Nanoporous carbon tunable resistor/transistor and methods of production thereof

This application claims priority to U.S. Provisional Patent Appl. No. 61/419,988, filed Dec. 6, 2010, which is herein incorporated by reference.

The United States Government has rights in this invention pursuant to Contract No. DE-AC52-07NA27344 between the United States Department of Energy and Lawrence Livermore National Security, LLC for the operation of Lawrence Livermore National Laboratory.

The present invention relates to an electrochemically controlled tunable resistor/transistor, and more particularly, to a nanoporous carbon tunable resistor/transistor.

It is known that the resistance of a thin metal film can be reversibly tuned via the application of an external charge in an electrolyte where the Helmholtz double layer is utilized as the gate electrode. In this context, the idea of an electrochemically gated all-metal device is related to the well known field-effect transistor (FET), where the electronic transport through a semiconductor channel is controlled by an external gate potential.

However, the effect, i.e., the change in resistance (ΔR/R) is only about 1-2% in the case of a pure metal due to a very large intrinsic carrier density and very short screening length resulting from the very large intrinsic carrier density. This restricts the selection of the pure metal to materials that are stable in an electrochemical environment, which are inert, noble metals and are extremely expensive and heavy. The size of the surface charge induced variation during electronic transport in a metallic conducting material increases with an increase in surface-to-volume ratio. However, most metallic materials with very high surface-to-volume ratio, such as de-alloyed metals, are not thermodynamically stable. Any supply of external energy thus reduces the surface area and the size of surface related effects. Accordingly, a material which could be used in these electrochemical environments and remain stable and do not suffer from the deficiencies of inert, noble metals would be very beneficial.

In one embodiment, a tunable transistor includes a porous material that is electrically coupled between a source electrode and a drain electrode, wherein the porous material acts as an active channel, an electrolyte solution saturating the active channel, the electrolyte solution being adapted for altering an electrical resistance of the active channel based on an applied electrochemical potential, wherein the active channel comprises nanoporous carbon arranged in a three-dimensional structure.

In another embodiment, a method for forming a tunable transistor includes forming a source electrode, forming a drain electrode, and forming a monolithic nanoporous carbon material that acts as an active channel and selectively couples the source electrode to the drain electrode electrically.

According to another embodiment, a method for tuning a tunable resistor/transistor includes altering an electrical resistance of a nanoporous carbon active channel by altering an applied electrochemical potential of an electrolyte, wherein the electrolyte saturates the nanoporous carbon active channel, and the nanoporous carbon active channel is selectively coupled to a source electrode and a drain electrode electrically.

In yet another embodiment, a tunable resistor/transistor includes a nanoporous carbon active channel electrically coupled between a source electrode and a drain electrode, the nanoporous carbon active channel having a monolithic three-dimensional structure such as a carbon aerogel, and an electrolyte solution saturating the nanoporous carbon active channel, the electrolyte solution being adapted for altering an electrical resistance of the nanoporous carbon active channel based on an applied electrochemical potential, wherein the carbon aerogel is doped with an element selected from nitrogen, oxygen, fluorine, and boron, the carbon aerogel has a surface area of greater than about 3000 m 2 /g, the electrolyte solution is an ionic liquid including at least one of: 1-Ethyl-3-methylimidazolium tetiafluoroborate (EMIMBF 4 ), 1-Butyl-3-methylimidazolium hexafluorophosphate (BMIMPF 6 ), and trioctylmethylammonium bis(trifluoromethyl-sulfonyl)imide or an aqueous or nonaqueous electrolyte solution including at least one of: LiF, NaF, KF, KOH, KCl, H 2 SO 4 , HClO 4 , LiClO 4 , NaClO 4 , Et 4 NPF 6 , Et 4 NBF 4 , Bt 4 NPF 6 , and Bt 4 NBF 4 .

Other aspects and embodiments of the present invention will become apparent from the following detailed description, which, when taken in conjunction with the drawings, illustrate by way of example the principles of the invention.

FIG. 1 shows morphology characteristic of a carbon aerogel, according to one embodiment.

FIG. 2 shows an experimental setup that includes a miniaturized electrochemical cell, according to one embodiment.

FIGS. 3A-3B show cyclovoltammograms obtained from activated carbon aerogels with different surface areas, according to one embodiment.

