## Abstract

Next-generation interconnects utilizing mechanically interlocking structures enable permanent and reworkable joints between micro-electronic devices. Mechanical metamaterials, specifically dry adhesives, are an active area of research which allows for the joining of objects without traditional fasteners or adhesives, and in the case of chip integration, without solder. This paper focuses on reworkable joints that enable chips to be removed from their substrates to support reusable device prototyping and packaging, creating the possibility for eventual pick-and-place mechanical bonding of chips with no additional bonding steps required. Analytical models are presented and are verified through finite element analysis (FEA) assuming pure elastic behavior. Sliding contact conditions in FEA simplify consideration of several design variations but contribute ∼10% uncertainty relative to experiment, analysis, and point-loaded FEA. Two designs are presented; arrays of flat cantilevers have a bond strength of 6.3 kPa, and nonflat cantilevers have a strength of 29 kPa. Interlocking designs present self-aligning in-plane forces that emerge from translational perturbation from perfect alignment. Stresses exceeding the material yield stress during adhesion operations present a greater concern for repeatable operation of compliant interlocking joints and will require further study quantifying and accommodating plastic deformation. Designs joining a rigid array with a complementary compliant cantilever array preserve the condition of reworkability for the surface presenting the rigid array. Eventual realization of interconnect technology based on this study will provide a great improvement of functionality and adaptability in heterogeneous integration and microdevice packaging.

## 1 Introduction

### 1.1 Heterogeneous Integration.

Integration of separately manufactured micro-electronic components into a larger assembly requires new strategies as devices have miniaturized and pushed operation to increasingly higher frequencies. These heterogeneous integration challenges include standard packaging concerns such as mechanical joining, rework, thermal expansion mismatch, thermal management, and electrical connections, and additional unique challenges such as alignment, coupling of radio frequency (RF) signals, accommodation of unique material constraints, small contact points, and assembly and manufacturing time [1]. Conventional approaches to chip integration such as wire bonding, solder, epoxy, and cold welding (Au–Au joints, etc.) or brazing face limitations of messiness, accuracy, temperature, signal loss, and process time, motivate the search for novel technologies for heterogeneous integration [1–11]. This work examines the potential for microfabricated interlocking structures to achieve manufacturing integration of heterogeneous components as seen in Fig. 1.

During the process of developing new packaging for micro-electronic devices, or in creation of specialized systems integrating several unique components, it is often desirable to remove and replace components. Such reworkability becomes a desirable feature because allows custom assemblies to be saved and reused in the event a bonded peripheral device fails. Using a traditional bonding method such as soldering and epoxy requires a tedious and difficult reworking tedious and difficult reworking process, which can result in damage to the components. A method of joining where components could be removed simply with mechanical force could be highly advantageous to prototyping. Mechanical interlocking poses one potential solution to these problems. Mechanical interlocking relies on small structures which join or “hook together” and bending of the interlocking structures is where strength and stiffness come from. This is akin to Velcro™ and related hook-and-loop materials where the hooks attach to some complementary attachment, and the hooks bend as the two opposing sides are separated. This differs from typical adhesives which rely on some form of chemical bonding, dry adhesive brushes using van der Waals bonding, or intermetallic bonding as with solder.

For chip attachment, there are two types of attachment tasks. There may be purely mechanical attachment; this would require patches of material on a chip to provide a mechanical joint in some unused portion of the chip footprint. Alternatively, bonding on chip contact pads adds electrical signal transfer to the mechanical attachment. In both cases, it is advantageous to consider the properties of some typical attachment patch as a means to draw abstract mathematical analysis into practical design choices.

### 1.2 Microfabricated Interlocking Structures.

In micro- and nanoscale systems, many compliant interface technologies, Table 1, have been considered for electrical [7,41] and mechanical interconnects. Our present study builds from earlier work [19], which demonstrated a proof-of-concept for mechanical interconnection using large-deflection cantilevers of nanometer-thickness films, bonding silicon chips together. In this initial demonstration, an interfacial mechanical bonding system joined heterogeneous microfabricated die with ∼34-nm thickness suspended structures fabricated using polyimide and atomic layer deposition, in combination with surface micromachining and lithography. For this study, we assume fabrication based on metal thin films with processing similar to this earlier experimental demonstration or other standard metal microfabrication processes. Earlier published work with related microfabricated designs [15–17] had not used compliant systems and had observed limited effectiveness due to brittle material failure. In contrast, our approach here enables controllable manufacturability, electrical conduction, and the potential for high bond strength. Many considerations guide an effective design. These are presented in Table 2, and to establish feasibility of a reworkable bonding system, this paper focuses particularly on the first three specifications: adhesion, alignment, and force asymmetry.

