Research Progress of Inverse Lithography Technology
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摘要: 反演光刻技术(ILT)相比传统的光学临近效应修正(OPC)，生成的掩模具有成像效果更好，工艺窗口更大等优点，在当前芯片制造的工艺尺寸不断减小的背景下，逐渐成为主流的光刻掩模修正技术。该文首先介绍了反演光刻算法的基本原理和几种主流实现方法；其次，调研了当前反演光刻技术应用在光刻掩模优化问题上的研究进展，分析了反演光刻技术的优势和存在的问题。以希望为计算光刻及相关研究领域的研究人员提供参考，为我国先进集成电路产业的发展提供技术支持。Abstract:
Objective Inverse Lithography Technology (ILT) provides improved imaging effects and a larger process window compared to traditional Optical Proximity Correction (OPC). As chip manufacturing continually reduces process dimensions, ILT has become the leading lithography mask correction technology. This paper first introduces the basic principles and several common implementation methods of the reverse lithography algorithm. It then reviews current research on using reverse lithography technology to optimize lithography masks, as well as analyzes the advantages and existing challenges of this technology. Methods The general process of generating mask patterns in ILT is exemplified using the level set method. First, the target graphics, light sources, and other inputs are identified. Then, the initial mask pattern is created and a pixelated model is constructed. A photolithography model is then established to calculate the aerial image. The general photoresist threshold model is represented by a sigmoid function, which helps derive the imaging pattern on the photoresist. The key element of the ILT algorithm is the cost function, which measures the difference between the wafer image and the target image. The optimization direction is determined based on the chosen cost function. For instance, the continuous cost function can calculate gradients, enabling the use of gradient descent to find the optimal solution. Finally, when the cost function reaches its minimum, the output mask is generated. Results and Discussions This paper systematically introduces several primary methods for implementing ILT. The level set method’s main concept is to convert a two-dimensional closed curve into a three-dimensional surface. Here, the closed curve is viewed as the set of intersection lines between the surface and the zero plane. During the ILT optimization process, the three-dimensional surface shape remains continuous. This continuity allows the ILT problem to be transformed into a multivariate optimization problem, solvable using gradient algorithms, machine learning, and other methods. Examples of the level set method’s application can be found in both mask optimization and light source optimization. The level set mathematical framework effectively addresses two-dimensional curve evolution. When designing the ILT algorithm, a lithography model determines the optimization direction and velocity for each