of the same order of, the lattice parameter of the crystal, diffraction will take place. From the diffraction angle one can derive the spacing of the crystal planes. Repeating this process for several incident and diffracted directions, one can completely reconstruct the crystal structure.

1.2.1

Bragg’s law

Bragg’s law is very simple: given a crystal lattice and assuming that a family of parallel lattice planes are separated by a distance d, and given that monochromatic X-rays with wavelength λ are impinging on the crystal at an angle θi with the planes, there will be a diffracted beam of X-rays at an angle θf = θi = θ if nλ = 2d sin θ. (1.4) Basically this is just the condition for constructive interference of X-rays reflected by different planes. The described geometry is displayed in Fig. 1.4 (a)). It is convenient at this point to introduce the reciprocal lattice. Each vector of the reciprocal lattice, indicated by qhkl and thus identified by the Miller indices h, k, l is orthogonal to a family of planes of the direct crystal lattice; moreover, the length of the reciprocal lattice vector is inversely proportional to the spacing of the planes: |qhkl | = 2π/dhkl . It can be easily


13

GONIOMETER

w

2Q

Y

F X-RAY TUBE BARTEL MONOCROMATOR

DETECTOR Figure 1.5: Schematic view of a four-axes diffractometer. The sample is colored in gray and all four controllable rotation angles are indicated.

seen that the Bragg condition (Eq. (1.4)) can be rewritten as kf − ki = qhkl

(1.5)

where ki and kf are the wavevectors of the incident and diffracted beams (|k| = 2π/λ) and qhkl is a reciprocal lattice vector (see Fig. 1.4 (b)). Thus, the Bragg reflections are indicated as (hkl) that identify univocally the reciprocal space vector and the family of lattice planes by which the Bragg condition is satisfied. By measuring an appropriate set of Bragg reflections, and thus a set of reciprocal lattice vectors, it is possible to completely reconstruct the structure of the observed crystal. In the particular case of a cubic crystal1 with lattice constant a, the spacing d for a family of planes with Miller indices hkl is given by a d= √ . (1.6) 2 h + k 2 + l2

1.2.2

Diffractometer

To measure accurately the incident and diffracted angle of the monochromatic X-ray radiation, a High Resolution X-Ray Diffractometer (HRXRD) is needed. Such an instrument is schematically depicted in figure 1.5 and is composed of four main elements: 1

GaAs, InAs, AlAs and their (unstrained) ternary alloys are all face centered cubic crystals.

14


• The X-ray tube. It generally consists of a metal anode hit by a beam of high energy (30-40 keV) electrons emitted by a nearby cathode; the core level electrons of the anode’s atoms are excited and when they recombine they emit X-ray photons at a discrete set of wavelengths, characteristic of the anode’s element. In our setup a copper anode tube has been used. • The monochromator selects only one of the emitted wavelengths. For high resolution diffraction measurements the wavelength has to be very precisely selected. To accomplish this, the X-rays undergo multiple Bragg reflections in appropriately chosen single crystals; this has also the appreciated side-effect of greatly reducing the angular divergence of the outcoming beam. For our measurements we have always used a so-called Bartel monochromator, in which the beam undergoes four times the 220 reflection in germanium single-crystals. This gives an uncertainty in the determination of the wavelength of less than one part in 100,000 and an angular divergence of 12 arcsec. • The goniometer is responsible for the measurements of the Bragg angles and the appropriate alignment of the crystal planes with respect to the incident beam. Four angles can be adjusted: 2θ is twice the diffraction angle while ω is the angle between the incident beam and the surface sample. Since these two angles alone are needed to extract the lattice plane spacing, their setting and determination has to be as accurate as possible. In our instrument their resolution is 10−5 degrees. The other two angles, Ψ and Φ, necessary only to align the sample crystal axes to the incident beam, are less important in determining the overall resolution of the measurements; in our diffractometer they have a resolution of 0.01 degree. • The detector collects and counts the X-ray photons. According to the resolution to be achieved it can be coupled to a receiving slit that simply limits the acceptance angle of the detector or to another germanium monochromator (which increases the resolution, but on the other hand reduces greatly the count rate, thus increasing the acquisition times). The diffractometer employed to perform the measurements presented in this thesis, and shown in figure 1.6, is a Phillips X’Pert-MRD using a Cu-Kα radiation with a wavelength λ = 1.54056 ˚ A.

1.2.3

Strain and alloy composition in epilayers

In this thesis X-ray diffraction has been used both for indium flux calibration and for indium concentration and residual strain measurements in


15

Figure 1.6: The Phillips X’Pert-MRD High Resolution X-Ray Diffractometer employed in this thesis.

Inx Ga1−x As/Inx Al1−x As quantum wells grown on GaAs substrates. This greatly simplifies the analysis of diffraction data since all indium containing layers are grown epitaxially on GaAs h001i, have the same crystal lattice (face-centered cubic), and have lattice constants of comparable size. This allows to measure the overlayer crystal structure by comparison to GaAs. Typically one only measures the difference ∆θ of the angle of the diffraction peak of the overlayer with respect to the peak of the substrate, performing a so called rocking curve or ω-2θ scan. A rocking curve consists in a simultaneous scan of the angles 2θ and ω so that one probes reciprocal lattice vectors of different length but same orientation. Before a rocking curve measurement the crystal plane of the substrate must be aligned with respect to the diffractometer setup, by maximizing the peak intensity of GaAs with respect the two angles Ψ and Φ. As pointed out by Hornstra and Bartels [8], the measured angular difference ∆θM between epilayer and substrate can be different from the real difference ∆θB due to tilting effects between the substrate and the overlayer. However, this tilt effect can be eliminated by measuring four rocking curves after successive 90◦ rotations of the sample around the Φ axis; the correct Bragg angle difference is then ∆θB =

0 + ∆θ 90 + ∆θ 180 + ∆θ 270 ∆θM M M M . 4

(1.7)

16


Figure 1.7: A (004) rocking curve for a 1µm-thick relaxed Inx Ga1−x As (x ∼ 0.10) layer on GaAs. The overlayer peak is much less intense and broader than the substrate peak due to its smaller thickness and crystal defects.

In a perfect cubic crystal a⊥ = aq = a. In principle it would be sufficient to measure only the (004) rocking curves and deduce from them the crystal structure of the grown material. However, when epitaxially growing an overlayer with a different lattice parameter than the substrate, the overlayer is tetragonally distorted to match the in plane lattice parameter of the substrate, compensating with an opposite distortion of the out of plane dimension. Even when the overlayer is grown way beyond the critical thickness, and even if care is taken to relax the strain due to lattice-mismatch and have a cubic lattice for the overlayer, one cannot exclude a slight tetragonal distortion. That is why both (004) and (224) rocking curves are needed to find the overlayer primitive cell. Since all samples have been grown on (001) oriented GaAs substrates, to measure a⊥ , the lattice parameter along the growth direction, we have taken the rocking curves in the vicinity of the symmetric (004) Bragg reflection of GaAs, while to evaluate aq , the lattice parameter in the h110i directions, we have recorded the rocking curves in the vicinity of the asymmetric (224) reflections of GaAs with both grazing incidence (224 ω + ) and grazing exit (224 ω − ) angles.


17

The [001] lattice parameter is then calculated as a⊥ =

λ (004) 2 sin(θB

(004)

,

+ ∆θB

)

v u 8 aq = u u (224) (224) t 2 sin(θB +∆θB ) 2

(1.8)

while the h110i lattice parameter is

λ

(004)

−

4 a⊥

2 .

(1.9)

(224)

In these expressions θB and θB are the Bragg angles of GaAs for the (004) (224) (004) and (224) reflections, and ∆θB and ∆θB are the angular distances of the overlayer peak with respect to the GaAs one. A typical (004) rocking curve is plotted in figure 1.7. To get the indium concentration x of the alloy layer knowing both the in plane and out of plane lattice parameters, one has to take into account the tetragonal distortion and rely on elasticity theory to derive the “unstrained” lattice parameter of the alloy [9]. Practically one has to self-consistently solve the following equation: ε⊥ (x) = −2

C12 (x) εq (x) , C11 (x)

(1.10)

0 (x) 0 (x) where ε⊥ (x) = a⊥a−a and εq (x) = aqa−a with a0 (x), C11 (x) and C12 (x) 0 (x) 0 (x) are the lattice parameter of the unstrained unit cell and the stiffness constants of the layer with In concentration x respectively. These values are obtained by linear interpolation of the binary compounds values as stated by Vergard’s law and confirmed by recent literature [10].

1.3

Atomic force microscopy

In 1986, Binnig, Quate, and Gerber invented a new type of microscope, the atomic force microscope (AFM), able to obtain high resolution images of both conductive and insulating samples. The AFM belongs to the family of the scanning probe microscopes (SPMs). The idea behind all SPMs is very simple: the sample to be investigated or the microscope probe (tip) is mounted onto a piezo-resistive crystal (piezo). The piezo is deformed with sub-nanometer accuracy in all three dimensions by applying voltages. The various SPMs differ in the parameter used to detect the tip-sample distance. For the AFM the monitoring parameter is the force between tip and sample.

18


Figure 1.8: Typical force versus tip-sample distance curve. Indicated are the force regimes of contact, non-contact, and intermittent-contact mode.

The main forces present between tip and sample arise from [11]: (i) electrostatic or Coulomb interactions; (ii) polarization forces; (iii) quantummechanical forces, which give rise to covalent bonding and repulsive exchange interactions; (iv) capillary forces present when the AFM is operating in a humid environment. Capillary forces are always attractive with a magnitude of about 10−8 N. When the tip is far away from the sample, the interaction is an attractive Van der Waals force. When the tip-sample distance becomes of the order of the interatomic distances in solids (a few ˚ Angstroms), repulsive forces start dominating and become extremely strong when the electronic clouds of the atoms of tip and sample start to overlap (see figure 1.8). The tip-sample interactions are usually described by a Lennard-Jones potential: σ 12 σ 6 U (d) = 4 − , (1.11) d d where d is the tip-sample distance, σ is the distance at which U(σ)=0, and √ - is the energy value in the equilibrium position deq =σ 6 2. The derivative of Eq. (1.11) represents, to a first approximation, the force between tip and sample (see Fig. 1.8).

1.3.1

AFM operation

All the experiments described in this thesis have been performed using a CP Research, VEECO AFM system. The key elements of an AFM are show in Fig. 1.9. The AFM tip, fixed to a cantilever, is mounted on a carrier chip and approaches the sample using the Z-motor. When the tip interacts with the sample, the cantilever bends. This bending is measured by means of a laser beam which is deflected by the back of the cantilever onto a Position Sensitive PhotoDetector (PSPD). When the force is changing, the cantilever deflection changes, and the laser spot on the PSPD moves. Thus,


19

Figure 1.9: Schematic setup of an AFM.

the intensity of the signal measured by the PSPD can be related to the force acting on the cantilever. An AFM can be operated in constant height as well as constant force mode. In constant height mode, the AFM tip is scanned on the sample surface at a fixed distance, and the topography is directly related to the cantilever bending. Constant height mode is more suitable when high scanning rates are needed but works only for surfaces which are not too corrugated. In constant force mode, the feedback loop acts on the piezo in order to keep the tip-sample force constant at a reference value. In this mode, the sample topography is given by the movements of the piezo. An AFM can be operated in three regimes: contact, non-contact, and intermittent contact or tapping. Each mode can be associated with a specific region in the force-distance curve as shown in Fig. 1.8. In contact mode (CM), the tip is touching the sample surface. The tipsample interactions are repulsive with a magnitude of about 10−6 to 10−9 N. The spatial resolution can be very high since the forces driving the feedback system are the short-range repulsive ones. The drawbacks are that since the tip is always in physical contact with the sample while scanning, very high (on the atomic scale!) lateral friction forces are present, and these can both damage the sample surface, the tip, and give rise to artifacts in the images. In non-contact mode (NCM), the tip-sample distance is 10 to 100 nm.

20


Figure 1.10: Changes of the vibration amplitude of an AFM cantilever vibrating above the sample surface in the case of (a) non-contact mode and (b) intermittent contact mode.

The tip-sample force, due to van der Waals interactions, is attractive with a typical magnitude of the order of 10−9 to 10−12 N and positive derivative. Far away from the sample, the cantilever is vibrating at a frequency ωr just above its resonance frequency ω0 . As the tip is approaching the sample, the interaction induces a decrease in the resonance frequency ωint according to r ωint =

ω02 −

1 ∂F ts · , m ∂r

(1.12)

where F ts is the tip-sample force. This induces a decrease of the amplitude of the oscillation (see Fig. 1.10(a)) which is monitored by the feedback loop to control the tip-sample distance. In non-contact mode, the strength of the tip-sample interaction is 103 to 106 times lower than in contact mode; thus this operation mode is indicated to image very delicate samples such as organic films. However, the spatial resolution that can be achieved is lower than in contact mode since non-contact AFM is based on long range van der Waals interactions. The intermittent contact mode (ICM) is based on the same principles of NCM, but in ICM the cantilever is vibrating at a frequency just below the natural frequency of the cantilever. Thus, as the tip is brought closer to the sample, the vibration amplitude increases (see Fig. 1.10(b)) up to the point where the tip touches the sample surface at the end of every oscillation. This induces a reduction of the vibration amplitude to the set value. As in NCM, the vibration amplitude is used to control the tip-sample distance. In ICM the spatial resolution is comparable with that of CM while the interaction strength is intermediate between CM and NCM; moreover lateral friction forces are virtually absent, allowing the imaging of delicate samples. All topographies shown in this theses are taken using ICM.


21

Figure 1.11: Layer sequence (on the top) and conduction band profiles (bottom) for two types of 2DEGs: on the left a Si MOSFET inversion layer and on the right a GaAs/AlGaAs modulation doped single heterostructure (from von Klitzing’s Nobel lecture [12]).

1.4

Two dimensional electron gases

The transport experiments described in this thesis have as starting point a two-dimensional electron gas (2DEG). This section describes the experimental realization of 2DEGs. The first experimentally obtained 2DEG has been formed in the inversion layer of a silicon MOSFET2 (on the left side of Fig. 1.11) [13]; successively great success has been obtained in MBE-grown remotely doped GaAs/AlGaAs single heterojunctions (on the right side of figure 1.11) and quantum wells (Fig. 1.13). For Si inversion layers a bulk p-doped Si sample is taken, an insulating oxide layer is grown on it and a metal layer is deposited on top of it. Applying a positive voltage to the metal gate, the Si bands are bent so that at the silicon-oxide interface the conduction band falls below the Fermi level, and a triangular potential well is formed (see left side of Fig. 1.11). Electrons are thus free to move in the two dimensions parallel to the surface, but are confined in a narrow well perpendicular to the surface. While this type of devices has allowed pioneering studies on two-dimensional systems, the fact that the electrons are confined at the (far from perfect) silicon-oxide interface 2

Metal-oxide-semiconductor field effect transistor.

22

Two dimensional electron gases

Figure 1.12: Sketch of the formation of a triangular potential well at the interface of two semiconductors with different band gaps.

reduces enormously the quality of the system and its potential applications. In the case of GaAs/AlGaAs single interface heterostructures, the formation of a 2DEG is achieved by appropriately doping a GaAs/AlGaAs heterostructure grown by MBE. A heterostructure consists of a layered sequence of two or more semiconductors with different band gaps, which are combined in a single crystal. In particular, GaAs and AlGaAs are ideal candidates for the fabrication of heterostructures, because they have almost the same lattice constants: aGaAs =5.653 ˚ A and aAlAs =5.661 ˚ A at 300 K [10]. Figure 1.12 shows in a schematic way the formation of a 2DEG in a GaAs/ AlGaAs heterostructure. Intrinsic GaAs has the Fermi energy EF in the middle of the gap between the valence band EV and conduction band EC . It is brought into contact with a doped layer of AlGaAs, which has a larger band gap. In equilibrium, EF has to be the same throughout the whole crystal, and the band structure aligns itself self-consistently. At the interface between the two semiconductors, the conduction band of the undoped GaAs is bent down while the conduction band of the doped AlGaAs is bent up (lower part of Fig. 1.12). This leads to the formation of a triangular potential well. A quantum well (QW) is formed by sandwiching a thin layer of lower band gap material (well) within two layers of higher band gap material (barriers) (see figure 1.13). The choice of the pair of materials for the well/ barrier structure is wide: GaAs/AlGaAs, InGaAs/InAlAs, InGaAs/InP, and InGaAs/GaAs, to cite only a few III-V combinations. Quantum wells have the advantage of being the most versatile two-dimensional system, since the well parameters are under good control and can be tailored as needed. As a first approximation, the well is a rectangular potential profile whose width is controlled with monolayer precision (it is the thickness of the low band gap material), and whose depth is given by the conduction band offset (CBO) between the well and the barrier materials. The doping in the barriers and the pinning of the Fermi energy at the surface of the semiconductor slightly

surface

uniformely n-doped In0.75Al0.25As barrier

In0.75Ga0.25As QW

In0.75Ga0.25As cap


uniformely n-doped In0.75Al0.25As barrier

23

substrate

350 300 Charge density

Ec-HF (meV)

250 200 150

Conduction Band

100 50

Fermi energy

0 -50 -100 0

50

100

150

200

distance from surface (nm)

Figure 1.13: In0.75 Ga0.25 As/In0.75 Al0.25 As quantum well. The top cartoon is a sketch of the layer sequence starting from the surface (left) down toward the substrate(right). The bottom graph is the profile of the calculated conduction band minimum along the growth direction (black curve), and the carrier density profile (red line); the horizontal red line is the Fermi level.

distort the shape of the well as can be seen in figure 1.13; in this figure a selfconsistent Poisson-Schrödinger calculation of the conduction band profile and the carrier density for one of the In0.75 Ga0.25 As samples characterized in this thesis are shown.

1.5

Device fabrication

1.5.1

Optical lithography

This section describes the procedure employed for the fabrication of semiconductor heterostructure devices containing a 2DEG. These devices allow to perform transport measurements on the 2DEG itself. In the first fabrication step, properly designed structures are patterned on the sample. The geometry of the devices used in this thesis, commonly called Hall bar geometry, is shown in Fig. 1.14. This geometry is particularly indicated to study the mobility and carrier density of 2DEGs taking advantage of the classical Hall effect. Hall bars are typically defined through common optical lithography and a wet chemical etching process. Fig. 1.15 shows the fundamental steps of device fabrication using optical lithography. First, the sample to be processed (a) is covered with a positive resist 3 layer 3

By positive resist we mean a polymer that after illumination with UV radiation be-

24

Device fabrication

Figure 1.14: Sketch of the Hall bar geometry.

(b). Then the regions of the sample that should not be exposed to the ultraviolet (UV) light are protected using a chromium mask which, in some cases, has to be aligned with structures already patterned on the sample (c). The exposed resist is removed by immersing the sample in the developer solution. Only the unexposed regions remain protected by the resist (d). At this point, the pattern is transferred from the resist layer to the sample through a wet chemical etching (e). In the last step, the resist is removed using acetone (f). After the patterning of the Hall bars via chemical etching, another lithographic step is required for the definition of the contact regions (large shaded regions in Fig. 1.14). After the removal of the resist, a metal is evaporated on the contact regions and eventually annealed for a short time at high temperature to favor the penetration of the metal into the semiconductor to the 2DEG layer. The procedure is described in detail in Appendix A.

