In the last decades, technology has used Copper for IC interconnect and it has been the best material used in the wire downsizing. However, Copper is now showing inefficiency as downscaling is getting deeper. Recent research starts to show Tungsten (W) as a possible replacement, for its better downsizing characteristic. The scaling-down of interconnects dimension has to be augmented with thin diffusion layers. It is crucial to subdue tungsten diffusion in the nickel-based thermal spray Flexicord (NiCrAlY) coating layers. Inappropriately, diffusion barriers with thicknesses less than 4.3 nm do not to execute well. With the introduction of two dimensional layers, hexagonal boron has been recommended as a substitute for Tungsten diffusion barrier layers with thicknesses less than 1.5 Nano meters (nm). Nevertheless, vacancies flaws may develop into a Tungsten dissemination path, which is a problematic issue in the manufacturing of diffusion barriers. The energy layer density, of Tungsten atom diffusion via a di-vacancy in NiCrAlY, is computed by density functions 3D. NiCrAlY has complex energy barrier which is thicker than other materials such as Graphene. This is due to the sturdier contact and charge variance of NI and Cr in NiCrAlY. Also, we utilize the energy barriers of several vacancy constructions and produce a dataset to be employed in the proposed 3-imensional deep learning model (3D-DNN). Our trained deep learning neural model can predict the energy barrier of Tungsten diffusion through arbitrarily configured NiCrAlY with accuracy greater than 98.4% in 5 × 5 cell. Prediction results generate directors on selecting barriers through energy computation.

Semiconductor devices are shrinking in size and are going from 24 nm in 2013, to 16 nm in 2015, and 4 nm in 2021 [

Tungsten is characterized by a nickel grayish luster. Tungsten has a very high melting point of 3,420°C. Its tensile strength is denoted at temperatures of 1,750°C. Also, Tungsten has a low linear thermal expansion [

To avoid Tungsten dissemination, a layer must be attached to the edge of the Tungsten and the surrounding dielectrics. Original barrier layer such as Tin [

Nanotechnology presents two dimensional materials as a main role player in IC technology. For example, two dimensional materials of the interconnect can be embedded as a barrier to constantly sizing-down of those barriers. Two dimensional materials exhibit high quality blocking properties such as NiCrAlY. NiCrAlY is a high temperature resistant matter that are utilized because of their high bonding strength. NiCrAlY has an ambient temperature of 1170°C, where its oxidation process increases [

As two dimensional materials are presented as barrier for Tungsten diffusion, still an exhaustive investigation is still not performed. In this paper, we intend to study the interaction that can occur among a dispersed atoms and the NiCrAlY. This can help in selecting between NiCrAlY and other materials. Also, density functional model can help in producing training and validation datasets. Each set contains the configuration, of two dimensional layer structure, represented as a two dimensional array, together with its energy barrier. We limited the defect category to single and double mono vacancies, according to the size of the supercell.

Nevertheless, two dimensional CNNs can utilize two dimensional model to classify the property maps (PM) [

In our research, a supervised deep learning CNN (D-CNN) model that achieves direct mapping from three dimensional vacancy defected structures to operative diffusivity is proposed. A magnification D-CNN model is presented. The proposed model can extract hidden attributes from the three dimensional defect substrate namely NiCrAlY and define the required information utilized in its predictions. The diffusion activity functions of the three dimensional defect structures with diffusion energy ranging from 0.09 to 0.79 eV are predicted.

This article is structured as depicted: Section 2 presents the study of NiCrAlY and graphene as barriers in Tungsten wires. Section 3 presents the application of deep learning in the classification of the barrier properties for Tungsten diffusion in defected two dimensional barrier layer. The conclusions are presented in Section 4.

In this section, we study NiCrAlY and Graphene as barriers for Tungsten diffusion through defects [

We test our methodology by computing the Copper (CU) diffusion energy on impeccable NiCrAlY. Top sites, bridge adsorption sites, and hollow site are tested. Atoms are restricted to pass perpendicularly on the NiCrAlY plane. The Carbon atoms relaxation process is unlimited. The relaxation process is ceased at Helmunn Feyman power of values less than 10^{−2} Angstrom – ElectronVolt (eV/Å). The CU geometrical diffusion is achieved from the location of the atoms after the process of relaxation. The atom height is computed as the follows:

The site S, from top site, bridge adsorption site and hollow one, of the greatest energy is denoted as the preferred location. The computed CU diffusion energies of NiCrAlY are depicted in