FIGS. 4A-4C show changes in resistance of a carbon aerogel working electrode while the electrochemical potential of the carbon aerogel working electrode is periodically changed in an aqueous electrolyte, according to various embodiments.

FIGS. 5A and 5B show pulse current charging and discharging of a carbon aerogel. over time, and the corresponding change in resistance, respectively.

FIG. 6 shows a tunable resistor/transistor, according to one embodiment.

FIG. 7 is a flow diagram of a method according to one embodiment.

FIG. 8 is a flow diagram of a method according to one embodiment.

The following description is made for the purpose of illustrating the general principles of the present invention and is not meant to limit the inventive concepts claimed herein. Further, particular features described herein can be used in combination with other described features in each of the various possible combinations and permutations.

Unless otherwise specifically defined herein, all terms are to be given their broadest possible interpretation including meanings implied from the specification as well as meanings understood by those skilled in the art and/or as defined in dictionaries, treatises, etc.

It must also be noted that, as used in the specification and the appended claims, the singular forms “a,” “an” and “the” include plural referents unless otherwise specified.

In one general embodiment, a tunable transistor includes a porous material that is electrically coupled between a source electrode and a drain electrode, wherein the porous material acts as an active channel, an electrolyte solution saturating the active channel, the electrolyte solution being adapted for altering an electrical resistance of the active channel based on an applied electrochemical potential, wherein the active channel comprises nanoporous carbon arranged in a three-dimensional structure.

In another general embodiment, a method for forming a tunable transistor includes forming a source electrode, forming a drain electrode, and forming a monolithic nanoporous carbon material that acts as an active channel and selectively couples the source electrode to the drain electrode electrically.

According to another general embodiment, a method for tuning a tunable resistor/transistor includes altering an electrical resistance of a nanoporous carbon active channel by altering an applied electrochemical potential of an electrolyte, wherein the electrolyte saturates the nanoporous carbon active channel, and the nanoporous carbon active channel is selectively coupled to a source electrode and a drain electrode electrically.

In yet another general embodiment, a tunable resistor/transistor includes a nanoporous carbon active channel electrically coupled between a source electrode and a drain electrode, the nanoporous carbon active channel having a monolithic three-dimensional structure including a carbon aerogel, and an electrolyte solution saturating the nanoporous carbon active channel, the electrolyte solution being adapted for altering an electrical resistance of the nanoporous carbon active channel based on an applied electrochemical potential, wherein the carbon aerogel is doped with an element selected from nitrogen, oxygen, fluorine, and boron, the carbon aerogel has a surface area of greater than about 3000 m 2 /g, the electrolyte solution is an ionic liquid including at least one of: 1-Ethyl-3-methylimidazolium tetrafluoroborate (EMIMBF 4 ), 1-Butyl-3-methylimidazolium hexafluorophosphate (BMIMPF 6 ), and trioctylmethylammonium bis(trifluoromethyl-sulfonyl)imide or an aqueous or nonaqueous electrolyte solution including at least one of: LiF, NaF, KF, KOH, KCl, H 2 SO 4 , HClO 4 , LiClO 4 , NaClO 4 , Et 4 NPF 6 , Et 4 NBF 4 , Bt 4 NPF 6 , and Bt 4 NBF 4 .

According to one embodiment, an electrochemically controlled tunable resistor/transistor is based on a bulk nanoporous carbon material with extreme surface area and electrochemical addressing of the surfaces to control the density and nature of charge carriers. The intrinsic two dimensional morphology of sp 2 hybridized graphene sheets that are a major constituent of highly activated carbon aerogels stabilizes this material although it has an extremely high (greater than about 3000 m 2 /g) surface-to-volume ratio.

Furthermore, according to some embodiments, the electrochemically controlled tunable resistor/transistor based on the nanoporous carbon material may have a three dimensional (3D) structure, and may be made to form monolithic 3D bodies. The nanoporous carbon is highly conductive due to its graphitic nature and its continuous ligament morphology. Specifically, carbon aerogels with extremely high mass-specific surface area (about 3000 m 2 /g, more or less) may be used as the nanoporous carbon in some embodiments.

Carbon aerogels (CAs) are mesoporous materials which combine many interesting properties, such as low mass densities, continuous porosities, high surface areas, high electrical conductivities, and excellent mechanical properties. The properties of carbon aerogels are derived from their microstructure, which is a network of interconnected primary particles with characteristic diameters between about 3 to 25 nm, in some approaches. The material forms macroscopic (for instance, centimeter-sized) monolithic bodies that can support high electrical currents.

