1 Introduction

Nanomanufacturing, in the last decades, has rapidly evolved with the development of material, control, and manufacturing sciences. Recent research regards the study of nanomanufacturing as falling into two categories [1]: (1) The dimensions of the components are at the nanometric or micrometric scale with nanometric scale feature size, and (2) the geometrical accuracy of the components is at the nanometric scale.

For the first category, research has shown that when the scale of specific components reaches the nanometric scale, the performance of components is greatly improved [2]. For example, the nanoparticles of metal material can improve light absorptivity when the particle dimensions match the light wavelength [3]. The superparamagnetic effects associated with nano-ferromagnetic particles improve magnet stability at low temperatures [4]. Additionally, carbon nanotubes enhance the characteristics of polymeric matrices by increasing their tensile and shear stiffness 23-fold [5]. In addition to nanoparticles and tubes, other nanomanufacturing products are also widely studied for industrial applications, including nanofilms and nanotubes for sensing elements [6,7,8,9,10,11], thermal energy harvesting [12], flame retardants [13], self-cleaning mechanisms [14, 15], polymers [16] and nanowires for energy saving [17, 18], and structural reinforcement [19, 20].

The second category refers to processes for producing products of high-dimensional tolerances, e.g., diamond machining of metals and ultrasonic-assisted machining, which enables the advanced stability and surface quality of the products [21, 22]. The scope of this paper mainly addresses the measurement methods for the first category of nanomanufacturing. Depending on the type of nanostructures and their applications, the parameters of interest for the nanomanufacturing products cover a wide spectrum. According to the physical nature of these parameters, they can be categorized into three major types:

  • Critical dimensions (CDs), also considered geometrical properties, are the geometric dimensions determining the overall performance of the products. Depending on the type of nanostructure being produced, CDs cover the structural length, width, thickness, roughness, and angle of the geometric features [23].

  • Mechanical properties mainly include Young’s modulus, hardness, and other material-related characteristics for the structures on the nanoscale.

  • Thermal properties consider the heat capacity and conductivity of nanowires and nanoparticles.

Compared to the macroscale manufacturing processes, determining properties in nanomanufacturing faces new metrological challenges due to the scale of nanoproducts being close to or even smaller than the wavelength of light, e.g., ranging from 100 to 1000 nm. On the other hand, measurement technologies developed over the past decades can be categorized into two groups: direct and indirect methods, as shown in Fig. 1. The direct methods obtain these properties through direct imaging of the nanostructure or measurement process using atomic force microscopy (AFM) [24,25,26], scanning electron microscopy (SEM) [27], photoinduced force microscopy (PiFM) [28], electrostatic force microscopy (EFM) [29], scanning transmission electron microscopy (STEM) [30,31,32], optical scatterometry (OSM) [33,34,35], X-ray imaging [36,37,38,39,40], and other hybrid methods [41,42,43,44,45]. Because of the capability of obtaining CDs through direct optical or physical interaction, the direct measurement methods can achieve a high measurement resolution of up to 0.1 nm.

Fig. 1
figure 1

Indirect nanoscale measurement methods (FDTD finite difference time domain method, FEA finite element analysis method)

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In addition to serving as a metrological approach, direct methods are also used to calibrate each other or validate other measurement methods. At the same time, direct methods are also subject to constraints due to the direct physical interaction between the probes or electron/light beams and the nanostructure being measured. Measuring CDs on the nanometer scale requires a high spatial resolution on the level of at least a sub-nanometer. This measurement would need to define small incremental steps to move the probe along the horizontal trajectories on the scanned surface (e.g., AFM, PiFM, and EFM) or apply high spatial resolution to the light detectors (e.g., SEM, OSM, STEM, and X-ray imaging). Considering the overall dimensions of the nanoproducts and the large amount produced by nanomanufacturing, applying such high-resolution scanning would result in a long measurement period. As an example, AFM scans 1 mm2 in 1200 s [46]. Therefore, the direct methods are generally used for offline sampling and are not intended for in-process monitoring.

In comparison, indirect measurement quantifies in-process parameters, such as temperature [47], pressure [48], and etching solution concentration [49, 50], during the nanomanufacturing processes. Through proper data analytical techniques, these parameters are interpreted to infer the CDs and various mechanical and thermal properties by referring to the physics of the underlying process and/or the historical data [51]. Because these parameters are measured in real time, they can be used as feedback to adjust the operation without stopping the process [52]. As an example, an indirect measurement technique using deposition time, fluid atom concentration, temperature, and electric field intensity to infer the thickness and surface roughness of a nanofilm has a reported sampling rate of up to 10 samples/s, making online monitoring of the nanofilm manufacturing process feasible [53].

The core challenge in indirect methods is to establish the relationship between the measurable parameters and the properties of the produced nanostructures. This relationship justifies the meaningfulness and effectiveness of an in-process measurement. As illustrated in Fig. 1, this paper categorizes the indirect measurement methods into two major classes: data-based and physics-based methods. The former method establishes the relationship between in-process parameters and properties of the nanostructures through machine learning algorithms, which do not involve physical models and may be considered a black box. As the data-based method relies on much data to train the model, it is mainly adopted for measuring the CDs of the nanostructure, which can be accessed by direct methods. The latter method, in comparison, depends on the underlying physics of the manufacturing process to establish the process–property correlation. According to the mathematical models being applied to the physical dynamics and chemical reactions, the physics-based methods developed in the literature can be divided into four major categories: molecular dynamics (MD) simulations, the finite difference time domain (FDTD) method, the Monte Carlo (MC) method, and finite element analysis (FEA). As these models can simulate material behavior in macro- or microscales, the physics-based method can extend the scope beyond the measurement of CDs, for instance, to predict the mechanical and thermal properties of the nanoproduct.

