Wafers are the basic components of the manufacturing semiconductor circuit^[1-6]. In many cases, the front of the wafer is in adequately Chemical Mechanical Polishing (CMP), and a rough appearance is left on the back. With the continuous development of integrated circuits, the manufacturing of many large-sized wafers adopts double-sided polishing technology to obtain higher flatness^[7-10]. The Total Thickness Variation (TTV) of the two-sided polished wafers will directly affect the thickness uniformity of the thin film in CMP processing. If the Bow and Warp are too large, it may cause problems such as film peeling from the wafer surface stack, wafer rupture, and poor lithography accuracy^[11-14]. Therefore, the parameters such as TTV, BOW, and Warp in the wafer must be detected before entering the next step.

Some researchers used scanning measurement methods to effectively measure surface-related parameters of double-sided polished wafers. For example, Tahara K et al..^[15] used two double heterodyne interferometers to perform spiral scanning on the wafer, and the noise from the light source can be eliminated by switching the two modulation frequencies of the interferometers, which has a high measurement accuracy but suffers from frequency aliasing. Park J et al..^[16] placed the wafers on a motor-driven two-dimensional scanning bench and measured the wafers using a reflectance spectrometer to obtain information on parameters such as TTV, Bow, and Warp of the wafers based on the relationship between refractive index and optical range difference. This method has high spatial resolution but is time-consuming due to the influence of stepper motors. Hideki M et al..^[17] used two capacitive sensors to perform spiral or radial scanning on the wafers, which has the advantage of lower cost and higher measurement accuracy. However, the conductive nature of wafers makes it necessary to calibrate the conductivity of wafers before each measurement, which reduces the detection efficiency. Other researchers use a full surface measurement method. CHEN J Q et al..^[18] invented a wafer geometry parameter measurement device using two Hartmann wavefront sensors for wafer inspection, which has a simple structure but low resolution. ZHAO ZH L et al..^[19] invented a vertical wafer TTV interferometric device, which flips the wafer and thus obtains the TTV of the wafer.

In this paper, an interferometric method with high resolution and high measurement accuracy is proposed for the measurement of surface-related parameters of double-sided polished wafers, which can obtain the full surface morphology and related parameters of the wafer at one time without scanning or flipping. Two phase-shifting Fizeau interferometers with reference mirrors can measure the front and back surfaces of the wafer simultaneously. Based on the combination of the measured wafer front and back surfaces and the cavity topography of the two interferometers, the surface-related parameters of a double-sided polished wafer can be obtained without the influence of the reference mirror accuracy. Since the two reference mirrors are difficult to align precisely in spatial position, the mapping error is introduced during the combination operation, which affects the measurement results of the relevant parameters. To address this problem, a three-point positioning device is fixed between the two reference mirrors before wafer measurement, and the position of the two reference mirrors is continuously adjusted according to the three-point circular theorem, which can make the mapping error extremely small and thus reduce the influence of the mapping error on the measurement results. After adjusting the position of the two reference mirrors, the described interferometry method can easily and quickly measure the loading and unloading process of double-sided polished wafers, which can simplify the measurement steps and improve the measurement efficiency of double-sided polished large-size wafers.

As shown in Fig. 1(a), TTV is defined as the difference between the maximum and minimum values of the wafer thickness distribution^[20]. As shown in Fig. 1(b), Warp is defined as the difference between the maximum and minimum distances of a free unclamped wafer central plane relative to a reference plane, where the central plane is the trajectory of equidistant points from the front and back surfaces of the wafer, and the reference plane is the least-squares fitting surface of the wafer central plane. As shown in Figure 1(c), Bow is defined as the deviation between the center point of the central plane of a free unclamped wafer and the central plane reference plane, where the central plane reference plane is a plane determined by three equidistant points on the circumference of the diameter specified as smaller than the nominal diameter of the wafer^[21].

It can be seen from the above definition that in order to obtain TTV, Bow and Warp of a wafer at one time, the front and back surface morphology of the wafer must be obtained first.

The measurement principle of a double-sided polished wafer is shown in Figure 2. The wafer under test is placed vertically between two phase-shifting Fizeau interferometers, each with a complete interferometric system, which can measure the single-sided morphology of the wafer independently.

Figure 3(a) shows the cross-section of the measurement cavity consisting of two reference mirrors in the principle, with a wafer in the middle. Since the two interferometers are placed opposite to each other, each point in the front and back surfaces of the wafer from the interferometric measurement do not satisfy the one-to-one mapping relationship, and to facilitate the calculation, a right-handed Cartesian coordinate system is established on the front and back surfaces of the wafer with the origin at the center of the front and back surfaces. According to the results of the front surface morphology and the back surface morphology of the wafer measured by two interferometers simultaneously and flipped along the y-axis, it can be seen that:

$ \phi_{\rm{B}}\left( {x,y} \right) = \frac{{4{\text{π}} }}{\lambda }\left[ {r_{\rm{B}}\left( {x,y} \right) - t_{\rm{B}}\left( {x,y} \right)} \right]\quad, $ (1)

$ \phi_{\rm{A}}\left( { - x,y} \right) = \frac{{4{\text{π}} }}{\lambda }\left[ {r_{\rm{A}}\left( { - x,y} \right) - t_{\rm{A}}\left( { - x,y} \right)} \right] \quad,$ (2)

where $\phi_{\rm{B}}\left( {x,y} \right)$ and $\phi _{\rm{A}}\left( { - x,y} \right)$ are the striped phase in the interferogram of front and back surface morphology of wafer obtained by different interferometers; $\lambda $ is the wavelength of laser; ${r_{\rm{B}}}(x,y)$ and ${r_{\rm{A}}}( - x,y)$ are the surface deviations of each point on the reference mirrors TF_B and TF_A, respectively; ${t_{\rm{B}}}(x,y)$ and ${t_{\rm{A}}}( - x,y)$ are the surface deviations of each point on the front and back of the wafer, respectively; and the negative sign in the Eq. (2) refers to a flipping of the wafer's negative morphology compared to the front surface morphology along the y-axis. Eqs. (1) and (2) are solved together to obtain the thickness variation expressions $f\left( {x,y} \right)$ of the wafer and the surface shape deviation expression $s\left( {x,y} \right)$ from central plane:

$ \begin{split} f(x,y) =& {t_{\rm{B}}}(x,y) + {t_{\rm{A}}}( - x,y) = {r_{\rm{B}}}(x,y) + \\ & {r_{\rm{A}}}( - x,y) - \frac{\lambda }{{4{\text{π}} }}\left[ {\phi_{\rm{B}}\left( {x,y} \right) + \phi_{\rm{A}}\left( { - x,y} \right)} \right] \quad, \\ \end{split} $ (3)

