JP2023067013A - Resistance rate measurement system - Google Patents

Abstract

To provide a resistance rate measurement system that can highly accurately and quickly measure a resistance rate of a measured object and a resistance rate distribution in the measured object by non-destruction.SOLUTION: A resistance rate measurement system comprises: a voltage generation unit that generates a voltage having a cyclical waveform between a protection electrode 23 and a substrate supporting body 31; a current voltage conversion unit that, when a voltage is applied between the protection electrode 23 and the substrate supporting body 31 in a state where a capacitance probing needle part 20 is put into close with a substrate W, and a gap G is formed between the capacitance probing needle part and the substrate, converts a current flowing to a probing needle electrode 21 to the voltage; a Fourier transformation unit that Fourier-transforms digital data obtained by converting an analog voltage to be output from the current voltage conversion unit; an amplitude phase calculation unit that calculates an amplitude component and phase component from a real number component and imaginary number component to be obtained by the Fourier transformation; and a resistance rate calculation unit that calculates a resistance rate on the basis of at least one frequency dependence of the real number component, imaginary number component, amplitude component and phase component.SELECTED DRAWING: Figure 1

Description

The present invention relates to resistivity measurement systems.

Among compound semiconductors such as GaAs, InP, and SiC, those having semi-insulating properties are widely used as semiconductor substrates for producing integrated circuits and power devices. Such a semi-insulating semiconductor substrate (hereinafter sometimes abbreviated as a substrate) is prepared by slicing a compound semiconductor ingot crystal-grown by a method such as a pulling method from a melt or a sublimation method, and then polishing the substrate. It is manufactured by The characteristics of electronic devices such as FETs formed on a substrate are greatly affected by the resistivity of the substrate. ing. For this reason, it is essential to measure the average value and standard deviation value of the resistivity of the substrate, as well as the resistivity distribution over the entire surface of the substrate.

As a conventional method for measuring the resistivity of a substrate, with the substrate placed on a conductive substrate support and the conductive probe in contact with the substrate surface, A method of measuring the current that flows when a voltage is applied to the substrate (see, for example, Patent Document 1), and a method of measuring the resistivity by a four-terminal method with a plurality of ohmic electrodes formed on the substrate surface (see, for example, Patent Document 1). Reference 2), and a method of measuring a current flowing between a plurality of conductive probes in contact with the substrate surface (see Patent Reference 3, for example) has been used.

However, these methods have the problem that the substrate surface is damaged or destroyed by contacting the substrate surface with a conductive contact or forming an ohmic electrode. In addition, due to electrical instability at the interface between the conductive contact and the substrate surface or the difficulty of forming an ohmic electrode on a semi-insulating semiconductor substrate, it is difficult to measure the resistivity with high accuracy and stability. There was a problem that it was not possible to Furthermore, when measuring the resistivity distribution over the entire surface of the substrate, there is also the problem that it takes a relatively long time.

On the other hand, as a measuring method that does not damage or destroy the substrate, there is a non-destructive measuring method using a capacitive probe (for example, see Non-Patent Document 1 and Non-Patent Document 2). In these non-destructive measurement methods, one side of the substrate is placed in contact with a conductive substrate support, and the conductive probe electrode and side surfaces of the probe electrode are isolated from the protective electrodes surrounding the probe electrode. A capacitive probe of the type is brought close to the other surface of the substrate, the protective electrode is maintained at the ground potential, and the probe electrode is connected to the terminal of the charge amplifier as a pseudo-ground potential, and a certain waveform is obtained. is applied to the substrate support and the output voltage at the output terminal of the charge amplifier is measured.

In particular, in Non-Patent Document 1, when a voltage whose waveform is a unipolar single step wave is applied to the substrate support, the charge amount Q(t) representing the output voltage of the charge amplifier is The equation Q(τ)=Q(∞)−(1/e)(Q(∞ )-Q(0)) is theoretically established, and the resistivity ρ is obtained from the dielectric relaxation time τ. However, in the actual measurement of the charge amount Q(t), the rise of the single staircase wave was not instantaneous but took a certain amount of time. In addition to being dependent, it is also affected by DC drift and the like. Therefore, the non-destructive measurement method disclosed in Non-Patent Document 1 has problems in measurement accuracy and measurement time. In particular, when the resistivity ρ is large, it takes a relatively long time to measure Q(∞), and the influence of the DC drift of the charge amplifier increases, resulting in a decrease in the measurement accuracy of Q(∞). On the other hand, when the resistivity ρ is small, there is a problem that the measurement accuracy of Q(0) decreases due to the rise time of the single-shot staircase waveform and the frequency dependence of the gain and phase of the charge amplifier.

In order to solve the problem of the above-mentioned non-destructive measurement method, as seen in Non-Patent Document 2, the present inventor applied a bipolar, periodically repeated square wave voltage to the substrate support instead of a unipolar single-shot staircase voltage. We propose applying a voltage and analyzing the frequency response of the output voltage from the charge amplifier instead of analyzing the time response. I have already shown that.

JP-A-63-33666 JP-A-02-24573 JP-A-05-164795

R. Stibal, J. Windscheif, W. Janz, Semicond. Sci. Technol. Vol. 6, 1991, pp. 995-1001. M. Fukuzawa, M. Yamada, Institute of Physics, Conference Series Number174, Paper presented at 29th International Symposium Compound Semiconductors, Lausanne, Switzerland, 7-10 October 2002, pp. 85-88

However, Non-Patent Document 2 does not disclose a specific method for solving the above problems, a method for measuring resistivity, or a specific configuration of a system for measuring resistivity.