This paper systematically reviews the indirect measurement methods and associated models and algorithms, focusing on the measurement principles and applications. Specifically, in the category of physics-based methods, four primary techniques that have been extensively investigated are discussed: MD, FDTD, MC, and FEA methods. The remaining paper is organized as follows: Sect. Data-based Methods discusses the data-based method and the associated mathematical algorithms. In Sect. Physics-based Methods, indirect measurement techniques based on physical principles are introduced, including MD simulations, the FDTD method, the MC method, and the finite element method. For each method, representative applications are introduced. Discussion of the presented work is presented in Sect. Discussion. In Sect. Conclusion, conclusions and future research directions are summarized.

2 Data-Based Methods

The data-based method is rooted in statistical modeling to build a relationship between input and output parameters. The rapid development of machine learning techniques has made data-based modeling increasingly applicable to estimating the properties of nanostructures.

2.1 In-Process Parameters in the Nanomanufacturing Techniques

According to how a nanostructure is fabricated, nanomanufacturing processes can be classified as “top-down” or “bottom-up” approaches, as shown in Fig. 2. The associated process variables in these approaches are listed in Table 1. In the top-down approach, the material is subtracted from the raw piece, which includes nanomachining, etching, and nanolithography. Their manufacturing processes are related to different process parameters. Nanomachining is defined as the material removal process in which a dimension accuracy of 100 nm or better, even toward the 1-nm level, can be achieved [5]. The common nanomachining includes cutting, grinding, polishing, and single-point diamond turning. The cutting parameters, such as cutting speed, cutting depth, and cutting rate, influence the CDs of manufactured components as the macroscale machining techniques. Etching is a type of chemical material removal technique. The patterns obtained by etching depend on the etching time, temperature, solution concentration, etc. Nanolithography applies an electron beam or light beam to etch, write, and print raw material and then obtain nanoscale components. The main parameters related to the nanolithography process include layer material, layer thickness, aperture shape, and light wavelength.

Fig. 2
figure 2

Nanomanufacturing techniques

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Table 1 Nanomanufacturing approaches and process variables
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In the bottom-up approach, the nanostructures are built from molecules or particles through mechanical assembly or chemical synthesis. The common bottom-up methods include nanoprinting, deposition, and self-assembly. In-process parameters, such as vapor pressure in deposition, solution viscosity, and injection velocity in printing, influence the performances of manufactured components for each bottom-up approach.

2.2 Effect of In-process Parameters on Nanostructure Properties

The relationship between the in-process parameters and nanostructure properties has been quantitatively studied in recent decades. Keong et al. studied the nanoscale cutting process for aluminum 7075 and found that the surface roughness of the product increases with the cutting depth [54]. Zhang [55] used elliptical vibration cutting as an amplitude compensation method to achieve ultra-precise nanostructure fabrication. Sreemanth et al. [56] found that the scanning speed in optical nanolithography determines the width and depth of etched line structures. Piner et al. [57] found that the width of nanolines by dip-pen nanolithography is sensitive to the scanning speed of the pen-tip and the transport rate of the solution from the tip to the substrate. An increased scanning speed and a few traces resulted in a decreased line width. Ahn et al. [47] found that the orientation of nanoparticle patterns depended on the applied magnetic field distribution in high-resolution magnetic printing. Wilson et al. [58] measured the grain size of inkjet-printed polymer films under different solution concentrations. Kazuya et al. [50] used wet etching to form nanoholes in different concentration solutions containing HF and H2O2 with silver nanoparticle catalysts. The shape of the silver particles was an essential factor in determining the direction of nanoholes, while the etching time and solution concentration determined the length and diameter of the nanoholes.

Yuki et al. [59] studied the influence factors on silicon nanowires formed by catalytic chemical etching in Ag2O/HF solution. This work found that solution concentration, etching time, and temperature can influence the length of the nanowires, as shown in Fig. 3. Zhao et al. [60] found that the microstructure and morphology of Ag catalyst film affect the silicon nanowire fabrication process in chemical etching. Xie et al. [61] fabricated a well-ordered silicon nanowire array using an Ag-assisted chemical etching technique and found a linear relationship between the length of the nanowire and the etching time.

Fig. 3
figure 3

Factors affecting Si nanowire formation in catalytic chemical etching. 1 Nanowire length vs. Ag2O concentration at 25 °C for 30 min. 2 Nanowire length versus etching time in Ag2O/HF solution at 25 and 50 °C. 3 Nanowire formation rate versus reciprocal solution temperature [59]

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2.3 Data-Based Model in Nanomanufacturing

From experimental results in nanomanufacturing, a certain relationship is shown to hold between in-process parameters and properties of nanoscale products. Data-based modeling trained by the historically sampled data from experiments can describe this relationship based on experimental approaches, particularly for complicated processes or nanostructures where the physical relationship is difficult to obtain through theories [62, 63].

Typical machine learning algorithms for building data-based models were summarized in [64]. According to the requirement on ground truth data (labeled data), machine learning algorithms are classified into three major categories: supervised, unsupervised, and reinforcement learning. Supervised learning takes the labeled data as the output of the model and correlates some of them with the process variables (input) for model training. The remaining labeled data are used to validate and test the trained models. Random forest (RF), support vector machine (SVM), decision tree, and Naïve Bayes are typical supervised learning algorithms [64]. In comparison, unsupervised learning algorithms do not require labeled data but only rely on the process variables. Typical unsupervised learning includes clustering, principal component analysis, and singular value decomposition. Compared with supervised learning, unsupervised learning does not require data labeling [64], which may take a significant amount of time and effort to establish the ground truth. Reinforcement learning is a relatively new category in machine learning that determines a data-based model through an iterative approach. In each iteration, reinforcement learning statistically adjusts the model parameters by referring to the “rewards” representing the estimation/prediction accuracy. Through this approach, the algorithm seeks an optimal solution for long-term and maximum overall rewards. Compared with the other two categories of machine learning methods, the architecture of reinforcement learning algorithms generally has a higher level of complexity and requires a larger amount of process variable data for model training. In applications, labeled data are optional for reinforcement learning. Typical reinforcement learning methods are realized by a deep neural network, recurrent neural network, deep belief network, etc. [64].

