In semiconductors, conduction occurs when electrons are excited across an energy gap, from the valence band to the conduction band. However, in 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.

In semiconductors, a forbidden energy gap exists between the valence energy band (with maximum energy E_v) and the conduction energy band (with minimum energy E_c). In most devices, current flows when electrons are excited from the valence band to the conduction band. In tunnel diodes, electrons can tunnel through a potential barrier without changing energy. This results in complex current-voltage curves that can be explained by energy gap diagrams (figures 1-4).

Another unusual feature of tunnel diodes is the ability to change energy bands indirectly by changing momentum. The plot of energy bands versus k (from momentum = hbar * k) shows the minimum distance between the valence and conduction bands does not correspond to strictly a change in energy, but rather an electron may enter the conduction band by a small change in energy accompanied by a change of momentum from emitting a phonon (figure 5). When the energy applied to the tunneling diode (eV) is exactly equal to the discrete energy of a phonon (hbar * w ), there is an increase in the current through the diode which corresponds to a "kink" in the I-V curve (figure 6). This is only noticeable at low temperatures, since the thermal energy of electrons at room temperature is sufficient for direct tunneling.

The intent of this project is to 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 one must look at the derivatives of the experimentally determined function. The indirect tunneling appears as bends in the dV/dI versus V curve and spikes in the d²I/dV² curve.

An op-amp or lock-in amplifier circuit will be constructed to convert the measured current signal to a voltage and electronically differentiate it. It will be interfaced with a personal computer so the data can be logged.

The tunnel diode’s I-V characteristics 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 provide a calibration to ensure linear gain in the circuit over the operating range.

Analysis

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.

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. The interpolation polynomial is easily differentiated to give representations of the derivative and second derivative.

Figure 1: The current versus voltage graph for a tunnel diode has an initial peak in its I-V characteristics, followed by a region of negative resistance, and finally a region of high conductivity. These states are easily explained by energy gap diagrams (figures 2-4). The focus of this project is measuring small "kinks" in the I-V curve caused by indirect tunneling (figure 5).

Figure 2: Since tunneling diodes are heavily doped p-n junctions, their Fermi energy level lies in the valence band on the P side of the junction and the conduction band on the N side. Therefore, electrons can tunnel from the valence band to the conduction band with changing energy. When there is no potential voltage across the diode, zero net current flows since tunneling may initially occur in either direction with the same probability.

Figure 3: As the voltage increases are the P-N junction, the Fermi energy level increases on the P side. This initially causes increased tunneling and a resulting current. As the number of empty states on the P side decreases, the current decreases which results in a negative resistance.

Figure 4: When a strong potential is applied across the diode, electrons can diffuse across the junction, similar to any P-N junction.

Figure 5: The minimum distance between the valence and conduction bands does not correspond to strictly a change in energy in tunnel diodes. Therefore, electrons indirectly tunnel by emitting a phonon to change momentum (k), coupled with a small change in energy.

Figure 6: Exaggerated view of the kink in the I-V curve. This is due to increases in current when the energy of a phonon emitted during indirect tunneling (hbar * w ) exactly equals the applied energy across the diode (eV).

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.