SUMMARY OF THE INVENTION It is an object of the present invention to provide a resistivity measuring system capable of measuring the resistivity of an object to be measured and the resistivity distribution of the object to be measured nondestructively, accurately, and at high speed. and

In order to achieve the above object, the resistivity measurement system according to the present invention comprises:
a conductive substrate support for supporting an object to be measured;
A columnar probe electrode, a cylindrical protective electrode arranged to surround the probe electrode, and a tubular insulator portion interposed between the probe electrode and the protective electrode. a capacitive probe portion having an end portion of the probe electrode on the side of the substrate support and an end portion of the protective electrode on the side of the substrate support arranged on the same virtual plane;
a voltage generating unit that generates a voltage having a periodic waveform with a preset set cycle with the protective electrode as a ground potential between the protective electrode and the substrate support;
The protective electrode and the substrate are supported by the voltage generating section in a state in which the capacitive probe portion is brought close to the object to be measured supported by the substrate support and a gap is formed between the object to be measured and the object to be measured. a current-voltage converter that converts into a voltage the current that flows into the probe electrode when the voltage having the periodic waveform is applied between the body and the probe electrode and the probe electrode is maintained at a pseudo-ground potential;
an analog/digital conversion unit that samples the analog voltage output from the current-voltage conversion unit in synchronization with the set cycle and at time intervals that are 1/N of the set cycle, and converts the analog voltage into digital data;
a Fourier transform unit for Fourier transforming the digital data from the analog/digital converter;
an amplitude/phase calculator that calculates an amplitude component and a phase component from the real number component and the imaginary number component obtained by the Fourier transform;
a resistivity calculator that calculates a resistivity based on frequency dependence of at least one of the real component, the imaginary component, the amplitude component, and the phase component.

Moreover, the resistivity measurement system according to the present invention includes:
the resistivity calculator based on a preset frequency transfer function of the current-voltage converter, based on the frequency dependence of at least one of the real component, the imaginary component, the amplitude component, and the phase component; to calculate the resistivity.

Moreover, the resistivity measurement system according to the present invention includes:
When the voltage having the periodic waveform is applied between the substrate support and the protection electrode in a state in which the object to be measured is not supported by the substrate support, the frequency transfer function is the It may be obtained based on the real number component and the imaginary number component obtained by Fourier transforming the digital data output from the analog/digital converter.

Further, the resistivity measurement system according to the present invention includes:
The periodic waveform may be a sawtooth wave that fluctuates at the set period.

Moreover, the resistivity measurement system according to the present invention includes:
The periodic waveform may be a rectangular wave that fluctuates at the set period.

Moreover, the resistivity measurement system according to the present invention includes:
The periodic waveform may be a triangular wave that fluctuates at the set period.

Moreover, the resistivity measurement system according to the present invention includes:
The object to be measured is a semi-insulating semiconductor substrate,
The set period may be longer than the dielectric relaxation time of the object to be measured.

Moreover, the resistivity measurement system according to the present invention includes:
a moving mechanism for relatively moving the capacitive probe portion and the substrate support;
A control unit that controls the movement mechanism,
The control section may control the moving mechanism so as to measure the resistivity at a plurality of locations on the object to be measured.

According to the present invention, the capacitive probe portion, the object to be measured, and the substrate support are arranged in order of the conductive probe electrode, the gap, the object to be measured, and the conductive substrate support. , the capacitive probe does not come into contact with the object to be measured. Therefore, problems such as damage or breakage of the object to be measured and the influence of electrical instability at the contact portion with the object to be measured do not occur. Therefore, the resistivity of the object to be measured and the resistivity distribution of the object to be measured can be stably measured non-destructively with high accuracy.

Further, according to the present invention, since the protective electrode of the capacitive probe section is set to the ground potential and the probe electrode is connected to the terminal of the current-voltage converter, the probe electrode has a pseudo-ground potential. The electric field distribution between the capacitive probe portion and the substrate support becomes non-uniform due to the edge effect at the peripheral portion of the protective electrode of the capacitive probe portion. However, the above-mentioned electric field distribution in the probe electrode portion, that is, the target area measured by the capacitive probe portion is uniform, and is not affected by undesirable currents such as leakage current flowing from the protective electrode to the probe electrode. Therefore, highly accurate resistivity measurement becomes possible.

Furthermore, according to the present invention, the current-voltage converter and the analog/digital converter are operated in synchronization with the set cycle of the voltage generated by the voltage generator. It is possible to reduce electrical noise components during measurement. Therefore, it is possible to suppress variations in resistivity measurement values caused by electrical noise during measurement.

Further, according to the present invention, since the digital data obtained at time intervals of 1/N of the set period of the voltage generated by the voltage generator is Fourier transformed, the real number component and the imaginary number component of the angular frequency, and the real number component and the imaginary component, the amplitude component and the phase component can be immediately calculated. Therefore, the frequency dependence of each of the real number component, the imaginary number component, the amplitude component and the phase component can be calculated at once at high speed. can be removed. Therefore, resistivity measurement can be performed at high speed and with high accuracy.

1 is a schematic configuration diagram of a mechanical system of a resistivity measurement system according to an embodiment; FIG. 4 is a schematic enlarged cross-sectional view showing the arrangement of the capacitive probe portion, substrate, substrate support, and insulating support according to the embodiment; FIG. 1 is an electrical equivalent circuit diagram formed by a probe electrode, a gap, a substrate, and a substrate support according to an embodiment; FIG. 1 is a block diagram of a measurement arithmetic control system of a resistivity measurement system according to an embodiment; FIG. FIG. 2 shows an example of waveforms of voltages generated by a voltage generator according to an embodiment, where (A) is a diagram showing a sawtooth wave, (B) is a diagram showing a rectangular wave, and (C) is a triangular wave. It is a diagram. FIG. 10 is a diagram showing the relationship between the sampling interval, the number of sampling times, and the period during rectangular wave response according to the embodiment;

Hereinafter, the embodiment of the present invention will be described in detail with reference to the drawings, with the mechanical system and the measurement control system of the resistivity measurement system of the present invention being separated.

FIG. 1 is a schematic diagram of the mechanical system of the resistivity measurement system of the present invention. The mechanical system of the resistivity measurement system includes a capacitive probe section 20 , a substrate W, a substrate support 31 , an insulating support 32 and a moving mechanism 90 . The moving mechanism 90 has an X-axis stage 91 , a Y-axis stage 92 and a Z-axis stage 93 . The Z-axis stage 93 raises and lowers a moving table (not shown) on which the capacitive probe portion 20 is attached along the Z-axis direction as indicated by an arrow AR13. That is, the capacitive probe portion 20 can be vertically moved by the Z-axis stage 93 . The X-axis stage 91 moves a movable table 91a to which the insulating support 32 is attached along the X-axis direction as indicated by an arrow AR11. The Y-axis stage 92 moves a moving table 92a to which the X-axis stage 91 is attached along the Y-axis direction as indicated by an arrow AR12. Y-axis stage 92 is fixed to a base (not shown).

