Abstract:
A method and apparatus generate a measured data set by: (i) providing a probe tip at a selected height from a doped region of a substrate, (ii) applying a probing signal to the probe tip, (iii) measuring a characteristic of an electrical interaction between the probe tip and the doped region of the substrate, and (iv) repeating steps (i) through (iii) for a plurality of different selected heights. A plurality of reference data sets are provided characterizing the electrical interaction between the probe tip and the doped region of the substrate as a function of height between the probe tip and the doped region of the substrate. Each data set corresponds to a different dopant density. The measured data set is compared to the plurality of reference data sets and based on the comparison, the dopant density of the doped region of the substrate is determined.

Description:
BACKGROUND 
     In semiconductor manufacturing, doping is the process of intentionally introducing impurities into a semiconductor material to change its electrical properties. The electrical performance of doped semiconductor devices will change depending on the doping density and profile. There are therefore a number of techniques to try to determine the dopant concentration in a solid consisting mainly of semiconductor material. 
     One class of techniques employs principles of atomic force microscopy (AFM). An atomic force microscope consists of a microscale cantilever with a sharp conductive tip (probe) at its end that is used to scan a specimen surface. The cantilever is typically silicon or silicon nitride with a tip that is covered with a conductive material and which typically has a radius of curvature on the order of nanometers. 
     One technique for detecting doping is scanning capacitance microscopy (SCM). Scanning capacitance microscopy (SCM) is a type of scanning probe microscopy in which a sharp probe electrode is held near or on the surface of a sample and scanned across the sample. SCM characterizes the surface of the sample using information obtained from the change in differential capacitance between the surface and the probe. More precisely SCM uses an ultra-sharp conducting probe (often Pt/Ir or Co/Cr metal covering an etched silicon probe) to form a metal-insulator-semiconductor (MIS/MOS) capacitor with a semiconductor sample if an oxide is present. When no oxide is present, a Schottky contact/capacitor is formed. When the probe and surface are in contact, an AC bias is applied, generating capacitance variations in the sample which can be detected using a GHz resonant capacitance sensor or other means. The tip is then scanned across the semiconductor&#39;s surface in 2D while the tip&#39;s height is controlled by conventional contact force feedback. 
     By applying an alternating bias to the metal-coated probe, carriers alternately accumulate and deplete within the semiconductor&#39;s surface, changing the tip-sample capacitance. The magnitude of this change in capacitance with the applied voltage gives information about the concentration of carriers (SCM amplitude data), whereas the difference in sign of the capacitance change relative to the applied, alternating bias carries information about the sign of the charge carriers. Because SCM functions even through an insulating layer, a finite conductivity is not required to measure the electrical properties. 
     When the SCM tip is brought into close proximity with the sample surface a Metal/Oxide/Semiconductor (MOS) capacitor is formed between them, where: M is the metal probe, S is the semiconductor material and O is a thin dielectric formed on the semiconductor surface. Free carriers within the sample are able to move under the influence of an AC electric field applied by the conductive probe (tip). The capacitance measured by the SCM sensor varies as the carriers move towards (accumulation) and away from (depletion) the probe. When the sample is fully depleted the measured capacitance is that of the oxide plus the depletion layer. When carriers are accumulated at the surface, the measured capacitance is that of the oxide layer. This capacitance variation in response to the tip-applied field forms the basis of the SCM measurement. Movement of free carriers and hence the amplitude of the capacitance variation is a function of the dopant level of the sample directly beneath the probe. For heavily doped materials the carriers do not move far. Hence, the measured capacitance variation between accumulation and depletion is small. The opposite is true for lightly doped semiconductors which yield a large capacitance change. 
     However in general these techniques can only provide indications of the relative dopant concentrations in a device, but they cannot measure the absolute dopant densities in semiconductor devices, particularly not in small regions when the dopant density varies over such regions of a wafer. 
     What is needed, therefore, are new methods for determining absolute dopant densities in semiconductor devices. 
     SUMMARY 
     In an example embodiment, a method includes generating a measured data set, by: (i) providing a probe tip at a selected height from a doped region of a substrate, (ii) applying a probing signal to the probe tip, (iii) measuring a characteristic of an electrical interaction between the probe tip and the doped region of the substrate, and (iv) repeating steps (i) through (iii) for a plurality of different selected heights. The method further includes: providing a plurality of reference data sets characterizing the electrical interaction between the probe tip and the doped region of the substrate as a function of height between the probe tip and the doped region of the substrate, each reference data set corresponding to a different dopant density; comparing the measured data set to the plurality of reference data sets; and determining a dopant density of the doped region of the substrate from the comparison. 
     In another example embodiment, an apparatus comprises: a probe tip provided on a cantilever arm; a control device configured to move the probe tip with respect to a doped region of a substrate; a signal processing device configured to apply a probing signal to the probe tip and to sense an electrical interaction between the probe tip and the doped region; and a processor configured to control the apparatus to perform an algorithm. The algorithm includes generating a measured data set by: (i) moving the probe tip at a selected height from a doped region of a substrate, (ii) applying the probing signal to the probe tip, (iii) measuring a characteristic of an electrical interaction between the probe tip and the doped region of the substrate, and (iv) repeating steps (i) through (iii) for a plurality of different selected heights. The algorithm further includes providing a plurality of reference data sets characterizing the electrical interaction between the probe tip and the doped region of the substrate as a function of height between the probe tip and the doped region of the substrate, each reference data set corresponding to a different dopant density; comparing the measured data set to the plurality of reference data sets; and determining a dopant density of the doped region of the substrate from the comparison. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The example embodiments are best understood from the following detailed description when read with the accompanying drawing figures. It is emphasized that the various features are not necessarily drawn to scale. In fact, the dimensions may be arbitrarily increased or decreased for clarity of discussion. Wherever applicable and practical, like reference numerals refer to like elements. 
         FIG. 1  illustrates principles of scanning capacitance microscopy (SCM) and scanning microwave microscopy (SMM). 
         FIG. 2  is a functional block diagram of one embodiment of a scanning microwave microscopy (SMM) instrument. 
         FIG. 3  is a flowchart of a first embodiment of a method of determining a dopant density. 
         FIG. 4  illustrates a series of normalized capacitance vs. height curves for various dopant densities. 
         FIG. 5  is a flowchart of a second embodiment of a method of determining a dopant density. 
         FIG. 6  illustrates a series of normalized change of capacitance vs. height curves for various dopant densities. 
         FIG. 7  is a flowchart of a third embodiment of a method of determining a dopant density. 
         FIG. 8  illustrates a series of normalized reflectance phase difference vs. height curves for various dopant densities. 
         FIG. 9  is a flowchart of a fourth embodiment of a method of determining a dopant density. 
         FIG. 10  illustrates a series of reflectance magnitude ratio vs. height curves for various dopant densities. 
         FIG. 11  is a flowchart of a fifth embodiment of a method of determining a dopant density. 
         FIG. 12  illustrates a series of change of reflectance magnitude vs. height curves for various dopant densities. 
     