$ \begin{split} &s(x,y) = \frac{{{t_{\rm{B}}}(x,y) - {t_{\rm{A}}}( - x,y)}}{2} = \\ & \frac{1}{2}{r_{\rm{B}}}(x,y)- \frac{1}{2}{r_{\rm{A}}}( - x,y) + \frac{\lambda }{{4{\text{π}} }}\left[ {\frac{{{\phi _{\rm{A}}}( - x,y) - \phi_{\rm{B}}\left( {x,y} \right)}}{2}} \right] \quad. \\ \end{split} $ (4)

In general, the magnitude of the surface deviation of the reference mirror is nanometers, and the magnitude of surface shape deviation from the central plane of the wafer is microns, so the influence of the reference mirror can be ignored in the calculation of surface shape deviation from the central plane of the wafer, and its expression is:

$ \begin{split} s(x,y) = &\frac{{{t_{\rm{B}}}(x,y) - {t_{\rm{A}}}( - x,y)}}{2}\approx \\ &\frac{\lambda }{{4{\text{π}} }}\left[ {\frac{{{\phi _{\rm{A}}}( - x,y) - \phi_{\rm{B}}\left( {x,y} \right)}}{2}} \right]\quad. \\ \end{split} $ (5)

Since the magnitudes of the wafer thickness variation and the reference mirror surface deviation are nanometers, the influence of the reference mirror needs to be considered in the wafer thickness variation calculation. For this reason, after the wafer measurement, the cavities of the two interferometers needs to be examined, and Figure 3(b) shows the cavity cross-sections measured by the two interferometers.

Using interferometer B to detect the cavity of the two interferometers yields the following relationship:

$ r_{\rm{B}}\left( {x,y} \right) + r_{\rm{A}}\left( { - x,y} \right) = \frac{\lambda }{{4{\text{π}} }}\phi_{\rm{C}}\left( {x,y} \right)\quad, $ (6)

where $\phi _{\rm{C}}\left( {x,y} \right)$ is the striped phase in the cavity interferogram of the two interferometers. Substituting equation (6) into equation (3) yields:

$ \begin{split} f(x,y) = &{t_{\rm{B}}}(x,y) + {t_{\rm{A}}}( - x,y) = \\ &\frac{\lambda }{{4{\text{π}} }}\left[ {\phi _{\rm{C}}\left( {x,y} \right) - \phi _{\rm{B}}\left( {x,y} \right) - \phi _{\rm{A}}\left( { - x,y} \right)} \right]\quad. \\ \end{split} $ (7)

According to Eqs. (5) and (7), the morphology of surface shape deviation from the central plane of the wafer can be obtained by subtracting the result of the wafer's back surface morphology flipping along the y-axis from the wafer's front surface morphology. The wafer's thickness variation morphology can be obtained by the results of the cavity topography of the two interferometers subtracting the wafer's front surface morphology, and wafer's back surface morphology flipping along the y-axis.

After obtaining the thickness variation morphology and the morphology of surface shape deviation from the central plane of the wafer, TTV, Bow and Warp are obtained as:

$ TTV = f\left( {x,y} \right)_{\max} - f\left( {x,y} \right)_{\min}\quad, $ (8)

$ Bow = s\left( {0,0} \right) - Z_{{\rm{ref}}3}\left( {0,0} \right)\quad, $ (9)

$ \begin{split} Wrap = &\left[ {s\left( {x,y} \right) - Z_{{\rm{ref}}}} \right]_{\max} - \\ &\left[ {s\left( {x,y} \right) - Z_{{\rm{ref}}}} \right]_{\min} \quad, \\ \end{split} $ (10)

where $Z_{{\rm{ref}}3}$ and $Z_{{\rm{ref}}}$ are the three-dimensional point fitting plane and the least-squares fitting plane of $s\left( {x,y} \right)$, respectively.

The thickness variation of the wafer and surface shape deviation from the central plane of the wafer are obtained by combining the addition or subtraction operation results obtained by flippinging the wafer front surface morphology, the cavity morphology map of the two interferometers and the wafer back morphology map along the y-axis. When there is no tilt or offset between the two reference mirrors, the corresponding pixel points in the three morphology maps recorded by the CCD during the operation should satisfy the one-to-one mapping relationship. However, in the actual measurement, because the relative positions of the two reference mirrors are difficult to align precisely, the pixel points in the measured wafer front and back profiles do not satisfy the one-to-one mapping relationship, which will introduce mapping errors and affect the measurement results. When there is almost no tilt between the two reference mirrors, the magnitude and direction of the mapping error are determined by the offset of the two reference mirrors.

In order to determine the direction of the mapping error, let the offset of the circle center $o'$ of the reference mirror TF_A relative to the circle center $o$ of the reference mirror TF_B in the lateral and vertical directions of the CCD be $\varepsilon $ and $\eta $. The mapping error analysis diagram is shown in Figure 4, where m and n are the number of pixels in the lateral and vertical directions of the CCD, respectively. Since the center points $o$ and $o'$ of the two reference mirrors may deviate from the center position of the CCD, it is difficult to determine the magnitude of the mapping error by $oo'$.

In order to accurately obtain the magnitude of the mapping error, the three-point positioning device is fixed between the two reference mirrors before wafer measurement. Since the three points can form a circular domain, the deviation degree of the two reference mirrors can be known according to the pixel coordinates of the circle center of the circular domain formed by the three points in the CCD corresponding to the different reference mirrors. The schematic diagram of the pixel coordinates of the three points in the CCD corresponding to different reference mirrors is shown in Figure 5.

Let the pixel coordinates of the three points in the TF_B and TF_A corresponding CCDs be $( X_{{\rm{B}}1}, Y_{{\rm{B}}1} )$, $\left( {X_{{\rm{B}}2},Y_{{\rm{B}}2}} \right)$, $\left( {X_{{\rm{B}}3},Y_{{\rm{B}}3}} \right)$ and $\left( {X_{{\rm{A}}1},Y_{{\rm{A}}1}} \right)$, $\left( {X_{{\rm{A}}2},Y_{{\rm{A}}2}} \right)$, $\left( {X_{{\rm{A}}3},Y_{{\rm{A}}3}} \right)$, respectively, and the pixel coordinates of the circle center of the formed circle domain $o_1$, $o_2$ be $\left( {X_0,Y_0} \right)$ and $ \left( {X_1,Y_1} \right) $, respectively. From the pixel coordinates of the three points in the two CCDs, we can see that