The insulating support 32 is formed in a flat plate shape from an insulating material. The substrate support 31 is formed of a conductive material in a flat plate shape and arranged vertically above the insulating support 32 . Thereby, the substrate support 31 can be moved in the horizontal plane by the X-axis stage 91 and the Y-axis stage 92 . The capacitive probe portion 20 can move in the horizontal plane while maintaining the gap G between the probe surface 24 and the upper surface of the substrate W.

A substrate W, which is an object to be measured, is supported by the substrate support 31 while being placed on the upper surface of the substrate support 31 . The shape of the substrate W is generally a disk-shaped thin plate, but it may be of other shapes.

FIG. 2 is a schematic enlarged cross-sectional view showing the arrangement of the capacitive probe portion 20, the substrate W, and the substrate support 31. As shown in FIG. The capacitive probe portion 20 includes a conductive cylindrical probe electrode 21 , a cylindrical protective electrode 23 having conductivity surrounding the probe electrode 21 , and the probe electrode 21 . and a cylindrical insulator portion 22 interposed between the protective electrode 23 and the protective electrode 23 . Here, the end portion of the probe electrode 21 on the side of the substrate support 31 and the end portion of the protection electrode 23 on the side of the substrate support 31 are arranged on the same virtual plane VP1. In other words, the capacitive probe portion 20 includes a conductive probe electrode 21, a conductive protective electrode 23, and an insulator portion 22 interposed between the probe electrode 21 and the protective electrode 23 to provide insulation. have. Then, the end surface of the probe electrode 21 on the side of the substrate support 31 and the end surface of the protection electrode 23 on the side of the substrate support 31 are arranged to form the probe surface 24 existing in the same virtual plane VP1, and the probe electrode is formed. 21, insulator portion 22 and protective electrode 23 are integrally formed. Here, the central axis of the probe electrode 21, the cylindrical axis of the insulator portion 22, and the cylindrical axis of the protective electrode 23 are aligned. In addition, the shape of the probe electrode 21 is not limited to a columnar shape, and may be, for example, a prismatic shape or other shapes. Also, the shapes of the insulator portion 22 and the protective electrode 23 are not limited to a cylindrical shape, and may be a rectangular tube shape or other shapes. The probe electrode 21, the protective electrode 23, and the substrate support 31 are connected to terminals a, b, and c via conductors, respectively.

As will be described later, the probe electrode 21 and the protective electrode 23 of the capacitive probe portion 20 are used with pseudo equipotentials. Therefore, no leakage current flows from the protection electrode 23 to the probe electrode 21 or in the opposite direction. When the gap G is narrowed and a voltage is applied between the substrate support 31 and the protective electrode 23, the lines of electric force between the capacitive probe portion 20 and the substrate W are as indicated by the arrows in FIG. . Although the lower portion of the outer circumference of the protective electrode 23 is non-uniform due to the edge effect, the measurement target region, which is the lower portion of the probe electrode 21, is uniform.

Assuming that there is a virtual electrode on the upper surface of the substrate W in the measurement target area, the capacitance and resistance formed by the virtual electrode and the substrate support 31 sandwiching the substrate W are C _s and R _s , respectively, Assuming that the capacitance formed by the virtual electrode and the probe electrode 21 with the air gap G therebetween is _Ca , the electric equivalent circuit diagram formed by the probe electrode 21, the air gap G, the substrate W, and the substrate support 31 is shown in FIG. becomes as shown in FIG. Note that the capacitance between the protective electrode 23 of the capacitive probe portion 20 and the virtual electrode or the probe electrode 21 is omitted.

As shown in FIG. 2, let the distance between the probe surface 24 and the substrate support 31 be d ₀ , the thickness of the substrate be d _s , and the distance of the air gap G be d _a . Let ρ and _{ε r} be the resistivity and relative permittivity of the semi-insulating semiconductor substrate, ε ₀ be the permittivity of vacuum, and A be the electrode area of the virtual electrode, then C _s = ε ₀ ε _r A/d _s , R _s =ρd _s /A. Also, if the dielectric relaxation time is τ _s , then τ _s =C _s R _s =ε ₀ ε _r ρ. Here, τ _s is material-specific and does not depend on geometry such as electrode area or electrode spacing. Also, the capacitance C _a is C _a =ε ₀ A/d _a . Let τ _a =C _a R _s to simplify the handling of the equations below. Note that τ _s and τ _a have units of time and are called time constants in the field of electrical circuit engineering. The measurement of the resistivity ρ, which is the object of the present invention, is equivalent to the measurement of the dielectric relaxation time τ _s if the dielectric constant ε _r is known.

FIG. 4 is a schematic diagram of the measurement arithmetic control system of the resistivity measurement system of the present invention. The measurement calculation control system of the resistivity measurement system includes a voltage generator 40, a current-voltage converter 50, an analog/digital converter 60, an X-axis driver 96, a Y-axis driver 97, and a Z-axis driver 98. and a personal computer (hereinafter referred to as “PC”) 100 .

The voltage generation unit 40 has a waveform memory 41, a digital/analog conversion circuit 42, and a power amplifier circuit 43, and generates a periodic waveform having a preset period with the protective electrode 23 as the ground potential. is generated between the guard electrode 23 and the substrate support 31 . The voltage generator 40 sequentially sends the waveform data recorded in the waveform memory 41 to the digital/analog conversion circuit 42 in synchronization with the waveform clock W _clk output from the control unit 95 of the PC 100 , and passes through the power amplifier circuit 43 . , to generate a periodic voltage V ⁱⁿ (t) of arbitrary waveform. The voltage generated by the voltage generator 40 is supplied to the terminal c' connected to the output of the power amplifier circuit 43 and the terminal b' of ground potential. Terminal c' and terminal b' are connected to terminal c and terminal b shown in FIG. 1, respectively. The voltage generating unit 40 generates a periodic voltage having the above-described set cycle such as a sawtooth wave, a rectangular wave, or a triangular wave as shown in FIGS. 5A to 5C in synchronization with the waveform clock _Wclk . can be done. In the following description, the set period is assumed to be T.

