Noncontact measuring system for electrical conductivity

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First Claim
1. A noncontact measuring system for electrical conductivity comprising:
 microwave transmitting and receiving means which transmits and receives the microwave;
measuring means which measures an amplitude ratio of the transmitted microwave and a receivedreflected wave; and
electrical conductivity calculating means which determines the electrical conductivity of a low frequency from the measured amplitude ratio.
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Abstract
The invention provides a noncontact measuring system for electrical conductivity, which uses a microwave. In the measuring system for electrical conductivity, the microwave generated in a network analyzer (NA) 110 is guided to a surface of a silicon wafer (sample) 160 through a waveguide 130 and a sensor 140. The surface of the silicon wafer 160 is irradiated with the microwave, and the sensor 140 receives the reflected microwave. The electrical conductivity of the silicon wafer 160 is measured in such a manner that a computer (personal computer) 120 calculates the electrical conductivity from an amplitude ratio A and phase difference θ to a reflected wave of the silicon wafer 160, which is determined with the network analyzer 110. The computer 120 performs not only the calculation of the measurement but also whole control of the measuring system such as positioning of the sample.
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2 Claims
 1. A noncontact measuring system for electrical conductivity comprising:
 microwave transmitting and receiving means which transmits and receives the microwave;
measuring means which measures an amplitude ratio of the transmitted microwave and a receivedreflected wave; and
electrical conductivity calculating means which determines the electrical conductivity of a low frequency from the measured amplitude ratio.
 microwave transmitting and receiving means which transmits and receives the microwave;
 2. A noncontact measuring system for electrical conductivity comprising:
 microwave transmitting and receiving means which transmits and receives the microwave;
measuring means which measures an amplitude ratio and phase difference of the transmitted microwave and a receivedreflected wave;
electrical conductivity calculating means which determines the apparent electrical conductivity from the measured amplitude ratio and phase difference; and
electrical conductivity calibration means which determines the electrical conductivity of a low frequency from the determined apparent electrical conductivity.
 microwave transmitting and receiving means which transmits and receives the microwave;
1 Specification
[0001] 1. Field of the Invention
[0002] The present invention relates to an electrical conductivity measuring system for a material, particularly relates to the electrical conductivity measuring system on a noncontact basis, in which a microwave is used.
[0003] 2. Description of the Prior Art
[0004] Measurement of the electrical conductivity is essential in quality control and performance evaluation of a silicon wafer which is a typical material used for electronic devices.
[0005] A fourpoint probe method is conventionally adopted as the measuring method for the electrical conductivity of the silicon wafer. Though the fourpoint probe method has an advantage of the quite easy measurement, there is a problem that a leading end of the probe hurts a surface of the wafer because it is necessary that the leading end of the probe is in contact with the wafer during the measurement. Calibration of a thickness of the wafer is also required in quantitative evaluation. Therefore, the fourpoint probe method can not be adopted for online inspection in a manufacturing process of the silicon wafer or nondestructive inspection prior to the manufacturing of the device, and a spot check of few pieces of the silicon wafer products is the only way to inspect the silicon wafer.
[0006] The noncontact measuring method for the electrical conductivity is required to realize the nondestructive inspection. In the noncontact measuring method for the electrical conductivity, there is a technique in which a coil is used. However, a change in impedance of an induction coil depends on not only the electrical conductivity of the wafer but also permittivity and the thickness of the wafer. Accordingly, an additional measurement, e.g. the measurement in which the thickness of the wafer is determined by other technique is required in the quantitative evaluation of the electrical conductivity in which the coil is used. The technique using the coil also has the problem of low spatial resolution, so that it is not suited for the nondestructive inspection of the wafer prior to the manufacturing of the device.
[0007] In recent years, a study concerning the nondestructive evaluation of the semiconductor wafer using the microwave receives attention as the new measuring method on a noncontact basis. The measurement of a lifetime of a minority carrier (see nonpatent reference 1) has been realized, and the evaluation of photoconductivity of the wafer (see nonpatent reference 2) and dependence on the electrical conductivity of an absorption peak of the microwave (see nonpatent reference 3) have been also reported.
[0008] However, the quantitative evaluation of the electrical conductivity of the semiconductor wafer has not been successful yet due to influence of the permittivity and thickness of the wafer, or the like.