mask point, employing the level set for mask evolution. Intel has proposed an algorithm that utilizes a pixelated model to optimize the entire chip. However, this approach incurs significant computational costs, necessitating larger mask pixel sizes. Notably, the pixelated model is consistently used throughout the process, with a defined pixelated cost function applicable to multi-color masks. The frequency domain method for calculating the ILT curve involves transforming the mask from the spatial domain into the frequency domain, followed by lithography model calculations. This approach generates a mask with continuous pixel values, which is then gradually converted into a binary mask through multiple steps. When modifying the cost function in frequency domain optimization, all symmetric and repetitive patterns are altered uniformly, preserving symmetry. The transition of complex convolution calculations into multiplication processes within the frequency domain significantly reduces computational complexity and can be accelerated using GPU technology. Due to the prevalent issue of high computational complexity in various lithography mask optimization algorithms, scholars have long pursued machine learning solutions for mask optimization. Early research often overlooked the physical model of photolithography technology, training neural networks solely based on optimized mask features. This oversight led to challenges such as narrow process windows. Recent studies have, however, integrated machine learning with other techniques, enabling the physical model of lithography technology to influence neural network training, resulting in improved optimization results. While the ILT-optimized mask lithography process window is relatively large, its high computational complexity limits widespread application. Therefore, combining machine learning with the ILT method represents a promising research direction. Conclusions Three primary techniques exist for optimizing masks using ILT: the Level Set Method, Intel Pixelated ILT Method, and Frequency Domain Calculation of Curve ILT. The Level Set Method reformulates the ILT challenge into a multivariate optimization issue, utilizing a continuous cost function. This approach allows for the application of established methods like gradient descent, which has attracted significant attention and is well-documented in the literature. In contrast, the Intel method relies entirely on pixelated models and pixelated cost functions, though relevant literature on this method is limited. Techniques in the frequency domain can leverage GPU operations to substantially enhance computational speed, and advanced algorithms also exist for converting curve masks into Manhattan masks. The integration