Microscopic View of a 2D Ising System

V. P. LaBella, D. W. Bullock, M. Anser, Z. Ding, C. Emery, L. Bellaiche, and P. M. Thibado

Department of Physics, The University of Arkansas, Fayetteville, Arkansas 72701

Published in Phys. Rev. Lett. 84(18) 4152 (2000) (PDF) abstract

What does a social movement, such as the Pokemon craze have to do with a magnet? What do these two things have in common with the surface of a perfect crystal? The answer is that they all can be understood using a simple model from physics called the Ising Model. This model describes a large collection of objects interacting with their neighbors. For example, the Pokeman craze can come about simply by word-of-mouth communication passed from one individual to the next. The Pokemon movement and other social movements can be thought of as the entire nation suddenly or spontaneously reaching a common agreement on an issue. This behavior is mimicked in physical systems as well. For example, when a magnet, like the one on your refrigerator, is heated it looses its magnetization. In this non-magnetic state the small magnetic domains inside the magnet point in random directions making the total magnetization zero. However, when the magnet is cooled from high temperatures the randomly oriented domains will spontaneously align themselves forming a permanent magnet once again.

Likewise, when a perfectly flat crystal surface is heated to high temperatures, individual atoms will spontaneously collect on the surface. These atoms form islands and can be seen using a powerful microscope called the scanning tunneling microscope (STM) as shown in Fig. 1. Since the islands are one atomic layer high the surface can be modeled in a straightforward way. This is done by dividing the surface into sites and marking each site as occupied by an atom or unoccupied by an atom. This method is similar to marking each person as liking Pokemon or not liking Pokemon. One benefit of the crystal surface is that each atom is identical to all the others, unlike humans. This makes the study of the Ising model more controlled. The purity of the experiment allows for a deeper understanding of the nature of spontaneous and collective events.

The Ising model is a cornerstone in the field Many-Body Physics. Originally, the Ising model was developed by H. Lenz and his student Earnst Ising to explain the spontaneous magnetization of a magnet as it is cooled from high temperatures, which was studied experimentally by Piere Curie. As a graduate student, Ising solved this model in one dimension (i.e. where the small magnets form a long chain), which is no small feat. Unfortunately, when solved in one dimension the model fails to account for the spontaneous magnetization.

FIG. 1.  STM images of the GaAs(001) surface prepared at different temperatures. The islands are one monolayer high (0.28 nm) and represent the occupied area of the lattice gas, while the other area represents the unoccupied area of the lattice gas. The area covered by the islands increases as the temperature increases

FIG. 2.  Animated fly-by of the site-site correlation function computed from the real space domains. The eliptical shaped peak is a result of the elongated islands which is a result of asymetric coupling energies.

Some 20 years later Lars Onsager solved the Ising model exactly in two dimensions and it accounted for the spontaneous behavior of a magnet. This solution has been considered one of the greatest theoretical achievements of the past century. It not only accurately predicts the properties of a magnet, but it completely revolutionized the field of many body physics. It was the first exact solution of any kind to a system composed of more than two particles.

In our recent paper, we imaged the occupied and unoccupied domains of an Ising system on a scale comparable to the individual atoms that make up the domains. The domains appear as islands as seen in Fig. 1. To achieve this we used the world’s most powerful microscope, the scanning tunneling microscope (STM) to image individual atoms on the surface. This type of insight into the domain structure of an Ising system has never been seen before. This allowed us to study the Ising model with the finest resolution ever and in a system composed of perfect, pure, and identical component, namely the atom.

Surprisingly, this discovery was made on the surface of a material called gallium arsenide (GaAs), which is commonly used in manufacturing devices for high-speed fiber-optic telecommunication equipment, lasers for CD players, and transistors for cell phones. These devices are made using a technique called epitaxy where layers of atoms are deposited on top of a single crystal surface. A fundamental understanding of the physics which governs the motion of atoms of these crystal surfaces is needed for production of future exotic devices. In addition, simulating the growth and of a device using a computer to optimize the fabrication procedure prior to actually making it in fabrication facility would greatly reduce the time and cost in making new devices. Our discovery shows that the well understood Ising model accurately describes the surface morphology and may help in developing simulations for device growth.

FIG. 2.  The behavior of the density of islands on the GaAs(001) surface as a functions of substrate temperature and pressure. The inset shows how the lattice gas Ising model behaves in theory. The three thresholded images in the lower right show the state of the surface under different temperatures.

The atoms on the surface as seen in Fig. 1. compose a gas, similar to the gas in our atmosphere. The STM images give a picture of were all the atoms in the gas are at an instant. From the images, the density of the gas is calculated by measuring the percentage of the surface that the islands occupy. The pressure and temperature of the gas are known from how the surface was prepared. The density, temperature and pressure give a complete physical description of a gas. We systematically prepared samples over a range of temperatures and pressures and imaged the surface to measure the density. The result of this study is shown to the left in Fig 2.

Excellent agreement between the theoretical prediction for the Ising model [shown in inset of Fig. 2 (a)] and our data is achieved. Each data point represents the average density measured from 5-10 images of the surface. The time to measure each data point ranged from five hours for the highest temperatures to 2 days for the lowest temperatures.

Density, temperature and pressure are a macroscopic picture of a gas which is composed of billions of tiny microscopic particles. A surrealist painter, Rene Magritte, is famous for saying “Everything we see hides something else”. Hidden from view while looking at the macroscopic data is information about the microscopic structure of the system. For example, the domains as seen in Fig. 1. are elongated in one direction but this information can not be attained from looking at the the graph above. To gain information about the microscopic structure one needs to look at the correlation function of the domains. The correlation function shows the probability as a function of distance of finding an occupied site next to another occupied site.

The correlation function taken form our data is shown to the right rendered in three dimensions floating above the underlying domain structure. It has a peak located in the center and falls off quickly with distance from the center. In addition, a cut taken at the half the maximum is elliptical (shown in the inset) with an aspect ratio of three. This elliptical nature of the correlation function is a direct result of the elongation of the domains in one direction. An astute reader may point out that the reason why the domains are elongated in one direction is because the coupling energies are different in the two principle directions, or they are asymmetric. This is true, however to calculate the values of the coupling energies we need the following two equations:

The first equation is the Onsager condition from his exact solution to the Ising model. The second equation comes form the exact solution of the correlation function of the Ising model by T.T. Wu and B. M. McCoy et al. From these two equations and measuring the critical temperature from the macroscopic data and the aspect ratio of the correlation function we compute the following two energies:

These energies are the coupling energies for this Ising system and the the energy cost for creating and edge site or step edge on the GaAs(001) surface.

FIG. 3.  The measured site-site correlation function, shown rendered in 3D floating above the domain structure from where is was calculated. The aspect ratio of the correlation function is 3 gives a measure of the average domain geometry and can be used to compute the coupling energies.

After Inspection of the STM images it is obvious that the system is two dimensional. The islands that are forming are always one monolayer high. However, when studying a system near the critical point, one typically measures critical exponents. The values of these exponents only depends upon the dimensionality of the system. In this way one can determine the dimensionality of the system. Of the seven fundamental critical exponents we are able to measure four and they are given in the table below and agree with 2D theoretical predictions.

In conclusion, we demonstrated the Ising model can be used to understand the technologically important GaAs(001) single crystal surface. With this understanding we were able to measure the asymmetric coupling energies for this system. These energies are the energy it costs to create a step on the GaAs(001) surface. In addition, four critical exponents were measured that agree with 2D theoretical predictions. Lastly, this insight may help in developing models for crystal growth by providing a Hamiltonian and equation of state that relates surface morphology with substrate temperature and arsenic pressure.