FIG. 1 shows morphology characteristic of a carbon aerogel before activation in CO 2 at about 950° C., according to one embodiment. This is just an example of the morphology of a typical carbon aerogel, and any morphology resulting from the embodiments and approaches described herein may be used, as would be understood by one of skill in the art upon reading the present descriptions.

The material may be synthesized through sol-gel polymerization of resorcinol with formaldehyde in aqueous solution to produce organic gels which may be supercritically dried and subsequently pyrolyzed in an inert atmosphere, in one approach. The material may also be activated to increase the mass specific surface area, for example in CO 2 at about 950° C. The resulting material has macropores to facilitate mass transport, and micropores which provide a high surface area.

The electrical resistance of carbon aerogel materials was tested in an experimental setup that included a miniaturized electrochemical cell that allows performing four point probe electrical I/V measurements on the nanoporous carbon working electrode as a function of the applied electrochemical potential. As shown in FIG. 2 , this electrochemical cell 200 includes a reference electrode 202 , a counter electrode 204 , and the carbon aerogel 206 for testing. In one embodiment, the reference electrode 202 may include silver and/or silver chloride (Ag/AgCl). The counter electrode 204 , in one approach, may comprise nanoporus carbon or amorphous carbon (kynol). Of course, any suitable material may be used for the reference electrode 202 and counter electrode 204 as would be apparent to one of skill in the art upon reading the present descriptions.

Similar to a field effect transistor (FET), both the density and the nature of the charge carriers in the CA electrode may be controlled through the electrochemical double layer. Aqueous, nonaqueous, and/or ionic electrolytes may be used. Aqueous electrolytes may include salt solutions in water such as LiF, NaF, KF, KOH, KCl, H 2 SO 4 , HClO 4 , LiClO 4 , NaClO 4 , Et 4 NPF 6 (tetraethylammonium hexafluorophosphate), Et 4 NBF 4 , Bt 4 NPF 6 (tetrabutylammonium hexafluorophosphate), Bt 4 NBF 4 , etc. Nonaqueous electrolytes may include any of acetonitrile, ethylene carbonate (EC), propylene carbonate (PC), dimethyl carbonate (DMC), ethyl acetate (EA), n-methyl formrmide (NMF), dimethyl formamide (DMF), dimethyl sulfoxide (DMSO) in a supporting electrolyte, such as LiF, NaF, HClO 4 , LiClO 4 , NaClO 4 , Et 4 NPF 6 , Et 4 NBF 4 , Bt 4 NPF 6 , Bt 4 NBF 4 , etc. Ionic electrolytes include 1-Ethyl-3-methylimidazolium tetrafluoroborate (EMIMBF 4 ), 1-Butyl-3-methylimidazolium hexafluorophosphate (BMIMPF 6 ), trioctylmethylammonium bis(trifluoromethyl-sulfonyl)imide, etc.

The electrochemical cell 200 that was used for the surface charged induced change in resistance measurement included gold electrodes that were used to have pressure contact to the CA sample 206 . Four contacts were used for four probe resistance measurements (I + , V + , V − , I − ) and the fifth contact was connected to the working electrode WE which is the nanoporous carbon active channel that is electrically coupled between a source electrode and a drain electrode. The capacitive window of the CA sample/electrolyte combination was determined by cyclic voltammetry.

FIGS. 3A-3B show cyclovoltammograms obtained from activated carbon aerogels with different surface areas ( FIG. 3A : 1900 m 2 /g CA sample at a scan rate of 0.1 mV/s started from the open circuit potential (OCP) of 0.1 V); FIG. 3B : 2800 m 2 /g CA sample at a scan rate of 0.5 mV/s). Note: The OCP was determined by the zero current chronopotentiometry. The capacitive double layer region was about 800 mV wide. The total charge build up may be calculated as 1.64 C. The mass of the sample was 23.8 mg. Therefore, the specific surface charge (considering the BET surface area=electrochemical active surface area)=3.62 μC/cm 2 , and the specific double layer capacitance was typically in an order of 4-5 μF/cm 2 .