Non[0009] J. Schmidt and A. G. Aberle: Accurate Method for the Determination of Bulk MinorityCarrier Lifetimes of Monoand Multicrystalline Silicon Wafers, Journal of Applied Physics, Vol. 81, No. 9, pp. 61866199, (1997)
Non[0010] J. R. Niklas, W. Siegel, M. Jurisch and U. Kretzer: GaAs Wafer Mapping by MicrowaveDetected Photoconductivity, Materials Science and Engineering B, Vol. 80, No. 13, pp. 206209, (2001).
Non[0011] R. M. Lewis and J. P. Carini: Frequency Scaling of Microwave Conductivity in the Integer Quantum Hall Effect Minima, Physical Review B, Vol. 64, No. 7, 073310, (2001).
SUMMARY OF THE INVENTION[0012] It is an object of the invention to provide an electrical conductivity measuring system on a noncontact basis, in which the microwave is used.
[0013] In order to achieve the abovedescribed object, the invention is the noncontact measuring system for the electrical conductivity characterized by including microwave transmitting/receiving means which transmits and receives the microwave, measuring means which measures an amplitude ratio of the transmitted microwave and a received reflectedwave, and electrical conductivity calculating means which determines the electrical conductivity of a low frequency from the measured amplitude ratio.
[0014] Further, the invention is the noncontact measuring system for the electrical conductivity characterized by including microwave transmitting/receiving means which transmits and receives the microwave, measuring means which measures an amplitude ratio and phase difference of the transmitted microwave and a received reflectedwave, electrical conductivity calculating means which determines the apparent electrical conductivity from the measured amplitude ratio and phase difference, and electrical conductivity calibration means which determines the electrical conductivity of the low frequency from the determined apparent electrical conductivity.
BRIEF DESCRIPTION OF THE DRAWINGS[0015] <crossreference>FIG. 1</crossreference> shows an example of a configuration of the electrical conductivity measuring system;
[0016] <crossreference>FIG. 2</crossreference> shows the configuration of a reflective type of microwave converging sensor which is used in the electrical conductivity measuring system;
[0017] <crossreference>FIG. 3</crossreference> is a graph showing a relationship between apparent electrical conductivity σ<highlight><subscript>m </subscript></highlight>and electrical conductivity ad which is obtained by a fourpoint probe method;
[0018] <crossreference>FIG. 4</crossreference> is a graph showing the relationship between the calibrated electrical conductivity σ<highlight><subscript>d </subscript></highlight>for a low frequency and the electrical conductivity σ<highlight><subscript>l </subscript></highlight>which is obtained by the fourpoint probe method;
[0019] <crossreference>FIG. 5</crossreference> shows a construction of a coaxial cable sensor which is used in the electrical conductivity measuring system;
[0020] <crossreference>FIG. 6</crossreference> is a graph showing the electrical conductivity σ which is obtained by measuring the silicon wafer of a test piece with the fourpoint probe method;
[0021] <crossreference>FIG. 7</crossreference> is a graph showing an amplitude of the reflectivity which is obtained by measuring the silicon wafer of the test piece with the electrical conductivity measuring system;
[0022] <crossreference>FIG. 8</crossreference> is a graph showing the relationship between the amplitude of the reflectivity and the electrical conductivity σ which is obtained by the fourpoint probe;
[0023] <crossreference>FIG. 9</crossreference> shows a microwave image depending on magnitude of the reflectivity of the silicon wafer; and
[0024] <crossreference>FIG. 10</crossreference> is a view showing a distribution of the electrical conductivity of the silicon wafer.
DETAILED DESCRIPTION OF THE INVENTION[0025] Preferred embodiments of the invention will be described referring the accompanying drawing.