of ILT with machine learning technologies shows considerable potential for development. Further research is necessary to effectively reduce computational complexity while ensuring optimal results. Currently, ILT technology faces challenges such as high computational demands and obstacles in full layout optimization. Collaboration among experts and scholars in integrated circuit design and related fields is essential to improve ILT computational speed and to integrate it with other technologies. We believe that as research on ILT-related technologies advances, it will play a crucial role in helping China’s chip industry overcome technological bottlenecks in the future. -
1. 引言
反演光刻技术(Inverse Lithography Technology, ILT)的基本思路是，先确定期望得到的曝光后的图形z(x,y)，然后找出光刻模型传输函数的逆变换T –1，将z(x,y)带入T –1，逆向推出掩模板的函数m(x,y)，如图1所示，其中z'(x,y)表示目标图形。对于反演光刻技术，有可能找到的局部最优解不是全局最优解，全局最优解也可能存在多个。这就需要采用一些方法寻找低复杂度、高保真度和高对比度的解[1]。
早在1990年，加利福尼亚大学的Liu等人[2, 3]就开始对ILT算法进行研究，提出了分枝定界法(branch-and-bound)、单纯形法(the simplex method)、“细菌”算法(“Bacteria” algorithm)等。
2001年，IBM的Rosenbluth等人[4]在光源掩模协同优化(Source Mask Optimization, SMO)优化中使用了ILT优化掩模。之后出现了很多对ILT的研究[5]。然而，这些早期的ILT算法通常需要消耗很长的时间进行优化，无法应用在芯片生产中。
Luminescent公司提出了基于level-set的方法，并发表了一系列论文[6]，将这种方法正式命名为“反演光刻技术” (ILT)。Luminescent公司及其合作伙伴发表了大量论文，证明了ILT优化在工艺窗口(Process Window, PW)上的优势，解决掩模制造相关问题，并探索ILT的应用，如设计规则优化[7]。
2007年，Schenker等人[8]提出了基于像素的算法，并在建模、计算、掩模制作和检查以及集成芯片制造技术等方面发表了一系列论文。
Gauda公司提出了在频域中计算光学临近效应修正(Optical Proximity Correction, OPC)的方法[9]，随后又把这种方法应用到ILT算法中[10]，通过利用GPU计算大幅提高了ILT的计算速度。
Poonawala等人[11]开发了一个基于像素的连续函数公式的ILT优化框架，非常适合基于梯度的算法。
国内相关学者针对ILT也做了很多研究。浙江大学的研究团队开发了基于像素的梯度方法解决ILT问题[12]。清华大学的一个研究团队开发了一种不依赖初始条件的方法，并探索了使用GPU进行ILT优化[13,14]。香港大学的研究团队，对ILT进行了广泛的研究[15]，特别是关于ILT掩模解决方案的正则化，以满足掩模制造要求[16]，并提出用遗传算法自动优化掩模和照明，还将机器学习应用到ILT优化中[17]。复旦大学的Sun等人[18]提出使用多级分辨率的方法提高ILT计算速度。
近几年，机器学习结合ILT的方法成为了研究的热点。2021年，Ciou等人[19]在先进的3D NAND闪存的Via层上研究了生成式对抗网络，分别使用生成对抗性网络(Generative Adversarial Network, GAN)模型pix2pix和cycleGAN合成ILT图像，显著提高了亚分辨率辅助图形(Sub-Resolution Assist Features, SRAF)插入效率。2024年，Xu等人[20]提出了基于Swin注意力机制(Swin Transformer, ST)的OPC框架SwinT-ILT，该框架利用Swin Transformer强大的特征提取能力，直接为给定的目标布局生成优化掩模，实现了快速生成掩模并有较好的可制造性。2024年，Ceepcikay等人[21]开发了一个端到端流程，通过多条件生成对抗网络(Poly-GAN)无缝地将模型训练和应用集成到全芯片基于模型的亚分辨率辅助图形(Model-Based SRAF, MB-SRAF)生成和优化中，证明了机器学习反演光刻(Machine Learning ILT, ML-ILT)解决方案能够复制ILT的光刻优化质量并且计算速度提高20倍。
目前，针对曲线掩模(curvilinear mask)的优化方法逐渐成为热门的研究方向[22–24]，因为曲线掩模能显著提高晶圆的工艺窗口并且减少可能出现的缺陷[25]。基于像素的ILT是生成曲线掩模的一种理想方法[26–29]。随着多波束掩模写入器等技术的发展，掩模规则检查(Mask Rule Check, MRC)等流程也可以是基于像素的[30]，ILT生成掩模的可制造性能够显著提高[31–33]，因此ILT技术的研究在芯片性能、面积、产量和成本等方面的发展上有很高的潜在效益[34,35]。
本文结合几种不同的ILT实现方法，对ILT的基本原理和研究进展进行了介绍，分析了反演光刻技术的优势和存在的问题，希望能为计算光刻及相关研究领域的研究人员提供参考。
2. 反演光刻技术概述
