Perturbative symmetry in the engineering of stacked graphene

Abstract

The graphene structure may include multiple graphene layers stacked to have perturbation symmetry. A first graphene layer may be positioned at a first rotation angle with respect to a rotation axis that extends perpendicularly through the first graphene layer, and a second graphene layer may be positioned on the first graphene layer at a second rotation angle with respect to the rotation axis. A third graphene layer may be positioned on the second graphene layer at a third rotation angle with respect to the rotation axis, the third rotation angle being different from the second rotation angle. Additional graphene layers may be subsequently stacked on the graphene structure, each layer being placed at a different rotation angle than the previous layer. A total of six graphene layers may be stacked. The ratio relationship between all the rotation angles may constitute an arithmetic, geometric, or Fibonacci progression, or other pattern.

Description

[Technical field]

[0001] The present disclosure relates generally to graphene technology, and more particularly to improving properties in structured graphene arrangements. [Background technology]

[0002] Graphene has emerged as a wonder material and is expected to play a key role in a variety of future technologies such as biosensors, radiation detectors, and next-generation quantum technologies such as qubits. Current research is directed towards controlling the electronic properties of graphene devices by stacking and rotating individual graphene flakes. In such devices, the interactions of electrons are controlled by the interference patterns formed when the graphene flakes are stacked. For example, electrons can form a superconducting phase at a stacking angle of 1.1 degrees.

[0003] Despite this remarkable discovery, not much research has been done towards finding better graphene structures, and how controlling these interference patterns can lead to tunable electronic properties remains poorly understood. Unfortunately, the superconducting properties of the graphene layers at a stacking angle of 1.1 degrees are still limited and have not currently led to viable, cost-effective applications.

[0004] Although conventional methods for forming graphene structures have worked well thus far, improvements are always useful, and in particular, there is a need for graphene structures with improved properties beyond those of previously constructed structures. Summary of the Invention [Problem to be solved by the invention]

[0005] It is an advantage of the present disclosure to provide graphene structures with improved properties over previously constructed structures. The disclosed features, apparatus, systems, and methods relate to stacking of graphene with perturbed symmetries and methods of forming graphene with perturbed symmetries. The disclosed apparatus, systems, and methods utilize known properties of simple graphene structures to provide more complex graphene structures with significantly improved properties. [Means for solving the problem]

[0006] In various embodiments of the present disclosure, a device, which may be, for example, a graphene structure, may include at least a first, a second, and a third graphene layer. The first graphene layer may be positioned at a first rotation angle with respect to a rotation axis extending vertically through the first graphene layer. The second graphene layer may be positioned above the first graphene layer at a second rotation angle with respect to the rotation axis. The third graphene layer may be positioned above the second graphene layer at a third rotation angle with respect to the rotation axis, the third rotation angle being different from the second rotation angle.

[0007] In various detailed embodiments, the device may further include a fourth, a fifth, and a sixth graphene layer. The fourth graphene layer may be positioned on the third graphene layer at a fourth rotation angle about the rotation axis, and the fourth rotation angle may be different from the third rotation angle. The fifth graphene layer may be positioned on the fourth graphene layer at a fifth rotation angle about the rotation axis, and the fifth rotation angle may be different from the fourth rotation angle. The sixth graphene layer may be positioned on the fifth graphene layer at a sixth rotation angle about the rotation axis, and the sixth rotation angle may be different from the fifth rotation angle. In various arrangements, each rotation angle from the second rotation angle to the sixth rotation angle may be greater than the previous rotation angle. Furthermore, the ratio relationship from the first rotation angle to the sixth rotation angle may constitute an arithmetic progression, a geometric progression, or a Fibonacci progression. In various configurations, the device may exhibit superconductivity, improved thermal conductivity, improved tensile strength, and / or improved photosensitivity, as well as other improved properties at elevated temperatures. Also, one or more additional graphene layers may be overlaid on the device. Each additional graphene layer may be positioned on top of the previous graphene layer with an additional rotation angle about the axis of rotation, and each additional rotation angle may be different from the previous rotation angle. In some configurations, each additional rotation angle may be greater than the previous rotation angle, and the ratio relationship of all the rotation angles may also constitute an arithmetic progression, a geometric progression, or a Fibonacci progression.