    
    
     DETAILED DESCRIPTION 
     In the following detailed description, for purposes of explanation and not limitation, example embodiments disclosing specific details are set forth in order to provide a thorough understanding of an embodiment according to the present teachings. However, it will be apparent to one having ordinary skill in the art having had the benefit of the present disclosure that other embodiments according to the present teachings that depart from the specific details disclosed herein remain within the scope of the appended claims. Moreover, descriptions of well-known apparati and methods may be omitted so as to not obscure the description of the example embodiments. Such methods and apparati are clearly within the scope of the present teachings. 
       FIG. 1  illustrates principles employed by scanning capacitance microscopy (SCM) and scanning microwave microscopy (SMM) to determine information about doping concentrations in semiconductor devices. 
       FIG. 1  illustrates a conductive probe tip applying a voltage across a semiconductor substrate having an oxide layer formed thereon, and carriers doped therein. As shown on the left hand side of  FIG. 1 , when the probe tip applies the voltage having a first polarity, the majority carriers are driven toward the probe tip at the top of the substrate. In contrast, as shown on the right hand side of  FIG. 1 , when the probe tip applies the voltage having a second polarity, the carriers are driven away from the probe tip into the substrate. In  FIG. 1 , a capacitance is created between the probe tip and the edge of the majority carriers, with the oxide layer serving as the capacitor dielectric in the accumulation case shown on the left hand side of  FIG. 1 , and the oxide layer and the top part of the semiconductor substrate serving as the capacitor dielectric in the depletion case shown on the right hand side of  FIG. 1 . 
     Meanwhile, the capacitance between two plates of a capacitor is given by:
 
 C=∈A/t   (1)
 
where: ∈ is the dielectric constant, A is the area of the capacitor, and t is the spacing between the plates, where t&lt;&lt;√A.
 
