224.

However, when the bimorph stack is reversed as in FIG. 2C, (aluminum oxide 222r on top of aluminum 220r), the beam bends downwards (towards the substrate) upon release. Note however that from a manufacturability perspective, it is quite complicated to create a freestanding structure and then create a beam that can move upside down, as that puts strain on the overhang and the structure may collapse. As such, an upward curving resulting from residual stress as in the cantilever 228 of FIG. 2B is described hereinafter in the examples.

To summarize, bimorph material such as crafted from an aluminum and aluminum oxide stack atop a silicon (Si) substrate, is initially supported by a sacrificial layer of silicon dioxide. A subsequent fabrication step involves selectively etching away the sacrificial layer beneath the inner middle ring (while not fully etching away the outer ring/disk, if not otherwise anchored, so that they remain fixed to the substrate). Upon release, residual stresses prompt the non-anchored portion of the middle ring to deform upwards (away from the substrate), forming a structure akin to a MEMS cantilever attached to the substrate by two fixed anchors. The anchors can also act as bias pads.

FIG. 3 shows a reconfigurable intelligent surface 330 that is a made of an array of unit cells; one such unit cell 302 is labeled. As can be seen from the enlarged views 330(e1) and 330(e2) (showing the residual stress gradient in the thin flexure ring), the bimorph metallization middle ring 308 of the bullseye structure is bent upwards due to residual stress post-fabrication.

When a voltage difference is applied across the anchor pads, e.g., as controlled by a controller 332, the resulting current through the bimorph metallization middle ring 308 causes the joule heating effect, resulting in out-of-plane displacement or curved motion of the middle ring's free end due to the inequal thermal expansion in both materials. Because aluminum oxide has a lower coefficient of thermal expansion relative to aluminum, a suspended structure (the non-anchored portion of the cantilever) bends downwards with increasing temperature, that is, the deformation in the actuated state causes the middle ring to “straighten out” relative to its non-actuated state. The extreme examples are shown in FIG. 4A (zero voltage, largest bend/vertical displacement) compared to FIG. 4B (highest applied voltage, lowest bend/vertical displacement). This structural deformation under the effect of temperature caused by electrical voltage is called electrothermal actuation.

Hence, in the post-sacrificial layer release state, the middle ring is bent upwards with an angle of θ as shown in FIGS. 5A-5C, showing the 0 V non-actuated state of the bimorph ring. The vertical displacement, δ, of the tip of the free portion of the bimorph ring under electrothermal actuation due to thermal expansion without any other external forces applied, is derived from the thermal expansion equation and the bending moment equation:

δ
=

3
⁢

W
1

⁢

W
2

⁢

E
1

⁢

E
2

⁢

t
1

⁢

t
2

(

t
1

+

t
2

)

⁢

(

α
2

-

α
1

)

⁢
Δ
⁢

T
·

D
2

(

E
1

⁢

W
1

⁢

t
1
2

)

2

+

(

E
2

⁢

W
2

⁢

t
2
2

)

2

+

2
⁢

W
1

⁢

W
2

⁢

E
1

⁢

E
2

⁢

t
1

⁢

t
2

(

2
⁢

t
1
2

+

3
⁢

t
1

⁢

t
2

+

2
⁢

t
2
2

)

where W₁is the width of Al; W₂is the width of Al₂O₃; t₁is the thickness of Al layer, t₂is the thickness of Al₂O₃layer; α₁is the coefficient of thermal expansion of Al (=23.1×10⁻⁶K⁻¹for Al); α₂is the coefficient of thermal expansion of Al₂O₃(=8.1×10−6 K−1 for Al₂O₃); E₁is the Young's modulus of Al (=70 GPa (gigapascals)); E₂is the Young's modulus of Al₂O₃(=530 GPa); ΔT is the change in temperature due to joule heating; and D is the outside diameter of the displacement ring. This is a simplified expression that does not incorporate all factors; the complete equation is significantly more complex and often requires numerical methods for precise solutions.

To summarize, when electrical voltage is applied to the aluminum anchor pads, the temperature of the middle ring increases due to joule heating. This causes aluminum and aluminum oxide to expand at different rates, and because aluminum oxide has a lower coefficient of thermal expansion relative to aluminum, the suspended ring bends downwards towards the substrate with increasing temperature; that is, the vertical displacement ò is reduced based on the applied voltage. This continuous structural deformation drives a shift in the resonance frequency of the unit cell towards lower frequencies. Consequently, an incident electromagnetic wave reflects off the two-dimensional array of such reconfigurable intelligent surface unit cells, exhibiting varying phase values determined by the structural displacement in each cell. These individually controllable phase values can thus be used for constructive/destructive interference to beamform the reflected electromagnetic wave.

Turning to an analysis, a finite element analysis tool (e.g., COMSOL MULTIPHYSICS) was used; note that the interactions between thermal expansion, material stiffness, and geometric dimensions lead to nonlinear effects that are not easily captured in a closed-form equation. In general, finite element analysis involves dividing a continuous domain into smaller, simpler elements, referred to as the meshing as shown in FIG. 6A. Finite element analysis solves partial differential equations, which are discretized over the elements of the mesh to approximate the behavior of the system; (note that without meshing, it would be highly impractical to solve these equations numerically). The finer the mesh, the more accurate the response generated, e.g., the substrate 112 has a coarse mesh, whereas the elements of the resonating pattern 104 have a finer mash.