Returning to FIG. 4, the current-voltage conversion unit 50 generates a voltage in a state in which the capacitive probe unit 20 is brought close to the substrate W supported by the substrate support 31 and a gap G is formed between the substrate W and the capacitive probe unit 20 . A voltage having a periodic waveform is applied between the protective electrode 23 and the substrate support 31 by the portion 40, and the current flowing into the probe electrode 21 when the probe electrode 21 is maintained at a pseudo-ground potential is converted into a voltage. do. The current-voltage converter 50 has an operational amplifier 52 and a feedback passive element 51 . The input terminals of the current-voltage converter 50 are the terminal a' and the terminal b'. The terminals a' and b' are connected to the terminals a and b of the capacitive probe portion 20 shown in FIG. 1, respectively. The feedback passive element 51 is sometimes called a charge amplifier when it has only a capacitive component, and the feedback passive element 51 is sometimes called a current amplifier when it has only a resistive component. The operational amplifier 52 preferably has a large gain, a small phase delay, a large input impedance, and a small input bias current or input bias voltage.

The current flowing into the terminals of the current-voltage converter 50 is the sum of the current flowing into the feedback passive element 51 and the current flowing into the negative terminal of the operational amplifier 52 . Since the input impedance of operational amplifier 52 is extremely large, the current flowing into the negative terminal of operational amplifier 52 is extremely small, and the current flowing into the terminal of current-voltage converter 50 is equal to the current flowing out of feedback passive element 51 . Since the positive terminal of the operational amplifier 52 is kept at the ground potential, the terminal of the current-voltage converter 50, ie the negative terminal of the operational amplifier 52, becomes the pseudo-ground potential. The gain and phase of operational amplifier 52 and the impedance of feedback passive element 51 generally have frequency dependence. The frequency dependence of the current-voltage converter 50 is represented by a frequency transfer function having real and imaginary components.

The analog/digital converting section 60 has a sampling/holding circuit 61 and an analog/digital converting circuit 62, and converts the analog voltage output from the current/voltage converting section 50 in synchronization with the above-described set period and in one of the set periods. It is sampled at time intervals of /N and converted into digital data. The input of the analog/digital converter 60 is connected to the voltage output of the current-voltage converter 50 and the terminal g. By disconnecting the output of the current-voltage converter 50 and connecting an external analog voltage output to the terminal g, analog voltages other than the output voltage of the current-voltage converter 50 can be measured. FIG. 6 shows the timing of sampling the voltage output from the current-voltage converter 50 while the voltage generator 40 is generating the rectangular wave voltage. As shown in FIG. 6, the control unit 95 controls, at each time interval Δt obtained by dividing the set cycle T of the voltage of the periodic waveform generated by the voltage generation unit 40 by N, that is, time t=nΔt (n=1, 2, 3, . . . ), the sampling clock S _clk is sent to the analog/digital converter 60 . The analog/digital converter 60 samples/holds the voltage output from the current/voltage converter 50 by the sampling/hold circuit 61 in synchronization with the output sampling clock S _clk , and then converts it by the analog/digital converter 62 . The time-series digital data D _n (n=1, 2, 3, .

The PC 100 is a general-purpose PC, and has a CPU (Central Processing Unit), a main memory section, an auxiliary memory section, an interface, and a bus connecting each section. The main memory is composed of volatile memory and used as a work area for the CPU. The auxiliary storage unit is composed of a non-volatile memory and stores programs executed by the CPU. Then, the CPU reads the program stored in the auxiliary storage unit into the main storage unit and executes it, so that the preprocessing unit 81, the Fourier transform unit 70, the amplitude phase calculation unit 82, the resistivity calculation unit 83, and the control unit 95 Function.

The control section 95 sends the waveform clock W _clk to the voltage generating section 40 and the sampling clock S _clk to the analog/digital converting section 60 in synchronization with the system clock. In addition, the control unit 95 sends position command pulses to the X-axis driver 96, the Y-axis driver 97, and the Z-axis driver 98, and controls the X-axis stage 91, the Y-axis stage 92, and the Z-axis stage 93. positioning.

The preprocessing unit 81 converts time-series digital data D _n (n=1, 2, 3, . . . , N), and the processing of adding and averaging for M cycles is executed. The time-series digital data D _n (n=1, 2, 3, . is. Time-series digital data D _n (n=1, 2, 3, . . . ) is divided into cycles (n=1, 2, 3, _{. , m} (n = 1, 2, 3, ..., N, m = 1, 2, 3, ..., M), that is, D _1,1 , D _2,1 , D _3,1 , · ..., DN _,1 , _D1,2 , _D2,2 , _D3,2 , ..., _{DN,2, D1,3} , _D2,3 , _D3,2 , ... , D _N,3 , . . . D _1,M , D _{2, M} , D _{3,M ,} . , 2, 3, . . . , N) _, E ₁ = (D _1,1 +D _1,2 +D _{1,3 + . 1} + _D2,2 + _D2,3 +...+D2 _,M )/M, _E3 =( _D3,1 + _D3,2 + _D3,3 +...+ _D3,M )/M, . . , E _N =(D _N,1 +D _N,2 +D _N,3 + . . . +D _N,M )/M. If this preprocessing is performed over multiple cycles, noise in the digital data can be reduced, that is, the accuracy of resistivity measurement can be improved, but the measurement time becomes longer. Therefore, the number of cycles to which this preprocessing is to be performed may be determined by a trade-off between high precision and high speed resistivity measurement. Then, the preprocessing unit 81 notifies the Fourier transform unit 70 of the digital data _En (n=1, 2, 3, . . . , N) generated by executing the above-described weighted average processing.

The Fourier transform unit 70 Fourier transforms the digital data input from the analog/digital converter 60 . More specifically, the Fourier transform unit 70 performs Fourier transform on the digital data input from the analog/digital converter 60 and weighted average processed by the preprocessing unit 81 . The Fourier transform unit 70 performs a Fourier transform on digital data E _n (n=1, 2, 3, . . . , N) using a fast Fourier transform (FFT) algorithm when N is a power of two. Perform the transform, and perform the Fourier transform using the Discrete Fourier Transform (DFT) algorithm when N is not a power of two. Note that the conversion speed of FFT is much faster than that of DFT. As a result of this Fourier transform, a real number component P _n and an imaginary number component Q _n of the complex output voltage are obtained for angular frequencies nω _n (n=1, 2, 3, . . . , N).

