446 Mark J. Hagmann et al.
contact with radius a, the spreading resistance is given approximately by RS = 1/4aσ,where σ is the electrical con- ductivity (Gelmont & Shur, 1993). There is also a spreading capacitance, but its effects may be neglected at the frequencies of theMFC. Scanning spreading resistance microscopy (SSRM) is a means for carrier profiling in semiconductors, which provides a spatial resolution as fine as 1nm. This method is based on measuring the spreading resistance by inserting heavily doped diamond nanoprobes into the semiconductor (Vandervorst et al., 2014). However, in our measurements, the minority and majority carriers at a sub-nanometer spot, near the tunneling junction, flow through the spreading resistance (~1MΩ), which may be determined because of its strong effect on the attenuation of the measured MFC. Figure 8 shows an equivalent circuit for determining the
Figure 7. Expanded graph of the mean power at the second harmonic for 24 consecutive scans with an n-type GaN sample in the scanning tunneling microscope, with error bars.
In these figures, the frequency is offset to make the values on the abscissa readable (i.e., 20Hz is added to 74.254MHz, etc.). Figures 4 and 6 show the power at the fundamental frequency and Figures 5 and 7 are for the second harmonic. Figures 6 and 7 are expanded and error bars are added to enable interpretation. In each case, the semiconductor had a bias of −9V relative to the tip for a forward-biased tunneling junction. The DC tunneling current was 1μA, the resolution bandwidth was 2 Hz, and the laser had an average power of
60mW.The error bars correspond to the mean plus or minus 1 SD, based on the 24 consecutive scans. Figures 6 and 7 show that the apparent linewidth is 2.6Hz at the two harmonics and this value is primarily due to the 2-Hz resolution bandwidth of the spectrum analyzer. Figures 6 and 7 show that the MFC with an n-type GaN
sample has a power at the second harmonic that is 9 dB less than that at the fundamental frequency. However, equation (1) shows that with a metal sample, the power at the second harmonic is 94% of that at the fundamental frequency. Furthermore, with the semiconductor the power at the fun- damental frequency is 25 dB less than that with a gold sam- ple, using the same apparatus and settings. These differences may be understood by considering the different phenomena with metal and semiconductor samples. High-frequency currents in theMFCflow on the surface of a metal sample, so it is possible to make measurements with a bias-T in the sample circuit as shown in Figure 1. However, it has already been noted that with a semiconductor, sample minority carriers are injected to create a sub-nanometer spot of sur- face charge that is neutralized, by dielectric relaxation, as the injected carriers diverge and interact with the converging majority carriers over a distance of ~20 μm. Whenever current is injected at a small spot on the
surface of an object, the “spreading resistance” at that spot is the dominant component of the resistance of any other contact on the
object.This effect is seen using a semiconductor sample in an STM (Flores & Garcia, 1984). For a circular
power at the harmonics of the MFC for a grounded semi- conductor sample in an STM. For the case of a metal sample, the circuit elements between points P1 and P2, which repre- sent the grounded semiconductor, are deleted. Thus, we would have the circuit we have already described with only the constant current source In, the shunting capacitance CS, and the load resistance RL, to obtain equation (1). However, point P1 represents the location of the tunneling junction and point P2 represents the location of the probe. In our measurements, the semiconductor with a thickness of 0.5mm was in a grounded sample holder. Resistances RS1 and RS2 are, respectively, the spreading resistance at the tunneling junction and at the probe, and they have small physical size. The resistance from P1 to ground is RS1 and the resistance from P2 to ground is RS2. The resistance from P1 to P2 equals RS1+RS2. The capacitance C12, which is distributed over the length of the semiconductor from P1 to P2, is that between the upper surface of the semiconductor and ground. For this equivalent circuit, the microwave power
PL = RLI2 0
2 2
2 + R3 RS2
R2 +RL2 ;
whereR3 RS2 + RS2RL RS2RL
: RS1 -ω2RS1R3CSC12 ðÞ+ω2 RS1CS +R3CS +R3C2 +R3C2 2 ð2Þ (3) I0 is the root-mean-square value of the current from the
generator, which represents the constant current source in the tunneling junction. For applications of interest in char- acterizing the semiconductors, RS1 = 1MΩ, RS2 = 1kΩ, RL = 50 Ω, R3 = 1kΩ, CS = 6.4 pF, and C12 = 12 pF. Thus, for frequencies near the first few harmonics, equation (2) may be simplified to obtain the following expression:
PL = RLI2 0 4 +ω2R2 S1C2 S +ω4R2 S1R2 S2C2 SC2 12
Figure 9 shows the microwave power as a function of the spreading resistance at the tunneling junction, where PL was
: (4)
delivered to the load, at a specific harmonic with the angular frequency ω, is given by the following expression:
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