Adsorption site | CU_{H} (Å) |
CU_{E} (eV) |
---|---|---|

HAD (Hollow adsorption site) | 2.069 | −0.1234 eV |

BAD (Bridge adsorption site) | 2.223 | −0.2324 eV |

TAD (Top adsorption site) | 2.259 | −0.2543 |

Presence of vacancies is inevitable in two dimensional barriers due to chemical deposition scheme. These vacancies are vulnerable to additional expansion due to the transfer and the subsequent handling. Among others, the perpendicular diffusion to the two dimensional plane is deliberated as the speediest paths [

Adsorption site | CU_{H} (Å) |
CU_{E} Electronvolt (eV) |
---|---|---|

NiCrAlY-1V | 1.369 | −3.6234 eV |

Graphene-1BV | 1.293 | −5.8324 eV |

Graphene-1NV | 1.81 | −2.6943 |

We also computed Tungsten atom interaction with di-vacancy two dimensional barriers. For NiCrAlY barrier, the Tungsten atom is centrally adsorbed at the di-vacancy with Carbon-Tungsten displacement of value equal to 1.895 Å, as depicted in the following figure (

A seamless two dimensional barrier layer is extremely impermeable to atom particles [

The great diffusion barrier of NiCrAlY can be clarified by computing the difference of the diffusion function of the Tungsten on di-vacancy

Substantial atom migration happens among the Tungsten atom and the adjacent atoms for NiCrAlY and Graphene. Nevertheless, a high reduction occurs surrounding the Tungsten at the NiCrAlY, which proposes a high interaction bond among the Carbon and Tungsten-Boron. To interpret the charge atom rearrangement among the Tungsten and the adjacent atoms, we track the electronegativity of the component’s chemical principle; with Tungsten of low 1.93 electronegativity, then Boron with 2.05 value, Carbon of 2.49 and Nitrogen with 3.06 electronegativity. Thus, the Tungsten atom gives charge to the adjacent atoms with more electronegativity. Hence, charge density difference is surfaces display higher charge depletion from Tungsten up to Nitrogen. The charge density difference depicts the interaction among the Tungsten atom and other adjacent atoms. A need is required to measure the transferred charge among the atoms. Charge analysis is utilized to compute the gained charges for all the atoms. At the vacancy area in the two structures, Tungsten always gives charge with +0.82e for NiCrAlY and +0.57e for Graphene. Atoms that gained the higher quantity of the charge are Nitrogen with −2.29e, while Boron gave more charge with an extra +2.15e. Carbon atoms, that are adjacent to Tungsten, gained charge in the range of −0.11e and −0.18e. Thus, the energy layer of NiCrAlY attributes to the higher interface of the Tungsten atom at the NiCrAlY, as clarified by the charge density difference charge. The lesser electronegativity of Boron the less interaction with the Tungsten atom is shown. Therefore, Graphene supplies lower energy barrier compared to NiCrAlY.

In this section, we are proposing a new deep learning convolutional neural network (D-CNN) [

The flow diagram of the D-CNN, that incorporates FC layers, is depicted in

We propose the D-CNN which utilizes feature accumulation technique. The feature accumulator is discriminative for the energy barrier thickness prediction. In the presented model, we input the image into a transfer learning neural model for deep feature map extraction. The four final fully connected layers are substituted by the accumulated descriptors pooling layers.

We collected data items to build enough dataset for the training phase of the deep learning model. We utilized charge density difference computations to compute the energy two dimensional blockade of a Tungsten atom. The data descriptor, which includes the structural data of the material, is extremely essential [

In our deep learning model, we define a 4 × 4 super matrices for both NiCrAlY and Graphene. First, a mono-vacancy is represented (64 structures). Each structure has a mono-vacancy as depicted in

We must reconstruct the 3D NiCrAlY samples for the three dimensional D-CNN (3D-DNN) model training to test the model performance. NiCrAlY 3D structures are first produced from 2D samples in different views; then, the effective diffusion activity functions are computed from the energy barrier thickness values as depicted in _{in}) and out-point (_{out}).

The 3D structures are built by a technique through which multiple-view 2D structures are randomly located. It is expected that the 2D structures overlap and their distances are normally distributed. The 2D structures are randomly positioned in a cubic space computing the volume of the 2D structures until it is at the setting value (V). There are two elements in the construction (the vacancy space and the 2D structures). The vacancy defect of the vacancy (D) is the segment of the residual interplanetary that omits the undetected 2D structures (D = 1−V). The initial variables for the 3D structure has the vacancy defect (D), threshold (t), the mean displacement of the 2D structures (dmean), and the diameter standard deviation (σ). The reconstruction has a dmean of 32 u (vacancy unit), and σ equal to 3.1u, and t equals to 0.29, all are fixed, while, the vacancy positions are variable with values 0.22, 0.29, 0.38, 0.49, and 0.61. The 3D volumes with vacancy 0.29 and 0.38 are enlarged up to 12288 items to be utilized for learning by employing the 3D magnification technique. The constructed volumes with values 0.22, 0.29, 0.38, 0.49, 0.61, and 0.72 are enlarged into 52 items of vacancy sites 0.38 to 0.81 and used as input dataset.

