Data was taken at room temperature, liquid Nitrogen temperatures, and liquid Helium temperatures.  The resulting I-V characteristics as well as the electronically differentiated signals are presented here.   The results of the cubic spline method for numerical differentiation are also shown.

Preliminary Details (to be expanded)

The plots to the right show data runs of about 250 points at three different temperatures.  At room temperature and 77K no phonon emissions are noticeable, but at 4K one can see peaks in the second derivative located at the energies of two different phonons.  Due to the low energies involved,  thermal smearing hides the quantum phonon emissions even at liquid nitrogen temperatures.

This data should allows identification of the emitted phonons based on their energies.   This work is in progress.  Also a full error analysis will be completed to provide an uncertainty value for the phonon energies.

Further data was taken for 2 other Germanium tunneling diodes.  Plots of these results are on a separate page.

The images to the right show various results of the cubic spline interpolation method.  The program is given only the current and voltage data and it outputs the interpolated function as well as the first two derivatives.  See analysis page for a complete description. 

The cubic spline method accurately reconstructs the original function (current vs voltage) even without averaging the results.  However, the derivatives are plagued by oscillations in the interpolating cubic functions due to random noise in the data.   The averaging discussed in the analysis page successfully eliminates the noise from the first derivative.  The plot to the right shows a nearly perfect correlation between the derivative obtained from the lock-in amplifier and the results of the cubic spline method with 20 data averages.

Unfortunately, the signal averaging does not save the second derivative. While the resulting second derivative is a continuous function as prescribed by the method, it is completely obscured by small errors in interpolation which manifest themselves as oscillations.   The plot shown here illustrates the problems with the interpolated second derivative.  Without averaging oscillations range from -100 to 100, generally alternating every point (corresponding to every computation zone in the method).   The averaging damps extrema to the range -0.1 to -0.5.  However, it changes the period of the oscillations so that the number of extrema equal the number of averages.  This dependence on a parameter of the numerical method is unacceptable.  

The images to the right show various results of the least squares curve fitting method.  The program is given only the current and voltage data and it outputs the function fit to an arbitrary degree polynomial as well as the exact first and second derivatives of the polynomial function.  See analysis page for a complete description. 

The images to the left is an 11th order polynomial fit to the i-v data, while each subsequent derivative looses a degree in the fitting function.  The Gauss-Jordan elimination technique was used in the least squares algorithm. This method fits the original function perfectly for all practical purposes.  As one can see, the first derivative also fits the data very accurately, but the second derivative is much smoother than the electronically differentiated signal.  This suggests that electronic differentiation is still superior for resolving the second derivative, although numerical differentiation does provide a reliable "sketch" of the second derivative.  In this experiment the locations of the extrema in the second derivative is critical.  The least squares method displaces these values slightly which is unacceptable.

Overall, the least squares method is highly susceptible to numerical errors.   Using Cramer's method to solving the linear system of equations produces intermediate numbers that overflow and/or underflow the precision of the computer, resulting in a fit that diverges from the data at the edges.  The code uses a "long double" data type of 10 bytes per number to allow the maximum range, but the algorithm stills fails when fitting polynomials above 7th degree.

Using Gauss-Jordan elimination improves the performance of the algorithm immensely, as well as eliminating some of the numerical problems.  However, The data shown to the right allows only a 11-12 degree polynomial before numerical errors destroy the results.

Finally, a similar numerical differentiation was completed with the software package "Origin" by Microcal.  It allows up to a 9th order polynomial fit, but it will not easily take the exact derivative of the function.  However, similar results were obtained using built in local differencing technique to estimate the derivative of the resulting polynomial fit.  (i.e. the derivative is estimated with by an interpolation, using a constant value in each zone). 

Please send comments, criticisms, queries or congratulations to Michael Enz at enzx0002@tc.umn.edu.   This page was created 5/4/98 and last modified on 5/18/98.  It will be updated as work progress through June 1998.