KR102707331B1 - Electroabsorption modulator comprising graphene based waveguide with additional slot - Google Patents

Abstract

The present invention relates to a graphene-based waveguide, which has a strong light-graphene interaction enhanced by a slot, can be implemented using optical lithography, and can be efficiently coupled to a silicon waveguide. The slot in the graphene-based waveguide is wider than that in other graphene-based waveguides. This contradicts the general belief that a narrow slot is preferred for strong light-graphene interaction, and can reduce the burden of fabricating a narrow slot. The present invention relates to an EAM with better performance than other EAMs including graphene-based waveguides. The EAM can be implemented using a silicon waveguide fabrication process, a graphene transfer process, and a lift-off process. The EAM can be well-embedded in a silicon photonic integrated circuit as a compact low-loss modulator.

Description

{ELECTROABSORPTION MODULATOR COMPRISING GRAPHENE BASED WAVEGUIDE WITH ADDITIONAL SLOT}

The present invention relates to an electroabsorption modulator that controls the amount of light absorption according to an electric signal by utilizing two graphene layers.

Taking advantage of the excellent optical properties of graphene, such as electrically tunable photoconductivity, various photonic and plasmonic devices using graphene have been developed. Much effort has been made to develop modulators that include high-speed, compact graphene-based waveguides. It is necessary to design a waveguide structure with strong light-graphene interaction, and strong light-graphene interaction can be achieved when graphene is positioned at a position where the in-plane electric field of the waveguide mode is highly enhanced.

Among various graphene-based waveguides, a graphene-covered slot waveguide may be a desirable structure because the smaller the slot size, the stronger the electric field confined within the slot. The strong light-graphene interaction of the slot-added graphene-based waveguide may require a precise manufacturing process based on electron beam lithography. In addition, the loss of the slot waveguide may increase as the size decreases. The coupling between the slot-added graphene-based waveguide and other silicon photonic waveguides may not be efficient. Therefore, the development of a slot-added graphene-based waveguide that has strong light-graphene interaction, can be implemented by deep UV lithography, and can be efficiently coupled to silicon photonic waveguides may be required.

In one embodiment, a graphene-based waveguide may include: a silicon layer; a first graphene layer positioned on one surface of the silicon layer; an insulating layer covering a portion of the first graphene layer and a portion of the silicon layer; a second graphene layer positioned on the insulating layer and spaced from the first graphene layer by the insulating layer; a first electrode connected to the first graphene layer; a second electrode connected to the second graphene layer, and a metal rail positioned between the first electrode and the second electrode on the second graphene layer.

The silicon layer may include a silicon strip having one side in contact with the first graphene layer and the remaining side covered by silicon oxide.

The width of the above silicon strip may be greater than the width of the slot defined between the second electrode and the metal rail.

The above silicon strip, the metal rail, and the second electrode may not overlap each other when viewed in a direction perpendicular to a plane corresponding to the one surface of the silicon layer.

The above insulating layer may include aluminum oxide.

When the potential of the first graphene layer and the potential of the second graphene layer are different, capacitive coupling can be formed between the first graphene layer and the second graphene layer.

The above first graphene layer and the above second graphene layer,

When viewed in a direction perpendicular to a plane corresponding to the one surface of the silicon layer, an area occupied by the first graphene layer and an area occupied by the second graphene layer in the plane can overlap.

When viewed in a direction perpendicular to a plane corresponding to the one surface of the silicon layer, the width of the area where the area occupied by the first graphene layer and the area occupied by the second graphene layer overlap may be equal to or greater than the first critical width.

The chemical potential of the first graphene layer may be different from the chemical potential of the second graphene layer.

An electric field limited between the first graphene layer and the second graphene layer can be strengthened through a slot defined between the second electrode and the metal rail.

The width of the above metal rail may be greater than or equal to the second critical width.

The height of the above metal rail may be the same as the height of the second electrode.

The height of the slot defined between the second electrode and the metal rail may be equal to or greater than a first threshold height.

The distance between one side of the above metal rail and the first electrode may be constant.

The distance between the other side of the above metal rail and the second electrode may be constant.

The distance between one side of the metal rail and the first electrode may be greater than or equal to a third critical width.

A modulator using graphene-based waveguides,

A graphene-based waveguide comprising: a silicon layer; a first graphene layer positioned on one surface of the silicon layer; an insulating layer covering a portion of the first graphene layer and a portion of the silicon layer; a second graphene layer positioned on the insulating layer and spaced from the first graphene layer by the insulating layer; a first electrode connected to the first graphene layer; a second electrode connected to the second graphene layer; and a metal rail positioned between the first electrode and the second electrode on the second graphene layer.

input coupler; and

May include an output coupler.

FIG. 1 is a perspective view of a modulator (100) using a graphene-based waveguide according to one embodiment, FIG. 2 is a top view of the modulator using the graphene-based waveguide, and FIG. 3 is a cross-sectional view taken along line AA' of the modulator using the graphene-based waveguide.
Figure 4 shows a profile of the electric field distribution of a graphene-based waveguide mode according to one embodiment.
FIG. 5 shows a curve of the confinement factor of the in-plane component of the electric field per unit length when the width (w _s ) of the silicon strip is 150 nm according to one embodiment.
FIG. 6 shows a curve of the confinement factor of the in-plane component of the electric field per unit length when the width (w _s ) of the silicon strip is 320 nm according to one embodiment.
Figure 7 shows a curve of the confinement factor of the in-plane component of the electric field per unit length when the width (w _s ) of the silicon strip is 450 nm according to one embodiment.
Figure 8 shows the relationship between the insertion loss (IL _G ) and the slot height (h _m ) of a graphene-based waveguide according to one embodiment.
FIG. 9 shows the width of a slot (w _m,opt ) that can maximize the modulation depth of a graphene-based waveguide according to the width (w _s ) of a silicon strip according to one embodiment, and the maximum modulation depth (MD _G ) of the graphene-based waveguide.
FIG. 10 shows the insertion loss (IL _G ) and figure of merit (FoM _G ) of the graphene-based waveguide according to the width (w _s ) of the silicon strip according to one embodiment.
Figure 11 shows the coupler length (l _c,opt ) that can minimize coupler loss according to the width (w _s ) of the silicon strip according to one embodiment.
Figure 12 shows coupler loss according to the width (w _s ) of a silicon strip according to one embodiment.
Figure 13 shows the length (l _M ) of the modulator and the insertion loss (IL _M ) of the modulator according to the width (w _s ) of the silicon strip according to one embodiment.
Figure 14 shows the figure of merit (FoM _M ) of the modulator according to the width (w _s ) of the silicon strip according to one embodiment.
FIG. 15 shows the electric field distribution in a plane passing through the center of the silicon strip and perpendicular to the height direction when the modulator designed according to one embodiment is in the off state.
FIG. 16 shows the electric field distribution in a plane passing through the center of the silicon strip and perpendicular to the height direction when the modulator designed according to one embodiment is in the on state.
Figure 17 shows the insertion loss (IL _M ) and extinction ratio of the modulator according to the distance (d) that the slot deviates from the ideal position in the width direction according to one embodiment.