Technology | References |
---|---|

Miniaturized hook-and-loop systems | [12–14] |

Interlocking barb-like structures | [15–18] |

Comment: brittle, 1.1 MPa yield strength | |

Interlocking tabs | [19] |

Comment: our approach | |

Bulk micromachined pillars with angular sidewalls | [20] |

Sliding channel-locks | [21] |

Comment: complex assembly | |

Carbon nanotubes, nanowires, fibrils, and frictional pillars | [22–29] |

Gecko-like van der Waals adhesive structures | [30–40] |

Comment: not electrically conducting, ∼1 MPa strength |

Technology | References |
---|---|

Miniaturized hook-and-loop systems | [12–14] |

Interlocking barb-like structures | [15–18] |

Comment: brittle, 1.1 MPa yield strength | |

Interlocking tabs | [19] |

Comment: our approach | |

Bulk micromachined pillars with angular sidewalls | [20] |

Sliding channel-locks | [21] |

Comment: complex assembly | |

Carbon nanotubes, nanowires, fibrils, and frictional pillars | [22–29] |

Gecko-like van der Waals adhesive structures | [30–40] |

Comment: not electrically conducting, ∼1 MPa strength |

Property | Performance targets and comments |
---|---|

Adhesion | • Pull strength target ≥ 10 MPa (∼10% of standard epoxies) |

• Pull strength | |

• Pull strength refers to a force perpendicular to the substrate, and shear is parallel to the substrate | |

• Shear strength | |

Alignment | • Misalignment tolerance large enough that the chips can be pushed together easily |

• Translational | |

• Rotational | • Tolerance ≥ 3 μm |

Force asymmetry ratio (insertion force : retention force) | • Reworkable joints require symmetric forces |

• Easy attachment and strong retention strength require strongly asymmetric forces | |

• With flat design, push-in strength will be the same as pull-out | |

• Permanent joints should require force to separate chips be higher than force to join them | |

Thermal tolerance | Must be able to operate at $200\u2009\xb0\u2009C$ without significant loss in adhesion strength |

Electrical contact | • Design will conduct electricity without significant increase in resistance and with little interference from RF |

• May be used as connections to periphery devices and replace traditional wire bonds | |

Rigid connection | Once chips are joined, there is no allowed movement between them |

Property | Performance targets and comments |
---|---|

Adhesion | • Pull strength target ≥ 10 MPa (∼10% of standard epoxies) |

• Pull strength | |

• Pull strength refers to a force perpendicular to the substrate, and shear is parallel to the substrate | |

• Shear strength | |

Alignment | • Misalignment tolerance large enough that the chips can be pushed together easily |

• Translational | |

• Rotational | • Tolerance ≥ 3 μm |

Force asymmetry ratio (insertion force : retention force) | • Reworkable joints require symmetric forces |

• Easy attachment and strong retention strength require strongly asymmetric forces | |

• With flat design, push-in strength will be the same as pull-out | |

• Permanent joints should require force to separate chips be higher than force to join them | |

Thermal tolerance | Must be able to operate at $200\u2009\xb0\u2009C$ without significant loss in adhesion strength |

Electrical contact | • Design will conduct electricity without significant increase in resistance and with little interference from RF |

• May be used as connections to periphery devices and replace traditional wire bonds | |

Rigid connection | Once chips are joined, there is no allowed movement between them |

Mechanical retention by interlocking compliant structures is a subset of “integral attachment” fastening systems. Integral attachments use mechanical parts built into assembling components. A classic example of such systems are snap-fit components such as hook and latch systems, and a wide variety of designs have been explored [42,43]. These components present advantages in mechanical design such as low insertion force, high retention force, simple insertion motion by pushing, and easy automation of assembly [44–46]. They have been suggested for joining polymer-matrix composite structures [47], explored for fabrication in three-dimensional printing [48], and explored for heat-activated modification as design for disassembly [49,50].