1.5.2

Electron beam lithography

Electron beam lithography (EBL) works with the same principles of optical lithography: a thin layer of polymer (generally poly-methyl-meta-acrylate or PMMA) is put on the sample surface. Exposing the polymer to a beam of high energy (generally 30-100 keV) electrons changes its chemical properties so that the exposed areas become soluble (for positive resists such as PMMA) or insoluble (for negative resists as such SAL) in an appropriate solution. After dissolving the soluble part of the resist, the parts of the surface that is still covered with resist is protected from the successive process steps that are either subtractive (like etching) or additive (like metal deposition). The peculiarity of this technique is that the maximum attainable resolution comes soluble in a suitable solution called developer. Negative resists, on the other hand, become insoluble after exposure.


25

Figure 1.15: Basic steps of optical lithography.

is much higher than for optical lithography, since the wavelength of 30 keV electrons is much smaller than that of UV light. Moreover, electron beam technology has been developed for imaging purposes for scanning electron microscopes (SEMs), producing electron-optical columns able to raster an electron beam with a diameter of 1 nm with nanometer precision over millimeter wide areas. This gives unprecedented flexibility in pattern design and shortens the time from the pattern design to the transfer of the pattern on the sample, since the electron beam is directly writing the desired pattern on the sample. As a drawback, it being a serial writing method, EBL is unsuited to large scale serial fabrication, where the parallel production of thousands of identical patterns at a time by optical lithography is preferred. In scientific research, though, where flexible design is a must and volume production is not needed, electron beam lithography comes very handy. In this thesis, EBL has been extensively used to define small (in the range 1 µm to ∼10 nm) metallic gates on the surface of the samples. The approach has always been additive, using a high sensitivity positive resist (950K PMMA), developing it in a solution of methyl-isobutyl-ketone (MIBK) and iso-propyl-alcohol (IPA) in a 1:3 ratio, successively evaporating a metal film (usually titanium/gold, chromium/gold or aluminum) and finally removing the metal from the unexposed areas by lift-off. Various recipes for EBL, developed to define the nanostructures of this thesis, are described in more detail in Appendix A.

26

1.6

Transport measurements


The study of the transport properties of the fabricated devices was performed in both a variable temperature 4 He cryostat for measurements from room temperature down to 1.4 K and magnetic field up to 7 T and a 3 He refrigerator for temperatures down to 250 mK and magnetic fields up to 12 T. The sample holders allow to perform measurements in a magnetic field either perpendicular or parallel to the sample. The cooling of the sample in the 4 He cryostat takes place by thermal exchange with cold 4 He vapors that come from the liquid 4 He reservoir of the cryostat. In this way it is possible to cool the sample down to 4.2 K. Further cooling can be achieved by pumping the chamber in which the sample is located down to a few mbar. In this way, the 4 He coming from the reservoir undergoes an adiabatic expansion reducing its temperature to ∼1.4 K. The 3 He refrigerator (a Heliox system manufactured by Oxford Instruments) works with a three stage cooling strategy: first the whole insert is dipped into liquid helium, reaching 4.2 K temperature; then inside a vacuum chamber in the liquid helium bath a pumped 4 He circuit cools a condensing plate down to ∼1.5 K. Here the 3 He gas within its closed-loop circuit condenses, accumulating in a pot in thermal contact with the sample. Once all the 3 He is liquefied, one reduces its vapor pressure through a sorb pump, lowering its temperature down to ∼250 mK. Carrier density n and mobility µ of a 2DEG are measured in a four wire setup, which is schematically described in Fig. 1.14. A current I is driven through the main channel of the Hall bar. At zero magnetic field (B=0), the potential drop Vxx induced by the current between two lateral contacts is measured. From Vxx it is possible to obtain the longitudinal resistivity, ρxx , of the 2DEG as Vxx w ρxx = · , (1.13) I L where w and L are the width and length of the Hall bar, respectively. The dimensions of the typical Hall bars used in this thesis are w=60 µm and L=260 µm. If a magnetic field B perpendicular to the plane of the 2DEG ~ induces a potential drop Vxy between is applied, the Lorentz force, I~ × B, two transverse contacts (classical Hall effect). From the Drude model [14] it is than possible to obtain the following relations: 1 , enµ

(1.14)

B·I , e · Vxy

(1.15)

ρxx =

n=

where n is the carrier density of the 2DEG. Combining Eqs. (1.14) and (1.15)


27

with Eq. (1.13) results in the mobility µ of the 2DEG, µ=

Vxy L · . Vxx B · w

(1.16)

The characterization of 2DEG mobility and carrier density is typically done at T=1.4 K with a magnetic field B of 0.3 T. These measurements are typically performed using a conventional four wire lock-in technique with an AC excitation current of 100 nA at a frequency of about 20 Hz. Many of the low temperature measurements of this thesis have instead been performed in a DC setup. For such measurements care must be taken in the measurement system to reduce the noise to a minimum. In particular, all signal wires pass through π-filters as they exit the cryostat and currents and voltages to be measured are readily amplified by low-noise electronics. Both the amplifying electronics and the current and voltage sources are powered by batteries and are driven by a computer program through an optical fiber link to avoid any kind of external noise. The amplified signals are then read by digital multimeters and acquired by software. A in-house designed additional filtering stage, composed of RC low-pass filters with a cutoff frequency of ∼5 kHz, can be optionally added to the 300 mK stage of the cryostat to avoid electron gas heating by high frequency external noise.

28


Chapter 2

InAlAs step-graded buffers As already pointed out in sections 1.1.3 and 1.2, the growth of good quality In0.75 Ga0.25 As quantum wells is made very difficult by the absence of suitable substrates. One of our goals was to grow high electron mobility 2DEGs in virtually unstrained In0.75 Ga0.25 As QWs. To reach this result we had to develop a virtual substrate, lattice-matched with In0.75 Ga0.25 As, bearing no crystal defects, and grown on a commercially available material. This chapter describes the properties of the Inx Al1−x As graded buffers that we have optimized to accommodate our In0.75 Ga0.25 As QWs on GaAs substrates. Section 2.1 reviews briefly the main problems encountered when trying to perform lattice-mismatched growths, particularly focusing on III-V semiconductor systems. In Sec. 2.2 we describe the structural properties of our Inx Al1−x As buffers analyzed by cross-sectional Transmission Electron Microscopy (TEM), X-Ray Diffraction (XRD) and Atomic Force Microscopy (AFM), while Sec. 2.3 shows how different buffer designs influence the low temperature transport characteristics of the 2DEG. Part of the results presented in Sec. 2.2 and 2.3 have been published in Ref. [15].

2.1

Lattice-mismatched growth

Apart from a few lucky exceptions, crystals of different semiconductors have different lattice parameters or even different crystal structures. This makes it very difficult to grow heterostructures without the formation of crystal defects. The most famous exception is that of GaAs, AlAs and their alloys. In fact not only both GaAs and AlAs crystals have a zincblende structure, but their lattice parameters differ so little that virtually any thickness of any alloy of these binaries can be grown without having to worry about strain buildup. Moreover, the band gaps of GaAs and AlAs are very different, so that conduction band engineering can be easily done in AlGaAs/GaAs systems. GaAs/AlGaAs multilayers have been used to fabricate (by a variety

30


Figure 2.1: Band gaps and lattice parameters of most binary semiconductors. GaAs, AlAs, InAs and their ternary alloys are brightly colored (re-elaborated from [16]).

of growth techniques) extremely high mobility 2DEGs in modulation doped single heterointerfaces [12], high mobility 2-Dimensional Hole Gases [17], 2DEGs in almost arbitrarily shaped quantum wells, coupled 2DEGs in multiple quantum wells, very high quality superlattices, and so on. As can be seen in Fig. 2.1, the situation for InAs (and thus Inx Ga1−x As alloys) is not as good: the lattice mismatch between InAs and the most common III-V commercial substrate, GaAs, is huge (almost 7%), and with InP (other commercially available substrate) it is more than 3%. The only possible lattice-matched growth of an Inx Ga1−x As alloy is with x = 0.53 on InP. This was not a suitable choice in our case for two reasons: first, our target was x & 0.75, so strain buildup would have been a problem anyhow; second, phosphorus is a contaminant for high mobility 2DEGs in GaAs/ AlGaAs, so InP can not be used as substrate in our MBE chamber. Thus the only choice has been using GaAs as a substrate, and find a way to relax the strain to grow In0.75 Ga0.25 As layer with a low defect density in the active region of the structure.

2.1.1

Heteroepitaxy

Whenever growing an epilayer with a different lattice parameter than the substrate, strain will build up in the epilayer. This strain can be either compressive, when the epilayer has a larger lattice parameter than the substrate (as is he case for Inx Ga1−x As grown on

InAlAs step-graded buffers

31

Figure 2.2: Growth of a mismatched layer (red) on a substrate (blue) beyond the critical thickness Tc . In (a) a representation of the “bulk” fully relaxed materials; (b)-(e) are the various phases of growth. Below the cartoons are shown the graphs of elastic energy per unit area versus epilayer thickness. The horizontal green line represents the critical energy for dislocation formation.

GaAs), or tensile, in the opposite case. In both cases the first atomic layers of the growing film tend to have the in-plane lattice parameter matched to that of the substrate, and to expand or compress the out of plane lattice parameter to accommodate the misfit (the so called pseudomorphic growth). This, however, has an energy cost payed in elastic energy accumulation within the crystal. When this elastic energy exceeds the energy cost of a crystal defect, the defect becomes energetically favorable and tends to form throughout the interface between the epilayer and the substrate. This process is depicted in Fig. 2.2. The thickness at which defects start forming is called critical thickness. There exist several equilibrium models to predict the critical thickness of a strained layer, but MBE growth is far from equilibrium, and critical thickness calculations have to rely on growth specific parameters (see Sec. 2.2). The growth of an epilayer beyond the critical thickness, with the formation of crystal defects that lead to the relaxation of the epilayer, is referred to as metamorphic. The most common crystal defect forming to release the built up strain is a misfit dislocation (MD), depicted in Fig. 2.3(a): a whole row of crystal sites, lying parallel to the growth plane, is missing. This kind of defect can move (glide) sideways in a direction perpendicular to both its own and the growth direction: such a direction is parallel to the surface, and thus the dislocation can not go deeper or move towards the surface; it is confined at the depth at which it is formed. Another defect commonly associated to strain relaxation is the so-called threading dislocation (TD), which is a MD which forms on a plane that is not the growth plane [19]. Figure 2.3(b)

32


Figure 2.3: Dislocations. (a) A misfit dislocation in a cubic lattice: the dislocation is highlighted in red and runs perpendicular to the page. Modified from [18]. (b) A threading dislocation: the TD (drawn in red) originates from a MD (green line) lying in the (001) plane and runs all the way to the free surface. Modified from [19].

schematically represents a TD running through a strained layer grown in the [001] direction; in this example the TD, running along the [0¯11] direction, originates at the interface between the substrate and the strained layer at the end of a normal MD aligned along [1¯10]. This kind of dislocations is to avoid, since they propagate all the way to the surface, and are detrimental for most semiconductor applications: they are sources of scattering for electrons and holes, non radiative recombination centers for optics, etc. This is true also for MDs, but the fact that they form at the interface between the substrate and the grown layer allows to keep them at a safe distance from the active layer, thus reducing their potential harm. As can be seen in Fig. 2.4, for a given thickness of the strained film the formation of dislocations is a function of the lattice mismatch; in practice the slope of the line of the graphs of Fig. 2.2 gets steeper as the lattice

Figure 2.4: A qualitative plot of lattice energy stored at a heterointerface as a function of lattice mismatch for a given epilayer thickness [20]; ε0 is the critical mismatch: below this value of mismatch the epilayer will have no dislocations, above it will.


33

Figure 2.5: A representative AFM topography of one of the In0.75 Ga0.25 As/ In0.75 Al0.25 As samples (HM617), showing the characteristic cross-hatch pattern of surface roughness. The crystallographic directions that form the perpendicular network of corrugations are indicated in the image. The image size is 15×15 µm2 , while the vertical scale is 10 nm.

mismatch increases, thus reducing the critical thickness. This means that in order to grow dislocation-free lattice mismatched layers there are only two strategies: • for small lattice mismatch: to grow pseudomorphic strained layers below the critical thickness (solution adopted for the devices of Chapter 4, with low indium content Inx Ga1−x As); • for large lattice mismatch: to grow first a buffer layer where strain relaxation through MD formation takes place, i.e. to obtain a virtual substrate lattice-matched to Inx Ga1−x As. The design and optimization of this buffer layer is the main matter of this chapter. The surface of a metamorphic layer always shows a big roughness. This roughness typically is organized in a cross-hatch pattern (see Fig. 2.5) and has root mean square (RMS) values from few nm to hundreds of nm, depending on the materials involved in the growth, the growth techniques and growth conditions. Such cross-hatch patterns have been observed on Si1−x Gex /Si [21], on Inx Ga1−x As/GaAs [22, 23], and on Inx Ga1−x As/InP [24]. Even though there have been many attempts to explain the formation of this roughness, there is no general agreement on the exact dynamics. The two mainly proposed mechanisms are: 1. dislocation formation and glide on {111} planes and lateral mass transport to eliminate the surface steps [25]; 2. locally enhanced or suppressed growth rate due to the nonconstant strain field generated by dislocation bunching [22].

34

Structural properties

Our results reported in chapter 3 seem to confirm the second mechanism, but no detailed analysis of this problem has been carried out in this thesis.

2.1.2

Previous results for the growth of InGaAs QWs

Several research studies have been carried out to understand the mechanisms of strain relaxation in Inx Ga1−x As or Inx Al1−x As layers in cases where a lattice mismatch to the substrate exists [26, 27]. It was demonstrated that a defect-free region with an arbitrary indium concentration can be obtained both on GaAs and InP substrates by inserting a step- or linear-graded buffer layer (BL) with increasing In composition in order to smoothly adapt the lattice constant of the substrate to the one of the topmost layer. This approach enables to relax the strain and to bury the resulting dislocations away from the topmost defect-free epitaxial layers [28]. The key factors to reach this goal appear to be the low temperature growth of the BL [29], and the insertion of a AlGaAs/GaAs superlattice between the GaAs substrate and the BL [30]. More recently, BLs were used to grow, on GaAs substrates, almost unstrained Inx Ga1−x As QWs, (with x ≥ 0.7) containing a 2DEG with an electron mobility higher than 20 m2 /(Vs) [31, 32, 33]. In these cases the BL indium concentration was at the most equal to that of the QW. On the other hand, increasing the indium concentration of the buffer up to values greater than the target concentration of the active layers has already proven to be beneficial in InAlAs/InGaAs metamorphic QWs, but with much smaller In content (x = 0.33) [34], and in the growth of unstrained Zn0.75 Cd0.25 Se layers on In0.33 Ga0.67 As/GaAs buffers [35].

2.2


To obtain defect free active regions of Inx Ga1−x As, we have grown stepgraded Inx Al1−x As BLs with increasing indium concentration x on GaAs (001) wafers. The growth procedure, schematically represented in figure 2.6, can be divided in three phases characterized by different growth temperatures: • Substrate preparation. The growth starts with an epiready GaAs (001) wafer degassed at 580◦ C to remove the surface oxide. Then a 200 nm thick undoped GaAs layer followed by a 20 period GaAs/ Al0.33 Ga0.67 As (8 nm/2 nm) superlattice and a second 200 nm undoped GaAs layer are grown at 600◦ C. These three layers have several purposes: to reduce the wafer roughness, the diffusion of interface impurities in the grown layer, and to reduce the risk of threading dislocation formation during the subsequent phase, as pointed out in


35

Figure 2.6: The growth sequence of the Inx Ga1−x As/Inx Al1−x As samples. In the middle, a schematic view of the three parts of the growth. Each part is expanded on the sides. On the left is the buffer layer composition: the vertical axis represents the distance from the top of the interfacial layer, and the horizontal axis is the indium concentration x.

Ref. [30]. They are grown at this temperature since it is the optimal growth temperature to obtain high quality GaAs.

• Buffer growth. The substrate temperature is decreased to 330◦ C and the Inx Al1−x As buffer is grown in 50 nm steps with increasing indium concentration, starting from In0.15 Al0.85 As. The increase in x between each step is not constant throughout the buffer, but is ∼0.035 up to ∼60% and then it is decreased to ∼0.025 up to the end of the buffer. The In concentration is varied by increasing the In cell temperature while maintaining the Ga flux constant; there is no growth interruption between the steps, so that the interface is not abrupt, since the increase of the In cell temperature takes several seconds.

• Active layer. The substrate temperature is increased to 500◦ C. Typically this layer consists of a lower 50-100 nm thick In0.75 Al0.25 As barrier, a 10-30 nm thick In0.75 Ga0.25 As quantum well, an upper 130 nm thick In0.75 Al0.25 As barrier covered by a 5-10 nm thick In0.75 Ga0.25 As cap layer. This capping layer is necessary to avoid extensive oxidation of the In0.75 Al0.25 As upper barrier. The details of the active layer are slightly different from sample to sample and will be described as needed.

36


Figure 2.7: A TEM cross-section of the buffer of sample HM617.

2.2.1

The buffer

The BL has been grown at the unusually low temperature of 330◦ C because, as already pointed out in Sec. 2.1.2, low growth temperatures decrease the defect density. This temperature has been optimized on the basis of transport measurements described in the next section. Indeed in all our samples no surface defects, such as threading dislocations (as in Ref. [23]), are visible by optical or AFM microscopy. The layer sequence has been modified from the work of Gozu and coworkers [31]. Their graded buffer is substantially the same as ours up to x = 0.75. At this indium concentration they end the buffer and start the growth of the active layer. However, according to a previous work [35] and to our own experimental findings together with the model described ahead (see section 2.2.2), this is insufficient to completely release the strain accumulated in the buffer. The result is a residual compressive strain to which the active layer is subject with consequent mobility reduction (see Sec. 2.3). Our buffer, instead, includes also some overshoot: the indium fraction x is increased up to values exceeding 0.75, with the aim of completely relaxing the active layer. Viewing a cross-section, presented in Fig. 2.7, of one of the samples (HM617) measured with a TEM, it is evident that the strain is released through misfit dislocations in the lower part of the BL, while the upper part is defect free (within the limits of TEM statistics). The image is taken in the


37

Figure 2.8: A TEM plan-view of the end of the MD network taken using the (004) reflection on HM617.

[110] direction, and shows a complicated network of dislocations extending from the beginning of the BL up to roughly 450 nm from the surface. Similar images taken in the [1¯10] direction show that the last dislocations form further away to the surface, at a distance of ∼550 nm. Figure 2.8 is a plan view image of the sample taken using the (004) reflection. Here the layer of dislocations closer to the surface is visible. This specimen has been thinned down to electron transparency by a Xe+ ion beam forming an angle of 4◦ with respect to the (001) plane. The result is a specimen of varying thickness with the thinnest parts closer to the surface of the sample while the thickest parts arrive in the BL. This is schematically represented in the drawing on the left. In the lower part of the image a network of MD is clearly visible, while in the upper part there are no visible dislocations. The contrast in the upper part of the image might be due to some strain or indium concentration modulation.