Experiments showed that reversible changes of electrical resistance up to several 100% may be realized from centimeter-sized CA electrodes as described according to various embodiments herein. Changes in resistance of a CA working electrode while the electrochemical potential of the CA working electrode is periodically changed in an aqueous electrolyte are shown in FIGS. 4A-4C , according to various embodiments.

FIG. 4B shows surface charge induced variation in resistance of a CA working electrode with a surface area of 1900 m 2 /g while the electrochemical potential of the CA working electrode was periodically changed between −100 and +700 mV as observed in FIG. 3A . Prior to the measurement, the samples were degassed inside the electrolyte in a vacuum desiccator for about 24 hours. Referring again to FIG. 4B , as can be seen, the reversible change in resistance (ΔR) is more than 100%.

Similar results were observed for a 400 m 2 /g CA sample while the electrochemical potential of the CA working electrode was periodically changed between −100 and +700 mV (as shown in FIG. 4A ) and a 2800 m 2 /g CA sample while the electrochemical potential of the CA working electrode was periodically changed between −600 and +300 mV (as shown in FIG. 4C ). A small drift of the ΔR/R plot is observed in the later case ( FIG. 4C ), but this may be related to a small amount of irreversible surface adsorption.

It can be seen from FIGS. 4A-4C that an increase in resistance is observed when the voltage/bias that is applied to the CA with respect to the electrolyte becomes more negative. If the applied bias becomes sufficiently negative, a resistivity maximum (conductivity minima) may be observed. Such a conductivity maximum is expected if the Fermi level is at the Dirac point of the graphene constituent of the CA electrode. For more positive bias the material is predominantly an electron conductor, for more negative bias the material is predominantly a hole conductor (bipolar). Thus both holes and electrons may be responsible for the conductivity changes described above.

Undoped graphite has a carrier concentration of 2×10 19 cm −3 . The high conductivity is a result of extremely large in-plane mobility in graphite (13,000 cm 2 /Vs). The Fermi level moves downward or upward depending on if the material is doped with holes or electrons, respectively. The material may be both unipolar and bipolar, depending on the applied potential range and the doping level. The double layer capacitance increases with carrier concentration which is higher in CAs (2×10 20 cm −3 ) than in undoped graphite (2×10 19 cm −3 ). Of course, other explanations are possible, and this theory is not meant to be limiting on the embodiments and approaches presented herein.

Now referring to FIGS. 5A-5B , a 1900 m 2 /g CA sample in an aqueous electrolyte was subjected to pulse charging and discharging as shown in FIG. 5A , with corresponding change in resistance shown in FIG. 5B , according to one embodiment. FIG. 5A demonstrate that highly reproducible charging and discharging cycles are possible with nearly zero leakage current. Furthermore, the calculated surface charge density is 6 μC/cm 2 or 125 F/g. As shown in FIG. 5B , the charging and discharging time is of an order of a minute, which may be reduced by proper selection of the electrolyte and modification of the pore morphology, as would be apparent to one of skill in the art upon reading the present descriptions.

The reversible changes of electrical resistance may be further increased, for example, by introducing a band gap by controlled doping using any suitable material. According to various embodiments, nitrogen, oxygen, fluorine, boron, or any other element that modifies the electronic structure of the nanoporous carbon may be used to dope the CA. In one example, an n-type graphene-based field effect transistor device may be produced by nitrogen doping of graphene nanoribbons. Of course, any suitable doping material may be used, such as dopants commonly used in the semiconductor industry, as would be appreciated by one of skill in the art upon reading the present descriptions.

Now referring to FIGS. 6A-6C , a tunable resistor/transistor 600 is shown according to one embodiment. In FIG. 6A , an isometric view is shown of one embodiment of the tunable resistor/transistor 600 . In FIG. 6B , a top view of the tunable resistor/transistor 600 is shown according to one embodiment. A simplified circuit diagram of the tunable resistor/transistor 600 is shown in FIG. 6C , according to one embodiment.

The tunable resistor/transistor comprises a porous material 602 that is contacted by a source electrode 604 and a drain electrode 606 , and an electrolyte solution 608 saturating the porous material 602 , the electrolyte solution 608 being adapted for altering an electrical resistance of the porous material 602 based on an applied electrochemical potential. The electrochemical gate is formed by charge accumulation of ions on the electrolyte side and electrons/holes on the solid side at the solid/electrolyte interface when a voltage is applied between the porous material 602 and a counter electrode. The porous material 602 comprises nanoporous carbon arranged in a three-dimensional structure. Here, the overall structure of the porous material 602 is shown as a polyhedron having a rectangular cross-section. However, any three-dimensional shape may be used, such as irregular shapes and/or regular shapes, like a sphere, pyramid, prism, spheroid, or any other shape as would be known to one of skill in the art or that suits the application for which the porous material 602 is intended.

