As device dimensions keep shrinking, overlay (OVL) metrology has become one of the most critical issues for high-volume semiconductor manufacturing (HVM). According to the International Roadmap for Devices and Systems (IRDS)^[1], the requirement for on-product OVL is reducing to nanometer scale.

The use of an electron beam for metrology often causes irreversible damage to semiconductor devices^[2]. In contrast, optical OVL metrology is widely used for in-line lithography performance monitoring and advanced process control (APC) due to its nondestructive properties, speed, and robustness^[3–5]. There are two methods of optical OVL metrology: image-based overlay (IBO) and diffraction-based overlay (DBO). For many years, IBO metrology has been the workhorse for OVL measurement in HVM. But as the size of semiconductor devices shrinks, it is becoming increasingly difficult for IBO to meet the OVL requirements of advanced technology nodes^[6]. Lens aberrations and illumination imperfections, as well as wafer process variations, can introduce an offset called tool-induced shift (TIS) in IBO^[7,8]. DBO metrology has become a promising method, as it has been shown to have significantly reduced total measurement uncertainty (TMU) compared to the industry standard IBO due to its unmeasurable TIS^[9–11]. It has been reported that DBO offers better precision than IBO^[12]. Moreover, DBO shows no limitation on measuring repeatability (precision)^[13]. Therefore, DBO is considered as the OVL metrology method for the future.

The OVL tolerances for advanced nodes are getting close to 2 nm^[1]. The major contribution is from process-induced error, e.g.,  chemical mechanical polishing (CMP) or etching. The method for optimizing OVL marks before lithography is introduced for generating robust OVL marks that can accommodate various process perturbations, including lithography processes^[14,15]. However, the above-mentioned method has taken place only before the exposure process, and process-induced asymmetrical deformation of the OVL mark frequently happens^[7]. Thus, the research on OVL metrology that focuses on how to improve measurement accuracy for asymmetrical gratings that occur after the exposure process is also essential.

In this work, we theoretically show that DBO metrology sensitivity can be enhanced further by matching the measuring light wavelength to effective optical thickness of the material between two gratings. Moreover, we demonstrate that collecting high-order diffracted light intensity differences can help reduce an asymmetrical mark’s impact on measurement error. These findings can help improve the accuracy, sensitivity, and robustness of the DBO method and equipment.

Diffraction gratings are widely used as optical components that allow periodic spatial modulation of the amplitude or phase (or both) of the incident light. In semiconductor manufacturing, DBO metrology is commonly used to measure the alignment of diffraction gratings on two layers located at different heights above a substrate by analyzing the intensity difference between the two diffracted light signals.

The wave-vector component of the light with the incident angle θin in the x direction is |k→in|sinθin, where |k→in|=2π/λ (λ is the wavelength). The relation between the incident and reflected light in the x direction is given in the grating equation^[16,17], |k→in|sinθin+|k→out|sinθout=n·2πp.(1)p is the grating period; n is the diffraction order. The incident and reflected angle with respect to the normal can be expressed as sinθin+sinθout=n·λp.(2)

The complex amplitude of the optical wave field can be described as U(r,t)=A(r)exp(±jφ),(3)where φ=k→·r→−ωt+φ0,(4)where k→·r→ is the space phase factor, ωt is the time phase factor, and φ0 is the initial phase.

The model using scalar diffraction theory to describe the signal formation of DBO metrology ignores polarization effects and the impact of multiple diffraction orders.

Take ±1 diffraction order as an example. A and B are the amplitudes of the reflected lights from the top and bottom gratings. It can be seen from Fig. 1 that α is the spatial phase factor, which can be expressed as α=2πΔlp,(5)where Δl is the OVL between two gratings in the x direction. If the detector collects the diffracted light of the two sets of grating structures in the vertical direction, the phase shift due to the optical path difference β can be expressed as β≈4πhλ,(6)where h is an effective optical thickness between the bottom and top gratings. I−1 and I+1 represent the −1, +1st diffraction orders from top and bottom gratings, which can be described as $$\begin{gathered} I−1/+1=[Aexp(∓jα)+Bexp(jβ)]·[Aexp(∓jα)+Bexp(jβ)]* \\ =A2+B2+ABexp(∓jα−jβ)+ABexp(±jα+jβ) \\ =A2+B2+2ABcos(β±α). \end{gathered}$$(7)

The intensity difference of the ±1 order diffraction light can be written as $$\begin{gathered} I+1−I−1=2AB[cos(β−α)−cos(α+β)] \\ =4ABsinαsinβ. \end{gathered}$$(8)