$ \begin{split} &\left( {{X_{{\rm{A}}1}} - {X_{{\rm{A}}2}}} \right)X_1 + \left( {{Y_{{\rm{A}}1}} - {Y_{{\rm{A}}2}}} \right)Y_1 = \\ & \frac{{X_{{\rm{A}}1}^2 - X_{{\rm{A}}2}^2 + Y_{{\rm{A}}1}^2 - Y_{{\rm{A}}2}^2}}{2}\quad, \end{split} $ (11)

$ \begin{split} &\left( {{X_{{\rm{A}}1}} - {X_{{\rm{A}}3}}} \right)X_1 + \left( {{Y_{{\rm{A}}1}} - {Y_{{\rm{A}}3}}} \right)Y_1 = \\ &\frac{{X_{{\rm{A}}1}^2 - X_{{\rm{A}}3}^2 + Y_{{\rm{A}}1}^2 - Y_{{\rm{A}}3}^2}}{2}\quad, \end{split} $ (12)

$ \begin{split} &\left( {{X_{{\rm{B}}1}} - {X_{{\rm{B}}2}}} \right)X_0 + \left( {Y_{{\rm{B}}1} - Y_{{\rm{B}}2}} \right)Y_0= \\ & \frac{{X_{{\rm{B}}1}^2 - X_{{\rm{B}}2}^2 + Y_{{\rm{B}}1}^2 - Y_{{\rm{B}}2}^2}}{2} \quad, \end{split} $ (13)

$ \begin{split} &\left( {{X_{{\rm{B}}1}} - {X_{{\rm{B}}3}}} \right)X_0 + \left( {{Y_{{\rm{B}}1}} - {Y_{{\rm{B}}3}}} \right)Y_0 =\\ &\frac{{X_{{\rm{B}}1}^2 - X_{{\rm{B}}3}^2 + Y_{{\rm{B}}1}^2 - Y_{{\rm{B}}3}^2}}{2} \quad. \end{split} $ (14)

The specific pixel coordinates of $o_1$, $o_2$ can be known according to the above equation. Because the central pixel position of the CCD is mapped to the origin of the wafer coordinate system, the vertical center line is mapped to the y-axis. According to the pixel coordinates of $o_1$ and $o_2$, the magnitude of the mapping error are obtained as:

$ \varepsilon = n - X_1 - X_0 \quad,$ (15)

$ \eta = Y_1 - Y_0 \quad.$ (16)

By continuously adjusting the positions of the two reference mirrors, the offset of the circle center of the circular domain formed by the three points in different reference mirrors can be extremely small, and thus a very small mapping error can be obtained.

In order to analyze the effect of the adjusted mapping errors on the measurement results, the mapping errors $\varepsilon $ and $\eta $ and the two-dimensional coordinates $\left( {x,y} \right)$ of the wafer are represented using vector symbols, and the surface deviation at each point on the front and back of the wafer can be given by

$ T_{\rm{B}}\left( {\vec x ,\vec y } \right) = s\left( {\vec x ,\vec y } \right) + \frac{1}{2}f\left( {\vec x ,\vec y } \right)\quad, $ (17)

$ \begin{split} T_{\rm{A}}\left( {{ -\vec x} ,\vec y } \right) =& s\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) - \\ & \frac{1}{2}f\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right)\quad. \end{split} $ (18)

According to the above equation, the thickness variation of the wafer and surface shape deviation from the central plane are

$ \begin{split} F\left( {\vec x ,\vec y } \right) = &s\left( {\vec x ,\vec y } \right) + \frac{1}{2}f\left( {\vec x ,\vec y } \right)- \\ &s\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) + \frac{1}{2}f\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right), \\ \end{split} $ (19)

$ \begin{split} S\left( {\vec x ,\vec y } \right) = &\frac{1}{2}s\left( {\vec x ,\vec y } \right) + \frac{1}{4}f\left( {\vec x ,\vec y } \right)+ \\ & \frac{1}{2}s\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) - \frac{1}{4}f\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) . \end{split} $ (20)

In order to obtain the amount of error, $ s\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) $ and $ f\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) $ are expanded to the first order by Taylor's formula, and the expression of the measurement error in the thickness variation of the wafer and surface shape deviation from the central plane of the wafer can be found as follows

$ \Delta F\left( {\vec x,\vec y} \right) \approx \vec \nabla s\left( {\vec x,\vec y} \right) \cdot \left[ {\vec \varepsilon ,\vec \eta } \right] + \frac{1}{2}\vec \nabla f\left( {\vec x,\vec y} \right) \cdot \left[ {\vec \varepsilon ,\vec \eta } \right] \quad, $ (21)

$ \Delta S\left( {\vec x,\vec y} \right) \approx \frac{1}{2}\Delta F\left( {\vec x,\vec y} \right)\quad. $ (22)

Except for the very edge of the wafer, most of the gradients of the thickness variation are several magnitudes lower than the surface shape deviation gradient from the central plane. Therefore, the amount of error in the thickness variation of the wafer and surface shape deviation from the central plane are determined by the surface shape deviation gradient from the central plane and mapping errors ε and η. The minimal mapping error reduces the impact of error amount on subsequent related parameters. Since the measurement error in surface shape deviation from the central plane is half of the measurement error in thickness variation, the impact of the mapping error in surface shape deviation from the central plane can be ignored for good thickness variation measurements.

In order to verify the correctness of the adjustment method, a two-inch double-sided polished wafer is selected as the test sample according to the conditions of the laboratory. The experimental measurement devices of the wafer surface related parameters are shown in Figure 6(a). The experimental measurement devices include two two-inch phase-shifting Fizeau interferometers with two reference mirrors (the light source wavelength used in the interferometer is 632.8 nm), a wafer clip and a two-dimensional holder (for fixing the wafer clip and three-point positioning device).

The device diagram of the wafer clip is shown in Figure 6(b). The wafer clip has an annular recess and three plastic presses with a slight upward tilt. The wafer under test is placed in the recess and the three plastic presses are used to hold the wafer vertically. The stress at the contact point between the press and the wafer is very low, and the device causes little deformation to the wafer during the measurement.

The actual diagram of the three-point positioning device is shown in Figure 6(c). The three points of the three-point positioning device are determined by the three very fine lines.

Before the wafer measurement, the tilt knob of the two reference mirrors is first adjusted to ensure the two reference mirrors are as parallel as possible in spatial position, and then the three-point positioning device is fixed on the two-dimensional holder, and the position of the two reference mirrors is continuously adjusted according to the equation in the mapping error analysis, which can make the offset of the two reference mirrors extremely small and thus obtain a very small mapping error. Figure 7 shows the pixel coordinates of the three points in the light intensity map obtained from the two reference mirrors corresponding to the CCD, where the number of pixels in the horizontal and vertical directions of the CCD are 960 and 1280, respectively, and the mapping error $\varepsilon = $ 21.592 μm and $\eta = 37.480$ μm of the wafer can be known according to the pixel coordinates of the three points in the Figure 7.