In recent years, machine learning methods have been applied to nanostructures [65,66,67,68,69,70,71,72], nanobiomedication [73, 74], nanoidentification [75], and nanomaterials [66]. Specifically, for nanomanufacturing, supervised and unsupervised learning methods have been widely applied to building data-based models for data analysis in indirect measurements. Table 2 lists the reported modeling algorithms together with the parameters and estimated CDs found in recent publications.

Table 2 Statistical models for estimating CDs of nanostructure
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Hou et al. [63] studied a data-based model for the nanoparticle wet milling process. The model was trained through the response surface method (RSM) to represent the relationship between the milled grain size and process parameters such as milling time and rotation velocity. The model was further optimized by the genetic algorithm to minimize the estimation error. Erdemir et al. [76] studied the effect of milling parameters on nanoparticle size during a high-energy ball milling process based on the RSM. A central composite design was implemented to determine the effects of milling parameters, i.e., milling time, the ball-to-powder ratio, and milling speed, on the produced nanoparticle diameter. Yordem et al. [51] studied the effect of the in-process parameters of electrospinning on polyacrylonitrile nanofiber diameter. The RSM was used to design the experiments at the setting of solution concentration, voltage, and collector distance. Ayu et al. [49] developed an RSM model to study the influence of metal-assisted chemical etching process variables on the nanopillar profiles. The process variables include solution and catalyst concentration, deposition time, and etching time. Experimental validation showed an estimated error of 1.8% by referring to the dimensions measured via SEM.

Research [77] studied the behavior of bias power and Cl2 concentration in silicon trench etching and their influence on the etching rate and the variations in sidewall profile. Huey et al. [78] built the RSM and artificial neural network models for the chemical etching process to correlate the etching variables, including etching temperature, etching duration, and solution concentration, with the surface roughness of the etched surfaces. The model shows that for individual process parameters, the relationship between the parameter and product surface roughness varies significantly for different materials. Compared to the experimental results, the estimated roughness achieves an accuracy of 95%. Lin et al. applied the SVM to correct errors in microscope photographs when evaluating nanostructure thickness [79]. Kim tested eight machine learning methods to estimate the thickness of TiO2 films. A comparative study of these methods with respect to the estimation accuracy has revealed that the decision tree algorithm achieved the highest accuracy of 82%, while K-nearest neighbor learning had the lowest accuracy of 70%. Du et al. estimated the mobility of aluminum atoms on Al2O3 nanofilm during ductile deformation based on the SVM with an accuracy of 69%–82% [80]. Notably, most of the algorithms in the literature used to build data-based models are of the supervised or unsupervised learning type. Broad-based applications of reinforcement learning for indirect measurement are yet to be seen.

Data-based models can be applied to nonlinear and complex systems at the cost of requiring a certain amount of historical data to train the models. Compared with the physics-based method, the data-based method is advanced in the short response time for a trained model to estimate CDs from the measured process variables. As data-based models entirely depend on the training data, the developed data-based model can hardly be extended to other applications with different model architectures, manufacturing platforms, or materials [81].

3 Physics-Based Methods

3.1 Molecular Dynamics

Nanomanufacturing occurs at the molecular level by manipulating materials at the atomic and molecular scales. Physics-based modeling methods are generally based on the well-understood physical or chemical dynamics of the specific nanomanufacturing processes. The particle behavior in the nanomanufacturing process can be understood from theoretical analysis, and the results of the theoretical analysis can be calculated through numerical simulation. On the basis of the analysis scale and the type of properties studied, four physics-based modeling techniques will be discussed to obtain the properties of nanoscale products with respect to their principles and application status.

MD modeling is a method that determines the motion of a group of molecules following Newton’s second law, correlating the movement and the force applied together. The force field that results in the movements can be calculated based on the initial positions of the molecules and their interaction with neighboring molecules. In the MD analysis, the molecular motion in each set time step can be simulated based on the theories, making the process transparent. This advantage makes it a good complement for experiments, visualizing the motion of molecules to study the mechanisms of the nanomanufacturing process from the microscopic perspective. As the MD method solves the trajectory of each molecule, it simulates the actual nanomanufacturing process with high spatial accuracy.

3.1.1 MD Principles

3.1.1.1 Force Field

As the force field determines the motion of molecules physically, modeling the force field is one of the keys to the MD method. The study of a force field for describing microscopic particle motion was started in the 1960s by Aligner’s group [82] to calculate the motion of hydrocarbon molecules. Thus far, the concept has been extended to study various types of particles, such as peptides [83], aluminum [84], germanium [85], and ethane [86]. In general, the force field resulting in the potential energy of a single molecule in the MD model can be expressed as given in Eq. (1) [87].

$$V = V_{\rm bond} + V_{\rm angle} + V_\text{torsion} + V_{\rm oop} + V_{\rm nonbond}$$
(1)

where Vbond is the energy change related to bond length variation in a molecule, as shown in Fig. 4 (1), where the solid lines/circles show the molecule at the equilibrium position, while shapes in dash lines/curves show the positions after the conditions changed; Vangle is the energy change related to bond angle change, as shown in Fig. 4 (2); Vtorsion is the energy change when a certain torsion is applied to the molecule, resulting in a rotation, as shown in Fig. 4 (3); Voop is the “out-of-plane” energy, representing the energy change when one part of the molecule is moving out of the plane, as shown in Fig. 4 (4); Vnon-bond is the noncovalent interaction energy, including van der Waals interactions and Coulombic interactions.