ILT生成掩模图形的一般流程如图2所示。首先确定目标图形、光源等输入，确定初始掩模图形，并构造像素化模型。建立光刻模型，计算出空间像图形，一般用sigmod函数表示光刻胶阈值模型，得到光刻胶上的成像图形，然后设定一个代价函数(cost function)，即成像区域内晶圆上的图形与目标图形的差值，根据代价函数的不同选择确定优化方向的算法，例如代价函数是连续函数，可以计算梯度，利用梯度下降的方法求最优解。梯度下降方法也有不同的实现方法，如最速下降法(steepest descent)、动量法(momentum)、自适应梯度(adaptive gradient)和自适应矩估计(adaptive moment estimation)等方法[36]。
利用迭代的方法，每次迭代计算优化方向后按照设定的步长对掩模进行修正，直到找出满足要求的掩模。代价函数中可以包括许多附加元素，如掩模制造误差及其对晶片印刷的影响等[37–39]。
初始掩模图形的选取会影响迭代次数和最终优化后的掩模图形，通常采用连续传输掩模(Continuous Transmission Mask, CTM)的方法确定初始掩模。CTM与ILT算法类似，但是每次迭代后不再得到二值化的掩模，而是得到每个像素值均为连续值的掩模，代入光刻模型中计算成像图形和梯度。相比ILT算法，对掩模的要求条件降低，因此收敛速度较快，但仍然需要较长的计算时间。
3. 反演光刻技术研究进展
3.1 水平集方法
水平集方法(level-set method)是应用数学的一个分支，由Stan Osher和James Sethian[40]于1988年提出。Luminescent于2005年提出了基于水平集方法的ILT，以提高掩模优化效率，降低复杂性，从而减少运行时间[6,41–46]。
level set方法的主要思路是将2维封闭曲线转换为3维曲面，将2维封闭曲线看做3维曲面与零平面交线的集合，因此称为水平集。ILT优化过程中3维曲面图形的函数一直连续，能够把ILT问题转化为多变量优化问题，应用梯度算法、机器学习等方法进行求解。在掩模优化与光源优化中都有使用Level set方法的例子[41,47]。level set这种数学方法解决了2维曲线演化的问题，设计ILT算法时依靠光刻模型确定掩模上每个点的优化方向和速度，利用level set实现掩模的演化[48]。如图3所示，掩模上的一个主图形在优化过程中出现2个SRAF，这个过程在2维曲线的演化上发生了突变，由1个封闭曲线变为3个封闭曲线，而构造的3维曲面始终是一个连续曲面。
水平集方法构造3维曲面，将由封闭曲线构成的光刻图形转换为曲面与零值平面的交线。SRAF的出现使2维函数变得不连续，利用level-set method转化成的3维曲面可以是连续的，就能够使ILT优化可被表述为一般的多变量优化问题，并使用标准优化算法(如共轭梯度法)进行求解。
生成掩模图形的一般流程如图4所示。将目标图形像素化，根据像素化的模型生成3维曲面，2维图形转化为曲面与零平面的交线，然后用正向方法计算成像到晶圆上的图形，定义代价函数，使用梯度等方法确定优化方向，使用level set方法找出最优掩模图形。
在目标图形非常密集的情况下，无法通过插入SRAF改进聚焦深度。Level set ILT方法可以通过优化主图形提高不同焦距下的图案保真度，帮助处理密集线/空间阵列中的不规则部分。如图5所示，在目标图形非常密集的情况下，使用OPC优化后，不同聚焦深度下成像图形会发生变形，ILT优化的结果优于OPC。
如图6所示，利用ILT插入SRAF可以大幅提高聚焦深度。
与OPC方法类似，ILT方法生成的掩模图形及优化后的成像效果，很大程度上取决于使用的光源。如图7所示，相同的目标图形，采用不同的光源，经过Level set ILT优化后，得到了不同的掩模图形，同时工艺窗口也有很大差别。
3.2 Intel像素化方法
Intel提出一种使用像素化模型优化整个芯片的算法，计算量比较大，因此掩模每个像素尺寸比较大(100 nm/pixel)，特点是全程使用像素化模型，定义像素化Cost函数，可以应用于多色掩模。
这种算法首先对目标图形进行像素化。初始目标图形由多边形组成，利用相位着色算法对多边形进行着色，相位着色算法可能会导致相位冲突，依靠像素优化来解决这些相位冲突。像素在迭代优化流中进行优化，得到聚合像素分布。Intel像素化ILT的基本流程如图8所示。
对目标掩模的设计有要求：(1)现实的并且可实现；(2)数值化光刻胶阈值；(3)空间像变化速度的控制参数化(不同的掩模类型导致不同的空间像变化速度)。因此目标掩模写为式(1)。
T(x,y)=f(tresist,B(x,y),δ,γ) (1) 其中，tresist表示成像轮廓对应的空间像二值化阈值，B(x,y)是二值化掩模图形，δ是保真度控制参数，γ是正对比度参数。
Intel像素化ILT方法的特点是全程使用像素化模型，将成像轮廓阈值参数和设计图像斜率的控制参数等写入到优化目标函数中，定义的代价函数为成像结果与优化目标的差值，同样是像素化模型，而不是连续的函数。
这种方法还可以利用相位着色算法对目标图形进行着色，生成3种或3种以上颜色的掩膜图形，利用更复杂的掩模制造技术降低ILT算法复杂度。
图9为采用双色掩模的优化结果，图9(a)为目标图形，图9(b)为优化后的掩模。
3.3 频域中计算曲线 ILT的方法
频域中计算曲线 ILT的方法，将掩模从空间域转换到频域，在频域中进行光刻模型计算，使用霍普金斯公式给出的闭合形式积分建模，将方程转换为光学系统传递函数与掩模函数M之间的4维卷积，利用时域卷积等价于频域相乘，将卷积运算转换为更适合并行运算的傅里叶变换和乘法运算[9]。如果在频域优化中修改代价函数，则所有对称模式和重复模式将以相同的方式进行修改，因此，将自然保持对称性。复杂的卷积计算转化为在频域中的乘法计算，能够大幅减少计算量，并且可以使用GPU加速。
这种方法首先生成像素值连续的掩模，然后再分多个步骤将连续掩模逐渐转化为二值化掩模。如图10所示，图10(b)为连续掩模，通过分步在目标函数中加入二值化项，使用频域光刻模型进行修正，将连续掩模依次转化为图10(c)和图10(d)所示逐渐接近二值化的掩模，最终输出图10(e)所示的完全二值化掩模。总目标函数可以表示为
Ftotal=(1−η)Fcont(M)+ηFbinary(M,Mc)+Fmask(M) (2) 其中，Fcont是连续掩模传输适应度函数，用于计算从掩模M生成的轮廓与目标图形L的相似程度。Fbinary是二值适应度函数，用于二值化掩模，是一个连续可微函数。Fmask是掩模适应度函数，用于实现完全二值化。