2.2.2

Strain relief model

To analyze the strain relief in the buffer and compare it to the experimental TEM and XRD results (presented in the next section), we have employed an isotropic model proposed by Romanato et al. [28]. This model assumes that, in a mismatched growth with given misfit profile f (t)1 , the density of misfit dislocations n(t) that relax the strain and 1

Assuming the validity of Vergard’s law (confirmed for Inx Al1−x As by Ref. [10]), the misfit is linearly related to the In concentration. In fact if aA and aI are the lattice parameters of AlAs and InAs, and a(x(t)) is the unstrained lattice parameter of Inx Al1−x As with a In concentration x(t) at the height t: a(x(t)) = x(t)aI + [(1 − x(t)]aA . Then f (t) = (a(x(t)) − a(x(t0 )))/a(x(t0 )), that is linear with x for a given t0 .

38


the residual perpendicular strain ε⊥ (t) are given by the following equations: n(t) = bef1 f dfdt(t) ε⊥ (t) = 0

)

n(t) = 0 2C12 ε⊥ (t) = C11 [f (t) − f (t0 )]

for 0 < t < t0 (2.1) for t0 < t < T

In Eq. (2.1) t is the distance from the substrate/buffer interface, T is the total growth thickness, t0 is the thickness below which the layer is strain free and the misfit dislocations are confined, bef f is the misfit component of the Burgers vector, and C11 and C12 are the elastic stiffness constants of the material. The main assumption of this model is that the only mechanism to relax the strain is the formation of misfit dislocations; to be valid, any other mechanism has to be negligible. This is indeed reasonable since: (a) border effects play no role: the thickness of the film (∼ 1 µm) is much smaller than the area of the samples (∼ 1 cm); (b) there is no significant tilt of the epilayer with respect to the substrate, as noted during the XRD measurements; (c) no other kind of dislocations, apart from MD, appears to form in the samples. The second assumption is that the grown buffer, underneath the last dislocation layer at a thickness t0 , is completely relaxed, i.e. the misfit dislocations have a perfect efficiency in relaxing the built up strain. I want to point out that if this model is accurate, the only way to change the in-plane lattice parameter on top of the buffer, and match it to that of the active layer, is to form additional MDs deep in the buffer, raising the value of t0 . Increasing the strain of the upper part of the buffer, without forming additional dislocations, has simply no effect on the in-plane lattice parameter. One thing to note in Eq. (2.1) is that the density of misfit dislocations n(t) is proportional to the misfit gradient df dt ; with this in mind, the buffers have been designed to have a higher misfit gradient in the first part, near the substrate, and a lower gradient in the last part, near the surface (see Fig. 2.9), to insure a lower density of dislocations in the part of the buffer that is closer to the active layer. Applying this model, the residual strain in the active layer is given as the difference between the misfit in the active layer and the misfit at t0 , the height of the last strain-free layer. We calculate this distance using the following equilibrium energetic condition between the excess of elastic energy in the defect-free region and the critical energy for MD generation: 2 (2.2) Y (T − t0 ) f¯(t0 , T ) − f (t0 ) = Y K = Ecr RT

f (t)dt 0 where f¯(t0 , T ) = t(T −t0 ) denotes the average misfit between t0 and T , and Y is the Young’s modulus. Ecr is the critical elastic energy that can not be


39

Figure 2.9: Concentration profile versus thickness for the buffer and the active layer. The average misfit gradient is calculated from the concentration for the first part of the buffer (red) and for the second one (blue).

exceeded without the nucleation of new misfit dislocations, and K = (3.7 ± 0.7) × 10−3 nm is a phenomenological constant determined for Inx Ga1−x As layers grown on GaAs in the concentration range 0.035 to 0.15 by Drigo et al. [36].

2.2.3

XRD measurements

To quantify the residual strain on the active layer we have performed High Resolution XRD measurements in a series of samples with different buffer designs. The four samples measured have the same structure, except for the final indium concentration xf at which the Inx Al1−x As buffer is stopped. The samples (see table 2.1) have nominal xf of 0.75 (sample A), 0.80 (sample B), 0.85 (sample C), and 0.90 (sample D). The first sample has thus the same structure of previous BLs (like those of Gozu and coworkers [31]), while in the others an overshoot is present. This is intended, as previously mentioned, to increase the stress in the buffer so to induce the formation of additional MDs to increase the in-plane lattice parameter at the top of the buffer. The key point is to induce the MD formation deep in the buffer, as far away as possible from the active layer itself. At the end of growth of this overshooting buffer, the top part of the BL is still strained, but has the same in-plane lattice parameter of the x = 0.75 active layer. In this way one has at his disposal a virtual substrate that is lattice-matched with In0.75 Ga0.25 As, and thus the possibility of growing an active layer of arbitrary thickness, without any constraints due to strain buildup. The four samples were intended to accurately tune this overshoot and compare the results with the quantitative model described in Sec. 2.2.2.

40


Figure 2.10: (004) rocking curves for the four samples described in table 2.1. The curves are shifted for clarity. The gray and yellow dotted lines are a guide to the eye to follow the peaks due to the Inx Al1−x As with x ∼ 0.60 and the In0.75 Al0.25 As active layer, respectively.

Sample number

HM1065 HM1071 HM1064 HM1072

name

xnom f (%)

xmeas f (%)

xmeas (%)

∆x (%)

⊥ (×10−4 )

OD M ⊥ (×10−4 )

T − t0 (nm)

A B C D

75 80 85 90

72±1 76±1 80±1 86±1

72±1 72±1 72±1 71±1

0 4±1 8±1 15±1

37±6 19±6 3±5 -14±9

40 26 13 -8

350 450 480 510

Table 2.1: Results of XRD on samples with different overshoot. The columns show the sample number, the short name of the sample, the nominal (nom) and measured (meas) value of xf (the final In concentration of the buffer), the measured concentration of the active layer xmeas (which nominally was always 0.75), the difference ∆x = xmeas − xmeas , and the out of plane residual strain of the f active layer. The last two columns are the values of the residual strain in the active layer and the depth of the last dislocation layer estimated with the model described in Sec. 2.2.2.


41

The four samples were analyzed performing rocking curve measurements for both the symmetric (004) and the asymmetric (224) reflections. As pointed out in Sec. 1.2.3, from these measurements both the residual strain and the indium concentration can be deduced. In the case of the BLs the analysis is complicated by the fact that broad structures and multiple peaks are present in the rocking curves. Careful comparison of the spectra with the nominal growth structure, however, allows to extract the desired parameters. The (004) rocking curves of the four samples are shown in figure 2.10. The buffer gives rise to a broad background with several peaks. The peak around 31.5◦ is due to the layers with indium concentration close to 0.60 (see Fig. 2.9), and has a constant distance to the substrate peak. This means that the degree of relaxation of that layer is constant in the four samples. The smaller peak farthest from the GaAs substrate is the contribution of the last layer of the buffer, with xf indium concentration, that is obviously changing position from sample to sample. Between these two, there is another peak which is decreasing its distance from GaAs peak with increasing xf . This peak is due to the active layer, composed mainly of In0.75 Al0.25 As with smaller layers of In0.75 Ga0.25 As. The residual strain of the active layers shown in table 2.1 have been determined by fitting these peaks with gaussian curves on both (004) and (224) scans (not shown). The results of the fits are shown in Fig. 2.11, compared to the values calculated with the model described in the previous section, and in Table 2.1. The data show that it is possible to fine tune the residual strain on the active layer through the introduction of an overshoot in the indium concentration of the buffer. This allows to change the strain from compressive to slightly tensile passing through virtually unstrained samples. Comparing the data with the model a good agreement is found, even though the constant K used in the calculation was derived from a very different indium concentration range. This permits to use the model to estimate other characteristics of the samples. In fact, from the calculation, one can extract also the depth T − t0 at which the last dislocations are visible, which is reported in the last column of Table 2.1. The calculated T − t0 of sample C can be compared to the TEM observations on sample HM617 shown before (since the two samples are nominally identical), and are found in good agreement. In fact the TEM observed distance of the last dislocation layer to the surface is 450±50nm for the [110] images and 550±50 for the [1¯10] images2 , to be compared with the figure of 480 nm resulting from the calculation. 2

The anisotropy of MD formation along the two h110i directions is a well known property of III-V materials [37, 38, 39] and is not taken into account by the described model, which deals with averages of both types of MD, and considers t0 the average position (in the two crystallographic directions) of the last MD layer.

42


Figure 2.11: Measured residual strains on the active layer compared with the calculated ones.

2.2.4

Surface and interfaces

As previously mentioned, the surface of the samples shows a distinct crosshatch pattern (CHP) of roughness which is observable with an AFM. Does this surface roughness give us some information on the interface roughness between QW and barriers in the active layer? The answer, as we will see, is yes. We have grown several samples interrupting the growth at different levels of the active layer: all the samples are identical up to the end of the buffer layer, which is the same as previously described for the samples measured by TEM and XRD. Above the last step of the buffer layer, with nominal 85% In content, we have grown In0.75 Al0.25 As layers of different thicknesses, capped by a thin (5 nm) layer of In0.75 Ga0.25 As. The thickness of the In0.75 Al0.25 As layer for each of the samples is reported in Table 2.2. In what follows I will refer to this top In0.75 Al0.25 As layer as “top layer” or TL. AFM topographies have been recorded with image sizes of 5, 15, and 30 µm. The acquired data have been flattened with a consistent protocol to remove long wavelength modulations that are artifacts due to AFM scanner nonlinearities. Figure 2.12 reports 30 × 30 µm2 topographies of the samples. Panels (a) through (g) refer to the samples whose TL thickness is indicated in table 2.2. Figure 2.13 is a plot of the RMS roughness of the samples versus TL thickness. The TL of sample HM1292 (panel (f)) consists of the full quantum well structure employed for low temperature transport measurements: 50 nm thick In0.75 Al0.25 As lower barrier, 30 nm thick In0.75 Ga0.25 As QW and 120 nm thick In0.75 Al0.25 As upper barrier ended by the 5 nm thick In0.75 Ga0.25 As capping. The graph shows also the roughness of other QW


43

AFM image (a) (b) (c) (d) (e) (f) (g) Sample HM1202 HM1203 HM1205 HM1207 HM1209 HM1292 HM1434 TL thickness (nm) 0 10 20 50 100 200a 400 RMS rough. (nm) 1.60 1.45 1.35 1.38 1.39 1.83 2.41 a

This sample consists of the full QW structure; see text for details.

Table 2.2: Sample numbers, top layer (TL) thickness and RMS roughness of the AFM topographies shown in Fig. 2.12

Figure 2.12: 30×30 µm2 AFM topographies of the samples described in table 2.2. The blue arrow indicates the [1¯ 10] direction in each image.

Figure 2.13: RMS roughnesses versus top layer thickness for the samples listed in table 2.2.

44


Figure 2.14: Electron mobility µ (black symbols) and charge density n (red symbols) versus buffer growth temperature.

samples (HM1327, HM1200, and HM889) which, as HM1292, have 200 nm total thickness above the buffer. From this data it is evident that the roughness is nearly independent from the thickness grown above the buffer layer, and that the roughness measured at the surface of a QW sample is approximately the same as that of the interfaces between the QW and the barriers. This information will be of particular value in the discussion of chapter 3.

2.3


Our interest is, however, focused on the low temperature transport properties of the 2DEG formed in the In0.75 Ga0.25 As QWs. Even though no intentional doping is introduced in the growth, the QWs are intrinsically n-type. This has already been shown by our group to be due to deep donor impurities in the In0.75 Al0.25 As layers [40]. The energy of these deep donor impurities is 120-170 meV below the conduction band edge of In0.75 Al0.25 As, and thus is within the conduction band discontinuity of the In0.75 Al0.25 As/ In0.75 Ga0.25 As junction, making it easy to populate the In0.75 Ga0.25 As QW through charge transfer. In this way low carrier density (∼ 2÷4×1015 m−2 ) 2DEGs are formed. Figure 2.14 shows how the buffer growth temperature influences the electron mobility of the 2DEG. The three samples (HM1184, HM1180 and HM1182) are identical except for the buffer growth temperature (310, 330


45

Figure 2.15: Electron mobility µ versus residual strain of the active layer. Positive ε⊥ means in-plane compressive strain.

and 350◦ C, respectively) varied in a small range around the already optimized temperatures reported by other authors [31]. The mobility has a maximum value at 330◦ C enhanced by the lower charge density. This has led us to grow the BL at 330◦ C throughout this work. As for the mobility dependence on the residual strain of the active layer, low temperature electron mobility measurements have been performed for all four samples described in Sec. 2.2.3. The base temperature for the measurements was 1.5 K, and they were probed using a 20 Hz AC excitation current of 2 µA employing the Van der Pauw configuration. The mobility (µ) of the four samples is reported in Fig. 2.15 as a function of the residual strain in the active layer region, as derived from XRD measurements. For the samples A, B and C we find an electron density n ∼ 4 × 1015 m−2 while for the sample D a lower density, n ∼ 3 × 1015 m−2 , is observed. For sample A, with high compressive strain in the QW, we measured µ = (9.9 ± 0.9) m2 /(Vs) while an increase of µ up to (28.1 ± 0.3) m2 /(Vs) and (29.0 ± 0.2) m2 /(Vs) is observed for samples B and C, respectively. For sample D, under moderate tensile strain, µ drops to (1.6 ± 0.2) m2 /(Vs). The dependence of the electron mobility on the residual strain shown in Fig. 2.15 suggests that, in the compressive regime, strain-related scattering mechanisms start to contribute significantly at values of ε⊥ higher than about 20×10−4 .

46


Figure 2.16: 30 × 30 µm2 AFM topography (top left) and a 5 × 5 µm2 magnification (top right) of sample D of table 2.1 (HM1072), under tensile strain, and a cross section of some of the grooves (bottom).

On the other hand, the transport properties deteriorate strongly as soon as the strain become tensile, likely due to the formation of extended defects, revealed by the presence of deep grooves on the surface observed by AFM. These defects are likely to be responsible also of the lower charge density of sample D, acting as trapping centers for the carriers. Figure 2.16 shows a 30 × 30 µm2 AFM topography with a close-up topography of some of the grooves, and a cross section of the grooves. The measured depth of the grooves can get as high as 50 nm, and is likely to be underestimated by the AFM due to the fact that the tip can not reach the bottom of narrow pits. The strain-induced scattering mechanisms influencing the mobility in the compressive regime could be qualitatively associated to the roughness induced by piezoelectric scattering at the interface between well and barrier. It was in fact demonstrated by Quang et al. [41] that the combination of strain and local fluctuations of interface roughness induces a piezoelectric field that reduces the electron mobility for strained systems, with respect to equivalent unstrained layers. A quantitative application of the model developed in Ref. [41] to a QW with the same thickness and interface roughness as ours (as inferred from AFM images of the top surface) would however yield piezoelectric scattering limited mobilities much higher than our experimental values, suggesting that in our case this mechanism is weaker than other scattering mechanisms. A thorough analysis of low temperature scattering


47

mechanisms is discussed in the next chapter. Another possible explanation for the observed reduced mobility in sample A is that, contrary to lattice-matched QWs, a high level of strain can induce clustering inside the QW, creating regions with In fluctuations, which can act as additional scattering centers [42]. A further factor that could explain the reduced mobility of sample A is the closer distance between the QW region and the last region with misfit dislocations in the BL [24]. This value can be estimated from the model as the difference between the total dislocation-free region (T − t0 in Table 2.1) and the distance of the QW to the surface. The calculated values are about 190 nm for sample A and about 300 nm for samples B and C. In fact, it was demonstrated by Woodall et al. [43] that dislocations can act as trap states and thus can create a random electric field inside the material. The closer distance to the well of this random electric field in sample A, with respect to samples B and C, could be a possible cause of the observed mobility reduction.

48


Chapter 3

Scattering mechanisms in In0.75Ga0.25As quantum wells High electron mobility in semiconductors is a goal pursued not only to be able to build more efficient transistors for the electronics industry, but also to study fundamental aspects of condensed matter physics. Indeed, to increase the electron mobility means to have available a “more ideal” system of freelike electrons to investigate their fundamental properties. However, to reach a higher electron mobility it is necessary, as a first step, to understand the mechanisms that are limiting it. For 2DEGs in GaAs/AlGaAs heterostructures, studies on the mobility limiting mechanisms, and successive refinements in the growth process, have given outstanding results in the last 25 years. The main limiting factor of the low temperature mobility has been identified to be the scattering due to ionized impurities; this has led to grow GaAs/AlGaAs heterostructures in extremely clean MBE systems to avoid the presence of unwanted impurities, and to confine the donors needed to supply charge to the 2DEG far away from the 2DEG itself. This, along with atomically flat interfaces given by the optimized MBE growth, has led to increases in mobility by many orders of magnitude, up to the 104 m2 /Vs range. We have used the same approach with our high indium content QW structures: first, to identify the main scattering mechanisms, then to reduce them. In the first section of this chapter, I will review a theoretical model that describes the main low temperature scattering mechanisms in semiconductor heterostructures, with special attention to Inx Ga1−x As alloys. In section 3.2 this theory will be applied to our samples allowing to deduce, mainly from low temperature transport measurements, the strongest sources of scattering. The knowledge acquired by this analysis has allowed us to improve the sample design to reach higher electron mobilities. The improved design samples are discussed in section 3.3, in which I explain how alloy disorder scattering can be suppressed with the insertion of thin, strained InAs

50

Low temperature scattering mechanisms

layers in the QW. In Sec. 3.4 new issues are addressed such as the electron mobility anisotropy and the model developed to explain it. This chapter ends with some final remarks on the opportunities opened by high mobility 2DEGs in InGaAs quantum wells. The results presented in section 3.2 have been published in Ref. [44], while those of section 3.4 have been reported at the ICPS2006 conference and published in Ref. [45].

3.1


Low temperature scattering mechanisms in two-dimensional electron systems have been discussed in a review by T. Ando and coworkers [46]. Later work by A. Gold [47, 48] added contributions peculiar to alloys, better suited to describe Inx Ga1−x As quantum wells. According to Gold, at low temperatures, there are three main sources of scattering in a 2DEG system with a single subband: 1. Ionized impurity scattering (II), due to Coulomb interaction between the conduction electrons and an ionized impurity background uniformly distributed in the material. 2. Interface roughness scattering (IR), due to the non-planarity of the interfaces defining the quantum well, which act as fluctuations in the width of the quantum well which confines the electrons. 3. Alloy disorder scattering (AD), due to the random distribution of the indium and gallium atoms inside the crystal matrix. This scattering potential is assumed to arise from the difference in electron affinity, band gap, and electronegativities of the two constituent binaries, InAs and GaAs in our case. These three scattering sources are modeled, within the relaxation time approximation, assuming that the scattering processes are elastic and that the scattering centers are randomly distributed. Additionally, the quantum well is modeled as square well with infinite barriers, and the resulting ground state 1-D wavefunction is a half-period sine wave. Figure 3.1 shows a comparison between the results of a self-consistent one-dimensional Schrödinger-Poisson calculation of the conduction band edge and the electron squared wavefunction along the growth direction for a 30 nm thick In0.75 Ga0.25 As/In0.75 Al0.25 As quantum well (red dashed line), compared to Gold’s analytical wavefunction for a well with infinite barriers (green solid line). The two wavefunctions, normalized, are very similar; moreover, the analytical form of the wavefunction is only weakly influencing the results of the model. This allows us to use this approximation with good accuracy.