Nano-indenter-based measurement of the cantilever spring constant in the earlier proof-of-concept paper [19] suggested that the deflection forces and bonding stresses of interlocking microfabricated joints could be up to 15 MPa, nearly an order of magnitude better than typical values for dry adhesives based on van der Waals force or earlier work based on rigid microbarbs [17,30]. This result strongly suggested the presence of additional stiffening effects acting on cantilever deflection such as through residual-stress-induced curvature of cantilevers. For the present analysis, we assume no residual stress and instead focus on the consequences of large-deflection bending of flat cantilevers.

### 1.3 Mathematics of Interlocking Cantilevers.

Recent developments in analytical modeling of compliant mechanisms have allowed a framework for modeling of interlocking systems, including our group's contribution, a general solution to the analysis of geometrically nonlinear elastic mechanics problems of large-deflection bending under contact boundary conditions, which create a statically indeterminate problem [51]. These advances based on elliptic integral calculations may be used in place of more cumbersome finite element or iterative methods in order to provide a theoretical prediction for this class of large-deflection mechanics problems. Here, we extend this mathematical theory to analysis and design of compliant retention systems for application in heterogeneous integration.

## 2 Modeling and Design

In exploration of this design problem, it became evident that reworkability did not require identical parts on mating surfaces, and particularly the areal density of compliant joints, and the strength of individual joints, could be maximized if one surface presented rigid parts or quasi-rigid parts, while the complementary surface retained compliant parts to enable snap-through interlocking. This insight drives the implementation of compliant joints as a surface adhesive metamaterial array, and the analysis presented below.

### 2.1 Small-Deflection Analysis.

*s*along the curve is a function of the bending moment at that point along the beam and is modulated by the flexural rigidity $EI$, Eq. (1). All analytical models here also assume that cantilever contact points are frictionless, and the cantilevers are inextensible, thus all deflections are due to bending. The flexural rigidity is assumed to be constant along the length, and the thickness of the beam is much smaller than the length. Solving Eq. (1) for a flat beam subjected to a point load at the end results in Eq. (2), which models small cantilever deflections, where the end displacement $\delta $ is proportional to the applied load

*P*. This model works well for small displacements, but is no longer valid once the end of the beam is deflected ∼$10\u2009deg$ or more

### 2.2 Large-Deflection Analysis.

Thin-film interlocking structures may deflect to an extent beyond the customary small angle assumption of a few degrees, thereby requiring a large displacement model. Previous work [51] presented an approach to modeling interlocking cantilevers subject to large deflections; this model was implemented here with specific geometric choices for device design. Comparison of the large-deflection and small-deflection models for interlocking horizontal cantilevers subject to vertical displacement is provided in Fig. 2.

Relatedly, the material stresses within the cantilever can be determined analytically [51], supplementary section (S.1).

### 2.3 Finite Element Modeling.

Finite element analysis (FEA) was performed using commercial comsolmultiphysics 5.3a software to verify the analytical methods as well as to enable analysis of more complicated geometries that may add tedious complication to a purely analytical approach. A stationary study was performed using the solid mechanics module, approximating a quasi-static testing of an elastic material. “Form assembly” was used to create a frictionless contact pair between the two cantilevers. The maximum von Mises stress and contact force were found from surface maxima in postprocessing of model results. In simple contacting flat cantilever studies, we observed a divergence, Fig. 2, of about 10% from the peak value of the large-deflection analytical model. In previous work, the analytical flat cantilever model was observed to match well with macroscale experimentation and FEA based on point loading perpendicular to the cantilever end [51]. We examined several potential sources for the error in the contacting cantilever FEA used in this study, with more detail provided in supplementary section (S.2), along with more details of the FEA model. Reproducing the previous point-load FEA [51] produced good agreement with the analytical model, indicating that the newer FEA is not intrinsically a source of error. Furthermore, mesh and cantilever aspect ratio showed no significant effect on the error. Therefore, the error was most likely due to implementation of the contact boundary condition in the present FEA model. From this, we anticipate that FEA using the contact condition may imply a 10% deviation from analytical and experimental results.