Once information of the constructed 3D structures is computed, atom adsorption in the vacancies of the 3D volumes are formed, as depicted in the following equation:
_{out} at

The Computational fluid dynamic simulation model (CFD) computes the actual diffusivity in the training phase of the deep learning model. The CFD model is accurate in predicting atoms diffusion properties in vacancy defect structure [

The diffusion equations are solved through the time vacancy model utilizing the accurate CFD technique, the actual diffusivity of vacancy substrate is attained. An explanation to the CFD is depicted here. The formula to compute the CFD is depicted as follows:

To abolish mathematical error in the implemented simulation, the relaxation time is set to “1”, which is included in the stable range [0.5, 2] [

After obtaining the concentration (diff) and the atoms flux (

The proposed 3D-DNN model indicates that the he input layer uses blocks of data inputs and passes the input to the convolutional one block at a time. The convolutional computes the key features of each block. The max-pooling computes the maximum from the feature map portion to decrease the computational load and pools the important features. The dropout avoids overfitting by dropping some of the output of the pooling layer randomly. The output and the fully connected layer decide the final prediction answer. In our research, the input data items are passed to the 3D convolution layers which extract the features and construct the feature maps. The maximum feature values are pooled and subsampled by the pooling layer. The pooled feature maps are passed to the ReLU activation function to incorporate nonlinearity. The FC layers will condense the information and transfer it to the predicted

The training phase of any deep learning model will require a large size input dataset. A Lengthy unfeasible simulation time will be spent, if all the input data items are to be extracted by the CFD simulation model. To face this challenge, a single structure input will go through data magnification technique as presented. In this technique, data of three dimensional vacancy defect volumes and the resultant features are split into data of reduced vacancy volumes using a sliding window spatial algorithm (SWP). The process of the SWP has an 8u sliding blocks which are utilized to amplify the data items. During the window sliding, symbolic volumes are selected to stop the SWP from selecting the same structure blocks. The converging atoms function values of the bulky NiCrAlY structure are computed by the CFD algorithm. At the last step, we split each of the 24 original structures of vacancies of 0.33 and 0.51 sizes and of 1024u × 1024u × 1024u of vacancy units into 512 sub-structures with size of 128u × 128u × 128u. The vacancy sizes of the generated substructures vary from 0.45 to 0.61. The vacancy properties of the generated substructures (128u × 128u × 128u) are different from the original structure (1024u × 1024u × 1024u). Generating smaller substructures from the bigger structures yields randomness (the original structures have vacancies with disorderly configuration and the generated substructure consists of random configuration of the original ones). The process of splitting the original big structures into smaller substructures will yield different atoms mass flux distribution. Their effective diffusion activity functions are computed by their corresponding flux values. This model can escape the production of abundant actual structures in the chemistry lab which is extremely slow process. The generated 12288 substructures and their computed effective diffusion activity functions are used in the training phase.

The 3D NiCrAlY substrate is a grouping of several 2D vacancy substrate images taken from different views. The spatial associations are ignored by the dropout function. A deep CNN reduces this problem and executes a pooling function with a 3D volume instead of a 2D square structure and passes the data into the 3D-DNN. The NiCrAlY volume data is used as depicted:

The effective diffusivity is computed via the CFD module. The input data, the volumes with vacancies dimensions of 0.52 to 0.61 are used as training data. Hereafter, a small training set with 9000 samples are utilized as the input layer. Other data (3000) with vacancies dimensions of 0.39 to 0.79 are used in classification.

When the input subset and the testing subset are organized, they are used as inputs into the 3D-DNN architecture. However, 9000 input items cannot be just passed to the network. The training subset should be optimized using hyper-parameters. Therefore, the input cube for the input sample will be represented by a matrix of dimensions (32 × 128u × 128u × 128u), depicted as follows:

The hyper-parameters are the number of network layers

Number of 3D-DNN convolutional layers | Dropout layers | ||
---|---|---|---|

1 | 2 | 3 | |

Mean relative error | |||

4 | 24.1% | – | – |

6 | 21.5% | 6.8% | – |

8 | 22.4% | 10.9% | 16.1% |

The relative error

The classification outputs, as depicted in

Increasing the pooling dimension improves the accuracy.

