Apparatus and Algorithm for Carrier Profiling In Scanning Frequency Comb Microscopy

Abstract

A semiconductor carrier profiling method utilizes a scanning tunneling microscope and shielded probe with an attached spectrum analyzer to measure power loss of a microwave frequency comb generated in a tunneling junction. From this power loss and by utilizing an equivalent circuit or other model, spreading resistance may be determined and carrier density from the spreading resistance. The methodology is non-destructive of the sample and allows scanning across the surface of the sample. By not being destructive, additional analysis methods, like deconvolution, are available for use.

Claims

1. A method to determine carrier concentration of a semiconductor sample, the method comprising: a. placing the semiconductor sample in a scanning tunneling microscope (“STM”), the STM comprising a tip circuit with a preamplifier and a shielded STM tip, the shielded STM tip being in proximity with, but not in contact with, the semiconductor sample, forming a tunneling junction; b. placing a probe in contact with the semiconductor sample in proximity to the STM tip, the probe having functional connection with a bias supply and a spectrum analyzer, forming a probe circuit, the probe circuit and tip circuit forming a system; c. irradiating the sample with a mode-locked ultrafast laser, generating a microwave frequency comb in the tunneling junction; d. measuring the power of a given harmonic of the microwave frequency comb with the spectrum analyzer and pausing to adequately measure the measured power; e. determining the localized spreading resistance of the semiconductor sample at the tunneling junction from the measured power; f. repeating the steps b-e while positioning the STM tip and the probe at another location relative to the semiconductor sample, maintaining the proximity between the STM tip and probe; g. using data from repeated measurements to determine carrier concentration of the semiconductor sample.

2. The method of claim 1, additionally providing a second spectrum analyzer in functional connection with the STM tip, the first and second spectrum analyzers being used to measure attenuation of the microwave frequency comb.

3. The method of claim 2, further comprising the step of determining attenuation caused by at least one other component in an equivalent circuit of the system.

4. The method of claim 3, the determination of attenuation being accomplished by utilizing a network analyzer in operative connection with the probe circuit.

5. The method of claim 1, further comprising the step of determining an impulse function by utilizing a known sample with a sharp boundary of two different dopant densities and deconvolving a measured carrier density, then using the impulse function for further measurements on unknown samples.

6. An apparatus to determine carrier concentration of a semiconductor sample, the apparatus comprising: a. A scanning tunneling microscope (“STM”), the STM further comprising a tip circuit with a preamplifier and a shielded STM tip, the shielded STM tip being in proximity with, but not in contact with, the semiconductor sample, forming a tunneling junction; b. A first spectrum analyzer in operative communication with the tip circuit; c. A sample circuit formed by a probe in contact with the semiconductor sample in proximity to the STM tip, the probe having functional connection with a bias supply and a second spectrum analyzer; and d. a mode-locked ultrafast laser; wherein, the mode locked ultrafast laser irradiates the sample, forming a microwave frequency comb within the tunneling junction.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0029] FIG. 1 is a prior art SSRM line-scan made with a “sandwich” sample.

[0030] FIG. 2 is a graph depicting effective probe radius vs. resistivity for SSRM measurements (prior art).

[0031] FIG. 3 is a chart depicting effective probe radius vs. resistivity for SSRM Measurements at four different pressures (prior art).

[0032] FIG. 4 is a schematic depicting an equivalent circuit for SFCM with a metal sample.

[0033] FIG. 5 is a schematic depicting an equivalent circuit for SFCM with a semiconductor sample.

[0034] FIG. 6 is a schematic depicting an equivalent circuit for determining the carrier density of a semiconductor

[0035] FIG. 7 is a graph depicting normalized power γ vs. spreading resistance at the tunneling junction for three values of R.sub.L1, the load resistance in the tip circuit.

[0036] FIG. 8 is a graph depicting power at the spectrum analyzer R.sub.L2 vs. spreading resistance at the tunneling junction for two values of R.sub.L1, the load resistance in the tip circuit.

[0037] FIG. 9 is a schematic depicting an apparatus suitable for carrying out one method of the invention.

[0038] FIG. 10 is a schematic of a simplified equivalent circuit for either of the circuits shown in either FIG. 5 or FIG. 6.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0039] With reference now to the drawings, the preferred embodiment of the method for carrier profiling and an apparatus for conducting the same is herein described. It should be noted that the articles “a”, “an”, and “the”, as used in this specification, include plural referents unless the content clearly dictates otherwise.