### 2.4 Misalignment and Self-Alignment.

The force and bond strength analyses above assume that the interlocking structures are perfectly aligned. When attempting to join the interlocking structures with one another, it would be reasonable to assume that there would be some deviation of the positioning of the chips from the ideal location. To assess effects of any deviations from this ideal, the model was modified to include a translational misalignment factor *μ*, whereby the amount of deviation from the ideal center would factor into the amount of force holding the cantilevers together. This factor is important to the performance of the structures and helps in understanding the requirements for the precision of equipment needed to assemble devices. Joining methods such as epoxy and solder this factor is not as important as a poorly misaligned chip will still function the same, whereas with these periodic structures, misalignment could have a drastic effect on the performance.

Even small translational misalignments may consume a large fraction of the cantilever interaction lengths, leading to significant deviation from the perfectly aligned model. Other misalignments are less important, and other design factors such as the specific shape of the cantilevers, pitch, material thickness, and residual stresses all play a role in the strength of system but are intrinsic to design and manufacturing, and relationships among these are considered throughout this paper. The implementation of fabricated structures in device assembly depends on the resilience to rotational and translational misalignments between joining surfaces, which affect the final assembly and have the potential to determine whether or not interlocking structures are a viable solution toward heterogeneous integration. The joining mechanism operates through out-of-plane motion; therefore, out-of-plane translational and rotational misalignments are accommodated through the joining mechanism. For the analysis presented here, these are not limiting factors to the viability of the mechanism. In-plane rotational misalignment is also not a significant factor for initial design considerations. Rectilinear objects such as microchips can be rotationally aligned to a good degree of accuracy through even simple techniques such as contact with a flat surface. Furthermore, the rotation giving a 10 *μ*m misalignment at the edge of a 1 cm^{2} chip is only about 0.11 deg ($=tan\u22121(10\u22125\u2009m/5\xd710\u22123\u2009m)$); this is not enough to significantly modify the cantilevers from basic rectangular geometry. Across an array of interlocking structures, small rotational misalignments would manifest locally as translational effects on the contact force, with only minor effects due to small rotation of the contacting cantilevers. It should be noted that any large rotational misalignment that would cause any of the structures to not align would mean the structure as a whole could not be inserted or it means damage to those structures.

*μ*. This results in the interaction distance between two cantilevers to either grow or shrink the amount

*μ*. The snap-through force for two pairs of cantilevers on the same interlocking structure with some misalignment can be found with Eq. (4). The maximum bonding strength can then be formulated as Eq. (5). It should be noted that these formulations are approximations, with misalignment one pair will slip before the other, at which point the entire structure will snap-through

From Fig. 3(d), the net maximum bonding strength increases as misalignment increases. It should be noted that percent change is relatively small, and the figure has been drawn to enable visualization of the relationship. The net horizontal force also increases as the misalignment increases, Fig. 3(e), which suggests that there is an inherent self-alignment behavior where the chips will be pushed toward the ideal center position.

## 3 Periodic Array Designs With Interlocking Cantilevers

### 3.1 Design Implementation for Reworkable Interlocking Structures.

As mentioned previously, there is a balance among the beam parameters for the structures to prevent permanent deformation but also maximize bond strength. Under loading, the bending stresses may quickly exceed the yield strength of the material, becoming permanently deformed, thus making it unsuitable for reusable attachment.

Design begins by first selecting a desired force to displace the cantilevers. In the large-deflection analysis from Sec. 2.2, it was assumed that the cantilevers would always be sufficiently long that the cantilevers would experience the peak nondimensional force of 0.417. Selecting a nondimensional force before reaching the peak will give similar performance with less deflection and internal stress occurring. In Fig. 4(a), this is shown with label (A) where a snap-through displacement is selected at 0.3, which produces a snap-through force of 0.36, this is nearly 80% of the maximum, but importantly necessitates only 63% of the displacement required for the peak force.