     Therefore the capacitance in  FIG. 1  is higher in the accumulation state shown on the left side when the “plates” are closer together, and lowest in the depletion state shown on the right side when the “plates” are further apart. For n-type material the measured capacitance is therefore highest when the applied voltage is positive. The capacitance decreases as the bias is shifted negative as a result of free carriers being pushed away from the surface, analogous to an increase in the plate separation. Movement of free carriers and hence the amplitude of the capacitance variation is a function of the dopant level of the region of the substrate directly beneath the probe tip. For heavily doped materials the carriers do not move far. Hence, the measured capacitance variation between accumulation and depletion is small. The opposite is true for lightly doped semiconductors which yield a large capacitance change. 
     The principles described above with respect to  FIG. 1  may be employed in scanning capacitance microscopy and scanning microwave microscopy to determine information about dopant concentration levels in a semiconductor substrate. 
     It can be understood from the above discussion that the capacitances in depletion state and the accumulation state are also a function of the distance or height between the probe tip and the doped region of the substrate with the surface oxide formed thereon. In particular, as the probe tip is moved further away from the doped region of the substrate, the capacitance decreases—in both accumulation and depletion modes. Also, the change in capacitance between the accumulation state and the depletion state, and the ratio between the accumulation capacitance and the depletion capacitance, will vary as a function of the distance or height between the probe tip and the surface of the substrate. In a typical implementation, the substrate being measured is oriented horizontally, and so the probe tip is positioned at a desired vertical distance—or height—above the substrate. However it is conceivable that other orientations may be employed. Nevertheless, for simplicity of explanation, throughout this disclosure the distance between the probe tip and the substrate will be referred to using the term “height,” with the understanding that this term is intended to pertain to other arrangements besides those where the substrate being measured is oriented horizontally and the probe tip is positioned vertically above it. 
     In general, an electrical interaction between the probe tip and the doped region of the substrate is dependent upon the height between the probe tip and the top surface of the substrate. Also, the dependency of this interaction on the height will change depending upon the dopant density in the doped region. This principle may be employed to make capacitance or SMM reflectance measurements with a probe tip at various heights to directly measure the local dopant density. Furthermore, this provides a way to normalize signal measurements to reduce significantly the dependence on the radius of the probe tip, the oxide thickness, and other parameters. 
       FIG. 2  is a functional block diagram of one embodiment of a measurement instrument  200  for measuring a doping density in a doped region  12  of a substrate  10 . As will be appreciated by those skilled in the art, one or more of the various “parts” shown in  FIG. 2  may be physically implemented using a software-controlled microprocessor, hard-wired logic circuits, or a combination thereof. Also, while the parts are functionally segregated in  FIG. 2  for explanation purposes, they may be combined variously in any physical implementation. 
     Instrument  200  includes a measurement system  210  and a cantilever arm  220  with a probe tip  222 . Instrument  200  is of course but one exemplary embodiment, and other structural arrangements besides a cantilever arm (e.g., a tuning fork) may be provided for holding a probe tip within a close vicinity of a surface of a substrate region whose doping density is to be measured. It should also be understood that it is important in instrument  200  to precisely position probe tip  222  at various relatively small heights above substrate  100 , including maintaining and measuring the position of probe tip  222 . Beneficially, instrument  200  also includes some form of displacement sensor, such as a laser diode and light detector, (not shown in  FIG. 2 ) for detecting the position or displacement of the probe tip from the surface of the substrate  10 . In some embodiments, such an arrangement may provide feedback to measurement system  210  for controlling the position of probe tip  222 . 
     Measurement system  210  includes an AC &amp; DC signal generator, sensor, and signal processor block  212 , a control processor  214 , memory  216 , and a movement mechanism (e.g., a motor and/or a piezoelectric transducer)  218  or other means for moving probe tip  222  perpendicularly with respect to the top surface or doped region  12  of a substrate  10 . 
     Processor  214  is configured to execute one or more software algorithms in conjunction with memory  216  to provide functionality for instrument  200 . Beneficially, processor  214  includes its own memory (e.g., nonvolatile memory) for storing executable software programming code that allows it to perform the various functions of instrument  200 . Alternatively, or additionally, executable code may be stored in designated memory locations within memory  216 . 
     Memory  216  stores data and/or software programming code used in operations of instrument  200 . 
       FIG. 3  is a flowchart of a first embodiment of a method  300  of determining a dopant density. Method  300  may be executed by instrument  200  under control of processor  214 . In method  300 , localized probe tip/sample capacitance versus height measurements are made under different biasing conditions. Capacitance ratio (C RATIO ) versus height (H) data, C RATIO (H), is then produced by dividing the absolute value of the measured probe tip capacitance with a depletion voltage applied, by the measured probe tip capacitance with an accumulation voltage applied:
 
 C   RATIO ( H )= C   DEP ( H )/ C   ACC ( H )  (2)
 