After adjusting the position of the two reference mirrors, the three-point positioning device on the two-dimensional holder is replaced with a wafer clip. The tilt knob of the two-dimensional holder is so adjusted that the wafer held by the wafer clip is in a smaller tilt. The wafer is placed almost vertically, and the deformation caused by its gravity is small. At this time, the front and back surfaces of the wafer are measured simultaneously using two interferometers, and the measured front and back surfaces of the wafer are shown in Figs. 8(a)~8(b) (color online). In Fig. 8(a), the front morphology of the wafer can be analyzed by color distribution, where the PV value (the difference between the highest peak and the lowest valley) of the front morphology is 1.503 μm and the RMS value (root mean square deviation of the examined wavefront with respect to the reference surface) is 0.262 μm. Fig. 8(b) shows the results of the back morphology of the wafer after flipping along the y-axis, where the PV value of the back morphology is 1.361 μm and the RMS value is 0.228 μm. The measured cavity topography is shown in Fig. 8(c) (color online). It can be seen that the PV value of cavity is 0.024 μm and the RMS value is 0.002 μm.

The three morphology maps in Figure 8 are combined according to the measurement principle. The thickness variation of the wafer and the morphology of surface shape deviation from the central plane are shown in Figure 9 (color online). Figure 9(a) shows the thickness variation of the wafer, and the TTV of the wafer is 0.198 μm. Figure 9(b) shows the morphology of surface shape deviation from the central plane, and the Bow and Warp of the wafer are −0.326 μm and 1.423 μm, respectively, according to the definition.

To further verify the effectiveness of the method, a single interferometer B was used to perform flip measurements on double-sided polished wafers, and a diagram of the flip measurement experimental setup is shown in Figure 10.

The double-sided polished wafer held by the wafer clip is fixed in the center of the two-dimensional holder, and the front surface of the wafer is first inspected using an interferometer, and the left and right edge positions of the two-dimensional holder are marked through the threaded holes on the optical table at this moment, and then the two-dimensional holder is flipped, and the flipped two-dimensional holder is placed between the two marked positions, and then the back surface of the wafer is inspected, and the measured the front surface morphology of the wafer and the results of the back surface morphology flipped along the y-axis are shown in Figs. 11(a) to 11(b) (color online) after flipping along the y-axis. Due to the change of the measurement principle, the expression of the thickness change of the wafer and surface shape deviation from the central plane becomes

$ \begin{split} f(x,y) =& {t_{\rm{B}}}(x,y) - {t_{{\rm{A}}'}}( - x,y) = \\ &\frac{\lambda }{{4{\text{π}} }}\left[ {\phi _{\rm{A'}}\left( { - x,y} \right) - \phi _{\rm{B}}\left( {x,y} \right)} \right] \quad, \end{split} $ (23)

$ \begin{split} s(x,y) = &\frac{{{t_{\rm{B}}}(x,y) + {t_{{\rm{A}}'}}( - x,y)}}{2} \approx \\ &- \frac{\lambda }{{4{\text{π}} }}\left[ {\frac{{{\phi _{{\rm{A}}'}}( - x,y) + \phi _{\rm{B}}\left( {x,y} \right)}}{2}} \right] \quad, \end{split} $ (24)

where $\phi_ {\rm{B}}\left( {x,y} \right)$ and $\phi _{\rm{A'}}\left( { - x,y} \right)$ are the striped phase in the interferogram of the wafer front and back surfaces measured by the same interferometer B, $\lambda $ is the wavelength of the laser, and $ {t_{\rm{B}}}(x,y) $and ${t_{{\rm{A}}'}}( - x,y)$ are the deviations of the surface shape at each point on the front and back surfaces of the wafer in the same coordinate system, respectively. From Eqs. (23)~(24), it is clear that the thickness variation of the wafers obtained by the flip wafer method and surface shape deviation from the central plane are not affected by the reference mirror accuracy error.

According to Eqs. (23), and (24), the combination operation is carried out on the front and back surface morphologys of water. The thickness variation and the morphology of surface shape deviation from the central plane of the wafer can be obtained, as shown in Figs. 11(c) and 11(d) (color online). Due to the drift of the wafer between the two measurements, there are fluctuations in the thickness variation of the wafer. The TTV, Bow and Warp of the wafer are 0.208 μm, −0.326 μm and 1.415 μm, respectively, according to the relevant definitions.

The TTV, Bow and Warp of the wafers measured by the two methods are compared to further verify the effectiveness of the proposed method.

The flip wafer method requires two measurements of the front and back surfaces of the wafer in the measurement, and it is difficult to precisely align the spatial position of the measured wafer front surface morphology and the measured wafer back surface morphology after flipping along the y-axis, which introduces alignment errors, the amount of which is influenced by the flipping operation. In addition, the wafer may drift between measurements, which introduces drift errors and affects the accuracy of the measurement results.

The proposed interferometric method can achieve the same effect of eliminating the reference mirror error without the flip operation, and also prevents the wafer from drifting between measurements, with a smaller amount of error introduced during the measurement.

When the two methods are used for inspection of large-volume double-sided polished wafers, it can be found that the flip wafer method requires flipping and aligning the two-dimensional holder during each measurement, and there may be uncontrollable drift errors and alignment errors in the results obtained from each measurement. For the proposed interferometric method, after adjusting the position of the two reference mirrors, it is easy and fast to load and unload and inspect large quantities of double-sided polished wafers, avoiding the tedious flipping and alignment steps, improving the inspection efficiency of the wafers, and the error amount is smaller during the measurement.

In this paper, an interferometric method is proposed for the measurement of TTV, Bow and Warp of double-sided polished wafers. Two phase-shifting Fizeau interferometers with reference mirrors is used to measure the front and back surfaces simultaneously. The surface-related parameters of double-sided polished wafers can be obtained without the influence of reference mirror accuracy by combining the measured wafer front and back profiles with the cavity morphology of the two interferometers. Since it is difficult to precisely align the two reference mirrors in spatial position, the mapping error will be introduced during the combination operation, which will affect the measurement results of the relevant parameters. In order to accurately adjust the relative positions of the two reference mirrors, a three-point positioning device is fixed between the two reference mirrors before wafer measurement, and the positions of the two reference mirrors are continuously adjusted according to the three-point circular theorem, which can make the mapping error extremely small and thus reduce the influence of the mapping error on the measurement results of the relevant parameters. The experimental results show that the mapping errors of the 50 mm wafers are 21.592 μm and 37.480 μm in the transverse and longitudinal directions, and 0.198 μm, −0.326 μm and 1.423 μm for TTV, Bow and Warp, respectively. The TTV, Bow and warp of water are 0.208 μm, −0.326 μm and 1.415 μm when a single interferometer was uesd to flip the wafer for measurement. The effectiveness of the adjusting method is further verified by comparing the two sets of data. The proposed interferometric method, after adjusting the positions of the two reference mirrors, can be easily and quickly used for loading and unloading inspection of large quantities of double-sided polished wafers, avoiding tedious flipping and alignment steps. There is only a small amount of error in measurement, indicating that this method has good application value.