Specific structural or functional descriptions of the embodiments are disclosed for illustrative purposes only and may be implemented in various forms. Therefore, the actual implemented form is not limited to the specific embodiments disclosed, and the scope of the present disclosure includes modifications, equivalents, or alternatives included in the technical idea described in the embodiments.

Although the terms first or second may be used to describe various components, such terms should be construed only for the purpose of distinguishing one component from another. For example, a first component may be referred to as a second component, and similarly, a second component may also be referred to as a first component.

When it is said that a component is "connected" to another component, it should be understood that it may be directly connected or connected to that other component, but there may also be other components in between.

Singular expressions include plural expressions unless the context clearly indicates otherwise. In this specification, the terms "comprises" or "has" and the like are intended to specify the presence of a described feature, number, step, operation, component, part, or combination thereof, but should be understood to not preclude the presence or addition of one or more other features, numbers, steps, operations, components, parts, or combinations thereof.

Unless otherwise defined, all terms used herein, including technical or scientific terms, have the same meaning as commonly understood by one of ordinary skill in the art. Terms defined in commonly used dictionaries should be interpreted as having a meaning consistent with the meaning they have in the context of the relevant art, and will not be interpreted in an idealized or overly formal sense unless explicitly defined herein.

Hereinafter, embodiments will be described in detail with reference to the attached drawings. When describing with reference to the attached drawings, identical components are given the same reference numerals regardless of the drawing numbers, and redundant descriptions thereof will be omitted.

Figures 1 to 3 illustrate exemplary geometries of a modulator using a graphene-based waveguide according to one embodiment.

FIG. 1 is a perspective view of a modulator (100) using a graphene-based waveguide according to one embodiment, FIG. 2 is a top view of the modulator using the graphene-based waveguide, and FIG. 3 is a cross-sectional view taken along line AA' of the modulator using the graphene-based waveguide. In FIGS. 1 to 3, the x-axis direction is described as the length direction, the y-axis direction is described as the width direction, and the z-axis direction is described as the height direction.

A modulator (100) using a graphene-based waveguide according to one embodiment may include a graphene-based waveguide (110), an input coupler (120), and an output coupler (130).

A graphene-based waveguide (110) may include a silicon layer (111), a first graphene layer (112), an insulating layer (113), a second graphene layer (114), a first electrode (115), a second electrode (116), and a metal rail (117).

The silicon layer (111) may include a silicon strip (111a) and silicon oxide (111b). The silicon strip (111a) may have one side in contact with the first graphene layer (112) and the remaining side covered by silicon oxide (111b). For example, the height of the silicon strip (111a) may be 250 nm.

The first graphene layer (112) may be positioned on one side of the silicon layer (111). The insulating layer (113) may cover a portion of the first graphene layer (112) and a portion of the silicon layer (111). The second graphene layer (114) may be positioned on the insulating layer (113) and separated from the first graphene layer (112) by the insulating layer (113). For example, the insulating layer (113) may include aluminum oxide (Al ₂ O ₃ ). For example, the height of the insulating layer (113) may be 10 nm.

When the potential of the first graphene layer (112) and the potential of the second graphene layer (114) are different, a capacitive coupling can be formed between the first graphene layer (112) and the second graphene layer (114). When the first graphene layer (112) and the second graphene layer (114) are viewed in a direction perpendicular to a plane corresponding to one side of the silicon layer (111) on which the first graphene layer (112) is positioned, an area occupied by the first graphene layer (112) and an area occupied by the second graphene layer (114) in the plane can overlap. As will be described in more detail below, since most of the electric field within the graphene-based waveguide (110) is confined to the slot defined between the second electrode (116) and the metal rail (117), the width (w o ) of the region where the region occupied by the first graphene layer (112) and the region occupied by the second graphene layer ( ₁₁₄ ) overlap may be equal to or greater than a first critical width. The first critical width may be 500 nm.

The first electrode (115) may be connected to the first graphene layer (112). The second electrode (116) may be connected to the second graphene layer (114). The chemical potential (μ _C1 ) of the first graphene layer (112) and the chemical potential (μ _C2 ) of the second graphene layer (114) may be different.

The metal rail (117) may be positioned between the first electrode (115) and the second electrode (116) on the second graphene layer (114). A slot may be defined between the second electrode (116) and the metal rail (117). Through the slot, an electric field defined between the first graphene layer (112) and the second graphene layer (114) may be strengthened. As will be described in more detail below, the width (w _{r )} of the metal rail (117) may be equal to or greater than the second critical width because it may have little effect on the performance of the graphene-based waveguide (110). The second critical width may be 300 nm. The height (h _m ) of the metal rail (117) may be equal to the height of the second electrode (116). As will be described in more detail below, the height (h _m ) of the metal rail (117) may be equal to or greater than the first critical height, since it has little effect on the performance of the graphene-based waveguide (110) when the height (h _m ) of the metal rail (117) is equal to or greater than the first critical height. The first critical height may be 150 nm.

The width (w _s ) of the silicon strip (111a) may be greater than or equal to the width (w _m ) of the slot. When viewed in a direction perpendicular to a plane corresponding to one side of the silicon layer (111) on which the first graphene layer (112) is positioned, the silicon strip (111a), the metal rail (117), and the second electrode (116) may not overlap each other.

The distance (w _g ) between one side of the metal rail (117) and the first electrode (115) may be constant. For example, the distance from each position on one surface of the metal rail (117) to the first electrode (115) may be constant. The distance between the other side of the metal rail (117) and the second electrode (116), for example, the width (w _m ) of the slot, may be constant. For example, the distance from each position on the other surface of the metal rail (117) to the second electrode (116) may be constant. As will be described in more detail below, when the distance (w _g ) between one side of the metal rail (117) and the first electrode (115) is equal to or greater than a third critical width, the parasitic capacitance is small enough to be ignored, so that the distance (w _g ) between one side of the metal rail (117) and the first electrode (115) may be equal to or greater than the third critical width. The third critical width can be 400 nm.