     The measured capacitance ratio versus height data, C RATIO (H), is then compared to capacitance ratio versus height reference data previously determined for a variety of different dopant densities to determine the absolute dopant density of the region of the sample substrate on which the measurements are performed. Using the capacitance ratio can significantly reduce the dependence of the measurements on tip radius, oxide thickness, probe top cone angle, applied voltage (if voltages are large enough), etc. 
     Specifically, in a first step  305 , capacitance ratio versus height reference data is obtained for various dopant densities upon a set of parameters which match parameters under the capacitance versus height data of a sample are to be measured. The reference data can be provided in the form of a series of data points, as a curve, as an equation, or in any other convenient form. 
     In one embodiment, the capacitance ratio versus height reference data may be determined from a software model of the interaction between the probe tip and the doped substrate region. Beneficially, the model employs parametric values for the radius for the probe tip, the cone angle of the probe tip, an applied accumulation voltage, an applied depletion voltage, and an oxide thickness between the probe tip and the doped region of the substrate. 
     In another embodiment, the capacitance ratio versus height reference data may be determined by making measurements of one or more reference substrates having a known doping density. For example, a large number of measurements may be made—at different heights—over different areas of a reference substrate having a substantially uniform, known, doping density, and the measurements may be averaged to produce capacitance ratio versus height reference data. 
       FIG. 4  illustrates an example series of capacitance ratio vs. height curves for various dopant densities that may be employed in method  300 . The parameter set employed in the calculating the data plotted in  FIG. 4  include: [Va, Na, tox, R, alpha], where Va is the amplitude of the applied accumulation and depletion voltage, Na is the log of the dopant density (in this example, p-type), tox is the oxide thickness above the doped region (dielectric constant k=3.8), R is the radius of the probe tip, and alpha is the probe cone angle. In the data sets plotted in  FIG. 4 , Va=5V, tox=3 nm, R=50 nm; and alpha=10 degrees.  FIG. 4  plots values for six different data sets corresponding to six different dopant densities where Na is 15, 16, 17, 18, 19 and 20. In  FIG. 4 , the horizontal axis is in nanometers, ranging from 0 to 20 nm, and the vertical axis is unitless, since all data is normalized. 
     As can be seen, there is a nice separation of the data between all six normalized data sets. Beneficially, the data plotted in  FIG. 4  could be stored in memory  216  of instrument  210 . 
     Turning back to  FIG. 3 , in a step  307 , a height/measurement index “X” is set to 1. 
     In a step  310 , the probe tip is moved by a probe tip movement mechanism (e.g., a motor and/or piezoelectric transducer) to a predetermined height from the doped region of a substrate to be measured, corresponding to the index (“X”). For example, the first height (X=1) may be set to 0 nm. 
     In a step  315 , a depletion voltage (e.g., +5V for p-type dopants) is applied to probe tip  222 , and the capacitance C DEP (H X ) between probe tip  222  and the “plate” formed by the doped carriers is measured by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     Then, in a step  320 , an accumulation voltage (e.g., −5V for p-type dopants) is applied to probe tip  222 , and the capacitance C A MH X ) between probe tip  222  and the “plate” formed by the doped carriers is measured by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     In step  325 , a capacitance ratio C RATIO (H X ) is determined (e.g., by processor  214 ) according to equation (2) above. 
     In step  330 , a check is made to determine whether measurements at the final height have been made. If not the process increments the height/measurement index “X” at step  335 , and returns to step  310 . In a beneficial arrangement, measurements may be made up to a final height of 20 nm. 
     Once measurements have been made at all desired heights, then the process proceeds to step  340  where the measured C RATIO (H) response is determined for the dopant density in the doped region of a substrate being measured. 
     In step  345 , the measured C RATIO (H) response is compared to the capacitance ratio vs. height reference data from step  310 , and in step  350 , the dopant density in the doped region of a substrate being measured is determined from the comparison. For example, the measured C RATIO (H) response could be plotted on the graph of  FIG. 4  to determine the doped density of the measured sample. Alternatively, the comparison and determination could be performed entirely numerically by processor  214  without generating any plots. 
     Of course details illustrated in method  300  are exemplary. For example, measurements over a range from 0 to 20 nm may not be necessary in all cases to determine the dopant density. In other cases, more measurements and/or a larger height range may be employed. It should also be understood that the order of some of the steps illustrated in  FIG. 3  can be rearranged in other orders for convenience. For example: step  305  could be performed after steps  307 - 335 ; the order of steps  315  and  320  could be interchanged; step  325  could be performed immediately prior to step  340 ; etc. 
     One challenge in the execution of method  300  is the separation out of stray capacitances, such as capacitance due to the probe tip shank and the cantilever, from the capacitance that is desired to be measured under the probe tip. If the changes in stray capacitance as a function of height are small compared to the changes in capacitance between the probe tip and the doped region as a function of height, then this challenge is not significant. In other cases, where the changes in stray capacitance as a function of height are of the same order or larger than the changes in capacitance between the probe tip and the doped region as a function of height, then an additional step is required for measuring the change in the stray capacitance as a function of height, and subtracting this value out of the measured total change in capacitance as a function of height, leaving only the change in capacitance between the probe tip and the doped region as a function of height. 
       FIG. 5  is a flowchart of a second embodiment  500  of a method of determining a dopant density. Method  500  may be executed by instrument  200  under control of processor  214 . Method  500  illustrates one example of an approach to avoid the stray capacitance challenge discussed. Method  500  operates by measuring the change in capacitance as the probe tip goes from a strong accumulation voltage to a strong depletion voltage, as a function of height. Since the stray capacitance in many cases does not strongly depend on the applied voltage, this isolates the signal ΔC to the capacitance between the probe tip and the doped region of the substrate that is being measured. In method  500 , strong accumulation and depletion voltages are typically applied with some frequency, e.g., 10-50 kHz, and the capacitance change between the accumulation and depletion states is measured, e.g., with a lock-in amplifier in AC &amp; DC signal generator, sensor, and signal processor block  212 . This change of capacitance, ΔC, is measured at several different heights between the probe tip and the doped region of the substrate that is being measured. These data samples are normalized by the value of ΔC when the height is zero, to produce normalized change of capacitance data, ΔC NORM (H):
 