Scattering mechanisms in In0.75 Ga0.25 As quantum wells

51

Figure 3.1: Analytical ground state wavefunction for a square well with infinite barriers (green solid line) compared to actual wavefunction along the growth direction for a 30 nm thick In0.75 Ga0.25 As/In0.75 Al0.25 As QW. The wavefunction is calculated with the aid of a self-consistent 1-D Schr¨ odinger-Poissonsolver [49, 50]. The blue dotted line is the conduction band edge calculated with the same solver.

The relaxation time τα for a single source of scattering can be computed as [47] 2kF Z h|U (q)|2 iα 1 1 q2 q u dq, (3.1) τα 2πεF 2 (q) 2 − q2 4k F 0 where kF and εF are the Fermi wavevector and the Fermi energy of the electrons in the 2DEG, respectively, and (q) is the dielectric function of the electron gas, whose analytical expression can be found in Ref [47]. h|U (q)|2 iα is the random scattering potential peculiar to each scattering mechanism, with α being ‘II’ for the ionized impurity scattering, ‘IR’ for the interface scattering, and ‘AD’ in the case of alloy disorder. The random scattering potential caused by ionized impurities, in the case of a homogeneous distribution of the impurities in the material, has the form 2 2πe2 1 2 h|U (q)| iII = NB L FB (q). (3.2) L q Here NB is the ionized impurity density, L is the width of the QW, e is the electron charge, L is the dielectric constant of the barriers, and FB (q) is a form factor taking into account the finite extension of the electron gas in the z direction, whose analytical form depends only on the geometry of the quantum well [48]. The random potential due to the alloy scattering is described as h|U (q)|2 iAD = x(1 − x)

31 a3 (δV )2 ; 4 2L

(3.3)

52


a is the lattice parameter of the alloy and δV is the spatial average of the fluctuation of the alloy potential over the alloy unit cell. At last, the interface roughness between the barriers and the well is characterized by its amplitude ∆ in the z direction and its coherence length Λ in the xy plane. Its random scattering potential is given by [51] 2

h|U (q)| iIR

π5 =2 2 mz

∆2 Λ2 L6

2

e−

q 2 Λ2 4

,

(3.4)

where mz is the electrons effective mass in the z direction. From the relaxation time of a single scattering mechanism, τα , one can calculate what the electron mobility, µα , would be if only that mechanism were active: eτα µα = ∗ , (3.5) m where m∗ is the effective mass of the electrons. The mobilities associated with each of the scattering mechanisms can be summed using Mathiessen’s rule to give the total mobility of the 2DEG: 1 1 1 1 = + + . µ µII µAD µIR

(3.6)

The key point of this discussion is to notice that the value of the mobility can be computed from known material properties (m∗ , L, (q), L , kF , εF , F (q), x, and a), and few scattering-type specific parameters: NB for the ionized impurity scattering, δV for alloy disorder scattering, and ∆ and Λ for interface roughness scattering. This allows, in principle, to estimate the scattering parameters knowing the measured mobility of the sample. Of course, with a single mobility value little conclusion can be drawn on four parameters. However, a deeper insight can be acquired if any of the material properties can be varied in a controlled way. The easiest property to vary within the same sample is the carrier density, n, which in turn determines both the Fermi wavevector, kF , and the Fermi energy, εF , according to the semiclassical relations for a two-dimensional system [14]: √ (3.7) kF = 2πn 2 2 ~ kF εF = (3.8) 2m∗ There are two main ways to change the carrier density of a 2DEG in a quantum well system. One is the “persistent photoconductivity” effect. This consists in cooling the sample in the dark from room to cryogenic temperature, in order to freeze the charge state of the donor impurities. Illuminating the sample with short pulses of light allows some of the donors to ionize supplying additional charge to the 2DEG. The result is an increase of carrier density at each light pulse, and has been used, for example, by


53

0.30 0.25

εF-Ec (eV)

0.20 0.15 0.10 0.05 0.00 -0.05 120

125

130

135

140

145

150

155

160

165

170

distance from surface (nm)

Figure 3.2: Schr¨ odinger-Poisson self-consistent calculation of the conduction band (dotted traces) and the normalized ground state wavefunction (solid traces) of a 30 nm thick In0.75 Ga0.25 As QW with a variable surface gate voltage. The gate voltage changes the charge density of the 2DEG from 4 (black trace) to 0.4 × 1015 m−2 (red traces).

Ramvall and coworkers to change the carrier density in an InGaAs/InAlAs system [52, 53]. The second way to modify the 2DEG charge is to deposit a metal gate on the sample surface and to apply a voltage to it. The main advantage of the persistent photoconductivity approach is that it needs no additional processing of the sample; the drawbacks are: (i) the minimum charge density attainable is limited by the charge density in the dark and (ii) photoionizing the donors changes the background ionized impurity density of the sample, thus changing their contribution to the scattering. On the other hand, in order to apply a voltage to a surface metal gate one needs to fabricate the gate itself. Moreover, the gate voltage can substantially modify the band profile altering the extension of the charge distribution of the 2DEG. We have decided to choose this latter approach for two main reasons: first, because our self-consistent Schrödinger-Poisson simulations for the conduction band and electrons wavefunction show that – by varying the gate voltage – the wavefunction is not substantially altered (see Fig. 3.2). Second, because it guarantees a higher reproducibility of the results: the charge density can be increased and decreased at will, thus it is possible to repeat the electron mobility measurements in the same cooldown. This allows to check the stability of the sample and the consistency of the results.

3.1.1

Ionized impurity scattering

The ionized impurity scattering term is probably the most important. It is characterized by the parameter NB , the three-dimensional ionized impurity density. The order of magnitude for this parameter can be estimated from

54


1000 22

NB = 0.5 x 10

m

= 0.1 eV

-2

100

2 -1 -1

µAD (m V s )

100

2 -1 -1

µII (m V s )

1000

22

10

NB = 4.5 x 10

m

-2

1

10

= 0.9 eV

1

Ionized impurity

alloy disorder

0.1

0.1

0

a)

2

4 6 15 -2 n (x 10 m )

8

10

b)

0

2

4

6 15

8

10

-2

n (x 10 m )

Figure 3.3: Mobility versus carrier density curves as given by Eq. (3.1) for a 20 nm thick In0.75 Ga0.25 As QW. a) Ionized impurity scattering: the impurity density parameter, NB , is increasing from 0.5 × 1022 m−2 for the top blue trace to 4.5 × 1022 m−2 for the bottom green trace in steps of 0.5 × 1022 m−2 . b) Alloy disorder scattering: the alloy potential fluctuation δV is increased, in 0.1 eV steps, from 0.1 eV in the top blue trace to 0.9 eV in the bottom green one.

our Schrödinger-Poisson simulations. In fact, to obtain accurate results for the low temperature charge density of the 2DEG, it is necessary to fine tune the donor density in the In0.75 Al0.25 As barriers. The donor density results always to be in the low 1022 m−3 range. Figure 3.3 (a) is a plot of the mobility µII for a 2DEG formed in a 20 nm wide In0.75 Ga0.25 As/In0.75 Al0.25 As QW. The mobility is derived (through Eq. (3.5)) from the τII values obtained by a numerical evaluation of Eq. (3.1) using the potential of Eq. (3.2). The different mobility versus charge curves are calculated varying the ionized impurity density parameter, NB , from 0.5 × 1022 m−3 to 4.5 × 1022 m−3 in steps of 0.5 × 1022 m−3 . The main feature to notice is that the ionized impurity scattering is less and less effective with increasing charge density, due to the more efficient screening capability of a higher electron density 2DEG. The resulting curves are monotonically increasing with n, and show a µII ∝ n1.4 behavior for charge densities above 1 × 1015 m−2 [47].

3.1.2

Alloy disorder scattering

The alloy disorder scattering can be of importance in our samples due to the fact that the 2DEG is fully contained in an alloy. The parameter that


55

characterizes it is the alloy potential fluctuation δV . The only theoretical predictions for its value are 0.53 eV, based on the difference in electronegativities of the alloying species, and 0.83 eV, based on the difference in electron affinity [54]. However, none of this values are based on first-principles calculations of atomic potentials in the alloy, and are to be regarded as rough qualitative estimates. Previously experimentally determined values for δV have been 0.6 eV for “bulk” In0.53 Ga0.47 As [55], 0.5 eV for In0.53 Ga0.47 As/ InP QWs [47], and 0.3 eV for strained In0.75 Ga0.25 As QWs with InP barriers [52]. As for the ionized impurity scattering, we have calculated the mobility µAD limited by the alloy disorder scattering using the potential of Eq. (3.3). Figure 3.3 (b) is a plot of µAD versus charge density, calculated with δV increasing from 0.1 eV (top blue curve) to 0.9 eV (bottom green curve) in steps of 0.1 eV. Contrary to the case of ionized impurity scattering, the alloy disorder limited mobility is monotonically decreasing with increasing charge density. It has to be expected that its contribution is negligible at low electron densities while it may have an appreciable effect in high density 2DEGs.

3.1.3

Interface roughness scattering

The interface roughness scattering is characterized by two parameters: ∆, the amplitude of the roughness in the growth direction, and Λ, its characteristic length in the interface plane. Figure 3.4 reports the mobility versus charge density curves calculated by numerical integration of Eq. (3.1) using the interface roughness scattering random potential of Eq. (3.4). In Fig. 3.4 (a), the curves are calculated using a constant value for Λ (300 nm), and varying ∆ over a wide range of values. By comparison with the results for alloy disorder and ionized impurity scattering, it is clear that this scattering mechanism can give a non negligible contribution, with Λ = 300 nm, only if the roughness amplitude is comparable to the QW width. In Fig. 3.4 (b), the curves are calculated using a constant value for ∆ (3 nm), and varying Λ: in this case, the contribution of the interface roughness scattering becomes important only when the coherence length Λ approaches the Fermi wavelength of the electrons (∼ 10 nm). But what is the range of “reasonable” values for these two parameters in our system? As was pointed out in Sec. 2.2.4, the sample surface topography exhibits the same cross-hatch pattern of roughness as the interfaces. AFM measurements of the surface topography show that the root mean square (RMS) amplitude of the roughness is in the 2-3 nm range while the average period of the undulations is ∼ 300 nm in the [110] direction and ∼ 1 µm in the [1¯10] direction. In Fig. 3.4, the curves corresponding to ∆ = 3 nm and Λ = 300 nm are therefore evidenced in both graphs. In fact, those are the parameters to be used in the calculation, and they yield a mobility limit

56

Scattering in 30 nm thick In0.75 Ga0.25 As QW

1000000

1000000

∆ = 0.5 nm ∆ = 1 nm

100000

Λ = 1000 nm

100000

Λ = 500 nm

∆ = 3 nm 10000

Λ = 300 nm

10000

∆ = 10 nm

2 -1 -1

2 -1 -1

1000

µIR (m V s )

µIR (m V s )

∆ = 5 nm

100

1000

Λ = 100 nm

100

Λ = 50 nm

10

10

1

1

Λ = 10 nm

Interface roughness with Λ = 300 nm

0.1 0

a)

2

4

6 15

-2

n (x 10 m )

8

Interface roughness with ∆ = 3 nm

0.1 10

b)

0

2

4

6 15

8

10

-2

n (x 10 m )

Figure 3.4: Mobility versus carrier density curves as given by Eq. (3.1) for a 20 nm thick In0.75 Ga0.25 As QW considering only interface roughness scattering. The thick red curve in both graphs corresponds to the most reasonable values for our quantum wells while the green dotted curve is another possible set of parameters describing our samples: ∆ = 0.25 nm and Λ = 10 nm (see text). a) curves with fixed Λ and various values of ∆; b) with fixed ∆ and various values of Λ.

that is orders of magnitude higher than those set by ionized impurity and alloy disorder scattering, making interface roughness scattering a negligible contribution to the total mobility of electrons. One could object that, together with the long wavelength and high amplitude undulations of the interfaces detected by AFM, there could also be a short period roughness present, due to MBE monolayer stepped growth. Therefore, we have computed the interface roughness scattering contribution of this kind of roughness, considering an amplitude of the order of a monolayer and a period of a few nanometers (∆ = 0.25 nm and Λ = 10 nm). The result is plotted in both graphs of Fig. 3.4 as a green dotted line. Also for this kind of roughness the mobility limit is much higher than the other scattering mechanisms. In what follows we will thus not consider the interface roughness scattering mechanism and concentrate only on the contributions from ionized impurities and alloy disorder.

3.2


In this section the low temperature transport measurements on sample HM617, a 30 nm wide In0.75 Ga0.25 As QW with In0.75 Al0.25 As barriers, are


57

presented. The sample is composed of the usual step-graded buffer up to a nominal 85% indium concentration which, as shown in the previous chapter, allows to grow strain-free In0.75 Ga0.25 As and In0.75 Al0.25 As layers. On top of the buffer, a 100 nm thick layer of In0.75 Al0.25 As is grown (lower barrier) followed by a 30 nm thick In0.75 Ga0.25 As quantum well and a 120 nm In0.75 Al0.25 As upper barrier, capped by a 10 nm thick In0.75 Ga0.25 As layer. Hall mobility and charge density, measured at 1.5 K, are µ ∼19 m2 /Vs and n ∼3.0 × 1015 m−2 , respectively. The thick green line of Fig. 3.5 shows the conduction band profile for the quantum well and the charge distribution (dotted green line) in the growth direction, resulting from the SchrödingerPoisson simulation, for a charge density of 3.0 × 1015 m−2 . Details on the simulation are given in Sec. 3.2.1. This sample has no intentional doping in any of the layers. The charge, as previously mentioned, is supplied to the 2DEG by deep donors in the In0.75 Al0.25 As barriers [40]. Hall bars have been fabricated on this sample as described in Appendix A. A ∼ 100 nm thick SiO2 insulating layer and a top, 100 nm thick, aluminum gate has been deposited on the Hall bar. Applying a voltage to this top gate, the band profile is modified allowing to vary the charge density of the 2DEG from 0.9 to 4.5 × 1015 m−2 . Hall measurements on this sample have been performed at a temperature of 1.5K using a four probes setup, as described in Sec. 1.6. The root mean square AC excitation current used was 100 nA, at a frequency of ∼ 17 Hz. For the charge density measurements, a magnetic field of ±0.3 T has been applied perpendicular to the 2DEG.

3.2.1

Schr¨ odinger-Poisson simulations

Schrödinger-Poisson simulations have been used intensively to estimate carrier densities, distribution of charge in the structures, and number of populated subbands for our quantum wells. Here I will briefly describe the main features of our simulations. The simulations have all been performed with the aid of a SchrödingerPoisson solver released as freeware by Prof. G. Snider [49, 50]. Material parameters have been chosen to match our system. In particular: • Quantum well and capping layer material is defined as unstrained In0.75 Ga0.25 As with all band parameters taken from a review by Vurgaftman et al. [10]. • Barrier material is unstrained In0.75 Al0.25 As with the band parameters taken from the same Ref. [10]. The donor states in the In0.75 Al0.25 As have been defined to be 120 meV below the conduction band [40]. The donor density in the barriers, set to be uniform, has been adjusted so that the total charge density in the structure matches the measured value.

58


Figure 3.5: Conduction band and change distributions in the growth direction for sample HM617 calculated with our Schr¨ odinger-Poisson simulator, as a function of gate voltage. Band energies (right axes) are relative to the Fermi energy.

• The buffer is defined as unstrained and undoped Inx Al1−x As. The top part of the buffer is actually subject to some strain. However, the effect on the QW band profile is negligible.

• The boundary conditions have been chosen to be flat-band on the substrate side, and at a fixed energy at the surface side. This energy (30 meV above the Fermi energy) has been measured by photoemission spectroscopy on a thin In0.75 Ga0.25 As layer, grown by MBE in the same conditions as the QW samples [40].

Figure 3.5 shows the result of the simulations for HM617, a 30 nm thick In0.75 Ga0.25 As QW with In0.75 Al0.25 As barriers. The continuous green line represents the conduction band profile along the growth direction, while the green dotted line is the charge distribution. The donor density in the barrier has been adjusted to obtain a total carrier density of 3.0 × 1015 m−2 . The simulation has been repeated, without changing the donor density, including a bias applied to a top gate. This bends the bands and alters the total carrier density n. The figure shows the resulting bandstructure and carrier distribution with the top gate tuned to obtain n =0.9 × 1015 m−2 (blue traces) and n = 4.6 × 1015 m−2 (red traces). These are the actual lower and upper bounds of charge density experimentally reached in our measurements. As can be noted, the charge distribution for the highest charge density is distinctively different from the other two. This is due to the fact that, according to the simulation, two subbands of the QW are populated.


59

Figure 3.6: Charge density versus top gate voltage for sample HM617.

3.2.2

Mobility versus carrier density measurements

Figure 3.6 shows the measured carrier density dependence on top gate voltage. The carrier density varies almost linearly in the shown range. If the gate voltage is swept outside this range, the density deviates from linearity. In particular, if the voltage is swept to values greater then +0.4 V, and then brought back to zero, the measured carrier density for zero applied gate voltage is increased and the mobility reduced. We attribute this behavior to changes in the ionized donors distribution. These changes are not easily controllable, so care has been taken not to exceed this positive limit value, cross-checking the consistency of carrier density and electron mobility at zero applied voltage after each voltage sweep. The left panel of Fig. 3.7 shows the electron mobility versus carrier density for sample HM617. The mobility increases monotonically up to densities of about 3 × 1015 m−2 . Above that value, a distinctive dip in mobility is present. A decrease in mobility at high carrier densities has been observed in InP/Inx Ga1−x As in Ref. [52] and was attributed to the onset of intersubband scattering between the first and the second QW energy levels. A similar mobility behavior has been measured and quantitatively explained as intersubband scattering in a GaAs/AlGaAs modulation doped heterostructure [56]. In our case, the hypothesis of the population of the second subband is confirmed by the analysis of Shubnikov-de Haas (SdH) oscillations shown in the right panels of Fig. 3.7. In fact, for n =3.1 × 1015 m−2 a single frequency is observed in the Fourier analysis of the magnetoresistance as a function of the inverse magnetic field (Fig. 3.7 (b)). On the contrary, when n is tuned to a value of 4.6 × 1015 m−2 , some beatings are visible and two frequencies are observed in the Fourier analysis (see inset of Fig. 3.7 (c)). These peaks

60


Figure 3.7: Magnetotransport measurements on sample HM617. (a) Electron mobility versus carrier density data. The carrier density is varied through a voltage applied to a top gate. (b) and (c) are Shubnikov-de Haas measurements with carrier densities (set by the gate voltage) of 3.5 × 1015 m−2 and 4.6 × 1015 m−2 , respectively. In the insets: the Fourier analysis of the Shubnikov-de Haas traces as a function of inverse magnetic field.

correspond to the total density in the QW and to the density in the first subband (3.5 × 1015 m−2 ). The second subband peak (at ∼1 × 1015 m−2 ) is hardly visible in the FFT spectrum, as an almost unresolved shoulder of the low-frequency background. The population of the second subband at total n higher than about 3.5 × 1015 m−2 is also confirmed by Schrödinger-Poisson simulations of the structure (see Sec. 3.2.1). When a single subband is occupied, it is possible to apply the theory described in Sec. 3.1. The experimental data in Fig. 3.8 are well reproduced for n 2 the ground state at zero field can be a spin triplet, due to Hund’s rule [72].

