research
Electron-electron interactions and disorder in a carbon flatland
I study the effects of electron interaction and disorder in topological materials in two dimensions. Here, I hope to explain what exactly such systems entail, why they are interesting theoretically, and how, if we understand them better, they might revolutionize the electronics industry.
First, what do I mean by electrons in two dimensions? Don’t we live in a three dimensional world? Actually, what matters is how the electron wavefunction behaves. And by using electric fields, one can confine electrons so that their wavefunctions are effectively squashed into zero, one or two dimensions. 2D electrons are actually quite common. There are more than one billion transistors in your laptop computer, and each one is a small puddle of electrons whose wavefunction looks two dimensional.
More recently, scientists have been able to cleave bulk materials, pealing off layers that are just one atom thick. To contemplate how thick is a single atomic monolayer, you can imagine that the diameter of a strand of human hair has more than 25,000 atomic layers. The first material to be cleaved in this way was graphene – a single atomic sheet of graphitic carbon – but this technique has now been extended to several materials. One exciting area of contemporary research is cleaving down monolayers of different materials and then reassembling them again like lego blocks into new materials that don’t exist naturally in nature. These new materials are called van der Waals heterostructures, and in most cases, the resulting electrons have their wavefunction behave two-dimensional. The theoretical challenge in this game is to predict which assemblies of these layers would have interesting properties before our experimental colleagues assemble and measure them. But for the purpose of this article, we will concentrate on graphene, where the electrons live in a two dimensional carbon flatland.
When electrons go flat
The carbon atoms that make up graphene form a honeycomb (a plane of hexagons with each carbon atom bonded to three other carbon atoms). Of the four valence electrons in carbon, three form strong sp2 bonds with other carbon atoms, and contributing the remaining itinerant electron to the two-dimensional electron gas. But this is no ordinary electron gas. The topology of the carbon honeycomb makes the energy of the resultant electron gas scale linearly with momentum (similar to photons in light) and we call such electrons Dirac electrons. The two dimensional electrons in your laptop are called Schrödinger electrons and their energy scales as the square of the momentum (similar to the classical billiard balls you studied in high-school physics). In our recent article in the Reviews of Modern Physics [1], we examined the similarities and differences between Dirac electrons and their more familiar Schrödinger counterparts. Some of the most striking differences occur in both in how Dirac electrons interact with each other (dubbed "interaction effects”) and how the electrons interact with accidental defects or external dopants (generically called “disorder”).
To conduct or not to conduct, for electrons, that is the question
In 1958, Philip Anderson noticed that electrons in two-dimensions had a rather peculiar property. Even with vanishingly small disorder, when treating the electrons as quantum objects, they got stuck and could not conduct. When awarded the Nobel Prize in physics about 20 years later for this work, Anderson noted [2]
Localization was a different matter: very few believed it at the time, and even fewer saw its
importance… It has yet to receive adequate mathematical treatment, and one has to resort to
the indignity of numerical simulations to settle even the simplest questions about it.
By the time that graphene and other Dirac electrons (like topological insulators) came on the scene in 2006, the question immediately arose as to whether these new kind of electrons will get stuck just like Schrödinger electrons, or if they would continue to conduct even in the presence of disorder. It still remained true that only numerical simulations could address even this simple question. And it came as a surprise when it was found that disorder actually made graphene more conductive rather than less. Dirac electrons in graphene were a special case where you could not stop the electrons from conducting unless you destroyed the hexagonal topology of the original carbon lattice [3].
Electrons far away from each other interact more strongly
In an advanced undergraduate course in solid-state physics, it is common to pose the following paradox to students: “For electrons in two dimensions, why are the effects of electron-electron interactions stronger when the electrons are further apart?” The resolution of this conundrum lies in understanding that it is not the absolute value of the Coulomb interactions between electrons that determine its importance, but that ratio of the electron interactions relative to their kinetic energy. Both effects depend on the average electron spacing, but for Schrödinger electrons in two dimensions, the kinetic energy grows faster with inverse spacing than do the Coulomb interactions, and so it dominates at small electron separations (and by similar reasoning, the Coulomb interaction dominates when the electrons are far apart). This highlights the important complication that for Schrödinger electrons, the density of electrons is inexorably linked to how strongly they interact. Dirac electrons do not have this complication and carrier density and interaction strength can both be tuned independently. So what happens if we make the graphene electrons strongly interacting? Well, without any disorder, we can anticipate theoretically [5] that the electrons will get stuck into a Coulomb lattice analogous to what was proposed for Schrödinger electrons by Eugene Wigner (another Nobel Laureate).
Disorder and Interactions – The Holy Grail
So what happens when we have both disorder and interactions? Experiments done in situations where both disorder and interactions are weak revealed that the interactions could mostly be ignored. Lev Landau outlined the theoretical underpinnings for this puzzle. He demonstrated that when electrons interact weakly with each other, the system could be described by new particles (called quasi-particles) that don’t interact with each other. But these quasi-particles interact with disorder in a manner similar to how non-interacting particles interact with disorder. This Landau paradigm is the now the starting point for understanding electrons in condensed matter physics. In one of my own works, I used the Landau paradigm to address how Dirac electrons behave in the presence of weak disorder and weak interactions providing an explanation for why experiments on graphene found a finite minimum conductivity, even when the average carrier density vanished. We now understand that this conductivity without carriers puzzle is explained by the very unusual situation where disorder in graphene creates puddles of electrons. These electrons, by interacting with each other, effectively weaken the effect of the impurity potential by rearranging to compensate for the disturbance caused by the disorder. Only by solving for the electron arrangement self-consistently [6] does one get the correct electron state in the presence of both weak disorder and weak interactions.
The biggest problem with the transistors in your computer today is that they heat up. As each quasi-particle jumps through the electron puddle it transfers its energy to the environment. Having strongly interacting electrons all behave collectively, one could reduce the energy dissipated by several orders of magnitude [7]. The only problem is that this transition is poorly understood theoretically when both interactions and disorder are present. Since in graphene the three knobs of carrier density, interaction strength and disorder can each be tuned independently, this makes it an ideal system to study and understand how this bit of physics works, and perhaps suggest a route to making more energy efficient computers.
References
1. S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi “Electron transport in two dimensional graphene” Reviews of Modern Physics, 83, 407 (2011).
2. P.W. Anderson “Local moments and localized states” in Nobel Lectures, Physics 1971-1980, Editor Stig Lundqvist, World Scientific Publishing Co., Singapore, (1992).
3. M. S. Fuhrer and S. Adam “Carbon conductor corrupted” Nature 458, 38 (2009).
4. C. Jang, S. Adam, J-H Chen et al. “Tuning the effective fine structure constant in graphene: opposing effects of dielectric screening on short- and long-range potential scattering” Physical Review Letters 101, 146805 (2008).
5. V. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A.C.H. Castro Neto, “Electron-electron interactions in graphene: current status and perspectives”, Reviews of Modern Physics, 84, 1067 (2012).
6. S. Adam, E. H. Hwang, V. M. Galitski, and S. Das Sarma “A self-consistent theory for graphene transport” Proceedings of the National Academy of Sciences USA 104, 18392 (2007);
7. “Prototype `Mott transistor’ developed” Physicsworld.com (Institute of Physics, UK) July 25, 2012, available online at physicsworld.com/cws/article/news/2012/jul/25/prototype-mott-transistor-developed
A sampling of some recent projects:
Measuring and Modeling Inhomogeneous Energy Landscapes in Graphene
Comprehensive Overview of Graphene Transport
Electronic Properties of Graphene Stacks