晶圆是制造半导体电路的基础元件^[1-6]，在许多情况下，晶圆的正面经过充分的化学机械抛光(CMP)，而背面仍是粗糙的外观。随着集成电路的不断发展，许多大尺寸晶圆的制造采用了双面抛光技术，以获得更高的平整度^[7-10]。双面抛光晶圆总厚度变化(TTV)的大小将直接影响CMP加工中薄膜厚度的均匀性。弯曲度(Bow)和翘曲度(Warp)如果过大，可能会造成晶圆表面堆叠层薄膜脱落、晶圆破裂以及光刻精度变差等问题^[11-14]。因此在晶圆进入到下一步工序前需要对其TTV、弯曲度和翘曲度等参数进行必要的检测。

为了有效测量双面抛光晶圆的表面相关参数，一部分研究人员使用扫描式测量方法。Tahara K等^[15]使用两个双外差干涉仪对晶圆进行螺旋形扫描，通过切换干涉仪的两个调制频率可以消除来自光学光源的噪声，该方法具有很高的测量精度，但存在频率混叠现象。Park J等^[16]将晶圆放置于电机驱动的二维扫描工作台上，使用反射式光谱仪对晶圆进行测量，根据折射率和光程差的关系，可获得晶圆的TTV、弯曲度和翘曲度等参数信息，该方法具有很高的空间分辨率，但由于步进电机的影响，耗时较长。Hideki M等^[17]使用两个电容传感器对晶圆进行螺旋式或放射式扫描，该方法成本较低并具有较高的测量精度，然而由于晶圆的导电特性，使得每次测量前，需要对晶圆的电导率做标定，降低了检测效率。另一部分研究人员使用全表面测量法。陈建强等^[18]发明了一种晶圆几何参数测量装置，使用两个哈特曼波前传感器对晶圆进行检测，该方法结构简单，但分辨率较低。赵智亮等^[19]发明了一种立式晶圆TTV干涉测量装置，该装置通过翻转晶圆从而得到晶圆的TTV。

针对双面抛光晶圆的表面相关参数的测量，本文提出了一种干涉测量方法，在不进行扫描或者翻转的情况下，可一次性获得晶圆的全表面形貌和相关参数信息，同时具有较高的分辨率和测量精度。使用两个带有标准镜的菲索式相移干涉仪对晶圆正反面同时进行测量，根据测量所得晶圆正反面形貌与两个干涉仪空腔形貌的组合运算，可得到不受标准镜精度误差影响的双面抛光晶圆的表面相关参数。由于两个标准镜在空间位置上很难精确对准，在组合运算过程中会引入映射误差，影响相关参数的测量结果。针对这一问题，在晶圆测量之前，将三点定位装置固定在两个标准镜之间，根据三点定圆定理不断调整两个标准镜的位置，可使映射误差极小，进而减小映射误差对测量结果的影响。所提干涉测量法在调整好两个标准镜的位置后，可以方便快速地对双面抛光晶圆进行装卸测量，简化了测量步骤，提高了双面抛光大尺寸晶圆的测量效率。

如图1(a)所示，TTV定义为晶圆厚度分布的最大值与最小值之间的差值^[20]。根据图1(b)可知，翘曲度定义为一个自由无夹持的晶圆中位面相对于参考平面的最大和最小距离之差，其中，中位面是与晶圆正反表面等距离点的轨迹，参考平面为晶圆中位面的最小二乘法拟合面。由图1(c)可知，弯曲度定义为一个自由无夹持晶圆中位面的中心点与中位面基准面之间的偏离，其中，中位面基准面是由指定的小于晶圆标称直径的直径圆周上的3个等距离点决定的平面^[21]。

由上述定义可知，要想一次性获得晶圆的TTV、翘曲度和弯曲度，首先必须获得晶圆的正反面形貌。

双面抛光晶圆的测量原理如图2所示，被测晶圆被垂直放置在两个菲索式相移干涉仪之间，每个干涉仪都有完整的干涉体系，可以独立测量晶圆的单面形貌。

图3(a)为原理中两个标准镜组成的测量腔横截面，中间是一个晶圆。由于两个干涉仪相对放置，干涉测量所得晶圆正反面形貌中的各点并不满足一一映射关系，为了便于计算，在晶圆正反表面建立右手笛卡尔坐标系，其原点都处于表面的中心位置。根据两个干涉仪同时测量所得晶圆正面形貌图以及反面形貌图沿y轴翻转后的结果可知：

$ \phi_{\rm{B}}\left( {x,y} \right) = \frac{{4{\text{π}} }}{\lambda }\left[ {r_{\rm{B}}\left( {x,y} \right) - t_{\rm{B}}\left( {x,y} \right)} \right]\quad, $ (1)

$ \phi_ {\rm{A}}\left( { - x,y} \right) = \frac{{4{\text{π}} }}{\lambda }\left[ {r_{\rm{A}}\left( { - x,y} \right) - t_{\rm{A}}\left( { - x,y} \right)} \right] \quad,$ (2)

其中 $\phi _{\rm{B}}\left( {x,y} \right)$、 $\phi _{\rm{A}}\left( { - x,y} \right)$分别为不同干涉仪所得晶圆正反面形貌干涉图中的条纹相位， $\lambda $为激光的波长， ${r_{\rm{B}}}(x,y)$、 ${r_{\rm{A}}}( - x,y)$分别为标准镜TF_B和TF_A上各点的面形偏差， ${t_{\rm{B}}}(x,y)$、 ${t_{\rm{A}}}( - x,y)$分别为晶圆正反面上各点的面形偏差，公式(2)中的负号是指将晶圆反面形貌图相对于正面形貌图沿y轴进行了一次翻转。将式(1)~式(2)进行组合运算，可知晶圆的厚度变化 $f\left( {x,y} \right)$以及中位面面形偏差 $s\left( {x,y} \right)$的表达式为：

