Method of carrier profiling utilizing dielectric relaxation

Abstract

A mode-locked laser injects pulses of minority carriers into a semiconductor sample. A microwave frequency comb is then generated by the currents formed in the movement of majority carriers native to the semiconductor and the injected minority carriers. These carriers move to cause dielectric relaxation in the sample, which can be used to determine carrier density within the sample. Measurements require close proximity of transmitter and receiver contacts with the sample and may profile a semi-conductor with a resolution of approximately 0.2 nm.

Claims

1. A method for determining carrier density in a semiconductor, the method comprising: a. positioning a semiconductor sample in a scanning tunneling microscope such that a tunneling junction is formed between the sample and a tip of the scanning tunneling microscope; b. electrically biasing the sample such that a tunneling current is generated and no depletion region is formed in the sample; c. positioning a receiving probe on the semiconductor sample proximate the tunneling junction at a distance l; d. focusing a mode-locked laser on the tunneling junction and generating surges of minority carrier surface charge on the semiconductor by modulating the tunneling current with the mode-locked laser, thereby creating a microwave frequency comb therein; e. characterizing decay of the microwave frequency comb due to dielectric relaxation by taking measurements of power from the microwave frequency comb at a plurality of different distances l; f. determining carrier density of the semiconductor from the characterized decay of the microwave frequency comb.

2. The method of claim 1, a distance between the tunneling junction and receiving probe being less than the mean-free-path of majority carriers in the sample, such that electrical resistance is not defined and sub-nanometer resolution of carrier density is made possible.

3. The method of claim 1, wherein the distances l are attained by moving the receiving probe relative to the tunneling junction certain fixed distances, and power measured at said fixed distances is used to extrapolate the point at which dielectric relaxation is complete.

4. The method of claim 1, wherein the distances l are attained by moving the receiving probe relative to the tunneling junction and measurements are made until the point at which dielectric relaxation is complete is found.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a schematic detailing an apparatus for the method.

(2) FIG. 2 is a schematic illustrating dielectric relaxation on an alternate apparatus.

(3) FIG. 3 is a graph depicting received power in dBm vs. radial distance from the transmitting tunneling junction.

(4) FIG. 4 is a graph depicting Contours for 9 different values of the carrier concentration as a function of the number of injected carriers per pulse and the measured microwave power for a tip-probe distance of 8 nm, as shown by line A in FIG. 3.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

(5) With reference now to the drawings, the preferred embodiment of the method 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.

(6) With reference to FIG. 1, a scanning tunneling microscope is set up similarly to what is described in U.S. Pat. No. 9,442,078 (2016), also by this Inventor. This Patent is incorporated by reference in its entirety herein. In one embodiment, an apparatus set-up 10 is a scanning tunneling microscope 20 positioned over a sample 30 onto which a mode-locked laser 40 is focused so that the laser emits a beam onto the tunneling junction between the STM 20 and the sample 30. The semiconductor sample 30 is forward biased 70, as is shown in FIG. 1 where a p-type semiconductor 30 is used and the voltage on the STM tip 25 is set to make the tip negative, thus allowing the carriers in the semiconductor sample to remain in the region directly beneath the tip. A spectrum analyzer 50 is connected into the circuit by means of a microwave probe 60 making ohmic contact with the sample 30 in close proximity to the tunneling junction.

(7) It is convenient to refer to the tunneling junction as the transmitter of the microwave frequency comb, and the second probe tip as the receiver or receiving probe. The receiving probe 65 may be any type receiver making contact, such as an ohmic contact, as is shown in FIG. 2. In some applications it may be possible to use a point-contact, such as with a fine gold wire as the receiver. It should also be noted that the sample 35 in FIG. 2 is an n-type semiconductor. Standard concerns and adaptations for polarity, for instance using a p-type semiconductor, are considered part of the invention.

(8) In the following analysis N.sub.0 is defined as the number of excess minority carriers injected into the semiconductor at time t=0, n is the local density of these excess minority carriers in the semiconductor, and c is the local density of majority carriers in the semiconductor.