[0040] A microwave frequency comb (MFC) having hundreds of harmonics that set the present state-of-the-art for a narrow linewidth microwave source may be generated by focusing a mode-locked ultrafast laser on the tunneling junction of a scanning tunneling microscope (STM) [M. J. Hagmann, A. J. Taylor and D. A. Yarotski, “Observation of 200th harmonic with fractional linewidth of 10.sup.−10 in a microwave frequency comb generated in a tunneling junction,” Appl. Phys. Lett. 101,241102 (2012)]. Quasi-periodic excitation of the tunneling junction by the laser superimposes a regular sequence of short (≈15 fs) current pulses on the DC tunneling current, and in the frequency-domain this is equivalent to a microwave frequency comb with harmonics at integer multiples of the pulse repetition frequency of the laser (≈74 MHz).

[0041] In Scanning Frequency Comb Microscopy (SFCM) the MFC may be measured as the STM is scanned over the surface of an electrically-conductive sample. FIG. 4 is an equivalent circuit for SFCM with a metal sample. In a metal sample, generation of the MFC acts as a constant current source (I.sub.0) in the tunneling junction at the sub-terahertz harmonics [M. J. Hagmann, F. S. Stenger and D. A. Yarotski, “Linewidth of the harmonics in a microwave frequency comb generated by focusing a mode-locked ultrafast laser on a tunneling junction,” j. Appl. Phys. 114, 223107 (2013)]. This appears to be a non-thermal process because the roll-off in the magnitude of the harmonics at increasing frequency is consistent with a time constant of 320 ps corresponding to shunting the constant current source with the impedance of the measurement instrument which is typically a spectrum analyzer (R.sub.L=50 0), and the capacitance (C.sub.S≈6.4 pF) that is associated with the tunneling junction and its connections. Typically, the harmonics are measured using a bias-T to couple the spectrum analyzer to the sample circuit of the STM.

[0042] In recent measurements where a semiconductor was used as the sample, different carrier processes require the harmonics to be measured with a probe at the surface of the semiconductor within 1 mm of the tunneling junction. Furthermore, the magnitudes of the harmonics fall off more rapidly with increasing frequency. The second harmonic is 9 dB below the fundamental and higher order harmonics fall off as the inverse fourth power of the frequency (12 dB/octave) instead of the inverse second power (6 dB/octave) with metallic samples. Measurements with semiconductors are consistent with the model in FIG. 5 where R.sub.S1 and R.sub.S2 are the spreading resistance of the semiconductor at the tunneling junction and the probe, respectively, and C.sub.2 is the capacitance to the grounded sample holder. Others have shown [B. Gelmont and M. Shur, “Spreading resistance of a round ohmic contact,” Solid-State Electron. Vol. 36, No. 2, 1993, pp. 143-146] that a round ohmic contact with radius a at the surface of a semiconductor having resistivity p causes a spreading resistance (R.sub.S=ρ/4a) that is the dominant part of the electrical resistance to a contact anywhere on the surface of the semiconductor when that area of that contact is much larger than that of the round ohmic contact. Thus, in FIG. 5, where the tunneling junction is at point P.sub.1 and the probe is at point P.sub.2, the spreading resistances R.sub.S1 and R.sub.S2 act as both series circuit elements and shunts to the grounded sample holder.

[0043] FIG. 6 shows an equivalent circuit of the apparatus used for the carrier profiling of semiconductors in the present invention. In scanning tunneling microscopy typically feedback control is used to maintain a constant DC tunneling current by changing the distance between the tip and sample electrodes. However, in SFCM fluctuations in the magnitude of the harmonics of the MFC occur when the DC tunneling current is relatively stable. I attribute these fluctuations to changes in the capacitance of the tunneling junction when the tip-sample distance is altered in feedback control. To mitigate this problem for greater accuracy, a second load resistance (R.sub.L1 in FIG. 6), may be added so the power in the harmonics may be sampled in the tip circuit. It is helpful to refer to measurements of the MFC at R.sub.L2 in the sample circuit as the “transmitted power” because at point P.sub.2 in FIG. 6 the MFC has propagated through the semiconductor. Measurements of the power at R.sub.L1 in the tip-circuit may be used to normalize the measurements of the transmitted power at R.sub.L2 to mitigate the effects of a number of variables including fluctuations in the DC tunneling current and changes in the capacitance associated with the tunneling junction.

