Indirect Electron Tunneling in Germanium Using a Current to Detect Resonant Phonon Energies

Matthew Coles and Samuel Glasenapp

University of Minnesota

Methods of Experimental Physics Spring 2013

Abstract

In this experiment, a germanium tunneling diode was cooled using liquid helium to a temperature of 4.2K, at which point a voltage potential was applied across the diode, and a measurement of the current moving through the diode was recorded. By looking at the I-V curve, the first derivative (dI/dV), and the second derivative (dI²/dV²) of the I-V curve, the LO and TA mode resonant energies of the tunneling diode for maximum phonon emission were determined to be E_TA=(4.8±0.9) meV, with a χ_R²=0.971, and E_LO=(30.2±1.1) meV, with a χ_R²=0.994.

Introduction

In a tunneling diode, when a voltage is applied across the PN junction, a current is produced where electrons in the semiconductor materials tunnel across the band gap at the junction indirectly by changing momentum through the emission of a phonon^[1]. The emission of a phonon requires a certain energy to be obtained by the electron, which is given to it by the voltage drop across the PN junction. Since phonons are pseudo-particles, that represent the quantization of the vibrational modes of an interacting structure of particles, the energies at which phonon emission will yield the largest current happen at discrete resonant energy levels. It is this process of the emission or absorption of a phonon to change the momentum of the electron that is called "indirect" tunneling^[1]. In order to observe this tunneling phenomenon, various voltage potentials in the range of 0-80 mV were applied to a 1N3712 tunnel diode in a bath of liquid helium at 4.2K as well as at room temperature (~296K). At room temperature, the increase in current due to phonon emission was hidden by thermal smearing. At lower temperatures, such as that produced by the liquid helium, thermal excitation of the electrons across the band gap was reduced, thus the indirect tunneling effect became the primary source of current at the resonant phonon energies through the diode. The current was recorded, and an I-V curve was plotted. By taking the first derivative (dI/dV), the increase due to the current became a small step in the curve, and in the second derivative (d²I/dV²) the increase in current at the resonant phonon energies resulted in a well defined peak. Using the location of these peaks, the resonant energies of the phonons were calculated and then compared to previously measured values obtained by neutron scattering experiments^[1][2]. A similar experiment was completed in 1998 by Michael Enz, using similar diodes of germanium^[3]. The goal of this experiment was to confirm and increase accuracy of his earlier measurements.

Theory

In semiconductor materials, an energy gap called the band gap separates the valence band energy, E_v, and the conduction band energy, E_c. Electrons will fill the material up to an energy called the Fermi level energy, E_F, determined using Boltzmann statistics. This energy, in equilibrium, is usually located at an energy between E_v and E_c when no voltage difference is applied across, and must be at a constant energy across the material^[4].

A semiconductor can be doped, creating either an n-type or p-type semiconductor. In a p-type semiconductor, the material is doped with atoms that form holes accepting electrons, called acceptors, and in a n-type semiconductor, the material is doped with atoms that donate electrons, called donors. This doping changes the magnitude of the Fermi level energy, either raising or lowering it dependent upon the type of doping applied to the material^[5].

A regular diode is made by putting a doped n-type semiconductor in physical contact with a p-type semiconductor, which causes a skew in the band gap diagram at the point of contact, the PN junction. The E_F still generally maintains its constant position between E_c and E_v in the absence of a voltage applied across the junction. For an electron to jump from the valence band to the conduction band of such a diode, an energy must be applied to the electrons in the valence band, exciting it. This happens due to thermal excitation of the electron. This excited electron is now free to move under a voltage difference applied to the material, creating a current in the material. At higher temperatures, more electrons are promoted to this excited energy state, and thus a higher current can potentially be produced from the material^[5][6].

Figure 1: In a tunneling diode, a heavily doped n-type semiconductor is paired with a heavily doped p-type semiconductor. This causes the Fermi level of each type to raise or lower to an equilibrium level, which falls below the valence band in the p-type material, and above the conduction band in the n-type material. This change across the two types of materials allow the electrons to tunnel across the band gap by changing momentum through the emission of a phonon. Original Figure.

