# Microwave spectroscopy of edge and bulk modes of two dimensional electrons in magnetic field

код для вставкиСкачатьTHE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES MICROWAVE SPECTROSCOPY OF EDGE AND BULK MODES OF TWO DIMENSIONAL ELECTRON SYSTEMS IN MAGNETIC FIELD By BRENDEN A. MAGILL A Dissertation submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Spring Semester, 2013 UMI Number: 3564920 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3564920 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 Brenden A. Magill defended this dissertation on November, 19, 2012. The members of supervisory committee were: Lloyd Engel Professor Directing Dissertation Geoﬀ Strauss University Representative Nick Bonesteel Professor Co-Directing Dissertation Peng Xiong Committee Member Ingo Wiedenhoever Committee Member The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements. ii This dissertation is dedicated to my father, who raised me to be persistent and taught me to be a gentleman. iii ACKNOWLEDGMENTS There were a great many people who helped me become the person who I am today. I want to start by thanking my parents Michael Magill, Mary Cooney, and my stepmother Lucinda Phobes. They have all been there when I have needed them and I have learned a great deal from them. From my Dad I learned how to be persistent and not quit. My mom kindled in me a love of math and taught me to back up my statements. I remember being ten and her asking me if I could document that statement and having to walk away only to return a week later with a statistic to back up my claim. Even though I do not recall now what claim I had made, the lesson stuck with me. I want to thank my stepmother for teaching me how important it is to learn the rules and work within a bureaucracy instead of just railing against it. Over the course of my education I have had the pleasure of meeting and learning from a lot of wonderful teachers. From my undergraduate I want to thank Saul Oseroﬀ who took me on as his assistant and then as a student He was responsible for me getting to do research on super-heavy fermion superconductors as an undergraduate and inculcating in me a love for lab work. From him I will never forget the lesson that the disorder in a system is never zero and often is of fundamental importance to the physics of the system. I want to thank Richard Morris who kindled in me a desire to be an experimentalist and taught me to set up a good experiment and showed me that I could respond to tough challenges. In graduate school I want to start by thanking my advisor Lloyd Engel who has gone above and beyond the call of duty in teaching me and molding me as a physicist. He has worked hard to teach me to not jump to conclusions and how to evaluate whether an experiment has succeeded or not. iv I also want to thank him for standing by me through some tough projects. I want to thank Nicholas Bonesteel for being my advisor on record with FSU and putting up with all the administrative hassles that brought. I want to thank the post docs, Rupert Lewis, Sanbandamurthy Granpathy, Pei-Hsun Jiang, Byoung Hee Moon, and Anthony Hatke who were wonderful to work with and learn from. Murthy I want to thank for teaching me what it takes to survive as an experimentalist. Byoung I want to thank for teaching me about cryogenics and the dilution refrigerator. I want to give special thanks to Anthony for helping edit my thesis and for helping me ﬁnish up some of the work included here. I would like to thank the students I have worked with in the Engel group: Yong Chen, Zhigong Zhang, Han Zhu, and Shantanu Chakraborty. Han I want to thank for the good times working with him and the company late in the night when we were the only people in C120. Shanto I want to thank for being an eager collaborator on the WQW experiment that the second half of this dissertation is about. Without him I would not have been able to get so much done so quickly and I likely would not be writing this dissertation this year. I also want to thank all of my friends and classmates. Aaron Wade who I took all of my classes with, worked with on many projects, and who I will be pumping on information about Terahertz measurements and self assembled metal nano rods. I want to thank Adrienne Serra, Katherine Eastman, and Melissa Hall for helping me edit and being there to say that paragraph looks really awkward can you try to make it a little less horrible. Finally, I would like to thank my wife Tadja. She has been my rock through many years of graduate school, she has supported me through hard work that had be coming home tired and confused, through technical problems that had me pulling my hair and gnashing my teeth, and through setback that had me moping and miserable. Without her I would not be the happy and presumably well adjusted person I am today. v TABLE OF CONTENTS List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction 1.1 Spectroscopy of two regimes . . . . . . . . . . . 1.2 Magnetic edge magnetoplasmons . . . . . . . . 1.3 A new solid phase in “wide” quantum well two systems . . . . . . . . . . . . . . . . . . . . . . 1.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dimensional . . . . . . . . . . . . . . 2 Magnetic edge magnetoplasmons 2.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.2 Plasmons and Magnetoplasmons . . . . . . . . . . 2.3 Edge Magnetoplasmons . . . . . . . . . . . . . . . 2.4 Inter-Edge Magnetoplasmon . . . . . . . . . . . . 2.5 A Semi-Classical Model of Magnetic Conﬁnement 2.6 Magnetic Edge Magnetoplasmon . . . . . . . . . . . . . . . . . . . . . . 3 Magnetic Edge Magnetoplasmons – Experimental sults 3.1 Introduction to hybrid ferromagnet 2DES devices . . 3.2 Microwaves connections and mounting . . . . . . . . 3.3 Introduction to gate controlled non-resonant devices . 3.4 Sample characteristics . . . . . . . . . . . . . . . . . 3.5 NRMEMP device fabrication and design . . . . . . . 3.6 NRMEMP devices . . . . . . . . . . . . . . . . . . . 3.7 NRMEMP device results . . . . . . . . . . . . . . . . 3.8 NRMEMP Summary . . . . . . . . . . . . . . . . . . 3.8.1 A simple EMP model with nonzero ∂Bz /∂x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 2 4 6 6 7 7 10 11 13 Setup and Re. . . . . . . . . 15 15 15 16 17 17 19 19 24 25 4 Wigner crystals and other electron solids 4.1 The Quantum Hall Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Wigner Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Pinned Wigner Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 29 29 31 32 vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 4.5 4.6 Integer quantum Hall Wigner crystals . . . . . . . . . . . . . . . . . . Other electronic solids . . . . . . . . . . . . . . . . . . . . . . . . . . The reentrant integer quantum Hall eﬀect in 2DES . . . . . . . . . . 5 Wide Quantum Wells 5.1 Introduction . . . . . . . . . . . . . . . 5.2 Two subbands . . . . . . . . . . . . . . 5.3 Measuring the subband separation (Δ) 5.4 Δ in the vicinity of ν = 1 . . . . . . . 5.5 A RIQHE near ν = 1 in WQW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 35 37 40 40 41 43 44 46 6 Microwave spectroscopy of the RIQHE around ν = 1 in WQW 2DES 6.1 Samples and Sample Preparation . . . . . . . . . . . . . . . . . . . . 6.2 Microwave Measurement Technique – CPW . . . . . . . . . . . . . . 6.3 54 nm WQW – microwave spectroscopy of pinning mode resonances around ν = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Sample 54-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Sample 54-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 54 nm WQW – summary . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 42 nm WQW – pinning mode resonances around ν = 1 . . . . . . . . 6.5.1 42 nm WQW – symmetric charge distribution . . . . . . . . . 6.5.2 42 nm WQW – asymmetric charge distribution . . . . . . . . 6.5.3 42 nm WQW – summary . . . . . . . . . . . . . . . . . . . . . 6.6 Summary and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 47 47 48 7 Signiﬁcance and future work 7.1 EMPs in magnetic ﬁeld gradients and MEMPs . . . . . . . . . . . . . 7.1.1 Signiﬁcance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Future work, EMPs in magnetic ﬁeld gradients and MEMP search 7.2 A new electron solid in a WQW . . . . . . . . . . . . . . . . . . . . . 7.2.1 Signiﬁcance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Future work in WQW 2DES . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 76 76 77 78 78 79 81 Biographical Sketch 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 51 51 56 61 63 64 65 68 71 LIST OF FIGURES 1.1 2.1 2.2 2.3 3.1 3.2 3.3 Example of a pinning mode resonance in an ultra high mobility 65 nm quantum well. Plotted are spectra showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency, with the density (n) for each spectrum labeled on the graph. Adapted from Ye et al. [1] . . . . . . . 3 Comparison of the EMP and magnetoplasmon modes for a 10 μm disk of GaAs semiconductor 2DES with n = 2.8 × 1011 cm−2 . The higher frequency branch on the graph is the magnetoplasmon mode (labeled MP), the lower frequency branch the EMP (labeled EMP), and the dashed line the cyclotron frequency (labeled CR). . . . . . . . . . . . . . . . . . . 8 “Skipping” orbit depicted as a series of broken cyclotron orbits conﬁned to the edge of a sample, n = 0. . . . . . . . . . . . . . . . . . . . . . . 12 a) “snake” orbit depicted as a succession of cyclotron orbits reversing direction where Bz inhomogenously changes from B0 to −B0 . b) “cycloid” orbit depicted as a series of cyclotron orbits tightening in radius in the region where Bz is increased by Bm . . . . . . . . . . . . . . . . . . . . 13 Illustration of NRMEMP device the 2DES mesa grey and the contacts dark grey, all of the features on the are device to scale except for the ends of the microwave antennas are truncated. Note: the slight overlap of the tips of the microwave antennas over the edge of the mesa containing the 2DES is not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 All four device conﬁgurations for positive B. For each conﬁguration the charged gates are colored and the grounded ones are white. The red dotted line shows the path of the EMP and the blue dashed line shows the intended path of the MEMP. . . . . . . . . . . . . . . . . . . . . . 20 All four device conﬁgurations for negative B. For each conﬁguration the charged gates are colored and the grounded ones are white. The red dotted line shows the path of the EMP. . . . . . . . . . . . . . . . . . 21 viii 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.6 5.1 Transmitted power in dB vs B taken at 4 K. Gate that the EMP is incident, Py (permalloy) for positive B and Au for negative B on the graph with the on voltage the gate relative to the 2DES in parentheses next to it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Transmission dip number j (i.e. harmonic) plotted vs B, gate conﬁguration number listed in the caption. The EMP launched from the transmission antenna is incident on the Au gate for B < 0 and the (Py) permalloy gate for B > 0 as marked on the graph, lines on the graph are least squares ﬁts. Conﬁguration numbers and frequencies are shown on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Density of states for a 2DES showing disorder-broadened LL spectrum. g the Lande g-factor and μB the Bohr magneton. The Fermi level is as shown for the LL ﬁlling factor ν = 3. . . . . . . . . . . . . . . . . . . . 30 Spectra showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency, f , at many ﬁlling factors ν oﬀset vertically proportional to ν. Successive spectra are separated by steps of 0.01 in ν ν for black spectra are marked at right, in a 30 nm wide quantum well 2DES, and at 30 mK. Adapted from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Example of S/fpk plotted vs ν for an IQHEWC pinning mode resonance in a 30 nm wide quantum well 2DES at 50 mK. The black lines are the prediction from the sum rule Eq. 4.2 for full quasi-carrier participation. Apdated from Chen et. al. [2]. . . . . . . . . . . . . . . . . . . . . . . 34 Cartoon of the progression from an IQHEWC to a bubble phase WC and then a stripe phase WC as ν ∗ increases, with M the number of electrons per lattice point in the WC. . . . . . . . . . . . . . . . . . . . . . . . . 35 Re[σxx ] vs f for a very high mobility, μ = 24 × 106 cm2 V −1 s−1 , n = 1.0 × 1011 cm−2 , 50 nm QW, and a temperature of 30 mK. Resonance A shows dispersion with wave vector. Adapted from Chen et. al. [3]. . . 36 Example of the RIQHE observed by Cooper et. al. [4] near ν = 4 as ν increases with the RIQHE marked by the arrow, a) Rxy in units of h/e2 , and b) Rxx in units of Ohms. . . . . . . . . . . . . . . . . . . . . . . . 38 Charge distribution (ρ) on top in red, ground (ΨS in green) and ﬁrst excited state (ΨAS in blue) wave functions on bottom. From simultaneous calculations of the Poisson and Schroedinger equations. Diﬀerence in density between the top half and the bottom half of the well (δn) on top. n in 1011 cm−2 and subband separation (Δ) in K on bottom. Provided by Yang Liu. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ix 5.2 Re[σxx ] in μS plotted vs B at a frequency of 0.5 GHz, from a 54 nm WQW, a temperature of 30 mK, and an as cooled n = 2.41 × 1011 cm−2 . 43 5.3 B frequency peaks from the Fourier transform of the SdH oscillations with the B frequencies of the lowest lying state, the ﬁrst excited state, the sum of both, and the diﬀerence of both marked on the ﬁgure. . . . 44 Fan diagram for a 54 nm WQW (left axis) for Δ = 27.6 K. The energies of the N = 0 Landau level from the lowest lying (S0) and ﬁrst excited (A0) subbands are plotted with their spin directions shown. On the right axis a Re[σxx ] vs B trace taken at 500 MHz is plotted for n = 2.43 × 1011 cm−2 and the same Δ. ν = 1 is labeled on the ﬁgure. . . . . . . . . . . 45 Schematic of a coplanar waveguide on the surface of a sample, with metal in black. The 2DES is a fraction of a micron under the surface of the sample. An Agilent network analyzer is the source and the receiver of the microwave signal. . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Gate conﬁgurations for the WQW samples. a) back gate only: samples 54-1 and 54-2. b) front and back gate with glass spacer and increased CPW slot width: sample 42-1. c) front and back gate with etched glass spacer: sample 54-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Real part of diagonal conductivity (Re[σxx ]) vs B for sample 54-1 taken at a frequency of 500 MHz. The temperature is 30 mK and the density is 2.41 × 1011 cm−2 . The sample has no front gate, and there was no voltage bias on the back gate. A selection of IQHE and FQHE minima are labeled with their Landau ﬁlling factors. . . . . . . . . . . . . . . . 