4.1.3

Kondo effect in quantum dots

In section 4.1.1 the only transport mechanism considered was single electron tunneling, in which first an electron tunnels in the dot from one reservoir and then tunnels out to the other reservoir. This first-order tunneling mechanism gives rise to a current only at the Coulomb peaks, with the number of electrons on the dot being fixed between the peaks. This description is quite accurate for a dot with very opaque tunnel barriers. However, when the dot is opened, so that the resistance of the tunnel barriers becomes comparable to the resistance quantum, RQ = h/e2 = 25.8 kΩ, higher-order tunneling processes have to be taken into account. These lead to quantum fluctuations in the electron number, even when the dot is in the Coulomb blockade regime. An example of such a higher-order tunneling event is shown in Fig. 4.6 (a). Energy conservation doesn’t allow the number of electrons to change, as this would cost an energy of order EC /2. Nevertheless, an electron can tunnel off the dot, leaving it temporarily in a classically forbidden ‘virtual’ state (middle diagram in Fig. 4.6(a)). This is allowed by virtue of Heisenberg’s energy-time uncertainty principle, as long as another electron tunnels back onto the dot immediately, so that the system returns the energy it borrowed. The final state then has the same energy as the initial one, but one elec-

84

Quantum dots

Figure 4.6: Higher-order tunneling events overcoming Coulomb blockade. (a) Elastic cotunneling. The N th electron on the dot jumps to the drain to be immediately replaced by an electron from the source. If a small bias is applied between source and drain, such events give rise to a net current. (b) Kondo cotunneling. The spin-up electron jumps out of the dot to be immediately replaced by a spindown electron. Many such higher-order spin-flip events together build up a spin singlet state consisting of electron spins in the reservoirs and the spin on the dot. Thus, the spin on the dot is screened.

tron has been transported through the dot. This process is known as elastic cotunneling [73]. If the electron spin is taken into account, then events such as shown in Fig. 4.6(b) can take place. Initially, the dot has a net spin up, but after the virtual intermediate state, the dot spin is flipped. Unexpectedly, it turns out that by adding many spin-flip events of higher orders coherently, the spin-flip rate diverges. The spin of the dot and of the reservoirs are no longer separate, they have become entangled. The result is the appearance of a new ground state of the system as a whole – a spin singlet. The spin ot the electrons in the dot is thus completely screened by the spin of the electrons in the reservoirs. This is analogous to the well-known Kondo effect which occurs in metals containing a small concentration of magnetic impurities (e.g. cobalt). It was observed already in the 1930s [74] that below a certain temperature (typically about 10 K), the resistance of such metals would grow. This anomalous behavior was not understood, until in 1964 the Japanese theorist Jun Kondo explained it as screening of the impurity spins by the spins of the conduction electrons in the host metal [75]. The screening is accompanied by a scattering resonance at the Fermi energy of the metal, resulting in an increased resistance. In 1988, it was realized that the same Kondo effect should occur (at low temperatures) in quantum dots with a net spin [76, 77], where the scattering resonance at the Fermi energy manifests as an increased

InGaAs few-electron QDs

85

Figure 4.7: The behavior of the Kondo effect in transport measurements: the Kondo resonance leads to a zero-bias peak in the differential conductance, dI/dVSD , versus bias voltage, VSD .

probability for scattering from the source to the drain reservoir, i.e. as an increased conductance through the dot. The Kondo effect appears below the so-called Kondo temperature, TK , which corresponds to the binding energy of the Kondo singlet state. For an odd number of electrons in the dot, the total spin S is necessarily non-zero, and in the simplest case S = 1/2. However, for an even electron number on the dot — again in the simplest scenario — all spins are paired, so that S = 0 and the Kondo effect is not expected to occur. The Kondo tunneling at the Fermi energy of the reservoirs is observable as a zero-bias resonance in the differential conductance, dI/dVSD , versus VSD , as shown in Fig. 4.7. The full width at half maximum of this resonance gives an estimate of the Kondo temperature.

4.2

In0.11 Ga0.89 As/GaAs structures

The sample structures that I describe in this section have been designed with the purpose of being used for QD transport measurements. The characteristics that we were pursuing were: • A 2DEG at a depth of ∼ 100 nm from the surface, with a carrier density of ∼2 × 1015 m−2 and a mobility not below 10 m2 /Vs. The 2DEG depth and carrier density are chosen to match those of AlGaAs/GaAs 2DEGs used in most lateral quantum dot experiments found in literature. This guarantees that the depletion length of the gates and the confinement strength of our dots will be comparable to those found in literature. The mobility, on the other hand, has to be high enough to have ballistic transport in the QPC that we are going to use for charge detection. • A 10 nm thick pseudomorphic In0.11 Ga0.89 As quantum well. The reason for such a thin quantum well is twofold: first, to confine the electrons in a thin layer will avoid the onset of orbital effects when a

86


Figure 4.8: Schematic growth sequence and Schr¨ odinger-Poisson band structure and carrier distribution calculations for the two heterostructures. In the growth sequence GaAs is brown-colored, Al0.33 Ga0.67 As is blue, and In0.11 Ga0.89 As is yellow. The vertical axis od the Schr¨ odinger-Poisson graphs shows the distance from the surface, while the horizontal axis has the scale for the conduction band energy (black trace) with respect to the Fermi level (red line). The green curves represent the carrier distributions.

magnetic field is applied parallel to the 2DEG; second, it will allow us to reach a relatively high indium concentration without formation of crystal defects due to strain relaxation. When MBE growing ternary alloys, one has to take into account the behaviour of the constituent binaries. The main difficulty here lies in the different optimal growth temperatures for GaAs and InAs. GaAs, to be of high quality and defect-free, are usually grown at a temperature of 600◦ C or above. However, such a temperature is very high for InAs growth, and the indium atoms tend to re-evaporate as soon as they reach the substrate, resulting in sticking factors much lower than unity. A compromise has been reached growing the sample at a temperature of 550◦ , intermediate between GaAs and InAs optimal ones. The x = 0.11 indium concentration of the Inx Ga1−x As quantum well was chosen by growing several heterostructures with increasing In concentration, and measuring the low temperature electron mobilities of the 2DEGs. An indium concentration higher than 0.11 causes dislocations to form in the QW: this in turn is observable as a sharp drop in mobility and an increase in carrier density, a behaviour analogous to that already described for the InAs inserted QWs of Sec. 3.3. Two different sample structures have been selected: • A single In0.11 Ga0.89 As quantum well (sample HM1879). • A double QW: a In0.11 Ga0.89 As square well and a GaAs/AlGaAs triangular one well (sample HM1882).


sample number HM1879 HM1882

sample description InGaAs single QW GaAs-InGaAs double QW

n ( × 1015 m−2 ) 3.1 2.1

87

µ ( m2 /Vs) 9.1 23

Table 4.1: Measured low-temperature carrier concentration, n, and electron mobility, µ, for the two structures grown.

The layer sequence and the Schrödinger-Poisson simulations of the conduction band of the samples are shown in Fig. 4.8. The indium concentration in the samples has been determined by X-ray diffraction measurements on a calibration sample. The Schrödinger-Poisson calculations of the bandstructure shown in Fig. 4.8 use, for the In0.11 Ga0.89 As layers, the band parameters of fully strained In0.11 Ga0.89 As taken from Ref. [10]. Carrier concentration and mobility of the samples, measured at 1.5 K, are reported in table 4.1. The single quantum well sample consists basically of a 10 nm thick In0.11 Ga0.89 As well grown on a GaAs substrate. Above the well, an AlGaAs barrier containing a silicon δ-doping provides both the carriers to the quantum well and the necessary confinement. The double quantum well sample consists of a a 10 nm thick In0.11 Ga0.89 As QW grown on GaAs, followed by 40 nm-thick layer of GaAs, and a δ-doped Al0.33 Ga0.67 As barrier. A triangular GaAs quantum well is formed at the GaAs/Al0.33 Ga0.67 As interface. The confinement for the InGaAs QW is guaranteed by the conduction band discontinuity between In0.11 Ga0.89 As and GaAs. Figure 4.9 (a-b) shows how the conduction band profile and the carrier density distribution for the double quantum well sample change with the application of a negative voltage to a metal gate on the surface. As shown in Fig. 4.9 (c), the carrier density diminishes much faster in the triangular GaAs well (dotted line) then in the In0.11 Ga0.89 As square well (dashed line). With no voltage applied to the gate, almost 80% of the carriers are contained in the GaAs triangular well (Fig. 4.9 (c)); as a negative voltage is applied to the gate the relative carrier population of the In0.11 Ga0.89 As well increases. Near complete depletion of the 2DEG, the situation is reversed, and almost 80% of the carriers are in the In0.11 Ga0.89 As well (Fig. 4.9 (d)). The distance of the In0.11 Ga0.89 As well from the AlGaAs/GaAs heterointerface (40 nm) has been chosen to allow this inversion in the relative carrier population of the two wells. This should allow us to form quantum dots in both configurations. A small, few-electron QD will have the carriers mostly confined in the In0.11 Ga0.89 As well, since it is near depletion. A larger dot, on the other hand, should have its many electrons mostly confined in the GaAs triangular well.

88


Figure 4.9: Results of the Schr¨ odinger-Poisson simulations on the double QW structure, sample HM1882. (a) Conduction band profiles with increasing negative voltages (Vg ) applied to a top gate. The thick black trace corresponds to a zero applied gate voltage, while the red one corresponds to Vg = −0.35 mV. The black dotted line is the Fermi level. (b) Carrier density profiles for the same gate voltages as in (a). In these two plots the same colors are assigned to conduction band traces and carrier density profile traces corresponding to the same gate voltage. (c) Plot of the total carrier density (continuous line), carrier density in the GaAs QW (dotted line) and in the InGaAs QW (dashed line), as a function of gate voltage. (d) Fraction of the total carrier density of the heterostructure contained in the GaAs QW (dotted) and in the InGaAs QW (dashed), as a function of gate voltage.


89

Figure 4.10: A SEM micrograph of the metal gates defining the QDs. The light colored area are the metal gates, while the dark background is the sample surface. The metal gates are numbered from one to ten, while four Ohmic contacts, labeled S1, S2, D1 and D2 are schematically represented by the crossed squares.

4.3

E-beam lithography of the gates

Figure 4.10 shows a SEM micrograph of the electron beam lithography (EBL) defined metal gates. The EBL process developed to fabricate the gates is described in Appendix B. The metal gates are marked with numbers from one to ten, while four ohmic contacts are schematically represented by the crossed squares labeled S1, S2, D1, and D2. Two different sized dots can be formed according to the choice of negatively biased gates. A small dot is defined by negatively polarizing gates 1, 6, 7, and 8, as shown in Fig. 4.11 (a). In the figure the white areas represent the regions of the 2DEG that are depleted by the negatively biased gates. This small dot has a lithographic size of ∼ 260 × 280 nm2 . In particular gates 6 and 8, together with gate 1, form the tunneling barriers separating the dot from the 2DEG in the right and left part of the picture (the leads); gate 7 only weakly affects the coupling of the dot with the leads, and mainly shifts the potentialof the dot, changing the number of electrons it contains. Gate 7 is typically referred to as “plunger” gate. A larger dot is defined by gates 1, 8, 9, and 10 (Fig. 4.11 (b)). Its lithographic dimension is ∼ 710 × 310 nm2 . Gates 8 and 10 are the tunneling barriers, while gate 9 is the plunger gate. Gate 2, 3 or 4 can be used to form a quantum point contact (QPC).

90

Measurement issues

Figure 4.11: Schematic representation of the regions of the 2DEG that are depleted (white areas) when a negative voltage is applied to the gates. (a) the red arrow points to the small dot location. (b) the red arrow points to the large dot location. This dot has an elongated shape. (c) the small dot with the QPC in the typical measurement setup: a common source (S1) and two independent drain contacts (D1 and D2).

When one of these three is polarized together with gate 1, they form a onedimensional constriction in the 2DEG, acting as a QPC for the electrons flowing from S2 to D2. A measure of the conductance through this QPC can be used to count the electrons in the nearby QD [78]. In our measurements we have used only gate 3 to form a QPC, leaving the other two unpolarized. Finally, a large negative voltage applied to gate 5 allows to decouple the transport measurements on the dots from those on the QPC. Two independent source-drain biases can be applied to the ohmic contacts S1 and S2, while measuring the current through the dot in D1 and the current through the QPC in D2. However, in our typical measurement setup we have used a common source for both the QD and the QPC, leaving gate 5 unpolarized. This is shown in Fig. 4.11 (c). A bias VSD is applied to the ohmic contact S1 with respect to the contacts D1 and D2, which are grounded. The current IQD flowing to ground through contact D1 is the current through the dot, while the current flowing through contact D2 is the current of the QPC.

4.4 4.4.1

Measurement issues Effective electronic temperature

When performing low temperature measurements care has to be taken to insure that the effective temperature of the electrons is as close as possible to the base temperature of the cryostat. In fact, the temperature sensor of the cryostat measures the temperature of the 3 He pot. A copper sample holder is firmly attached to the 3 He pot, thus the sample is in good thermal contact with it. Also all the wirings for the electrical measurements need to be thermalized from room temperature to the base temperature of the cryostat. This is accomplished by winding them many times at the various low temperature stages, including at the temperature of the 3 He pot. The electron gas is not only heated by direct heat transport from the room temperature stages of the measurement equipment. For example, the high


91

Figure 4.12: A Coulomb blockade peak of a quantum dot measured without (black trace) and with (red trace) low-pass RC filters applied to the leads. The improvement given by filtering out the high-frequency noise is evident.

frequency part of the electromagnetic spectrum is very difficult to shield and high frequency noise will likely be introduced in the measurement leads, for example, by the voltage source supplying the source-drain bias. Furthermore, since electron-phonon coupling is weak at very low temperatures, electrons are not easily brought in thermal equilibrium with the crystal that contains them, and high-frequency noise can dramatically raise the effective electronic temperature. While measuring a quantum dot, however, it is relatively easy to directly measure the effective temperature of the electron gas. It has been shown by Beenakker [79] that the conductance peaks of a quantum dot in the linear regime, in the limit of very low transparency of the barriers, can be described by the curve ∆E G(∆E) −2 = cosh . (4.7) Gmax 2kB Tef f Here ∆E is the energy shift of the level in the dot with respect to the chemical potential of source and drain, and Gmax is the maximum intensity of the peak. The low transparency of the barriers insures that there is no broadening of the discrete levels of the dot. Thus, the curve described by Eq. (4.7) is basically the convolution of the thermal broadening of the chemical potentials of source and drain. Tef f is the effective temperature of the electrons. The full width at half maximum (FWHM) of such a curve is ∼ 3.5 KB Tef f . The minimum attainable energy width of a Coulomb blockade peak is thus a reliable estimate of the effective electronic temperature. We have measured our sample at base cryostat temperatures of ∼ 250 mK without low-pass RC filters, and we have measured very broad Coulomb blockade peaks (see Fig. 4.12, black trace). Their width indicates an electronic effective temperature of ∼ 850 mK. After inserting low-pass RC filters in all the leads in the 300 mK stage of the cryostat, the situation has changed drastically, and we routinely measure an effective electronic tem-

92

Measurement issues

500

-1150

Vg3 (mV)

120

-350

3

5x10 IQPC (pA)

300

4

100

3 2 1 0

80 200

dIQPC/dVSD (a.u.)

dIQD/dVSD (a.u.)

400

60

100

0 -900

-800

-700

-600

-500

-400

Vg (mV)

Figure 4.13: Differential conductance measurement (blue markers) in the linear regime (VSD = 0). The red markers show the simultaneous measurement of the conductance of the nearby QPC used as charge detector. The first electron enters in the dot at a plunger gate of -752 mV, marked by the green arrow. In the inset, a trace of the current through the QPC as gate 1 voltage is fixed and gate 3 voltage (Vg3 ) is swept (gate numbers are as in Fig. 4.10). The arrow marks the Vg3 value used for the measurement shown in the main graph.

perature comparable to the base temperature of the cryostat (red trace in Fig. 4.12).

4.4.2

Few-electron regime

A QD containing a single electron is an almost ideal sistem to work with: in fact the interpretation of the transport measurements in terms of single particle energy levels is straightforward, and since no electron-electron interactions come into play, the CI model holds very well. It is thus of fundamental importance to reach the so called few-electron regime, that is, to have the QD containing a known, small number of electrons, being at the same time able to perform transport measurements through it. This is not a trivial task, since to contain zero or one electron the dot has to be made very small, and the gates defining it have to be set to large negative voltages. This means that the coupling of the dot to the leads, and thus the tunneling currents through the dot itself, are small and difficult to measure. To overcome these difficulties, the charge-sensing QPC is used to count the number of electrons in the QD even if no current flowing through the dot itself is measurable. Figure 4.13 shows a differential conductance measurement of the CB peaks of the dot (blue markers) in the linear regime (i.e. VSD = 0). The red


93

Figure 4.14: Equivalent circuit of the QD and the QPC in common source configuration. All capacitances are ignored.

markers show the simultaneous measurement of the conductance of the QPC. By sweeping the plunger gate towards more negative voltages the broad peaks in the dot conductance become narrower and clearly separated by zeroconductance regions: the Coulomb blockade regime. The QPC conductance shows an upward step at every CB peak, indicating that the dot contains one electron less. At a gate voltage lower than the last visible CB peak, the QPC conductance shows another step, and then becomes smooth. This indicates that the last visible peak of the conductance of the dot does not correspond to the last electron being extracted. In fact, the last electron is extracted at Vg ∼ −760 mV, corresponding to the last QPC conductance step. The last peak of the QD conductance corresponds in this case to the second electron tunneling through the dot. By adjusting the transparency of the barriers (modifying the voltage applied to gates 6 and 8 in Fig. 4.11 (c)) one can increase the current tunneling through the dot to make the conductance peak at the N = 0 → 1 transition measurable. This allows to perform transport experiments on a one-electron quantum dot.