In a tunneling diode, the semiconductor materials are heavily doped such that across the PN junction, in order to maintain a constant level, E_F must be located at a level such that it is above E_c in the n-type region, and below E_v in the p-type region, as shown in figure 1 above. In these type of diodes, not only can thermal excitation of electrons cause electrons to jump from the valence band to the conduction band, but electrons can also tunnel across the band gap in the PN junction of the diode. In order for the electron to tunnel across the PN junction from the valence band, the minimum energy needed may require not only a change in energy as in a direct tunneling semiconductor, but also a change in momentum, leading to an "indirect" tunneling effect. When an electron changes its momentum to tunnel across the band gap, it does so by releasing a phonon, which is a quasi-particle representing a quantization of the excited vibrational modes of a lattice or interacting structure of particles. In a direct tunneling semiconductor, the electrons emit a photon to change its energy. A comparison of an indirect band gap versus direct band gap is shown in figure 2 below. In order to excite these quantized vibrational modes and emit a phonon, an extra source of energy must be supplied to the diode in the form of a voltage drop across the PN junction^[4][5].

Figure 2: Comparison of the energy band gap in k-space of a direct band gap versus an indirect band gap. In direct electron tunneling, the electron tunnels solely by changing energy. In indirect electron tunneling, the electron not only changes energy, but also changes its momentum. Original Figure.

Electron momentum can be quantized using the following formula:

p= ħk (1)

where p is the momentum of the electron, k is the wavevector, and ħ is the reduced Plank's constant. As ħ is a constant, the momentum is often referred to by using the wavevector k. By Fourier transforming the energy band from position space into momentum, or k-space, the lowest energy levels in the conduction band will appear at a different k-vector than those in the valence band. As the electron changes its momentum, the phonon emitted has discrete energy levels E described by the following formula:

E= ħω (2)

where is the phonon's frequency and is comprised of discrete values. When a voltage is applied across the PN junction of the tunneling diode, the electrons will start to tunnel across the band gap by the emission of phonon. At a voltage such that the energy of the phonon produced by one of the discrete frequencies ω is equal to eV, resonance is achieved, and we then have from substitution into equation (2) the following:

eV= ħω (3)

where e is the electron charge, and V is the strength of the voltage drop across the diode. It is at this voltage that the maximum number of phonons are emitted. As such, a maximum current from this emission will be present. On an I-V curve, this increase will show up as a bend in the plot^[1][3][5][6].

The bend in the I-V curve can be viewed by using differentiation techniques on the graph. When the first derivative of the I-V curve is viewed, (dI/dV), the resonant energies of the emitted phonons will appear as a step in the plot. In the second derivative plot, (d²I/dV²), the resonant energies will appear as a peak in the plot.

Experimental Setup

The setup for the experiment added a reference sine wave signal of amplitude 1 mV and frequency 1.5 kHz from a Stanford Research 830 lock-in amplifier with a DC signal from an external power supply. The power supply was used to supply between 1 mV and 100 mV to our germanium tunnel diode. These two signals were added together with a basic summing amplifier circuit shown in figure 4, using an LF411 operational amplifier as the primary component in the circuitry. The output of this circuit was then applied to a 1N3712 tunnel diode inside of a cryostat filled with liquid helium at 4.2K. The DC voltage across the diode was directly measured using an HP 34410A multimeter. The signal coming from the tunnel diode was then sent into a Stanford Research 570 current pre-amplifier to be changed into a voltage proportional to the amplitude of the current, which was read by another HP 34410A multimeter. This voltage signal was then fed into the lock-in amplifier to be electronically differentiated. This setup is shown in figure 3 below.

Figure 3: Diagram of experimental setup. A reference signal is added to a variable power supply which supplies a voltage to the tunnel diode in a cryostat. The resulting signal is sent through an op-amp, whose output is added to the signal. The voltage and current of the signal is measured, and the sent through a current pre-amplifier and into a lock-in amplifier. The lock-in amplifier electronically differentiates the signal. Original Figure.

Lock-in amplifiers are often used to measure signals that are obscured by noise using phase-sensitive detection to single out the component of the signal at a specific frequency and phase, determined by a periodic reference signal. In this experiment, this reference signal was added to the DC variable voltage signal, causing the DC voltage to have the same frequency. The current output from the tunnel diode also had this same reference frequency applied to it, and when input to the lock-in amplifier is sent through a band pass filter with a quality factor Q=100. The lock-in amplifier allows for adjustment of the RC constant to match the applied reference frequency. The signal in the lock-in amplifier also passes through a phase-sensitive detector (PSD) after the band-pass filter. This PSD will only allow signal that matches a phase equal to that of the reference signal. Since most noise will not have a phase equal to that of the reference signal, only the wanted signal is allowed to pass through the PSD, allowing for accurate measurement^[7].