The Fourier series analysis performed by the Fourier transform unit 70 will now be described in detail. The periodic voltage V ⁱⁿ (t) output from the voltage generator 40 can be expanded into a Fourier series, and given the period T and the angular frequency ω _n =2πn/T, it can be expressed by the following equation (1): be done.

（ｎ＝１，２，３，・・・，∞）
・・・式（１）

(n = 1, 2, 3, ..., ∞)
... formula (1)

Moreover, the Fourier coefficients a _n and b _n in the equation (1) are represented by the following equations (2) and (3). a ₀ is the average value of the voltage V ⁱⁿ (t), ie, the DC component. In the following, a ₀ =0 is assumed to simplify the handling of the formula. This corresponds to the case where V ⁱⁿ (t) is a bipolar positive and negative symmetrical waveform.

（ｎ＝１，２，３，・・・，∞）
・・・式（２）

(n = 1, 2, 3, ..., ∞)
... formula (2)

（ｎ＝１，２，３，・・・，∞）
・・・式（３）

(n = 1, 2, 3, ..., ∞)
... formula (3)

A Fourier series using a sine wave and a cosine wave as in equations (1) to (3) may be represented by a complex Fourier series and complex Fourier coefficients using complex numbers, which are different expressions. Here, showing specific formulas is omitted.

Since the voltage actually output from the voltage generator 40 has a finite rise time and fall time, it is appropriate to use partial sums up to N terms in equations (1) to (3). be. In the case of partial sums, an oscillating phenomenon called Gibbs _' phenomenon occurs in the vicinity of the point of discontinuity. b _n should be corrected. Alternatively, as will be described later, the voltage output from the voltage generator 40 may be actually measured and the coefficients a _n and b _n of each term may be calculated in advance.

As shown in FIGS. 5(A) to 5(C), the voltage output from the voltage generator 40 is considered to be of three types: a sawtooth wave voltage, a rectangular wave voltage, or a triangular wave voltage with an amplitude of |V ⁱⁿ |. . When the waveform of the voltage output from the voltage generator 40 is a sawtooth wave as shown in FIG. 5A, the above equation (1) becomes the following equation (4).

（ｎ＝１，２，３，・・・，∞）
・・・式（４）

(n = 1, 2, 3, ..., ∞)
... formula (4)

From equation (4), the sawtooth voltage consists only of the sum of sinusoidal voltages, whose frequency has odd and even components, the amplitude of the odd component being positive, the amplitude of the even component being negative, and the frequency It can be seen that the higher the , the more the absolute value of the amplitude decreases by an odd number.

When the waveform of the voltage output from the voltage generator 40 is a rectangular wave as shown in FIG. 5B, Equation (1) becomes Equation (5) below.

（ｎは正の奇数）
・・・式（５）

(n is a positive odd number)
... formula (5)

From equation (5), it can be seen that the square-wave voltage is composed only of the sum of sine-wave voltages, has only odd-numbered wave components in its frequency, and decreases in amplitude by odd-numbered times as the frequency increases.

When the waveform of the voltage output from the voltage generator 40 is a triangular wave as shown in FIG. 5C, Equation (1) becomes Equation (6) below.

（ｎは正の奇数）
・・・式（６）

(n is a positive odd number)
... formula (6)

From equation (6), the triangular wave voltage consists only of the sum of cosine wave voltages, its frequency has only odd wave components, and the amplitude decreases to the square of an odd multiple as the frequency increases, that is, the harmonic components are It can be seen that it decreases rapidly.

Here, consider the case where a complex voltage V ⁱⁿ _n (ω _n ) with angular frequency ω _n is applied between terminals a and c in the electrical equivalent circuit diagram shown in FIG. Let Z _n (ω _n ) be the complex impedance between terminal a and c, and let ^j be _the imaginary unit _. expressed.

・・・式（７）

... formula (7)

Assuming that In(ωn) is the complex current flowing between the terminal a and the terminal c, the following relational expression (8) is obtained.

・・・式（８）

... formula (8)

The complex voltage of the n-th item when the voltage ^Vin (t) is expanded into a complex Fourier series is Vin ⁿ (ω _n ), the real component _of the frequency transfer function of the current-voltage converter 50 is G(ω _n ), and the imaginary component is F(ω _n ), the complex output voltage V _n (ω _n ) output from the current-voltage conversion unit 50 when the complex current I _n (ω _n ) is input to the current-voltage conversion unit 50 is given by By substituting the formula (8), the following formula (9) is obtained.

・・・式（９）

... formula (9)

Also, the real component Re[V _n (ω _n )] and the imaginary component Im[V _n (ω _n )] of the complex output voltage are expressed by the following equations (10) and (11), respectively.

・・・式（１０）

... formula (10)

・・・式（１１）

... formula (11)

Returning to FIG. 4, the amplitude/phase calculator 82 calculates the amplitude component and the phase component from the real number component P _n and the imaginary number component Q _n obtained by Fourier transform. Specifically, the amplitude-phase calculator 82 uses the orthogonality of each angular frequency component to calculate for each angular frequency ω _n (n = 1, 2, 3, ..., N) . In the case of the real number component P _n and the imaginary number component Q _n obtained by the Fourier transform, the amplitude/phase calculator 82 calculates the amplitude component as _Sn =[P _n ² +Q _n ² ] ^1/2 and the phase component as U _n =arctan( Q _n /P _n ).

Here, the amplitude component |V _n (ω _n )| and the phase component φ _n (ω _n ) of the complex output voltage are expressed by the following equations (12) and (13), respectively.

・・・式（１２）

... formula (12)

・・・式（１３）

... formula (13)

Note that γ=τ _a /τ _s = _ds /ε _{rd a} in equations (10) to (13).