[0008] In various further embodiments of the present disclosure, a method of forming a graphene structure is provided. The associated process steps may include forming a first graphene layer using a forming fixture, rotating the forming fixture a first rotational amount, forming a second graphene layer on the first graphene layer, rotating the forming fixture a second rotational amount, and forming a third graphene layer on the second graphene layer. The first graphene layer may be formed at a first rotational angle with respect to a rotational axis extending perpendicularly through the first graphene layer. The second graphene layer may be formed at a second rotational angle equal to the first rotational angle plus the first rotational amount. The second rotational amount may be different from the first rotational amount. The third graphene layer may be formed at a third rotational angle equal to the second rotational angle plus the second rotational amount.

[0009] In various detailed embodiments, the additional steps may include rotating the formation facility a third rotational amount, forming a fourth graphene layer on the third graphene layer, rotating the formation facility a fourth rotational amount, forming a fifth graphene layer on the fourth graphene layer, rotating the formation facility a fifth rotational amount, and forming a sixth graphene layer on the fifth graphene layer. Each rotational amount may be different from the previous rotational amount, and each graphene layer may be formed at a rotational angle equal to the previous rotational angle plus the most recent rotational amount. Each rotational angle from the second rotational angle through the sixth rotational angle may be greater than the previous rotational angle.

[0010] Further process steps may include rotating the forming fixture through one or more additional rotations, each rotation amount being different from the previous rotation amount, and forming one or more additional graphene layers after each rotation of the forming fixture. Each additional graphene layer may be positioned on top of the previous graphene layer at an additional rotation angle about the axis of rotation, each additional rotation angle being equal to the previous rotation angle plus the most recent rotation amount. In some configurations, the ratio relationship of all the rotation angles may constitute an arithmetic, geometric, or Fibonacci progression, or other pattern. Each of the various possible graphene structures may exhibit superconductivity, enhanced thermal conductivity, enhanced tensile strength, and / or enhanced photosensitivity, as well as other enhanced properties at elevated temperatures.

[0011] In various further detailed embodiments, further process steps may include preparing a precursor graphene structure, pressing a coated substrate onto the precursor graphene, and heating the substrate to peel the graphene flakes off the coated substrate. The precursor graphene structure may be bilayer graphene comprising a single layer of hexagonal boron nitride, and the coated substrate may be glass coated with polydimethylsiloxane.

[0012] Other devices, methods, features, and advantages of the present disclosure will be or become apparent to one with skill in the art upon examination of the following figures and detailed description, and all such additional devices, methods, features, and advantages are intended to be included within this specification, be within the scope of the disclosure, and be protected by the accompanying claims. [Brief description of the drawings]

[0013] The included figures are illustrative and are intended only to provide examples of possible structures, arrangements, and methods for forming the disclosed stacked graphene with perturbative symmetry technology, and are in no way intended to limit the changes in form and detail that may be made to the present disclosure by those skilled in the art without departing from the spirit and scope of the present disclosure.

[0014] FIG. 1 shows a top view of an example graphene layer.

[0015] FIG. 2A illustrates a top view of an example structure having two stacked rotated graphene layers according to one embodiment of the present disclosure.

[0016] FIG. 2B illustrates a larger scale top view of the structure of FIG. 2A according to one embodiment of the present disclosure.

[0017] FIG. 3A illustrates a top view of an example structure having six stacked graphene layers at different angles according to one embodiment of the present disclosure.

[0018] FIG. 3B illustrates a larger scale top view of the structure of FIG. 3A according to one embodiment of the present disclosure.

[0019] FIG. 4 illustrates a flow chart of an example of a high level method for forming a layered graphene structure in accordance with one embodiment of the present disclosure.

[0020] FIG. 5 illustrates a flow chart of an example of a detailed method for forming a layered graphene structure with perturbative symmetry according to an embodiment of the present disclosure.