Δ C   NORM ( H )=Δ C ( H )/Δ C (0)  (3)
 
     Measured normalized change-of-capacitance (ΔC) versus height (H) data, ΔC NORM (H), is then compared to normalized ΔC vs. H reference data for a variety of different dopant densities to determine the absolute dopant density of the region of the sample substrate on which the measurements are being performed. 
     Specifically, in a first step  505 , normalized change of capacitance versus height reference data is obtained for various dopant densities. The reference data can be provided in the form of a series of data points, as a curve, as an equation, or in any other convenient form. The reference data may be obtained using techniques similar to those described above with respect to  FIG. 3 . 
       FIG. 6  illustrates an example series of normalized change of capacitance vs. height curves for various dopant densities that may be employed in method  500 . The parameter set employed in the calculating the data plotted in  FIG. 6  include: [Va, Na, tox, R, alpha], where Va is the amplitude of the applied accumulation and depletion voltage, Na is the log of the dopant density (in this example, p-type), tox is the oxide thickness above the doped region (dielectric constant k=3.8), R is the radius of the probe tip, and alpha is the probe cone angle. In the data sets plotted in  FIG. 6 , Va=5V, tox=1.5 nm, R=30 nm; and alpha=10 degrees.  FIG. 6  plots values for six different data sets corresponding to six different dopant densities where Na is 15, 16, 17, 18, 19 and 20. In  FIG. 6 , the horizontal axis is in nanometers, ranging from 0 to 10 nm, and the vertical axis is unitless, since all data is normalized. 
     Turning back to  FIG. 5 , in a step  507 , a height/measurement index “X” is set to 1. 
     In a step  510 , the probe tip is moved by a probe tip movement mechanism (e.g., a motor and/or piezoelectric transducer) to a predetermined height from the doped region of a substrate to be measured, corresponding to the index (“X”). For example, the first height (X=1) may be set to 0 nm. 
     In a step  515 , a depletion voltage (e.g., +5V) and an accumulation voltage (e.g., −5V) are alternatingly applied to probe tip  222  at a selected frequency (e.g., 10-50 kHz) by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     Then, in a step  520 , the change of capacitance ΔC (H X ) between probe tip  222  and the “plate” formed by the carriers in doped region  12 , between the strong accumulation state to the deep depletion state, is measured by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     In step  525 , the normalized change of capacitance ΔC NORM (H X ) is determined (e.g., by processor  214 ) according to equation (3) above. 
     In step  530 , a check is made to determine whether measurements at the final height have been made, if not the process increments the height/measurement index “X” at step  535  and returns to step  510 . In a beneficial arrangement, measurements may be made up to a final height of 10 nm. 
     Once measurements have been made at all desired heights, then the process proceeds to step  540  where the measured ΔC NORM (H) response is determined for the dopant density in the doped region of a substrate being measured. 
     In step  545 , the measured ΔC NORM (H) response is compared to the normalized capacitance vs. height reference data from step  505 , and in step  550 , the dopant density in the doped region of a substrate being measured is determined from the comparison. For example, the measured ΔC NORM (H) response could be plotted on the graph of  FIG. 6  to determine the doped density of the measured sample. Alternatively, the comparison and determination could be performed entirely numerically by processor  214  without generating any plots. 
     Of course details illustrated in method  500  are exemplary. For example, measurements over a range from 0 to 10 nm may not be necessary in all cases to determine the dopant density. In other cases, more measurements and/or a larger height range may be employed. Also, instead of applying the alternating depletion/accumulation voltage and measuring the change in capacitance using a lock-in amplifier, separate accumulation and depletion voltages may be applied and the capacitance may be separately measured in each state. It should also be understood that the order of some of the steps illustrated in  FIG. 3  can be rearranged in other orders for convenience. For example: step  505  could be performed after steps  507 - 535 ; step  525  could be performed immediately prior to step  540 ; etc. 
     The inventor has recognized that the approaches described above with respect to methods  300  and  500  may be applied to SMM reflectance measurements with probe tip  222 . In particular, SMM reflectance versus height measurements, or SMM reflectance versus height measurements, may be employed to determine the dopant density in a sample. Furthermore, either the phase or the magnitude of the reflectance at the probe tip due to interaction with the doped region of the substrate may be used to determine dopant density. 
       FIG. 7  is a flowchart of a third embodiment of a method  700  of determining a dopant density. Method  700  may be executed by instrument  200  under control of processor  214 . Method  700  determines dopant density by measuring a normalized change of reflectance phase at the probe tip, due to an interaction with the doped region, between a strong accumulation voltage and a strong depletion voltage. The normalized change of reflectance phase, ΔRPHASE NORM (H), may be calculated as:
 
ΔRPHASE NORM ( H )=[RPHASE ACC ( H )−RPHASE DEP ( H )]/[RPHASE ACC (0)−RPHASE DEP (0)]  (4)
 