[0026] <crossreference>FIG. 1</crossreference> shows the configuration of the electrical conductivity measuring system using the microwave according to the embodiment of the invention. The measuring system includes a network analyzer (NA) <highlight><bold>110</bold></highlight>, a computer (personal computer) <highlight><bold>120</bold></highlight>, a waveguide <highlight><bold>130</bold></highlight>, a sensor <highlight><bold>140</bold></highlight>, and an xyz stage <highlight><bold>170</bold></highlight>. The microwave generated in the network analyzer <highlight><bold>110</bold></highlight> is guided to the surface of a silicon wafer (sample) <highlight><bold>160</bold></highlight> through the waveguide <highlight><bold>130</bold></highlight> and the sensor <highlight><bold>140</bold></highlight>. The surface of the silicon wafer <highlight><bold>160</bold></highlight> is irradiated with the microwave, and the sensor <highlight><bold>140</bold></highlight> receives the reflected microwave. The electrical conductivity of the silicon wafer <highlight><bold>160</bold></highlight> is measured in such a manner that the computer <highlight><bold>120</bold></highlight> calculates the electrical conductivity from an amplitude ratio A and phase difference θ to the reflected wave of the silicon wafer <highlight><bold>160</bold></highlight>, which is determined with the network analyzer <highlight><bold>110</bold></highlight>.
[0027] The sensor <highlight><bold>140</bold></highlight> positions the sample <highlight><bold>160</bold></highlight> by relative motion between a sensor support <highlight><bold>150</bold></highlight> and the xyz stage <highlight><bold>170</bold></highlight>. The computer <highlight><bold>120</bold></highlight> performs not only the calculation of the measurement but also whole control of the measuring system such as the positioning of the sample. Though the network analyzer is used for measurement of the amplitude ratio and phase difference of the microwave in the abovedescribed measuring system, it is not limited to the network analyzer.
[0028] <crossreference>FIG. 2</crossreference> shows the configuration of the reflective type of microwave converging sensor which is developed.
[0029] A shape of reflecting mirrors <highlight><bold>141</bold></highlight> and <highlight><bold>142</bold></highlight> of the used microwave is an ellipsoidal plane. In the optics, when a light source is placed at one focal point A in the ellipsoid, incident light is reflected on the ellipsoidal plane and condensed at the other focal point B. Further, the phases are coherent because an optical path from the focal point A to the focal point B is constant. In the embodiment of the invention, by adopting this principle, the two reflecting mirrors <highlight><bold>141</bold></highlight> and <highlight><bold>142</bold></highlight> are combined so as to share the focal point B, standoff distance is formed as much as possible, and the microwave from the focal point A converges into a focal point C.
[0030] Actually combination of the reflecting mirrors <highlight><bold>141</bold></highlight> and <highlight><bold>142</bold></highlight> (solid line parts in <crossreference>FIG. 2</crossreference>), which is two quarters of the ellipsoid, and a horn antenna (not shown) are integrally fixed, and it is used as one sensor <highlight><bold>140</bold></highlight>.
[0031] By using the sensor <highlight><bold>140</bold></highlight>, a minuter region can be irradiated with the converged microwave compared with the conventional nonconvergence sensor, so that silicon wafer can be evaluated on a nondestructive basis with a larger standoff distance, higher resolution, and higher sensitivity.
[0032] Though <crossreference>FIG. 2</crossreference> shows the example of the reflective type of microwave converging sensor in which the two reflecting mirrors having the ellipsoidal plane are combined, three or more reflecting mirrors may be combined in the same way.
PRINCIPLE OF MEASUREMENT[0033] In the abovedescribed system, the microwave to be used is an electromagnetic wave having a frequency of 300 MHz to 300 GHz (wavelength of 1 m to 1 mm). Because the microwave propagates through the air without a transmitter medium, the material can be evaluated on noncontact and nondestructive bases with the microwave. Further, because response of the microwave depends on the permittivity, permeability, and the electrical conductivity of the material, the change in the material can be measured with high sensitivity.
[0034] The reflection measuring method using the microwave is adopted in the measuring system of the invention. This is the method in which the material is irradiated with the microwave and the amplitude and the phase of the reflectivity, which is the ratio of the reflected wave to the incident wave, are measured.