[0021] FIG. 6 illustrates a top view of an example structure having six stacked graphene layers rotated at Fibonacci angles according to one embodiment of the present disclosure. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0022] Exemplary uses of the devices, systems, and methods according to the present disclosure are described in this section. These examples are provided only to add background information and aid in the understanding of the present disclosure. Thus, it will be apparent to one of ordinary skill in the art that the present disclosure may be practiced without some or all of those specific details set forth herein. In some examples, well-known processing steps have not been described in detail to avoid unnecessarily obscuring the present disclosure. The examples described below should not be taken as limiting, and other applications may be realized. In the following detailed description, reference is made to the accompanying drawings, which form a part of the specification, and which are shown therein by way of example of specific embodiments of the present disclosure. These embodiments are described in sufficient detail to enable one of ordinary skill in the art to practice the present disclosure, but it will be understood that these examples are not limiting, and that other embodiments may be used and changes may be made without departing from the spirit and scope of the present disclosure.

[0023] The present disclosure, in various embodiments thereof, relates to features, apparatus, systems, and methods for forming graphene arrangements and graphene structures with enhanced properties. Disclosed embodiments include constructing and using graphene arrangements having stacked graphene layers arranged at a rotation angle relative to one another. In various embodiments, more than two stacked graphene layers are formed. In some structures, six or more stacked graphene layers may be formed. The stacked graphene layers may be rotated by different rotation angles relative to one another to have perturbation symmetry to improve the overall properties of the stacked graphene structure. In some configurations, the relative rotation ratio between stacked layers may follow an arithmetic, geometric, or Fibonacci progression, or other pattern.

[0024] Although the various embodiments disclosed herein describe stacked graphene layers rotated at different angles to each other to have perturbation symmetry, it will be readily understood that the disclosed features, apparatus, systems, and methods may similarly use any suitable alternatives or materials that utilize the disclosed features. Similarly, other forms of different rotation patterns may be used other than arithmetic, geometric, or Fibonacci progressions. Of course, other numbers of stacks, more or less than six layers, and other suitable methods of forming these stacks may also be used. Other applications, arrangements, and estimates beyond the illustrated embodiments are also contemplated.

[0025] Referring first to FIG. 1, an example of a graphene layer is shown in a top view. As is commonly known, graphene is an amazing two-dimensional material composed entirely of carbon atoms. A graphene layer 100 may include a single layer of carbon atoms 102 arranged in a repeating hexagonal pattern 104 to form a flat two-dimensional honeycomb lattice. The graphene layer 100 may extend laterally as far as needed in both the X and Y dimensions, but is limited to a single atomic thickness in the Z direction. Each carbon atom 102 may be covalently bonded to its three nearest neighbors and may donate one electron to a conduction band that extends throughout the graphene layer 100. The six-fold symmetry of the atomic lattice, combined with the small nuclear mass of the carbon atom, ensures that electrons behave as weakly bound, nearly massless particles, allowing them to be conducted at near relativistic velocities. This combined effect results in a low-dimensional quantum matter system whose properties are directly related to its geometric configuration or geometry. Other characteristics and properties of the graphene layer 100 are generally well understood by those of skill in the art.

[0026] 2A and 2B, an example of a structure having two stacked and rotated graphene layers is shown in a top view. The graphene structure 200 may include a first graphene layer 100 and a second graphene layer 110, both of which may include the same pattern of carbon atoms arranged in a repeating hexagonal pattern 104. The first graphene layer 100 may be positioned at a first rotation angle with respect to a rotation axis that extends vertically (i.e., along the Z-axis into the page) through the first graphene layer 100. Such a first rotation angle may be a reference angle of 0 degrees for purposes of illustration. The second graphene layer 110 may be positioned above the first graphene layer 100 such that the layers are stacked and positioned at a second rotation angle with respect to the rotation axis. This second rotation angle may be small or slight, but not zero. For example, if the first rotation angle is zero, the second rotation angle may be 1 degree. Of course, other rotation amounts are also contemplated.