4.4.3

QPC curve interpretation

The measurements with the QPC have all been performed in the “common source” configuration shown in Fig. 4.11 (c). This gives some artifacts in the QPC trace when high currents are flowing through the QD. In fact the whole device can be modeled in terms of simple circuital elements as shown in Fig. 4.14. The current flowing through the QPC (IQP C ), measured by the current meter, is determined by several series and parallel resistances. From the voltage source, which supplies a voltage VSD , the current first flows to the leads running from the electronics to the low temperature stage, then to

94

Measurement issues

the low-pass RC filters, and finally through the ohmic contact (S1) and the 2DEG to the right side of the gates (see Fig. 4.11 (c)). This is all represented by the series resistance RL (where ‘L’ stands for leads). Then the current is split in two parallel paths, one through the QD and one through the QPC. On each path it encounters again a series resistance RL (approximately equal to the first) that takes into account the 2DEG to the left of the devices, the RC filter and the leads from the low temperature stage out to the current meter. In a given configuration of the gates, the conductance through the QPC and through the QD are fixed and correspond to the resistances RQP C and RQD , respectively. So the current through the QPC is IQP C =

VQP C , RQP C + RL

(4.8)

where VQP C = VQD depends on what amount of VSD drops in the first, series resistance RL . This in turn depends on the total resistance of the circuit, and thus on the resistance of the dot. If we write the conductance of the QD (upper) and of the QPC (lower) arms of the circuit as GD = 1/(RQD + RL ) and GQ = (RQP C +RL ), respectively, then the total resistance of the circuit, RT OT , is 1 RT OT = RL + , (4.9) GQ + G D and the voltage drop across the QPC is: VQP C = VQD

RL VSD RL = VSD 1 − . = VSD − RT OT RT OT

Finally, the current through the QPC has the value  IQP C = GQ VQP C = GQ VSD 1 −

RL +

(4.10)



R L

1 GQ +GD



(4.11)

This expression is difficult to simplify due to the comparable values of the resistances involved. In fact RL . RQP C < RQD or, showing the actual values, RL ∼ 10 KΩ, RQP C ∼ 20-40 KΩ, and RQD ranges from infinity (when the dot is Coulomb blockaded) down to ∼ 50 kΩ for some of the intense CB peaks. It is evident, however, that a peak in the conductance of the dot causes a dip in the current of the QPC, superimposed to the normal QPC trace. This condition is quite peculiar of our sample and setup for three concurrent reasons: • Our 2DEGs have relatively high sheet resistances, compared to GaAs/AlGaAs high mobility samples, due to the low growth temperature of the heterostructures. This results in a higher RL .


95

• To filter out high frequency noise that raises the effective electronic temperature we have used low-pass RC filters, made with a resistance of ∼ 3 kΩ. This again increases RL . • Since we have only one clean voltage source to provide the source-drain bias, we are forced to use the common source configuration instead of decoupling the QPC and the dot. This can (and will, in future experiments) be done with the aid of the gate numbered 5 in Fig 4.10. This discussion was intended only to convince the reader that the QPC is properly behaving, and that for each change in the number of electrons in the dot a step in the QPC conductance is observed.

4.5

A few-electron QD in the single QW sample

This section describes the low temperature transport measurements on the small quantum dot fabricated on the single In0.11 Ga0.89 As quantum well structure (HM1879). In this sample, most of the carriers are confined in the In0.11 Ga0.89 As QW, until complete depletion of the 2DEG. This QD has been measured at a base temperature of ∼ 260 mK in a He3 refrigerator. All measurements have been performed with the gates polarized as in Fig. 4.11 (c), with gate 3 used to form a QPC. Although the conductance of this QPC is not ideal (see inset of Fig. 4.13) it is possible, near pinch-off, to set it in the regime for charge sensing of the nearby dot.

4.5.1

Stability diagram

By appropriately tuning the tunneling barriers with the source and drain reservoirs (respectively gates 6 and 8 in Fig. 4.11 (c)), it is possible to set this dot to work in the few-electron regime. Figure 4.15 (a) shows the stability diagram of the QD focused on the first few electrons. Blue areas in the diagram correspond to constant current through the dot, while white and red areas indicate a non zero differential conductance, i.e., a change in the current. A single horizontal trace taken at VSD = 0, is shown in Fig.4.15 (b) (blue trace), and represents the conductance in the linear regime . In the large diamond shaped regions centered at zero source-drain voltage, the number of electrons in the dot is fixed. In the area at more negative gate voltage, labeled ‘N=0’, the dot is empty: the lowest chemical potential level of the dot is at energies higher than both the source and drain chemical potentials, and no current flows. We know this by looking at the conductance in the linear regime (blue trace) together with the conductance of the QPC (red trace) shown in Fig. Fig.4.15 (b): the peak at Vg ∼ −1005 mV causes the last step in the QPC conductance, so it corresponds to the transition from an empty dot to a one electron dot. Moving to more positive gate voltage, the next diamond, labeled ‘N=1’, corresponds to having one electron

96


(b) Vg (mV) -1100

-1000

-900 2 1 0

50 N=0

0

(a)

N=1

N=2

N=3

VSD (mV)

100

-1 -2

Figure 4.15: (a) Stability diagram for the quantum dot in the single QW sample. The colors of the image indicate the intensity of the differential conductance, dI/dVSD , varying both the plunger gate voltage (horizontal axis) and the sourcedrain voltage (vertical axis). In the color scale, blue means zero (or less than zero) differential conductance, while white and red indicate positive differential conductance. The labels in the diamond-shaped zero conductance regions indicate the number of electrons in the dot. The rectangle shows the region of the magnetic field measurements described in Sec. 4.5.2. In (b) is shown a single differential conductance trace taken at VSD = 0 (blue curve), along with the QPC conductance trace (red curve). A line has been subtracted to the QPC conductance to enhance the step visibility. The Vg scaleis the same for both (a) and (b).

in the dot. Going at even higher plunger gate voltage one encounters additional diamonds indicating the gradual filling of the dot with an increasing number of electrons. From the stability diagram it is possible to extract a wealth of information about energy levels in the dot. The addition energies for the second and third electron (Eadd (1) and Eadd (2), respectively) can be read off directly, by measuring the VSD half-width of the N = 1 and N = 2 diamonds. The resulting values are Eadd (1) ' 1.5 meV and Eadd (2) ' 2.0 meV. In the SET regions of the 0 → 1 and the 1 → 2 transitions, many lines parallel to the diamond edges are present (for negative source-drain voltage they are pointed at by black arrows in the 0 → 1 transition). These may be due to modulations in the density of states of the source and the drain reservoirs, which give rise to changes in conductance. It is not trivial to isolate the exited state transitions from this fine structure. It should be the


97

more intense pair of lines, parallel to the edges of the N = 0 region, crossing the N = 1 diamond edge at VSD ' ±0.5 mV. On the positive VSD side, this line is very well defined; its intersection with the N = 1 diamond is marked by the green arrow. The source-drain voltage at the crossing point corresponds to the energy of the first excited state in the one-electron dot, with respect to the ground state. In the case of N = 1, this energy is ∆E, the single-particle level spacing between the first and the second level of the dot, and thus ∆E ' 0.5 meV. Since the second electron occupies the same single-particle energy level of the first, but with opposite spin, at zero magnetic field Eadd (1) = Ec . The third electron, on the other hand, will populate the second single-particle energy level, so that Eadd (1) = Ec +∆E. The energy values derived from the measured stability diagram are compatible with this description, indicating that the CI model holds well for this dot. As previously mentioned, the stability diagram allows also to convert the plunger gate voltage in units of energy, through the leverage factor α. For the N = 1 diamond we have α ' 45 µeV/mV.

4.5.2

Land´ e g-factor

There are two ways to measure the g-factor of the electrons in the dot. The first is through excited state spectroscopy, and is a direct measurement of the Zeeman splitting. It is performed by taking stability diagram measurements at increasing magnetic field. The Zeeman splitting of the lowest single-particle energy level of the dot introduces a new excited state. In the stability diagram two new lines (one at positive and the other at negative VSD ), parallel to the edges of the N=0 region, should split from the edges and increase their distance to it. The distance should increase linearly with magnetic field [80]. The same splitting should occur for the first excited state lines. Figure 4.16 shows several measurements of the N = 0 → 1 transition with increasing magnetic field. The field has been applied parallel to the plane of the 2DEG to limit orbital effects in the dot and avoid quantum hall effects in the 2DEG. In our sample, the presence of fine structure resembling excited states makes it hard to identificate a new excited state line. However, we have not observed a clear pattern of splitting with magnetic field neither of the ground state or of the excited state lines. This would indicate that there is no detectable Zeeman splitting, implying a near-zero g-factor. Figure 4.17 shows two cross-sections of the B = 12 T stability diagram taken at different gate voltages, as indicated by the two colored lines in (a). The same ground state peak has been fit with a cosh−2 curve in both traces. An eventual Zeeman splitting would —at least— broaden the ground state peak of the trace in (b) but not that of the trace in (c), where the splitting does not occur. The comparison of the width of the peaks

98

A few-electron QD in the single QW sample 1.0

VSD (mV)

0.5 0.0 -0.5 B=0 T

B= 2 T

B= 4 T

B= 6 T

-1.0 -1044

-1034

-1043

-1033

-1042

-1032

-1040

-1030

1.0 dI/dVSD (a.u.)

VSD (mV)

0.5 0.0 -0.5 B= 8 T

B= 10 T

60 40 20 0

B= 12 T

-1.0 -1040

-1030

-1036

-1026

-1032

-1022

Figure 4.16: Several stability diagram measurements of the N = 0 → 1 transition for the dot (the area enclosed by the rectangle in Fig. 4.15), taken at increasing parallel magnetic field. The horizontal axis in each plot is the plunger gate voltage Vg . The applied magnetic field is indicated in each plot.

has been repeated for many cross-sections at several magnetic field values, resulting always in comparable FWHM of the lines. This sets in an upper limit for the Zeeman splitting, equal to the thermal energy (at the effective temperature of the electrons) derived from the FWHM of the peaks, i.e. g < 0.1. In order to confirm this finding, we have used also the second method of detecting the presence of a Zeeman splitting. This consists in measuring, in the linear regime, the magnetic field dependence of the relative distance (in plunger gate voltage) of the Coulomb peaks. The plunger gate voltage differences can then be converted in an energy with the previously determined α factor. Figure 4.18 shows such measurements, where the parallel magnetic field has been varied from 0 to 12T. Four CB peaks are clearly resolved, and their positions and intensities vary slightly with B. We have fitted the four peaks with cosh−2 curves and computed the distances between adjacent pairs of peaks. The distances are shown in Fig. 4.19(a). The filled black symbols are the differences between the first and second peak, the empty red symbols refer to the second and third peak, and the green crosses to the third and the fourth. We concentrate on the first two peaks, labeled “peak 1” and “peak 2”, corresponding to the N = 0 → 1 and N = 1 → 2 transitions, respectively. Their distance has been converted in energy through the leverage factor α = 45 µeV/mV, and is plotted in Fig. 4.19(b) as a function


(b) dI/dVSD (a.u.)

1.0

0.5

99

20 15 10

0.22 mV

5

0.0

-0.5

-1.0

(a)

-0.4

(c) dI/dVSD (a.u.)

VSD (mV)

0 -5 -0.2

0.0

0.2

0.4

0.0 0.2 VSD (mV)

0.4

12 8 4

0.21 mV

0

-1032

-1028 -1024 Vg (mV)

-1020

-0.4

-0.2

Figure 4.17: (a) Stability diagram around the N = 0 → 1 transition with B=12 T. (b)(c) Cross sections of (a) taken at Vg = −1029.5 mV (orange symbols) and Vg = −1025.5 mV (green symbols), respectively. The continous lines are fit of the indicated peaks with cosh−2 curves. The FWHMs resulting from the fit are indicated in the graphs.

of magnetic field. At zero bias, the distance between the peaks represents the energy needed to add an additional electron in the dot. So the difference between the first and second peak is the energy needed to add a second electron in the one-electron dot. This second electron will occupy (at zero magnetic field) the same, spin-degenerate single-particle level as the first electron. Thus the peak distance is due only to the charging energy of the dot. At finite magnetic field, however, the second electron will occupy the Zeeman split level, and thus its addition energy is the zero field energy (the charging energy) plus the energy difference between the Zeeman split first singleparticle level: Eadd (1) = EC + |g|µB B.

(4.12)

This equation implies two hypothesis: one is that that the charging energy does not depend on the magnetic field and the other is that the ground state of the two-electron dot is a singlet. Our measurements show no evidence of a dependence of charging energy on magnetic field, satisfying the first hypothesis. As for the singlet ground state, at zero magnetic field we rely on a fundamental theorem of quantum mechanics stating that a two-electron system, whose Hamiltonian does not contain spin dependent terms, has a spin singlet ground state [14]. Singlet-triplet energy crossing at finite magnetic field would be revealed by a sharp change in the slope of the peak distance versus B curves [71] of Fig. 4.19 (a), which are not present in our measurement. This indicats that the ground state of the QD is a singlet for all values of magnetic field, and Eq. 4.12 holds.

100


peak 1

peak 2

peak 3

peak 4

N=4 N=0

N=3

N=2

N=1

dIQD/dVSD (a.u.)

500 400 300 200 100 0 -1000

-950

-900 Vg (mV)

-850

-800

Figure 4.18: Differential conductance in the linear regime (i.e. VSD = 0) with varying magnetic field. The magnetic field is zero for the lowest trace and 12 T for the upper one. The curves are shifted for clarity. The number of electrons, N , in each Coulomb blockaded region is indicated by the labels.

70

1.465

peak 2-3

Eadd(1) (meV)

peak distance in Vg (mV)

1.460 60 peak 3-4

50

1.455

1.450

1.445 40

1.440

y = (1.442 ± 0.002) + (0.0017 ± 0.0003)•x

peak 1-2

0

2

4

6 B (T)

8

10

12

0

2

4

6 B (T)

8

10

12

Figure 4.19: Differential conductance peak distance as a function of magnetic field. (a) The difference in gate voltage between pairs of peaks of a trace at fixed magnetic field, as a function of magnetic field. The black dots are the distances between the peak of the N = 0 → 1 (at ∼ −1005 mV in Fig. 4.18) and that of the N = 1 → 2 transition (∼ −975 mV); the red circles are the distances between the N = 1 → 2 and N = 2 → 3 (∼ −900 mV) transition, and the green crosses are the distances of the N = 2 → 3 and the N = 3 → 4 (∼ −850 mV) transition. (b) The “peak 1-2” data of (a) converted in energy using α = 45 µeV/mV. The continous line is a linear fit, whose parameters are shown in the figure.


101

Figure 4.19 (b) shows also a linear fit of the data. The intercept of the line is the charging energy, Ec = 1442 ± 2 µeV, while the slope is |g|µB = 1.7 ± 0.3 µeV/T, yielding a g-factor value |g| = 0.030 ± 0.005.

(4.13)

The electrons in the first single-particle energy level of the dot are the easiest to interpret, and by employing two indipendent techniques to measure the Zeeman splitting of this level the splitting results to be very close to zero in our system. This finding is quite unexpected since the bulk gfactor of In0.11 Ga0.89 As is greater than that of GaAs and recently measured g-factors in GaAs lateral QDs report values that are an order of magnitude greater than our result [80, 81]. This behavior might be due to confinement effects in the dot, that alter the effective g-factor of its electrons. In fact, it has been calculated that for self-assembled few-electron quantum dots or dots confined in nanocrystals, the g-factor should tend to the atom-like value of +2 [82], or even more positive [83]. Also measurements in QDs formed in InAs nanowires have shown a considerable shift of the g-factor towards less negative values due to confinement effects [84]. However, these effects are not expected to be strong in the case of electrostatically defined lateral QDs like the one studied here. Further investigations at lower temperature would thus be needed to acquire better understanding of this result.

4.6

A few-electron QD in the double QW sample

Even if its origin is unclear, a nearly-zero g-factor dot is a very unique system, that can be exploited in a number of ways. Of special interest would be again to couple the InGaAs small QD to a dot in which the levels do split in magnetic field, such as a GaAs one. For this reason we have measured the double quantum well heterostructure (sample HM1882). The sample structure and gate design are aimed at having a In0.11 Ga0.89 As few-electron dot coupled to an almost GaAs large (N ∼ 50) one. In this thesis we have limited ourselves to the analysis of the small dot, focusing on spin-related issues.

4.6.1

Few electron regime

Figure 4.20 shows a measurement of the differential conductance of the dot (at VSD = 0, i.e. in the linear regime) as a function of plunger gate voltage (blue trace) taken at a base temperature of 255 mK. The red trace is the simultaneous measurement of the conductance through the nearby QPC, used as charge detector. The behavior is analogous to the single quantum well sample. Going towards more negative plunger gate several conductance peaks are separated by the Coulomb blockade regions. The peaks

102


dIQD/dVSD

300

200

1.75 mV

-658

Vg(mV)

-640

100

0 -750

-700

-650 Vg (mV)

-600

-550

Figure 4.20: Differential conductance measurement (blue trace) in the linear regime (VSD = 0) for the small dot of sample HM1882. The red trace show the simultaneous measurement of the conductance of the nearby QPC, used as charge detector. In the inset: a close-up plot of the peak of the N = 0 → 1 transition at Vg ∼ 650 mV. Its FWHM corresponds to a Tef f of ∼ 250 mK.

become less intense and narrower with more negative gate voltages due to the decrease of the coupling of the dot with the leads. The last visible peak (shown in the inset) has a width of 1.75 mV that corresponds, considering the leverage factor α = 47 µeV/mV (estimated from the stability diagram of Fig. 4.21), to an effective electron temperature of ∼ 250 mK. This peak corresponds to the N = 0 → 1 transition in the dot, since no more steps appear at more negative gate voltages in the QPC conductance. Figure 4.21 is the stability diagram of the dot. All gates are polarized as for the linear regime curve and QPC trace of Fig. 4.20; this allows us to identify the number of electrons contained in the dot within each diamond. The single electron tunneling regions at finite bias show a fine structure similar to that observed in the QD described in Sec. 4.5. The dot, in this measurement, is strongly coupled to the source and drain leads, as can be inferred by the large width of the lines definig all transitions except the first one (N = 0 → 1). The coupling of the dot to the leads is controlled by the voltages applied to the barrier gates (gates 6 and 8 in Fig. 4.11 (c)), and can be decreased making them more negative. To avoid changes in the number of electrons in the dot, the increased polarization of the barriers has to be balanced by a reduction of the polarization of the plunger gate. We have characterized the dot also in a configuration with a lower coupling to the leads. The stability diagram of the “less coupled” QD is shown in Fig. 4.22. Comparing the two diagrams, it is clear that the variation of the voltages


103

2

VSD (mV)

1 N=0

N=1

N=2

N=3

N=4

0 -1 0 200 400 dI/dVSD (a.u.)

-2

-650

-600

-550

-500

Vg (mV)

Figure 4.21: Stability diagram for the small dot of sample HM1882 with high coupling to the leads. The number of electrons, N , in each diamond is indicated with the labels.

2

VSD (mV)

1 N=0

N=1

N=2

N=3

N=4

0 -1 0

-2

-550

-500

-450 Vg (mV)

-400

200 400 dI/dV (a.u.) -350

Figure 4.22: Stability diagram for the small dot of sample HM1882 with low coupling to the leads. The number of electrons, N , in each diamond is indicated with the labels.