(9) Consider a spherical model where N.sub.0 excess minority carriers are injected to a small region centered at the origin at time t=0. Electrical forces cause the injected carriers to move radially outward and the majority carriers in the semiconductor, having density c, to move radially inward to complete dielectric relaxation within a sphere of radius R, where

(10) $\begin{array}{cc}R={\left(\frac{3N_0}{4c}\right)}^{\frac{1}{3}}& \left(1\right)\end{array}$

(11) These two sets of carriers with opposite signs move in opposite directions to contribute to a flow of current having the same direction. This current may be approximated as a Gaussian pulse where the local density of the excess minority carriers is given by the following expression which allows for the effects of spreading as well as termination at r=R. Here is a normalization constant:

(12) $\begin{array}{cc}n=\frac{N_0}{r^2}\left[1-{\left(\frac{r}{R}\right)}^3\right]e^{-{k\left(r-\mathrm{vt}\right)}^2}& \left(2\right)\end{array}$
At a specific time t.sub.1, when the peak for this pulse is at r.sub.1=vt.sub.1, let xrr.sub.1so that local density of excess majority carriers is given by

(13) $\begin{array}{cc}n=\frac{N_0}{r^2}\left[1-{\left(\frac{r}{R}\right)}^3\right]e^{-{\mathrm{kx}}^2}& \left(3\right)\end{array}$

(14) The short initial duration for the pulse may be assumed to cause the coefficient k to be large so that the pulse will have a short radial extent. Now this approximation is used to evaluate the normalization constant by equating two expressions for the total number of excess carriers with a specific value for the radius r at the peak.

(15) $\begin{array}{cc}{N}_0\left[1-{\left(\frac{r}{R}\right)}^3\right]=\frac{N_0}{r^2}\left[1-{\left(\frac{r}{R}\right)}^3\right]4r^2{}_{-}^{}{e}^{-{\mathrm{kx}}^2}\mathrm{dx}& \left(4\right)\\ \mathrm{But}{}_{-}^{}{e}^{-{\mathrm{kx}}^2}\mathrm{dx}=\sqrt{\frac{}{k}}& \left(5\right)\\ \mathrm{Thus}=\frac{1}{4}\sqrt{\frac{k}{{}^3}}& \left(6\right)\\ \mathrm{so}\mathrm{that}n=\sqrt{\frac{k}{}}\frac{N_0}{4r^2}\left[1-{\left(\frac{r}{R}\right)}^3\right]e^{-{k\left(r-\mathrm{vt}\right)}^2}& \left(7\right)\end{array}$
Thus, the current received by a probe with area A<<4r.sup.2 located at a fixed value of r is given by

(16) $\begin{array}{cc}I\left(r,t\right)=\mathrm{evAn}& \left(8\right)\\ I\left(r,t\right)=\sqrt{\frac{k}{}}\frac{{\mathrm{evAN}}_0}{4r^2}\left[1-{\left(\frac{r}{R}\right)}^3\right]e^{-{k\left(r-\mathrm{vt}\right)}^2}& \left(9\right)\end{array}$
Assume that the probe is connected to an ideal load, having resistance R.sub.L but no capacitance or inductance. During each pulse, the power that is delivered to the load is given by

(17) $\begin{array}{cc}P\left(r,t\right)=\frac{{\mathrm{kR}}_L}{}{{\left(\frac{{\mathrm{evAN}}_0}{4r^2}\right)}^2\left[1-{\left(\frac{r}{R}\right)}^3\right]}^2{e}^{-2{k\left(r-\mathrm{vt}\right)}^2}& \left(10\right)\end{array}$
Thus, the energy that is delivered to the load during each pulse is given by

(18) $\begin{array}{cc}E\left(r\right)=\frac{{\mathrm{kR}}_L}{}{{\left(\frac{{\mathrm{evAN}}_0}{4r^2}\right)}^2\left[1-{\left(\frac{r}{R}\right)}^3\right]}^2{}_{-}^{}{e}^{-2{\mathrm{kv}}^2{t\text{'}}^2}\mathrm{dt}\text{'}& \left(11\right)\\ E\left(r\right)=\frac{{\mathrm{kR}}_L}{}{{\left(\frac{{\mathrm{evAN}}_0}{4r^2}\right)}^2\left[1-{\left(\frac{r}{R}\right)}^3\right]}^2\sqrt{\frac{}{2{\mathrm{kv}}^2}}& \left(12\right)\\ E\left(r\right)=R_{L}v\sqrt{\frac{k}{2}}{{\left(\frac{{\mathrm{eAN}}_0}{4r^2}\right)}^2\left[1-{\left(\frac{r}{R}\right)}^3\right]}^2& \left(13\right)\end{array}$
Finally, the power that is delivered to the load is given by the following expression where f.sub.p is the pulse repetition frequency of the laser:

(19) $\begin{array}{cc}P\left(r\right)=R_L{f}_{p}v\sqrt{\frac{k}{2}}{{\left(\frac{{\mathrm{eAN}}_0}{4r^2}\right)}^2\left[1-{\left(\frac{r}{R}\right)}^3\right]}^2& \left(14\right)\end{array}$

(20) The power spectral density may be determined by taking a Fourier transform to evaluate the power that would be received at each harmonic of the microwave frequency comb. Alternatively a Fourier series may be used for this calculation because of the quasi-periodic nature of the excitation by a mode-locked laser.

(21) Equations (15) and (16) give the peak value of the current as the pulse crosses a sphere with radius r, and the total power that would be measured in the microwave frequency comb at the distance r. Here Eq. (17) defines F, the fraction of the total current that is subtended by the probe.

(22) 0$\begin{array}{cc}{I}_p\left(r\right)=\mathrm{evF}\sqrt{\frac{k}{}}\left(N_0-\frac{4r^3}{3}c\right)& \left(15\right)\\ P\left(r\right)=e^2{\mathrm{vF}}^2{f}_p{R}_L\sqrt{\frac{k}{2}}{\left(N_0-\frac{4r^3}{3}c\right)}^2& \left(16\right)\\ F=\frac{A}{4r^2}& \left(17\right)\end{array}$
FIG. 3 shows the received power calculated in dBm as a function of the radial distance from the tunneling junction which is the source of the microwave frequency comb. These calculations were made using Eq. (16) with the following parameters: k=210.sup.8/m, v=1.1610.sup.6 m/s corresponding to 1 eV electrons in silicon, F=10.sup.3, f.sub.p=74.254 MHz, and R.sub.L=50. The values for the pulse repetition rate, load resistance, and the range in sensitivity that are required are consistent with those already used in other measurements [M. J. Hagmann, A. J. Taylor and D. A. Yarotski, Observation of 200.sup.th harmonic with fractional linewidth of 10.sup.10 in a microwave frequency comb generated in a tunneling junction, Appl. Phys. Lett. 101 (2012) 241102].

(23) These calculations were made for four values of N.sub.0, as shown in FIG. 3. The four separate curves for each value of N.sub.0 correspond to c=510.sup.18, 110.sup.19, 210.sup.19, and 510.sup.19/cm.sup.3, where the curves fall off faster as the value of c is increased because a smaller volume of the semiconductor is required for neutralization.

(24) It is possible to determine the local value of the carrier density c in the semiconductor by making measurements when the tunneling junction is at one or more points on the semiconductor. This is possible by measuring the total microwave power and its spectrum in the microwave frequency comb. It is possible to obtain additional information by measuring the microwave frequency comb as a function of the distance from the tunneling junction but it is preferable to only make measurements when this distance is held constant. This may be understood because of the errors that are caused by temperature dependence, hysteresis, and nonlinearity in various means for positioning the probe and the time required for the additional measurements. By contrast, it is a relatively simple matter to accurately change the value of N.sub.0 by varying the power flux density of the laser. FIG. 4 shows that at a fixed distance the contour of N.sub.0 vs. microwave power is a unique function of the carrier concentration.

(25) It stands to reason that the distance between the transmitter and receiver l must be less than a given r in order to accurately measure the dielectric relaxation phenomena before the carriers in the semiconductor neutralize the space charge. Therefore, it is suitable to have both transmitter and receiver mobile in relation to each other and the sample. The distance e may be on the order of 8-10 nm or even closer as semiconductors with higher dopant concentrations are used. Adjustments to power output and location of the transmitter and receiver may also be used to accomplish 3D carrier profiling of a given sample.

(26) Characterization of the carrier density may be made by taking measurements of the power at specific fixed distances l.sub.1, l.sub.2, l.sub.3 . . . and curve fitting to extrapolate at which distance l.sub.f the power will be fully attenuated. Alternatively, power may be measured by increasing or decreasing distances l.sub.1, l.sub.2, l.sub.3 . . . until the null point l.sub.f is found. Distances l may be achieved by relative motion of the receiver to the transmitter, which may involve either or both transmitter and receiver moving.

(27) 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. No limitation with respect to the specific embodiments disclosed herein is intended or should be inferred.