The lock-in amplifier also allows for differentiation. When the output signal from the tunnel diode V_sig is passed through the PSD, it is multiplied by the reference signal V_ref. The output V_PSD of the PSD is as follows:

where ω_sig is the signal frequency, ω_ref is the reference frequency, θ_sig is the signal phase, and θ_ref is the reference phase^[7]. Ideally, it can be assumed that the lock-in amplifier is a linear device, and since ω_sig equals ω_ref, the expression simplifies to the following:

However, generally this output voltage is a function over some outside stimulus of which we would have control over. This produces a nonlinear response in the PSD of the lock-in due to modulation, which means there is an added sine term in equation (2), as follows:

where ω₀ is the modulation response frequency, which oscillates about ω_ref. If it is assumed the modulation amplitude is small, then by expanding the added nonlinear term in equation (3) using a Taylor's series about ω_0, the following equation comes about:

giving the first derivative^[8]. The series can be expanded further to obtain the second derivative as well. This process allowed us to obtain the first and second derivatives for the I-V curve for our diode in real time to assess the accuracy of our experiment and remove the need to calculate the derivatives afterwards. Michael Enz also showed in 1998 that simply calculating the 1st and 2nd derivatives for the IV curves are too inaccurate to be made useful for polynomial fits as high as 11th order, thus introducing a need for electronic differentiation using a lock-in amplifier^[3].

Results

The data recorded for this experiment was done using a germanium 1N3712 tunnel diode at both room temperature (296K) and in a liquid helium bath at 4.2K. In the room temperature data, thermal smearing covers all resonant phonon energy peaks. The IV curve shows the known features of the tunnel diode, including a region at low voltages of negative resistance, as shown in figure 4. This room temperature measurement was completed to determine that the equipment was functioning correctly, and that our first and second derivatives were accurate. Scaling of the first and second derivatives in the plots are arbitrary. We looked for peaks in the second derivative of the IV curve, which happen at the same voltage reading and have the same width at any scale. At room temperature, as expected, there were no peaks in the second derivative as thermal smearing due to the higher temperatures had a larger effect on the data than the tunneling of the electrons.

In the liquid helium data at 4.2K, the known resonant phonon energies all happen at energies lower than 60 meV, so data was collected between 0 and 80 mV across the diode. In this range, we expected to see 2 to 3 phonon modes present in the diode: the longitudinal optical mode (LO mode), the transverse acoustic mode (TA) and the transverse optical mode (TO). Our second derivative data in figure 5 however clearly shows 2 separate peaks - one shown as a bump at less than 10 mV, and another larger peak around the 30 mV region. Due to the strong upward increase in the magnitude of the second derivative, it is difficult to see the location of the peaks. Isolating the peaks from this background data was done through later analysis.

Figure 4: Current vs Voltage reading, and its derivatives, of the tunnel diode at room temperature. This IV characteristic clearly shows the expected negative resistance region of the diode at lower voltages, showing our setup worked correctly. Scales of the y-axes of the first and second derivative plots are arbitrary.

Figure 5: Current vs voltage readings at 4.2K from liquid helium. All phonon modes that we were interested in occur at less than 60 mV, so data was taken from 0-60 mV. The IV characteristic and the 1st derivative look similar to that of the room temperature curves at these voltage ranges, however, the 2nd derivative clearly shows 2 peaks due to phonon emission. It is the location of these peaks that we were interested in.

Analysis

In the second derivative of the IV characteristic curve at 4.2K as shown in figure 5, 2 peaks are clearly visible. In order to isolate the peaks to determine their location, a second order polynomial was fit to the curve at locations outside of the peaks, and subsequently subtracted off from the data. To accomplish this, a polynomial was fit to the overall complete data. After this fit was completed, a chi-squared for the plot was calculated. Points were then subtracted from the fit, starting from the locations of the peaks, and working outwards from them. After each point was subtracted, the fit was recalculated, with an accompanying chi-squared value. This was continued until the chi-squared value reached a value less than one. This fit line was then graphed over the data for the second derivative, to be sure that the fit line was fit to the correct part of the data, as shown in figure 6. After the fit was verified, the fit was subtracted from the data, leaving behind the two peaks allowing for further analysis, as shown in figure 7.

Figure 6: Plot of the liquid helium 2nd derivative data (in blue) with a second-order polynomial fit to the background data (shown in red). This fit line was subsequently subtracted from the data to isolate the individual peaks to determine their voltages, and thus their energies.