52 Δ vs n for the 54 nm WQW, illustrating how the subband separation changes as we change n with the back gate. n = 2.41 × 1011 cm−2 is the as-cooled density with no gate bias. . . . . . . . . . . . . . . . . . . . . 53 Re[σxx ] vs B for n = 1.94 × 1011 cm−2 to n = 3.20 × 1011 cm−2 , each trace vertically oﬀset for clarity. Taken at ∼ 30 mK and a frequency of 0.5 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 a) Spectra from sample 54-1 showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency taken at many ν. ν of blue traces is marked on the right, at a temperature of 30 mK, n as-cooled, and normalized to ν = 1 with each spectrum proportionally oﬀset vertically from the last and the ﬁlling factor step between each spectra is ν = 0.005. b) The peak frequency of the pinning mode resonance (fpk ) vs. ν. c) S/fpk vs ν. The black lines are the theoretical prediction for full participation in the resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4 6.1 6.2 6.3 6.4 6.5 6.6 x 6.7 Waterfalls of spectra from sample 54-1 show the real part of the diagonal conductivity (Re[σxx ]) vs frequency taken at many ν. ν of blue traces marked on the right, normalized using a ν = 1 spectrum, n marked at the top of each waterfall, and each is spectrum is proportionally oﬀset vertically. a) and b) The step in ν between spectra is 0.01. c) The step in ν between spectra is 0.005. There was no data taken above ν = 1.05 for n = 2.17 × 1011 cm−2 . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Waterfalls of spectra from sample 54-1 showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency, taken at many ν. ν of blue traces marked on the right, 30 mK, normalized using a ν = 1 spectrum, n marked at the top of each waterfall, and each spectrum is proportionally oﬀset vertically. a) and b) The step in ν between spectra is 0.01. c) The step in ν between spectra is 0.005. . . . . . . . . . . . . 57 6.9 fpk in GHz vs ν as n is increased, from Figs. 6.7 and 6.8. . . . . . . . . 58 6.10 Sample 54-1, S/fpk vs. LL ﬁlling factor for all measured n. (See section 4.3.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 a-c) Re[σxx ] vs f at many ν and T = 30 mK. ν of black traces marked on the right. a) Spectra from ν = 0.8 to 1.2 with a ν step between each spectrum of ν = 0.0143. b) Spectra from ν = 0.8 to 1.0 with a step of ν = 0.0111 between spectrum. c) Spectra from ν = 1.0 to 1.2 with a step of ν = 0.0091 between each spectrum. . . . . . . . . . . . . . . . . 60 Image plots of f vs ν with the color representing Re[σxx ] for a variety of n. n listed on in image plot in units of 1011 cm−2 . . . . . . . . . . . 61 fpk vs ν, n is listed along the right of the plot in units of 1011 cm−2 . All fp k taken from spectra taken while maintaining symmetric charge distribution in the well. . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Location of the transition to a higher frequency range in ν vs n for fpk vs ν for samples 54-1 and 54-3 for ν < 1. . . . . . . . . . . . . . . . . . 63 Sample 42-1, 50 mK, n = 3.04 × 1011 cm−2 , and no bias on either gate. a) Many spectra taken at a diﬀerent ν normalized to ν = 1. ν is marked on the left of the graph for black spectra, spectra are proportionally vertically oﬀset, and the step between spectra is ν = 0.00625. b) fpk vs. ν. c) S/fpk vs. ν. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.8 6.11 6.12 6.13 6.14 6.15 xi 6.16 6.17 6.18 6.19 Sample 42-1, waterfalls of spectra taken at many ν. ν of black traces are shown on the left, oﬀset vertically proportional to ν, taken at 50 mK, normalized to ν = 1, at similar Δ, with n at the top of each waterfall, and a step of ν = 0.00625 between spectra. . . . . . . . . . . . . . . . . 66 Sample 42-1 fpk vs. ν, traces are oﬀset vertically by 0.5 GHz per trace with n = 3.58 not oﬀset, and n is in units of 1011 cm−2 . . . . . . . . . . 67 Sample 42-1, waterfalls of spectra taken at 50 mK, normalized to a ν = 1 spectrum. ν of black traces is shown on the left, oﬀset vertically proportional to ν, a step of ν = 0.00625 between spectra, and n 3.25 × 1011 cm−2 . a) Asymmetric charge distribution, with gates set to give δn = 0.22 × 1011 cm−2 . b) Symmetric charge distribution, gates balanced. c) Asymmetric charge distribution, gates set to give δn = −0.22 × 1011 cm−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Image plots of f vs. ν with the intensity of the plot representing Re[σxx ] from sample 42-1. n and δ in units of 1011 cm−2 as marked on each plot; plots on the left have symmetric charge distribution in the well (δn 0); plots right of the thick vertical line have asymmetric charge distribution with postive δn corresponding to more charge added by the front gate. 69 6.20 Sample 42-1, with fpk vs. ν on the left, and S/fpk vs. ν on the right. Symmetric states are black; non-symmetric with the voltage biased more on the back gate in blue, non-symmetric with the voltage biased towards the front gate in red; with the densities on the graph in units of 1011 cm−2 . 70 6.21 Sample 54-1, (Re[σxx ] vs f ) taken at n = 2.41 × 1011 cm−2 , and various temperatures as marked on graph. a) taken from a region of ν near ν = 1 where fpk is not enhanced. b) from a region of ν where fpk is enhanced. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 fpk vs. ν of the pinning mode for samples 54-1 (circles) and 54-2 (squares). Both from spectra taken at n = 2.41 × 1011 cm−2 . . . . . . 75 6.22 xii ABSTRACT Edge magnetoplasmons [5] (EMPs) and pinning mode resonances [6] in two dimensional electron systems (2DESs) can both be thought of as lower hybrid modes of cyclotron and plasma resonances. This dissertation describes low temperature microwave spectroscopy of both of these modes. EMPs have oscillating charge conﬁned at the 2DES edge by the combination of the perpendicular magnetic ﬁeld and the electrostatic potential that produces the edge. Pinning mode resonances are from electron solids oscillating against conﬁnement provided by disorder in the bulk of the 2DES. The ﬁrst part of this dissertation concerns the search for a mode similar to an EMP but conﬁned solely by a linear magnetic inhomogeneity in the perpendicular magnetic ﬁeld (Bz ). While we do not observe such an excitation, we do observe a marked reduction in the velocity of an EMP in the presence of a Bz -inhomogeneity. In the second part of this dissertation, we investigate pinning modes in “wide” quantum well samples, for which the eﬀective electron-electron interaction is softened at short range due to the vertical extent of the wavefunction. We observe a pinning mode resonance whose peak frequency (fpk ) vs Landau level ﬁlling (ν) shows an anomalous increase as ν moves away from ν = 1 under roughly the same conditions as anomalous quantum Hall eﬀects observed previously in DC transport [7]. A region of ν with enhanced fpk is interpreted as evidence for a new electron solid phase. xiii CHAPTER 1 INTRODUCTION 1.1 Spectroscopy of two regimes This dissertation covers two principal topics, linked by an experimental technique, low temperature microwave spectroscopy of two-dimensional electron systems (2DES) in magnetic ﬁeld. The ﬁrst topic, concerning the edges of a 2DES, is a study of charge density (plasma) excitations (edge magnetoplasmons, EMPs) at the edge of a 2DES in a spatially varying perpendicular magnetic ﬁeld (Bz ). This arises from a search for a charge density excitation bound to an inhomogeneity in Bz (a “magnetic” edge magnetoplasmon or MEMP). The second topic focuses on electron solid phases found at higher magnetic ﬁeld and milikelvin temperatures, in the 2DES bulk in comparatively wide quantum wells. Since the two principal topics in this dissertation are diﬀerent, each section will include its own literature review. 1.2 Magnetic edge magnetoplasmons Charge density excitations in 2D electron or hole systems include plasmons, magnetoplasmons, and of most relevance to this dissertation, edge magnetoplasmons (EMPs). EMPs are charge density excitations (with a frequency below the cyclotron frequency) that propagate along the edge of a 2DES [5] in a perpendicular magnetic ﬁeld (Bz ). The oscillating charge density of an EMP is conﬁned by the combination of Bz and the static electric potential that causes the electron density in a 2DES to 1 go to zero at its edge. Steps or gradients in Bz (x) can also conﬁne electron orbits without any electrostatic potential [8]. In this dissertation, we describe a search for “magnetic edge magnetoplasmons” (MEMPs), charge density excitations analogous to EMPs, but conﬁned by only such a Bz -inhomogeneity. To create a magnetic inhomogeneity at the 2DES we pattern ferromagnetic ﬁlm onto the surface of the sample. These hybrid ferromagnet-semiconductor devices are designed to allow us both to launch EMPs and to search for MEMPs. While we found no evidence of a MEMP, which by deﬁnition would be conﬁned solely by a magnetic inhomogeneity, we observed a marked decrease in the velocity of an EMP moving along an edge with both vanishing density and a Bz gradient. The magnitude of the velocity decrease is reasonable when compared to the result of a simple EMP model, considering what we know about the magnetic properties of the ﬁlm. 1.3 A new solid phase in “wide” quantum well two dimensional electron systems An electron solid, such as a Wigner crystal [9, 10], is a lattice formed by carriers to minimize their mutual repulsion. Such solids are pinned by disorder and are insulators. At high magnetic ﬁelds the quantum Hall eﬀect series of a 2DES is terminated at high Bz by an insulating phase [11]. In samples of suﬃciently low disorder, the microwave spectrum of this insulating phase exhibits a resonance that is interpreted as a pinning mode of an electron solid [12]. The resonance frequency (fpk ) is increased by increasing the eﬀective strength of the disorder pinning the solid, and decreased by increasing the stiﬀness (shear modulus) of the electron solid [13]. Fig. 1.1 shows an example of the peak frequency of the pinning mode resonance changing as the density of the 2DES is increased. Pinning modes have also been observed for Wigner solids in the regime of the integer quantum Hall eﬀect [2], for which they have been interpreted as due to a Wigner crystal in the Landau ﬁlling (ν) range of the integer quantum Hall 2 eﬀect, an integer quantum Hall eﬀect Wigner crystal (IQHEWC). Figure 1.1: Example of a pinning mode resonance in an ultra high mobility 65 nm quantum well. Plotted are spectra showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency, with the density (n) for each spectrum labeled on the graph. Adapted from Ye et al. [1] In “wide” quantum wells (WQW), the increased vertical extent of the electron wavefunction reduces the eﬀective electron-electron interaction at short range. The softening of the short range interactions aﬀects the competition between states in which this interaction plays a crucial role, including electron solids and also the fractional quantum Hall eﬀects (FQHE) [14]. A reentrant integer quantum Hall eﬀect (RIQHE), in which an integer quantum Hall plateau vanishes then reappears as ν changes, has been observed by Liu et al. [7] in DC magnetotransport measurements. They observed the ν of this RIQHE move in toward ν = 1 as they increased the density of the electrons in the 2DES. They proposed that this RIQHE is a manifestation 3 of a new electron solid. In WQW 2DES we observe a transition in fpk vs ν, with a region of enhanced fpk appearing at ν and densities similar to those found for the RIQHE in Ref. [7]. We ascribe the enhanced fpk to a new electron solid distinct from the IQHEWC previously studied in samples at lower density [2]. We consider two possibilities for the composition of this unknown electron solid. The ﬁrst is an electron solid that has some vertical order similar to bilayer electron solids [15–17]. The second, and more likely possibility is that it is an electron solid composed of composite fermions (CFs). A CF is an exotic particle that ﬁgures importantly into descriptions of the fractional quantum Hall eﬀects and that can be thought of as an electron bound to an even number of magnetic ﬂux quanta [18–20]. 1.4 Thesis overview Chapter 2 is the introduction to our search for a charge density excitation bound to an inhomogeneity in the perpendicular magnetic ﬁeld. In this section, we will introduce edge magnetoplasmons, also touching on plasmons and magnetoplasmons in 2DES. We will also mention previous DC transport work done on devices that combine ferromagnets and 2DES. Chapter 3 presents the sample design, experimental methods, and the results of our search for a MEMP. Though we did not observe a MEMP, the same experimental setup allowed us to characterize EMPs at an edge with a Bz inhomogeneity. We found a reduction in EMP velocity in a Bz inhomogeneity that is at least plausible according to classical EMP models. Chapter 4 introduces pinned solids in systems that show quantum Hall eﬀects. These solids include the terminal insulating phase at high Bz and solids composed of quasiparticles under conditions of the integer quantum Hall eﬀect. We also discuss 4 more exotic electron solids that may be relevant to our results. Finally, we introduce the reentrant quantum Hall eﬀect. Chapter 5 is an introduction to WQW 2DES. We focus on using the Shubnikov-de Hass (SdH) oscillations to get information on the vertical charge distribution in the wells. Also, we describe the speciﬁc DC transport measurements that are directly related to experiments described in the next chapter. Chapter 6 covers our microwave measurements of WQW 2DES. We give an overview of our microwave measurement technique and present data taken from samples with 42 nm and 54 nm WQWs. We interpret the data as evidence of a new electron solid and discuss the possible nature of this solid. 