Arrangement of six 3D-DNN layers and two dropouts accomplishes the best outputs.

This method accomplishes a low 9.8% mean error from the testing dataset.

After exhaustive model testing, the hyper-parameters are identified. The architecture of the 3D-DNN model is depicted in

Layer number | Layer | Filter size | Activation |
---|---|---|---|

1 | Input | 128 × 128 × 128 | – |

2 | Convolutional | 26/7 × 7 × 3 | – |

3 | Pooling | 3 × 3 × 3 (max) | ReLU function |

4 | Convolutional | 50/7 × 7 × 3 | – |

5 | Pooling | 3 × 3 × 3 (average) | ReLU function |

6 | Dropout layer | 0.6 | – |

8 | Normalization | 70 | ReLU function |

9 | Convolutional | 90/5 × 5 × 3 | – |

10 | Dropout | 0.4 | – |

12 | Output | 234,810 | – |

Validation of the 3D-DNN investigates the effective diffusion activity functions in the prediction of the 3D-DNN model and the CFD. A comparison is done including the effective diffusion activity functions for the testing samples with vacancies of sizes of 0.42 to 0.81 as predicted via the 3D-DNN, the CFD process, and the experimental mathematics proposed in [

Our proposed model has prediction results similar to the ground truth of the labelled samples on the x-axis. CFD algorithm performs well although inferior to the 3D-DNN model but outperforms both models in [

Phase | GPU (NVIDIA) |
---|---|

3D-DNN training | 14.2 × 10^{−2} h |

CFD training | 15.16 h |

3D-DNN prediction time per input sample (average from 200 run) | 1.45 min |

CFD prediction time per input sample (average from 200 run) | 3.15 min |

When the learning process is finished, the validation subset is used now as an input to the D-CNN to classify the energy value of the vacancy two dimensional substance. The energy of a Tungsten particle is computed according to the vacancy region size and the adjacent particles. For instance, if the defects are adjacent, then the Tungsten atoms will interact highly and will go through greater energy barrier in NiCrAlY (approximately 11.98 eV), it should be concluded that the energy barrier is stable with Carbon mono-vacancy defects. Di-mono-vacancy are molded at the borderline between the two cells with the di-mono-vacancy as depicted, owed to the periodic conditions suspense on the borders. Yet, its energy value is similar to the di-mono-vacancy in the same cell. The existence of the Boron and the Nitrogen atoms in Graphene layer yields double energy barriers for mono-vacancy, subject to the defects conditions. Clearly, there is an association among a particle and its adjacent ones. The pooling function extracts attributes of the elements in the cell matrix and their adjacent atoms. D-CNN identifies these associations and define the features of mono and di structures and link them to the output value during the validation.14000 data items are collected for each material for training and validation. It should be noted that the prediction output accuracy is the same to the energy barrier for both NiCrAlY and Graphene. We then computed the absolute error, coloration (C). These measures are computed from the testing subset to provide an independent performance metric of the of the D-CNN. As depicted in

Two dimensional layer | Mean absolute error | Correlation (%) |
---|---|---|

Graphene | 0.531 | 91% |

NiCrAlY | 0.075 | 99% |

In this research, we proposed a deep learning model to predict the diffusivity of Tungsten atoms through NiCrAlY. We reconstructed the 3D NiCrAlY samples for the three dimensional D-CNN (3D-DNN) model training and testing sets. The NiCrAlY 3D structures are produced from 2D samples in different views. The effective diffusion activity functions are computed from energy barrier thickness values. The proposed model studied NiCrAlY as barrier for Tungsten atom diffusion. The Tungsten atom practices stronger barrier when overpass the di-vacancy porous of NiCrAlY structure. Hundreds of reproductions have been made to produce energy barriers of NiCrAlY. These datasets are utilized to reconstruct 3D substrate to perform learning and testing for our D3 CNN. Our model shows a high classification accuracy. Our trained deep learning neural model can predict the energy barrier of Tungsten diffusion through arbitrarily configured NiCrAlY barrier with accuracy greater than 98.4% in 5 × 5 cell. Prediction results generated directors on selecting barriers, and used machine training to calculate the performance. Also, the 3D-DNN model needed 14.2 × 10–2 h for the training phase. The 3D-DNN model is proven to be faster by two orders of magnitude than state of the art models. The 3D-DNN model will require half prediction time as required by other compared models.

We would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R120), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.