[0027] With just two graphene layers 100, 110 stacked and rotated relative to one another, at least some Moiré interference patterns can be observed in the overlapping hexagons 104 throughout the structure 200. These long period interference patterns can affect the electronic interactions between the graphene layers 100 and 110, which can result in a superconducting phase at a desired temperature under the right circumstances. Other property enhancements can include unique semiconductivity and / or magnetism, improved thermal conductivity, improved tensile strength, and / or improved photosensitivity. While much research has focused on single layers of graphene, it is noted that stacked layers of graphene can result in a variety of interesting phenomena. This can be due to interactions between the layers that affect the movement of electrons in each layer. For example, two layers of graphene flakes twisted (i.e., rotated) at a relative angle can produce a superconducting transition and can also exhibit magnetic and other strongly correlated phase transitions. Thus, by controlling the rotation angle between the graphene layers, the electronic properties of the entire structure can be tuned. FIG. 2B shows a simplified representation of a larger scale graphene structure 200.

[0028] In various embodiments of the present disclosure, new and unexplored patterns are used in forming structures having more than two stacked graphene layers. Turning to Figures 3A and 3B, an example of a structure of six stacked graphene layers rotated at a relative angle is also shown in top view. The graphene structure 300 may include first through sixth graphene layers 100, 110, 120, 130, 140, 150, 160 stacked on top of each other. As with the graphene structure 200 shown above, each of these graphene layers may be structurally identical or substantially identical, and each layer may extend as far as necessary in both the X and Y directions. The primary difference between these graphene layers 100, 110, 120, 130, 140, 150, 160 may be the amount of relative rotation between each layer with respect to the layer above and / or below it. 3, the relative rotation between each of the graphene layers 100, 110, 120, 130, 140, 150, 160 can be about 1 degree. In this rotated arrangement of six stacked graphene layers, more Moiré interference patterns can be seen between the overlapping hexagons in all layers, and an improved desirable effect of the overall structure can be observed.

[0029] While benefits may be observed using six stacked rotated graphene layers such as the graphene structure 300 shown in FIG. 3A, further benefits may be observed by carefully controlling and varying the amount of relative rotation or relative twist between each graphene layer. Patterns other than a simple linear pattern of the graphene structure 300 may provide further enhancement of the desired effect in the overall graphene structure. For example, a progression of increasing relative rotation (i.e., relative "twist angle") from one layer to the next may be better than a simple linear pattern. Rather than having a completely or partially symmetrical relative rotation arrangement, perturbed symmetry of the relative rotation between layers may be used. FIG. 3B shows a simplified view of the graphene structure 300 on a larger scale.

[0030] In various embodiments, the pattern of relative rotation between the graphene layers may follow an arithmetic progression, a geometric progression, or a Fibonacci progression. For example, in the case of a Fibonacci progression, the first graphene layer may be arranged at a rotation angle of 0 degrees, the second graphene layer may be arranged at 1 degree, the third graphene layer may be arranged at 1 degree, the fourth layer may be arranged at 2 degrees, the fifth layer may be arranged at 3 degrees, and the sixth layer may be arranged at 5 degrees. If additional layers are used, they may be arranged at rotation angles of 8, 13, 21, 34 degrees, etc. Alternatively, the first graphene layer may be arranged at 1 degree, and the remaining layers may be arranged at 1, 2, 3, 5, and 8 degrees. Other starting points in the Fibonacci progression may also be used. Other types of arithmetic or geometric progressions may also be used.

[0031] Furthermore, the initial rotation amount is not limited to 0 degrees or 1 degree. Rather, the relative rotation amount may follow a ratio pattern that matches the Fibonacci sequence. For example, if a first graphene layer has a first rotation angle of 0 degrees and a second graphene layer has a second rotation angle of 5 degrees, then the first rotation amount between the first graphene layer and the second graphene layer is 5 degrees. The second rotation amount between the second graphene layer and the third graphene layer may be 5 degrees, and subsequent rotation amounts between subsequent layers may be 10, 15, 25, and 40 degrees, etc., following the ratio pattern of the Fibonacci sequence. Other starting amounts in degrees or radians for the first rotation angle and / or first rotation amount are also contemplated.

[0032] To establish the effect of perturbation symmetry in the stacking and relative rotation of these layers, transport measurements can be performed that focus specifically on establishing the relationship between superconductivity, critical temperature, and stacking angle configurations. This can be done for certain relative rotations between stacked layers, such as arithmetic, geometric, or Fibonacci ratio patterns, among other possible sequences or patterns.