[0044] As one skilled in the art should see, it is practical to have R.sub.L2 be spectrum analyzers with a bias-T so that a DC bias may be applied to the probe, and for R.sub.L1 to also be a spectrum analyzer including a bias-T to separate the low-frequency preamplifier and control electronics of the STM from the microwave circuit. This may be accomplished by directly attaching a second spectrum analyzer as R.sub.L1, or attaching a load resistance in series with a directional coupler to which the second spectrum analyzer is connected.

Approximation of the equivalent circuit from measurements with a single spectrum analyzer as in FIG. 5:

[0045] 1. While capacitance C.sub.1 in FIG. 6 includes the effect of the tip-sample distance, it is predominantly caused by the connections to the tunneling junction. Thus, a suitable value for a specific STM may be determined from measurements using a metallic sample as already described in Hagmann, et al. (Appl. Phys. Lett. 101, supra), so the measured value of 6.4 pF may be used.

[0046] 2. I have used a sample of intrinsic n-type GaN with σ≈200 S/m. Assuming r≈1 nm at the tunneling junction, I use 1 MΩ as an estimate for R.sub.S1.

[0047] 3. Simulations of the equivalent circuit in FIG. 2 were used to determine approximate values of R.sub.S2 and C.sub.2 to cause the observed 9 dB roll-off of the second harmonic. These values are consistent with the material properties and dimensions that were used. R.sub.S2=120Ω and C.sub.2=10 pF.

[0048] 4. Using the semiconductor sample, I have measured a power of −117 dBm at the fundamental so I have used I.sub.0=50 μA as the peak value of the current.

[0049] This completes determining the properties for the equivalent circuit in FIG. 5.

Extension to the Model in FIG. 3 to Predict the Effects of Including the Second Spectrum Analyzer:

[0050] From Eq. (A1), the normalized power, γ, defined as the ratio of the average power measured at R.sub.L2 to that measured at R.sub.L1, is given by the following expression:

[00001] $\begin{matrix} γ = \frac{R_{L .Math. .Math. 1} .Math. R_{L .Math. .Math. 2}}{{(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})}^{2} [{[(\frac{R_{S .Math. .Math. 1}}{R_{S .Math. .Math. 2}}) .Math. \frac{(R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2})}{(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})} + 1]}^{2} + {(ω .Math. .Math. R_{S .Math. .Math. 1} .Math. C_{2})}^{2}]} & (1) \end{matrix}$

[0051] Table I gives the spreading resistance that would be measured with a semiconductor for typical values of the radius of the contact (r) and the conductivity of the semiconductor (σ). This table shows that the spreading resistance is typically from 100 kΩ to 100 MΩ. By contrast, the spreading resistance at the probe is typically 100Ω because of the much larger size for the contact. The load resistance R.sub.L2 for the spectrum analyzer is typically 50Ω. Thus, Eq. (1) may be simplified to obtain Eq. (2). Furthermore, C.sub.2 is typically about 10 pF so at a frequency of 74 MHz the second term in the large brackets is comparable to the first term. Note that Eq. (2) shows that the normalized power γ is inversely proportional to the square of the spreading resistance at the tunneling junction R.sub.S1.

TABLE-US-00001 TABLE Typical values for the spreading resistance. a, nm ρ = .1 Ωm σ = .01 Ωm σ = .001 Ωm .1 2.5 × 10.sup.8 Ω 2.5 × 10.sup.7 Ω 2.5 × 10.sup.6 Ω 1 2.5 × 10.sup.7 Ω 2.5 × 10.sup.6 Ω 2.5 × 10.sup.5 Ω 10 2.5 × 10.sup.6 Ω 2.5 × 10.sup.5 Ω 2.5 × 10.sup.4 Ω 100 2.5 × 10.sup.5 Ω 2.5 × 10.sup.4 Ω 2.5 × 10.sup.3 Ω