^{Next, using the resistivity measurement system 10 according to the present embodiment, the complex voltage V in n} ₍ ω _n ) will be described. The output of the current-voltage converter 50 shown in FIG. The analog/digital converter 60 converts the voltage output from the voltage generator 40 into digital data, and the Fourier transform 70 performs Fourier transform on the digital data to obtain a complex voltage ^Vin _n (ω _n ).

Next, a method of measuring the real number component G(ω _n ) and the imaginary number component F(ω _n ) of the frequency transfer function of the current-voltage converter 50 using the resistivity measurement system according to the present embodiment will be described. 4 is connected to the terminal a' of the current-voltage converter 50, and the voltage output from the voltage generator 40 is converted by the analog/digital converter 60. When the digital data is converted into digital data and Fourier-transformed by the Fourier transform unit 70, the real component G(ω _n ) and the imaginary component F(ω _n ) of the frequency transfer function of the current-voltage transform unit 50 are obtained directly.

In addition, when the substrate W is not placed on the substrate support 31 in FIGS. 2 and 3, that is, when d _s =0 and d a =d ₀ , C _{a =} C ₀ =ε ₀ A/d ₀ , τ _s →∞, and the voltage output from the voltage generator 40 is converted into digital data by the analog/digital converter 60, and the digital data is Fourier-transformed by the Fourier transformer 70, the above-mentioned equations (8) and equations The following formula (14) is obtained from (9).

（ｎ＝１，２，３，・・・，Ｎ）
・・・式（１４）

(n = 1, 2, 3, ..., N)
... formula (14)

Therefore, if jω _n C ₀ V ⁱⁿⁿ (ω _{n )} is measured in advance, the real component G(ω _n ) and the imaginary component F(ω _n ) of the frequency transfer function of the current-voltage converter 50 can be obtained from the equation (14). ) is obtained.

In the resistivity measurement system according to the present embodiment, when the substrate W is not placed on the substrate support 31, C _a =C ₀ and τ _s when d _s =0 and d _a =d ₀ . → becomes ∞. 4 is connected to the input side of the sampling/hold circuit 61 of the analog/digital converter 60, and the voltage output from the voltage generator 40 is analog/digital. After being converted into digital data by the digital conversion unit 60, _the digital data is ^Fourier transformed by the Fourier transform unit 70 to obtain _jωnC0Vinn ( _{ωn )} .

The resistivity calculator 83 calculates the resistivity based on the frequency dependence of at least one of the real component, the imaginary component, the amplitude component and the phase component of the complex output voltage. As described above, let the distance between the probe surface 24 of the capacitive probe portion 20 and the substrate support 31 be d ₀ , the thickness of the substrate be _ds , and the distance of the gap G be _da . Let ρ and ε _r be the resistivity and relative permittivity of the semi-insulating semiconductor substrate to be measured, ε ₀ be the permittivity of vacuum, and A be the electrode area of the virtual electrode. For example, C _s =ε ₀ ε _r A/d _s and R _s =ρd _s /A. When the dielectric relaxation time is τ _s , τ _s =C _s R _s =ε ₀ ε _r ρ. Therefore, the resistivity calculator 83 calculates the dielectric relaxation time τ _s described above, and calculates the resistivity ρ from the calculated τ _s using the relational expression ρ=ε ₀ ε _r /τ _s .

Here, a method for calculating the dielectric relaxation time τ _s by the resistivity calculator 83 will be described. Real number components P _n obtained by Fourier transform unit 70 performing fast Fourier transform or discrete Fourier transform on periodic digital data E _n (n=1, 2, 3, . . . , N), The imaginary component Q _n , the amplitude component _Sn = [P _n ² + Q _n ² ] ^1/2 , and the phase component U _n = arctan(Q _n /P _n ) are angular frequencies ω _n (n = 1, 2, 3, . . Then, when the angular frequency dependence of V ⁱⁿ _n (ω _n ), G(ω _n ), and F(ω _n ) can be ignored, or when it is measured in advance and known, equation (10), equation ( 11) and equation (12) are simultaneous equations with τ _s , γ and _Ca as unknowns, and equation (13) is a simultaneous equation with τ _s and γ as unknowns. Therefore, the resistivity calculator 83 calculates _τs using these continuity equations. The resistivity calculator 83 calculates γ from the geometric dimensions d _s and d _a and the dielectric constant ε _r of the object to be measured, and calculates _Ca from the geometric dimensions A and d _a and dielectric constant ε ₀ in vacuum. Further, the resistivity calculator 83 calculates the real number obtained from the result of performing the fast Fourier transform or the discrete Fourier transform on the periodic digital data E _n (n=1, 2, 3, . . . , N). The component P _n , the imaginary component Q _n , the amplitude component _Sn = [P _n ² + Q _n ² ] ^1/2 , and the phase component U _n = arctan(Q _n /P _n ) are expressed by _Equation (10 ), τ _s may be calculated so as to satisfy all of equations (11), (12), or (13).

The resistivity calculator 83 calculates the dielectric relaxation time τ _s from |V _n (ω _n )| given by Equation (12) and the amplitude component _Sn obtained by the Fourier transform. In the following, |V _n (ω _n )| is represented by |V _n ( _{ω n} : τ _{s ,} γ, _{Ca ) |} is represented as The resistivity calculator ₈₃ calculates the mean square of the difference between |V _n (ω _n : τ _s , _γ , _Ca )| That is, we use an error function that expresses the variance s ² .

（ｎ＝１，２，３，・・・，Ｎ）
・・・式（１５）

(n = 1, 2, 3, ..., N)
... formula (15)

The resistivity calculator 83 calculates the parameters τ _s , γ, and _Ca that make the variance s ² represented by the error function of Equation (15) the maximum likelihood estimated value. Here, the resistivity calculator 83 calculates the parameters τ _s , γ, and _Ca using nonlinear least-squares methods such as Gauss-Newton method and Levenberg-Marquardt method, which are iterative methods. When calculating approximately using the least squares method, if |V _n (ω _n )| given by Equation (12) or the amplitude component S _n includes an outlier or an abnormal value, There is a risk that the plausibility of the approximation will be extremely reduced. Therefore, for the parameters γ and _Ca , the previously measured geometric dimensions and the relative permittivity _εr of the object to be measured are used, or the search range for the values of the parameters γ and _Ca in the least-squares method is limited. preferably. Then, the resistivity calculator 83 calculates the resistivity ρ from the calculated τ _s using the relational expression ρ=ε ₀ ε _r /τ _s .