Fig. 4
figure 4

Molecular movement model. 1 Translation of molecule. 2 Rotation of molecule. 3 Torsion of molecule. 4 “Out-of-plane” motion

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According to the physics of each component, Eq. (1) can be extended as [82, 87]:

$$V(r) = \sum\limits_{\rm bonds} {\frac{{k_{b} }}{2}} (b - b_{0} )^{2} + \sum\limits_{\rm angles} {\frac{{k_{\theta } }}{2}} (\theta - \theta_{0} )^{2} + \sum\limits_{\rm torsions} {\frac{{k_{\phi } }}{2}} [\text{cos}(n\phi + \delta ) + 1] + \sum\limits_{\rm oop} {\frac{{k_{\psi } }}{2}} (\psi - \psi_{0} )^{2} + \sum\limits_{\begin{subarray}{l} \text{nonbond} \\ \text{pairs} \end{subarray} } {4\varepsilon_{ij} [(\frac{{\sigma_{ij} }}{{r_{ij} }})^{12} + (\frac{{\sigma_{ij} }}{{r_{ij} }})^{6} ]} + \frac{{q_{i} q_{j} }}{{r_{ij} }}$$
(2)

where \(k_{b} ,k_{\theta } ,k_{\phi } ,k_{\psi }\) are the stiffness coefficient of the bond length, angle, torsion, and out-of-plane motion, respectively; b is the distance between atoms, b0 is the equilibrium distance; θ is the angle between two bond vectors, θ0 is the equilibrium angle; \(\phi\) is the torsion angle, \(\delta\) is the phase, n is the dihedral potential, \(\psi\) is the out-of-plane displacement, and \(\psi_{0}\) is the equilibrium displacement. The last summation term of Eq. 2 represents the van der Waals interaction, composed of a 12–6 Lennard–Jones potential and Coulombic interaction. In this term, \(\sigma_{ij}\) is the size parameter of atom i and atom j,\(\varepsilon_{ij}\) is the energy parameter, rij is the distance between atoms i and j [88], and qi and qj are the effective charges on atom i and atom j, respectively.

Based on the application and the type of molecules, the detailed parameters measured or approximated can be substituted into the force field formation equations (Eqs. 1 ~ 2) to generate different force field models. Examples of these models include the first force field developed by Allinger’s group, MM2 [89], MM3 [90], and MM4 [91], which are used to study hydrocarbons. The force field concept has been further extended in recent decades to study complex systems with large molecules in nanomanufacturing. Dreiding and universal force fields are two examples that can be used to model all molecules [92]. AMBER [93] is a force field model that can be applied to biomolecules, canonical/noncanonical nucleic acids, and proteins in water. The CHARMM force field [94] studies small and large biological macromolecules, such as lipids and carbohydrates [82]. GROMOS [95] was developed to simulate condensed matter.

In addition, other types of force field models, such as OPLS [96], COMPASS [97], CFF [98], and MMFF [99], have been developed to describe the behavior of specific molecules or particles. The application scope of every force field is introduced in [82]. The performance of a force field strongly depends on the system and particle type being simulated. Each force field has its strengths and weakness related to the molecular formation procedure and environment, while some general force field models can generate reasonable results comparable with experimental results for a wide range of molecules and various environments [100].

3.1.1.2 Motion Algorithm

The force field model in Eqs. 1–2 is applied to each molecule in a material, and the motion of each molecule can be described by a second-order partial derivative function based on Newton’s second law [92]:

$$m_{i} \frac{{\text{d}^{2} r_{i} }}{{\text{d}t^{2} }} = f_{i} = - \frac{\partial }{{\partial r_{i} }}V(r_{1} ,r_{2} ,...,r_{N} )$$
(3)

where mi and ri are the mass and location of atom i, respectively, and \(V(r_{1} ,r_{2} ,...,r_{N} )\) is the sum of the potential energy of atoms 1–N in their initial locations.

The molecular trajectories are calculated in n consecutive time steps, t0, t0 + Δt, …, t0 + nΔt. In each discrete time step, a solution to the second-order partial differential equation, Eq. (3), is obtained through numerical methods. The most used numerical method includes the Verlet integrator (leap-frog algorithm) and velocity-Verlet algorithm. In the Verlet integrator method, the solution is obtained through iterative approximations, expressed as [101]:

$$\left\{ \begin{gathered} r_{i} (t_{0} + \Delta t) = r_{i} (t_{0} ) + v_{i} (t_{0} + \frac{\Delta t}{2})\Delta t \hfill \\ v_{i} (t_{0} + \frac{\Delta t}{2}) = v_{i} (t_{0} - \frac{\Delta t}{2}) + a_{i} (t_{0} )\Delta t \hfill \\ \end{gathered} \right.$$
(4)

where vi and ai represent the velocity and acceleration of the molecule in the studied space, respectively. The velocity-Verlet algorithm further considers the second-order integration of acceleration in calculating the molecule’s position and accordingly solves Eq. (3) as follows:

$$\left\{ \begin{gathered} r_{i} (t_{0} + \Delta t) = r_{i} (t_{0} ) + v_{i} (t_{0} )\Delta t + \frac{1}{2}a_{i} (t_{0} )\Delta t^{2} \hfill \\ v_{i} (t_{0} + \Delta t) = v_{i} (t_{0} ) + \frac{1}{2}[a_{i} (t_{0} ) + a_{i} (t_{0} + \Delta t)]\Delta t \hfill \\ \end{gathered} \right.$$
(5)

In addition, other methods, such as the Euler method [102] and the Crank–Nicolson method [103], are also reported in the MD modeling research for nanomanufacturing. Although complicated algorithms that include more terms in the expression for the molecule trajectory will improve the accuracy of the model, considering the many molecules within a nanostructure, such algorithms generally result in an increased computational load [104, 105]. In practical applications, selecting numerical algorithms for solving the trajectory in the MD method generally considers the balance between accuracy and computational efficiency.

