Electronic Liquid Crystals

Systems dominated by long-range repulsive forces often exhibit a uniform phase, while those dominated by short-range attractive forces tend to exhibit phase separation into compact structures. Competition between these two kinds of forces often leads to spatially inhomogeneous and anisotropic phases (such as ‘stripe’ or ‘bubble’ phases, depending on the area fraction of the sample) in a number of physical situations. Classical examples abound, such as blockcopolymers, ‘pasta phases’ of the crusts of neutron stars or DNA-intercalated lipid bilayers. One of my favorites is stripe formation in ferrofluids (due to competition between dipole-dipole repulsions and the attractive surface tension):

The image and a really neat movie of the stripes forming can be found at Ken Cooper’s website at Caltech. The experiment strikes me as being remarkably simple - just press an immiscible mixture of ferrofluid and IPA between two glass disks, and apply a magnetic field! The really interesting thing is that recent work has indicated such phases in strongly correlated electron systems as well (such as stripe phases of cuprate superconductors or manganates/nickelates) in which repulsive Coulomb interactions compete with the effective short range attractive Pauli exchange interaction.

A striking example is in the transport measurements of figure 1 (click ‘Read on’ below to see it), indicating significant anisotropy in the resistivity of a two-dimensional electron gas (2DEG) in very clean MBE-fabricated GaAs/AlGaAs heterostructure samples in the presence of moderately large magnetic fields. I’ll only focus on the results of Lilly et al. [1] here, but this has been confirmed independently by another group as well [2]. Recent theoretical work by Dorsey, Fradkin, Kivelson, Oganesyan, Radzihovsky and undoubtedly many others [3-4] explains this phenomenon via considerations of an orientationally ordered Quantum Hall Nematic (QHN). This naturally gives rise to a broader notion of ‘electronic liquid crystals’ [5] – that is, quantum mechanical analogues of classical liquid crystal phases in which rotation and/or translational symmetry (as dictated by the point-group symmetry of the crystal) is spontaneously broken.

Figure 1 (click for larger version)
(a) Transport anisotropy in 2DEG sample in moderately large magnetic fields. Solid and dashed lines represent different measurement directions across the sample, while arrows indicate different values of the Landau level filling fraction ν (from [1]).
(b) Monte Carlo simulation of classical 2D XY model with coupling J and external field h for a 100 × 100 lattice, fit to the data of [1]. The best fit is found for J = 73.5 mK, h = 0.05J = 0.5mK, T_c = 65mK (from [4]). Systems dominated by long-range repulsive forces often exhibit a uniform phase, while those dominated by short-range attractive forces tend to exhibit phase separation into compact structures.

Experimental Results
2DEGs have clearly been the source of very rich physics for over twenty years. When a GaAs/AlGaAs heterostructure (doped with Si) is produced using MBE, electrons are confined to the two-dimensional interface between the two, the confining potential of which is due to the band gap mismatch between the two materials. When placed in a magnetic field perpendicular to the 2DEG, the energy spectrum of the electrons is discretized into Landau levels (LLs) with a filling fraction per spin of v=hn/eB (where n is the electron density), and the conductance of the sample is quantized. At very high magnetic fields, only the lowest Landau level (N = 0) is occupied, leading to the fractional quantum Hall effect (FQH) regime in which the physics is dominated by electron-electron interactions. At intermediate magnetic fields (e.g. ~ 2–3 Tesla), more than one Landau level is occupied, and other interesting effects may arise.

In their work, Lilly et al. [1, 6-8] studied high mobility (~ 10⁷ cm²/Vs) MBE-grown (rotated during growth) samples in the intermediate magnetic field regime (i.e. N ~ 2). Among their numerous results, perhaps the first is the most striking, and is indicated in fig. 1a. Without rotating the sample, the researchers measured the resistivity of the sample across two perpendicular in-plane directions at T = 25 mK, and found a large anisotropy in the measurements, with a resistance of nearly up to 100 times larger in one direction than another for v>4 (i.e. N > 2 ). This anisotropy aligns with the GaAs crystal axis, and is clearly temperature dependent, as indicated in fig. 1b.

Electronic Liquid Crystals: The Quantum Hall Nematic
(I’ll only briefly summarize the theory here, because this post is getting long.)
As early as 1979, Fukuyama, Platzman and Anderson [9] proposed interesting phase transitions in a 2DEG due to the existence of charge-density-waves (CDWs) as the ground state of a partially filled Landau level, within the Hartree-Fock (HF) approximation. This treatment was then extended to higher Landau levels (for example, within the HF approximation by Koulakov, Fogler et al. [10] and shown to be exact by Moessner and Chalker [11], both in 1996). Near half-filling of the LL, the CDW is a uniformly directed quasi-one dimensional stripe phase. The electron density of the highest LL in such stripe states alternates between no filling and full filling, and the wavelength is of order the cyclotron radius – that is, such stripe states can be thought of as arrays of ‘sliding’ Luttinger liquids. The principle behind this is a general theme for such systems: if the two competing interactions (Coulomb/exchange) balance themselves out in just the right way, interesting phases of matter (e.g. bubbles or stripes) can form.