[0033] Generally, compared to the bilayer graphene structures discussed above, certain graphene structures with three or more layers, especially six layers, may provide increased superconducting critical temperatures and other improved structural properties. Such structures may provide precisely tunable electronic properties that are beneficial for pushing these technologies to greater maturity, and may provide implications for a variety of technologies, including superconducting dislocation edge sensors and bolometer detectors used in space exploration, as well as qubits and quantum sensors used in solid-state quantum computers, among other possible technologies.

[0034] 4, a flow chart of an example high level method 400 of forming a layered graphene structure is shown. After a start step 402, a first process step 404 may include forming a first graphene layer. This may be achieved by any suitable graphene formation process, and the first graphene layer may be positioned at a first rotational angle with respect to an axis of rotation extending perpendicularly through the first graphene layer.

[0035] In a subsequent process step 406, the fabrication equipment used to form the graphene layer may be rotated a first rotational amount, which may provide a relative rotation between the already formed first graphene layer and the next graphene layer to be formed. In a next process step 408, a second graphene layer may be formed on the first graphene layer, which may include the same fabrication equipment and techniques used to form the first graphene layer in process step 404.

[0036] In a subsequent process step 410, the formation equipment may be rotated a second amount of rotation, which may differ from the first amount of rotation such that the relative rotation between the stacked layers is different. In a subsequent process step 412, a third graphene layer may be formed on the second graphene layer, which again may involve the same formation equipment and formation techniques, and the method ends in an end step 414. If desired, additional graphene layers may be formed using similar rotation and formation steps for each new additional graphene layer.

[0037] Stacked graphene structures or devices can be fabricated or formed by using commonly established techniques. For example, a cleavage method can be used where a polymer stamp is used to cleave the graphene, which is then pressed onto a substrate with predefined electrodes. Subsequent layers can be peeled off and rotated by specific angles using a microprobe station. Conventional lithography processes can be used to pattern the electrodes and gate structures required for testing and control of the structure or device.

[0038] The resulting structures or devices can be characterized and studied at cryogenic temperatures to ensure that the desired quantum effects are observed. This can include, for example, dilution refrigeration at temperatures below 1K. Systems that can apply a magnetic field to the device under test conditions can also be used. Basic device testing can include measuring the resistance as the temperature drops below 1K, and the superconducting phase can be observed as the resistance drops sharply to zero as the temperature approaches the critical point. By measuring the transitions and the purported and perturbed symmetries across samples of different twist angle configurations, a correlation between the relative rotation angle and the critical temperature can be established.

[0039] In various configurations, tension can be applied to different layers to stretch one or more layers to the same or different extents. Alternatively, or in addition, the entire two-dimensional structure can be bent. In some embodiments, one or more layers can be doped with another material, such as hydrogen. Additionally, each layer can have different two-dimensional dimensions. For example, a second layer can be 90% of the dimensions of a first layer and / or other layers can have different dimensions. These and other features can further improve the properties of the overall structure.

[0040] FIG. 5 illustrates a flow chart of an example of a detailed method 500 for forming a layered graphene structure with perturbed symmetry. After a start step 502, a first process step 504 may include the preparation of a bilayer graphene and hexagonal boron nitride ("hBN") configuration. This precursor structure may be prepared using the "Scotch tape method," which may include placing adhesive tape on electronic grade graphite or hBN and peeling the layer off the surface. The peeled layer may be rubbed onto a substrate such as silicon dioxide and silicon, and the process is repeated until micrometer flakes of several layers of graphene are obtained. The thickness and quality of these flakes may be verified, such as by using optical and atomic force microscopy.

[0041] In the next process step 506, a substrate may be pressed onto the layered material. The substrate may be glass coated with a polymer such as polydimethylsiloxane ("PDMS"). In a subsequent process step 508, the substrate may be heated to peel the graphene flakes off the substrate. Heating may be, for example, up to 100 degrees Celsius. The transparent glass and PDMS layers allow visualization of the hBN flakes attached to them. Van der Waals interactions between hBN and graphene allow the graphene flakes to be peeled off the silicon dioxide and silicon substrate, resulting in a layer of graphene.