A modulator (100) utilizing a graphene-based waveguide may include a graphene-based waveguide (110), an input coupler (120) connecting the graphene-based waveguide (110) and a silicon photonic waveguide (140), and an output coupler (130).

The input coupler can convert the mode of the optical signal from the fundamental TE (Transverse Electric) mode of the silicon photonic waveguide to the graphene-based waveguide mode. The mode power can be modulated by the strong light-graphene interaction existing in the graphene-based waveguide. The output coupler can convert the mode of the optical signal from the graphene-based waveguide mode back to the fundamental TE mode of the silicon photonic waveguide.

The input coupler (120) may include an insulating layer extension portion (123) extending from the insulating layer (113) to one side, a first electrode extension portion (125) extending from the first electrode (115) to one side, a second electrode extension portion (126) extending from the second electrode (116) to one side, and a metal rail extension portion (127) extending from the metal rail (117) to one side. The output coupler (130) may include an insulating layer extension portion (133) extending from the insulating layer (113) to the other side, a first electrode extension portion (135) extending from the first electrode (115) to the other side, a second electrode extension portion (136) extending from the second electrode (116) to the other side, and a metal rail extension portion (137) extending from the metal rail (117) to the other side.

The input coupler (120) may have a shape in which the distance between the second electrode extension portion (126) and the metal rail extension portion (127) increases as it gets farther away from the waveguide along the longitudinal axis. For example, the second point may be a point further from the graphene-based waveguide (110) along the longitudinal axis than the first point. The distance between the second electrode extension portion (126) and the metal rail extension portion (127) at the second point may be greater than the distance between the second electrode extension portion (126) and the metal rail extension portion (127) at the first point.

The output coupler (130) may have a shape in which the distance between the second electrode extension portion (136) and the metal rail extension portion (137) increases as it gets farther away from the waveguide along the longitudinal axis. For example, the fourth point may be a point further away from the graphene-based waveguide (110) along the longitudinal axis than the third point. The distance between the second electrode extension portion (136) and the metal rail extension portion (137) at the fourth point may be greater than the distance between the second electrode extension portion (136) and the metal rail extension portion (137) at the third point.

The input coupler (120) may include a silicon strip extension portion extending from the silicon strip (111a) to one side. The output coupler (130) may include a silicon strip extension portion extending from the silicon strip (111a) to the other side.

The width of the extended portion of the silicon strip of the input coupler (120) may linearly increase from the width (w _s ) of the silicon strip (111a) to the silicon strip width (e.g., 450 nm) of the silicon photonic waveguide as the position gets farther away from the graphene-based waveguide (110) along the longitudinal axis. The width of the extended portion of the silicon strip of the output coupler (130) may linearly increase from the width (w _s ) of the silicon strip (111a) to the silicon strip width (e.g., 450 nm) of the silicon photonic waveguide as the position gets farther away from the graphene-based waveguide (110) along the longitudinal axis.

The distance at which the second electrode extension portion (126) and the metal rail extension portion (127) of the input coupler (120) are spaced apart may linearly increase from the width (w _m ) of the slot as the position gets farther away from the graphene-based waveguide (110) along the longitudinal axis. The distance at which the second electrode extension portion (136) and the metal rail extension portion (137) of the output coupler (130) are spaced apart may linearly increase from the width (w _m ) of the slot as the position gets farther away from the graphene-based waveguide (110) along the longitudinal axis.

In a graphene-based waveguide (110) included in a modulator (100), when the potential of the first graphene layer (112) and the potential of the second graphene layer (114) are different, capacitive coupling can be formed between the first graphene layer (112) and the second graphene layer (114).

The graphene-based waveguide (110) included in the modulator (100) can strengthen the electric field limited between the first graphene layer (112) and the second graphene layer (114) through a slot defined between the second electrode (116) and the metal rail (117).

To analyze the graphene-based waveguide, the eigenmode solver based on the finite difference method of Lumerical Inc. can be used. The refractive indices of silicon (Si), silicon oxide (SiO ₂ ), aluminum oxide (Al ₂ O ₃ ), and gold can be 3.45, 1.44, 1.74, and 0.559+i9.81 at a wavelength of 1550 nm, respectively. Graphene can be treated as a conducting boundary with optical conductivity (σ _g ). Assuming that the relaxation time of graphene is set to 0.1 ps, the analytical expression derived from the Kubo formula can be used for the optical conductivity (σ _g ).

Under the assumption that the width (w _o ) of the overlapping region occupied by the first graphene and the area occupied by the second graphene and the width (w _r ) of the metal rail are infinite and the height (h _m ) of the slot is 150 nm for various values of the width (w _m ) of the slot and the width (w _s ) of the silicon strip, the graphene-based waveguide mode polarized mainly in the width direction (y direction) can be determined. The electric and magnetic fields of the mode can be normalized so that the mode power becomes 1 mW. Since the graphene layer interacts with the in-plane component (E _y ) of the electric field (E), it may be necessary to determine how much of the in-plane component (E _y ) of the electric field exists at the location of the graphene layer. To this end, the confinement factor (Г(z)) of the in-plane component (E _y ) of the electric field per unit length can be defined as shown in the following mathematical expression 1.

Here, L _Z represents a horizontal straight line with height z, and A _∞ can represent an infinite cross-section of the waveguide. The limiting factor (G(z)) of the in-plane component of the electric field can be calculated for various values of the slot width (w _m ) and the silicon strip width (w _s ).

FIG. 4 shows profiles of mode electric fields. Profile (411) is a profile of a mode electric field when the width (w _s ) of a silicon strip is 150 nm and the width (w _m ) of a slot is 150 nm. Profile (412) is a profile of a mode electric field when the width (w _s ) of a silicon strip is 150 nm and the width (w _m ) of a slot is 200 nm. Profile (413) is a profile of a mode electric field when the width (w _s ) of a silicon strip is 150 nm and the width (w _m ) of a slot is 250 nm. Profile (421) is a profile of a mode electric field when the width (w _s ) of a silicon strip is 320 nm and the width (w _m ) of a slot is 250 nm. Profile (422) is a profile of a mode electric field when the width (w _s ) of the silicon strip is 320 nm and the width (w _m ) of the slot is 320 nm. Profile (423) is a profile of a mode electric field when the width (w _s ) of the silicon strip is 320 nm and the width (w _m ) of the slot is 450 nm. Profile (431) is a profile of a mode electric field when the width (w _s ) of the silicon strip is 450 nm and the width (w _m ) of the slot is 250 nm. Profile (432) is a profile of a mode electric field when the width (w _s ) of the silicon strip is 450 nm and the width (w _m ) of the slot is 320 nm. Profile (433) is a profile of a mode electric field when the width (w _s ) of the silicon strip is 450 nm and the width (w _m ) of the slot is 450 nm.