[00002] $\begin{matrix} γ = \frac{R_{L .Math. .Math. 1} .Math. R_{L .Math. .Math. 2}}{{[1 + 2 .Math. (\frac{R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2}})]}^{2} [{(\frac{(R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2})}{(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})})}^{2} + {(ω .Math. .Math. R_{S .Math. .Math. 2} .Math. C_{2})}^{2}] .Math. R_{S .Math. .Math. 1}^{2}} & (2) \end{matrix}$

[0052] Numerical simulations for the equivalent circuit in FIG. 3 were made by assigning values to all of the components and then using complex notation to determine the impedance presented to the constant current source. Then the voltage across this source was calculated, and values for the currents and voltages in each part of the circuit were determined. To be consistent with the above calculations which are based on the measurements the values of the input parameters were as follows: I.sub.0=50 μA peak, C.sub.1=6.4 pF, C.sub.2=10 pF, R.sub.L2=50Ω, R.sub.S2=120Ω, and R.sub.S1 and R.sub.L1 were varied as described in the following two figures. The frequency used was 74.254 MHz to be consistent with the measurements.

[0053] FIG. 7 shows the normalized power γ as a function of R.sub.S1, the spreading resistance of the semiconductor at the tunneling junction, for load resistances R.sub.L1 of 100Ω, 10 kΩ, and 1 MΩ in the tip circuit. The values shown in FIG. 7 from the simulations are equal to those determined separately by using Eq. (2), as a verification of that equation. Again, note that the normalized power γ is inversely proportional to the square of the spreading resistance at the tunneling junction R.sub.S1, which makes measurements with this method highly sensitive to the spreading resistance even though the magnitude of this resistance may be quite large.

[0054] FIG. 8 shows the power at the spectrum analyzer R.sub.L2 as a function of the spreading resistance of the semiconductor at the tunneling junction for load resistances R.sub.L1 of 100Ω and 10 kΩ in the tip circuit. Values of 10 kΩ or greater for R.sub.L1 have no significant effect on the power that is transmitted through the semiconductor and measured at R.sub.L2 so curves for higher values of R.sub.L1 would coincide with the curve for 10 kΩ in FIG. 8. Thus, R.sub.L1 could be a series resistor of 10 kΩ or greater with a spectrum analyzer attached to a directional coupler to measure the power in the tip circuit. FIG. 8 also shows that a spectrum analyzer with a 50Ω input impedance could be attached directly as R.sub.L1. However, values of R.sub.L1 lower than 50Ω would make it necessary to increase the power in the MFC to have the measurements at R.sub.L2 above the noise level for the system.

[0055] Changes in the spreading resistance at the tunneling junction have a small effect on the power that is absorbed by the reference load R.sub.L1. Variation of the spreading resistance R.sub.S1 over the range from 10.sup.4 to 10.sup.8Ω would cause the load resistance R.sub.L1 equal to 100Ω, 10 kΩ and 1 MΩ to have powers with mean values of 114 nW, 14 nW, and 140 pW and standard deviations that are 1.6%, 0.41%, and 0.22% of each respective mean. This is one example of the high stability for the reference power in R.sub.L1.

[0056] FIG. 9 is a sketch of one possible implementation of the apparatus for practicing the art of this invention. In this sketch R.sub.L1 of the equivalent circuit in FIG. 6 is shown as a spectrum analyzer but I have also described how a directional coupler and a load resistor could also be used with a spectrum analyzer as R.sub.L1. In order to make accurate microwave measurements it is necessary that the microwave energy be coupled to the semiconductor using a microwave transmission line, such as semi-rigid miniature coaxial cable. This is not generally done in a scanning tunneling microscope, but the art for doing this has already been described in U.S. Pat. No. 9,075,081, which is incorporated herein by reference in its entirety.

[0057] It is recommended that capacitance C.sub.2 and spreading resistance R.sub.Ω be measured using a network analyzer attached to the surface probe before the microwave frequency comb is enabled for scanning frequency comb microscopy. Other means of obtaining these measurements are possible, but not preferred.

[0058] Measurements of the microwave frequency comb may be used to determine the normalized power γ at two or more harmonics to test for consistency with the previously measured values of R.sub.2 and C.sub.2. Consistency may be tested since R.sub.L2 (typically 50Ω) and the measurement frequencies are known, so that Eq. (A2) should be satisfied by using the measured values of R.sub.2 and C.sub.2 with the values of γ for each pair of the measured harmonics. By following this procedure, it is possible to reduce or perhaps eliminate the need for calibration—which would require making repeated measurements with standard samples.