$ \begin{split} f(x,y) =& {t_{\rm{B}}}(x,y) + {t_{\rm{A}}}( - x,y) = {r_{\rm{B}}}(x,y) + \\ & {r_{\rm{A}}}( - x,y) - \frac{\lambda }{{4{\text{π}} }}\left[ {\phi _{\rm{B}}\left( {x,y} \right) + \phi _{\rm{A}}\left( { - x,y} \right)} \right] \quad, \\ \end{split} $ (3)

$ \begin{split} &s(x,y) = \frac{{{t_{\rm{B}}}(x,y) - {t_{\rm{A}}}( - x,y)}}{2} = \\ & \frac{1}{2}{r_{\rm{B}}}(x,y)- \frac{1}{2}{r_{\rm{A}}}( - x,y) + \frac{\lambda }{{4{\text{π}} }}\left[ {\frac{{{\phi _{\rm{A}}}( - x,y) - \phi_ {\rm{B}}\left( {x,y} \right)}}{2}} \right] \quad. \\ \end{split} $ (4)

一般来说，标准镜的面形偏差量级为纳米，晶圆的中位面面形偏差量级为微米，因此在晶圆中位面面形偏差计算中可以忽略标准镜的影响，其表达式为：

$ \begin{split} s(x,y) = &\frac{{{t_{\rm{B}}}(x,y) - {t_{\rm{A}}}( - x,y)}}{2}\approx \\ &\frac{\lambda }{{4{\text{π}} }}\left[ {\frac{{{\phi _{\rm{A}}}( - x,y) - \phi _{\rm{B}}\left( {x,y} \right)}}{2}} \right]\quad. \\ \end{split} $ (5)

由于晶圆的厚度变化的量级和标准镜面形偏差的量级都为纳米级，在晶圆厚度变化计算中需要考虑标准镜的影响。为此在晶圆测量之后，需要对两个干涉仪的空腔进行检测，图3(b)为两个干涉仪的测量空腔横截面。

利用干涉仪B对两个干涉仪的空腔进行检测，可得到以下关系式：

$ r_{\rm{B}}\left( {x,y} \right) + r_{\rm{A}}\left( { - x,y} \right) = \frac{\lambda }{{4{\text{π}} }}\phi_ {\rm{C}}\left( {x,y} \right)\quad,$ (6)

其中 $ \phi _{\rm{C}}\left( {x,y} \right) $为两个干涉仪的空腔干涉图中的条纹相位。将式(6)代入式(3)可得：

$ \begin{split} f(x,y) = &{t_{\rm{B}}}(x,y) + {t_{\rm{A}}}( - x,y) = \\ &\frac{\lambda }{{4{\text{π}} }}\left[ {\phi _{\rm{C}}\left( {x,y} \right) - \phi _{\rm{B}}\left( {x,y} \right) - \phi _{\rm{A}}\left( { - x,y} \right)} \right]\quad. \\ \end{split}$ (7)

根据公式(5)和公式(7)可知，将晶圆反面形貌图沿y轴翻转后的结果与晶圆正面形貌图进行减法运算，便可得到晶圆的中位面面形偏差形貌图；而将两个干涉仪的空腔形貌图与晶圆正面形貌图以及晶圆反面形貌图沿y轴翻转后的结果进行减法运算，可得到晶圆的厚度变化形貌图。

在获得晶圆厚度变化形貌图和中位面面形偏差形貌图后，根据TTV、Bow和Warp的定义可知：

$ TTV = f\left( {x,y} \right)_{\rm{\max}} - f\left( {x,y} \right)_{\rm{\min}}\quad, $ (8)

$ Bow = s\left( {0,0} \right) - Z_{{\rm{ref}}3}\left( {0,0} \right)\quad, $ (9)

$ \begin{split} Wrap = &\left[ {s\left( {x,y} \right) - Z_{{\rm{ref}}}} \right]_{\rm{\max}} - \\ &\left[ {s\left( {x,y} \right) - Z_{{\rm{ref}}}} \right]_{\rm{\min}} \quad, \\ \end{split} $ (10)

其中 $Z_{{\rm{ref}}3}$、 $Z_{{\rm{ref}}}$分别为 $s\left( {x,y} \right)$的三点拟合平面和最小二乘法拟合平面。

晶圆的厚度变化与中位面面形偏差是通过将晶圆正面形貌图、两个干涉仪的空腔形貌图以及晶圆反面形貌图沿y轴翻转后的结果经过加法或减法运算组合获得的。当两个标准镜之间不存在任何倾斜和偏移时，运算过程中由图像控制器(CCD)记录的三幅形貌图中对应的像素点应该满足一一映射关系。然而在实际测量过程中，由于两个标准镜的相对位置很难精确对准，测量所得晶圆正反面形貌图中对应的像素点并不满足一一映射关系，会引入映射误差，影响测量结果。当两个标准镜之间几乎不存在倾斜时，映射误差的大小与方向由两个标准镜的偏移量所决定。

为了确定映射误差的方向，设标准镜TF_A的圆心 $o'$相对于标准镜TF_B的圆心 $o$在CCD横向和纵向的偏移量分别为 $\varepsilon $、 $\eta $，映射误差分析图如图4所示，其中 $m$和 $n$分别为CCD横向和纵向的像素数目。由于两个标准镜的中心点 $o$、 $o'$ 可能偏离CCD的中心位置，很难通过 $oo'$来确定映射误差的大小。

为了准确获得映射误差的大小，在晶圆测量之前，将三点定位装置固定在两个标准镜之间，由于三点可以形成一个圆域，根据三点在不同标准镜对应CCD中形成的圆域圆心像素坐标，可知两个标准镜的偏离程度。三点在不同标准镜对应CCD中的像素坐标示意图如图5所示。

设三点在TF_B和TF_A对应CCD中的像素坐标分别为 $\left( {X_{{\rm{B}}1},Y_{{\rm{B}}1}} \right)$、 $\left( {X_{{\rm{B}}2},Y_{{\rm{B}}2}} \right)$、 $\left( {X_{{\rm{B}}3},Y_{{\rm{B}}3}} \right)$和 $\left( {X_{{\rm{A}}1},Y_{{\rm{A}}1}} \right)$、 $\left( {X_{{\rm{A}}2},Y_{{\rm{A}}2}} \right)$、 $\left( {X_{{\rm{A}}3},Y_{{\rm{A}}3}} \right)$，形成的圆域圆心像素坐标 $o_1$、 $o_2$分别为 $\left( {X_0,Y_0} \right)$和 $ \left( {X_1,Y_1} \right) $。由三点在两个CCD中的像素坐标可知：