Figures 5 to 7 show curves of the confinement factor G(z) of the in-plane component of the electric field per unit length. Figure 5 shows a curve (501) of the confinement factor G(z) when the width (w _s ) of the silicon strip is 150 nm and the width (w _m ) of the slot is 150 nm, a curve (502) of the confinement factor G(z) when the width (w _m ) of the slot is 200 nm, and a curve (503) of the confinement factor G(z) when the width (w _m ) of the slot is 250 nm. FIG. 6 shows a curve (601) of the confinement factor G(z) when the width (w _s ) of the silicon strip is 320 nm and the width (w _m ) of the slot is 250 nm, a curve (602) of the confinement factor G( _{z )} when the width (w m ) of the slot is 320 nm, and a curve (603) of the confinement factor G(z) when the width (w _m ) of the slot is 450 nm. FIG. 7 shows a curve (701) of the confinement factor G(z) when the width (w _m ) of the silicon strip is 450 nm and the width (w _m ) of the slot is 250 nm, a curve (702) of the confinement factor G(z) when the width (w m ) of the slot is 320 nm, and a curve (703) of the confinement factor G(z) when the width (w _m ) of the slot is 450 nm. In Figures 5 to 7, vertical dotted lines indicate the positions of the two graphene layers.

When the width of the slot (w _m ) or the width of the silicon strip (w _s ) is less than the critical width, most of the electric field can be mainly confined in the slot, especially strongly confined in the region between the corner of the slot and the corner of the silicon strip. It can be confirmed that the peak of the confinement factor (G(z)) appears at the position of the graphene layer, indicating that the graphene-based waveguide can have strong light-graphene interaction. For the silicon strip including silicon oxide, the fundamental TE mode can be supported at a wavelength of 1550 nm when the silicon strip width is larger than 210 nm.

As shown in the profiles of the mode electric fields (411, 412, and 413) when the width (w _s ) of the silicon strip is 150 nm, the slot can support the graphene-based waveguide mode.

As shown in Fig. 5, as the width (w _m ) of the slot increases, the electric field confined to the slot may weaken and the confinement factor (G(z)) may decrease. As shown in the profiles of the mode electric fields (421, 422, and 423) when the width (w _s ) of the silicon strip is 320 nm, as the width (w _m ) of the slot increases, the electric field confined to the silicon strip may increase.

As the width of the slot (w _m ) increases in the range of heights in which the silicon strips exist in FIG. 6 (e.g., the range where z is equal to or greater than -0.125 μm and equal to or less than 0.125 μm), the confinement factor (G(z)) may increase. When the width of the slot (w _m ) is equal to or less than the width of the silicon strip (w _s ), the confinement factor (G(z)) at the position of the graphene layer may be larger than the confinement factor (G(z)) in the range of heights in which the silicon strips exist. As shown in the profiles of the mode electric fields (431, 432, and 433) when the width of the silicon strip (w _s ) is 450 nm, when the width of the slot (w _m ) exceeds a critical width (e.g., 220 nm), the silicon strips may mainly support the graphene-based waveguide mode.

As shown in Fig. 7, when the width of the slot (w _m ) and the width of the silicon strip (w _s ) are the same, the confinement factor (G(z)) in the range of heights where the silicon strip exists can be at a similar level to the confinement factor (G(z)) at the position of the graphene layer. Regardless of the width of the silicon strip (w _s ), when the width of the slot (w _m ) and the width of the silicon strip (w _s ) are the same, the confinement factor (G(z)) can have two peaks. When the width of the silicon strip (w _s ) is 150 nm or 320 nm, the confinement factor (G(z)) can be very large at the position of the graphene layer. When the width of the silicon strip (w _s ) is 450 nm, the two peaks of the confinement factor (G(z)) can be similar to the value of the confinement factor (G(z)) in the range of heights where the silicon strip exists.

As a result, the light-graphene interaction can be very strong when the width of the slot (w _m ) is equal to the width of the silicon strip (w _s ) when the width of the silicon strip (w _s ) is 150 nm or 320 nm. In contrast, the light-graphene interaction can be strongest when the width of the slot (w _m ) is 220 nm when the width of the silicon strip (w _s ) is 450 nm. There may be a method of selecting the width of the slot (w _m ) such that the width of the slot (w _m ) is equal to the width of the silicon strip (w _s ) when the width of the silicon strip (w _s ) is 150 nm or 320 nm, and a method of selecting the width of the slot (w _m ) when the width of the silicon strip (w _s ) is 450 nm. There may be two different methods of selecting the width of the slot (w _m ).

The output power of the modulator can be controlled by adjusting the voltage applied between the first electrode and the second electrode. When the modulator is in an on state where the chemical potential (μ _C1 ) of the first graphene layer can be defined as 0.6 eV and the chemical potential (μ _C2 ) of the second graphene layer can be defined as -0.6 eV, the output power of the modulator can be high. When the modulator is in an off state where the chemical potential (μ _C1 ) of the first graphene layer can be defined as 0.2 eV and the chemical potential (μ _C2 ) of the second graphene layer can be defined as -0.2 eV, the output power of the modulator can be low.

The modulation depth (MD _G ) of a graphene-based waveguide can be defined as the difference in propagation loss between the off-state and the on-state (e.g., the unit is dB/μm). The insertion loss (IL _G ) of the graphene-based waveguide can be defined as the on-state propagation loss. In the design of the graphene-based waveguide, the values of the width of the silicon strip (w _s ), the width of the slot (w _m ), and the height of the slot (h _m ) mentioned above can be determined as numerical values that make the modulation depth (MD _G ) equal to or greater than a critical depth and the insertion loss (IL _G ) of the graphene-based waveguide equal to or less than a critical loss. By changing the values of the width of the silicon strip (w _s ), the width of the slot (w _m ), and the height of the slot (h _m ), the modulation depth (MD _G ) and the insertion loss (IL _G ) of the graphene-based waveguide can be calculated.

