«Yong P. Chen A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for ...»
Quantum Solids of Two Dimensional
Electrons in Magnetic Fields
Yong P. Chen
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
c Copyright by Yong P. Chen, 2006.
All Rights Reserved
I certify that I have read this thesis and that
in my opinion it is fully adequate, in scope and in quality,
as a dissertation for the degree of Doctor of Philosophy.
Daniel C. Tsui (Principal Adviser) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy.
Lloyd W. Engel (National High Magnetic Field Lab.) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy.
Herbert A. Fertig (Indiana University)
Approved for the Princeton University Graduate School:
Dean of the Graduate School iii Abstract This thesis studies the solid phases of two-dimensional electrons subject to a perpen- dicular magnetic ﬁeld (i.e., the “quantum Hall system”). Traditionally, such a solid, known as “Wigner cystal” (WC), is believed to be the ground state of a two dimen- sional electron system (2DES) when the Landau level (LL) ﬁlling factor ν=nh/eB (n being the electron density and B the magnetic ﬁeld) is suﬃciently small (thus follow- ing the termination of quantum Hall states). Due to disorder in realistic samples, the solid is pinned, therefore insulating. Collective oscillation of crystalline domains of the solid around disorder gives rise to a “pinning mode” resonance in the frequency dependent conductivity, which we measure with rf/microwave spectroscopy.
The resonance has interesting behaviors in its dependence on samples, n, B and temperature (T ) and contains valuable information about disorder. For example, we are able to show that the most relevant disorder that pins the solid comes from the interface that vertically conﬁnes the 2DES, with a (sample dependent) disorder correlation length that can become shorter than 10 nm.
Most importantly, the resonance is a characteristic signature of pinned electron solids, as well as a tool to study their physical properties. We show that many such solid phases can exist, in diﬀerent regimes of ν; and their properties also depend largely on ν, which captures the quantum correlation between electrons.
Among the new solid phases that we have discovered in the state-of-the-art 2DES samples are the Wigner crystal phases formed in the partially ﬁlled top LL around integer Landau ﬁllings. In high LLs, these “integer quantum Hall Wigner crystals” (IQHWC) join with other phases, such as the bubble and stripe phases, to form a rich array of charge density wave phases.
In the lowest Landau level (LLL), we have observed two distinct solid phases, which we name as solid “A” and “B” phases respectively. The “A” phase is observable for ν2/9 (but reentrant around the ν=1/5 fractional quantum Hall liquid iv (FQHL)) and transitions to the “B” phase which dominates at suﬃciently low ν. The two phases coexist in intermediate ν (0.18 ∼ ν ∼ 0.12). Moreover, the resonance of “A” phase is found to show dispersion with respect to the size of transmission line, indicating that “A” phase has a large correlation length. Many-body quantum correlations appear to play an important role in giving rise to the diﬀerent solids. In particular, “A” phase appears to be a solid intimately related to FQHE. Possible interpretations involving a composite fermion crystal and/or a FQHL-WC mixed phase are discussed.
We have also studied the T -dependence of the pinning mode resonance of a Wigner cystal (in high magnetic ﬁelds) and in particular its melting behavior. In a given sample, the melting temperature (Tm ) is found to be mainly determined by ν, in contrast to the case for any other known solids (including, particularly, a classical 2D electron solid) whose Tm is determined by the solid density n. This not only attests to the quantum solid nature of the Wigner cystal in our samples; but also constitutes, to our best knowledge, the only example of a solid whose Tm has been shown to mainly depend on inter-particle quantum correlation (here through ν).
The appendices of the thesis contain more background/supporting material, as well as some theoretical results. In one appendix we develop a model for pinned bilayer Wigner crystals, helped by the knowledge of pinning and disorder that we have earlier learned from our experiments on single layer 2DES. We propose that pinning mode resonance can distinguish a (pesudospin) antiferromagnetic WC (AFMWC) from a ferromagnetic WC (FMWC), the latter of which can be viewed as a supersolidlike phase. Our model shows that pinning is enhanced in a FMWC, which possesses interlayer coherence (IC), compared to an AFMWC without IC and predicts a decreasing pinning mode frequency (fpk ) with the eﬀective layer separation in a FMWC, opposite to the behavior in an AFMWC, and an abrupt drop of fpk at a FMWC to AFMWC transition.