3.1.2 MD Applied to Nanomanufacturing

3.1.2.1 Mechanical Properties

The MD method has been applied to simulating nanoscale components from which mechanical properties are derived. The effectiveness of the method has been verified experimentally, as reported in the literature. Tang et al. applied transmission electron microscopy (TEM) and MD simulations to study the deformation and fracture mechanism of Si nanowires with a diameter of 9 nm [106]. The nanostructure behavior under tensile and bending loads was obtained through MD simulations and experiments, as shown in Figs. 5 and 6. In the tensile test (Fig. 5), the nanowire breaking process during increasing normal stretch stress on the two ends of the nanowire was simulated. The simulated linear relationship (approximate) between stress and strain well matches the experimental results. Figure 6 shows the observed nanowire deformation during the bending test. The nanowire was imaged by TEM in the experiments when the bending moment was loaded to the nanowire. A crack was visually observed on the image when the bending strain exceeded 21.5%. This result matches well with the MD modeling results, as shown in Fig. 6 (2) [106].

Fig. 5
figure 5

Simulated nanowires under tensile test. 1 Simulated nanowire structure in the side view and cross-sectional view. 2 Simulated stress–strain relationship [106]

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Fig. 6
figure 6

Comparison of nanowire behavior in the bending test under experiments and simulation. 1 Experimental image obtained via TEM. 2 Simulated nanowire [106]

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Li et al. simulated physical vapor deposition and the nanoindentation process by MD to study the hardness and defect of nanofilm under different deposition temperatures [107]. A piece of Cu/Au thin film of 7.2 × 7.2 × 30 nm3 was simulated in the study. An ideal spherical indenter with a radius of 2 nm at a constant velocity of 50 m/s was built to simulate the nanoindentation process by indenting the depth from 1.6 nm to 2.2 nm. The force–depth curve was calculated during the indentation process [107]. Li et al. studied via the MD method the process in which the nanofilm thickness changes the glass transition temperature, Young’s modulus, and yield stress [86]. Lai et al. conducted a 3D MD simulation to study the nanometric cutting of germanium, as shown in Fig. 7. The studied workpiece dimension was set as 45 × 27 × 12 nm3. The uncut thickness and cutting force with different cutting depths on different crystal orientation faces were calculated based on this MD model. This work can provide references for selecting the cutting parameters and modeling the nanometric cutting mechanisms [85].

Fig. 7
figure 7

Simulated nanometric cutting process 1 MD simulation model. 2 Simulated uncut thickness at different cutting depths on the (010) and (111) crystal orientation face. 3 Simulated cutting force at different cutting depths [85]

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3.1.2.2 Critical dimensions

Chen et al. built a kinetic competition model with kinetic parameters from in situ TEM to estimate the diameter of silicon nanowire [108]. Sindhu et al. studied nanoparticle formation in an inter-gas evaporation technique, and the particle size and distributions were predicted by solving the general dynamic equations through the nodal approach under different ambient conditions. When the injecting energy is higher than the vaporization energy, the dimensions of the produced particles are reduced [109]. Xu et al. studied the supramolecular structure of surfactant and carbon nanotube aggregates in aqueous solution using atomic-level MD simulation [110]. The surfactant adsorption on a nanotube with a hemimicellar structure was found to be consistent with the observations under TEM in the structure periodicity (4.5 ± 0.5 nm). The average height of the hemimicelle is also in good agreement with the experimentally measured height [110]. The MD method was also applied to simulate the deposition and annealing of a Ni–Al wire on a Ni substrate [84]. Sputtering parameters (including the incident energy, incident angle, and substrate temperature) and subsequent heat treatment were considered in the model.

The MD method has a wide application in analyzing the nanomanufacturing process and estimating the mechanical properties of nanoscale products from the initial and in-process working conditions. The MD method can build a relationship between the in-process parameters and nanoscale product. It can also predict the mechanical properties of nanoscale products. However, as the simulated elements are generally at the molecular dimension level, modeling the nanostructure generally includes many elements. In addition, since the MD needs to be sampled every 10–100 fs, simulating the transients of the nanomanufacturing process requires at least trillions of time steps, which further increases the computational load [82]. In general, on the current computation platforms, such a constraint limits the MD method to simulating processes shorter than a few seconds.

3.2 Finite Difference Time Domain Method

The FDTD method is a numerical method for modeling an electromagnetic field and its propagation in 1D, 2D, or 3D space. The idea of FDTD was first developed by Yee [111] in the 1960s. The technique was fully developed by Umashankar and Taflove in the 1970s [112]. Since then, the FDTD method has become a fundamental tool for studying microwaves and optics [113]. In recent decades, the FDTD method has been used to model the optical properties of nanostructures [114] and the nanomanufacturing process, such as nanolithography [115] and surface plasmon polarization processes [116]. In these processes, the light waves or photons dynamically interact with the molecules of the photosensitive material, change the material characteristics, and determine the nanostructure CDs in consecutive procedures. To mathematically model the electromagnetic waves, the FDTD method discretizes the space and time to calculate a time-varying electromagnetic field in each time step based on Faraday’s law and Ampere’s law [117].

In nanomanufacturing, FDTD has been used to determine the relationship between the applied light intensity and the geometry/shape of the nanostructure. Shao et al. built a two-dimensional FDTD model to study fabricating high-density arbitrary patterns on a sub-100-nm scale in the ultraviolet nanolithography process, as shown in Fig. 8 [113]. Figure 8 (1) shows the modeled setup, with UV light applied to react with the photoresist film under the metal mask. Simulation results from the FDTD model are compared with the theoretical results (analytical) in Fig. 8 (2), where a good matching between the two types of data is observed. The model was also used to simulate the electric field distribution in the photoresist layer, as shown in Fig. 8 (3). When the aperture in the metal mask is reduced from 50 to 20 nm, the light transmission is significantly reduced to 60%. This relationship provides a reference for manufacturers to determine the aperture size and light source intensity according to the preferred nanostructure CDs.