[0059] Normalized power (γ) may be determined in the following manner, based on the equivalent circuit shown in FIG. 10:

Two branches with impedances Z, and Z, are in parallel across the constant current source:

[00003] $Z_{1} = R_{2} + \frac{R_{3}}{1 + j .Math. .Math. ω .Math. .Math. R_{3} .Math. C_{2}} = \frac{(R_{2} + R_{3}) + j .Math. .Math. ω .Math. .Math. R_{2} .Math. R_{3} .Math. C_{2}}{1 + j .Math. .Math. ω .Math. .Math. R_{3} .Math. C_{2}};$ $Z_{2} = \frac{R_{1}}{1 + j .Math. .Math. ω .Math. .Math. R_{1} .Math. C_{1}}$ $Z_{effective} = \frac{R_{1} (R_{2} + R_{3}) + j .Math. .Math. ω .Math. .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{2}}{[\begin{matrix} (R_{1} + R_{2} + R_{3} - ω^{2} .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{1} .Math. C_{2}) + \\ j .Math. .Math. ω (R_{1} .Math. R_{2} .Math. C_{1} + R_{1} .Math. R_{3} .Math. C_{1} + R_{1} .Math. R_{3} .Math. C_{2} + R_{2} .Math. R_{3} .Math. C_{2}) \end{matrix}]}$

Voltage across the constant current source:

[00004] $V_{CCS} = \frac{[R_{1} (R_{2} + R_{3}) + j .Math. .Math. ω .Math. .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{2}] .Math. I_{0}}{[\begin{matrix} (R_{1} + R_{2} + R_{3} - ω^{2} .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{1} .Math. C_{2}) + \\ j .Math. .Math. ω (R_{1} .Math. R_{2} .Math. C_{1} + R_{1} .Math. R_{3} .Math. C_{1} + R_{2} .Math. R_{3} .Math. C_{2} + R_{1} .Math. R_{3} .Math. C_{2}) \end{matrix}]}$ $V_{CCS} = \frac{[\begin{matrix} [\begin{matrix} (R_{1} + R_{2} + R_{3} - ω^{2} .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{1} .Math. C_{2}) - \\ j .Math. .Math. ω (R_{1} .Math. R_{2} .Math. C_{1} + R_{1} .Math. R_{3} .Math. C_{1} + R_{2} .Math. R_{3} .Math. C_{2} + R_{1} .Math. R_{3} .Math. C_{2}) \end{matrix}] \\ [R_{1} (R_{2} + R_{3}) + j .Math. .Math. ω .Math. .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{2}] .Math. I_{0} \end{matrix}]}{\begin{matrix} {(R_{1} + R_{2} + R_{3} - ω^{2} .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{1} .Math. C_{2})}^{2} + \\ {ω^{2} (R_{1} .Math. R_{2} .Math. C_{1} + R_{1} .Math. R_{3} .Math. C_{1} + R_{2} .Math. R_{3} .Math. C_{2} + R_{1} .Math. R_{3} .Math. C_{2})}^{2} \end{matrix}}$ $V_{CCS} = \frac{[\begin{matrix} [(R_{2} + R_{3}) .Math. (R_{1} + R_{2} + R_{3}) + ω^{2} .Math. R_{2} .Math. R_{3}^{2} .Math. C_{2}^{2} (R_{1} + R_{2})] - \\ j .Math. .Math. ω .Math. .Math. R_{1} (R_{2}^{2} .Math. C_{1} + 2 .Math. R_{2} .Math. R_{3} .Math. C_{1} + R_{3}^{2 .Math.} .Math. C_{1} + R_{3}^{2} .Math. C_{2} + .Math. {ω .Math.}^{2} .Math. R_{2}^{2} .Math. R_{3}^{2} .Math. C_{1} .Math. C_{2}] \end{matrix}] .Math. R_{1} .Math. I_{0}}{[\begin{matrix} {(R_{1} + R_{2} + R_{3} - ω^{2} .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{1} .Math. C_{2})}^{2} + \\ {ω^{2} (R_{1} .Math. R_{2} .Math. C_{1} + R_{1} .Math. R_{3} .Math. C_{1} + R_{2} .Math. R_{3} .Math. C_{2} + R_{1} .Math. R_{3} .Math. C_{2})}^{2} \end{matrix}]}$