A new nondimensional term $L*=L/L0$ is then introduced, which is the arc length $L$ of the beam from the anchor point to the loading point, as drawn in Fig. 2(a), divided by the horizontal distance $L0$ of the loading point to the anchor point. Another nondimensional term $Ar=L/t$ is introduced; this is the aspect ratio and is defined as the dimensionless measure of the total cantilever length $L$ (which is defined by the arc length at snap-through) to its thickness $t$. This term is important for further analysis and becomes one of the most important parameters that will determine many parameters in the design.

*L** can be found with the deflection from Fig. 4(a), as indicated with label (B). Next,

*A*can be found using Fig. 4(c). Here, plots of the maximum material stress at given displacements as functions of

_{r}*A*are plotted. These lines are Eq. (6) evaluated at the end angle $\theta B$ at a given dimensionless displacement $\delta B$ [51]. In Fig. 4(c), these lines are shown by label (C)

_{r}The yield strength of the material is plotted as a horizontal line. At the intersection of the stress plots (C) with the yield strength, the minimum *A _{r}* is obtained. Selecting an

*A*lower than this value will result in the bending stresses exceeding the yield strength and will result in permanent deformation of the structures.

_{r}The aspect ratio constraint interacts with constraints of lithography and fabrication processes to define the geometry for a repeating unit in an array of interlocking cantilevers, illustrated in Fig. 5. Geometric parameters in the unit cell are *D*, Δ, *ω*, *L*, and *L*_{0}, where *D* is the width of the pillar that suspends the cantilevers in free space, Δ is the width of the rigid pillar (here set equal to *D*, for simplicity), and *ω* is the length of the rigid cantilever that extends from the rigid pillar. Unit cell pitch $\rho =2(L0+\omega +D)$ is determined by the sum of other parameters as shown in Fig. 5(c).

*D*can be obtained by plotting interfacial strength $\sigma m$ as a function of

*D*, Eq. (7), Fig. 5(d). Doing so will result in a graph that peaks at some value of

*D*, then decrease toward 0 as

*D*continues to increase. The peak of this graph is the maximum possible bond strength for the given parameters. Following these steps, the optimal interlocking structure geometry is obtained

As a specific example following this procedure, consider titanium as a fabrication material, due to compatibility with common materials in micro-electronics coupled with high stiffness and high yield strength. With $\delta B=0.30$ and corresponding snap-through nondimensional force $C1=0.36$, *L** is then 1.05 and $Ar=250$ (Fig. 4(c)) to maintain operation under elastic behavior. Applying Eq. (7) with the parameters from above, and selecting an *ω* value of 4 *μ*m, *D* is selected to be 20 *μ*m and leads to a $\rho $ of 42 *μ*m. This configuration then leads to a maximum bond strength of 250 Pa as shown in Fig. 5(d).

It is clear from this analysis that designing interlocking structures that remain within the elastic regime of its material will lead a low-performing material. Pure elastic operation is required of patterned surfaces that can be separated and reattached repeatedly, but this comes at the price of adhesion strength. The condition of reworkability can be preserved if the die bearing the compliant cantilevers is afforded some plastic deformation and treated as a single-use component. In this case, the surface of patterned rigid structures enables attachment, removal, and replacement of components.

Following the design and optimization strategy above while allowing some plastic deformation, a design for interlocking flat cantilevers shows the possibility of significantly better performance. First, *L* and *L*_{0} are selected to be 10 *μ*m and 8 *μ*m, respectively. This gives $L*=1.25$, which means it will reach the maximum $C1=0.417$ and $Ar=100$. These design parameters feed into the relations above all to generate the resulting parameters in Table 3, which are illustrated as the specific models in Figs. 5(a) and 5(b). These parameters produce a snap-through force per cantilever of 0.81 *μ*N, which leads to a bond strength of 6.3 kPa, which is a theoretical maximum comparable to the performance of commercially available hook-and-loop materials. This shows that these micro-interlocking structures have great promise in improving integration methods of chips, but more work is required to better refine their design through improved modeling of plastic behavior coupled with physical testing of the metallic films that will comprise these structures.