Building on previous work on high-temperature superconductors, Fradkin and Kivelson have constructed a general framework for these systems that incorporates this treatment. They have introduced and developed the notion of ‘electronic liquid crystals’, quantum mechanical analogues of classical liquid crystal phases as a general feature of strongly correlated electron systems, in which repulsive Coulombic interactions compete with attractive exchange interactions (such as the competing Hartree and Fock terms in the HF treatment).

These phases can be classified according to the degree of broken symmetry:
- Insulating electron crystals break translational symmetry (in both in-plane directions), rotational symmetry, and time-reversal symmetry (which is broken by the external magnetic field), thus effectively restricting an integer number of electrons to each unit cell – for example, Wigner crystals or ‘bubble’ phases.

• If translational symmetry is only broken in one direction, one has what is known as a quantum smectic liquid crystal.
• If neither translational symmetry is broken, the 2DEG is characterized as being a quantum nematic liquid crystal
• If only time-reversal symmetry is broken, the system is known as an isotropic quantum fluid. The incompressible (i.e. lacking low-energy excitations) quantum Hall plateau states (characterized by zero resistivity) are an example of this.

However, complexities arise. For example, in the context of the measured transport anisotropy in 2DEGs, simple considerations of striped states do not give the whole picture - considering transport in a static (that is, non-fluctuating) CDW striped state gives a prediction that is overly anisotropic. Dorsey, Radzihovsky, and Wexler (among other people) have studied some of the complexities involved in several papers [3, 13] and have found a number of interesting results:

• If one includes fluctuations of these stripe states, the system can be thought of as a quantum Hall smectic [5, 12], the elastic properties of which have been studied by Wexler and Dorsey [13]. They found that the low-energy perturbations of the CDW striped state (within the HF approximation) are equivalent to those of a conventional smectic liquid crystal. In addition (in the spirit of Toner and Nelson [14]) if one considers dislocations, the quantum Hall smectic ‘melts’ and the system is better thought of as a two-dimensional quantum Hall nematic. Since the energy of a disclocation is finite, the low-temperature phase of such 2DEGs is best described as a quantum Hall nematic.
• Given this, one may additionally consider the possibility of the nematic to isotropic transition via dislocation binding – that is, a Kosterlitz-Thouless transition [15]. Building on work by Nelson and Pelcovitz [16] using the 2D XY model, Wexler and Dorsey [13] have estimated dislocation-unbinding transition temperatures in qualitative agreement with the experimental results. In addition, further comparisons with experimental measurements of the transport anisotropy suggest that quantum fluctuations are important at low temperatures, and Radzihovsky and Dorsey [3] have studied this by considering the dynamics of the local smectic layers and developing a microscopic/hydrodynamic model of the quantum Hall nematic.

In addition, working off the fact that the 2D nematic only possesses quasi-long-range order, Fradkin and Kivelson et al. [4] have constructed a Monte Carlo simulation (fig. 1b) of a model of a classical nematic in a symmetry breaking field which shows a good deal of agreement with the experimental data.

Clearly a huge amount of work is being skimmed over here, and my goal in this post has been to outline how I got interested in this subject. To me, the work I’ve mentioned here really draws connections between these 2DEG samples and ‘conventional’ liquid crystals. This is, after all, a phenomenon which must necessarily be understood by mutual considerations of quantum/correlated-electron physics and soft matter physics.

There are a lot of resources available online for further exploration, such as a really nice review article by Kivelson et al. [17] that deals with many other related ideas from superconductivity as well. In addition, I’ve based a lot of this post on a talk by Fradkin given at KITP and one by Dorsey given at ITP Utrecht, which can be accessed from their webpages. In his talk, Dorsey presents a nice diagram illustrating the various concepts. I’ve tweaked it and added some more stuff to it, and I think my version is a concise way of summarizing the various considerations that have led to increased understanding of this measurement, at least to me (click for larger image):

A good deal of research is still underway dealing with the general subject of novel electronic liquid crystal phases – for example, in the context of cuprate superconductors, and striped phases of other correlated electron systems. Research is still underway on electronic liquid crystals in 2DEGs as well, and open questions and new directions abound. For example:

• What are the effects of an in-plane magnetic field on such anisotropic transport measurements?
• How exactly does the crystalline orientation affect this system?
• Would coupling to quasiparticles cause dampening of the Goldstone mode in these electronic liquid crystal systems?
• How exactly is the anisotropy in the transport characteristics related to the quantum Hall nematic order parameter?
• What other experimental probes can be used to study these systems to reveal new physics?