5 CHAPTER 2 MAGNETIC EDGE MAGNETOPLASMONS 2.1 Introduction Charge density excitations in 2D electron or hole systems have long been a subject of considerable theoretical [5, 21–30] and experimental [31–49] interest. Such charge density excitations include plasmons, magnetoplasmons, and of most relevance to this dissertation, edge magnetoplasmons (EMPs). EMPs are charge density excitations the propagate along the edge of a 2DES. In this dissertation, we describe a search for excitations analogous to EMPs but which are conﬁned by an inhomogeneity in the magnetic ﬁeld, rather than by the physical edge of the 2DES. Such magnetically conﬁned excitations are of interest as novel physical phenomena, and as possible probes of individual and collective electron physics in low temperature 2DESs. This chapter, and the next, concern the search for a charge density excitation bound to a linear inhomogeneity in the perpendicular ﬁeld Bz . We will refer to such an excitation as a magnetic edge magnetoplasmon, or MEMP. We will ﬁrst give some background on plasmons, magnetoplasmons, and EMPs, followed by a discussion of a few previous experiments involving 2DESs in the presence of magnetic inhomogeneities. Lastly we will introduce MEMPs as an analog to EMPs. 6 2.2 Plasmons and Magnetoplasmons A change in charge density from equilibrium will create an electric ﬁeld which tends to restore that equilibrium. If the scattering by disorder is small enough, the inertia of the electrons will keep them moving past the position where they would cancel out the original ﬂuctuation, so the charge density ﬂuctuation propagates through the 2DES. Absent nearby metals, such a propagating charge density ﬂuctuation, has been termed a plasmon [50]. Absent nearby metal surfaces the dispersion of a plasmon in a 2DES is given by [5]: ω 2p = 2πne2 k , 0 m∗ (2.1) where n is the electron density, m∗ is the eﬀective electron mass of the charge carriers, the relative permittivity of the medium, 0 the permittivity of free space, and k = 2π/λ with λ the wavelength of the plasmon. In the presence of a ﬁnite perpendicular magnetic ﬁeld the plasmon hybridizes with the cyclotron mode resulting in a magnetoplasmon whose dispersion is [50]: ω 2mp = ω p (k)2 + ωc2 , (2.2) where ω c = eB/m∗ is the cyclotron frequency. 2.3 Edge Magnetoplasmons EMPs [5, 21–23, 25–32, 37, 40, 44, 51] can be thought of as the low frequency branch of the magnetoplasmon. A 2DES structure can exhibit EMP resonances in its electromagnetic spectrum when the perimeter of the structure is a integer number of EMP wavelengths. These resonance frequencies are of lower frequency than the magnetoplasmon. EMP resonances were ﬁrst observed in semiconductor heterostructure dots of 4 μm diameter by Stormer et. al. [31]. Fig. 2.1 shows calculated fundamental EMP and magnetoplasmon resonance frequencies versus Bz for a 10 μm dot. Soon 7 3500 3000 2500 f(GHz) MP 2000 CR 1500 1000 500 EMP 0 2 4 B(T) 6 8 Figure 2.1: Comparison of the EMP and magnetoplasmon modes for a 10 μm disk of GaAs semiconductor 2DES with n = 2.8 × 1011 cm−2 . The higher frequency branch on the graph is the magnetoplasmon mode (labeled MP), the lower frequency branch the EMP (labeled EMP), and the dashed line the cyclotron frequency (labeled CR). 8 after, Mast et. al. [40] and Glattli et. al. [37] found EMP resonances on the surface of liquid helium. Grodnensky et. al. [38] in 1991 observed EMPs at radio frequencies in large (∼ 1 cm square) 2DES GaAs heterostructures. EMPs have low damping in strong magnetic ﬁelds and have been used to probe 2DES in the quantum Hall eﬀect regime [30, 32, 36, 52]. Experiments involving EMPs have been done in a wide variety of systems. Some examples are: arrays of quantum dots in semiconductor 2DESs [31, 33, 34], arrays of anti-dots in semiconductor 2DESs [33], along quantum wires [35], electrons on the surface of liquid helium [37, 40], and in larger-size semiconductor 2DES structures [38, 42, 45, 46] at radio frequencies. The ﬁrst theoretical work on EMPs was done by Fetter [25–27, 40] and by Volkov and Mikailov [28, 29, 51]. Fetter examined the problem of EMPs on the surface of liquid helium and obtained a solution using a classical hydrodynamic approach [25–27, 40]. Volkov and Mikailov addressed the problem of EMPs on the surface of liquid helium, in GaAs/Alx Ga1−x As semiconductor heterostructures, and in MOSFETs [28, 29, 51]. Various treatments [21, 25, 29] all obtain ωEMP = 2 ln e−γ 2ka ne2 k, 0 m∗ ωc (2.3) where a = 10aB [aB is the Bohr radius and 11.6 nm in GaAs] is the width of the strip the EMP is conﬁned in at the edge of the sample, γ is the Euler constant (γ = .577), and the relative permittivity of the medium is often ﬁt from the data as an eﬀective permittivity to take into account the dielectric environment. The width of the strip the EMP is conﬁned in (a) is the distance over which the energy of the electronelectron interaction is comparable with the kinetic energy of the individual electrons and a = aB assumes B such that the Landau ﬁlling factor (ν) is less then or equal to 10 [53]. For low ﬁeld (and if we assume ω ωc ) a = e2 ν/0 ωc [5]. Then, Eq. 2.3 gives a ωemp = 2.86 GHz at Bz = 1 T for a 1 cm square of 2DES with the same 9 density (n = 2.8 × 1011 cm−2 ) as the devices for which we will present data for in the next chapter. Metal surfaces near the 2DES will change the charge distribution of the EMP. For a “screened” EMP near a metal surface the dispersion is [42]: ωEMP = 2ne2 d k, 0 m∗ ωc a (2.4) where d is the distance between the metal surface and the 2DES. It is useful to mention that the natural log factor in the frequency dependence of the EMP is absent for diﬀerent assumptions about the static distribution of charges near the edge of the 2DES. To summarize this section and prepare the reader for the rest of this and the next chapter, here are a few useful things to keep in mind about EMPs. 1. The EMP frequency is proportional to the Hall conductivity σxy , with ω EMP ∝ (1/Bz ) [5]. 2. EMPs are chiral, i. e. they propagate in only one direction with their propagation direction dependent on the sign of Bz [5]. 3. The velocity of an EMP can be, and generally is, higher than the Fermi velocity [5]. 4. Having metal on or near the surface of the sample will change the dielectric environment of the EMP, damping the EMP [45], and driving the peak frequency down. 5. It is an experimental fact that grounded contacts can absorb an EMP [32]; a sample with a grounded contact along an edge will not show a resonance peak from circulation of the EMP around the sample. 2.4 Inter-Edge Magnetoplasmon EMPs have also been observed propagating along a change in charge density in a 2DES [43, 44, 47]. So-called inter-edge magnetoplasmons [5], IEMPs, travel along 10 a boundary deﬁned by a change in charge carrier density between two diﬀerent but non-zero n, instead of along a change between n of some ﬁnite value and n = 0 at the edge of the sample. This IEMP shares some similarity with our theorized MEMP since it exists in the bulk of the 2D gas not along a physical edge of the sample. The presence of metal gates, used to create the density step, on the sample will screen the IEMP. The dispersion of a screened IEMP is predicted to be [5]: R σ xy − σ Lxy d k, ω IEMP = 2π0 a (2.5) L where σ R xy and σ xy are the Hall conductivities of the regions on either side of the boundary on which the IEMP propagates. In typical 2DES in GaAs with evaporated metal ﬁlm gates d/a is of order unity. The essential diﬀerence between the frequency dependence of the screened EMP and a screened IEMP is the change from ω EMP ∝ 1/B to ω IEMP ∝ (1/BR − 1/BL ) 2.5 A Semi-Classical Model of Magnetic Conﬁnement There have been many papers involving DC transport through 2DESs decorated with ferromagnetic materials [8, 54, 55] showing increases in resistance based on the magnetization of the ferromagnetic material. To give a model of how a magnetic inhomogeneity can conﬁne an excitation, we will use a semi-classical approach starting with a model of an EMP conﬁned near a physical edge, and then generalize to a charge density ﬂuctuation conﬁned by a magnetic inhomogeneity. In Fig. 2.2, we depict a succession of cyclotron orbits which are broken at the edge of the sample, forming a chain of semicircular orbits where the path of individual electrons rebound at the edge of the sample. In the semi-classical model the channel that conﬁnes the charge density ﬂuctuation that composes the EMP is composed of many of these “skipping” orbits overlapping on top of each other. Then the EMP can 11 n = n₀ n=0 Figure 2.2: “Skipping” orbit depicted as a series of broken cyclotron orbits conﬁned to the edge of a sample, n = 0. be thought of as an excitation within these “skipping” orbits. This captures the chiral nature of the EMP. To extend this semi-classical model to electrons conﬁned along a change in Bz , we introduce two simple examples of idealized Bz inhomogeneities, namely linear steps in Bz . In the ﬁrst example Bz reverses in sign upon crossing the step, which results in “snake” orbits like those shown in Fig. 2.3a. If Bz changes in magnitude, but not in sign, on crossing the step then the electrons travel in “cycloid” orbits along the magnetic inhomogeneity; as shown in Fig. 2.3b. In both cases, the trajectories of electrons incident on the step in Bz are bent with the electrons becoming trapped along the step in Bz , assuming the step is large enough to curve the paths of electrons incident so that the electrons encounter the step again in a distance on the order of the mean free path [56]. “Cycloid” orbits at magnetic ﬁeld steps have been predicted to provide additional DC magnetoresistance [54, 57]. Ensslin et. al. [8] used a setup in which a ferromagnetic cobalt ﬁlm was deposited across a narrow Hall bar which was magnetized along the ﬁlm using an in plane magnetic ﬁeld. The result was a large inhomogeneity in Bz under the edge of the strip in the 2DES. The DC resistance of the device increased, 12 a) Bz = B0 Bz = -B 0 b) Bz = B0 Bz = B0 + B m Figure 2.3: a) “snake” orbit depicted as a succession of cyclotron orbits reversing direction where Bz inhomogenously changes from B0 to −B0 . b) “cycloid” orbit depicted as a series of cyclotron orbits tightening in radius in the region where Bz is increased by Bm . following the hysteresis curve of the ferromagnetic strip. This magnetoresistance arose from the charge carrier trajectories being curved under the edge of the magnet, where the Bz inhomogeneity was located, and being scattered under the edge of the cobalt ﬁlm. These previous DC results were part of the inspiration for searching for MEMPs. 2.6 Magnetic Edge Magnetoplasmon In this dissertation we have hypothesized that there is a charge density excitation similar to an EMP that travels along a linear magnetic inhomogeneity in Bz instead of along a change in electron density. To introduce the expected properties of this 13 theorized MEMP, let us consider Eq. 2.5, where, neglecting the quantum Hall eﬀect, the frequency of an IMEMP is given by ω emp L k σR xy − σ xy = , 2π0 (2.6) Δne e , Bz (2.7) where Δσ xy (IEMP) = and Δne is the diﬀerence of electron densities of the regions in the 2DES on either side of the boundary. Instead of a density change, if there is a diﬀerence in Bz we can rewrite Eq. 2.7 as: Δσ xy (IEMP) = ne eΔ with Δ 1 Bz = 1 Bz 1 1 − , BzR BzL , (2.8) (2.9) where BzR and BzL are the perpendicular magnetic ﬁelds under either side of the boundary along which the MEMP is propagating. To prepare for the next chapter, let us summarize the properties of our hypothesized MEMP: 1. They are conﬁned by a change in Bz . 2. In the IEMP analogy the MEMP frequency is proportional to the change in Hall conductivity Δσxy between regions of 2DES with diﬀerent Bz , ω MEMP ∝ Δ(1/Bz ). 3. Like EMPs, MEMPs are expected to have a gapless spectrum with ωMEMP ∝ k like an IEMP. 14 CHAPTER 3 MAGNETIC EDGE MAGNETOPLASMONS – EXPERIMENTAL SETUP AND RESULTS 3.1 Introduction to hybrid ferromagnet 2DES devices This chapter is about hybrid 2DES-ferromagnet devices. These devices use a ferromagnetic ﬁlm on the surface of the semiconductor wafer to provide a magnetic inhomogeneity Bz at the 2DES. We intended this inhomogeneity to support a MEMP. These devices are intended not to support a resonant mode and we refer to these devices as non resonant MEMP (NRMEMP) devices. 3.2 Microwaves connections and mounting We measure the NRMEMP devices immersed in liquid helium at 4K. An Agilent network analyzer acts as the source and receiver for the microwave signal. Semi-rigid metallic coax transmit microwaves from room temperature to the sample then back up to room temperature. We use a pre-ampliﬁer to increase the magnitude of the returning signal before the receiver. 15 3.3 Introduction to gate controlled non-resonant devices For these devices, a ferromagnetic permalloy ﬁlm is deposited on the surface of the device to provide a magnetic inhomogeneity in Bz . A diagram of the device is show in Fig. 3.1. The ferromagnet ﬁlm can also be biased to deplete the charge carriers in the area under the it so we often refer to the permalloy as a gate. This fully depleted region of 2DES creates an edge that EMPs can propagate along though there is a Bz inhomogeneity as well as a density step at the edge of the biased ferromagnetic ﬁlm. The 2DES under a Au gate opposite to the permalloy can also be depleted to create an edge that will support an EMP. Two grounded contacts block transmission around the device if the gates are not charged and there is no MEMP along the permalloy gate. The source antenna excites an EMP traveling along the edge of the 2DES mesa. The direction of the external ﬁeld controls the propagation direction of this EMP, directing it towards the permalloy gate (B > 0) and the Au gate (B < 0). We will refer to these hybrid ferromagnet 2DES devices as non-resonant magnetic edge magnetoplasmon (NRMEMP) devices. In NRMEMP devices microwaves couple capacitively to the 2DES in the small region where the antennas overlap the the 2DES mesa. We optimized the size of this overlap to be large enough for EMP measurements but not so large that the EMP is damped by the measuring circuits. Transmission of the signal via an EMP (or EMP-MEMP combination) can be observed as either an increase or decrease in total transmitted power depending on the phase diﬀerence between the EMP/EMP-MEMP signal and the “crosstalk” which travels through the 2DES bulk or by capacitive coupling between the source and receiver antennas. From the phase diﬀerence we can ﬁnd the time diﬀerence in propagation between the EMP (EMP-MEMP) and the crosstalk. For there to be dip in the B trace: τm = τc ± j/2f 16 (3.1) where j an integer, τm is the edge mode propagation delay, and τc is crosstalk propagation delay. For any series of Np EMP/MEMP on paths p of length lp Np lp τm = vp p (3.2) If EMP/MEMP velocities are given by αp /B (and for MEMPs they well might not be) and if we deﬁne a Cm then Np lp = BCm . τm = B α p p (3.3) Now the condition for a dip is: −1 B j = Cm (τc ± j/2f ), (3.4) and if we plot j vs Bj the slope is 2Cm f . 