### 3.2 Design of Interlocking Arrays of Nonflat Interlocking Cantilevers.

While exploring the mechanical behavior of design variations seeking to reduce internal material stresses, we observed the development of force asymmetry in interlocking “L” shapes similar to Fig. 5(c) while allowing compliance in the vertical support of the compliant cantilever. Unfortunately, the result was opposite of ideal for the attachment problem: L shapes created high insertion force and low retention force, with corresponding high and low probabilities of exceeding the yield stress. We hypothesized that this result could be applied to improve performance by flipping the L structure and attaching it to a rigid support; this resulted in a concept for a nonflat cantilever design. A model of the repeating unit cell of the proposed design can be seen in Fig. 6(a). Finite element simulation confirmed that the added bend allows a low push-in force, and relatively higher force required to separate the components. For implementation in a specific design, a rigid permanent structure is again provided similar to above. With the added shape, it is necessary to include additional parameters for design, seen in Fig. 6(b) and specified in Table 4.

L_{c} | 3 μm |

L_{0} | 11 μm |

t | 100 nm |

D | 12 μm |

Δ | 12 μm |

H | 15 μm |

H_{B} | 5 μm |

$\rho $ | 50 μm |

ω | 4 μm |

L_{c} | 3 μm |

L_{0} | 11 μm |

t | 100 nm |

D | 12 μm |

Δ | 12 μm |

H | 15 μm |

H_{B} | 5 μm |

$\rho $ | 50 μm |

ω | 4 μm |

The performance of this design was evaluated with FEA, under pure elastic conditions. The maximum von Mises material stresses are shown in Fig. 6(c). Plastic deformation is expected to occur as the yield strength of titanium is 140 MPa is exceeded quickly. The maximum force required to interlock a pair of cantilevers was 2.5 *μ*N and to separate required a force of 9 *μ*N, Fig. 6(d). This corresponds to push-in and pull-out strengths of 8 kPa and 29 kPa, respectively, higher than the maximum 6.3 kPa pull-out strength found earlier for arrays of simple flat cantilevers. This design is promising for future work as it gives potentially high retention strength in a reworkable design, but much more optimization is required.

## 4 Discussion

Traditional methods of joining chips such as epoxy and solder can be problematic because of material cleanup, failure under thermal cycling, and reworkability requiring elevated temperatures or chemical solvents to remove the bonding material. Compliant attachment presents the potential for die to expand freely without creating high thermal stresses that could cause failure of the joint. Electrical connections can also be potentially made using these structures, meaning techniques like wire bonding can be avoided, reducing packaging complexity and potentially improving performance of devices such as RF devices which operate at high frequencies.

The analysis and design efforts in Sec. 3 support the potential for compliant mechanical die attachment systems, but consideration of the internal bending stresses in the cantilever material is critical for successful design. The yield stress is quickly exceeded for most materials; designs which rely on purely elastic bending are may be expected only to have weak performance. Much more work will need to be performed to include plastic deformation and other considerations such as fatigue studies and nonflat or curved designs, and further design optimization may be possible through sensitivity analysis and virtual design-of-experiments modeling considering material and geometry variability. Interlocking cantilever array metamaterial attachment systems show promise for mechanical connection, but further studies must be performed to show acceptable electrical and thermal performance. In RF applications, it must also be shown that they can outperform other methods for electrical connections, and that signals do not degrade and experience little to no interference.

The challenge then is to improve the performance to match more permanent attachment methods. Exploration of different materials which can sustain large displacements without permanent deformations is one way that performance can be increased. For example, certain formulations of shape memory alloys such as Nitinol display hyperelastic behavior, where the elastic region of the material is much higher than in typical engineering materials. To reduce the bending stresses, one approach is to process the films such that the sharp corners will be smoothed out into curves. Once the interlocking surfaces have been joined, another concern is the free movement of the chips, i.e., whether the joint experiences any “play.” To stop this free movement, the cantilevers can be designed so that their lengths are longer than the interaction distance *D*. This would imply the cantilevers would always be in contact with the opposing pillar.