     Measured normalized change of reflectance phase (ΔRPHASE) versus height (H) data, ΔRPHASE NORM (H), is then compared to normalized ΔRPHASE vs. H reference data for a variety of different dopant densities to determine the absolute dopant density of the region of the sample substrate on which the measurements are performed. 
     Specifically, in a first step  705 , normalized change of reflectance phase versus height reference data is obtained for various dopant densities. The reference data can be provided in the form of a series of data points, as a curve, as an equation, or in any other convenient form. The reference data may be obtained using similar techniques to those described above with respect to  FIG. 3 . 
       FIG. 8  illustrates a series of normalized change of reflectance phase vs. height curves for various dopant densities. The parameter set employed in the calculating the data plotted in  FIG. 8  include: [Va, Na, tox, R, alpha], where Va is the amplitude of the applied accumulation and depletion voltage, Na is the log of the dopant density (in this example, p-type), tox is the oxide thickness above the doped region (dielectric constant k=3.8), R is the radius of the probe tip, and alpha is the probe cone angle. In the data sets plotted in  FIG. 8 , Va=5V, fox=3 nm, R=50 nm; and alpha=10 degrees.  FIG. 8  plots values for six different data sets corresponding to six different dopant densities where Na is 15, 16, 17, 18, 19 and 20. In  FIG. 8 , the horizontal axis is in nanometers, ranging from 0 to 20 nm, and the vertical axis is unitless, since all data is normalized. 
     Turning back to  FIG. 7 , in a step  707 , a height/measurement index “X” is set to 1. 
     In a step  710 , the probe tip is moved by a probe tip movement mechanism (e.g., a motor and/or piezoelectric transducer) to a predetermined height from the doped region of a substrate to be measured, corresponding to the index (“X”). For example, the first height (X=1) may be set to 0 nm. 
     In a step  715 , a depletion voltage (e.g., +5V) and an accumulation voltage (e.g., −5V) are alternatingly applied to probe tip  222  at a selected frequency (e.g., 10-50 kHz) by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     Then, in a step  720 , the change of reflectance phase, ΔRPHASE(H X )=|PHASE ACC (H X )−PHASE DEP (H X )|, between the strong accumulation state and the deep depletion state, due to the interaction between probe tip  222  and the doped carriers is measured by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     In step  725 , the normalized change of reflectance phase ΔRPHASE NORM (H X ) is determined (e.g., by processor  214 ) according to equation (4) above. 
     In step  730 , a check is made to determine whether measurements at the final height have been made, if not the process increments the height/measurement index “X” at step  735 , and returns to step  710 . In a beneficial arrangement, measurements may be made up to a final height of 20 nm. 
     Once measurements have been made at all desired heights, then the process proceeds to step  740  where the measured ΔRPHASE NORM (H) response is determined for the dopant density in the doped region of a substrate being measured. 
     In step  745 , the measured ΔRPHASE NORM (H) response is compared to the normalized reflectance phase difference vs. height reference data from step  705 , and in step  750 , the dopant density in the doped region of a substrate being measured is determined from the comparison. For example, the measured ΔRPHASE NORM (H) response could be plotted on the graph of  FIG. 8  to determine the doped density of the measured sample. Alternatively, the comparison and determination could be performed entirely numerically by processor  214  without generating any plots. 
     Of course details illustrated in method  700  are exemplary. For example, measurements over a range from 0 to 20 nm may not be necessary in all cases to determine the dopant density. In other cases, more measurements and/or a larger height range may be employed. Also, instead of applying the alternating depletion/accumulation voltage and measuring the change in reflectance phase using a lock-in amplifier, separate accumulation and depletion voltages may be applied and the reflectance phase may be separately measured in each state. It should also be understood that the order of some of the steps illustrated in  FIG. 7  can be rearranged in other orders for convenience. For example: step  705  could be performed after steps  707 - 735 ; the order of steps  715  and  720  could be interchanged; step  725  could be performed immediately prior to step  740 ; etc. 
       FIG. 9  is a flowchart of a fourth embodiment of a method  900  of determining a dopant density. Method  900  may be executed by instrument  200  under control of processor  214 . Method  900  determines dopant density by measuring reflectance magnitude ratio at the probe tip, due to an interaction with the doped region, between a strong accumulation voltage and a strong depletion voltage. The reflectance magnitude ratio, RMAG RATIO (H), may be calculated as:
 
RMAG RATIO ( H )=RMAG DEP ( H )/RMAG ACC ( H )  (5)
 