Furthermore, each of the source electrode 604 and/or the drain electrode 606 may comprise any suitable material as would be known to one of skill in the art, such as active carbon, platinum, electrically conductive metals, alloys and/or polymers, etc. In addition, the source electrode 604 and/or the drain electrode 606 may have any shape suitable for use in the resistor/transistor and compatible with the shape of the porous material 602 , as would be known to one of skill in the art.

Also, the orientation and positioning of each of the source electrode 604 , drain electrode 606 , and/or the porous material 602 may be altered as would be understood by one of skill in the art upon reading the present descriptions. For example, the source electrode 604 may be positioned opposite the drain electrode 606 on longitudinal ends of the porous material 602 , they may be positioned on the same side of the porous material 602 , they may be smaller or larger than one another, they may be connected to the porous material 602 using vias, wires, channels, or any other electrical connections, they may be positioned on a side and a top of the porous material 602 , etc.

In one embodiment, the nanoporous carbon may comprise a carbon aerogel. The carbon aerogel may be procured through any method of production. In one exemplary embodiment, the carbon aerogel may be formed by polymerizing a sol-gel of resorcinol and formaldehyde (or any other suitable materials as known in the art) in an aqueous solution to produce an organic gel, supercritically drying the organic gel, pyrolizing the dried organic gel in an inert atmosphere and activating in carbon dioxide (or some other suitable gas as known in the art) at a temperature of greater than about 900° C. to form the carbon aerogel.

In another embodiment, the carbon aerogel may be doped with an element suitable for use in the semiconductor industry, such as nitrogen, oxygen, fluorine, boron, or any other suitable material as would be understood by one of skill in the art upon reading the present descriptions. Oxygen, nitrogen, and boron are readily available and much experience exists in using them as dopants.

In some approaches, the nanoporous carbon may have a surface area of greater than about 2800 m 2 /g, preferably greater than about 3000 m 2 /g, and even more preferably greater than about 3200 m 2 /g, such as in a range from about 2500 m 2 /g to about 3500 m 2 /g, according to various embodiments.

The electrolyte solution may be in a liquid, solid, or semi-solid form (such as a gel, extrusion, etc.), as would be understood by one of skill in the art. In some approaches, the electrolyte solution may be an aqueous electrolyte. Some examples of suitable aqueous electrolyte solutions include one or more of LiF, NaF, KF, KOH, KCl, H 2 SO 4 , HClO 4 , LiClO 4 , NaClO 4 , Et 4 NPF 6 , Et 4 NBF 4 , Bt 4 NPF 6 , Bt 4 NBF 4 , etc. In other approaches, the electrolyte solution may be a nonaqueous electrolyte which is one or more electrolytes in a solvent. Some exemplary electrolytes suitable for use include one or more of LiF, NaF, HClO 4 , LiClO 4 , NaClO 4 , Et 4 NPF 6 , Et 4 NBF 4 , Bt 4 NPF 6 , Bt 4 NBF 4 , etc. Some exemplary solvents suitable for use include one or more of acetonitrile, ethylene carbonate (EC), propylene carbonate (PC), dimethyl carbonate (DMC), ethyl acetate (EA), n-methyl formamide (NMF), dimethyl formamide (DMF), dimethyl sulfoxide (DMSO), etc. In alternate approaches, the electrolyte solution may be an ionic liquid. Some exemplary ionic liquids that may be used include one or more of 1-Ethyl-3-methylimidazolium tetrafluoroborate (EMIMBF 4 ), 1-Butyl-3-methylimidazolium hexafluorophosphate (BMIMPF 6 ), trioctylmethylammonium bis(trifluoromethyl-sulfonyl)imide, etc.

Now referring to FIG. 7 , a method 700 for forming a tunable resistor/transistor is shown according to one embodiment. The method 700 may be carried out in any desired environment, including those shown and described in FIGS. 1-6C , according to various embodiments. Also, more or less operations than those specifically described in FIG. 7 may be included in method 700 in several approaches.