[0042] At decision step 510, an inquiry may be made as to whether additional graphene layers are needed in the stacked graphene structure. If additional graphene layers are needed, the method proceeds to process step 512 where the substrate may be rotated. As described above, the rotation may be to a required angle or amount, which may follow an arithmetic, geometric, or Fibonacci pattern, or other relative pattern, to the previous angle or amount of the previous layer. The process returns to process step 504, and steps 504-510 are repeated. This sequence of steps may be repeated until the required number of graphene layers have been built. Again, the amount of rotation in step 512 may be different for each iteration.

[0043] However, if no additional graphene layers are required at decision step 510, the method may instead proceed to process step 514 where the final stack structure may be released onto the heated gate device. This may include a gate device structure that may be pre-heated to, for example, about 170 degrees Celsius, reinforced with selected palladium and gold.

[0044] In the next process step 516, electrical contacts and a top gate can be formed on the final structure. This can involve the use of electron beam lithography and reactive ion etching. The electrical contacts and top gate can be deposited, for example, by thermal evaporation of chromium and gold, which forms edge contacts to the encapsulated graphene.

[0045] In a subsequent step 518, the final structure may be characterized. This may be performed, for example, by using a transport measurement system in a dilution refrigerator with a superconducting magnet. In various configurations, this process may involve a low frequency lock-in method for data acquisition using a lock-in amplifier. Resistance measurements may involve the use of a voltage excitation, for example, below 100 μV or a current excitation, below 10 nA. The method then ends in an end step 520.

[0046] It will be understood that not all process steps of the above methods 400 and 500 are required, and other process steps may be added in some configurations. Additionally, the order of steps may be changed in some cases, and some steps may be performed simultaneously. For example, step 514 may be performed earlier in the process in some configurations. Although known process steps are provided for the various formation techniques in method 500, it will be understood that other suitable methods for forming and depositing the graphene layers may be used. Additionally, other variations and extrapolations of the disclosed methods will be readily apparent to those of skill in the art.

[0047] As mentioned above, the above described structures and formation techniques using perturbation symmetry between stacked two-dimensional layers are not limited to graphene. Similar types of structures and techniques as disclosed herein may also be applied to other two-dimensional materials, such as transition metal dichalcogenides. Such materials exhibit a variety of phenomena that are suitable for next generation technologies such as light emitters, detectors, valleytronic devices, spintronic devices, and the like. These materials may be composed of large atoms such as tungsten combined with chalcogenides (e.g., sulfur, tellurium, selenium, etc.), and the resulting two-dimensional layers may also be stacked and twisted to form Moiré interference patterns, as in the stacked graphene structures disclosed herein. Again, other materials, rotation patterns, and assumptions are also contemplated.

[0048] 6 illustrates a top view of an example structure having six stacked graphene layers rotated at angles in the Fibonacci sequence. Graphene structure 600 is similar to graphene structure 300 described above, except that the relative rotation angles from one layer to the next follow the Fibonacci sequence.

[0049] Although the foregoing disclosure has been described in detail by way of illustration and example for purposes of clarity and understanding, it will be appreciated that the foregoing disclosure may be embodied in numerous other specific variations and embodiments without departing from the spirit or essential characteristics of the disclosure. It will be understood that certain changes and modifications may be practiced and that the disclosure should not be limited by the foregoing details, but rather should be defined by the appended claims.

Claims

1. a first graphene layer positioned at a first rotation angle relative to an axis of rotation extending perpendicularly through the first graphene layer; a second graphene layer positioned on the first graphene layer at a second rotation angle relative to the rotation axis; a third graphene layer positioned on the second graphene layer at a third rotation angle with respect to the rotation axis; Equipped with the third rotation angle is different from the second rotation angle; Device.

2. The device of claim 1 , wherein the device exhibits superconducting properties at high temperatures.

3. a fourth graphene layer positioned on the third graphene layer at a fourth rotation angle relative to the rotation axis, the fourth rotation angle being different from the third rotation angle; a fifth graphene layer positioned on the fourth graphene layer at a fifth rotation angle relative to the rotation axis that is different from the fourth rotation angle; a sixth graphene layer positioned on the fifth graphene layer at a sixth rotation angle different from the fifth rotation angle with respect to the rotation axis; and The apparatus of claim 1 further comprising:

4. The apparatus of claim 3 , wherein each rotation angle from the second rotation angle to the sixth rotation angle is greater than the previous rotation angle.