FIG. 8 shows the relationship between the insertion loss (IL _G ) of a graphene-based waveguide and the height of a slot (h _m ). Curve (801) is a curve (801) of the insertion loss (IL _G ) of the graphene-based waveguide when the width (w _s ) of the silicon strip is 150 nm and the width (w _m ) of the slot is 150 nm. Curve (802) is a curve (802) of the insertion loss (IL _G ) of the graphene-based waveguide when the width (w _s ) of the silicon strip is 300 nm and the width (w _m ) of the slot is 300 nm. Curve (803) is a curve (803) of the insertion loss (IL _G ) of the graphene-based waveguide when the width (w _s ) of the silicon strip is 450 nm and the width (w _m ) of the slot is 220 nm.

As shown in Fig. 8, as the height of the slot (h _m ) increases, the insertion loss (IL _G ) of the graphene-based waveguide decreases, and when the height of the slot (h _m ) is equal to or greater than a first critical height, the insertion loss (IL _G ) of the graphene-based waveguide can approach a constant value. When the height of the slot (h _m ) is less than the first critical height, the electric field of the graphene-based waveguide mode can leak to the top surface of the slot, so that not only the bottom edge but also the top edge of the slot can affect the mode, and the insertion loss (IL _G ) of the graphene-based waveguide can be equal to or greater than the critical insertion loss. When the height of the slot (h _m ) is equal to or greater than the first critical height, the electric field is well confined to the lower region of the slot, and the insertion loss (IL _G ) of the graphene-based waveguide can be independent of the height of the slot (h _m ). The height of the slot (h _m ) can be determined as a first critical height so that the insertion loss (IL _G ) of the graphene-based waveguide is less than the critical loss. The first critical height can be 150 nm.

Fig. 9 shows the slot width (w _m,opt ) that can maximize the modulation depth of a graphene-based waveguide according to the width (w _s ) of a silicon strip, and the maximum modulation depth (MD _G ) of the graphene-based waveguide. By keeping the width (w _s ) of the silicon strip constant and changing the slot width (w _m ), the modulation depth (MD _G ) can be calculated, and the optimal slot width (w _m,opt ) that maximizes the modulation depth (MD _G ) can be determined. As shown in Fig. 9, the optimal slot width (w _m,opt ) can be expressed as a function of the width (w _s ) of the silicon strip.

As shown in Fig. 9, when the width (w _s ) of the silicon strip is 300 nm or less, the width of the optimal slot (w _m,opt ) may be the width of the silicon strip (w _s ). When the width (w _s ) of the silicon strip is 300 nm or more and 340 nm or less, the width of the optimal slot (w _m ,opt ) may be slightly smaller than the width (w _s ) of the silicon strip. As the width (w _s ) of the silicon strip increases from 340 nm to 380 nm, the width of the optimal slot (w _m,opt ) may decrease rapidly, and when the width (w _s ) of the silicon strip exceeds 380 nm, the width of the optimal slot (w _m,opt ) hardly changes. As shown in FIGS. 5 to 7, when the width of the silicon strip (w _s ) is less than 340 nm and the width of the silicon strip (w _s ) and the width of the slot (w _m ) are similar, G(z) has two peaks at the position of the graphene layer, so the modulation depth (MD _G ) can be maximum.

Through the relationship between the optimal slot width (w _m,opt ) and the silicon strip width (w _s ), the slot width (w _m ) can be selected differently depending on the range of the silicon strip width (w _s ). When the silicon strip width (w _s ) is 340 nm or less, the slot width (w _m ) can be selected to be similar to the silicon strip width (w _s ), and when the silicon strip width (w _s ) exceeds 340 nm, the slot width (w _m ) can be selected to be similar to 220 nm. According to Fig. 9, as the silicon strip width (w _s ) increases to 340 nm, the optimal value of the slot width (w _m,opt ) increases, the two peaks of the confinement factor (G(z)) of the in-plane component of the electric field at the position of the graphene layer can decrease, and the maximum modulation depth (MD _G ) can decrease. As the width (w _s ) of the silicon strip increases from 340 nm, the electric field confined to the silicon strip becomes significant, the confinement factor (G(z)) peak at the position of the second graphene layer can decrease, and the maximum modulation depth (MD _G ) can continue to decrease.

Figure 10 shows the insertion loss (IL _G ) of the graphene-based waveguide calculated with the slot width (w _m ) set to the optimal slot width (w _m,opt ) as a function of the width of the silicon strip (w _s ).

Similar to the modulation depth (MD _G ), the insertion loss (IL _G ) of the graphene-based waveguide decreases as the width of the silicon strip (w _s ) increases up to 330 nm. As the width of the silicon strip (w _s ) increases from 330 nm, the insertion loss (IL _G ) of the graphene-based waveguide increases rapidly and can be saturated. When the chemical potentials of the two graphene layers are +0.6 eV and -0.6 eV, respectively, the light absorption at 1550 nm by the graphene layers can be very small. Therefore, the insertion loss (IL _G ) of the graphene-based waveguide can be mainly determined by the absorption by the slot. When the slot width (w _m ) is larger than the critical width, the insertion loss (IL _G ) of the graphene-based waveguide can be below the critical value. As the width of the silicon strip (w _s ) increases from 340 nm, the width of the optimal slot (w _m,opt ) rapidly decreases to 220 nm, which may increase the insertion loss (IL _G ) of the graphene-based waveguide.

The figure of merit (FoM _G ) of a graphene-based waveguide can be defined as the ratio of the modulation depth (MD _G ) to the insertion loss (IL _G ) of the graphene-based waveguide. The value of the width (w _s ) of the silicon strip can be selected based on the figure of merit (FoM _G ) of the graphene-based waveguide. Fig. 10 shows the insertion loss (IL _G ) of the graphene-based waveguide and the figure of merit (FoM _G ) of the graphene-based waveguide according to the width (w _s ) of the silicon strip. The figure of merit (FoM _G ) of the graphene-based waveguide can be expressed as a function of the width (w _s ) of the silicon strip. When the width (w _s ) of the silicon strip is 330 nm and the width (w _m ) of the slot is 324 nm, the figure of merit (FoM _G ) of the graphene-based waveguide can have a maximum value of 3.90. At this time, the modulation depth (MD _G ) can be 0.682 dB/μm, and the insertion loss (IL _G ) of the graphene-based waveguide can be 0.175/μm.

Another waveguide structurally similar to the graphene-based waveguide (110) may have a slot width of 200 nm, a silicon strip width of 150 nm, a modulation depth of 0.316 dB/μm, an insertion loss of 0.087 dB/μm, and a figure of merit of 3.63. The figure of merit of the other waveguide may be smaller than that of the graphene-based waveguide (110) of the present invention. In addition, as will be described below, a narrower graphene-based waveguide may require a longer coupler, and the other waveguide may have higher coupler loss.