3.4 Sample characteristics The NRMEMP devices use modulation doped GaAs/Alx Ga1−x As with a quantum well of width 30 nm that is centered 100 nm below the surface of the wafer. This wafer was grown by molecular beam epitaxy, MBE, by D. Larouche, M. P. Lilly, and J. L. Reno at Sandia National Laboratory. Samples from the wafer have have mobility μ = 9.1 × 105 cm2 V−1 s−1 an as-cooled density n = 2.8 × 1011 cm−2 . This wafer was used for all the data presented for NRMEMP devices. 3.5 NRMEMP device fabrication and design We prepare NRMEMP devices by cleaving 2DES wafer into 4 mm x 3 mm rectangles. Then we etch away all the 2DES away except for a mesa of dimensions of 3 mm x 0.5 mm. We evaporate layers of Ge (40 nm), Au (80 nm), Ni (10 nm), and Au (200 nm), in that order, onto the both ends of the mesa and then anneal the sample 17 Microwave Antenna 1mm Contact Gold 2DES Microwave Antenna Contact Permalloy Figure 3.1: Illustration of NRMEMP device the 2DES mesa grey and the contacts dark grey, all of the features on the are device to scale except for the ends of the microwave antennas are truncated. Note: the slight overlap of the tips of the microwave antennas over the edge of the mesa containing the 2DES is not shown. 18 in a reducing atmosphere (H2 :N2 ) to make ohmic contacts. Next microwave antennas and the Au gate are evaporated onto the surface with a shallow (10 nm) layer of Cr under the thick (200 nm) Au layer. Last, a 250 nm thick ferromagnetic permalloy gate is evaporated onto the surface of the device opposite to the Cr/Au gate. Figure 3.1 depicts the ﬁnal design conﬁguration. 3.6 NRMEMP devices We designed these devices to allow EMPs to couple to MEMPs. We work with four conﬁgurations for the NRMEMP device: 1. Both gates grounded. 2. Both gates at -1 V. 3. Au gate grounded, permalloy gate at -1 V. 4. Au gate at -1 V, permalloy gate grounded. Figs. 3.2 and 3.3 show the expected paths for the EMP and EMP-MEMP through the device for positive BZ and negative Bz . The antennas are marked in the ﬁgures for what we call forward transmission. For “reverse” transmission the transmitter and receiver are interchanged. 3.7 NRMEMP device results Data for this section was taken with the sample at 4 K in an 8 T superconducting magnet, using an Agilent network analyzer as the source and receiver for the microwaves. In this and the next section of the dissertation we will report traces where the microwave frequency is held constant and B is swept. The traces show transmitted power as a ratio with incident power, in dB. 19 Receive ϐ 1 Contact Gold B Contact 2DES Transmit Permalloy Receive ϐ 2 Contact -1V Gold B Contact 2DES -1V Transmit Permalloy Receive ϐ 3 Contact B Contact 2DES -1V Gold Transmit Permalloy Receive ϐ 4 Contact -1V Gold 2DES Transmit B Contact Permalloy Figure 3.2: All four device conﬁgurations for positive B. For each conﬁguration the charged gates are colored and the grounded ones are white. The red dotted line shows the path of the EMP and the blue dashed line shows the intended path of the MEMP. In Figs. 3.2 and 3.3 the EMP is incident on the permalloy gate for B > 0 and the Au gate for B < 0. Dips are present when the EMP is incident is directed toward a gate that is biased to deplete the 2DES. 20 Receive ϐ 1 Contact Gold Contact -1V Gold Contact X B X B X Contact 2DES -1V Transmit Permalloy Receive ϐ 3 B Permalloy Receive ϐ 2 X Contact 2DES Transmit B Contact 2DES -1V Gold Transmit Permalloy Receive ϐ 4 Contact -1V Gold 2DES Transmit Contact Permalloy Figure 3.3: All four device conﬁgurations for negative B. For each conﬁguration the charged gates are colored and the grounded ones are white. The red dotted line shows the path of the EMP. The lack of dips on the Au side when both gates are grounded conﬁrms that a grounded contact absorbs an incident EMP. The lack of dips on the permalloy side when the permalloy gate is grounded tells us either there is no MEMP traveling along the permalloy gate or that the power transmitted by MEMP is very small in 21 Configuration 1 2 GHz 4 GHz -20 -40 Configuration 2 Au (0 V) Transmitted power (dB) Transmitted power (dB) 0 Py (0 V) -60 -80 -100 -3 -2 -1 0 B Z (T) 1 2 3 2 GHz 4 GHz -40 Au (-1 V) -120 -160 -3 -2 -1 Au (0 V) Transmitted power (dB) Transmitted power (dB) -60 Py (-1 V) -80 -100 -120 -3 -2 -1 0 B Z (T) 1 1 -20 2 GHz 4 GHz -40 0 B Z (T) 2 3 Configuration 4 Configuration 3 -20 Py (-1 V) -80 2 2 GHz 4 GHz -40 -60 Au (-1 V) -80 -100 -120 -140 -3 3 Py (0 V) -2 -1 0 B Z (T) 1 2 3 Figure 3.4: Transmitted power in dB vs B taken at 4 K. Gate that the EMP is incident, Py (permalloy) for positive B and Au for negative B on the graph with the on voltage the gate relative to the 2DES in parentheses next to it. comparison to the power transmitted by the crosstalk. Fig. 3.5 shows the B of the dips from Fig. 3.4 plotted vs the harmonic index (j). We can ﬁt j vs B in Fig. 3.5 to ﬁnd the slope and from the slope ﬁnd an average velocity for the mode connecting source and receiver. We can rewrite eq. 3.2 as: τm = lEM P lmg + vEM P vmg (3.5) where lEM P = 2 mm (the length of the edge the EMP propagates along not near a gate), vEM P is the velocity of the unscreened EMP, lmg = 0.5 mm (the length of the Au or permalloy gate), and vmg is the velocity of the mode traveling along the metal gate (Au or permalloy). The unscreened EMP traveling along the edge of the 2DES 22 Figure 3.5: Transmission dip number j (i.e. harmonic) plotted vs B, gate conﬁguration number listed in the caption. The EMP launched from the transmission antenna is incident on the Au gate for B < 0 and the (Py) permalloy gate for B > 0 as marked on the graph, lines on the graph are least squares ﬁts. Conﬁguration numbers and frequencies are shown on the right. 23 mesa has a vB of 1.16 × 107 Tm/s according to Eq. 2.3 using the average relative permittivity of GaAs (12.9) and vacuum (1) equal to 6.95. We will determine τm for each observed series of dips from the ﬁt of j vs B using Eq. 3.4, then get vmg from Eq. 3.5 using the calculated vEM P . Any error in vmg owing to the calculated value of vEM P will be small because vEM P is approximately 5 times larger than vmg . The ﬁt for the B < 0 dips in conﬁguration 2 (the EMP along the Au gate) from Fig. 3.4 give vB = 2.53 × 106 Tm/s for the 2 GHz trace and vB = 2.46 × 106 Tm/s for the 4 GHz trace. When we ﬁt the dips from conﬁguration 4 for B < 0 (the EMP along the Au gate) we get vB = 2.52 × 106 Tm/s for the 2 GHz trace and a vB = 2.56 × 106 Tm/s for the 4 GHz trace. The data for vB for the EMP along the Au gate can ﬁt Eq. 2.4 which gives vB = 2.71 × 106 Tm/s when is set to 6.95, the average of the vacuum and GaAs permittivity and the width (a) of the strip on which the EMP propagates (a) is 10 Bohr radii (aB ) in GaAs, a = 116 nm. The depth of the 2DES from the top surface in this wafer is d = 100 nm. When we ﬁt the dips for in Fig. 3.5 for conﬁguration 2 and B > 0 (when the EMP is incident on the permalloy gate) we get vB = 1.04 × 106 Tm/s for the 2 GHz trace and vB = 1.07 × 106 Tm/s for the 4 GHz trace. For conﬁguration 3 and B > 0 (when the EMP is incident on the permalloy gate) we get vB = 1.25 × 106 Tm/s at 1 T for the 2 GHz trace and vB = 1.27 × 106 Tm/s for the 4 GHz trace. The main result for this section is that this velocity is more than a factor of two slower than velocities observed for the EMPs along the Au gate. 3.8 NRMEMP Summary The velocities of the modes along the Au gate agree with calculation from Eq. 2.4 for a screened EMP. The modes propagating along the biased permalloy gate have their velocity reduced compared to the EMP on the biased Au gate. There are no dips in the traces for transmission along the permalloy gate or the Au gate when 24 they are grounded. The lack of dips from conﬁgurations with the permalloy grounded suggests that the magnetic inhomogeneity along the permalloy gate does not support a MEMP. The origin of the increase in delay times for modes along the permalloy gate when the Au gate is biased is likely from a mode circulating around the whole device when the Au gate is biased. A small component of the EMP that has made 1.5 revolutions around the entire device could add an additional component to the detected signal and change the observed delay time, hence the calculated velocity reported in the last section. The extra delay from the circulating mode may explain why the velocity calculated from the slope in Fig. 3.5 is lower for the mode along the permalloy when both gates are biased. While we have obtained no evidence for a MEMP, we do see a notable diﬀerence in the EMP velocities along the Au and the permalloy gates. We use a simpliﬁed EMP model to argue that this result can be plausibly interpreted as due to the Bz inhomogeneity due to the permalloy, though a quantitative analysis will require theoretical development [58], and a more detailed knowledge than is currently available to us of the magnetic properties of the permalloy ﬁlm near its edge. 3.8.1 A simple EMP model with nonzero ∂Bz /∂x An EMP wave equation can be obtained [5, 42, 51, 59] starting from the continuity equation, Ohm’s Law. With ñ the deviation of the density equilibrium from equilibrium, ṅ = −∇ · (σE) (3.6) where σ is the conductivity tensor, and E = −∇ϕ is the electric ﬁeld. An integral of the Green’s function relating the potential (ϕ) to the deviation (ñ) of the local carrier density from equilibrium, is substituted to give the equation of the EMP. Here we roughly obtain an expression for the velocity in the local capacitance approximation 25 following ref. [42]. This approximation requires the gate to be much closer to the 2DES than one EMP wavelength. More complete theories for EMPs in Bz gradients (and for MEMPs) are under development [58], which will deﬁne the proﬁle of the strip of oscillating charge density. For EMPs in the absence of magnetic ﬁeld gradients, and fully screened by a gate, the strip width [42] is of order the separation between the gate and the 2DES. The potential is then ϕ= ñ , C (3.7) where C is the capacitance per unit area between the 2DES and the gate, ñ is the time-varying part of the carrier density, and the x-axis perpendicular to the edge. For x near the maximum of the oscillating potential at the 2DES edge, the EMP velocity becomes vp = ∂σxy . C∂x (3.8) Taking σxy ∝ ne/Bz , and evaluating the derivative, including the x-dependence of n gives e vp = CBz n ∂Bz ∂n − ∂x Bz ∂x (3.9) The ﬁrst term in the parenthesis is due to the density gradient, the second to the Bz gradient. The ratio of velocities with and without the Bz gradient is n ∂Bz η = 1− Bz ∂x ∂n , ∂x (3.10) in the strip of oscillating charge at the 2DES edge. The measured value of this ratio is about 0.46. The ﬁrst term in the parenthesis in (3.9) involves only the density of the 2DES at the edge. Chklovskii and coworkers [53] calculated the density proﬁle without considering the depth, d = 100 nm, of the 2DES below the surface, but as discussed in ref. [60] that should be a reasonable approximation for the the gate voltage we 26 applied, which is about 2.5 times that required to deplete the 2DES well under the gate. Ref. [53] calculated the horizontal distance L between edges of the gate and edge of the 2DES (at which the equilibrium density n(x) goes to 0, as L= 2Vg 0 πn0 e (3.11) where n0 is the density far from the edge, and for GaAs = 12.8. For Vg = 1 V and n0 = 2.8 × 1011 cm −2 , L ≈ 160 nm. For the equilibrium density the reference obtained neq (x) = L 1− x 1/2 n0 , (3.12) where the x-axis is perpendicular to the edge, and x = 0 at the edge of the gate. The second term involves the magnetic ﬁeld gradient. With the external ﬁeld perpendicular to the permalloy ﬁlm, according to ref. [61] the permalloy magnetization M vs H is roughly linear up to μ0 H = 0.96 T. The susceptibility χ in that direction is about 1. From continuity of the normal B, the ﬁlm magnetization is μ0 M = χ/(χ + 1)B0 ∼ B0 /2, where B0 is the applied ﬁeld. Neglecting edge eﬀects and taking M to be uniform, (which will tend to overestimate ﬁelds near the gate edge) we then have [62] Bz = B0 2χhx 1− (χ + 1)(x2 + d2 ) , (3.13) where x = 0 just under the gate edge and h is the height of the magnet. Grain size in permalloy ﬁlms of thickness around ∼ 100 nm can be as large as 1 μm [61, 63], so when the ﬁlm is not fully saturated, variations in Bz of around that spatial extent, at around that distance from the edge, may exist. Simply using equations (3.11),(3.12) and (3.13) to evaluate η, we ﬁnd that the magnetic term is much too large for the observed velocity change due to the Bz gradient. The most natural explanation is that (3.13) is overestimating ∂Bz /∂x, which is expected since the uniform magnetization model would be expected to be 27 incorrect for B0 below saturation. Setting the ﬁeld in equation (3.13) 1 μm further away from the 2DES gives a result in which the two terms are comparable, and would give the measured η for x = 700 nm. 28 CHAPTER 4 WIGNER CRYSTALS AND OTHER ELECTRON SOLIDS 4.1 The Quantum Hall Eﬀect At low temperatures, suﬃciently low disorder 2DESs exhibit the quantum Hall eﬀect (QHE) [64, 65]. The QHE is characterized by vanishing diagonal resistance, Rxx , concomitant with plateaus in Hall resistance Rxy with integer [65] or rational fractional [14] multiples of h/e2 in certain ranges of magnetic ﬁelds. The integer quantum Hall eﬀect (IQHE) is an eﬀect of a resolved Landau level (LL) spectrum, and occurs when the Fermi energy lies between the LLs, as shown in the Fig. 4.1. We will refer to this gap in the density of states between LLs as a “Landau vacuum”. This Landau vacuum is responsible for the zero observed in Rxx , and the plateaus observed in Rxy . The these LLs are separated by the cyclotron energy ωc = eB/m∗ , where m∗ is the band mass of the charge carriers [66], or by the Zeeman splitting gμB B, which in GaAs is generally much less than ωc . The QHE plateau centered at ν = νc has a quantized Rxy = h/e2 νc . Each of the LLs has a degeneracy of the density of ﬂux quanta, nφ = 2eB/h, and the LL ﬁlling factor is: ν= nh n . = nφ 2eB (4.1) As depicted in Fig. 4.1, in the presence of disorder each of these LLs is broadened. The IQHE plateaus are centered at integers of ν and have a Rxy = h/e2 . 