With peak values for the voltage and current the average power from the constant current source

[00005] $\begin{matrix} {\overline{P}}_{CCS} = .Math. Re (V_{CCS} .Math. I_{0}^{*}) \\ = .Math. \frac{[\begin{matrix} (R_{2} + R_{3}) .Math. (R_{1} + R_{2} + R_{3}) + \\ ω^{2} .Math. R_{2} .Math. R_{3}^{2} .Math. C_{2}^{2} (R_{1} + R_{2}) \end{matrix}] .Math. R_{1} .Math. I_{0}^{2}}{2 [\begin{matrix} {(R_{1} + R_{2} + R_{3} - ω^{2} .Math. R_{1} .Math. R_{2} .Math. R_{3} .Math. C_{1} .Math. C_{2})}^{2} + \\ {ω^{2} (R_{1} .Math. R_{2} .Math. C_{1} + R_{1} .Math. R_{3} .Math. C_{1} + R_{2} .Math. R_{3} .Math. C_{2} + R_{1} .Math. R_{3} .Math. C_{2})}^{2} \end{matrix}]} \end{matrix}$

Transformation between the equivalent circuit in FIG. 3 and the above equations for FIG. 7 requires:

[00006] $R_{S .Math. .Math. 2} + \frac{R_{S .Math. .Math. 2} .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}} .Math. R_{3}, \frac{R_{S .Math. .Math. 1} .Math. R_{L .Math. .Math. 1}}{R_{S .Math. .Math. 1} + R_{L .Math. .Math. 1}} .Math. R_{1}, and$ $R_{S .Math. .Math. 1} .Math. R_{2}$

The voltage across resistor R.sub.L1 is V.sub.CCS, so

[00007] ${\overline{P}}_{RL .Math. .Math. 1} = \frac{{.Math. V_{CCS} .Math.}^{2}}{2 .Math. R_{L .Math. .Math. 1}}$

The current in R.sub.3 is given by

[00008] $I_{3} = (1 - \frac{R_{2}}{Z_{1}}) .Math. \frac{V_{CCS}}{R_{3}}$

Thus, in FIG. 3 the current in R.sub.S2 is

[00009] $I_{RS .Math. .Math. 2} = (1 - \frac{R_{S .Math. .Math. 1}}{Z_{1}}) .Math. \frac{V_{CCS}}{(R_{S .Math. .Math. 2} + \frac{R_{S .Math. .Math. 2} .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}})}$

so the current in R.sub.L2 is

[00010] $\begin{matrix} I_{RL .Math. .Math. 2} = .Math. (1 - \frac{R_{S .Math. .Math. 1}}{Z_{1}}) .Math. \frac{V_{CCS}}{(R_{S .Math. .Math. 2} + \frac{R_{S .Math. .Math. 2} .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}}) .Math. (1 + \frac{R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2}})} \\ = .Math. (1 - \frac{R_{S .Math. .Math. 1}}{Z_{1}}) .Math. \frac{V_{CCS}}{(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})} \end{matrix}$

Thus,

[0060] [00011] ${\overline{P}}_{RL .Math. .Math. 2} = (1 - \frac{R_{S .Math. .Math. 1}}{Z_{1}}) .Math. (1 - \frac{R_{S .Math. .Math. 1}}{Z_{1}^{*}}) .Math. \frac{R_{L .Math. .Math. 2} .Math. {.Math. V_{CCS} .Math.}^{2}}{2 .Math. {(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})}^{2}}$

Normalizing, we define

[00012] $γ \equiv \frac{{\overline{P}}_{RL .Math. .Math. 2}}{{\overline{P}}_{RL .Math. .Math. 1}} = (1 - \frac{R_{S .Math. .Math. 1}}{Z_{1}}) .Math. (1 - \frac{R_{S .Math. .Math. 1}}{Z_{1}^{*}}) .Math. \frac{R_{L .Math. .Math. 1} .Math. R_{L .Math. .Math. 2}}{{(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})}^{2}}$ $γ = \frac{R_{L .Math. .Math. 1} .Math. R_{L .Math. .Math. 2} [{(R_{S .Math. .Math. 1} - R_{2} - R_{3})}^{2} + ω^{2} .Math. R_{3}^{2} .Math. {C_{2}^{2} (R_{S .Math. .Math. 1} - R_{2})}^{2}]}{{(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})}^{2} [{(R_{2} + R_{3})}^{2} + ω^{2} .Math. R_{2}^{2} .Math. R_{3}^{2} .Math. C_{2}^{2}]}$