It can be seen from Eq. (8) that the intensity difference between the ±1 diffraction orders is proportional to sinβ. The largest intensity difference is obtained when the absolute value of sinβ equals to 1, meaning maximum OVL sensitivity. The condition for β can be found at β=π2+qπ,q∈z.(9)

Therefore, the optimal light wavelength λ for obtaining maximum OVL sensitivity can be calculated as λ≈8h2q+1,q∈z.(10)

From Eq. (10), one can select multiple optimal measuring wavelength for a certain effective optical thickness to make sinβ=1. However, the amplitude of light diffracted from the top and bottom gratings [A and B in Eq. (8)] also affects the measurement sensitivity, which is wavelength-dependent. In order to maximize the intensity difference in Eq. (8), one can choose the measuring wavelength that has the smallest absorption for the material. In this case, the product of A and B in Eq. (8) can also reach its maximum. Therefore, the optimal light wavelength for measuring certain material with a confined thickness can be selected by taking into account the two above considerations.

For ADI OVL metrology, it is generally believed that the top grating profile is made of photoresist or antireflection coating, which is uniform. However, the bottom grating profile processed after CMP, etch, and other processes, is usually nonuniform and asymmetrical^[21]. We use mathematical methods to investigate how asymmetrical gratings affect DBO metrology.

OVL marks have a period of p1 and a width of p1/2. The top grating is uniform, but the bottom grating is deformed (the outline is curved). The OVL between the two marks is 0.

There is no ideal phase grating in reality due to absorption, especially for thick intermediate layers^[22,23]. Therefore, we made a mixed grating setting of amplitude and phase grating to keep the possibility of extending the model for universal applications when absorption cannot be ignored. The amplitude between x∈[−p1/4,p1/4] is normalized to 1. The amplitude elsewhere can be set as a nonzero value when the exact absorption is known, which does not change the result but only provides an offset. Here, we set it to 0 for clarity consideration. Thus, the amplitude AM(x) and phase ϕM(x) of the OVL mark function are defined as follows: $$\begin{gathered} AM(x)={1,x∈[−p14,p14]0,others, \\ ϕM(x)={ax2+bx+cp1,x∈[−p14,p14]0,others, \\ Asymmetric mark,b≠0. \end{gathered}$$(11)

Here we ignored the absorption difference due to the thickness differences (d in Fig. 2) at the asymmetrical deformation because d is normally only a few nanometers.

It is expanded by Fourier series in the period p1. The nth-order component is examined, as follows: An(x)=Cnexp(j2πnxp1)+C−nexp(−j2πnxp1),(12)C±n=1p1∫−p1/2p1/2AM(x′)exp[jϕM(x′)]exp(∓j2πnx′p1)dx′.(13)

By substituting the AM(x) and ϕM(x) into Eq. (13), $$\begin{gathered} C±n=1p1∫−p1/4p1/4exp(jax2+bx+cp1)exp(∓j2πnxp1)dx \\ =1p1∫−p1/4p1/4exp[jax2+(b∓2πn)x+cp1]dx \\ =1p1∫−p1/4p1/4exp[j(ax+b∓2πn2a)2+c−(b∓2πn2a)2p1]dx \\ =exp[jc−(b∓2πn2a)2p1]p1∫−p1/4p1/4exp[j(ax+b∓2πn2a)2p1]dx \\ =exp[jc−(b∓2πn2a)2p1]p1p1ja∫−p1/4p1/4exp[jp1(ax+b∓2πn2a)]2 \\ ⁢d[jp1(ax+b∓2πn2a)], \end{gathered}$$(14)

and C+n and C−n can be written as $$\begin{gathered} C±n=exp[jc−(b∓2πn2a)2p1]p1p1ja∫−p1/4p1/4exp[jp1(ax+b∓2πn2a)]2 \\ ⁢d[jp1(ax+b∓2πn2a)]. \end{gathered}$$(15)

The square difference between C+n and C−n is defined as the error signal. It can be seen from the equation that the error signal is nonzero if the grating has an asymmetrical deformation, which induces OVL measurement error. When b=0, regardless of the value of a, the grating profile is symmetrical, which means no error signal. Here, the quadratic term of ax2 is kept because the model can also be applied in investigating the asymmetrical deformation of applications, e.g.,  the roughness metrology of the microstructure of the extreme ultra violet (EUV) collector mirror when diamond turning is used.

Asymmetrical bottom grating causes intensity difference between the positive and negative orders even if the OVL is 0. We consider the following two cases to see how to overcome this issue.