[0035] Suppose that an angular frequency of the microwave is ω, the permeability of the material is σ, the permittivity of the material is ε, and the electrical conductivity is σ, characteristic impedance μa7 of the material and characteristic impedance μa7<highlight><subscript>o </subscript></highlight>in the air are determined by the following equations.<mathcwu><number>1</number><math><mtable><mtr><mtd><mrow><mrow><mo>[</mo><mrow><mi>EQUATION</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>1</mn></mrow><mo>]</mo></mrow><mo></mo><mstyle><mtext> </mtext></mstyle></mrow></mtd><mtd><mstyle><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mrow><mi>η</mi><mo>=</mo><msqrt><mfrac><mi>μ</mi><mrow><mi>ϵ</mi><mo></mo><mrow><mi>j</mi><mo></mo><mfrac><mi>σ</mi><mi>ω</mi></mfrac></mrow></mrow></mfrac></msqrt></mrow></mtd><mtd><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>η</mi><mi>o</mi></msub><mo>=</mo><msqrt><mfrac><msub><mi>μ</mi><mi>o</mi></msub><msub><mi>ϵ</mi><mi>o</mi></msub></mfrac></msqrt></mrow></mtd><mtd><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mtd></mtr></mtable></math><mathematicafile></mathematicafile><image></image></mathcwu>
[0036] Where μ<highlight><subscript>o </subscript></highlight>and ε<highlight><subscript>o </subscript></highlight>indicate permeability of vacuum magnetic constant and permittivity of vacuum electric constant respectively. The reflectivity Γ<highlight><subscript>t </subscript></highlight>on the surface of the material can be represented by the following equation.<mathcwu><number>2</number><math><mtable><mtr><mtd><mrow><mrow><mo>[</mo><mrow><mi>EQUATION</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>2</mn></mrow><mo>]</mo></mrow><mo></mo><mstyle><mtext> </mtext></mstyle></mrow></mtd><mtd><mstyle><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mrow><msub><mi>Γ</mi><mi>t</mi></msub><mo>=</mo><mfrac><mrow><mi>η</mi><mo></mo><msub><mi>η</mi><mi>o</mi></msub></mrow><mrow><mi>η</mi><mo>+</mo><msub><mi>η</mi><mi>o</mi></msub></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mtd></mtr></mtable></math><mathematicafile></mathematicafile><image></image></mathcwu>
[0037] When the silicon wafer is irradiated with the highfrequency microwave, a component of the microwave transmitted by the surface of the wafer is largely attenuated inside the silicon wafer. When terms of the electrical conductivity and thickness of the wafer satisfy given conditions, the amount of microwave reflected from a bottom surface of the wafer to the sensor is as small as negligible. For example, when the microwave having the frequency of 110 GHz is used, the reflectivity of the microwave reflected from the bottom surface of the silicon wafer having the thickness of 500 μm and the electrical conductivity of 200 S/m becomes about 1/10<highlight><superscript>4 </superscript></highlight>of Γ<highlight><subscript>t</subscript></highlight>. Accordingly, only Γ<highlight><subscript>t </subscript></highlight>on the surface of the material may be taken into consideration.
[0038] In the measurement concerning the silicon wafer, the wafer is a nonmagnetic material, by substituting equations (1) and (2) for the equation (3), Γ<highlight><subscript>t </subscript></highlight>can be expressed as follows.<mathcwu><number>3</number><math><mtable><mtr><mtd><mrow><mrow><mo>[</mo><mrow><mi>EQUATION</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>3</mn></mrow><mo>]</mo></mrow><mo></mo><mstyle><mtext> </mtext></mstyle></mrow></mtd><mtd><mstyle><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mrow><msub><mi>Γ</mi><mi>t</mi></msub><mo>=</mo><mrow><mrow><mi>a</mi><mo>+</mo><mi>jb</mi></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo></mo><msqrt><mrow><mfrac><mi>ϵ</mi><msub><mi>ϵ</mi><mi>o</mi></msub></mfrac><mo></mo><mrow><mi>j</mi><mo></mo><mfrac><mi>σ</mi><mrow><mi>ω</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><msub><mi>ϵ</mi><mi>o</mi></msub></mrow></mfrac></mrow></mrow></msqrt></mrow><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mfrac><mi>ϵ</mi><msub><mi>ϵ</mi><mi>o</mi></msub></mfrac><mo></mo><mrow><mi>j</mi><mo></mo><mfrac><mi>σ</mi><mrow><mi>ω</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><msub><mi>ϵ</mi><mi>o</mi></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mtd></mtr></mtable></math><mathematicafile></mathematicafile><image></image></mathcwu>