104


2

VSD (mV)

1 0 -1 -2

0

-550

-500

-450 Vg (mV)

-400

200 400 dI/dV (a.u.) -350

Figure 4.23: Quantum dot energies deduced from the stability diagram in the case with low coupling to the leads. The green continous lines mark the edges of the diamonds, while the dashed lines indicate the excited states and the elastic cotunneling transitions.

of the three gates forming the dot also changes the energy levels within the dot itself. In fact, the VSD widths of the diamonds change, as well as the energies of the excited states. This is due to changes in the shape of the confinig potential. From both diagrams the addition energies and some of the excited state energies can be deduced and compared. How these values are derived from the stability diagram is explicitly shown in Fig. 4.23 for the low coupling case. The continous green lines mark the edges of the diamonds, while the dashed lines correspond to the excited state transitions and to the elastic cotunneling steps. Table 4.2 shows the values deduced from both diagrams. Even if the confining potential changes, the dot shows some common features for both configurations. In particular, the presence of a finite differential conductance line at VSD = 0 in the N = 3 diamond is evident in both diagrams (in the higly coupled dot, a finite conductance line at zero bias is visible also in the N = 2 diamond). Such finite conductance at zero bias in a Coulomb blockaded region is typical of Kondo effect and will be discussed in detail in Sec. 4.6.3. The other common feature of both configurations is the fact that the addition energy of the second electron (Eadd (1)) is larger than the addition energy of the third (Eadd (2)). This is in contrast to the simple CI model according to which Eadd (1) = Ec and Eadd (2) = Ec + ∆E1 . The failure of the CI model for this dot can have two possible explanations. One is that the charging energy Ec is not constant with respect to the plunger gate voltage, but decreases as the gate is made less negative. This happens if the confinig potential changes enough to significatly alter the total capacitance of the dot. The second explanation is related to the


Eadd (1) Eadd (2) Eadd (3) ∆E1 ∆E2 ∆E3

high coupling 1.5 ∼1.3 ∼1.2 0.5 (0.25?) (0.2?)

105

low coupling 2.4 1.7 1.7 0.8 0.6 0.25

Table 4.2: Addition energies, Eadd (N ), and first excited state energies, ∆n , for the first three diamonds of the QD in the “high coupling to the leads” (Fig. 4.21) and in the “low coupling to the leads” (Fig. 4.22) configurations. In the stability diagram for the “high coupling” QD the features are so broad and smeared that the energies of the excited states ∆E2 and ∆E3 are almost impossible to determine. All values in mV.

20

ΨA ΨB

15

EC ρ

En (meV)

10

5

εF

0

∆ ~ 500 µeV

-5 -300

-250

-200

-150

-100

-50

0

Vg (mV)

Figure 4.24: Energies of the lowest two-dimensional subbands in the heterostructure, as a function of top gate voltage. The values have been calculated with a Schr¨ odinger-Poisson simulator [49] on the base of our sample structure. Blue (red) is the bonding (antibonding) state energy. In the insets, the profiles in the growth direction of the conduction band minimum (EC , black), the carrier density distribution (ρ, green) and the bonding (ΨB , blue trace) and antibonding (ΨA , red trace) normalized wavefunctions. These are calculated (from left to right) at gate voltages of -225 mV, -140 mV, and 0 mV.

106


peculiarities of the confinig potential in the growth direction, i.e. perpendicular to the plane of the 2DEG. The sample consists, in fact, of two coupled quantum wells. The energies of the double quantum well system depend on the bias applied to a surface gate, and in particular the charge layer can be shifted from one quantum well to the other, as already descrbed in Sec. 4.2. Far from the extremes, when the charge is evenly distributed between the two wells, two subbands are occupied, and they are almost degenerate in energy. The lowest energy state is the so called “bonding” state, while the the other is the “antibonding” state. The energies of the two states, as calculated through a Schrödinger-Poisson simulation varying the gate voltage, are plotted in Fig. 4.24. The bonding and antibonding subband energies show an anticrossing behaviour at Vg ∼ −140 mV. In QDs made from a single quantum well structure, the confinement along the growth direction is much stronger than that provided by the gates on the 2DEG plane, so that the 2D subband spacing is much larger than the spacing between the levels in the dot. So one geneally deals with a single 2D subband that is split in the quantized levels of the dot by the confinement on the plane. On the contrary, in our sample the minuimum energy gap, ∆, at the anticrossing point is a little less than 500 µeV, of the same order of the excited state energies deduced from the stability diagrams. Thus the energy separation of the lowest lying 2D subbands is comparable to the single-particle level spacing of the dot, and this mixing can significantly deviate the energy spectrum of the dot from the simple case of the CI model.

4.6.2

g-factor

We have performed excited state spectroscopy in a magnetic field for the N = 0 → 1 transition, to have an estimate of the Landè g-factor of the one-electron dot, and possibly confirm the near-zero g-factor value obtained in the single quantum well sample. The measurement shown in Fig. 4.25 is analogous to that described in Sec. 4.5.2. It is a series of stability diagram measurements around the N = 0 → 1 transition, taken at several magnetic field values. The dot is set in the high coupling regime. The Zeeman splitting of the ground state should manifest as a split line associated with the ground state lines at the N = 0 diamond edge, with a separation roughly linear in magnetic field. No splitting seems to take place, nor there is broadening of those lines. This is confirmed also by a second measurement taken at a constant, finite, source-drain bias, while sweeping both the gate voltage and the magnetic field, shown in Fig. 4.26. The peak shown, corresponding to the transition between the empty dot and the single electron tunneling region, is not split nor broadened by the magnetic field. Above 9 T (the red trace of (b)), the peak seems to split, but this splitting is too fast to be due to the Zeeman effect: it is due to one of the fine structure peaks interfering with the SET boundary. Both these measurements indicate that


107

1.0

VSD (mV)

0.5 0.0

N=0

N=1

-0.5 B=0 T

B= 2 T

B= 4 T

B= 6 T

-1.0 -645

-635

-645

-635

-645

-635

-645

-635

1.0 2

VSD (mV)

0.5 0.0

0

-0.5 B= 8 T

B= 10 T

B= 12 T -2

-1.0 -645

-635

-640

-630

-640

-630

-670

-620

Figure 4.25: Several stability diagram measurements of the N = 0 → 1 transition for the dot (the area enclosed by the rectangle in the bottom right plot, corresponding to the stability diagram of Fig. 4.21, with the dot in the regime of strong coupling with the leads), taken at increasing parallel magnetic field. The horizontal axis in each plot is the plunger gate voltage Vg . The applied magnetic field is indicated in each plot.

the Zeeman splitting is smaller than the thermal broadening of the lines, kB Tef f = 80 µeV, even with a 12 T magnetic field. This implies |g| < 0.12. Figure 4.27 shows the magnetic field dependence of the differential conductance in the linear regime. The first two peaks (labeled ‘pk1’ and ‘pk2’) correspond to the transitions N = 0 → 1 and N = 1 → 2, respectively. Their spacing, once converted in energy through the leverage factor α = 53 µeV/mV, is the addition energy of the second electron in the dot, which corresponds to the charging energy plus the Zeeman splitting (Eq. (4.12)). A linear fit of this spacing with magnetic field yealds a Zeeman energy of 1.7 ± 0.6 µeV/T, which corresponds to an absolute value of the g-factor of 0.03 ± 0.01. The magnetic field dependence of the first single-particle energy level of this QD shows that the Zeeman splitting is negligeable. This finding is in agreement with the result obtained for the quantum dot formed in the single quantum well sample described in Sec. 4.5. The origin of this behaviour is still unclear, but might be related to confinement effects that strongly reduce the absolute value of the g-factor in the QD with respect to the bulk value.

108


-544 -546

8 4 0

15

B = 12 T

Vg (mV)

dI/dVSD (a.u.)

SET 0->1

-548 -550 -552 -554

10

5

0

-556

B=0T

N=0

-558 0

2

4

6 B (T)

8

10

12

-565

-560

-555 -550 Vg (mV)

-545

Figure 4.26: (a) Differential conductance at a fixed source drain bias, VSD = +1.20 mV, with varying magnetic field. The color of the pixels indicates the value of the conductance for each pair of plunger gate (left axis) and magnetic field (bottom axis) values. The peak shown is the transition from the empty dot to the SET region. (b) Some traces of (a) at several magnetic fields from B = 0 (bottom trace) to B = 12 T (top trace). The traces have been shifted in gate voltage to align the peaks, and vertically spread for clarity.

-350

N=5

y = ( 2403 ± 4 ) + ( 1.7 ± 0.6 ) B

pk5

N=4

-400

pk3 N=2

pk2

-500 Eadd(1) eα

2420

2400

N=1

pk1

-550

0

Eadd(1) (µeV)

Vg (mV)

N=3

-450

500

2440

pk4

N=0

0

4

8 B (T)

12

0

4

8

12

B (T)

Figure 4.27: [??? DA SISTEMARE ???](a) Differential conductance in the linear regime with varying magnetic field. The color of the pixels indicates the value of the conductance for each pair of plunger gate (left axis) and magnetic field (bottom axis) values. The number of electrons in the dot in the zero conductance regions is indicated by the labels. (b) Spacing between the first and second peak, converted in energy with the leverage factor, as a function of the magnetic field. The red trace is a linear fit of the experimental data.


4.6.3

109

Kondo effect

The absence of Zeeman splitting can have interesting implications in the strongly coupled regime, where high order tunneling effects, like cotunneling and the Kondo effect, come into play. In particular, the Kondo effect has been most commonly observed in semiconductor quantum dots having a total spin S = 1/2 [85, 86, 87, 88]. In these works, the application of a magnetic field causes the zero-bias Kondo resonance to split in two separate resonance peaks at finite bias. This is due to the Zeeman splitting of the level in which the Kondo tunneling is taking place. For the tunneling and spin-flipping of the Kondo effect to occur in a Zeeman-split level, an external source (the source-drain bias) must supply the required energy difference. Thus, under an applied magnetic field, two resonances are visible at Vsd = ±gµB B. When the Zeeman energy becomes very high, the systam has to be too far from equilibrium for the many-body Kondo effect to take place: the resonance is suppressed and the double peak fades into a pair of elastic cotunneling steps. We have already mentioned that a Kondo resonance is clearly visible in the N = 3 diamond of the stability diagrams of Figs. 4.21 and 4.22. In this section, we have investigated what happens to the resonance if the level in the dot, bearing a zero g-factor, does not split with magnetic field. Unfortunately, no Kondo effect was visible in the N = 1 diamond, the simplest spin-1/2 state of the dot, due to the difficulty in forming a oneelectron dot with strong coupling to the leads 3 . The next most favorable spin-1/2 configuration is the N = 3 state, for which the dot is more coupled to the leads. Figure 4.28 (a) is a magnified view of the N = 3 diamond of Fig. 4.21. The enlargement allows to distinguish clearly the resonance at zero source-drain bias. The colorplot of Fig. 4.28 (b) represents the magnetic field evolution of this resonance. Each vertical line represents a differential conductance trace taken by sweeping the source-drain voltage at a fixed value of the plunger gate along the black line in the middle of the diamond, drawn in (a). From trace to trace, only the magnetic field, applied parallel to the plane of the 2DEG, is changed4 . The Kondo resonance completely disappears at a magnetic field B ∼ 2 T, without showing any splitting. This is evident in the plot of Fig. 4.28 (c). As previously stated, in non-zero g-factor, spin-1/2 quantum dots, the res3

To have a single electron in the dot, all gates defining the dot are brought to large negative voltages, and the area of the dot is squeezed so that only one electron can fit in. This process makes the barriers between the dot and the source and drain reservoirs very wide, reducing the coupling and the tunneling rates. High coupling between dot and leads is a necessary condition for the Kondo effect to take place. 4 Actually, since the diamond position in plunger gate voltage drifts with magnetic field (see for example Fig. 4.27), also the plunger gate voltage has been slowly varied to follow the center of the diamond. In this way each trace is taken at a plunger gate that is midway between the two transitions N = 2 → 3 and N = 3 → 4.

110


1.0

m178 dIQD/dVSD

120

m175 dIQD/dVSD

0.4

0.5

0.0

100

dI/dVSD

VSD (mV)

VSD (mV)

0.2

0.0

80

-0.2 -0.5 250 200 150 100 50 0

100 80 60 40 20

-0.4

60

-1.0 -580

-560 -540 Vg (mV)

-520

0

1

2 B (T)

3

4

-0.2

-0.1

0.0 0.1 VSD (mV)

0.2

Figure 4.28: Kondo effect for N = 3 in magnetic field. (a) The stability diagram of the N=3 diamond of the dot. The intense peak running horizontally at VSD ∼ 0 is the Kondo resonance. (b) the magnetic field evolution of the Kondo resonance. The colorplot shows the measured differential conductance as a function of VSD (swept a5 fixed Vg along the black line of (a)) and the magnetic field. (c) some traces from (b) at magnetic fields values from 0 T (upper trace) to 2T (lower trace) in steps of 0.25 T. The traces are vertically offset for clarity.

onance is split by the magnetic field and then (at higher field values) disappears due to the increasing energy difference between the Zeeman-split levels of the dot. In a zero g-factor dot like ours, this mechanism cannot be responsible for the suppression of the Kondo effect. In fact, the resonance does not split but simply fades out with increasing magnetic field. This is a quite unique feature of this zero g-factor quantum dot. Our experimental observations show an evidence of the fact that an applied magnetic field can suppress the many-body Kondo effect not only acting on the energy levels within the quantum dot, but also in different ways, for example through a spin polarization in the leads. However, to confirm this hypothesis, additional experimental work and a robust theoretical framework are needed. As already pointed out during the discussion on the energy levels of the dot, a resonance at zero source-drain bias is present also in the N = 2 diamond of the stability diagram. Figure 4.29 (a) shows the stability diagram of the dot in the high coupling regime, limited to the first two diamonds. To enhance the visibility of the resonance, the area enclosed in the rectangle is magnified in Fig. 4.29 (b), which also has a higher contrast. As previously mentioned, such a finite differential conductance at zero bias is a signature of the Kondo effect. However, to the best of our knowledge, the Kondo effect has never been observed for a two-electron dot at zero magnetic field. In fact, the ground state for a two-electron system with no spin dependent terms in the Hamioltonian is always a singlet [14], and for a singlet state no spin-flip processes (and thus no Kondo effect) can occur. To have a better insight on the physical origin of the Kondo resonance, we have studied its evolution under changes in coupling strength and in external magnetic field. In particular, modifying the coupling between the dot and


111

0.6

2

0.4 1 VSD (mV)

VSD (mV)

0.2 0

0.0 -0.2

-1

(a)

-0.4

300 200 100 0

-2

-680

-660

-640

-620 -600 Vg (mV)

-580

-560

80 60 40 20 0

-0.6

(b)

-620

-610

-600 Vg (mV)

-590

-580

Figure 4.29: Stability diagram of the QD in the configuration in which it is strongly coupled to the leads. (a) The first two diamonds. (b) A close up view of the N = 2 diamond with enhanced contrast to highlight the Kondo resonance.

50

30 decrease coupling

dIQD/dVSD (a.u.)

40

20

10

-0.2

-0.1

0.0 0.1 VSD (mV)

0.2

0.3

Figure 4.30: Evolution of the Kondo resonance from stronger coupling (upper trace) to weaker coupling (lower trace) of the dot with the source and drain reservoirs.

112


the reservoirs seems to have a strong effect on the spacing of the energy levels in the QD, as was already shown in Sec. 4.6.1. The coupling strength has been decreased by applying more negative voltages to the ‘barrier’ gates (gates 6 and 8 in Fig. 4.10). A more negative voltage on this pair of gates has three simultaneous effects: • Increase the confinement of the dot. • Decrease the number of electrons in the dot. • Reduce the coupling to the leads. The increased confinement changes the energy spacing of the singleparticle levels of the dot. To balance the more negative voltage applied to the barrier gates, and still have two electrons in the dot, the plunger gate electrode (gate 7 in Fig. 4.10) has to be made less negative. These variations can have an effect also on the confinig potential of the quantum wells and, as previously observed, change the energy spacing of the first and second two-dimensional subbands. Thus the interpretation of the variations in gate voltages on the energy spectrum is not straightforward. The third effect, the reduction of the coupling of the dot with the leads, will reduce also the Kondo effect strength. In the limit of very weak coupling, the Kondo effect disappears completely. Thus a balance must be found between increased confinement and decreased tunneling rate to still be able to measure the Kondo current. Figure 4.30 shows several differential conductance traces vs. VSD . The dot-reservoirs coupling is gradually lowered from the top orange trace to the bottom blue one. All traces have been taken at plunger gate voltages midway between the N = 2 → 3 and N = 3 → 4 transitions, to insure that they are consistently in the center of the Coulomb diamond. The top trace has been taken at a coupling strength corresponding to the stability diagram of Fig. 4.21, while the bottom one is in the same conditions as in Fig. 4.22. The Kondo resonance decreases in intensity and gradually evolves in a double peak structure when the coupling to the leads is decreased. Additionally decreasing the coupling leads to the complete disapperance of the peaks. The magnetic field evolution of the resonances is shown in Figs. 4.31 and 4.32 for the high coupling and the low coupling regime, respectively. In both figures the left panel is a magnified view of the N = 2 Coulomb diamond plotted with enhanced contrast to clearly show the resonance. The central panels of both figures are colorplots of the differential conductance as a function of source-drain bias (left axis) and magnetic field (bottom axis). In the strongly coupled case of Fig. 4.31 the single peak intensity initially decreases with magnetic field, and then (above 2 T) splits in a double peak whose separation increases with B. The peak at positive bias is much more


113

0.4

0.8

60

0.6 50

VSD (mV)

VSD (mV)

0.2 0.0

0.0

25 20 15 10

-0.2 -0.4

dI/dVSD (mV)

0.2

0.4

-0.2

50 0

-0.8

10

-0.4

-620

-610

-600 -590 Vg (mV)

-580

30

20

100 -0.6

40

0

2

4

6 B (T)

8

10

12

-0.4

-0.2

0.0 0.2 VSD (mV)

0.4

Figure 4.31: Evolution of the two-electron dot Kondo resonance with increasing magnetic field, when the dot is strongly coupled to the leads. (a) the N = 2 diamond of the stability diagram. (b) Source-drain voltage sweeps taken along the black line of (a), as a function of magnetic field. The dotted lines are a guide to the eye to follow the peaks. (c) Traces taken from (b) from 0 to 7 T with 1 T step. The dotted lines are a guide to the eye.