     Measured reflectance magnitude ratio (RMAG RATIO ) versus height (H) data, RMAG RATIO (H), is then compared to RMAG RATIO  vs. H reference data for a variety of different dopant densities to determine the absolute dopant density of the region of the sample substrate on which the measurements are performed. 
     Specifically, in a first step  905 , reflectance magnitude ratio versus height reference data is obtained for various dopant densities. The reference data can be provided in the form of a series of data points, as a curve, as an equation, or in any other convenient form. The reference data may be obtained using similar techniques to those described above with respect to  FIG. 3 . 
       FIG. 10  illustrates a series of reflectance magnitude ratio vs. height curves for various dopant densities. The parameter set employed in the calculating the data plotted in  FIG. 10  include: [Va, Na, tox, R, alpha], where Va is the amplitude of the applied accumulation and depletion voltage, Na is the log of the dopant density (in this example, p-type), tox is the oxide thickness above the doped region (dielectric constant k=3.8), R is the radius of the probe tip, and alpha is the probe cone angle. In the data sets plotted in  FIG. 10 , Va=5V, tox=3 nm, R=50 nm; and alpha=10 degrees.  FIG. 10  plots values for six different data sets corresponding to six different dopant densities where Na is 15, 16, 17, 18, 19 and 20. In  FIG. 10 , the horizontal axis is in nanometers, ranging from 0 to 20 nm, and the vertical axis is unitless, since all data is normalized. 
     Turning back to  FIG. 9 , in a step  907 , a height/measurement index “X” is set to 1. 
     In a step  910 , the probe tip is moved by a probe tip movement mechanism (e.g., a motor and/or piezoelectric transducer) to a predetermined height from the doped region of a substrate to be measured, corresponding to the index (“X”). For example, the first height (X=1) may be set to 0 nm. 
     In a step  915 , a depletion voltage (e.g., +5V) is applied to probe tip  222 , and the reflectance magnitude RMAG DEP (H X ) due to the interaction between probe tip  222  and the doped carriers is measured by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     Then, in a step  920 , an accumulation voltage (e.g., −5V) is applied to probe tip  222 , and the reflectance magnitude RMAG ACC (H X ) due to the interaction between probe tip  222  and the doped carriers is measured by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     In step  925 , the reflectance magnitude ratio RMAG RATIO (H X ) is determined (e.g., by processor  214 ) according to equation (5) above. 
     In step  930 , a check is made to determine whether measurements at the final height have been made, if not the process increments the height/measurement index “X” at step  935 , and returns to step  910 . In a beneficial arrangement, measurements may be made up to a final height of 20 nm. 
     Once measurements have been made at all desired heights, then the process proceeds to step  940  where the measured RMAG RATIO (H) response is determined for the dopant density in the doped region of a substrate being measured. 
     In step  945 , the measured RMAG RATIO (H) response is compared to the reflectance magnitude ratio vs. height reference data from step  905 , and in step  950 , the dopant density in the doped region of a substrate being measured is determined from the comparison. For example, the measured RMAG RATIO (H) response could be plotted on the graph of  FIG. 10  to determine the doped density of the measured sample. Alternatively, the comparison and determination could be performed entirely numerically by processor  914  without generating any plots. 
     Of course details illustrated in method  900  are exemplary. For example, measurements over a range from 0 to 20 nm may not be necessary in all cases to determine the dopant density. In other cases, more measurements and/or a larger height range may be employed. It should also be understood that the order of some of the steps illustrated in  FIG. 9  can be rearranged in other orders for convenience. For example: step  905  could be performed after steps  907 - 935 ; the order of steps  915  and  920  could be interchanged; step  925  could be performed immediately prior to step  940 ; etc. 
       FIG. 11  is a flowchart of a fifth embodiment of a method of determining a dopant density. Method  1100  may be executed by instrument  200  under control of processor  214 . Method  1100  determines dopant density by measuring normalized change of reflectance magnitude at the probe tip, due to an interaction with the doped region, between a strong accumulation voltage and a strong depletion voltage. The normalized change of reflectance magnitude, ΔRMAG NORM (H), may be calculated as:
 
ΔRMAG NORM ( H )=[RMAG ACC ( H )−RMAG DEP ( H )]/[RMAG ACC (0)−RMAG DEP (0)]  (6)
 