Comparing FIGS. 3 and 7, R.sub.2.fwdarw.R.sub.S1 and

[00013] $R_{3} .fwdarw. R_{S .Math. .Math. 2} + \frac{R_{S .Math. .Math. 2} .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}}$

so that

[00014] $\begin{matrix} γ = \frac{R_{L .Math. .Math. 1} .Math. {R_{L .Math. .Math. 2} (R_{S .Math. .Math. 2} + \frac{R_{S .Math. .Math. 2} .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}})}^{2}}{{(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})}^{2} [\begin{matrix} {(R_{S .Math. .Math. 1} + R_{S .Math. .Math. 2} + \frac{R_{S .Math. .Math. 2} .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}})}^{2} + \\ ω^{2} .Math. {R_{S .Math. .Math. 1}^{2} (R_{S .Math. .Math. 2} + \frac{R_{S .Math. .Math. 2} .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}})}^{2} .Math. C_{2}^{2} \end{matrix}]} .Math. .Math. γ = \frac{R_{L .Math. .Math. 1} .Math. R_{L .Math. .Math. 2}}{{(R_{S .Math. .Math. 2} + 2 .Math. .Math. R_{L .Math. .Math. 2})}^{2} [{((\frac{R_{S .Math. .Math. 1}}{R_{S .Math. .Math. 2}}) .Math. \frac{(R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2})}{(R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2})} + 1)}^{2} + {(ω .Math. .Math. R_{S .Math. .Math. 1} .Math. C_{2})}^{2}]} & (A .Math. .Math. 1) \end{matrix}$

When R.sub.S1 is large,

[00015] $\begin{matrix} \frac{γ (f_{2})}{γ (f_{1})} = \frac{1 + {(2 .Math. π .Math. .Math. f_{1} .Math. R_{S .Math. .Math. 2} .Math. C_{2})}^{2} .Math. {(\frac{R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}})}^{2}}{1 + {(2 .Math. π .Math. .Math. f_{2} .Math. R_{S .Math. .Math. 2} .Math. C_{2})}^{2} .Math. {(\frac{R_{S .Math. .Math. 2} + 2 .Math. R_{L .Math. .Math. 2}}{R_{S .Math. .Math. 2} + R_{L .Math. .Math. 2}})}^{2}} & (A .Math. .Math. 2) \end{matrix}$

[0061] The advantages over the prior art are not only finer resolution, but a non-destructive process which allows for repeatable testing the same sample over many areas of the sample, and lessened time in tip preparation as the use of larger tips is feasible in the methodology.

Applications of the Methodology

[0062] Because the methodology is non-destructive, deconvolution strategies may be employed. As an example, a known sample with a defined variation in dopant density may be measured. The measurement may then be deconvolved to determine the impulse function of the STM. Once this impulse function is known, it may be used to deconvolve measurements taken from unknown samples.

[0063] Likewise, the equivalent circuits may be used with measurements to determine the spreading resistance of a semi-conductor sample at the tunneling junction, and therefore the carrier density. By measuring known sources of attenuation, such as shunting capacitance of the semi-conductor sample and spreading resistance from the probe, the remaining attenuation may be evaluated to determine spreading resistance at the tunneling junction. These known sources of attenuation may be measured by any means known now or later developed, including using a network analyzer connected to the probe.

[0064] Although the present invention has been described with reference to preferred embodiments, numerous modifications and variations can be made and still the result will come within the scope of the invention. As an example, the equivalent circuit model may be replaced with a field model, utilizing Maxwell's equations, by one skilled in the art. However, because the wavelengths in the MFC are much larger than the separation between the probe and the tunneling junction, it is generally found to be more convenient to utilize equivalent circuits to model the methodology. No limitation with respect to the specific embodiments disclosed herein is intended or should be inferred.

Apparatus and Algorithm for Carrier Profiling In Scanning Frequency Comb Microscopy

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Cpc classification

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G01Q60/10

PHYSICS

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G01Q60/12

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G01Q10/04

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International classification

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G01Q10/04

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G01Q60/10

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Abstract

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