As described above, according to the resistivity measurement system according to the present embodiment, the capacitive probe portion 20, the substrate W, and the substrate support 31 are composed of conductor (probe electrode 21)-insulator (gap G)- The semi-insulator (substrate W) and the conductor (substrate support 31) are arranged in this order from vertically upward to downward, and the capacitive probe portion 20 does not come into contact with the substrate W. FIG. Therefore, problems such as damage or breakage of the substrate W and influence of electrical instability at the contact portion with the substrate W do not occur. Therefore, the resistivity of the substrate W and the resistivity distribution of the substrate W can be stably measured nondestructively with high accuracy.

Further, according to the resistivity measuring system according to the present embodiment, since the protective electrode 23 of the capacitive probe section 20 is set to the ground potential and the probe electrode 21 is connected to the terminal a' of the current-voltage conversion section 50, The probe electrode 21 becomes a pseudo-ground potential, and the electric field distribution between the capacitive probe portion 20 and the substrate support 31 becomes nonuniform due to the edge effect at the outer peripheral portion of the protective electrode 23 of the capacitive probe portion 20 . . However, the electric field distribution is uniform in the probe electrode 21 portion, that is, the target region measured by the capacitive probe portion 20, and the influence of undesirable current such as leakage current flowing from the protective electrode 23 to the probe electrode 21 is suppressed. no longer receive. Therefore, the resistivity measurement accuracy can be improved.

Furthermore, according to the resistivity measurement system according to the present embodiment, the current-voltage converter 50 and the analog/digital converter 60 are operated in synchronization with the set period of the voltage generated by the voltage generator 40, so that the set period The electrical noise component at the time of measurement can be reduced by adding the digital data every time. Therefore, it is possible to suppress variations in resistivity measurement values caused by electrical noise during measurement.

Further, according to the resistivity measurement system according to the present embodiment, the digital data obtained at the time interval ΔT (=T/N), which is 1/N of the set period T of the voltage generated by the voltage generator 40, is subjected to Fourier processing. By converting, the voltage generated by the voltage generator 40 has a real number component and an imaginary number component at the angular frequency ω _n (=2πn/T, n=0, 1, 2, . . . , N), and a real number component and the imaginary component, the amplitude component and the phase component can be immediately calculated. Therefore, since the frequency dependence of the real number component, the imaginary number component, the amplitude component and the phase component can be obtained at once at high speed, the influence of undesirable current generation such as DC drift occurring in the substrate W can be eliminated. Therefore, resistivity measurement can be performed at high speed and with high accuracy.

Furthermore, according to the resistivity measurement system according to the present embodiment, since the real number component G(ω _n ) and the imaginary number component F(ω _n ) of the frequency transfer function of the current-voltage converter 50 are obtained in advance, the voltage The real number component and the imaginary number component obtained by Fourier transform of the digital data obtained at the time interval ΔT (=T/N) which is 1/N of the set period T of the voltage generated by the generator 40 can be easily corrected. Therefore, the influence of the frequency dependence of the gain and phase of the current-voltage converter 50 can be eliminated, so that highly accurate resistivity measurement is possible.

Further, according to the resistivity measurement system of the present embodiment, the digital data obtained by measuring the capacitive probe portion 20 close to the substrate support 31 without placing the substrate W on the substrate support 31 is subjected to Fourier transform. By doing so, the real component G(ω _n ) and the imaginary component F(ω _n ) of the frequency transfer function of the current-voltage converter 50 are obtained. Therefore, the real component G(ω _n ) and the imaginary component F(ω _n ) of the frequency transfer function of the current-voltage converter 50 can be obtained relatively easily.

Furthermore, according to the resistivity measurement system of the present embodiment, the waveform of the voltage generated by the voltage generator 40 is a sawtooth wave, a rectangular wave, or a triangular wave with the set period T. Accordingly, if the voltage waveform is expanded into a Fourier series, it is represented by the sum of sine waves or cosine waves of angular frequency ω _n (=2πn/T, n=0, 1, 2, . . . , N). That is, the voltage generator 40 applies a voltage having a waveform represented by the sum of sine waves or cosine waves from the fundamental angular frequency _ω1 to the ultra-high angular frequency _ω∞ . Can speed up processing.

Further, according to the resistivity measurement system of the present embodiment, the set period T of the voltage generated by the voltage generator 40 is longer than the dielectric relaxation time τ _s of the substrate W. As shown in FIG. As a result, the dielectric relaxation frequency _ωs , which is the reciprocal of the dielectric relaxation time _τs , can be set within the angular frequency range from the fundamental angular frequency _ω1 to the super-high angular frequency _ωM . Therefore, the resistivity can be calculated with high accuracy from the frequency dependence of at least one of the real number component, the imaginary number component, the amplitude component, and the phase component after the Fourier transform of the voltage generated by the voltage generating section 40. can.

Furthermore, according to the resistivity measurement system of the present embodiment, the capacitive probe 20 is placed against the substrate support 31 with the gap G formed between the capacitive probe 20 and the substrate W. can be moved relative to each other. As a result, the resistivity can be measured at a plurality of positions on the substrate W at high speed and with high accuracy without damaging the substrate W, so that the resistivity distribution can be measured at high speed and with high accuracy.

Although the embodiments of the present invention have been described above, the present invention is not limited to the configurations of the above-described embodiments. For example, the resistivity calculator 83 calculates Re[V _n (ω _n )] and Im[V _n (ω _n )] represented by the above-described equations (10) and (11) and the Fourier transform. The dielectric relaxation time _τs may be calculated from the real number component _Pn and the imaginary number component _Qn . Hereinafter, Re[V _n (ω _n )] and Im[V _n (ω _{n )} ] are expressed as Re[V _n (ω _n : τ _{s ,} γ, _Ca )] and Im[V _n (ω _n : τ _s, γ, C _a )]. Specifically, the resistivity calculator 83 calculates the relationship between Re[V _n (ω _n : τ _s, γ, _Ca )] at each angular frequency ω _n and P _n as shown in the following equation (16). Error representing the variance ^s2 obtained by weighting and summing the mean squared difference and the mean squared difference between Im[V _n (ω _n : τ _s, γ, C _a )] and Q _n A function may be used.