[0039] The reflectivity Γ<highlight><subscript>t </subscript></highlight>is a complex number, its real part is set to a and its imaginary part is set to b. ω is constant on the right side of the equation (4). Therefore, when ε is eliminated from the expression of the real part and the imaginary part of the equation (4), σ is derived as shown in the following equation (5), and σ becomes a function having variables a and b. As a result, the electrical conductivity of the wafer and the response of the microwave can be derived by a onetoone relational expression.<mathcwu><number>4</number><math><mtable><mtr><mtd><mrow><mrow><mo>[</mo><mrow><mi>EQUATION</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>4</mn></mrow><mo>]</mo></mrow><mo></mo><mstyle><mtext> </mtext></mstyle></mrow></mtd><mtd><mstyle><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mrow><mi>σ</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo></mo><mi>ω</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><msub><mi>ϵ</mi><mi>o</mi></msub><mo></mo><mrow><mi>b</mi><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo></mo><msup><mi>a</mi><mn>2</mn></msup><mo></mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow><msup><mrow><mo>{</mo><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>}</mo></mrow><mn>2</mn></msup></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr></mtable></math><mathematicafile></mathematicafile><image></image></mathcwu>
CONCRETE EXAMPLE[0040] The ntype of silicon wafer having a plane direction <<highlight><bold>100</bold></highlight>> was used as a test piece. In a size of the wafer, a diameter was 100 mm and the thickness was 525 μm. Three groups of the wafer, which have the electrical conductivity of 9 to 11, 50 to 110, and 166 to 333 S/m respectively, were used. The reflective type of microwave converging sensor shown in <crossreference>FIG. 2</crossreference> was used and the standoff distance between the sensor and the test piece was constantly set to 35 mm. The frequency was set to 110 GHz, the amplitude ratio A and the phase difference θ of the microwave which was transmitted and received were measured. Paying attention to temperature dependence of the electrical conductivity in the measurement, all the measurements were performed under the condition of a room temperature of 21° C.
[0041] Substituting the amplitude ratio A and the phase difference θ of the microwave which was transmitted and received for the following equation, the reflectivity Γ<highlight><subscript>m </subscript></highlight>of the microwave was determined from the measurement.<mathcwu><number>5</number><math><mtable><mtr><mtd><mrow><mrow><mo>[</mo><mrow><mi>EQUATION</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>5</mn></mrow><mo>]</mo></mrow><mo></mo><mstyle><mtext> </mtext></mstyle></mrow></mtd><mtd><mstyle><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mrow><msub><mi>Γ</mi><mi>m</mi></msub><mo>=</mo><mrow><msup><mn>10</mn><mfrac><mi>A</mi><mn>20</mn></mfrac></msup><mo>*</mo><msup><mi>ⅇ</mi><mrow><mi>j</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mi>θ</mi></mrow></msup></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mtd></mtr></mtable></math><mathematicafile></mathematicafile><image></image></mathcwu>
[0042] σ<highlight><subscript>m </subscript></highlight>by the microwave measurement was determined by substituting a real part am and an imaginary part b<highlight><subscript>m </subscript></highlight>of Γ<highlight><subscript>m </subscript></highlight>for the equation (5). Where σ<highlight><subscript>m </subscript></highlight>is the apparent electrical conductivity, and σ<highlight><subscript>m </subscript></highlight>includes an error caused by the measuring apparatus and the sensor and influence of the high frequency. The calibration is required to remove the error and the influence.
[0043] At this point, the electrical conductivity of the wafer test piece is measured by the fourpoint probe method and its value is set to σ<highlight><subscript>d</subscript></highlight>. Comparing σ<highlight><subscript>m </subscript></highlight>obtained by the microwave to σ<highlight><subscript>d </subscript></highlight>obtained by the fourpoint probe method, a calibration equation is derived. By adopting the calibration equation, the electrical conductivity σ<highlight><subscript>l </subscript></highlight>for the low frequency of the test piece can be measured with the microwave.
[0044] In order to derive the calibration equation, the test pieces of each group were measured by the noncontact measuring method using the microwave and the fourpoint probe method. <crossreference>FIG. 3</crossreference> shows the apparent electrical conductivity σ<highlight><subscript>m </subscript></highlight>obtained by the noncontact measuring method using the microwave and electrical conductivity σ<highlight><subscript>d </subscript></highlight>obtained by the fourpoint probe method.
[0045] From the result shown in <crossreference>FIG. 3</crossreference>, the following calibration equation was obtained by replacing σ<highlight><subscript>d </subscript></highlight>by the electrical conductivity σ<highlight><subscript>l </subscript></highlight>for the low frequency.