Figure 7: Plot of the second derivative after the background fit subtraction. The location of the 2 peaks are clearly evident in this plot.

After the fit subtraction, two peaks in the data are clearly visible. To normalize the data, narrow the peaks and further refine the data for analysis, the second derivative curve was then divided by the cube of the first derivative data^[1][3]. A Gaussian curve was then fit to each of these peaks to determine the location of the center of each peak by using the curve fitting toolbox in MATLAB. By fitting the peaks to a Gaussian, the location of the first peak was determined to be at an energy of E=4.8±0.9 meV, with a reduced χ² of 0.971. This corresponds with the known location of the TA phonon mode in germanium. In 1998, Michael Enz recorded a value of 9.13±0.57 meV for the TA mode in the 1N3712 tunnel diode^[3]. Our determined value for the resonant energy of the TA mode is thus 4.8σ away from Enz's value. The accepted value for this mode in germanium is at an energy E=7.6±0.2 meV^[2], which is 3.1σ away from our measured value.

The location of the second peak was determined to be at an energy E=30.2±1.1 meV. This corresponds with the known location of the LO phonon mode in germanium. Michael Enz recorded a value of E=30.34±1.38 meV for the LO phonon mode in the 1N3712 tunnel diode^[3]. Our determined value for the resonant energy of the LO mode is thus 0.04σ away from Enz's value. The accepted value for this mode in germanium is at an energy of E=31.1±0.2 meV^[2], which is 0.73σ away from our measured value.

One possible source of error for our experiment was in the measuring of the IV characteristic and its derivatives in the liquid helium. While doing the run, a block of ice formed on top of the liquid helium from moisture in the air, in which our tunnel diode was frozen into. This could have resulted in our tunnel diode not reaching the expected 4.2K, and thus more thermal smearing than expected could have occurred. This could be a source for the discrepancy between our recorded measurement for the TA phonon mode and Enz's measurement.

Uncertainties in our measurements include an uncertainty in our current measurements of ΔI=±1µA. The uncertainty in our voltage measurements are ΔV=±0.1mV.

Future Directions

We spent a lot of time building our circuit. We used something around 40-46 gauge wire for our pole that was inserted into the helium dewar, and solder does not stick to that very well, as well as the wire breaks very easily. We could have easily used something like 30-32 gauge wire, which is thicker, stronger, and easier to work with. By the time our circuit was done with the wiring wrapped up with nothing shorting or broken, we were already halfway done with the time alotted for the project.

We spent too much time on the LabView program. It doesn't have to be fancy or anything, just enough to get the job done. In the end, the program was basically the same thing as in our lab manuals for the Verdet constant/diffraction labs mixed together, so if those still are used as labs, just follow those as a framework.

Due to our issues just getting the project together, we didn't get any data until the last week of class, and we had to really rush to do analysis and write our papers. We would have liked to do a test with a couple other different tunnel diodes to see if that made a difference, as well as we would've liked to do another run with the tunnel diode we did use since, as mentioned above, the diode got frozen into water ice a few inches above the helium, so it definitely didn't get as cool as we wanted, hence the added background we had to subtract to show the peaks. We also could've just done a doped germanium wafer, not a diode, to see how the PN junction in the diode affects the resonant energies.

Another thing a future group could do that we wanted to do with this lab is maybe try hooking up a vacuum pump to the dewar, and hit the lambda point (~2.7K, where helium goes from a 'normal' liquid into a superfluid state) and try the run there.

Conclusion

In germanium tunnel diodes, electrons will tunnel across the band gap to produce a current. This current can be measured in an IV characteristic curve, and used to determine at what energies the phonons emitted from this tunneling effect resonate at. By looking at the second derivative of the IV characteristic curve, the resonant energies show up as distinct peaks on the plot, due to an increased current at these resonant energies. By comparing our data with the known location of specific phonon modes determined from neutron scattering, we determined a measured TA phonon mode of 4.8±0.9 meV, which is 4.8σ from Michael Enz's data^[3], and 3.1σ away from neutron scattering data^[2]. We also determined an LO phonon mode energy of 30.2±1.1 meV, which is 0.04σ away from Michael Enz's recorded data^[3], and 0.73σ away from neutron scattering data^[2].

Acknowledgements

Project Advisor Lee Wienkes

Professor Greg Palowski

Professor Clement Pryke

Kurt Wick

References