Fig. 8
figure 8

Photolithography modeled using the FDTD method to study the effect of aperture size on the nanostructure CDs. 1 Schematic view of the photolithography process. 2 Comparison of the electric field distribution of the FDTD simulation and theoretical solution. 3 Electric field distribution in the photoresist with (a) 50-nm aperture and 60-nm resist thickness (b) 20-nm aperture and 50-nm resist thickness [113]

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In the work presented by Wang et al. [118], the FDTD method was used to simulate an electric field distribution through square, rectangular, and bowtie apertures. As shown in Fig. 9 [118], the simulation results were consistent with experimental data. Chang et al. applied an FDTD simulation to obtain the morphology at different penetration depths in the photoresist film during 3D nanolithography, as shown in Fig. 10 (1) [119]. Consistent morphologies were observed by comparing the simulation results and the SEM image at penetration depths from 40 to 650 nm, as shown in Fig. 10 (2). The FDTD simulation results were used to predict the CDs of nanostructure produced by nanolithography [119].

Fig. 9
figure 9

Electric field distribution with different apertures. 1 Square, 2 rectangular, 3 bowtie [118]

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Fig. 10
figure 10

Penetration depth in nanolithography. 1 Schematic of the nanolithography process. 2 SEM images of the fabricated structure versus depth from 40 to 650 nm [119]

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Kim et al. developed a simulation code, AMADEUS (advanced modeling and design environment for sputter process), to simulate the topography in ion beam-induced nanofabrication based on the ion beam milling process. The ion beam milling hole depth and diameter were predicted [120]. Dragos et al. analyzed nanosphere lithography progress and the relationship between nanosphere shapes and electric fields in photolithography. A shape-prediction model was built based on the FDTD method [121]. Sreemanth et al. studied the influence of laser scanning speed on the width of nanolines [56]. FDTD was used to numerically design the bowtie apertures to estimate the nanoline width under a scan speed of 0.2 to 0.9 µm/s with an error of 21% [56]. Mishra and Palai presented a technique for measuring the intensity of reflection, transition, and absorbed light to compute the percentage of porosity in porous silicon. A linear relationship between the transmitted light intensity and the percentage of porosity was calculated using an FDTD model [114].

Barth et al. designed a hybrid nanophotonic device that can achieve metal nanoparticles nanoassembling in the cavity because metal nanoparticles localize themselves in positions with high electric field intensity, as shown in Fig. 11. FDTD was adopted to analyze the electric field in the designed cavity. FDTD simulation results showed that when a gold nanorod was inside the designed cavity, the electric field maximum moved from the cavity surface to the nanorod surface [122]. Lee and Hahn applied FDTD to obtain a localized field distribution on the photoresist under C-shaped and bowtie-shaped apertures during nanolithography. The calculated field distribution was used to predict the profile of the nanopatterns and optimize the initial parameters of nanolithography [123].

Fig. 11
figure 11

Nanoparticle assembly process modeled by FDTD. 1 AFM image of a cavity with a gold nanorod. 2 Electric field distribution of a cavity without a gold nanorod. 3 Electric field distribution of a cavity with a gold nanorod [122]

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As the light transportation through the nanostructure can be simulated by FDTD, this function makes FDTD a candidate model for optimizing the parameters for controlling the nanolithography process, which can hardly be accessed by direct measurement methods. The FDTD method can also simulate the internal variation in the nanostructure. This capability makes it a powerful tool for understanding the physical variation inside the raw material. However, as one of the nanoscale modeling methods, FDTD shares the same problem as the MD method of a high computational load. The discrete spatial sections and time steps must be sufficiently small to satisfy the requirement of the Nyquist theorem. In addition, constrained by the core theories of electromagnetic waves, FDTD can only be applied in nanolithography or other procedures affected by microwave, UV, visible, or infrared light.

3.3 Monte Carlo Method (or Monte Carlo Molecular Method)

The MC method is a numerical approach to solving deterministic problems with random trials. It was first developed in the 1940s at Los Alamos National Laboratory [124]. In simulating molecule behavior in a complex macromolecular system, the method generates a molecular ensemble under specific thermodynamic conditions by assuming stochastic distribution in the molecule displacement [125]. Compared with the MD method, the MC method does not provide information about time evolution [126]. The rationality of updating the molecular configuration is determined by the molecular distribution probability variation and the resulting change in potential/dynamic energy. The MC method has been applied to studying heat conduction characteristics in nanostructures [127], gas absorption ratios of nanomaterial [128, 129], critical dimension formation during nanomanufacturing [130, 131], and cutting forces/temperatures in nanometric cutting processes [132].

3.3.1 Working Principle

The MC method assumes the number of particles, system volume, and temperature as constants in each thermodynamic system. Accordingly, it models the system ensemble as a representative of all molecular states [131]. The system ensemble is expressed as a function of the particle variables (position, momentum) and the material properties [132]:

$$\left\langle A \right\rangle = \frac{{\int_{ - \infty }^{ + \infty } {\exp ( - \beta E(\vec{r}^{N} ,\vec{p}^{N} ))A(\vec{r}^{N} ,\vec{p}^{N} )} \text{d}\vec{r}^{N} \text{d}\vec{p}^{N} }}{{\int_{ - \infty }^{ + \infty } {\exp ( - \beta E(\vec{r}^{N} ,\vec{p}^{N} ))} \text{d}\vec{r}^{N} \text{d}\vec{p}^{N} }},\,\beta = \frac{1}{{k_{B} T}}$$
(6)

where \(k_{B}\) is Boltzmann’s constant, T is the system temperature, \(E(\vec{r}^{N} ,\vec{p}^{N} )\) is the total potential energy of the N particles at position \(\vec{r}^{N}\) with momentum \(\vec{p}^{N}\)(\(\vec{r}^{N}\) and \(\vec{p}^{N}\) are N-dimensional arrays). \(A(\vec{r}^{N} ,\vec{p}^{N} )\) is a generic property representing the potential energy, kinetic energy, local density, coordination number, or radius of gyration. Because of the nonlinear terms in Eq. (6), the solution is generally approximated through numerical methods. In the system ensemble, the space is discretized into grid cells to calculate the behavior over the entire space. To reduce the computational load, the MC method adopts the metropolis acceptance criterion to determine the states. The metropolis acceptance criterion always accepts the state where the potential energy is smaller than the previous state. Otherwise, the system rejects the new state by following a certain probability that is determined by the Boltzmann distribution [126, 133, 134]:

$$\Pr (\vec{r}^{N} ,\vec{p}^{N} ) = \frac{{\exp ( - \beta H(\vec{r}^{N} ,\vec{p}^{N} ))}}{{\int {\rm{d}p^{N} \rm{d}r^{N} \exp ( - \beta H(\vec{r}^{N} ,\vec{p}^{N} ))} }}$$
(7)

where \(H{(}\vec{r}^{N} ,\vec{p}^{N} )\) is the Hamiltonian value of the system with position \(\vec{r}^{N}\) and momenta \(\vec{p}^{N}\), determined by Hamilton’s equations. For each grid or molecule, if the molecule transfers from state m to state n, the mathematical expression of the metropolis acceptance criterion in the MC molecular modeling can be expressed as [133]:

$$p_{mn} = \left\{ \begin{array}{ll} \exp ( - \beta (E(n) - E(m))) & \quad {{E}(n) - E(m) > 0} \\ 1 &\quad {{E}(n) - E(m) < 0} \\ \end{array} \right.$$
(8)

where E is the potential energy of the particle. Based on the principle of the MC method, the ensemble in a specific thermodynamic condition can be calculated using the following iterative algorithm:

  1. 1.

    Select a particle randomly, and calculate the potential energy E(m).

  2. 2.

    Add a random displacement to the selected particle:

    $$\vec{r}(n) = \vec{r}(m) + \Delta$$
    (9)
  3. 3.

    Calculate the new potential energy E(n).

  4. 4.

    Determine the new particle to accept or reject by Eq. (8). If \(\Delta E < 0\), the new particle is accepted; if \(\Delta E > 0\), draw a random number \(\gamma\) on the interval (0,1) and accept the new particle when \(\exp ( - \beta \Delta E) > \gamma\).

  5. 5.

    If the new particle is accepted, the particle movement and potential energy calculation are based on the new particle; if the new particle is rejected, the system update is based on the old particle’s position.

3.3.2 Applications of the MC Method in Nanomanufacturing

In nanomanufacturing, the MC method has been used to model or simulate thermodynamic systems. Platzgummer et al. developed a model to simulate the ion beam nanoimprinting process [130]. The developed model included an MC module that simulated sputter yield under different ion beam angles [130]. Tang et al. studied the heat of a nanofilm based on the MC method [135]. Physically, the system heats one side of the nanofilm with a laser beam and measures the temperature difference across the film, as shown in Fig. 12 (1) [127]. The simulation reveals that the porosity, film thickness, and pore radius codetermine the thermal conductivity, as shown in Figs. 12 (2, 3). The simulation results also indicate that the MC method estimates the thermal conductivity more accurately than the conventional effective medium approach model in reference to the experimental results [127]. Leila compared the MC method and Langevin dynamic simulation in a study of SnO2 nanofilm absorbance when it is used as a sensor to detect ethanol [127]. The data show that the two models produce consistent results in simulating the sensing process [129].

Fig. 12
figure 12

Nanofilm thermal conductivity modeled with the MC method. 1 Schematic for cross-plane heat conduction in nanoporous silicon film. 2 Thermal conductivity of nanoporous film versus pore radius (KnR: pore radius; keff/kbulk: effective thermal conductivity). 3 Thermal conductivity of nanoporous film at different porosity levels (ɛ: porosity) [127]

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Komanduri et al. studied the local temperate in the cutting zone of the nanometric cutting process based on the MC method and compared the computational load with that of the MD method, as shown in Figs. 13 (1, 2) [133]. Figure 13 (1) shows that when the number of atoms increases, the difference in computation time between MD and MC increases exponentially. The relationship is observed when the cutting velocity is under 500 m/s, as seen in Fig. 13 (2) [133]. Zhu et al. built a semiempirical model based on density functional theory combined with the MC method to study the relationship between the catalyst particle and the carbon nanotube diameters and compared the results with that from the MD model [131, 136]. A good agreement was shown between the results of the two models.

Fig. 13
figure 13

Computational time comparison between the MC and MD methods [133]

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According to the literature, MC can be used to simulate thermodynamic systems. As the MC method considers state transitions instead of continuous time steps, it only discretizes the space or volume of the object. As a result, the computational load is lighter than that of the MD and FDTD methods. However, the principle of the MC method also determines that the dynamics in the time domain cannot be provided in the results [134].

3.4 Finite Element Method

Finite element (FE) analysis is a numerical method for solving continuum mechanics problems by dividing a continuous structure into a finite number of structural elements. The conventional FE method assumes a linear relationship among the elements, thereby transforming a nonlinear problem into a multi-degree-of-freedom linear problem [137, 138]. Over the last 20 years, the FE method has been applied in nanomanufacturing to model nanostructures. As the FE method (FEM) does not need to model each molecule, as in the MD, FDTD, or MC methods, the meshed element only needs to satisfy the dimensional requirement of the specific nanostructure. Consequentially, the computational load of the FE method is generally much lower than that of the other methods. The FEM has been used to calculate the interfacial adhesion on top of nanofilms [139], the stiffness tensor of nanocomposite material [140], the tomography of nanostructure [141], the tensile modulus [142], and the cutting force during the nanometric cutting process [143].

Figure 14 shows a study of the interfacial adhesive characteristic of nanofilm deposited on a Cu/Si substrate using FE analysis and experiments [139]. The FE model simulated the process of a tool indenting and scratching the surface of the nanofilm and accordingly determined the critical force for different nanofilm thicknesses, as shown in Fig. 14 (1). The results show good agreement between the experimental and FE simulation results [Fig. 14 (2)]. Li et al. modeled silicon nanofilms decorated with nanopillar arrays to calculate the light absorbance of the nanofilms [144]. The simulation results were verified via the transfer matrix method [145], as shown in Fig. 15 (1, 2). The difference between the results from the two methods is shown to be below 0.25%. Liu et al. simulated the nanoimprinting process with the FE method on the ABAQUS platform and validated the results experimentally [146]. The simulation model achieves an error of 8.3% in the ratio of formation height to mold depth relative to the experimental data [141].