The impact of interlocking structures on nanoscale and microscale designs will be to enable greater interfacing and adaptability of sensors within microsystem and nanosystem packaging. It could be possible to scale this technology down from the microscale to the nanoscale, possibly even to atomically thin films such as two-dimensional nanomaterials like graphene or boron nitride. With reduction to the nanoscale, surface effects such as van der Waals bonding and cold welding arise and may require consideration in design. Other areas which can be explored include the mechanisms of load, phonon propagation, electron transfer, and scaling effects which can affect larger systems.

## 5 Conclusion

Mechanically interlocking structures present a promising technology for heterogeneous integration. The ability to remove microdevices from larger assemblies has the possibility to make microdevices simpler to service and reuse when prototyping or when replacing dead components on a final product. This paper explored the elastic constraints on design of arrays of mechanically interlocking cantilevers, forming complementary metamaterial surfaces for mechanical adhesion. Interlocking structures with flat cantilevers may have a theoretical bond strength up to 6.3 kPa. Structures with nonflat cantilevers are proposed which require 8 kPa to join chips and require 29 kPa to separate them. Applications in RF and reworkability are the main areas where this technology provides the most distinct advantage over current state-of-the-art. Remaining issues with this technology are low bonding strengths and accommodation of plastic deformation from internal stresses during die attachment and in modeling. Designs which operate in the purely elastic regime will allow reuse at the cost of low performance. If plastic deformation is allowed to occur on the interlocking surface supported on a replaceable component, the performance may increase significantly to the point of being competitive with other surface bonding technologies.

## Acknowledgment

The authors gratefully acknowledge the following individuals for helpful discussions, feedback, and insight in the pursuit of the ideas presented in this paper: Dr. Eric Kreit, Dr. Harris Hall, and Dr. David Torres-Reyes, U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base, OH; Professor Marcelo Kobayashi, University of Hawai‘i at Mānoa. The authors also thank the following individuals for review of drafts of this paper: Dr. Tyler Ray and Dr. Lloyd Hihara, University of Hawai‘i at Mānoa.

## Funding Data

Air Force Office of Scientific Research (Award No. FA9550-20-1-0256; Funder ID: 10.13039/100000181).

Air Force Research Laboratory Summer Faculty Fellowship Program (AFRL SFFP; Funder ID: 10.13039/100006602), summer 2020.

## Nomenclature

*A*=area of a unit cell of an interlocking array

*A*=_{r}aspect ratio, nondimensional measure of total cantilever length divided by cantilever thickness

*b*=cantilever width

*d*=cantilever interaction distance, distance between the two fixed ends of a pair of interlocking cantilevers

*D*=pillar width, width of interlocking cantilever

*E*=Young's modulus of the material

*EI*=flexural rigidity, the product of the Young's modulus and the moment of inertia

- $f$ =
a function of elliptic integrals and $\theta B$, defined in Ref. [51]

- FEM =
finite element modeling

*H*=pillar height

*H*=_{B}height of bend in nonflat cantilever

*I*=moment of inertia of the beam cross section

*L*=arc length of cantilever between anchor point and contact point

*L*=distance from loading point to cantilever fixed end

*L** =dimensionless measure of

*L*and*L*_{0}*L*=_{c}length of nonflat cantilever extending from anchor point before vertical wall downwards

*L*_{0}=horizontal distance from cantilever end to contact point

*M*=bending moment at point

*s*along cantilever*N*=number of cantilever pairs in a unit cell

*P*=vertical force applied to the tip of the horizontal cantilever

*P*_{1}=interlocking cantilever pair with an increasing interaction distance as misalignment increases

*P*_{2}=interlocking cantilever pair with a decreasing interaction distance as misalignment increases

- RF =
radio frequency

*s*=arc length, used to parametrize the points of the beam

*t*=cantilever thickness

- $\delta $ =
vertical displacement of the tip of a horizontal cantilever

- Δ =
width of pillar of rigid interlocking structure

*θ*=tangent angle at point

*s*along cantilever- $\theta B$ =
cantilever end angle

- $\kappa $ =
curvature

*μ*=measure of array misalignment from ideal center

*ρ*=pitch

- $\sigma b$ =
maximum bending stress within the cantilever

- $\sigma $
=_{m} maximum snap-through strength

*ω*=length of overhang of rigid interlocking structure