Mapping the Density States of a Superconductor as a Function of Energy

Summary

The Seamus Davis research group at the University of California at Berkeley and at Cornell University, has performed research related to the study of experimental condensed matter physics. Using a scanning tunneling microscope, graduate student Kyle McElroy took an image of a high temperature superconductor called Bi₂Sr₂CaCu₂O_8+x and mapped the density of states of the superconductor as a function of energy. In layman's terms, he made a map of the energy levels of the electrons at different locations in the crystal.

0.1 0 nm

Fig 1: Atomic resolution image of the surface of the high-temperature superconductor BSCCO. This image was acquired with a sctunneling microscope. Each white dot is an individual bismuth atom. The horizontal waves are an incommensurate reconstruction due to stresses in the crystal.

Although a scanning tunneling microscope can map the energy levels of the electrons as a function of position, it cannot directly access the momentum of the electrons, which is also very important for understanding the way the crystal works. But with some pretty clever analysis, the momentum information was backed out of the spatial information which was acquired directly.

Details

Another graduate student, Jenny Hoffman, took the two dimensional Fourier Transform of the electron-density maps, as a function of energy. The Fourier transform shows that there are numerous modulations of the density of states, at multiple frequencies and directions. There are too many modulations to see clearly by eye in real space, but the different modulations are easily distinguished in k-space.

These modulations are actually standing waves of electrons (an electron is both a particle and a wave!). The exact wavevectors of these standing waves were inverted to indicate the momentum of the electrons. However, each k-space map has about sixteen 2-dimensional peaks which had to be fit accurately, and Jenny had k-space maps at dozens of energies. Unfortunately, much of the data was too noisy for a 2-dimensional peak-fitting routine to converge reliably, so the data had to be extracted in lines along all different angles out from the zero-wavevector center of the k-space image, and then the peaks in these lines had to be fitted. In fact, 16 peaks x 30 or more energies per dataset had to be analyzed! Once analyzed, the parameter results from all fits had to be organized and used to back out the crucial momentum information. This is where Origin came in.

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Fig 2: A map of the local electronic density of states at energy -16 meV, in the same field of view as Fig 1. This is effectively a density map of the number of electrons at each spatial location which have energy -16 meV. There are a number of different modulations and periodic structures in several different directions: vertically and horizontally and diagonally. Each of these modulations is an electron standing wave.

Analysis in Origin

Jenny worked extensively with Origin C programming to automate many aspects of her analysis in Origin. Functions were created to assist her in importing the data, preparing the worksheet for graphing and analysis, fitting the data, and visualizing the data and results in graphical form. These functions were hooked up to buttons placed on a custom worksheet template.

Fig 4: The buttons on the custom worksheet template.

Bringing in the data

An "Import" button was created and placed on the worksheet template. The button uses LabTalk™ to bring up a dialog for selecting the file to import. The script also calls an Origin C function that handles the import once the file is selected. The C function processes the data file, looking at and parsing header information along the way, and then dumps the data into the worksheet template.

Setting up the worksheet

A separate "Set XYXY" button was created to update the worksheet column designations. This button also uses LabTalk script to call an Origin C function.

Visualizing the data before analysis

The "Scatter Waterfall", "Line Waterfall", and "LineSymb Waterfall" buttons were created to allow you to plot the data as scatter, line, or line and symbol plots prior to analysis. As with the other buttons, these three use LabTalk to make calls to Origin C functions, which in turn perform the necessary operations to plot the data and annotate the graph with information from the selected columns of data.

Fitting the data

The analysis of the data took a lot of initial setup time, but once this was completed, it was a snap. Jenny first created the fitting functions to be used during the fitting process. She used Origin's nonlinear curve fitter to do this.

Fig 5: The Nonlinear Curve Fitter showing a fitting function formula used in the analysis.

The fitting functions were then called from various Origin C functions she created.

The C functions were in turn called by the "Fit Selected" button on the custom worksheet template.

// bring up attention box 
break.style=1;
break.open("Loading and compiling Origin C function...");

// now attempt to load and compile function
if(run.LoadOC("%yOriginC\JEHprograms\JEHworkspace.ocw") != 0)
{
	// report if failure
	break.close();
	type -b "Error trying to load and compile the Origin C file.";      
	break;
}
break.close();

//now call C function
prepareForFit(%H);

In addition, several worksheet and graph templates were utilized during the fitting process. In all, the LabTalk script and Origin C code behind the Fit Selected button was designed to automatically perform the following operations:

• Fit all peaks automatically to a sum of several user-defined functions
  
• Plot each fit on a graph template using the information from the imported file (i.e. the file name and header information) in order to label the axes
  
• Create an internal folder structure in the project to organize each fit (specifically, put all the fits from a single energy into a single folder)
  
• After all the fitting is done, plot each fit parameter as a function of energy and plot one specific fit parameter as a function of another
  
• Label all graphs correctly with information from the input file names and headers
  
  

Fig 7: A final result graph including the raw data plotted
as scatter plots and the fit lines plotted as black line plots

Acknowledgements and Biographies
This work was performed in the research group of Seamus Davis, a professor of physics at Cornell University who specializes in experimental studies of condensed matter.

Kyle McElroy is a physics graduate student at the University of California, Berkeley, currently working with Professor Davis at Cornell University.

Dr. Jenny Hoffman performed this analysis while she was a graduate student with Professor Davis at the University of California at Berkeley. She is currently a post-doc at Stanford University and will be starting as a professor at Harvard University in January, 2005.

The scanning tunneling microscope used in this experiment was built in the lab of Professor Davis by Dr. Shuheng Pan (now a professor at the University of Houston) and Dr. Eric Hudson (now a professor at Massachusetts Institute of Technology).

The BSCCO crystals used in this experiment were grown by Dr. Hiroshi Eisaki (now at AIST-Tsukuba in Japan) and Professor Shin-ichi Uchida at the University of Tokyo.

Parts of this work were funded by the New Energy and Industrial Technology Development Organization of Japan, the Office of Naval Research, Lawrence Berkeley National Laboratory, the National Science Foundation, and Cornell University.