An electro-absorption modulator (EAM) can have a subcritical size and a subcritical loss. The width of the slot (w _c ) of the coupler can be defined as the distance between the second electrode extension (126) and the metal rail extension (127) at the farthest point from the coupler length (l _c ) and the graphene-based waveguide. The width of the slot (w _c ) of the coupler should be determined so that the EAM can have a subcritical size and a subcritical loss, and the graphene-based waveguide can be fine-tuned if required. The modulator can be simulated using the finite difference time domain (FDTD) method of Lumerical Inc., and the coupler loss can be calculated. For simplicity, a graphene-based waveguide without a graphene layer can be considered because the field profile of a graphene-based waveguide mode can be little affected by the graphene layer.

The coupler loss can be defined as the power loss (dB) between the input silicon photonic waveguide and the output silicon photonic waveguide minus the loss (dB) of the graphene-based waveguide divided by 2. For a given silicon strip width (w _s ), the width of the slot (w _m,opt ) can be set as the optimal slot width, and _the length (l _c,opt ) of the optimal coupler with the minimum coupler loss can be determined.

Fig. 11 shows the coupler length (l _c,opt ) that can minimize the coupler loss according to the width (w _s ) of the silicon strip. Fig. 12 shows the coupler loss according to the width (w _s ) of the silicon strip. As the width (w _s ) of the silicon strip increases to 340 nm, the width of the slot (w _m ) may increase, and both the optimal coupler length (l _c,opt ) and the coupler loss may decrease. As the width (w _s ) of the silicon strip increases from 340 nm, the width (w _m ) of the slot may decrease, and both the optimal coupler length (l _c,opt ) and the coupler loss may increase. Therefore, when the width (w _s ) of the silicon strip at which the figure of merit (FoM _G ) of the graphene-based waveguide is maximum is 340 nm, the coupler loss can be minimized. The length (l _c,opt ) of the optimal coupler increases as the width (w _c ) of the coupler slot increases, but the coupler loss when the width (w _c ) of the coupler slot is 500 nm can be smaller than the coupler loss when the width (w _c ) of the coupler slot is 450 nm and 550 nm.

The optimal values of the width of the silicon strip (w _s ) and the width of the slot of the coupler (w _c ) can be determined as follows. The extinction ratio of an intensity modulator typically used for short-range data communications is several dB. Therefore, the extinction ratio obtained from the length of the graphene-based waveguide (l _G ) can be determined as 3 dB (i.e., the length of the graphene-based waveguide (l _G ) = 3/modulation depth (MD _G )).

The length of the modulator (l _M ) can be given by the length of the graphene-based waveguide (l _G ) plus twice the length of the optimal coupler (l _c,opt ). The insertion loss of the modulator (IL _M ) can be calculated by the product of twice the coupler loss plus the length of the graphene-based waveguide (l _G ) plus the insertion loss of the graphene-based waveguide (IL _G ). Figure 13 shows the relationship between the calculated length of the modulator (l _M ) and the width of the silicon strip (w _s ), and the relationship between the insertion loss of the modulator (IL _M ) and the width of the silicon strip (w _s ).

The relationship between the calculated modulator length (l _M ) and the silicon strip width (w _s ) is shown by the curve (1301a) of the calculated modulator length (l _M ) when the slot width (w _c ) of the coupler is 450 nm, the curve (1302a) of the calculated modulator length (l _M ) when the slot width (w _c ) of the coupler is 500 nm, and the curve (1303a) of the calculated modulator length (l _M ) when the slot width (w _c ) of the coupler is 550 nm. In addition, the relationship between the insertion loss (IL _M ) of the modulator and the width (w _s ) of the silicon strip is expressed by the curve (1301b) of the insertion loss (IL _M ) of the modulator when the width (w _c ) of the slot of the coupler is 450 nm, the curve (1302b) of the insertion loss (IL _M ) of the modulator when the width (w _c ) of the slot of the coupler is 500 nm, and the curve (1303b) of the insertion loss (IL _M ) of the modulator when the width (w _c ) of the slot of the coupler is 550 nm.

As the width of the silicon strip (w _s ) increases to 340 nm, the length of the graphene-based waveguide (l _G ) can increase, but the length of the optimal coupler (l _c,opt ) can decrease. As the width of the silicon strip (w _s ) increases from 340 nm, both the length of the graphene-based waveguide (l _G ) and the length of the optimal coupler (l _c,opt ) can increase, and the length of the modulator (l _M ) can increase rapidly. The curve of the insertion loss (IL _M ) of the modulator can be similar to the curves of the insertion loss (IL _G ) and coupler loss of the graphene-based waveguide. The figure of merit (FoM _M ) of the modulator can be defined as the inverse of the product of the length of the modulator (l _M ) and the insertion loss of the modulator (IL _M ).

Fig. 14 shows the relationship between the figure of merit (FoM _M ) of the modulator and the width (w _s ) of the silicon strip. Curve (1401) is a curve of the figure of merit (FoM _M ) of the modulator when the width (w _c ) of the slot of the coupler is 450 nm. Curve (1402) is a curve of the figure of merit (FoM _M ) of the modulator when the width (w _c ) of the slot of the coupler is 500 nm. Curve (1403) is a curve of the figure of merit (FoM _M ) of the modulator when the width (w _c ) of the slot of the coupler is 550 nm. The figure of merit (FoM _M ) of the modulator can be at its maximum when the width (w _s ) of the silicon strip is 320 nm and the width (w _c ) of the slot of the coupler is 500 nm.

A modulator can be designed in which the width of the silicon strip (w _s ) is 320 nm, the width of the slot (w _m ) is 316 nm, the width of the coupler slot (w _c ) is 500 nm, the length of the coupler (l _c ) is 1.05 μm, and the length of the graphene-based waveguide (l _G ) is 4.13 μm, and the designed modulator can have a modulation depth (MD _G ) of 0.729 dB/μm and an insertion loss (IL _G ) of the graphene-based waveguide of 0.187 dB/μm. The width (w _s ) of the silicon strip having the maximum figure of merit (FoM _M ) of the modulator may be different from the width (w _s ) of the silicon strip having the maximum figure of merit (FoM _G ) of the graphene-based waveguide. The length (l _M ) of the designed modulator can be 6.23 μm, and the insertion loss (IL _M ) of the modulator can be 1.01 dB.