A large number of people have kindly helped me in one way or another in relation to this thesis, the bulk work of which was conducted in two places (Princeton and Tallahassee) and between 2002 and 2005.
First, I thank my thesis advisor Daniel Tsui, who has been largely responsible for my transition from a pure mathematician to an experimenting physicist since I started this PhD in 1999. I am particularly grateful to him for introducing me the opportunity to work on the problem of electron solid (the subject of this thesis) and work in National High Magnetic Field Laboratory (NHMFL) in Tallahassee, both of which turn out to be extremely scientiﬁcally fulﬁlling. His encouragement, trust as well as critics, his scientiﬁc insights, taste and judgment, and his style of inspirational education have all been blessings for my entire PhD and will continue inﬂuencing me profoundly. Thank you, Dan, for teaching me the “dao” of science.
I am also very fortunate to have worked with another superior teacher, Lloyd Engel, who is my co-advisor and host in NHMFL. Without the cutting edge microwave technology that Lloyd had developed in the earlier years, all experimental discoveries described in this thesis would remain as theorists’ fantasies at best. Lloyd made his seemingly inﬁnite supply of knowledge in experimental physics accessible to me even in the most weird hours, and burnt many late nights’ oil helping edit my papers.
Thank you, Lloyd, for also making a life in science so entertaining.
Two of my co-workers, Rupert Lewis and Sambandamurthy (”Murthy”) Ganapathy, both postdocs based in Tallahassee, were directly involved with the research in this thesis. Rupert not only started me up in the microwave measurements, he is my major collaborator in a large part of my thesis research (especially on integer Landau level Wigner crystals and “A and “B” solid phases in lowest Landau level).
I also thank Rupert for sending me many of his data on bubble and stripe phases.
Murthy has been my major collaborator on Wigner crystal melting as well as oﬀering vi great philosophical inﬂuence on data analysis. Their participation and help in the measurements during many of my “magnet times” lead to maximum utilization of the magnet resources and many data that I would not have been able to measure if running alone. Their contributions and roles will be more speciﬁcally acknowledged later in the individual chapters. Above all, thank you Rupert, and Murthy, for making our lab such a fun fraternity.
I also thank fellow graduate students in the microwave lab Zhi-hai Wang and Brenden Magill for assistance in some of the data acquisition and Zhi-hai for discussion and communication about his bilayer Wigner crystal experiments (which motivated the pinned bilayer Wigner crystal model in Appendix K).
Over the years, I have greatly beneﬁted from many other former or current postdocs in the group. I thank Peide Ye, who taught me the fabrication of microwave samples and provided many valuable comments on my research; Gabor Csathy, who taught me a great deal on low temperature experiments, provided me many his DC measurement data and was always ready to help in many other ways; Guillaume Gervais, for a source of scientiﬁc excitement; Michael Hilke, who mentored my ﬁrst low temperature research project in Princeton; Wei Pan and Zhi-gang Jiang, for many interesting discussions; Jin-jin Li for help on thin-ﬁlm measurements as well as Leonid Rokhinson and Hwa-yong Noh for general lab help earlier-on in my PhD.
I have also received kind help and learned a great deal from many of the fellow graduate students in Prof.Tsui’s group, especially Keji Lai and Ravi Pillarisetty on clean room processing, Wanli Li for providing valuable sample information and I also thank Amlan Majumdar and Jie Yao for earlier help in the lab.
The high quality GaAs/AlGaAs samples for studying electron solids were all provided by Loren Pfeiﬀer and Ken West from Bell Labs. They were processed in the fabrication facilities at Princeton EE department. I am particularly grateful for generous help from Eric Shaner and Guillaume Sabouret in Lyon’s group as well as Nathan vii Bishop and Oki Gunawan in Shayegan’s group.