     Measured normalized change of reflectance magnitude (ΔRMAG NORM ) versus height (H) data, ΔRMAG NORM (H), is then compared to ΔRMAG NORM  vs. H reference data for a variety of different dopant densities to determine the absolute dopant density of the region of the sample substrate on which the measurements are performed. 
     Specifically, in a first step  1105 , normalized change of reflectance magnitude versus height reference data is obtained for various dopant densities. The reference data can be provided in the form of a series of data points, as a curve, as an equation, or in any other convenient form. The reference data may be obtained using similar techniques to those described above with respect to  FIG. 3 . 
       FIG. 12  illustrates a series of normalized change of reflectance magnitude vs. height curves for various dopant densities. The parameter set employed in the calculating the data plotted in  FIG. 12  include: [Va, Na, tox, R, alpha], where Va is the amplitude of the applied accumulation and depletion voltage, Na is the log of the dopant density (in this example, p-type), tox is the oxide thickness above the doped region (dielectric constant k=3.8), R is the radius of the probe tip, and alpha is the probe cone angle. In the data sets plotted in  FIG. 12 , Va=5V, tox=3 nm, R=50 nm; and alpha=10 degrees.  FIG. 12  plots values for six different data sets corresponding to six different dopant densities where Na is 15, 16, 17, 18, 19 and 20. In  FIG. 12 , the horizontal axis is in nanometers, ranging from 0 to 10 nm, and the vertical axis is unitless, since all data is normalized. 
     Turning back to  FIG. 11 , in a step  1107 , a height/measurement index “X” is set to 1. 
     In a step  1110 , the probe tip is moved by a probe tip movement mechanism (e.g., a motor and/or piezoelectric transducer) to a predetermined height from the doped region of a substrate to be measured, corresponding to the index (“X”). For example, the first height (X=1) may be set to 0 nm. 
     In a step  1115 , a depletion voltage (e.g., +5V) and an accumulation voltage (e.g., −5V) are alternatingly applied to probe tip  222  at a selected frequency (e.g., 10-50 kHz) by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     Then, in a step  1120 , the change of reflectance phase, ΔRMAG(H X )=|RMAG ACC (H X )−RMAG DEP (H X )|, between the strong accumulation state and the deep depletion state, due to the interaction between probe tip  222  and the doped carriers is measured by AC &amp; DC signal generator, sensor, and signal processor block  212 . 
     In step  1125 , the normalized change of reflectance magnitude ΔRMAG NORM (H X ) is determined (e.g., by processor  214 ) according to equation (6) above. 
     In step  1130 , a check is made to determine whether measurements at the final height have been made, if not the process increments the height/measurement index “X” at step  1135 , and returns to step  910 . In a beneficial arrangement, measurements may be made up to a final height of 20 nm. 
     Once measurements have been made at all desired heights, then the process proceeds to step  1140  where the measured normalized change of reflectance magnitude ΔRMAG NORM (H) response is determined for the dopant density in the doped region of a substrate being measured. 
     In step  1145 , the measured normalized change of reflectance magnitude ΔRMAG NORM (H) response is compared to the normalized change of reflectance magnitude vs. height reference data from step  1105 , and in step  1150 , the dopant density in the doped region of a substrate being measured is determined from the comparison. For example, the measured normalized change of reflectance magnitude ΔRMAG NORM (H) response could be plotted on the graph of  FIG. 12  to determine the doped density of the measured sample. Alternatively, the comparison and determination could be performed entirely numerically by processor  1114  without generating any plots. 
     Of course details illustrated in method  1100  are exemplary. For example, measurements over a range from 0 to 10 nm may not be necessary in all cases to determine the dopant density. In other cases, more measurements and/or a larger height range may be employed. Also, instead of applying the alternating depletion/accumulation voltage and measuring the change in reflectance magnitude using a lock-in amplifier, separate accumulation and depletion voltages may be applied and the reflectance magnitude may be separately measured in each state. It should also be understood that the order of some of the steps illustrated in  FIG. 11  can be rearranged in other orders for convenience. For example: step  1105  could be performed after steps  1107 - 1135 ; the order of steps  1115  and  1120  could be interchanged; step  1125  could be performed immediately prior to step  1140 ; etc. 
     Methods  300 ,  500 ,  700 ,  900  and  1100  are merely a few exemplary embodiments of methods that measure the dependency of an electrical interaction between the probe tip and the doped region of the substrate on the height between the probe tip and the top surface or doped region of the substrate in order to determine the dopant density in the doped region. However it should be understood that other measurements and calculations based on the dependency of an electrical interaction between the probe tip and the doped region of the substrate on the height between the probe tip and the top surface or doped region of the substrate could be employed to determine the dopant density in the doped region. 
     For example, in addition to the methods described in detail above, the following four methods may in principle be employed to characterize an electrical interaction between the probe tip and the doped region of the substrate as a function of height between the probe tip and the doped region of the substrate. 
     Normalized depletion capacitance versus height:
 
 C -DEP NORM ( H )= C   DEP ( H )/ C   ACC (0)  (7)
 
     Normalized reflectance phase ratio versus height:
 
RPHASE RATIO ( H )=RPHASE DEP ( H )/RPHASE ACC ( H )  (8)
 
     Normalized depletion reflectance phase versus height:
 
RPHASE-DEP NORM ( H )=RPHASE DEP ( H )/RPHASE ACC (0)  (9)
 
     Normalized depletion reflectance magnitude versus height:
 
RMAG-DEP NORM ( H )=RMAG DEP ( H )/RMAG ACC (0)  (10)
 
     Also, some example embodiments above measured utilized the magnitude or phase of the reflectance. However, instead of using magnitude or phase, the real or imaginary parts of the reflectance could be used, as it is well known how to convert back and forth between magnitude/phase and real and imaginary parts. 
     Therefore the invention should not be limited to the particular example embodiments described in detail above. 
     While example embodiments are disclosed herein, one of ordinary skill in the art appreciates that many variations that are in accordance with the present teachings are possible and remain within the scope of the appended claims. The invention therefore is not to be restricted except within the scope of the appended claims.