$ \begin{split} &\left( {{X_{{\rm{A}}1}} - {X_{{\rm{A}}2}}} \right)X_1 + \left( {{Y_{{\rm{A}}1}} - {Y_{{\rm{A}}2}}} \right)Y_1 = \\ & \frac{{X_{{\rm{A}}1}^2 - X_{{\rm{A}}2}^2 + Y_{{\rm{A}}1}^2 - Y_{{\rm{A}}2}^2}}{2}\quad, \end{split} $ (11)

$ \begin{split} &\left( {{X_{{\rm{A}}1}} - {X_{{\rm{A}}3}}} \right)X_1 + \left( {{Y_{{\rm{A}}1}} - {Y_{{\rm{A}}3}}} \right)Y_1 = \\ &\frac{{X_{{\rm{A}}1}^2 - X_{{\rm{A}}3}^2 + Y_{{\rm{A}}1}^2 - Y_{{\rm{A}}3}^2}}{2}\quad, \end{split}$ (12)

$ \begin{split} &\left( {{X_{{\rm{B}}1}} - {X_{{\rm{B}}2}}} \right)X_0 + \left( {Y_{{\rm{B}}1} - Y_{{\rm{B}}2}} \right)Y_0= \\ & \frac{{X_{{\rm{B}}1}^2 - X_{{\rm{B}}2}^2 + Y_{{\rm{B}}1}^2 - Y_{{\rm{B}}2}^2}}{2} \quad, \end{split} $ (13)

$ \begin{split} &\left( {{X_{{\rm{B}}1}} - {X_{{\rm{B}}3}}} \right)X_0 + \left( {{Y_{{\rm{B}}1}} - {Y_{{\rm{B}}3}}} \right)Y_0 =\\ &\frac{{X_{{\rm{B}}1}^2 - X_{{\rm{B}}3}^2 + Y_{{\rm{B}}1}^2 - Y_{{\rm{B}}3}^2}}{2} \quad. \end{split}$ (14)

根据上述公式可知 $o_1$、 $o_2$的具体像素坐标，由于CCD的中心像素位置映射于晶圆的坐标系原点，纵向中心线映射于y轴，根据 $o_1$、 $o_2$的像素坐标可知映射误差的大小为：

$ \varepsilon = n - X_1 - X_0 \quad, $ (15)

$ \eta = Y_1 - Y_0 \quad. $ (16)

通过不断调整两个标准镜的位置，可使三点在不同标准镜中形成的圆域圆心的偏移量极小，进而获得极小的映射误差。

为了分析调整后的映射误差对测量结果的影响，使用向量符号表示映射误差 $\varepsilon $、 $\eta $以及晶圆的二维坐标 $\left( {x,y} \right)$，此时晶圆正反面上各点的面形偏差可由以下公式给出：

$ T_{\rm{B}}\left( {\vec x ,\vec y } \right) = s\left( {\vec x ,\vec y } \right) + \frac{1}{2}f\left( {\vec x ,\vec y } \right)\quad, $ (17)

$\begin{split} T_{\rm{A}}\left( {{ -\vec x} ,\vec y } \right) =& s\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) - \\ & \frac{1}{2}f\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right)\quad. \end{split} $ (18)

根据上述公式可知晶圆的厚度变化和中位面面形偏差为：

$ \begin{split} F\left( {\vec x ,\vec y } \right) = &s\left( {\vec x ,\vec y } \right) + \frac{1}{2}f\left( {\vec x ,\vec y } \right)- \\ &s\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) + \frac{1}{2}f\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right), \\ \end{split} $ (19)

$ \begin{split} S\left( {\vec x ,\vec y } \right) = &\frac{1}{2}s\left( {\vec x ,\vec y } \right) + \frac{1}{4}f\left( {\vec x ,\vec y } \right)+ \\ & \frac{1}{2}s\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) - \frac{1}{4}f\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) . \end{split} $ (20)

为了便于得到误差量，将 $ s\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) $、 $ f\left( {\vec x + \vec \varepsilon ,\vec y + \vec \eta } \right) $经过泰勒公式展开到一阶，可知晶圆的厚度变化和中位面面形偏差中的测量误差表达式为：

$ \Delta F\left( {\vec x,\vec y} \right) \approx \vec \nabla s\left( {\vec x,\vec y} \right) \cdot \left[ {\vec \varepsilon ,\vec \eta } \right] + \frac{1}{2}\vec \nabla f\left( {\vec x,\vec y} \right) \cdot \left[ {\vec \varepsilon ,\vec \eta } \right] \quad, $ (21)

$ \Delta S\left( {\vec x,\vec y} \right) \approx \frac{1}{2}\Delta F\left( {\vec x,\vec y} \right) \quad.$ (22)

除了在晶圆的最边缘，厚度变化梯度大多比中位面面形偏差梯度低好几个数量级，因此，晶圆的厚度变化与中位面面形偏差中的误差量主要由中位面面形偏差梯度和映射误差 $\varepsilon $、 $\eta $来决定，极小的映射误差减小了误差量对后续相关参数的影响。由于中位面面形偏差中的测量误差是厚度变化中测量误差的一半，对于良好的厚度变化测量，可以忽略中位面面形偏差中映射误差的影响。

为了验证调整方法的正确性，根据实验室条件，选择一片两寸双面抛光晶圆作为被测样品。晶圆表面相关参数实验测量装置如图6(a)所示，实验测量装置使用两台带有标准镜的两寸菲索式相移干涉仪（干涉仪使用的光源波长为632.8 nm）、一个晶圆夹持器和一个二维支架(用于固定晶圆夹持器和三点定位装置)。

晶圆夹持器的装置图如图6(b)所示，晶圆夹持器中存在一个环形的凹槽以及三个稍微向上倾斜的塑料压片，被测晶圆放置于凹槽中，采用三个塑料压片对晶圆进行竖直夹持，压片与晶圆接触的位置存在很小的应力，该装置在测量过程中对晶圆造成的变形量较小。

三点定位装置实物图如图6(c)所示，三点定位装置中不共线的三点通过三根极细的线所确定。

在晶圆测量之前，首先通过调整两个标准镜的倾斜旋钮来保证两个标准镜在空间位置上尽可能平行，然后将三点定位装置固定在二维支架上，根据映射误差分析中的公式不断调整两个标准镜的位置，可使两个标准镜的偏移程度极小，进而获得极小的映射误差。图7显示了三点在两个标准镜对应CCD所得光强图中的像素坐标，其中CCD横向和纵向的像素数目分别为960和1280，根据图中的三点像素坐标可知晶圆的映射误差 $\varepsilon = 21.592\;{\text{μm}}$、 $\eta = 37.480\;{\text{μm}}$。