All microwave measurements were performed at NHMFL (supported by NSF Cooperative Agreement No. DMR-0084173 and by the State of Florida). Most high ﬁeld experiments were conducted in the user magnets with technical assistance from many scientists and engineers in the NHMFL operations department. I particularly thank Eric Palm, Tim Murphy, Glover Jones (and earlier April Teske) for their assistance and patience with my experiments, as well as Scott Hannahs (especially for his help on Labview) and Bruce Brandt for help on magnet times. I also thank Bob Smith and John Pucci for assistance in operating the 45T hybrid magnet and for honoring many of my urgent helium request, Mike Davidson for assistance in sample microscopy, Andy Powell and Lee Bonninghausen in electronics shop and numerous staﬀ in the magnet control room.
Many support and secretarial staﬀ have helped made easy the administrational stuﬀ related to my research. I particularly thank Alice Hobbs in the condensed matter group of NHMFL, Jamie Kubian in Princeton EE department, as well as many others that have worked over the years as program assistants for Tsui/Chou groups in Princeton.
Financial supports for our research were provided by Air Force Oﬃce of Scientiﬁc Research, Department of Energy, National Science Foundation and NHMFL in-house research program. I also acknowledge a Gordon Wu fellowship during the earlier part of my PhD.
I thank also a number of faculty members in Princeton or NHMFL for inspirational interactions: Ravin Bhatt, Steve Lyon and Mansour Shayegan; Nick Bonesteel, Jan Jaroszynsky and Kun Yang. I particularly thank Prof. Bhatt for lending me a panoramic oﬃce in his group during my last year in Princeton. I have also beneﬁted greatly from numerous discussions with Cheng-gang Zhou and Dima Novikov in Bhatt group, Emanuel Tutuc in Shayegan group and Xin Wan and Akakii Melikidze in Kun viii Yang’s group. Of course I also thank Professors Shayegan and Lyon for being in my ﬁnal public oral examination committee (in addition to my advisor).
I also acknowledge discussions that have inﬂuenced my work or writing of this thesis with Alan Bishop, Moses Chan, Tat Chui, Rene Cote, Herb Fertig, Misha Fogler, Bert Halperin, Ganpathy Murthy, Woowon Kang, Michael Lilly, Leonid Levitov, Nick Read, Boris Spivak, Philip Stiles, Xiao-Gang Wen, Bob Willett, Clare Yu and many others in the academic community who have kindly shared with me their knowledge and enthusiasm about relevant physics subjects. I am particularly grateful to Prof.
Herb Fertig who read my thesis and gave me feedbacks at a short notice.
I thank all my friends in Princeton, as well as many Seminoles and even gators, who, with all the help, inspiration and pleasant distractions, have made my PhD and the three years of residence in Florida particularly memorable time. I also thank my old friend Bo in Institute of Advanced Studies, who kindly provided accommodation during some of my sample processing trips to Princeton, and interesting comments to my work from a string theorist’s perspective.
Now at the end of my PhD, I want to thank the people who kindly recommended me to enter Princeton: Wanda Andreoni in IBM Zurich Lab, Paul Schechter in MIT physics department, and my late MIT mathematics advisor Gian-Carlo Rota, who unfortunately passed away shortly after writing his recommendation for me, and has been a unique source of inspirations in all the years.
A large part of my thesis, believe it or not, was actually written after I started working as a postdoc in Randy Hulet’s group in Rice University. I thank all my colleagues there for their understanding of this apparent violation of Lorentz invariance.
Finally I thank my parents Chen Xing-hua and Li Xiang-yuan, for everything.
Although knowing little about my research, they have always educated me since my childhood to look for truths and beauties underneath the surface, which is particularly ﬁtting in the case of studying 2D electrons! To them I dedicate this thesis.
Introduction The stories that we are going to tell in this thesis are about the solid phases formed by two dimensional electrons whose 2D motion is quantized by a perpendicular magnetic ﬁeld. Realized in semiconductor structures, these electron solids embody eﬀects from both electron-electron interaction and disorder, two of the fundamental themes in modern condensed matter physics. Quantum correlations often reveal themselves as the underlying driving force in forming a variety of solid phases as well as enabling many of their fascinating yet intriguing properties.