29 Figure 4.1: Density of states for a 2DES showing disorder-broadened LL spectrum. g the Lande g-factor and μB the Bohr magneton. The Fermi level is as shown for the LL ﬁlling factor ν = 3. 30 The fractional quantum Hall eﬀect (FQHE) is a manifestation of a many-body quantum liquid which has fractionally charged quasi-particles, and was ﬁrst described by Laughlin [67]. The composite fermion (CF) model [18–20] encompasses most of the physics of the FQHE. A CF can be thought of as an electron bound to an even number of magnetic ﬂux quanta. In the CF description, the FQHE is an analog of the IQHE with LLs of CFs instead of electrons; here the simplest FQHE states arise at odd denominator rational fractional ν for which the CF Fermi level lies between CF LL. 4.2 Wigner Crystals In 1934, Eugene Wigner [9] theorized that the ground state of a low density electron gas in a uniform neutralizing background was a regular lattice, or crystal, of electrons. He pointed out that an electron gas in a neutralizing background has two principal energy scales: the Coulomb energy, EC , and the kinetic energy, Ek . The Coulomb energy scales as EC ∝ 1/r where r is the inter-electron spacing. As T goes to 0, the EK is taken to be the the Fermi energy, EF , which is proportional to kF2 where kF is the Fermi wavevector. In 2DES, KF2 is proportional to the density n. So EF goes as 1/r2 Hence at suﬃciently low density, EC will be larger than EK . Under such conditions electrons arrange themselves into a crystal lattice to minimize EC . The lattice is expected to be triangular in 2D [68]. This regular lattice of electrons is referred to as a Wigner crystal (WC). While 2D Wigner crystals hosted in semiconductor systems at B = 0 have been considered theoretically [69, 70], they have yet to be deﬁnitively observed. The addition of a perpendicular magnetic ﬁeld (B) changes the shape of free electron wave functions from plane waves to Landau orbitals. Experimental evidence of a WC stabilized by the presence of a perpendicular magnetic ﬁeld in semiconductor 2DES does exist, for examples see reviews in [17, 68]. As B is increased, the area that 31 an electron in a Landau orbital takes up decreases. We can think of ν as a measure of the overlap between electron wavefunctions. This is because ν = 2(lB /r)2 where lB = /eB is the magnetic length and is the size of the lowest LL single particle wavefunction and r = 1/πn the average electron separation. Landau quantization allows for WCs of arbitrarily high n as long as suﬃcient B is available. The WC is the predicted ground state of a 2DES, with no disorder, for ν 1/7 [11, 71, 72]. The existence of WCs in this “ low ν” regime at the high magnetic ﬁeld termination of the QHE series is corroborated by microwave measurements as described in the next section [3, 12, 73–76]. 4.3 Pinned Wigner Crystals In the presence of weak disorder, the crystalline order of the WC will have a ﬁnite correlation length, or equivalently the disorder breaks the crystal up into domains. This correlation length is a consequence of the electrons moving to reduce their total energy, the sum of the electron-electron energy, and the electron-disorder energy [77, 78]. Also in the presence of disorder, the WC is pinned since it can no longer slide as a whole, and thus is an insulator [6]. The disordered WC can oscillate collectively within the disorder potential. This collective oscillation in 2DES in high B turns out to have a frequency in the microwave range or RF range, and it is referred to as a pinning mode resonance. The frequency of a pinning mode resonance is a function of the collective restoring force from disorder on the pinned WCs. If the strength of the pinning potential increases the peak frequency of the pinning mode also increases. For example, if the electron density (n) is increased, then the individual electrons will not be as closely associated with the disorder pinning potentials, and the frequency of the pinning mode decreases [13]. Fukuyama and Lee [13] derived a sum rule for the integrated intensity of a pinning 32 Figure 4.2: Spectra showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency, f , at many ﬁlling factors ν oﬀset vertically proportional to ν. Successive spectra are separated by steps of 0.01 in ν ν for black spectra are marked at right, in a 30 nm wide quantum well 2DES, and at 30 mK. Adapted from [2]. mode resonance if all available charge carriers are participating in the resonance: S/fpk = neπ/2B, where S is the integrated pinning mode resonance conductivity S(Re[σxx ]) = (4.2) Re[σxx ]df . We can use Eq, 4.2 to evaluate the carrier density in a pinning mode resonance. 33 4.4 Integer quantum Hall Wigner crystals In ultra high mobility 2DESs pinning mode resonances have been observed near IQHE plateaus [2]. As ν moves away from the central ﬁlling, νc , of some IQHE plateau, a population of quasi-carriers is created, quasi-holes for ν < νc , and quasielectrons for ν > νc . These quasi-carriers interact with each other. The population of these quasi-carriers is n∗ = n|ν − νc |/ν. This dilute population of quasi-carriers crystallizes, similar to the low ν WC. fpk , the peak frequency of the pinning mode decreases as n∗ increases [2, 79], as discussed in the previous section. Figure 4.3: Example of S/fpk plotted vs ν for an IQHEWC pinning mode resonance in a 30 nm wide quantum well 2DES at 50 mK. The black lines are the prediction from the sum rule Eq. 4.2 for full quasi-carrier participation. Apdated from Chen et. al. [2]. Using the sum rule, Eq. 4.2, the IQHEWC pinning mode appears to involve nearly all the quasi-carriers available near the integer ﬁlling as shown in Fig. 4.3. 34 4.5 Other electronic solids Electronic solids can have more complex carrier arrangements than the single carrier-per-site arrangement of the WC we introduced earlier. A WC can be regarded as a particular example of an electron solid. We will refer to any of these other “exotic” crystals as an electronic solid, electron solid. In general electron solids have a regular crystal lattice as a consequence of Coulomb repulsion, and in the presence of disorder have a ﬁnite correlation length and are pinned. v* M=1 M=2 IQHEWC Wigner Crystal Bubble Phases Stripes Figure 4.4: Cartoon of the progression from an IQHEWC to a bubble phase WC and then a stripe phase WC as ν ∗ increases, with M the number of electrons per lattice point in the WC. Now we will give a few examples of previously observed electron solids. One example of an electron solid is a so-called “bubble phase” in certain ν ranges in the N ≥ 2 LL in ultra high mobility 2DESs [79, 80] that are ascribed to bubble phase electron solids. These bubble phase electron solids are thought to be clusters of M electrons, or holes, in a triangular lattice [12]. They are also predicted to be pinned by disorder, and pinning mode resonances of these modes have been studied [79] 35 Figure 4.5: Re[σxx ] vs f for a very high mobility, μ = 24 × 106 cm2 V −1 s−1 , n = 1.0 × 1011 cm−2 , 50 nm QW, and a temperature of 30 mK. Resonance A shows dispersion with wave vector. Adapted from Chen et. al. [3]. Another example is a crystal composed of skyrmions. Skyrmions are particles composed of a single charge in a spin texture containing multiple ﬂipped spins, spread out in space to reduce exchange energy. For suﬃciently low Zeeman energy, skyrmions (anti-skyrmions) are predicted to be the quasi-carriers around ν = 1 [81]. The pinning mode frequency is shown experimentally to shift upward with the addition of an inplane magnetic ﬁeld, consistent with the expectation that increased Zeeman energy due to the in-plane ﬁeld would cause the skyrmions to shrink into regular Landau quasi-carriers [82–84]. 36 In very high mobility, μ = 24 × 106 cm2 V −1 s−1 , 50 nm wide quantum well 2DES at low ν just above ν = 1/5 a pinning mode resonance was observed [3], and as was done by Chen et. al. we will refer to it as the A-phase. This A-phase initially coexists with the low ν pinning mode resonance (B-phase), with the A-phase vanishing by ν ∝ 0.12 and only the B-phase (low ν WC) remaining. This A-phase pinning mode resonance shows dispersion with respect to the width of the microwave transmission line used to measure it (see Fig. 4.5, indicating that it has a larger correlation length compared to previously measured electron solids). The FQHE has been suggested to play a major role in the A-phase electron solid, and it has been proposed that this electron solid is composed of CFs. 4.6 The reentrant integer quantum Hall eﬀect in 2DES The reentrant integer quantum Hall eﬀect, RIQHE, is characterized by the vanishing and then appearance of an IQHE, both the plateau in Rxy and the zero (or minimum) in Rxx [4, 74, 79, 85]. The RIQHE was ﬁrst observed by R. R. Du et. al. and M. P. Lilly et. al. 1998 [86, 87]. Both groups observed this reentrance of the quantum Hall eﬀect in a GaAs/Alx Ga1−x As semiconductor 2DESs between ν = 4 and ν = 5 at low temperature (∼ 20 mK), as shown in Fig. 4.6. Fig. 4.6a shows Rxy and Fig 4.6b shows Rxx , the three regions marked on them are the IQHEWC near ν = 4 on the left, the bubble phase electron solid on the right, and the region where they compete with each other in the middle. Concurrent with the RQHE observed between ν = 4 and ν = 5, two diﬀerent pinning mode resonances have also been observed for the same range in ν [79]. These regions were identiﬁed by a combination of DC measurements [4, 79] and microwave spectroscopy [74]. The pinning mode resonance observed near the IQHE plateaus grows weaker as the ﬁlling factor moves away from the plateau, where a second pinning mode resonance appears 37 Bubble Phase Competing Phases IQHEWC WC a) 0.250 R xy 0.249 RIQHE b) 20 RIQHE R xx 0 2.6 B (T) 2.7 Figure 4.6: Example of the RIQHE observed by Cooper et. al. [4] near ν = 4 as ν increases with the RIQHE marked by the arrow, a) Rxy in units of h/e2 , and b) Rxx in units of Ohms. 38 in the same region of ν where in DC the IQHE zero in Rxx vanishes. The zero in Rxx then returns as the ﬁrst resonance vanishes, leaving only the second resonance. The ﬁrst resonance, near the integer value of ν, has been identiﬁed as an IQHEWC, and the second resonance as a bubble phase Wigner solid [79]. As the “domains” of the bubble phase WC increase in size there will be a critical ν where the individual domains link up and their walls merge, forming a path across the 2DES and allowing electrons to move again, short circuiting the ν = 4 IQHE plateau. Another example of the RIQHE occurs at very low T, < 15 mK, in ultra high mobility 30 nm quantum well 2DESs. Multiple RQHE have been observed in the ﬁrst excited Landau level, between ν = 2 and ν = 3 with both ν = 2 and ν = 3, reentering multiple times [85, 88]. This RIQHE is likely related to multiple transitions between bubble and stripe phases. A RIQHE has been observed in 42nm wide quantum well samples on either side of ν = 1 as ν moves away from ν = 1 [7]. The location of this RIQHE in ν has been observed to move in toward ν = 1 from either side of ν = 1 and eventually merge with the ν = 1 plateau, as n is increased. This RIQHE has also been observed to vanish below a critical n. The DC experiment that observed this speciﬁc example of the RIQHE was one of the principle inspirations for the experiments described in Ch. 6. 39 CHAPTER 5 WIDE QUANTUM WELLS 5.1 Introduction Mansour Shayegan‘s group in Princeton has recently been investigating 2DES QW samples with wider wells [7, 89–94]. In comparison with narrower QW 2DESs, “wide” quantum well (WQW) 2DESs have reduced electron-electron interaction at short range from the larger extent of the electron wavefunction in the vertical direction (in the growth direction, perpendicular to the 2D plane) and they have an electron distribution that can be tuned from a single layer to a bilayer like distribution [95]. These WQW 2DES exhibit a wide variety of phenomena due to the addition of the subband degree of freedom. Stabilization and enhancement of the q/2 FQHE states [90, 91, 93, 96] (where q is an integer) and enhancement or weakening of the q/3 FQHE states [92, 94] has been observed based on which speciﬁc subband-split LL is below the Fermi energy. While level crossings between the subbands, have been observed to cause the IQHE plateaus to vanish [97]. These WQW 2DES also display quantum Hall ferromagnetism [95, 98, 99] when there are crossings between subbands with the same spin orientation. A possible bilayer electron solid [15], where the positions of individual charges are correlated between the top and bottom layers of charge in the well, has also been observed in WQW 2DES. Finally, an RIQHE has been observed around ν = 1 by Liu et al. [7] in 31, 42, and 44 nm WQWs. This instance of the RIQHE is notable as the RIQHE has never before been observed in 40 the lowest LL (ν < 2) in high mobility 2DESs, though disordered 2DES can show an RIQHE in the lowest LL [100, 101]. 5.2 Two subbands Figure 5.1: Charge distribution (ρ) on top in red, ground (ΨS in green) and ﬁrst excited state (ΨAS in blue) wave functions on bottom. From simultaneous calculations of the Poisson and Schroedinger equations. Diﬀerence in density between the top half and the bottom half of the well (δn) on top. n in 1011 cm−2 and subband separation (Δ) in K on bottom. Provided by Yang Liu. 41 If we consider a 1D quantum well that is symmetric about the origin and solve for the wavefunction of a single electron in that well, the ground state is symmetric about the origin and the ﬁrst excited state is antisymmetric. We will refer to these states as ΨS and ΨAS . In semiconductor QWs, the addition of carriers will change the potential of the well and we will need to simultaneously solve the Poisson and the Schroedinger equations to calculate ΨS and ΨAS . As we increase the number of carriers in the well, the ground state (ΨS ) will ﬁll up and we will begin to populate the ﬁrst excited state (ΨAS ). The electron charge distribution in the well then is ρ = e(nS |ΨS |2 + nAS |ΨAS |2 ) where nS is the population in the ground state and nAS is the population in the ﬁrst excited state. As show on the left in Fig. 5.1 ρ is symmetric about the center of the well and double peaked. We can visualize this charge distribution as a pseudo-bilayer, each peak representing a layer of 2DES. If we apply a bias to a front and back gate on the sample we can tune the symmetry of the electron distribution as well as the overall density (n). Increasing n while keeping the charge distribution symmetric decreases the gap (Δ) between the ground and the ﬁrst excited states, and moves the peaks in the charge distribution toward the walls of the well [102, 103]. We can also operate the gates to move charge from one side of the well to the other, as shown in the center and on the right of Fig. 5.1. The uneven charge distribution in the well then increases Δ [102]. The well and its ground state are then no longer symmetric. We will keep referring to the lowest lying state with the subscript S and the ﬁrst excited state with the subscript AS one even though the ground state is no longer symmetric and the ﬁrst excited state is no longer antisymmetric. As can be seen in the simulations shown in Fig. 5.1, there is a relationship between δn, the diﬀerence between nt (n of the top half of the well) and nb (n of the bottom half of the well), and the subband separation Δ [103]. 