在调整好两个标准镜的位置后，将二维支架上的三点定位装置更换为晶圆夹持器。调整二维支架的倾斜旋钮可使由晶圆夹持器夹持的晶圆的倾斜量较小，晶圆几乎处于垂直放置状态，其重力导致的变形量较小，此时利用两个干涉仪对晶圆的正反面同时进行测量，测量所得晶圆正反面形貌图如图8(a)~8(b)(彩图见期刊电子版)所示。在图8(a)中可通过颜色分布分析晶圆的正面形貌，其中正面形貌的PV值（最高峰与最低谷之间的差值）为1.503 μm，RMS值(被检波面相对标准面的均方根偏离)为0.262 μm；图8(b)为晶圆反面形貌图沿y轴翻转后的结果，其中反面形貌的PV值为1.361 μm，RMS值为0.228 μm。

由于实验所得晶圆正反面形貌图中包含标准镜的面形偏差，为了获得更高的测量精度，在晶圆测量之后，利用干涉仪B对两个干涉仪的空腔进行检测，测量所得空腔形貌图如图8(c)(彩图见期刊电子版)所示，其中空腔形貌的PV值为0.024 μm，RMS值为0.002 μm。

根据测量原理将图8中的三幅形貌图进行组合运算，由运算结果可知晶圆的厚度变化和中位面面形偏差形貌图如图9(彩图见期刊电子版)所示。图9(a)为晶圆的厚度变化形貌图，根据定义可知晶圆的TTV为0.198 μm；图9(b)为晶圆的中位面面形偏差形貌图，根据相关定义可知晶圆的弯曲度和翘曲度分别为−0.326 μm和1.423 μm。

为了进一步验证该方法的有效性，利用单个干涉仪B对双面抛光晶圆进行翻转测量，翻转测量实验装置图如图10所示。

由晶圆夹持器夹持的双面抛光晶圆固定在二维支架的中心位置，首先利用干涉仪对晶圆的正面进行检测，通过光学平台上的螺纹孔标记此刻二维支架的左右边缘位置，然后将二维支架进行翻转，翻转后的二维支架放置于两个标记位置之间，再对晶圆的反面进行检测，测量所得晶圆正面形貌图以及反面形貌图沿y轴翻转后的结果如图11(a)~11(b)(彩图见期刊电子版)所示。由于测量原理发生了变化，此时晶圆的厚度变化与中位面面形偏差的表达式变为：

$ \begin{split} f(x,y) =& {t_{\rm{B}}}(x,y) - {t_{{\rm{A}}'}}( - x,y) = \\ &\frac{\lambda }{{4{\text{π}} }}\left[ {\phi _{\rm{A}'}\left( { - x,y} \right) - \phi _{\rm{B}}\left( {x,y} \right)} \right] \quad, \end{split} $ (23)

$ \begin{split} s(x,y) = &\frac{{{t_{\rm{B}}}(x,y) + {t_{{\rm{A'}}}}( - x,y)}}{2} \approx \\ &- \frac{\lambda }{{4{\text{π}} }}\left[ {\frac{{{\phi _{{\rm{A}}'}}( - x,y) + \phi _{\rm{B}}\left( {x,y} \right)}}{2}} \right] \quad, \end{split} $ (24)

其中 $ \phi _{\rm{B}}\left( {x,y} \right) $、 $\phi _{\rm{A}'}\left( { - x,y} \right)$分别为同一干涉仪B测量所得晶圆正反面形貌干涉图中的条纹相位， $\lambda $为激光的波长， $ {t_{\rm{B}}}(x,y) $、 ${t_{{\rm{A}}'}}( - x,y)$分别为同一坐标系下晶圆正反面上各点的面形偏差。由公式(23)~(24)可知，翻转晶圆法所得晶圆的厚度变化与中位面面形偏差不受标准镜精度误差的影响。

将晶圆正反面形貌图根据公式(23)~(24)进行组合运算可知晶圆的厚度变化和中位面面形偏差形貌图如图11(c)~11(d)(彩图见期刊电子版)所示。由于晶圆在两次测量之间发生了漂移，晶圆的厚度变化形貌图中存在波动。根据相关定义可知晶圆的TTV、弯曲度和翘曲度分别为0.208 μm、−0.326 μm和1.415 μm。

将两种方法测量所得晶圆的TTV、弯曲度以及翘曲度进行对比，进一步验证了调整方法的有效性。

翻转晶圆法在测量过程中需要对晶圆的正反面先后进行两次测量，测量所得晶圆正面形貌图与翻转后所得晶圆反面形貌图沿y轴翻转后的结果在空间位置上很难精确对准，会引入对准误差，其误差量受翻转操作的影响。此外，晶圆在两次测量之间还可能发生漂移，从而引入漂移误差，影响测量结果的精度。

所提干涉法在不需要翻转操作的情况下可同样达到消除标准镜误差的作用，还可以防止晶圆在两次测量之间发生漂移，测量过程中引入的误差量较小。

采用两种方法对大批量双面抛光晶圆进行检测时，可以发现：翻转晶圆法在每次测量过程中都需要对二维支架进行翻转和对准，每次测量所得结果中都可能存在不可控的漂移误差和对准误差。而所提干涉法在调整好两个标准镜的位置后，可以方便快速地对大批量双面抛光晶圆进行装卸检测，避免了繁琐的翻转和对准步骤，提高了晶圆的检测效率，同时测量过程中存在较小的误差量。

针对双面抛光晶圆的TTV、弯曲度和翘曲度的测量，本文提出了一种干涉测量方法。使用两个带有标准镜的菲索式相移干涉仪对晶圆正反面同时进行测量，根据测量所得晶圆正反面形貌与两个干涉仪空腔形貌的组合运算，可得到不受标准镜精度误差影响的双面抛光晶圆的表面相关参数。由于两个标准镜在空间位置上很难精确对准，在组合运算过程中会引入映射误差，影响相关参数的测量结果。为了精确调整两个标准镜的相对位置，在晶圆测量之前，将三点定位装置固定在两个标准镜之间，根据三点定圆定理不断调整两个标准镜的位置，可使映射误差极小，进而减小映射误差对相关参数测量结果的影响。实验结果表明，50 mm晶圆横向和纵向的映射误差分别为21.592 μm和37.480 μm，TTV、弯曲度和翘曲度分别为0.198 μm、−0.326 μm和1.423 μm；利用单个干涉仪对晶圆进行翻转测量可知晶圆的TTV、弯曲度和翘曲度分别为0.208 μm、−0.326 μm和1.415 μm。将两组数据进行对比，进一步验证了调整方法的有效性。所提干涉法在调整好两个标准镜的位置后，可以方便快速地对大批量双面抛光晶圆进行装卸检测，避免了繁琐的翻转和对准步骤，同时测量过程中存在较小的误差量，具有良好的应用价值。