These (and other questions) remain unresolved. For example, Doug Natelson kindly pointed out a paper of his [18] that correlates the surface roughness of MBE-grown GaAs/AlGaAs samples (studied using AFM) to anisotropies in the FQH regime due to striped states in the 2DEG. The paper is with Bob Willett at Bell Labs, who has a nice KITP talk on experiments at 5/2 filling factor online that I didn’t fully understand, but hope to someday (he talks about experiments using ’small structures’ such as quantum point contacts to study this regime, and quantum computing applications of such devices). At about the same time, Jim Eisenstein’s group (the people who authored [1]) released a paper [19] reporting that “neither micron-scale surface roughness features nor the precise symmetry of the quantum well potential confining the 2D system are important factors” in the anisotropy. To my knowledge, the relation between the surface roughness and the transport anisotropy hasn’t yet been definitively answered, but as Prof. Natelson pointed out, new insights may be on the horizon due to new clean samples (e.g. [20], which reports anisotropic transport in 2D hole systems at v=7/2 and 11/2 versus isotropic transport at v=9/2).

More recently, as I’ve mentioned before, scientists in Europe have measured a large magnetoresistive anisotropy in the correlated electron oxide strontium ruthenate (Sr₃Ru₂O₇) near the ‘metamagnetic quantum critical point’, indicating the formation of a new quantum nematic phase [21]. This is strikingly similar to the tranport anisotropy in 2DEGs I’ve been talking about in this post - in particular, both show strong sensitivity to disorder - and the authors claim that the formation of this phase is tuned by the divergence in the quasiparticle effective mass near this critical point. One can only wonder what other kinds of systems could yield such behavior as well.

So, in summary: research on electronic liquid crystals has been flourishing for over eight years, building on significant developments in hard and soft condensed matter physics, and work like this indicates promising future developments. Interesting questions are being raised, interesting phenomena seen, and exciting new ways of thinking about these are being developed.

Addendum: some very recent work on the subject in Science:

“Electron Nematic Phase in a Transition Metal Oxide”, E. Fradkin, S. A. Kivelson and V. Oganesyan, Science 315, 196 (2007)

which is a ‘Perspectives’ article by three of the leaders in the field of electronic liquid crystals (their work is pretty much what my term paper was based on) on the following paper by a group of researchers mainly at the University of St. Andrews in the UK:

“Formation of a Nematic Fluid at High Fields in Sr₃Ru₂O₇“, R. A. Borzi et al., Science 315, 214 (2007).

Here’s the abstract:

In principle, a complex assembly of strongly interacting electrons can self-organize into a wide variety of collective states, but relatively few such states have been identified in practice. We report that, in the close vicinity of a metamagnetic quantum critical point, high-purity strontium ruthenate Sr₃Ru₂O₇ possesses a large magnetoresistive anisotropy, consistent with the existence of an electronic nematic fluid. We discuss a striking phenomenological similarity between our observations and those made in high-purity two-dimensional electron fluids in gallium arsenide devices.

References:
[1] M. P. Lilly et al., Phys. Rev. Lett. 82, 394 (1999) and K. B. Cooper, M. P. Lilly et al., Phys. Rev. B 65, 241313 (2002).
[2] R. R. Du et al., Solid State Commun. 109, 389 (1999).
[3] L. Radzihovsky and A. T. Dorsey, Phys. Rev. Lett. 88, 216802 (2002).
[4] E. Fradkin, S. A. Kivelson et al., Phys. Rev. Lett. 84, 1982 (2000).
[5] E. Fradkin and S. A. Kivelson, Phys. Rev. B 59, 8065 (1999).
[6] M. P. Lilly et al., Phys. Rev. Lett. 83, 824 (1999).
[7] K. B. Cooper et al., Phys. Rev. B 60, 11285 (1999).
[8] K. B. Cooper et al., Solid State Commun. 119, 89 (2001).
[9] H. Fukuyama et al., Phys. Rev. B 19, 5211 (1979).
[10] For example, M. M. Fogler et al., Phys. Rev. B 54, 1853 (1996).
[11] R. Moessner and J. T. Chalker, Phys. Rev. B 54, 5006 (1996).
[12] A. H. MacDonald and M. P. A. Fisher, Phys. Rev. B 61, 5724 (2000).
[13] C. Wexler and A. T. Dorsey, Phys. Rev. B 64, 115312 (2001).
[14] J. Toner and D. R. Nelson, Phys. Rev. B 23, 316 (1982).
[15] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973).
[16] D. R. Nelson and R. A. Pelcovits, Phys. Rev. B 16, 2191 (1977).
[17] S. A. Kivelson et al., Rev. Mod. Phys. 75, 1201 (2003).
[18] R. L. Willett et al., Phys. Rev. Lett. 87, 126803 (2001).
[19] K. B. Cooper et al., Solid State Commun. 119, 89 (2001).
[20] M. J. Manfra et al., arxiv:cond-mat/0603173 (2006).
[21] R. A. Borzi et al., Science 315, 214 (2007).