0.8 0.6

20 15 10 5 0

0.4

0.4

7 6 5 4 3

14

12

0.2

0.0

dI/dVSD

10

VSD (mV)

VSD (mV)

0.2 0.0

8

-0.2 6

-0.2

-0.4 -0.6

4

-0.4

-0.8 -510 -500 -490 -480 -470 -460 Vg (mV)

0

2

4

6 B (T)

8

10

12

-0.4

-0.2

0.0 VSD (mV)

0.2

0.4

Figure 4.32: Magnetic field evolution of the split Kondo resonance of the dot when weakly coupled to the leads. (a) The N = 2 diamond. (b) Source-drain voltage sweeps taken along the black line of (a), as a function of magnetic field. The dotted lines are a guide to the eye to follow the peaks. (c) Traces taken from (b) from 0 (bottom trace) to 10 T (top trace) with 1 T step. The traces have been offset vertically for clarity. The dotted lines are a guide to the eye.

intense than that at negative bias. This could be due to asymmetries in the coupling of the dot states with the source and drain leads. In the weakly coupled case of Fig. 4.32, the two resonances are very weak, but appear to increase their separation with B as in the strongly coupled regime. An explanation of the presence of a Kondo resonance in a two-electron system can be found in the electronic structure of our few-electron quantum dot. The spacing, ∆E1 , between the first two single-particle level in the coupled regime is ∼ 500 µeV (see Table 4.2). This value is rather small if compared to the typical values (& 1 meV) found in GaAs lateral QDs [85, 86, 87, 88]. The energy difference between singlet and triplet states is ∆E −Eex , where Eex is the exchange energy. The typical exchange energies between two electrons in a dot is in the few hundreds µeV range [70]. Assuming that in our dot the exchange energy is only slightly smaller than ∆E, then the

114

Summary of quantum dots results

triplet state could be quasi-degenerate with the singlet ground state. Under this conditions all four states (three triplet and one singlet) could give raise to a Kondo effect, as it was shown by S. Sasaki and coworkers in a recent experiment for a six-electron quantum dot at finite B [71]. In the work by Sasaki et al., the Kondo resonance due to singlet-triplet degeneracy was split by a variation of ∆E with B. In our two-electron dot the B-evolution of the singlet-triplet Kondo resonance follows a similar trend. With the dot set in the lower coupling regime, the excited state in the N = 0 → 1 transition has an energy of ∆E1 ∼ 800 µeV, with an increase of ∼ 300 µeV with respect to the less confined state. This increased level spacing is not balanced by the exchange energy any more, so the singlettriplet degeneracy is partially lifted. The spacing of the double Kondo peaks of Fig. 4.30 is a measure of the singlet-triplet energy difference. This spacing is approximately 130 µeV and corresponds to twice the quasi-degenerate level spacing. Thus the singlet-triplet energy spacing for the weakly coupled regime, if our hypothesis are true, is ∼ 60 µeV.

4.7

Summary of quantum dots results

The main features we have observed in our quantum dots are: • An extremely small –if any– Zeeman splitting of the single-particle levels in few-electron quantum dots formed in both the single and double quantum well structures. • Due to the absence of Zeeman splitting, the spin-1/2 Kondo resonance for three electrons on the dot does not split in magnetic field but it is nonetheless suppressed. • A Kondo resonance in a two-electron quantum dot: this result is explained in terms of singlet-triplet quasi-degeneracy. This quasidegeneracy in our sample is due to the weak confinement of the dot leading to a relatively small single-particle level spacing, comparable with the exchange energy. This contrasts with previous results on lateral GaAs QDs.


115

Figure 4.33: Schematic draw of a top-down view of a Hall bar mesa (blue) with ohmic contacts (yellow), crossed by a metal gate (grey). Left and right drawings are schematic cross sections of the mesa without and with the gate, respectively. The red circles highlight where the 2DEG in a In0.75 Ga0.25 As QW comes in electrical contact with the metal gate.

4.8

Schottky gates on In0.75 Ga0.25 As samples

Even though the quantum dots fabricated in In0.11 Ga0.89 As QW samples have given extremely interesting results, we have found no increase in the electron g-factor. It is important, thus, to be able to fabricate few-electron quantum dots in the high indium content QWs whose optimization has been described in chapter 3 and 4 of this thesis. This would clarify the role of confinement in the determination of the spin dynamics in few-electron QD. However, the definition of the surface gates on a In0.75 Ga0.25 As heterostructure requires that metal fingers are brought from outside the mesa on top of it. This means that, even if the upper In0.75 Al0.25 As barrier gives enough isolation between the metal gate and the 2DEG, on the mesa border the In0.75 Ga0.25 As quantum well gets in close proximity to the metal gates. It has already been pointed out, in chapter 3, that the Schottky barrier of In0.75 Ga0.25 As is extremely low, of the order of a few meV at the most. The advantages of this are clear: no need of alloying the metal ohmic contacts, very clean interfaces between the 2DEG and the metal or superconducting contacts, and so on. When it comes to gate fabrication, however, the low Schottky barrier is a big problem. Figure 4.33 shows schematically an etched mesa of a Hall bar with its ohmic contacts. A metal gate running from its big bonding pad on the left all the way to the top of the mesa is also schematically depicted. On the sides, two cross-sections are shown, one with no metal gate (left) and the other where the metal gate crosses the mesa (right). The green line represents the location of the quantum well where the 2DEG is formed. In the right cross-section the red circles mark the areas where the metal of the gate is shorted to the 2DEG in the absence of a Schottky barrier. To avoid this shorting, one could deposit a dielectric layer between the sample surface and the gates, but this would increase the separation between the gates and the 2DEG. Typical thickness for such dielectric would be of the

116


Figure 4.34: Cartoon showing the fabrication steps needed to deposit a gate on the surface of a In0.75 Ga0.25 As mesa without shorting it to the 2DEG (see text).

order of 100 nm in the case of SiO2 , effectively doubling the distance of the gates to the 2DEG for a ∼ 100 nm deep QW. This results in a unacceptable loss of resolution in the definition of the depleted regions of the 2DEG by the top metal gates. So we decided to try to bring the gates on the surface of the hall bar without touching the mesa border. The first thing that we have controlled was that the insulation of a top gate from the 2DEG guaranteed by the In0.75 Al0.25 As upper barrier was sufficient. To do this we have prepared some samples as schematically depicted in Fig. 4.34: • a) Fabricate Hall bars with ohmic contacts to the 2DEG. • b) Protect the mesa sides with a thick layer of insulator (vitrified photoresist, see Appendix A). • c) Deposit an aluminum gate reaching the Hall bar mesa. The low temperature leakage current from such metal gates to the 2DEG is below our detection threshold (∼ 1 pA) from -3 to +2 Volts of applied bias. This is satisfactory since the typical negative bias that we apply to the gates to define our nanostructures is well below 2 Volts. This is however not a satisfactory solution, since under the insulating layer the gates are too far from the 2DEG to deplete it.

4.8.1

Suspended bridges

The metal gate, to properly deplete the 2DEG and insulate the two ends of the Hall bar, needs to be in contact with the surface of the heterostructure across the whole width of the mesa. To this aim we have thus decided to drastically change the sample fabrication process, and bring the metal gates to the surface through suspended metal bridges. The basic idea is, prior to mesa etching, to fabricate the top gates with an etch-resistant metal. The gates are shaped as in figure 4.35, in which a short section has a narrow bottleneck. The width of the narrow part has to be smaller than about twice the depth of the 2DEG. After the gate fabrication, the hall bar mesa is defined as usual through optical lithography. Prior to etching the optical resist covers the Hall bar region while the areas


117

Figure 4.35: Schematic drawing of the suspended bridge fabrication. In (a) the bottleneck-shaped gate is fabricated through e-beam lithography, Cr/Au (5/40 nm) evaporation and lift-off. The top drawing is a side view, while the bottom one is a top-down view. Grey is metal bilayer and orange is the sample. In (b) the Hall bar mesa is defined with photoresist (red), through optical lithography. In (c) is shown the result after the isotropic etching. The darker regions have been etched. Panel (d) shows two cross-sections corresponding to dash-dotted lines of panel (c). The upper one is taken in the narrow part of the gate (red line), while the lower one refers to the wide part of the gate (blue line). Going from left to right the etching time increases. The green horizontal line represents the QW with the 2DEG.

to be etched are left uncovered. When performing the isotropic etching with the usual H3 PO4 :H2 O2 :H2 0 solution (see Appendix A), the sample surface is masked both by the resist and by the deposited metal gates. As shown in Fig. 4.35 (d), under the narrow part of the gates the heterostructure is etched well below the 2DEG, while the larger parts are still firmly standing on the mesa. At the end of the process, the left arm (in Fig. 4.35) of the gate reaches the far-away bonding pad, while the right arm arrives in the center of the Hall bar. In this way the metal gate is freely suspended and does not come in contact with the mesa side, and is electrically insulated from the 2DEG. Figure 4.36 shows SEM micrographs taken at the end of the fabrication

118


Figure 4.36: SEM micrographs of the suspended metal bridges. In (a) a wide view of the Hall bar mesa, A, the EBL defined metal gates, B, and the metal fingers leading to the bonding pads defined by optical lithography, C. The arrow marks the area magnified in (b)-(d), which are close-ups of one of the suspended bridges. In (b) the mesa (A) is visible. With this high tilt angle the surface corrugations due to the cross-hatch pattern (see Sec. 2.2.4) are clearly resolved. From the topdown view (d) it is possible to compare the real width and length of the bridge. In (e) is shown an image of the whole, bonded Hall bar device; the imaged area is about 1.5 mm wide.

process, at different magnification levels. As can be seen, the bridges withstand all the processing steps, the capillary forces of the drying water after the wet etching, and the strain that may accumulate in the metal deposition. They are actually suspended and do not touch the mesa border. We have tested the electrical behavior of a device like those shown in the SEM micrographs: a 3 µm wide metal stripe running across a Hall bar. By negatively polarizing the gate we have been able to completely deplete the underlying 2DEG without having any measurable leakage from the gate. Figure 4.37 shows such measurements. A fixed 100 µV bias is applied between the source and the drain of the 2DEG, at the two ends of the Hall bar. A varying negative voltage is applied to the top gate (the metal stripe), while measuring both the source-drain current through the 2DEG (red trace) and the leakage current from the top gate to the drain (green trace). For a gate voltage of ∼ −900 mV the source-drain current falls to zero indicating that the 2DEG underneath the gate is completely


119

Figure 4.37: Gate leakage current (green squares) and 2DEG source-drain current (red circles) as a finction of the votage applied to the gate. On the left, a schematic representation of the measurement setup.

depleted. At the same time no measurable current flows from the gate to the 2DEG, even at more negative gate voltages, showing that the suspended bridge design effectively insulates the metal gate from the 2DEG. This demonstrates that it is possible to fabricate top metal gates on In0.75 Ga0.25 As/In0.75 Al0.25 As quantum wells through electron beam lithography. Such gates do not show detectable leakings, and are capable of depleting the underlying 2DEGs.

120


Conclusions In this thesis we have followed a route leading to the fabrication of mesoscopic devices on Inx Ga1−x As-based two dimensional electron gases. A great effort has been made in the investigation of the structural and transport properties of unintentionally doped In0.75 Ga0.25 As/In0.75 Al0.25 As quantum wells grown by molecular beam epitaxy on GaAs (001) substrates, in order to realize two dimensional electron gases with high electron mobility at low temperature. Due to the large lattice mismatch between the active In0.75 Ga0.25 As layer and the GaAs substrate, a step-graded buffer layer structure was employed to adapt the two different lattice parameters. In chapter 2 we have tested different buffer structures. We have found that the presence of an overshooting layer with suitable indium content on the top of the buffer can strongly reduce the residual strain in the In0.75 Ga0.25 As quantum well region. Corresponding to this strain reduction, an increase of the electron mobility up to 29 m2 /Vs is reached. A three-fold decrease of the mobility is, instead, observed for a structure subject to a compressive strain higher than 1.9 × 10−3 , while a much stronger deterioration of the low temperature transport properties is measured in case of tensile residual strain. In this last case, the strong decrease in the low temperature mobility is associated to the formation of extended defects, revealed by the presence of deep grooves on the surface. In order to improve the low temperature transport properties, the scattering processes limiting the electron mobility in our quantum wells are inferred from the dependence of the low temperature electron mobility on the carrier density. In chapter 3, we have shown that for carrier densities lower than 2 × 1015 m−2 the dominant scattering source is represented by the Coulomb field caused by an ionized impurity background. For higher carrier density, alloy disorder scattering is no more negligible and reduces the electron mobility. By eliminating intersubband scattering and inserting a thin (4 nm-thick) strained InAs layer inside the InGaAs well to reduce the effect of alloy disorder, we have been able to reach electron mobilities in excess of 50 m2 /Vs, with carrier densities well below 5 × 1015 m−2 . A large mobility anisotropy has been found in the highest mobility samples, and its origin has been investigated with the aid of numerical simulations of a model system.

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Conclusions

The results on transport measurements on In0.11 Ga0.89 As few-electron quantum dots have been shown in chapter 4. The MBE growth of two structures containing a In0.11 Ga0.89 As quantum well has been optimized to allow the definition of few-electron lateral quantum dots. We have fabricated, by electron beam lithography, the metal gates necessary to define small quantum dots on our samples. Low temperature transport measurements on the dots in the few-electron regime have been carried out, focusing on the oneelectron state, with the aid of a quantum point contact used as a charge detector. Under an applied magnetic field the single particle energy levels of the dots exhibit a nearly-zero Zeeman splitting. This has given us the opportunity to study, in the strongly coupled regime, the uncommon suppression of a spin-1/2 Kondo resonance by the magnetic field in the absence of Zeeman splitting. Moreover, we have observed the Kondo effect in a twoelectron dot. This had never been previously reported. We have ascribed the presence of the Kondo resonance to a singlet-triplet quasi-degeneracy for the two-electron state in zero magnetic field. Besides the work on the In0.11 Ga0.89 As quantum dots, we have developed a process that allows the fabrication of lateral quantum dots also in the In0.75 Ga0.25 As samples. The work on In0.11 Ga0.89 As few-electron quantum dots is still ongoing. In particular, measurements of the transport characteristics of a few-electron In0.11 Ga0.89 As quantum dot coupled to a large, many-electron dot formed in a GaAs triangular well are proceeding. Given the nearly-zero g-factor of the In0.11 Ga0.89 As dot – and the expected nonzero one of the GaAs dot – to study the coupling of the spin states of the two dots is expected to yield interesting results. Moreover, we are further investigating the mechanism responsible for the suppression of the spin-1/2 Kondo resonance under magnetic field. Furthermore, we are starting to fabricate devices on In0.75 Ga0.25 As, to observe the behavior of the electron spin in magnetic field in a high indium content Inx Ga1−x As alloy.

Appendix A

Optical lithography recipes This appendix describes the recipes that our group generally uses for defining Hall bars and other structures by optical lithography on our InGaAs samples. They have been developed over the years and it is hard to give credits to specific persons. It is more of an oral cleanroom tradition passed on from generation to generation. Cleaning in what follows means rinsing sample in running acetone, wash acetone away with alcohol (either methanol or isopropanol), and blowing away the alcohol by blowing dry nitrogen gas on the sample.

A.1

Mesa etching

• Clean sample • Spin a ∼1.5 µm thick layer of Shipley S1818 photoresist at 5000 rpm for 1 minute • Bake the resist at 90◦ for 10 min on a covered hot plate • Expose the pattern to UV • Develop for ∼30 sec in Shipley MF319 at room temperature • Rinse in de-ionized (DI) water and blow dry with nitrogen gas • Etch in H3 PO4 :H2 O2 :H2 O 3:1:50 at room temperature (22-24◦ ) gently stirring the sample • Rinse in DI water and dry • Clean sample (to remove photoresist) The etching solution has the following etching rates:

124

Ohmic contacts

Material GaAs Alx Ga1−x As (x 1 hour), and then flushig energycally with acetone.

B.2

Suspended metal bridges lithography

The starting point for this process is a sample with ohmic contact pads and side bonding pads for the gates, defined by optical lithography. Alloyed gold marks allow to align the EBL to the optically patterned pads. No mesa is etched before EBL. Sample cleaning is as already described in the optical lithography description in Appendix A. Resist As resist I have used a 5% solution of 950K PMMA diluted in chlorobenzene. The sample is pre-baked at 100◦ C for 5 min. on a hot plate to favour adhesion of the polymer, then it is spin-coated at 5000 RPM. The solvent is evaporated by a 5 minutes bake on a hot plate at 175◦ C. The resulting thickness of the PMMA layer is ∼ 400 nm.

E-beam lithography recipes

129

Exposure The resist is exposed to a dose of 150 µC/cm− 2 using a 30 keV electron beam. The smallest fenditure (5 µm, yelding a current of ∼ 30 pA) has been used for the whole pattern. Development Development was 40 seconds in MIBK:IPA 1:3 at ∼ 20◦ C, followed by a 30 second rinse in IPA, dried with nitrogen gas. No plasma cleaning or descumming was performed. Metal evaporation and lift-off The metal has been evaporated with electron-beam heated sources, at pressures lower than 5 × 10−6 mbar. A bilayer of Cr and Au (50 nm) has been ˙ deposited at rates ∼ 2 A/sec. Both metals resist well to a H3 PO4 :H2 O2 :H2 O wet etch. Chromium has good adhesion on In0.75 Ga0.25 As and Au is an excellent conductor. Lift-off was done by keeping the sample in acetone for a long time (> 1 hour), and then flushig energycally with acetone.

130

Suspended metal bridges lithography

Appendix C

Other publications During my PhD, along with the work described in this thesis and focused on transport experiments in InGaAs QWs, I have been involved in a line of research that has stemmed from my undergraduate thesis on local anodic oxidation of GaAs samples. This work has focused on the chemical and structural characterization of nanometer sized oxide patterns fabricated by locally applying a bias voltage between the tip of an AFM and a GaAs or AlGaAs sample. This is a widely used technique for defining mesoscopic devices (quantum point contacts, quantum dots, Aranov-Bohm rings, etc.) on GaAs/AlGaAs high mobility 2DEG samples. The main steps of this work have been: • Fabricate a working QPC through local anodic oxidation, and measure its low temperature transport properties in high magnetic fields [90]. • Chemically characterize the fabricated oxide through spatially resolved photoelectron emission spectroscopy experiments using synchrotron radiation. We discovered a pronounced instability of the oxides when exposed to an intense flux of extreme ultraviolet photons. This instability causes a fast oxide desorption. We ascribed this behaviour to the weakening of the chemical bonds due to an Auger-like two-electron mechanism [91]. • Characterize the dynamics of the oxide desorption process and acquire chemical information on the oxides at the early stages of desorption through a time resolved photoemission electron spectroscopy experiment [92]. • Test the currently accepted model for oxide formation during the local anodic oxidation process through photoemission electron spectroscopy, revising the model and proposing an alternative model consistent with our results [93, 94].

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• Characterize the oxides fabricated on silicon and silicon dioxide substrates with the same local oxidation but with reversed bias, showing, also in this case, the inadequatedness of the currently accepted model of oxide formation, and pointing out that also silicon oxides show an instability under extreme ultraviolet photon fluxes [95, 96]. My contribution to this project has been, in the first stages, determinant: preparation and AFM characterization of the samples, participating during the synchrotron beam-times in the measurements shifts, analysis of the acquired data and discussion of the results. As I got more involved in the InGaAs project, others have taken the lead in the oxide characterization, but I kept participating in the synchrotron measurements and in the discussion of the results.

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