Figures 15 and 16 show the electric field distribution in a plane perpendicular to the height direction and passing through the center of the silicon strip when the designed modulator is in the off-state and on-state. The designed modulator can be simulated using the FDTD method. As shown in Figure 15, when the chemical potential of the graphene layer is 0.2 eV (for example, the chemical potential (μ _C1 ) of the first graphene layer is 0.2 eV and the chemical potential (μ _C2 ) of the second graphene layer is -0.2 eV), the electric field may be weakened while passing through the modulator, and the electric field may be weak at the output of the silicon strip. On the other hand, as shown in Fig. 16, when the chemical potential of the graphene layer is 0.6 eV (for example, the chemical potential of the first graphene layer (μ _C1 ) is 0.6 eV and the chemical potential of the second graphene layer (μ _C2 ) is -0.6 eV), the electric field passing through the modulator may be less weak than when the chemical potential of the graphene layer is 0.2 eV, and the electric field at the output of the silicon strip may be stronger than when the chemical potential of the graphene layer is 0.2 eV.

In the design processes described above, the overlap width (w _o ) of the first graphene layer and the second graphene layer and the width (w _r ) of the metal rail can be assumed to be infinite. In a graphene-based waveguide with a silicon strip width (w _s ) of 320 nm, a slot width (w _m ) of 316 nm, and an infinite metal rail width (w _r ), the modulation depth (MD _G ) and the insertion loss (IL _G ) of the graphene-based waveguide can be calculated according to the overlap width (w _o ) of the graphene layers. The modulation depth (MD _G ) and the insertion loss (IL _G ) of the graphene-based waveguide can be almost constant when the overlap width (w _o ) of the graphene layers is greater than or equal to a first critical width because most of the electric field in the graphene-based waveguide is confined to the slot. The overlap width (w _o ) of the graphene layers can be set to be greater than or equal to the first critical width. The first critical width can be 500 nm.

Assuming that the distance (w _g ) between one side of the metal rail (117) and the first electrode (115) is infinite, the modulation depth (MD _G ) and the insertion loss (IL _G ) of the graphene-based waveguide can be calculated with respect to the width (w _r ) of the metal rail. When the width (w _r ) of the metal rail is equal to or greater than a second critical width, the modulation depth (MD _G ) and the insertion loss (IL _G ) of the graphene-based waveguide can be almost independent of the width (w _r ) of the metal rail. The width (w _r ) of the metal rail can be equal to or greater than the second critical width. The second critical width can be 300 nm.

Fig. 17 shows the insertion loss (IL _M ) and extinction ratio of the modulator according to the distance (d) that the slot is displaced in the width direction from the ideal position. During the fabrication of the EAM, the slot may not be perfectly aligned with the silicon strip. The FDTD method can be used to identify the effect of this fabrication error. The EAM can be simulated in which the slot is displaced in the width direction (y direction) from the ideal position. The difference between the on-state and off-state for the optical power (dB) transmitted by the input silicon photonic waveguide and the output silicon photonic waveguide can be calculated. From the difference, the extinction ratio of the EAM and the insertion loss (IL _M ) of the modulator can be determined according to the distance (d) that the slot is displaced in the width direction from the ideal position.

As shown in Fig. 17, the extinction ratio and the insertion loss (IL M ) of the modulator may not deviate significantly from the ideal values, except for the case where the slot is deviated from the ideal position in the width direction (d) by 40 nm while the distance ( _d ) that the slot deviates from the ideal position in the width direction increases up to 70 nm. When the slot is deviated from the ideal position in the width direction by 40 nm, the graphene-based waveguide mode may not be well converted into the silicon waveguide mode by the output coupler and may be reflected at the coupler. This may be because when the slot is deviated from the ideal position in the width direction by 40 nm, the graphene-based waveguide mode is almost antisymmetric and does not match with the symmetric silicon waveguide mode. The tolerance of the alignment process may be 30 nm, but an alignment error of 70 nm can be tolerated.

Finally, the 3 dB bandwidth and energy consumption of the EAM can be analyzed. For this purpose, the device capacitance and resistance of the EAM can be determined. The device capacitance and resistance of the EAM can be affected by the distance (w _g ) between one side of the metal rail (117) and the first electrode (115). If the distance (w _g ) between one side of the metal rail (117) and the first electrode (115) is less than the third critical width, the parasitic capacitance between the first electrode and the metal rail cannot be ignored and can affect the capacitance of the device. If the distance (w _g ) between one side of the metal rail (117) and the first electrode (115) is greater than the third critical width, the device resistance can increase. As a result of calculating the parasitic capacitance using commercial software Device of Lumerical Inc., it can be ignored compared to the device capacitance when the distance (w _g ) between one side of the metal rail (117) and the first electrode (115) is equal to or greater than the third critical width. The third critical width can be 400 nm.

The device capacitance (C _d ) can be determined by the parallel-plate capacitance (C _g ) between the two graphene layers and the quantum capacitance of graphene (C _q ). When ε ₀ is the vacuum permittivity, ε _a is the dielectric constant of the insulating layer, and t _a is the thickness of the insulating layer, the parallel-plate capacitance (C _g ) between the two graphene layers can be calculated using the following mathematical expression 2. The insulating layer can be aluminum oxide, and the dielectric constant of the insulating layer (ε _a ) can be 10.3.

e is the electronic charge, is the reduced Planck constant, v _F is the Fermi velocity of graphene, w ₁ is the sum of the width of the metal rail (w _r ), the overlap width of the graphene layers (w _o ) divided by 2, and the width of the slot (w _m ) divided by 2, and w ₂ is the overlap width of the graphene layers (w _o ), then the quantum capacitance of the first graphene layer (C _q1 ) and the quantum capacitance of the second graphene layer (C _q2 ) can be calculated using the following mathematical equation 3.

The parallel plate capacitance (C _g ) between the graphene layers can be 26.6 fF, the quantum capacitance (C _q1 ) of the first graphene layer can be 0.275 pF, and the quantum capacitance (C _q2 ) of the second graphene layer can be 0.194 pF. The device capacitance (C _d ) can be calculated through the following mathematical expression 4. C _d can be 21.6 fF.

With respect to the width of the metal rail (w _r ), the overlapping width of the graphene layer (w _o ), the width of the slot (w _m ), the distance (w _g ) between one side of the metal rail (117) and the first electrode (115), the contact resistance (r _c ) of the graphene layer, and the surface resistance (sheet resistance) (r _s ) of the graphene layer, the device resistance (R _d ) can be calculated using the following mathematical expression 5. When the contact resistance (r _c ) of the graphene layer is 100Ωμm and the surface resistance (r _s ) of the graphene layer is 125Ω/sq, the device resistance (R _d ) can be calculated as 94.5Ω. The 3dB bandwidth (f _3dB ) can be given by 1/(2πR _d C _d ), and the 3dB bandwidth (f _3dB ) can be calculated as 78.1GHz.