图11为OPC与频域中计算曲线 ILT方法的对比，工艺窗口标为绿色。可以看到，与OPC相比，ILT可以将工艺窗口增加100%以上。
Torunoglu等人[10]利用频域中计算曲线 ILT方法优化一个45 nm工艺尺寸，10 mm×10 mm的芯片，将版图划分为多个15 360 nm×15 360 nm的小区域，波长为λ=193 nm, NA=1.35，光源为C-quad, σin=0.53, σout=0.98, defocus: ±100 nm, 5% intensity variation。采用17核Intel CPU, 17 Nvidia GTX 295 GPUs, 16 GB 内存，1 TB硬盘，运行系统为Linux Open Suse 11.1。对整个芯片的ILT优化需要122.4 h。
3.4 结合机器学习的ILT方法
由于不同的光刻掩模优化算法普遍存在计算量较大的问题，相关学者很早就开始研究利用机器学习的方法解决掩模优化问题[53–57]。早期的研究并没有考虑光刻技术的物理模型，仅从优化后掩模的特征上对神经网络进行训练，因此会存在工艺窗口较低等问题[58–61]。近年来已经出现将机器学习方法与其他方法相结合的研究，从而使光刻技术的物理模型影响神经网络的训练模型，达到更好的优化效果。ILT方法优化得到的掩模光刻工艺窗口较大，但计算量太大始终难以大规模应用，因此结合机器学习的ILT方法是一个很好的研究方向[20,21,62–65]。
Ma等人[66]提出的模型驱动神经网络(Model-driven Convolution Neural Network, MCNN)[67]可以提高ILT的计算速度并降低图形误差(Pattern Error, PE)。MCNN并不是继承自现有深度学习架构，而是从通用的ILT模型中派生而来。基于梯度的ILT算法计算流程如图12所示，MCNN是通过展开和截断ILT的迭代来构建的，将ILT的每次迭代作为MCNN的一层，一层的输出作为下一层的输入，如图13所示。光刻成像模型被用作解码器，这允许以无监督的方式训练MCNN。
GAN[54,68,69]以及条件生成对抗性网络(Conditional GAN, CGAN)是实现机器学习与ILT结合的一种常见方法。GAN直接将随机输入向量映射到重建图像，使用对抗性学习生成图像。一种使用CGAN实现ILT的算法框架如图14所示[54,70]。GAN包含一个生成器(Generator)和一个鉴别器(Discriminator)。生成器用于基于输入目标图像生成像素化ILT掩模，鉴别器用于鉴别是使用传统ILT导出的ILT掩模还是生成器生成的掩模。在CGAN中，鉴别器基于特定的目标片段来计算生成的掩模是传统ILT掩模的概率，将训练神经网络问题转换为极小极大问题。
在GAN和CGAN基础上直接生成最终输出掩模，在计算速度上有很大优势，但完全不考虑光刻成像的物理模型，因此对于部分案例可能无法达到传统ILT的成像效果。
Zhang等人[71]提出了一种模型驱动的图形卷积网络(Model-driven Graph Convolutional Network, MGCN)框架，编码器使用图形卷积网络(GCN)，而解码器基于光刻成像物理过程，通过编码器和解码器之间的协作，实现无监督的训练策略。Ma等人[72]提出了一种基于逆向光刻物理的深度神经水平集方法(Inverse Lithography physics-informed Deep neural Level Set, ILDLS)，如图15所示，他们通过链式规则使用基于水平集的逆向光刻校正层为深度学习模型的神经元添加权重。在机器学习的基础上考虑光刻的物理模型无疑是实现机器学习与ILT方法相结合的一个很好的研究方向。
4. ILT的优势和存在的问题
由于ILT非常复杂，计算量庞大，早期ILT技术只能用于对处理芯片版图的局部进行处理。对于整个芯片，先使用OPC技术以及插入SRAF，完成掩模版数据的处理，然后找出不符合要求的部分，把这些部分截取出来，局部做ILT处理。
即使ILT在全芯片的应用上存在很多困难，对ILT的研究一直是计算光刻领域的研究热点，因为ILT技术能够为芯片提供优越的工艺窗口。一个主要原因是ILT优化得到的掩模图形不受目标图形的限制，可能会得到和目标图形差别非常大的掩模，因此有可能找到比OPC优化得到的掩模更好的解。
前面介绍了3种实现ILT的方法，同时给出了部分参考文献中对于ILT和OPC技术的对比，可以看出，应用ILT可以大幅提高光刻技术的聚焦深度，减小成像图形与目标图形的差距[50,73]。其中，D2S, Inc. 于2019年SPIE会议上提出的利用GPU加速的方法，首次实现了对全芯片的ILT优化，对于全芯片的优化能够在一两天的实际运行时间内完成，并使工艺窗口与OPC相比提高了100%[52]。
目前，ILT存在的主要问题仍然是如何尽可能提高计算速度。相关的研究中解决这一问题主要有两个研究方向：一个是利用GPU加速等方法尽可能提高计算速度；另一个是尽可能优化算法，提高迭代的收敛速度，减少计算量，例如与机器学习等方法相结合[74,75]。
另外，还有与压缩感知(Compressive sensing)[76]等相结合的方法提高ILT计算速度。
英伟达在GTC 2023大会上推出了一个突破性的光刻计算库(cuLitho)，能够将计算光刻(包括OPC和ILT)的速度加速40倍以上，使得2 nm及更先进芯片的生产成为可能。
5. 结束语
使用ILT优化掩模主要有3种主流技术：Level set方法、Intel像素化ILT方法和频域中计算曲线 ILT的方法。其中，Level set方法把ILT问题转化为多变量优化问题，并且代价函数是连续的，可以使用梯度下降法等成熟的数学方法，受到广泛关注，相关文献较多。Intel像素化ILT方法的许多关键技术在文献中没有具体说明，相关文献也比较少。频域中的方法可以利用GPU运算，从而大幅提高计算速度，并且由曲线掩模转化为曼哈顿掩模也有比较完善的算法。ILT与机器学习等技术相结合仍有很大的发展潜力，在有效降低计算量同时保持较好优化效果等方面仍需深入研究。目前，ILT技术仍然存在计算量较大，难以应用于全版图优化等问题，这需要集成电路设计及相关领域的专家学者共同携手，寻找提高ILT计算速度及ILT与其他技术相结合的手段。相信随着ILT相关技术的研究的不断深入，将成为未来我国芯片产业技术突破“卡脖子”问题的一个重要途径。
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