42 Re[σxx(μS)] 30 20 10 0 0.1 0.2 0.3 B(T) 0.4 0.5 0.6 Figure 5.2: Re[σxx ] in μS plotted vs B at a frequency of 0.5 GHz, from a 54 nm WQW, a temperature of 30 mK, and an as cooled n = 2.41 × 1011 cm−2 . 5.3 Measuring the subband separation (Δ) Δ can be experimentally determined from Fourier analysis of the Shubnikov de Haas (SdH) oscillations. An example of SdH oscillations (Re[σxx ] vs B) in a WQW is shown in Fig. 5.2. We ﬁnd a “B spectrum” of the SdH from the “Fast” Fourier transform (FFT) of (Re[σxx ] vs 1/B) in the SdH regime. The FFT has peaks whose position is (in Tesla) depends on the subband populations. Fig. 5.3 shows the B spectrum from the data in Fig. 5.2. There are four peaks corresponding to the total density(n), the density of electrons in the lowest lying subband (ns ), the ﬁrst excited subband (nAS ), and the diﬀerence between the two subband densities (nS − nAS ). The population density of the lowest lying and ﬁrst excited subbands is given by: e nS = 2BS , h (5.1) e nAS = 2BAS , h (5.2) and 43 Fourier transform intensity (a.u.) BAS BS - BAS BS + BAS BS 1 2 3 frequency (T) 4 5 6 Figure 5.3: B frequency peaks from the Fourier transform of the SdH oscillations with the B frequencies of the lowest lying state, the ﬁrst excited state, the sum of both, and the diﬀerence of both marked on the ﬁgure. where BS and BAS are the SdH frequencies of the the lowest lying and ﬁrst excited subbands. Once we have the subband densities we can calculate the subband separation as a temperature: 1 (nS − nAS )π2 Δ= kB m∗ (5.3) where kB is the Boltzmann constant. For GaAs, with m∗ = 0.067 free electron masses, Eq. 5.3 becomes: Δ = 20.03|BS − BAS |, (5.4) using BS and BAS in Tesla and producing Δ in units of K. 5.4 Δ in the vicinity of ν = 1 Fig. 5.4 shows a fan diagram of the LL energies vs B in a 54 nm WQW in the lowest LL and a Δ = 27.6 K superimposed over a Re[σxx ] vs B trace from the same 44 Figure 5.4: Fan diagram for a 54 nm WQW (left axis) for Δ = 27.6 K. The energies of the N = 0 Landau level from the lowest lying (S0) and ﬁrst excited (A0) subbands are plotted with their spin directions shown. On the right axis a Re[σxx ] vs B trace taken at 500 MHz is plotted for n = 2.43×1011 cm−2 and the same Δ. ν = 1 is labeled on the ﬁgure. well with n = 2.43 × 1011 cm−2 . The energies of the LLs of each of the subbands is: ES = (N + 1)eB/m∗ ± gμB B, (5.5) EAS = Δ + (N + 1)eB/m∗ ± gμB B, (5.6) ES is for the lowest lying subband, EAS is for the ﬁrst excited subband, and N is 0 for the lowest LL. At ν = 1 there is one ﬁlled LL under the Fermi energy. For the samples we studied, there are no level crossings between the symmetric and antisymmetric LLs around ν = 1. 45 5.5 A RIQHE near ν = 1 in WQW Shayegan’s group observed an RIQHE for ﬁllings both above and below ν = 1 in 31, 42, and 44 nm WQWs [7]. In the 42 nm WQW, as n is increased while keeping the well symmetric, the RIQHE appears at ν ≈ 0.81 as n reaches 2.05 × 1011 cm−2 . The ﬁlling factor of the RIQHE then increases toward the main ν = 1 plateau region as n is increased further, eventually merging with the main plateau around n = 3.14 × 1011 cm−2 . At n = 2.87 × 1011 cm−2 another RIQHE appears, this time at ν ≈ 1.18. As n is increased, this RIQHE, like the one for ν < 1 moves in toward, then merges with, the plateau around ν = 1. In Yang Liu et al. [7] it was proposed that both of these RIQHEs were due to electron solids composed of CFs. Our experimental data to be presented in the next chapter addresses the nature of these RIQHE states and will discussed in comparison with data of Ref. [7]. 46 CHAPTER 6 MICROWAVE SPECTROSCOPY OF THE RIQHE AROUND ν = 1 IN WQW 2DES The origin of the RIQHE around ν = 1 in WQW 2DES is not currently clear [7]. In this chapter we present microwave measurements of WQW 2DESs. We will start by presenting our measurement techniques. Next, we will present data from microwave transmission measurements from 54 and 42 nm WQW 2DES. Finally, we will interpret this data in the context of multiple competing electron solids. All data for the samples in this chapter was taken with a bottom loading dilution refrigerator with a base temperature of 30 mK. 6.1 Samples and Sample Preparation The microwave measurements reported in this chapter are all of modulation doped GaAs/Alx Ga1−x As samples. The speciﬁcs of each individual sample are shown in Table 6.1. These samples are grown by molecular beam epitaxy (MBE) by Loren Pfeiﬀer and Ken West at Princeton University. To prepare the samples we ﬁrst cleaved a rectangle of wafer approximately 4 mm x 3 mm in size. We then fabricated contacts of thermally evaporated Ge/Au/Ni, which we then annealed at 440 C in a reducing atmosphere of H2 : N2 for 10 minutes. After that we deﬁne a microwave waveguide on the surface of the sample using a 200 nm thick Au layer on top of a 47 Table 6.1: Sample properties sample # ID # well width 54-1 8-17-10.1 54 nm 54-2 8-17-10.1 54 nm 54-3 8-17-10.1 54 nm 42-1 2-9-11.1 42 nm mobility (μ) 4 × 106 cm2 V−1 4 × 106 cm2 V−1 4 × 106 cm2 V−1 9 × 106 cm2 V−1 −1 s s−1 s−1 s−1 CPW 30 μm, 80 μm, 30 μm, 40 μm, meander slot meander meander thin 5 nm Cr layer. 6.2 Microwave Measurement Technique – CPW Ohmic Contacts Sample 50 : Microwave Source Detector 3 mm Figure 6.1: Schematic of a coplanar waveguide on the surface of a sample, with metal in black. The 2DES is a fraction of a micron under the surface of the sample. An Agilent network analyzer is the source and the receiver of the microwave signal. To obtain the diagonal ac conductivity of the 2DES we measure the microwave attenuation through a coplanar waveguide (CPW transmission line) patterned on top of the sample as shown in Fig. 6.1. CPW, invented by C. P. Wen in 1969 [104], is used widely in planar integrated circuit devices. In our setup, an Agilent network analyzer acts as both the source and receiver of the microwave signal The signal from the network analyzer goes through attenuators 48 before traveling down coaxial cable (coax) to the sample in the cryostat. From the sample, the signal is carried by coax to the top of the cryostat and pre-ampliﬁed before being returned to the network analyzer. The center conductor of the CPW is connected to the center conductor of the coax, and the planes on either side of the center conductor are grounded. The electric ﬁeld from the driven center conductor to the ground planes interacts mainly with the 2DEG lying under the slots separating the ground planes from the center conductor. For the samples considered here, Table 6.1 shows the slot width (W ) of the CPW, and whether the CPW is straight or if it meanders. The meander CPW, like that shown in Fig. 6.1, increases the sensitivity of the transmission line and is not intended to couple to wavevectors related to the meander pitch. The characteristic impedance of the CPW on the sample with σxx set to 0, Z0 , is determined by the ratio of the width of the center conductor of the CPW to the width of the slot of the CPW. We designed the CPW so that the Z0 = 50 Ω, matching the impedance of the coax in the experiment. We model the 2DES conductivity as a perturbation of the transmission line, assuming the following [105]: 1. The measuring system has no reﬂections at the end of the lines. 2. The per-unit-area coupling capacitance (CC ) between the 2DES and CPW ﬁlm, is suﬃciently large that the RF current is well conﬁned in the 2DES under the slots |σxx |/πf CC W . 3. The measurement is done at a suﬃciently high frequency that |σxx |v0 Z0 /πf W , where v0 is the propagation velocity of the RF signal through the 2DES. When the above assumptions are satisﬁed, the diagonal conductivity is approximated by: W Re[σxx (f )] = ln 2lZ0 P P0 , (6.1) where P0 is the transmitted power with zero 2DES conductivity (usually found using a trace taken on a quantum Hall plateau where Re[σxx ] is vanishing), P is the 49 Figure 6.2: Gate conﬁgurations for the WQW samples. a) back gate only: samples 54-1 and 54-2. b) front and back gate with glass spacer and increased CPW slot width: sample 42-1. c) front and back gate with etched glass spacer: sample 54-3. 50 transmitted power, and l the total length of the CPW on the sample. We verify the result of Eq. 6.1 using more complicated calculations [106] that include higher loss, greater conductivity through the 2DES, and the distributed nature of the CPW-2DES coupling. Fig. 6.2 shows the gate conﬁgurations for all the WQW samples. To keep the top gate from capacitively shorting out the CPW, a spacer is needed between the top gate and the CPW. All of our samples with top gates have a spacer made out of glass. For sample 42-1, the CPW has a wider slot width to take into account the increase in the eﬀective dielectric constant from the glass spacer. Sample 54-3 has a glass slide with the area over the CPW etched away so that the eﬀective dielectric constant near the CPW is less aﬀected by the spacer, and strain due to diﬀerential contraction is reduced. 6.3 6.3.1 54 nm WQW – microwave spectroscopy of pinning mode resonances around ν = 1 Sample 54-1 Sample 54-1 has a back gate, and no front gate. The lack of a front gate on sample 54-1 prevents us from controlling Δ when we vary n. Δ vs n for sample 54-1, as assessed from SdH measurements, is shown in Fig. 6.4. Fig. 6.3 shows Re[σxx ] vs B for n = 2.41 × 1011 cm−2 with a selection of IQHE and FQHE minima marked on the graph. Fig. 6.5 shows traces of Re[σxx ] vs B, taken at a frequency of 0.5 GHz, a temperature of 30 mK, and several n which are shown on the right in units of 1011 cm−2 . We calculated the densities for the spectra we present in the section from the traces in Fig. 6.5. Fig. 6.6 shows data from sample 54-1 in its as-cooled state (i.e. with no gate voltage). Fig. 6.6a displays spectra, Re[σxx ] vs f , of 54-1 for many ν. These spectra show a pinning mode resonance for ν on either side of ν = 1. The small (∼300 51 10PS Re[ σ xx ](μS) 3 1 2 8/3 6/3 8/5 6/5 4/5 5/3 4/3 2 4 6 8 10 12 14 B(T) Figure 6.3: Real part of diagonal conductivity (Re[σxx ]) vs B for sample 54-1 taken at a frequency of 500 MHz. The temperature is 30 mK and the density is 2.41 × 1011 cm−2 . The sample has no front gate, and there was no voltage bias on the back gate. A selection of IQHE and FQHE minima are labeled with their Landau ﬁlling factors. MHz period) peaks on the spectra are from reﬂections arising from the increased conductivity of the pinning mode resonance, causing an impedance mismatch. The resonance is absent at ν = 1, and vanishes for ν just above 6/5 and just below 4/5. Fig 6.6b shows fpk vs ν from the data in Fig 6.6a. There is a prominent kink in the curve between ν = 0.91 and 0.87 Fig. 6.6c displays S/fpk vs ν from the spectra in Fig. 6.6a, with the black lines showing the theoretical prediction for full participation of the quasicarriers at density n∗ = nν ∗ /ν, as explained in Section 4.3 (see Eq. 4.2). The drop oﬀ in participation seen in Fig. 6.6c near ν = 4/5 and ν = 6/5 is from the pinning mode resonance dying oﬀ as ν approaches the 4/5 and 6/5 FQHE states. In Fig. 6.6c, we do not observe any changes in the number of quasicarriers participating in the resonance for 0.87 ≤ ν ≤ 0.91, where where there is a kink in fpk vs ν in Fig. 6.6b. The kink is of central importance; the rest of this section will detail its dependence on n. 52 Figure 6.4: Δ vs n for the 54 nm WQW, illustrating how the subband separation changes as we change n with the back gate. n = 2.41 × 1011 cm−2 is the as-cooled density with no gate bias. Fig. 6.7 shows spectra taken for many ν (0.8 < ν < 1.2) for n = 2.83 × 1011 cm−2 , 3.00 × 1011 cm−2 , and 3.20 × 1011 cm−2 . The smaller range in ν for Figs. 6.7b and 6.7c is due to the 14 Tesla limit of the superconducting magnet that was used to take these spectra. Relative to the n = 2.41×1011 cm−2 spectra of 6.6a the n = 2.83×1011 cm−2 spectra in Fig. 6.7b are shifted to higher frequency for ν between 1.1 and 1.15. The transition to a mode with enhanced frequency moves in towards ν = 1 as n is increased as shown in Figs. 6.7b and 6.7c. At n = 3.20 × 1011 cm−2 the resonance in almost its whole range for ν > 1 has moved to higher frequency. The eﬀect of reducing n on the pinning mode resonance is not as dramatic as 53 Figure 6.5: Re[σxx ] vs B for n = 1.94 × 1011 cm−2 to n = 3.20 × 1011 cm−2 , each trace vertically oﬀset for clarity. Taken at ∼ 30 mK and a frequency of 0.5 GHz. increasing n. Fig. 6.8 shows spectra for many ν (0.8 < ν < 1.2) for n = 2.17 × 1011 cm−2 , 2.08 × 1011 cm−2 , and 1.94 × 1011 cm−2 . We do not observe any changes in the pinning mode for ν > 1 as n is decreased. The transition observed in the n = 2.43 × 1011 cm−2 spectra for ν < 1 weakens and moves away from ν = 1 as n is decreased and has almost completely vanished by n = 1.94 × 1011 cm−2 . Fig. ?? shows how fpk vs. ν evolves as n is changed. For ν > 1, if n < 2.83 × 1011 cm−2 fpk fpk smoothly decreases as ν moves away from 1, but a region of enhanced fpk appears for n ≥ 2.83 × 1011 cm−2 . The ν of this transition between normal and enhanced fpk s moves in toward ν = 1 as n is increased. For ν < 1, a ν-region of enhanced fpk is visible for all n. Similar to the transition in fpk for ν > 1, the transition in fpk seen for ν < 1 also moves in toward ν = 1 as n is increased. Fig. 6.10 is S/fpk vs ν for all measured n for sample 54-1. The black lines show the theoretical prediction for full participation of the quasicarriers at density n∗ = nν ∗ /ν 54 Figure 6.6: a) Spectra from sample 54-1 showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency taken at many ν. ν of blue traces is marked on the right, at a temperature of 30 mK, n as-cooled, and normalized to ν = 1 with each spectrum proportionally oﬀset vertically from the last and the ﬁlling factor step between each spectra is ν = 0.005. b) The peak frequency of the pinning mode resonance (fpk ) vs. ν. c) S/fpk vs ν. The black lines are the theoretical prediction for full participation in the resonance. as in Eqs. 4.2. As shown in Fig. 6.10, there does not appear to be any change in participation in the resonance that corresponds to the feature we see in fpk vs ν. 55 Figure 6.7: Waterfalls of spectra from sample 54-1 show the real part of the diagonal conductivity (Re[σxx ]) vs frequency taken at many ν. ν of blue traces marked on the right, normalized using a ν = 1 spectrum, n marked at the top of each waterfall, and each is spectrum is proportionally oﬀset vertically. a) and b) The step in ν between spectra is 0.01. c) The step in ν between spectra is 0.005. There was no data taken above ν = 1.05 for n = 2.17 × 1011 cm−2 . 