In operation 702 , a source electrode is formed, such as above a substrate, in one approach. The source electrode may comprise any suitable material, such as active carbon, platinum, electrically conductive metals, alloys and/or polymers, etc.

Any suitable material may be used for the substrate, as would be known to one of skill in the art, such as silicon, silicon composites and alloys, oxides such as alumina, glass, etc.

In operation 704 , a drain electrode is formed, such as above the substrate, in one approach. The drain electrode may comprise any suitable material, such as active carbon, platinum, electrically conductive metals, alloys and/or polymers, etc.

The source and drain electrodes may be formed in a single operation, simultaneously, using the same material or different materials, on different substrates, at different times, etc.

In operation 706 , a monolithic nanoporous carbon material that acts as an active channel is formed, such as above the substrate, or one or more of the source and/or drain electrodes. The nanoporous carbon material selectively couples the source electrode to the drain electrode electrically. There may be a direct connection between the active channel and either or both of the drain electrode and/or source electrode, or any vias, wires, channels, or any other electrical connections may be used to form the connection.

In one embodiment, the nanoporous carbon material may comprise a carbon aerogel. The carbon aerogel may be formed as described herein using a sol-gel method or using any other suitable method as would be appreciated by one of skill in the art.

In one such embodiment, the carbon aerogel may be formed by polymerizing a sol-gel of resorcinol and formaldehyde (or some other suitable chemistry) in an aqueous solution to produce an organic gel, supercritically drying the organic gel, pyrolizing the dried organic gel in an inert atmosphere to form the carbon aerogel, and activating the carbon aerogel in carbon dioxide at a temperature of greater than about 900° C. to increase surface area of the carbon aerogel to above about 2000 m2/g.

In a further embodiment, the carbon aerogel may be doped with a suitable material, such as nitrogen, oxygen, fluorine, and/or boron.

In more approaches, the nanoporous carbon may have a surface area of greater than about 2800 m 2 /g, preferably greater than about 3000 m 2 /g, and even more preferably greater than about 3200 m 2 /g, such as in a range from about 2500 m 2 /g to about 3500 m 2 /g, according to various embodiments.

In one embodiment, the method 700 may further comprise applying an electrolyte to the nanoporous carbon active channel and altering an electrical resistance of the nanoporous carbon active channel based on an applied electrochemical potential of the electrolyte.

In further embodiments, the electrolyte may have physical characteristics as described herein and may comprise any suitable material, as described herein, according to various embodiments. For example, the electrolyte may be an aqueous electrolyte solution, a nonaqueous electrolyte solution, an ionic liquid, etc.

Now referring to FIG. 8 , a method 800 for tuning a tunable resistor/transistor is shown according to one embodiment. The method 800 may be carried out in any desired environment, including those shown and described in FIGS. 1-6C , according to various embodiments. Also, more or less operations than those specifically described in FIG. 8 may be included in method 800 in several approaches.

In operation 802 , an electrical resistance of a nanoporous carbon active channel is altered by altering an applied electrochemical potential of an electrolyte. The nanoporous carbon active channel may be saturated by the electrolyte in some approaches. However, any method of conveying electrical properties of the electrolyte to the nanoporous carbon active channel may be used as would be known to one of skill in the art.

In addition, the nanoporous carbon active channel is selectively coupled to a source electrode and a drain electrode electrically, in some embodiments, to form a tunable resistor/transistor.

In one approach, the nanoporous carbon may comprise a carbon aerogel doped with a suitable material, such as nitrogen, oxygen, fluorine, and/or boron. Of course, any other embodiments and/or approaches described herein may be used in the context of functionality of FIG. 8 , in various implementations.

Illustrative applications of various embodiments include tunable resistors for high power DC applications, sensing, electrochemically controlled all-metal field effect transistors, use in low voltage high power electrical circuits, etc.

An additional advantage is that the various embodiments allow fabrication and/or use of bulk transistors/resistors, in contrast to the thin film technology of the semiconductor industry. Moreover, some embodiments may include all-carbon transistor/resistor technology, where the source and drain electrodes include carbon, as does the CA active channel.

While various embodiments have been described above, it should be understood that they have been presented by way of example only, and not limitation. Thus, the breadth and scope of a preferred embodiment should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.