5. The apparatus of claim 4 , wherein the ratio relationships from the first rotation angle to the sixth rotation angle form an arithmetic progression, a geometric progression, or a Fibonacci progression.

6. The device of claim 5 , wherein the device exhibits superconducting properties at high temperatures.

7. The device of claim 5 , wherein the device exhibits increased thermal conductivity, increased tensile strength, and / or increased light sensitivity at relatively high temperatures.

8. 4. The apparatus of claim 3, further comprising one or more additional graphene layers, each additional graphene layer positioned on a previous graphene layer at an additional rotation angle relative to the axis of rotation, each additional rotation angle different from the previous rotation angle.

9. The apparatus of claim 8 , wherein each additional angle of rotation is greater than the previous angle of rotation.

10. 10. The apparatus of claim 9, wherein the relationship of the ratios of all rotation angles constitutes an arithmetic progression, a geometric progression, or a Fibonacci progression.

11. 1. A method of forming a graphene structure, comprising: forming a first graphene layer using a forming facility at a first rotation angle about an axis of rotation extending perpendicularly through the layer; rotating the forming fixture a first rotational amount; using the formation facility to form a second graphene layer on the first graphene layer at a second rotation angle relative to the rotation axis, the second rotation angle being equal to the first rotation angle plus the first rotation amount; rotating the forming fixture a second amount of rotation, the second amount of rotation being different from the first amount of rotation; using the formation facility to form a third graphene layer on the second graphene layer at a third rotation angle relative to the rotation axis, the third rotation angle being equal to the second rotation angle plus the second rotation amount; method.

12. 12. The method of claim 11, wherein the graphene structure exhibits superconducting properties at high temperatures.

13. rotating the forming fixture a third amount of rotation different from the second amount of rotation; using the formation facility to form a fourth graphene layer on the third graphene layer at a fourth rotation angle relative to the rotation axis, the fourth rotation angle being equal to the third rotation angle plus the third rotation amount; rotating the forming equipment a fourth rotation amount different from the third rotation amount; using the formation facility to form a fifth graphene layer on the fourth graphene layer at a fifth rotation angle relative to the rotation axis, the fifth rotation angle being equal to the fourth rotation angle plus the fourth rotation amount; Rotating the forming equipment a fifth rotation amount different from the fourth rotation amount; using the formation facility to form a sixth graphene layer on the fifth graphene layer at a sixth rotation angle relative to the rotation axis, the sixth rotation angle being equal to the fifth rotation angle plus the fifth rotation amount; The method of claim 11.

14. The method of claim 13 , wherein each rotation angle from the second rotation angle through the sixth rotation angle is greater than the previous rotation angle.

15. 15. The method of claim 14, wherein the relationship of the ratios from the first rotation angle to the sixth rotation angle constitutes an arithmetic progression, a geometric progression, or a Fibonacci progression.

16. 16. The method of claim 15, wherein the graphene structure exhibits superconducting properties at high temperatures.

17. rotating the forming equipment one or more additional rotational amounts, each additional rotational amount being different from the immediately preceding rotational amount; forming one or more additional graphene layers after each rotation of the forming facility, each additional graphene layer being positioned on the previous graphene layer at an additional rotation angle relative to the rotation axis, each additional rotation angle being equal to the previous rotation angle plus a most recent rotation amount; The method of claim 13.

18. 18. The method of claim 17, wherein the relationship of the ratios of all rotation angles constitutes an arithmetic progression, a geometric progression, or a Fibonacci progression.

19. The method for forming each graphene layer is as follows: Preparing a precursor graphene structure; pressing the coated substrate onto the precursor graphene structure; Heating the substrate to peel the graphene flakes from the coated substrate The method of claim 11 , comprising:

20. 20. The method of claim 19, wherein the precursor graphene structure is bilayer graphene comprising a single layer of hexagonal boron nitride and the coated substrate is polydimethylsiloxane-coated glass.

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