Fig. 14
figure 14

Nanofilm scratching modeled based on the FE method. 1 Setup of a nanoscratch on the nanofilm. 2 Comparison of experimental and FEM results with 80-nm and 120-nm film thicknesses (h: penetrations of indenter; Fz: critical force) [139]

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Fig. 15
figure 15

Nanofilm with pillar array modeled in FE software. 1 Schematic of light incident into nanofilm decorated by nanopillars. 2 Efficiency function with array periodicity with exact values of 2330-nm film calculated by TMM and FEM [144]

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Sun and Cheng presented a combined FE–MD model to simulate crystal aluminum processed by nanometric cutting [143, 147]. The developed simulation model employs the MD method to model the cutting zone on the nanometer scale and then uses the FE method to model the rest of the workpiece to reduce the computational load while maintaining the accuracy, as shown in Fig. 16 [143]. Liu and Jiang developed an atomic-scale FE (AFE) method in which the atoms of the processed material are modeled as nodes for calculating the process parameters, such as displacement, strain, stress, and temperature [146]. The CPU time and the number of iterations required for the developed method were compared with those of the MD method. The CPU time for calculating 3200 atoms by the AFE method is 65 s in 31 iterations and 4000 s by the MD method in 9000 iterations, which indicates that AFE reduces the computation time approximately 60-fold [148].

Fig. 16
figure 16

Multiscale simulation model of the nanomatrix cutting process [143]

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4 Discussion

Compared to direct measurement methods, such as AFM and EBM, indirect measurement methods, particularly the data-based methods, are advanced in fast response to estimate the CDs from the measured process variables. Although training a data-based model can take hours, using the trained model to estimate the CDs takes less than a second [53]. The physics-based methods generally have a longer response time than data-based methods because of the heavy computational load to simulate the behavior of particles or structures on the nanoscale. The actual processing time depends on the scale of the simulation in specific applications. Table 3 summarizes the four physics-based modeling methods in terms of computational efficiency, application, and limitation. Because of the constraint of the working principles, the FDTD method is generally adopted in an application involving microwaves, such as nanolithography and polarization processes. The MC method is advanced in simulating thermodynamics without considering each time step. The MD and FE methods are universal tools for modeling most nanomanufacturing processes. The selection and application of these physics-based models can be determined by the nature of the manufacturing process and the available computational capability [81]. All physics-based methods take seconds to hours to finish simulating a model. Although they can hardly be used for online control based on the present computational hardware, the simulated chemical reactions and physical behavior of the particles provide a theoretical basis for indirect measurement.

Table 3 Computational efficiency and application of physics-based methods
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Uncertainty is an essential characteristic that quantifies the reliability of measurement results. In the direct measurement methods, a measurement result is determined by the physical structure of the probes or light/X-ray beams, and the uncertainty is closely related to the system hardware. In comparison, the uncertainty associated with the indirect methods can result from the following four factors [48, 62, 63, 77, 78, 118,119,120,121,122,123,124,125,126,127,128,129,130, 149,150,151]:

  1. (1)

    The physical limitation of the instrument used when measuring process variables associated with the nanomanufacturing process;

  2. (2)

    Simplifications and assumptions made when interpreting the measured parameters using empirical or regression models;

  3. (3)

    The disturbance experienced during the measurement process, such as noise, environmental conditions, and electromagnetic pulses;

  4. (4)

    Variation in the execution of the instrument operator when performing the measurement.

Because indirect measurement methods use the ground truth provided by direct measurement methods for model training, calibration, and validation, the uncertainty of the direct measurement method will be incorporated into the outcome of the indirect methods. As a result, indirect methods may have a higher level of uncertainty and a lower level of accuracy than direct methods.

Table 4 summarizes the validation methods in the literature. The AFM and SEM methods are most used to establish the ground truth of the CDs of nanostructures because of their high accuracy in quantifying the dimensional measurement and good adaptability to work with different types of nanostructures, such as fins, channels, films, and tubes.

Table 4 Validation of indirect measurement methods
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5 Conclusions

This paper reviewed indirect measurement methods for monitoring the nanomanufacturing process and measuring the properties of nanoscale products. Compared with direct measurement methods, such as AFM, SEM, PiFM, and EFM, which use probes or electron/light beams to measure the geometry of nanostructures, indirect measurement methods exploit measuring the CDs in situ without interfering with the manufacturing process. This paper categorizes the indirect measurement methods into two major classes: data-based and physics-based methods. The data-based methods establish the relationship between in-process parameters and properties of the nanostructures through machine learning models. These machine learning models are trained by historical data from experimental studies and provide high accuracy when much data are collected from various conditions. The physics-based modeling method, in comparison, depends on the underlying physics of the manufacturing process to establish the process–property correlation. The numerical models in the physics-based modeling method can simulate the physical or chemical dynamics during the nanomanufacturing process, allowing accurate predictions to be provided with limited historical data. However, the computational load of the numerical simulation quickly increases with increasing model complexity and spatial resolution. As the indirect methods infer the CDs of nanoproducts from the in-process parameters accessible by the sensors during the nanomanufacturing process, they can predict the nanoproduct quality and provide feedback to fine-tune the control parameters in nanomanufacturing. In recent years, indirect methods have received increased attention due to the increased demand for real-time control for improved accuracy and stability in nanomanufacturing. Questions such as how to build computational data-based models with improved accuracy, how to fully understand the physics of the nanomanufacturing process, and how to combine the physics- and data-based models to address the generality and accuracy are still major challenges that require further research.