To evaluate the displacement of the middle ring that acts like a cantilever, a voltage difference of 20 V is applied on the anchors. FIG. 6B shows the voltage difference in one example simulation setup. The corresponding normalized electric field distribution is shown in FIG. 7A.

Under no external bias, the bimorph ring is at room temperature and curved upwards due to residual stress as shown in FIG. 7B. When electrical current passes through the thin aluminum ring, the temperature of the entire unit cell increases due to joule heating. This rise in temperature led to the downward deformation of bimorph middle ring (towards the substrate) as shown in FIG. 8A.

As the voltage is decreased (the device starts to cool down), the ring starts coming back upwards, highlighting the repeatability of the operation. The curve in FIG. 8B shows the normalized displacement at the tip of the ring achieved with the applied voltage. Full wave analysis can be done in 3D electromagnetic simulation software (e.g., Ansys HFSS) to extract the reflection characteristics of the unit cell for the tuning range. Periodic boundary conditions can be used during simulation of the cell, which assume an infinite structure. A parametric simulation can be done for different displacement levels of the ring cantilever by adjusting the applied voltage. The resonance frequency is at 42 GHz when the ring is suspended after the sacrificial layer etching; this non-actuated state corresponds to a voltage supply of 0 V. Continuous tuning is observed in the resonant frequency with respect to the input voltage. When the voltage is 20 V, the resonant frequency is at 34.5 GHz as shown in FIGS. 9A and 9B.

The reason for this phenomenon is that the resonant frequency of the reconfigurable intelligent surface structure can be given as:

ω
=

1

L
⁢
C

,

where L and C refer to the effective inductance and capacitance in the reconfigurable intelligent surface structure that enables coupling of the incident electromagnetic wave. It can be quantitatively seen that as the bimorph ring bends towards the substrate, the gap between the metal and the substrate decreases. This increases the capacitance, whereby the resonance frequency decreases. Compared to an electrostatic actuation mechanism, a larger range of motion can be achieved by using electrothermal actuation, which causes a larger shift in the resonance frequency and the corresponding output phase.

This change in phase from individual unit cell leads to constructive/destructive interference in the desired direction when combined into an array to form a reconfigurable intelligent surface panel. Such beam steering is shown in FIGS. 10A and 10B in which the reflected beam can be steered in a desired direction based on the voltages applied to the cantilever rings of the unit cells. More particularly, for redirecting impinging signals, the reconfigurable intelligent surface 1030 is coupled to or otherwise incorporates the controller 1032 that controls the phase shifts of the unit cells designed for signal redirection by applying appropriate voltages (e.g., from zero volts to the maximum voltage) to the unit cells to deform respective bimorph middle rings to change their respective phases, facilitating constructive (or destructive interference) for beamforming. Beamforming allows the incoming electromagnetic wave/signal to be redirected (reflected or refracted) as a beam that can be shaped and steered in a desired direction, as shown in FIG. 10A (one set of respective voltages for the respective unit cells), and 10B (another set of respective voltages for the respective unit cells).

To summarize, the MEMS-based bimorph actuator for a reconfigurable unit cell offers a higher tuning range, larger stroke, and enhanced repeatability compared to other unit cell technologies, achieving a maximum tuning range of approximately 8 GHz with a 20 V input voltage. This electrothermal approach is particularly effective for applications needing substantial displacements and force outputs at the millimeter/microscale device level.

One or more example embodiments can be embodied in a unit cell device, such as described and represented herein. The unit cell device can include a microelectromechanical systems (MEMS)-based resonating pattern on a substrate, which can include a fixed resonating portion, and a bimorph MEMS cantilever; the bimorph MEMS cantilever can include an anchored portion and a non-anchored portion. The bimorph MEMS cantilever can have a first vertical displacement relative to the substrate at a tip of the non-anchored portion of the bimorph MEMS cantilever as a result of residual stress, in response to the bimorph MEMS cantilever being in a non-actuated state. The unit cell device further can include electrical contact pads electrically coupled to the bimorph MEMS cantilever, in which energy applied via the electrical contact pads can changes the non-actuated state of the bimorph MEMS cantilever to an actuated state that strains the non-anchored portion of the bimorph MEMS cantilever to change the first vertical displacement distance at the tip to a second vertical displacement distance that is based on an amount of the energy applied. In response to an impinging electromagnetic wave on the unit cell device, the resonating pattern resonates to redirect an instance of the electromagnetic based on a phase shift determined by the first vertical displacement distance in response to the bimorph MEMS cantilever being in the non-actuated state, and the second vertical displacement distance in response to the bimorph MEMS cantilever being in the actuated state.

In the non-actuated state, the bimorph MEMS cantilever can be curved upward, and the first vertical displacement distance can be greater than the second vertical displacement distance.

In the non-actuated state, the bimorph MEMS cantilever can be curved downward, and the first vertical displacement distance can be less than the second vertical displacement distance.

The energy applied via the electrical contact pads can include a bias voltage applied across the electrical contact pads, and the amount of the energy applied can be based on a bias voltage level.