（ｎ＝１，２，３，・・・，Ｎ）
・・・式（１６）

(n = 1, 2, 3, ..., N)
... formula (16)

Here, the weighting factor of the real number component and the weighting factor of the imaginary number component are set to w and v, respectively. The resistivity calculator 83 calculates the parameters τ _s , γ, and _Ca that make the variance s ² represented by the error function of Equation (16) the maximum likelihood estimated value, by the method of least squares.

Further, the resistivity calculator 83 calculates the dielectric relaxation time τ _s from φ _n (ω _n ) given by the above equation (13) and the phase component U _n obtained by Fourier transform. may Hereinafter, φ _{n (} ω _{n )} is expressed as φ _n (ω _n : τ _{s ,} γ) with τ s and γ as parameters. Specifically, the resistivity calculation unit 83 calculates the mean square of the difference between φ _n (ω _n : τ _s, γ) and U _n at each angular frequency ω _n as shown in the following equation (17), That is, an error function representing the variance ^s2 may be used.

（ｎ＝１，２，３，・・・，Ｎ）
・・・式（１７）

(n = 1, 2, 3, ..., N)
... formula (17)

The resistivity calculator 83 calculates the parameters τ _s and γ at which the variance s ² represented by the error function of Equation (17) becomes the maximum likelihood estimated value by the least squares method. Here, the variance ^s2 represented by the error function of Equation (17) does not depend on the parameter _C a unlike Equation (15) or Equation (16) above. Therefore, the resistivity calculator 83 can quickly calculate the parameters τ _{s and} γ by the method of least squares.

The present invention is capable of various embodiments and modifications without departing from the broader spirit and scope of the invention. Moreover, the embodiment described above is for explaining the present invention, and does not limit the scope of the present invention. That is, the scope of the present invention is indicated by the claims rather than the embodiments. Various modifications made within the scope of the claims and within the meaning of the invention equivalent thereto are considered to be within the scope of the present invention.

INDUSTRIAL APPLICABILITY The present invention is suitable as a resistivity measurement system for measuring the resistivity and resistivity distribution of a semi-insulating semiconductor substrate.

20: capacitive probe portion, 21: probe electrode, 22: insulator portion, 23: protective electrode, 24: probe surface, 30: substrate, 31: substrate support, 32: insulating support, 40: voltage generation Section 41: Waveform Memory 42: Digital/Analog Conversion Circuit 43: Power Amplifier Circuit 50: Current Voltage Conversion Section 51: Feedback Passive Element 52: Operational Amplifier 60: Analog/Digital Conversion Section 61: Sampling /hold circuit, 62: analog/digital conversion circuit, 70: Fourier transform unit, 81: preprocessing unit, 82: amplitude phase calculation unit, 83: resistivity calculation unit, 90: moving mechanism, 91: X-axis stage, 91a , 92a: moving table, 92: Y-axis stage, 93: Z-axis stage, 95: control unit, 96: X-axis driver, 97: Y-axis driver, 98: Z-axis driver, 100: PC, a, a', b, b', c, c', g: terminal, G: gap, VP1: virtual plane

Claims (8)

a conductive substrate support for supporting an object to be measured;
A columnar probe electrode, a cylindrical protective electrode arranged to surround the probe electrode, and a tubular insulator portion interposed between the probe electrode and the protective electrode. a capacitive probe portion having an end portion of the probe electrode on the side of the substrate support and an end portion of the protective electrode on the side of the substrate support arranged on the same virtual plane;
a voltage generating unit that generates a voltage having a periodic waveform with a preset set cycle with the protective electrode as a ground potential between the protective electrode and the substrate support;
The protective electrode and the substrate are supported by the voltage generating section in a state in which the capacitive probe portion is brought close to the object to be measured supported by the substrate support and a gap is formed between the object to be measured and the object to be measured. a current-voltage converter that converts into a voltage the current that flows into the probe electrode when the voltage having the periodic waveform is applied between the body and the probe electrode and the probe electrode is maintained at a pseudo-ground potential;
an analog/digital conversion unit that samples the analog voltage output from the current-voltage conversion unit in synchronization with the set period and at time intervals that are 1/N of the set period, and converts the analog voltage into digital data;
a Fourier transform unit for Fourier transforming the digital data from the analog/digital converter;
an amplitude/phase calculator that calculates an amplitude component and a phase component from the real number component and the imaginary number component obtained by the Fourier transform;
a resistivity calculator that calculates a resistivity based on the frequency dependence of at least one of the real component, the imaginary component, the amplitude component, and the phase component;
Resistivity measurement system. The resistivity calculator calculates frequency dependence of at least one of the real component, the imaginary component, the amplitude component, and the phase component based on a preset frequency transfer function of the current-voltage converter. to calculate the resistivity;
A resistivity measurement system according to claim 1 . The frequency transfer function is obtained when the voltage having the periodic waveform is applied between the substrate support and the protection electrode while the object to be measured is not supported by the substrate support. Based on the real number component and the imaginary number component obtained by Fourier transforming the digital data output by the analog / digital conversion unit,
3. A resistivity measurement system according to claim 2. The periodic waveform is a sawtooth wave that fluctuates at the set period,
A resistivity measurement system according to any one of claims 1 to 3. The periodic waveform is a rectangular wave that fluctuates at the set period,
A resistivity measurement system according to any one of claims 1 to 3. The periodic waveform is a triangular wave that fluctuates at the set period,
A resistivity measurement system according to any one of claims 1 to 3. The object to be measured is a semi-insulating semiconductor substrate,
the set period is longer than the dielectric relaxation time of the object under test;
A resistivity measurement system according to any one of claims 1 to 6. a moving mechanism for relatively moving the capacitive probe portion and the substrate support;
A control unit that controls the movement mechanism,
The control unit controls the moving mechanism so as to measure the resistivity at a plurality of locations on the object to be measured.
A resistivity measurement system according to any one of claims 1 to 7.

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