6.3.2 Sample 54-3 Sample 54-3 is sample 54-1 with the addition of a front gate. In principal the front and back gates let us vary n while maintaining a symmetric charge distribution in the well. However, on applying voltage to the front gate SdH oscillations were no longer well developed. In practice, we kept the charge distribution nearly symmetric 56 Figure 6.8: Waterfalls of spectra from sample 54-1 showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency, taken at many ν. ν of blue traces marked on the right, 30 mK, normalized using a ν = 1 spectrum, n marked at the top of each waterfall, and each spectrum is proportionally oﬀset vertically. a) and b) The step in ν between spectra is 0.01. c) The step in ν between spectra is 0.005. as took the data on sample 54-3. Study of the gate voltage dependence of the density of 54-3 indicates the gate voltage-density ratios are about the same for the back gate and front gate. The data presented in this section was taken using equal front gate and back gate voltages. Simulations indicate that the sample as-cooled (both gate voltages zero) is near its balanced state, δn/n < 0.1. The data with the sample 54-3 as cooled (n = 2.43 × 1011 cm−2 and zero gate 57 Figure 6.9: fpk in GHz vs ν as n is increased, from Figs. 6.7 and 6.8. voltages) is essentially the same as sample 54-1. Fig. 6.11a is a waterfall showing (Re[σxx ]) vs B. As for sample 54-1 there is a resonance peak for ν on either side of one. The resonance frequency decreases uniformly as ν moves away from ν = 1 as is typical for an IQHEWC resonance [2]. In Fig. 6.11c, we plot a waterfall of spectra from ν = 1 to ν = 0.8 for n = 2.43 × 1011 cm−2 , and symmetric charge distribution in the well. The resonance in 58 Figure 6.10: Sample 54-1, S/fpk vs. LL ﬁlling factor for all measured n. (See section 4.3.) Fig. 6.11c initially appears similar to the one in Fig. 6.11b, but the frequency of the resonance peak then increases near ν = 0.87. In Fig. 6.11d we plot a waterfall of spectra from higher ν (1 to 1.2) to illustrate that as n is increased, the resonance peak observed at higher ν does not appear to change. Fig. 6.12 is composed of six image plots of Re[σxx ] spectra on the ν-f plane, with the color representing Re[σxx ] as n is increased and Δ is minimized. As n increases 59 Figure 6.11: a-c) Re[σxx ] vs f at many ν and T = 30 mK. ν of black traces marked on the right. a) Spectra from ν = 0.8 to 1.2 with a ν step between each spectrum of ν = 0.0143. b) Spectra from ν = 0.8 to 1.0 with a step of ν = 0.0111 between spectrum. c) Spectra from ν = 1.0 to 1.2 with a step of ν = 0.0091 between each spectrum. we can see a region of enhanced resonance frequency develop for ν < 1. This region moves in from low ν towards ν = 1 as n increases. For this sample there is no similar region of enhanced f for ν > 1. Fig. 6.13 shows fpk vs ν for ν = 0.8 to 1.0 and n from 1.7 to 3.0 × 1011 cm−2 . As n is increased an enhancement in fpk appears ﬁrst at a ν = 0.87 then the ν of the transition to enhanced fpk moving in towards ν = 1 as n is increased. 60 Figure 6.12: Image plots of f vs ν with the color representing Re[σxx ] for a variety of n. n listed on in image plot in units of 1011 cm−2 . 6.4 54 nm WQW – summary The data from the 54 nm WQW samples have a number of common features. The 54 nm WQW samples (54-1, 54-2, 54-3) have spectra that exhibit a pinning mode 61 Figure 6.13: fpk vs ν, n is listed along the right of the plot in units of 1011 cm−2 . All fp k taken from spectra taken while maintaining symmetric charge distribution in the well. resonance. At suﬃcient n all these 54 nm WQW samples exhibit an n dependent transition to a region of increased frequency in fpk vs ν for ν < 1. For samples in which n was changed only with back gate bias, there is also a transition to a region of enhanced f that appears for ν > 1. Deﬁning a ﬁlling factor (νc ) of the transition to enhanced fpk we summarize the evolution of the transition with n in Fig. 6.14, the ﬁgure shows that νc increases toward with n. Deﬁning the transition ﬁlling factor (νc ) 62 as that of the peak in fpk vs ν. For all of the 54 nm samples transitions to regions of enhanced fpk move in toward ν = 1 as n is increased. Figure 6.14: Location of the transition to a higher frequency range in ν vs n for fpk vs ν for samples 54-1 and 54-3 for ν < 1. 6.5 42 nm WQW – pinning mode resonances around ν = 1 This section presents data for sample 42-1, a 42 nm WQW with both a front and a back gate, taken in a top-loading dilution refrigerator at a temperature of 50 mK. The data were taken in the milikelvin facility at the National High Magnetic Field Laboratory, in superconducting magnet number one (SCM1). For this sample as for sample 54-3 we could measure clean SdH oscillations. Again when we were trying to keep the charge distribution symmetric, we apply gate voltages so that the change in overall density due to each gate is the same. 63 Figure 6.15: Sample 42-1, 50 mK, n = 3.04 × 1011 cm−2 , and no bias on either gate. a) Many spectra taken at a diﬀerent ν normalized to ν = 1. ν is marked on the left of the graph for black spectra, spectra are proportionally vertically oﬀset, and the step between spectra is ν = 0.00625. b) fpk vs. ν. c) S/fpk vs. ν. 6.5.1 42 nm WQW – symmetric charge distribution In this section, we present data taken while keeping the charge distribution approximately symmetric, similar to the data for sample 54-3. Fig. 6.15a shows spectra taken at many ν, n as cooled, and both gates at 0 V. There is a resonance peak on 64 either side of ν = 1, its peak frequency decreases as ν moves away from ν = 1, and it vanishes as ν approaches ν = 4/5 and ν = 6/5. If we plot fpk vs ν, as shown in Fig. 6.15b, there is a slight bump at ν = 0.84. When we plot S/fpk vs. ν, as shown in Fig. 6.15c, we observe that S/fpk falls near the line for full participation, and that it does not have a feature at ν = 0.84 where we observe a bump in fpk vs ν. The spectra in Fig. 6.16b and 6.16c show a small range of weakly enhanced resonance frequency around ν = 0.84. This small range of increased frequency becomes more visible as n is increased. Fig. 6.17 shows fpk vs ν, for n = lower spectra were insuﬃciently well developed for ν greater than about ν = 1.12 for us to discern a resonance. There are two features in fpk vs. ν; a small bump occurs at ν ∼ 0.84 for n ≥ 3.26 × 1011 cm−2 , and a small bump occurs for ν ∼ 1.15 for n ≥ 3.49 × 1011 cm−2 . 6.5.2 42 nm WQW – asymmetric charge distribution We can explore the eﬀect of Δ on the pinning mode resonance if we hold n constant while making the charge distribution asymmetric, by using combinations of gate voltages such that δn = 0. Fig. 6.18 shows three sets of spectra taken at many ν, normalized to ν = 1, taken at 50 mK, with n held constant near 3.25 × 1011 cm−2 . Fig. 6.18a and Fig. 6.18c both show spectra taken at increased Δ compared to Fig. 6.18b to explore the eﬀect of making the charge distribution assymetric. To get a broad overview of the eﬀect of changing the symmetry of the well on the pinning mode for a variety of n, we can use a matrix of image plots as shown in Fig. 6.19. All of the image plots in Fig. 6.19 of f vs ν (color representing Re[σxx ]) are for n where we observe a kink in fpk vs ν. Plots where the well is kept symmetric on the left, plots of asymmetric charge distributions to the right of the thick black line, and plots of the same n are separated by the dashed vertical lines. Examining Fig. 6.19, we can see that changing the symmetry of the charge distribution in the well does not seem to aﬀect the fpk of the pinning mode resonance. Increasing Δ appears to 65 Figure 6.16: Sample 42-1, waterfalls of spectra taken at many ν. ν of black traces are shown on the left, oﬀset vertically proportional to ν, taken at 50 mK, normalized to ν = 1, at similar Δ, with n at the top of each waterfall, and a step of ν = 0.00625 between spectra. 66 Figure 6.17: Sample 42-1 fpk vs. ν, traces are oﬀset vertically by 0.5 GHz per trace with n = 3.58 not oﬀset, and n is in units of 1011 cm−2 . increase the width and decrease the total conductivity of the pinning mode resonance. On the left in Fig. 6.19, plots of S/fpk vs ν for various n show that the bump in fpk is associated with a change in participation in the pinning mode resonance, but that there does not seem to be any aﬀect on the bump from making the charge distribution in the well asymmetric other than to decrease the overall participation at the highest n. 67 Figure 6.18: Sample 42-1, waterfalls of spectra taken at 50 mK, normalized to a ν = 1 spectrum. ν of black traces is shown on the left, oﬀset vertically proportional to ν, a step of ν = 0.00625 between spectra, and n 3.25 × 1011 cm−2 . a) Asymmetric charge distribution, with gates set to give δn = 0.22 × 1011 cm−2 . b) Symmetric charge distribution, gates balanced. c) Asymmetric charge distribution, gates set to give δn = −0.22 × 1011 cm−2 . 6.5.3 42 nm WQW – summary To summarize the 42 nm WQW data, we have observed a pinning mode on either side of ν = 1 that changes only slightly as n is changed. As in the 54 nm WQW 68 Figure 6.19: Image plots of f vs. ν with the intensity of the plot representing Re[σxx ] from sample 42-1. n and δ in units of 1011 cm−2 as marked on each plot; plots on the left have symmetric charge distribution in the well (δn 0); plots right of the thick vertical line have asymmetric charge distribution with postive δn corresponding to more charge added by the front gate. 69 Figure 6.20: Sample 42-1, with fpk vs. ν on the left, and S/fpk vs. ν on the right. Symmetric states are black; non-symmetric with the voltage biased more on the back gate in blue, non-symmetric with the voltage biased towards the front gate in red; with the densities on the graph in units of 1011 cm−2 . spectra, we identify the pinning mode resonance (minus the additional feature seen in fpk vs ν for n ≥ 3.26 × 1011 cm−2 ) as a pinned IQHEWC. The bump we observed in fpk vs ν appears on both sides of ν = 1, at least at the largest n. Unlike sample 54-1 or 54-3 there is little sensitivity of the ν of the bump to changing density. It is 70 unclear if the small bump in fpk vs ν seen in sample 42-1 is related to the large kink fpk vs ν seen in the 54 nm samples. 6.6 Summary and analysis The 42 nm WQW (42-1) and the 54 nm WQW (54-1, 54-2, 54-3) have spectra that exhibit a pinning mode resonance, and these pinning mode resonances imply the existence of electron solids in our samples. This pinning mode resonance has features that evolve as n is changed. The transition we observe in the 54 nm WQWs spectra from a region in ν of lower fpk to a region of higher fpk , we interpret as a transition between an IQHEWC and an unknown electron solid. The location in ν of this transition ﬁlling factor moves in toward ν = 1 as n is increased. As shown in Fig 6.14 for ν < 1, samples 54-1 and 54-3 have νc vs n in agreement. For ν > 1 sample 54-1 shows a transition for n ≥ 2.83 × 1011 cm−2 . Sample 54-3 does not show the same transition for ν > 1. The diﬀerence between the samples is that for sample 54-3, n is changed with front and back gate voltages set to at least roughly preserve a symmetric charge distribution in the well. From the thesis of Shabani [103], we know that an asymmetry in the charge distribution can stabilize insulating states. Thus the charge distribution asymmetry in sample 54-1 is likely responsible for making the unknown electron solid state preferable for ν > 1 as n is increased. We do not understand why charge asymmetry would stabilize the unknown electron solid for lower n for ν > 1, but does not change νc vs n for ν < 1. We can draw a parallel to the RIQHE that Liu et al. [7] observed in a piece of 42 nm WQW from the same wafer as our sample. The RIQHE he observed appeared ﬁrst for ν < 1, then moved in toward ν = 1 as n was increased, similarly to the behavior of the transition in fpk versus ν (for ν < 1) in the 54 nm samples we measured. As they increased n further, they observed an RIQHE appear for ν > 1 and move in toward ν = 1, again similar to the behavior of the transition in fpk vs ν that we report in our 71 samples. Liu et al. proposed that the RIQHE in the 42 nm WQW they measured was from a transition between the ν = 1 IQHEWC and an electron solid comprised of CFs [7]. Our results build on and expand those of Liu et al. While they are only able to infer the existence of multiple electron solid phases, we observe pinning mode resonance that has ν-regions of enhanced fpk that we identify as a new electron solid phase. We will now focus on the electron solid with enhanced fpk in 54 nm WQW, and discuss the possibilities for its composition. The enhancement of fpk is either from a decrease in the eﬀective electron-electron interaction between quasi-carriers in the electron solid, an increase in the strength of the eﬀective disorder on the quasi-carriers composing the electron solid, or a combination of both. Fig. 6.21 shows spectra (Re[σxx ] vs f ) taken at n = 2.41 × 1011 cm−2 , various temperatures, and two diﬀerent ν from sample 54-1. If we compare Fig. 6.21a (taken from a region of ν where the IQHEWC is) and Fig. 6.21b (taken from a region of ν where the unknown electron solid is), we can see that the pinning mode resonance in the region where the unknown electron solid is the dominant phase melts earlier. While the enhanced-fpk region at ν = 0.85 shows the resonance disappear at lower temperature then the unenhanced-fpk resonance at