EQUATION<inlineformula>σ<highlight><subscript>l</subscript></highlight><highlight><italic>=C</italic></highlight><highlight><subscript>1</subscript></highlight>σ<highlight><subscript>m</subscript></highlight><highlight><italic>+C</italic></highlight><highlight><subscript>2 </subscript></highlight> (7) </inlineformula>
[0046] Where C<highlight><subscript>1</subscript></highlight>=−63.69 and C<highlight><subscript>2</subscript></highlight>=−400.63 S/m. By adopting the calibration equation (7), the electrical conductivity σ<highlight><subscript>l </subscript></highlight>for the low frequency can be determined by the noncontact measuring method using the microwave.
[0047] <crossreference>FIG. 4</crossreference> is a graph showing comparison between the electrical conductivity, in which the apparent electrical conductivity σ<highlight><subscript>m </subscript></highlight>obtained by the noncontact measuring method using the microwave is calibrated to the electrical conductivity σ<highlight><subscript>l </subscript></highlight>for the low frequency with the calibration equation (7), and the electrical conductivity σd obtained by the fourpoint probe method.
[0048] As shown in <crossreference>FIG. 4</crossreference>, the electrical conductivity σ<highlight><subscript>l </subscript></highlight>obtained by the noncontact measuring method using the microwave well conforms to the electrical conductivity σ<highlight><subscript>d </subscript></highlight>measured by the fourpoint probe method.
[0049] When the measuring condition or a kind of the wafer is changed, since C<highlight><subscript>1 </subscript></highlight>and C<highlight><subscript>2 </subscript></highlight>of the calibration equation are also changed, it is necessary to determine C<highlight><subscript>1 </subscript></highlight>and C<highlight><subscript>2</subscript></highlight>. It can be easily obtained in a manner that measures two reference samples by the noncontact measuring method using the microwave and the fourpoint probe method.
[0050] Though the calibration equation shows the linear equation in the abovedescribed embodiment, since the calibration equation is the relationship between the electrical conductivity obtained by the noncontact measuring method using the microwave and the electrical conductivity obtained by the fourpoint probe method, it is not limited to the linear equation.
[0051] As described above, by adopting the measuring system, when the wafer has the thickness in which the reflected wave from the bottom surface of the wafer can be negligible, the electrical conductivity of the semiconductor wafer can be measured on a noncontact basis without depending on the carrier type.
[0052] When σ is eliminated from the equation (4) of the real part and imaginary part, ε is derived as shown below and becomes the function having the variables a and b. This also allows the permittivity of the semiconductor wafer to be measured on a noncontact basis from the response of the microwave, similarly to the abovedescribed way.<mathcwu><number>6</number><math><mtable><mtr><mtd><mrow><mrow><mo>[</mo><mrow><mi>EQUATION</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>7</mn></mrow><mo>]</mo></mrow><mo></mo><mstyle><mtext> </mtext></mstyle></mrow></mtd><mtd><mstyle><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mrow><mi>ϵ</mi><mo>=</mo><mrow><mfrac><mrow><msup><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo></mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo></mo><mrow><mn>4</mn><mo></mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mrow><msup><mrow><mo>{</mo><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>a</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>}</mo></mrow><mn>2</mn></msup></mfrac><mo></mo><msub><mi>ϵ</mi><mi>o</mi></msub></mrow></mrow></mtd><mtd><mstyle><mtext> </mtext></mstyle></mtd></mtr></mtable></math><mathematicafile></mathematicafile><image></image></mathcwu>
ANOTHER EMBODIMENTS[0053] In the measuring system having the same configuration as that of <crossreference>FIG. 1</crossreference>, the electrical conductivity of the semiconductor wafer can be also measured in such a manner that a highresolution coaxial cable sensor <highlight><bold>140</bold></highlight>′ shown in <crossreference>FIG. 5</crossreference> is used as the sensor and the amplitude of the reflectivity, i.e. only the amplitude ratio of the transmitted microwave to the received reflected wave is measured.
[0054] In the nondestructive evaluation, which uses the microwave, of the material, the waveguide is usually used as the sensor. However, in the waveguide, there is a cutoff frequency indicating the lower limit of the frequency, where the microwave can not be transmitted lower than the frequency. It is necessary to enlarge the size of the waveguide in order to lower the cutoff frequency below the using frequency, which results in the problem that the spatial resolution is decreased. On the other hand, when the coaxial cable sensor which the inventor has developed is used, the cutoff frequency is eliminated, the wider frequency range can be handled, and the spatial resolution can be improved.
