In semiconductors, conduction occurs when electrons are excited across an energy gap, from the valence band to the conduction band. However, in certain silicon and germanium tunneling diodes, the minimum in the conduction band does not occur at the same point in k-space (momentum-space) as the maximum in the valence band. Therefore, electrons can indirectly tunnel by emission of a phonon to conserve momentum.

This project will consist of measuring the indirect tunneling of electrons in diodes at various temperatures. The phonons involved in the interaction will be identified and their energies will be compared to theoretical calculations.

Two distinct energy bands provide the unique characteristics of semiconductor devices. For current to flow through a semiconductor electrons must be excited from the valence energy band, across a forbidden energy gap, into the conduction band. When no voltage is applied to the junction, electrons will fill up to the Fermi energy level. Tunneling diodes are heavily doped p-n junctions which have a Fermi energy level which lies in the valence band on the P side of the junction and the conduction band on the N side (figure 1)[1-2]. Therefore, electrons are free to "tunnel" across the junction without changing energy. However, the net current through the diode is zero until a potential voltage is applied, since tunneling may initially occur in either direction with the same probability.

Figure 1: Diagram showing the Fermi energy (E_F) falls between the conduction energy (E_C) and valance energy (E_V) levels in a tunneling P-N junction.

Figure 2: Sketch of energy levels versus momentum which results in indirect tunneling.

This experiment will measure phonon emissions which occur during indirect energy gap transitions in a silicon and/or germanium tunneling diode. To examine small changes in the I-V characteristics that occur when the input energy corresponds to the exact quantum of energy necessary for phonon emission, once must look at the derivatives of the I-V function. The indirect tunneling appears as a bend in the dV/dI versus V curve and a spike in the d²I/dV² curve [4]. One can convert the current signal to a voltage and electronically differentiate it using a differentiating op-amp circuit. This also amplifies the derivative, which allows for closer inspection of the I-V characteristics.

The circuit will be interfaced with a personal computer running Windows NT and the LabWindows programming environment. With the proper circuitry, the current and derivatives will be converted to a binary voltage value and logged to a file as the input voltage sweeps the entire operating range of the tunnel diode. The interface will immediately graph the current data, as well as the first two derivatives.

The tunnel diode’s I-V curve will be measured at 293K, 77K, and 4.2K, corresponding to the room temperature, liquid Nitrogen, and liquid Helium. This will show the effects of thermal "smearing" on the quantum interaction.

To test the differentiating circuitry, a resistor (with an expected linear I-V curve) will be tested at all three temperatures. This will ensure the constant gain of the op-amps, as well as indicating possible errors in circuit design.

Analysis of the d²I/dV² versus V plot will show peaks at locations where the energy of the emitted phonon equals the energy applied across the diode, or hbar * w = eV. These emissions will be correlated with the type of phonon emitted, using neutron scattering data from Brockhouse’s neutron scattering experiment [5]. The same data can be used for calculating the energy of emitted phonons, which can be directly compared to the measured values in the lab since hbar * w = eV.

To investigate the accuracy of electronic differentiation, the current versus voltage data will also be interpolated. The function will be approximated as a cubic polynomial between each two data points (which is defined as a single zone). This is easily differentiated to give representations of the derivative and second derivative, although the differentiation reduces the order of the interpolation polynomial within each zone (and the accuracy decreases).

Conclusion

This project will confirm experimental results of Brockhouse and Merrill concerning the energies of various phonon’s emitted during indirect tunneling. It will also provide the means for the experimenter to explore the integration of computers and data acquisition circuit design. Finally, the project will allow for comparisons between electronic differentiation and interpolation algorithms.

Time Table

• At least one tunneling Silicon or Germanium diode or rectifier (such as the obsolete Hoffman HU-5)
• Several small containers of liquid nitrogen and helium for immersing the diode
• Several op-amps and various electrical components
• Prototype card and a computer

1. Kittel, Charles. Elementary Solid State Physics: A Short Course. John Wiley & Sons, New York, 1962.
2. Eisberg, Robert and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. John Wiley & Sons, New York, 1985.
3. Mellema, Jim and Gary Sjolander. Phonon Assisted Tunneling in Silicon. Methods Lab Final Report, 1970.
4. Merrill, J.R. American Journal of Physics. 37, 269 (1969).
5. Brockhouse, B.N. Physical Review Letters. 2, 256 (1959).

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