If the non-return-to-zero modulation format is used, where ΔV _d is the difference between the driving voltages required to make the chemical potential of the first graphene layer (μ _C1 ) 0.2 eV and 0.6 eV, the energy consumption (E _b ) can be given by E _b = C _b (ΔV _d ) ² . The change in driving voltage (ΔV _d ) can be calculated by using the mathematical expression for the driving voltage (V _d ). The change in driving voltage (ΔV _d ) is 4.93 V, and the energy consumption (E _b ) can be 131 fJ/bit.

In many previous studies, graphene-based nanoplasmonic waveguides were considered without considering integration into a silicon photonics platform. Even if the graphene-based waveguide has better performance, the coupler required to connect the graphene-based waveguide to the silicon photonic waveguide may have large loss. The EAM of the present invention can have better performance than EAMs using other graphene-based waveguides. The EAM of the present invention can be compared with EAMs using other graphene-based waveguides in terms of modulator length (l _M ), modulator insertion loss (IL _M ), modulator figure of merit (FoM _M ), 3 dB bandwidth, and minimum feature size required for EAM implementation. Among various graphene-based waveguide-using EAMs, the EAM of the present invention can have a minimum modulator length (l _M ), intermediate modulator insertion loss (IL _M ), maximum modulator figure of merit (FoM _M ), and maximum minimum feature size. The EAM of the present invention can be promising as a compact modulator that can be implemented relatively easily and can be well embedded in a silicon photonic direct circuit.

Although the embodiments have been described with limited drawings as described above, those skilled in the art can apply various technical modifications and variations based on the described embodiments. For example, even if the described techniques are performed in a different order than the described method, and/or the components of the described system, structure, device, circuit, etc. are combined or combined in a different form than the described method, or are replaced or substituted by other components or equivalents, appropriate results can be achieved.

Therefore, other implementations, other embodiments, and equivalents to the claims are also included in the scope of the claims described below.

100: Modulator
110: Graphene-based waveguide
111: Silicon layer
111a: Silicone strip
111b: Silicon oxide
112: First graphene layer
113: Insulating layer
114: Second graphene layer
115: First electrode
116: Second electrode
117: Metal Rail
120: Input Coupler
123: Insulation layer extension
125: First electrode extension
126: Second electrode extension part
127: Metal rail extension
130: Output Coupler
133: Insulation layer extension
135: First electrode extension part
136: Second electrode extension part
137: Metal rail extension
140: Silicon photonic waveguide

Claims (21)

A graphene-based waveguide comprising: a silicon layer, a first graphene layer positioned on one surface of the silicon layer, an insulating layer covering a portion of the first graphene layer and a portion of the silicon layer, a second graphene layer positioned on the insulating layer and spaced apart from the first graphene layer by the insulating layer, a first electrode connected to the first graphene layer, a second electrode connected to the second graphene layer, and a metal rail positioned between the first electrode and the second electrode on the second graphene layer;
An input coupler including an insulating layer extension portion extending to one side from the insulating layer, a first electrode extension portion extending to one side from the first electrode, a second electrode extension portion extending to one side from the second electrode, and a metal rail extension portion extending to one side from the metal rail; and
An output coupler including an insulating layer extension portion extending from the insulating layer to the other side, a first electrode extension portion extending from the first electrode to the other side, a second electrode extension portion extending from the second electrode to the other side, and a metal rail extension portion extending from the metal rail to the other side.
A modulator using a graphene-based waveguide including: In the first paragraph,
The above silicon layer,
A silicon strip comprising one side in contact with the first graphene layer and the other side covered by silicon oxide;
Modulator using graphene-based waveguides. In the second paragraph,
The width of the above silicone strip is,
The width of the slot defined between the second electrode and the metal rail is greater than or equal to that of the second electrode.
Modulator using graphene-based waveguides. In the second paragraph,
The above silicon strip, the metal rail, and the second electrode,
When viewed in a direction perpendicular to the plane corresponding to the one surface of the above silicon layer, they do not overlap each other,
Modulator using graphene-based waveguides. In the first paragraph,
The above insulating layer comprises aluminum oxide.
Modulator using graphene-based waveguides. In the first paragraph,
When the potential of the first graphene layer and the potential of the second graphene layer are different, capacitive coupling is formed between the first graphene layer and the second graphene layer.
Modulator using graphene-based waveguides. In the first paragraph,
The above first graphene layer and the above second graphene layer,
When viewed in a direction perpendicular to a plane corresponding to the one side of the silicon layer, the area occupied by the first graphene layer and the area occupied by the second graphene layer in the plane overlap,
Modulator using graphene-based waveguides. In the first paragraph,
The width of the area where the area occupied by the first graphene layer and the area occupied by the second graphene layer overlap when viewed in a direction perpendicular to the plane corresponding to the one side of the silicon layer is
greater than or equal to the first critical width,
Modulator using graphene-based waveguides. In the first paragraph,
The chemical potential of the first graphene layer is different from the chemical potential of the second graphene layer.
Modulator using graphene-based waveguides. In the first paragraph,
Through the slot defined between the second electrode and the metal rail, the electric field limited between the first graphene layer and the second graphene layer is strengthened.
Modulator using graphene-based waveguides. In the first paragraph,
The width of the above metal rail is greater than or equal to the second critical width,
Modulator using graphene-based waveguides. In the first paragraph,
The height of the above metal rail is the same as the height of the second electrode.
Modulator using graphene-based waveguides. In the first paragraph,
The height of the slot defined between the second electrode and the metal rail is equal to or greater than the first threshold height.
Modulator using graphene-based waveguides. In the first paragraph,
The distance between one side of the metal rail and the first electrode is constant.
Modulator using graphene-based waveguides. In Article 7,
The distance between the other side of the metal rail and the second electrode is constant.
Modulator using graphene-based waveguides. In the first paragraph,
The distance between one side of the metal rail and the first electrode is greater than or equal to a third critical width.
Modulator using graphene-based waveguides. delete delete In the first paragraph,
The above input coupler and the above output coupler,
The second electrode extension portion and the metal rail extension portion have a shape in which the distance between them increases as they get farther away from the waveguide along the longitudinal axis.
Modulator using graphene-based waveguides. delete delete

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