[0055] The coaxial cable sensor sends the microwave transmitted from a signal source to the test piece and receives the reflected wave from the test piece. <crossreference>FIG. 5</crossreference> shows a section of the coaxial cable sensor. The sensor includes a coaxial cable portion having an open end and including a central conductor <highlight><bold>148</bold></highlight> and an external conductor <highlight><bold>146</bold></highlight>, and a coaxial connector <highlight><bold>145</bold></highlight> connected to the coaxial cable.
CONCRETE EXAMPLE[0056] Concretely the measurement was performed by using the ntype of silicon wafer having the diameter of 100 mm and the thickness of 525 μm as the test piece. A radius of the central conductor <highlight><bold>148</bold></highlight> in the used coaxial cable sensor was 0.15 mm and an inner radius of the external conductor <highlight><bold>146</bold></highlight> was 0.47 mm.
[0057] The average electrical conductivity of the wafer measured by the fourpoint probe method was 14.73 S/m. The standoff distance between the sensor and the test piece was constantly set to 200 μm. The frequency was set to 110 GHz, and the amplitude A of the reflectivity was measured. Paying attention to the temperature dependence of the electrical conductivity, all the measurements were performed under the condition of the room temperature of 25° C.
[0058] <crossreference>FIG. 6</crossreference> is the measurement result of the electrical conductivity along a diameter direction of the wafer, which is obtained by the fourpoint probe method. It is found that the central portion has a slightly larger value compared with the outer portion in the electrical conductivity of the wafer. <crossreference>FIG. 7</crossreference> is the result measured with the microwave along the same diameter direction as that of <crossreference>FIG. 6</crossreference>. A scanning range is 100 mm and a distance is 2 mm. It is found that the response of the microwave measured from <crossreference>FIG. 7</crossreference> has the same distribution as that of the electrical conductivity of the wafer shown in <crossreference>FIG. 6</crossreference>. <crossreference>FIG. 8</crossreference> shows the relationship between the amplitude A of the reflectivity of the microwave and the electrical conductivity of the silicon wafer, which are obtained from <crossreference>FIGS. 6 and 7</crossreference>. The following equation can be obtained from <crossreference>FIG. 8</crossreference>.
EQUATION<inlineformula>σ<highlight><subscript>l</subscript></highlight><highlight><italic>=C</italic></highlight><highlight><subscript>1</subscript></highlight><highlight><italic>A+C</italic></highlight><highlight><subscript>2 </subscript></highlight> (8) </inlineformula>
[0059] Where C<highlight><subscript>1</subscript></highlight>=−9.1821 S/mdB and C<highlight><subscript>2</subscript></highlight>=−26.696 S/m.
[0060] <crossreference>FIG. 9</crossreference> is the microwave image depending on the reflectivity, which is obtained by scanning the silicon wafer while a scanning distance of 2 mm and the diameter of a scanning range is 100 mm in the measuring system of the invention. A twodimensional distribution map of the electrical conductivity for the silicon wafer, shown in <crossreference>FIG. 10</crossreference>, can be obtained by performing the abovedescribed calculation from <crossreference>FIG. 9</crossreference>. To obtain the precise distribution map of the electrical conductivity shown in <crossreference>FIG. 10</crossreference> exhibits the high, precise resolution of the electrical conductivity of the measuring system. Though values are colorcoded in <crossreference>FIGS. 9 and 10</crossreference>, here the colorcoded values are indicated in monochrome.
[0061] The reflective type of microwave converging sensor in which the reflecting mirrors of the ellipsoid are combined or the coaxial cable sensor is used in the abovedescribed measuring system, the reflective type of microwave converging sensor has an advantage of enlarging the standoff distance between the sensor and the wafer, and the coaxial cable sensor has the advantage of the high resolution, so that the using sensor can be selected considering those advantages. The sensor used in the measuring system is not limited to those sensors.
[0062] In the concrete example, though the single frequency of 110 GHz was used, other frequencies and multifrequencies may be used.
[0063] The electrical conductivity of the semiconductor or the like can be precisely measured on a noncontact basis by using the abovedescribed electrical conductivity measuring system of the invention.