ν = 0.95 we can not ascribe it to a diﬀerent solid phase at ν = 0.85. This is because the melting point of a Wigner solid depends upon the ﬁlling factor (ν ∗ ), in this case as described in Ref. melting. A decrease in quasi-carrier density of the IQHEWC would reduce the strength of the electron-electron interaction and increase fpk [12]. If there is a change in quasicarrier density, we would expect a corresponding dip in S/fpk vs. ν since Fukuyama’s sum rule gives us a measure of the density of chargecarriers are participating. From Fig. 6.10, we can observe that S/fpk vs ν indicates participation at the full quasi carrier density n∗ . The unknown electron solid is probably not a bubble or stripe phase. Bubbles and 72 Figure 6.21: Sample 54-1, (Re[σxx ] vs f ) taken at n = 2.41 × 1011 cm−2 , and various temperatures as marked on graph. a) taken from a region of ν near ν = 1 where fpk is not enhanced. b) from a region of ν where fpk is enhanced. stripes are not predicted to be stable in the lowest LL, and have never been observed in the lowest LL [12, 79, 80]. Another possibility is an electron solid composed of skyrmions [83, 84]. These electron solid have been seen around ν = 1 [82]. The unknown electron solid in our samples is unlikely to be an electron solid composed of skyrmions, as skyrmions will shrink when the Zeeman energy increases and our samples have a g-factor 4 to 5 times higher than the 30 nm QW in which the skyrmion electron solid was observed. It is conceivable that under the experimental conditions, a two-component bilayer solid could occur. Narasimhan and Ho [16] have described bilayer solid phases in73 cluding the eﬀects of interlayer tunneling with phase diagrams in the space of the interlayer separation d/a, and the tunneling Δ/EC , where a is a typical intercarrier spacing (a2 = 2/n∗ ), and EC is the Coulomb energy of point charges at that separation. For large tunneling, one-component solids (no interlayer staggering) are predicted, since charge transfer has an energy cost related to the tunneling (subband) energy. In our experiments taking νc = 0.9, and n = 2.8 × 1011 cm−2 , a = 80 nm and d/a = 0.4, giving EC = 16 K. From simulations Δ ∼ 25 K at this n, so Δ/EC ∼ 1.5. For d/a ∼ 0.4, Narasimhan and Ho [16] calculate a transition to two-component rectangular lattice occurs at Δ/EC ∼ 0.6, so we expect the transition to be well within the one component regime. It is not clear how Δ increased by unbalance will aﬀect the bilayer solid phase diagram. In a low density (n = 1.0 × 1011 cm−2 ) 50 nm WQW 2DES system, a transition between a phase (referred to as the A-phase) with a pinning mode that shows dependence on wavevector and the terminal WC (referred to as the B-phase) has been observed [3]. This A-phase was observed below ν = 2/9, was reentrant around the 1/5 FQHE state, and then crossed over into the B-phase which dominates for ν ≤ 0.12. One of the possibilities for the A-phase is a CF WC or a charge density wave composed of CF. The ν range of this A-phase (speciﬁcally 0.2 < ν < 0.12) appears similar to the ν ∗ range in which we see our unknown electron solid. A transition between an IQHEWC and an electron solid composed of CFs has been proposed as the source of a n dependent RIQHE observed near the ν = 1 plateau [7]. Some calculations indicate that various types of CF Wigner crystal are the applicable ground states for the low ν (ν 1/5) solid phase [107–109]. To see if the new electron solid we observed in our WQW samples was related to this A-phase, we prepared sample 54-2 with an 80 μm wide slot CPW on its surface. fpk vs ν in Fig. 6.22 shows no change between sample 54-1 (30 μm) and sample 54-2 (80 μm). Hence, the new electron solid in our WQW does not show the same 74 Figure 6.22: fpk vs. ν of the pinning mode for samples 54-1 (circles) and 54-2 (squares). Both from spectra taken at n = 2.41 × 1011 cm−2 wavevector dependence as this A-phase of Ref. [3]. Even though we do not observe any wavevector dependence in our unknown electron solid, an electron solid composed of CF still a possibility. 75 CHAPTER 7 SIGNIFICANCE AND FUTURE WORK This chapter places the results presented in this dissertation into broader context and discusses some possibilities for future work. 7.1 7.1.1 EMPs in magnetic ﬁeld gradients and MEMPs Signiﬁcance In systems that show the quantum Hall eﬀects, edges are of crucial, fundamental importance. The picture of the edge of these systems has developed over the long history of the study of 2DES in high ﬁeld, starting from the original Halperin picture [110] of the edge, and continuing with the use of the Landauer-Büttiker formalism [111] to describe non-local transport eﬀects. The nature of the edge states and of the reconstruction of the edges has been illuminated by calculations [53, 112], and by experiments on tunneling between edge states [113–117], and on tunneling between edge states and bulk material [118]. In particular, experiments addressing fundamental physics of the fractional quantum Hall eﬀect, including fractional charge [119, 120], and more recently the nature of non-Abelian states [121, 122] have relied on edge-state transport and tunneling. EMPs, while capable exhibiting features due to edge reconstruction [44], can be mostly explained by simple hydrodynamic models [21, 24, 29, 48], which combine 76 the electrostatics near the edge with the Hall conductivity. On one level the large eﬀort we expended in studying magnetic edges, was an attempt to access 2DES on a small length scale without a density step, and so to study edge states with altered balance between electrostatics, exchange and the spatially varying compressibility of the 2DES. Less remotely, EMPs in magnetic ﬁeld gradients and MEMPs are novel plasma excitations, which are clearly related to the way the 2DES is conﬁned, as shown in our work on the EMP in gradient ﬁeld. They are diﬀerent from normal EMPs and also from the well-known [50] (magneto)plasmon excitations. Coupling to spin at magnetic edges with only an in-plane external ﬁeld has been reported [54]. Particularly in low magnetic ﬁelds (with the sort of magnetic ﬁlm we used), it is possible to envision magnetic quantum dots [123], in which conﬁnement is exclusively magnetic, or augmented by the gradient ﬁeld of a ferromagnet. MEMPs or EMPs in magnetic ﬁeld gradient are closely related to the lower hybrid modes that such magnetic dots would exhibit, and microwave measurements could provide a contactless means of coupling to such dots. 7.1.2 Future work, EMPs in magnetic ﬁeld gradients and MEMP search In light of the work we present in this dissertation, the approach to better understanding of EMPs in magnetic ﬁeld gradients is reasonably clear. First, we would like to improve our understanding of the ferromagnetic permalloy ﬁlm, either by measuring the magnetization of the ﬁlm directly (using a superconducting quantum interference detector or a magnetic force microscope) or by measuring it indirectly through the DC magnetoresistance of the 2DES under the magnet. A better understanding of the magnetization of the permalloy ﬁlm will allow us to evaluate if the ∂Bz /∂x term in Eq. 3.9 is correct. 77 Second, we would like to tilt the sample to provide an in-plane ﬁeld to allow us to change the inhomogeneity from a dip to a spike based on the direction of the in-plane ﬁeld as well as increase the overall magnitude of the magnetic inhomogeneity from the permalloy ﬁlm. This will allow us to change the sign and value of the ∂Bz /∂x term and see if the velocity for the EMP we measure changes comparably. Third, we would like to vary the gate bias to move where the density gradient is located in relation to the magnetic ﬁeld inhomogeneity. Varying the location of the density gradient is another way to probe the dependence of vp on Bz and ∂Bz /∂x and explore the accuracy of our model of an EMP propagating in a magnetic ﬁeld gradient. With improved understanding of the EMP in magnetic ﬁeld gradient, it may be that the parameters (dispersion, damping, chirality etc.) of a MEMP will be predicted [124]. In that case a more focused search for MEMPs can be undertaken. 7.2 7.2.1 A new electron solid in a WQW Signiﬁcance Electron solid phases stabilized by the electron-electron interaction, as discussed at length in Chapter 4, exist in many forms in systems that show the quantum Hall eﬀect. The shift to enhanced fpk that we have seen substantiates the suggestion in ref. [7], that the WQW samples exhibit a previously unknown solid phase. Besides demonstrating the existence of the pinning mode, and its sensitivity to the transition between solids, our work essentially extends ref. [7] to n for which it shows the νrange of the IQHE centered at ν = 1 is signiﬁcantly extended. Hence we studied a transition between solids within the IQHE around ν = 1. The most striking feature of the new solid is the dependence of the transition ﬁlling νc on n, and the close of approach of νc to ν = 1 is increased. 78 It is likely that the region nearest ν = 1, without enhanced fpk , is a solid of quasicarriers similar to those found in a narrower (30 nm) QW of similar n [2] and in a 50 nm QW of lower n [82]. This follows since narrower QWs require larger n [7, 125] to show the RIQHE features in dc that were reported in ref [7], and at low n the fpk vs ν curves do not show an enhanced region. The conclusion of Chapter 6 compared the various known electron solids with the observed new solid phase, but there is as yet no clear identiﬁcation of the nature of the new solid. It is clear that the combination of a wide well and suﬃcient n is required to see the new phase, though even in the WQWs we studied the ν = 1 IQHE gap is due to spin, and not to subband energy. The softening of the short-range eﬀective interaction is likely of importance, and Jastrow correlations as in a CF WC [107–109], may play a role, possibly in combination with order from spin or some modulation or correlation involving the vertical wave function but more subtle than a two component crystal. The key fact that we have elucidated here is that the transition can approach ν = 1 so closely, and so correspond to small quasicarrier density n∗ . 7.2.2 Future work in WQW 2DES Future work at higher ﬁelds (∼ 30 T) will allow us to see if there are also multiple solid phases in the WQW 2DES at the low ν termination of the FQH series. If so, we can compare these phases with the behavior of the new electron solid we have observed around ν = 1, and also with the “A and B” phases [3] discussed in Section 4.5. Our collaborators on the WQW project already plan high ﬁeld DC magnetotransport measurements of similar WQW 2DES, and we hope to use their results as a starting point for higher ﬁeld measurements. For future work in exploring this new electron solid at B < 15 T there are three approaches. The ﬁrst is to measure WQW 2DES with similar mobility and n, but with diﬀerent well widths. Larger well widths may have other eﬀects than reducing the density 79 required to see the new phase [7, 125] We could search for a relationship between the width of the well and νc vs n. The magnitude of the enhancment in fpk shift may be dependent on the well width either through the altered sheer modulus of the solid or through altered disorder. In particular disorder due to interface roughness [78] well width is likely to be sensitive to well width [126] . The second (related) approach is to vary the charge asymmetry in the well systematically to see if νc , as a function of n, changes. We have already seen that increasing the charge asymmetry causes our new electron solid to appear for ν > 1, and we would like systematically study the eﬀect of δn on νc . Third, we would like to apply an in-plane magnetic ﬁeld. The addition of an in-plane ﬁeld has been observed to stabilize electron solids at higher ν [127]. 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Florida State University University (08/2003-06/2006) Teaching Assistant Performed teaching duties and laboratory assistance in undergraduate physics labs. San Diego State University (01/2001-05/2003) Undergraduate Research Assistant. Undergraduate Research Assistant. 93 Current Research I am currently involved in a variety of research that can be separated into three broad categories. These categories are: pump probe measurements of narrow gap semiconductors, using surface plasmon resonance to functionalize surfaces for nano-assembly, and setting up a cyrostat and probe to do optically detected magnetic resonance measurements. References Dr. Lloyd W. Engel: Scholar/Scientist, National High Magnetic Field Laboratory, 1800 E. Paul Dirac Dr., Tallahassee, Fl, 32310, engel@magnet.fsu.edu, 850-644-6980 Dr. Nick Bonesteel: Professor of Physics, Florida State University, 315 Keen Building, Tallahassee, Fl, 32310, bonestee@magnet.fsu.edu, 850-644-7805 Dr. Stephen McGill: Assistant Scholar/Scientist, National High Magnetic Field Laboratory, 1800 E. Paul Dirac Dr., Tallahassee, Fl, 32310, mcgill@magnet.fsu.edu, 850-644-5890 Presentations/Posters Moon, B.H.; Magill, B.A.; Engel, L.W.; Tsui, D.C.; Pfeiﬀer, L.N. and West K.W., Pinning mode of 2D electron system with short-range alloy disorder, American Physical Society March Meeting, Dallas, TX, March 21-25 (2011) Magill, B.A.; Polyanskii, A.A.; Engel, L.W.; Lilly, M.P.; Simmons, J.A.; Reno, J.L., Microwave absorsoption of a 2D electron system in a spatially varying magnetic ﬁeld, American Physical Society March Meeting, Portland, OR, March 17-19 (2010) Magill, B.A.; Engel, L.W.; Lilly, M.P.; Simmons, J.A.; Reno, J.L., Microwave modes of a two dimensional electron system in a spatially varying magnetic ﬁeld, American Physical Society March Meeting, Pittsburg, PA, March 16-20 (2009) 94 Magill, B.A.; Engel, L.W.; Lilly, M.P.; Simmons, J.A.; Reno, J.L., Microwave modes of a two dimensional electron systems near macroscopic ferromagnets, American Physical Society March Meeting, New Orleans, LA, March 10-14 (2008) Magill, B.A.; Engel, L.W.; Lilly, M.P.; Simmons, J.A.; Reno, J.L., Microwave modes of a two dimensional electron system in a spatially varying magnetic ﬁeld, American Physical Society March Meeting, Denver, CO, March 5-9 (2007) Urbano, R.R.; Pires, M.A.; Bittar, E.M.; Rettori, C.; Pagliuso, P.G.; Magill, B.A.; Oseroﬀ, S., ESR of Gd3+ in the Intermediate Valence Y bAl3 and its Reference Compound LuAl3 , American Physical Society March Meeting, Los Angelas, CA, March 21-25 (2005) 95

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