The CMP process usually creates linear deformation. Thus, a linear model is used to describe the situation in ADI and AEI. Assuming an OVL mark with a period of p2 and a width of p2/2, the bottom grating has a linear deformation, and the OVL sets to 0, as shown in Fig. 3.

The amplitude AM1(x) and phase ϕM1(x) of the OVL mark function are defined as $$\begin{gathered} AM1(x)={1,x∈[−p24,p24]0,others, \\ ϕM1(x)={wxp2,x∈[−p24,p24]0,others. \end{gathered}$$(16)

It is expanded by Fourier series in the period p2. And the nth-order component is examined, combined with Eqs. (12) and (13). Substituting the AM1(x) and ϕM1(x) into Eq. (13), $$\begin{gathered} C±n=1p2∫−p2/4p2/4exp(jwx′p2)exp(∓j2πnx′p2)dx′ \\ =1j(w∓2πn)[exp(jw∓2πn4)−exp(−jw∓2πn4)] \\ =2w∓2πnsin(w∓2πn4), \end{gathered}$$(17)

C±n may be written as C±n=2w∓2πnsin(w∓2πn4),(18)so that the error signal can be formulated as |C+n|2−|C−n|2=4(w−2πn)2sin2(w−2πn4)−4(w+2πn)2sin2(w+2πn4).(19)

Note that no matter what n is, the values of the two sine function in Eq. (19) are equal, so $$\begin{gathered} |C+n|2−|C−n|2=[4(w−2πn)2−4(w+2πn)2]sin2(w−2πn4) \\ =32πwn(w2−4π2n2)2sin2(w−2πn4) \\ =32πw3n−8π2wn+16π4wn3sin2(w−2πn4). \end{gathered}$$(20)

For odd order n, the error signal can be described as |C+n|2−|C−n|2=32πw3n−8π2wn+16π4wn3cos2(w4).(21)

For different odd orders (n=1,3,5), the error signal can be written as {|C+1|2−|C−1|2=32πw3−8π2w+16π4wcos2(w4)|C+3|2−|C−3|2=32πw33−24π2w+432π4wcos2(w4)|C+5|2−|C−5|2=32πw35−40π2w+2000π4wcos2(w4).(22)

For even order n, the error signal can be described as |C+n|2−|C−n|2=32πw3n−8π2wn+16π4wn3sin2(w4).(23)

For different even orders (n=2,4,6), the error signal can be written as {|C+2|2−|C−2|2=32πw32−16π2w+128π4wsin2(w4)|C+4|2−|C−4|2=32πw34−32π2w+1024π4wsin2(w4)|C+6|2−|C−6|2=32πw36−48π2w+3456π4wsin2(w4).(24)

In semiconductor manufacturing, the typical absolute value of w is less than 0.1. As shown in Fig. 4, the error signal decreases rapidly with the increase of diffraction order. It is remarkable to see that the employment of even orders would help reduce asymmetrical mark impact on measurement error significantly compared to the odd orders. Collecting the second-order diffraction signal to calculate the OVL is suitable in order to meet the diffraction efficiency requirements of measurement as well to improve the accuracy and robustness of DBO metrology. Moreover, even-order diffraction signals are more sensitive to the change of w, which can be used to monitor subtle changes in the asymmetrical deformation during the manufacturing process.

For AEI OVL metrology, both the top and bottom gratings are affected after process steps (such as CMP and etching), and the grating profiles of both layers are asymmetrical. Due to the absorption of the photoresist layer and the antireflection coating, the amplitude of the top layer diffracted light is larger than that of the bottom layer diffracted light. So η=Atop/Abottom, η>1. Assuming an OVL mark with a period of p₃ and a width of p₃/2, both top and bottom grating profiles are linearly deformed, as described in Fig. 5. The OVL error sets to 0.

The amplitude AM2(x) and phase ϕM2(x) of the OVL mark function are defined as $$\begin{gathered} AM2bottom(x)={1,x∈[−p34,p34]0,others,AM2top(x)={η,x∈[−p34,p34]0,others. \\ ϕM2bottom(x)={γxp3,x∈[−p34,p34]0,others,ϕM2top(x)={εxp3,x∈[−p34,p34]0,others. \end{gathered}$$(25)

For the deformation of the top grating, it is expanded by Fourier series in the period p3, and the nth-order component is examined and it is brought into Eq. (13) to obtain $$\begin{gathered} C±ntop=ηp3∫−p3/4p3/4exp(jεx′p3)exp(∓j2πnx′p3)dx′ \\ =ηj(ε∓2πn)[exp(jε∓2πn4)−exp(−jε∓2πn4)] \\ =ηε∓2πnsin(ε∓2πn4). \end{gathered}$$(26)

C±ntop can be expressed as C±ntop=2ηε∓2πnsin(ε∓2πn4),(27)

C±nbottom can be expressed as C±nbottom=2γ∓2πnsin(γ∓2πn4),(28)so that the light intensity can be formulated as $$\begin{gathered} I±n(x)=[C±ntopexp(jϕ1)+C±nbottomexp(jϕ2)]·[C±ntopexp(−jϕ1)+C±nbottomexp(−jϕ2)] \\ =|C±ntop|2+|C±nbottom|2+C±ntopC±nbottom[exp(jϕ1−jϕ2)+exp(jϕ2−jϕ1)] \\ =4η2(ε∓2πn)2sin2(ε∓2πn4)+4(γ∓2πn)2sin2(γ∓2πn4) \\ +8η·sin(ε∓2πn4)sin(γ∓2πn4)cos(ϕ1−ϕ2)(ε∓2πn)(γ∓2πn). \end{gathered}$$(29)

It is known from Eqs. (5) and (6) that ϕ1 is equal to 0 due to OVL error of 0. The value of ϕ2 depends on the relationship between the wavelength of the measuring light and the thickness between two layers, $$\begin{gathered} I+n−I−n=32πη2ε3n−8π2εn+16π4εn3sin2(ε−2πn4)+32πγ3n−8π2γn+16π4γn3sin2(γ−2πn4) \\ +32ηcosϕ2sin(ε−2πn4)sin(γ−2πn4)⁢(ε+γ)πn(ε2−4π2n2)(γ2−4π2n2). \end{gathered}$$(30)

In order to increase the sensitivity of the OVL signal, the following conditions usually need to be satisfied: ϕ2=π2+qπ,q∈z.(31)

Equation (30) turns out to be I+n−I−n=32πη2ε3n−8π2εn+16π4εn3sin2(ε−2πn4)+32πγ3n−8π2γn+16π4γn3sin2(γ−2πn4).(32)

As shown in Fig. 6(a), the error signals are all decreasing and then increasing with increasing amplitude ratio η. When the error signal is minimum, the amplitude ratio for the even order is larger than that for the odd order, which means that even signals are more suitable for measuring thicker AEI samples that cause absorption of the reflected light from the bottom layer. As shown in Fig. 6(b), the error signal increases with increasing amplitude ratio η. Even orders have a smaller error signal compared to odd orders. But odd order diffraction signals are more sensitive to the change of η.

In general, cosϕ2 is not equal to 0 because the wavelength of the measuring light is not optimal to match the thickness between the two layers. In this situation, F(ε,γ)=sin(ε−2πn4)sin(γ−2πn4)(ε+γ)πn(ε2−4π2n2)(γ2−4π2n2).(33)

If ε,γ are opposite numbers, the function F(ε,γ) obtains a minimum. It can be seen in Fig. 7 that there is always a minimum value for the error signal in Eq. (30).

If ε,γ are the same sign, it is known from Figs. 6(b) and 8 that the first two and the third terms in Eq. (30) are approximately of the same order of magnitude. Thus, the third term can be adjusted to produce a minimum of Eq. (30), which can be done by adjusting the wavelength of the measuring light.

In principle, the method proposed in this work is a general method of dealing with any possible asymmetrical deformation (e.g.,  linear, quadratic, cubic) in the manufacturing process while scalar theory is valid (the period of grating and the wavelength of measuring light meet the requirement of p>5λ). Therefore, the theory would be suitable for short light wavelength OVL metrology methods and equipment, which is the research trend of this field.

In this work, we use mathematical methods to improve measurement sensitivity and reduce the error signal for DBO metrology. The DBO method has the advantages of high precision, deep sampling inside the circuit, and low TMU. However, processes such as CMP and etching can cause an asymmetrical profile of the OVL mark, which can affect measurement accuracy. In this work, it is shown that measurement sensitivity is affected by the measured light wavelengths and the thicknesses of the two layers. This means that high measurement sensitivity can be achieved by adjusting the wavelength of the measuring light. In some cases, the effect of the error signal caused by the asymmetrical profile of the OVL mark can also be reduced by adjusting the wavelength of the measuring light. Additionally, we found theoretically that the error signal induced by the asymmetrical grating decreases rapidly with the increase of the diffraction light order. Making the correct choice of measuring odd or even orders depends on the user; this choice can significantly reduce the effect of the asymmetrical mark-induced error. This opens up a new way to improve the accuracy of DBO.