A device structure including a thin single-crystalline bismuth film and a method of forming the device structure

Abstract

A device structure and a method of forming the device structure are provided. The device structure includes for example a thin single-crystalline bismuth film having a first surface, a thickness of the bismuth film being between 25-120nm; and multiple ohmic contacts formed on the first surface. Also provided are temperature sensors, magnetic field sensors and dual sensors as well as electromagnetic isolators, electromagnetic circulators, thermal isolators, thermal circulators, thermoelectric isolators and thermoelectric circulators comprising the device structure.

Description

A DEVICE STRUCTURE INCLUDING A THIN SINGLE-CRYSTALLINE BISMUTH FILM AND A METHOD OF FORMING THE DEVICE STRUCTURECROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of United States Provisional Patent Application No. 63 / 615,905 filed December 29, 2023, and the entire contents of United States Provisional Patent Application No. 63 / 615,905 is hereby incorporated by reference in its entirety.FIELD

[0002] This application relates to a device structure and a method of forming the device structure.BACKGROUND

[0003] The Anomalous Hall Effect (AHE) was discovered by Edwin Hall only a year after his discovery of the ordinary Hall Effect. However, unlike the classical Hall effect that was immediately rationalized, the mechanism for AHE remained the subject of debate for nearly a century. Today, it is believed that AHE can have two types of contributions [1 , 2]: an intrinsic, scattering-free, mechanism originally proposed by Karplus and Luttinger [3] that can also be reconciled with the presence of a Berry curvature (see, e.g., FIG. 1A, which is a schematic diagram showing intrinsic AHE arises from time reversal symmetry breaking intrinsic to electronic structure), as well as scattering-dependent mechanisms known as extrinsic contributions (see e.g., FIG. 1 B, which is a schematic diagram showing extrinsic AHE arises from impurity scattering mechanisms that break time reversal symmetry). Materials that exhibit AHE provide advantages compared with other materials (that do not exhibit AHE) in electromagnetic and thermal sensors / devices designed to exploit the AHE or other material properties related to AHE.

[0004] Bismuth, the heaviest group V element (Z = 83), has been extensively studied for its transport properties. Effects such as the Shubnikov-de Haas (SdH) effect, the de Haas- van Alphen effect and the Nernst-Ettingshausen effect were first discovered in bismuth.In 2017, bismuthene, the 2D limit of bismuth, was grown for the first time on SiC substrate by F. Reis et al. [4], and the unsupported and free-standing bismuthene was successfully grown in 2020 [5], Bismuthene has a honeycomb crystal structure that is similar to graphene, except that it is buckled as shown in FIG. 1 C, which is a schematic diagram showing a buckled honeycomb crystal structure of bismuthene, a 2-dimensional form of bismuth.SUMMARY

[0005] The following summary is provided to introduce the reader to the more detailed discussion to follow. The summary is not intended to limit or define any claimed or as yet unclaimed invention. One or more inventions may reside in any combination or sub- combination of the elements or process steps disclosed in any part of this document including its claims and figures.

[0006] According to some aspects, a device structure is provided comprising: a thin single-crystalline bismuth film having a first surface, a thickness of the bismuth film being between 25-120nm; and multiple ohmic contacts formed on the first surface.

[0007] In some embodiments, purity of the bismuth film is greater than 98% or 99%.

[0008] In some embodiments, the multiple ohmic contacts are arranged in a Hall effect geometrical configuration.

[0009] In some embodiments, the Hall effect geometrical configuration is a van der Pauw configuration, a comb configuration, a Hall bar configuration, or a Corbino configuration.

[0010] In some embodiments, the device structure further comprises a capping layer on the first surface, for example, wherein the capping layer is a polymethyl methacrylate (PMMA) layer of thickness between 150-300nm.

[0011] In some embodiments, the thickness of the bismuth film is between 50-100nm.

[0012] In some embodiments, the bismuth film is formed by mechanically exfoliating thin bismuth flakes.

[0013] In some embodiments, the multiple ohmic contacts are formed by: depositing one or more metals on contact areas of the first surface; and annealing the deposited one or more metals, the annealing being a thermal annealing or a current annealing.

[0014] In some embodiments, the one or more metals includes Ti and Au.

[0015] In some embodiments, the bismuth film has a positive temperature coefficient of resistance in a temperature range between 10K and 300K.

[0016] In some embodiments, the first surface has a (1 ,1 ,1 ) orientation.

[0017] In some embodiments, the device structure further comprises a substrate layer contacting the bismuth film.

[0018] In some embodiments, the substrate layer is a Si / SiO2 substrate.

[0019] Also provided in other aspects, is method of forming a device structure, wherein the device structure as described herein.

[0020] Also provided in other aspects, is a temperature sensor comprising the device structure as described herein, wherein the temperature sensor is configured to provide temperature data based on a resistance between a pair of the multiple ohmic contacts and a calibration parameter.

[0021] Also provided in other aspects, is a magnetic field sensor comprising the device structure described herein, wherein the multiple ohmic contacts are arranged in a Hall effect geometrical configuration and the magnetic field sensor is configured to provide magnetic field data based on a resistance between a pair of the multiple ohmic contacts.

[0022] Also provided in other aspects, is a dual temperature and magnetic field sensor comprising the device structure as described herein, wherein the multiple ohmic contacts are arranged in a Hall effect geometrical configuration and the dual temperature and magnetic field sensor is configured to: provide temperature data based on a calibration parameter and a first resistance between a first pair of the multiple ohmic contacts; and provide magnetic field data based on a second resistance between a second pair of the multiple ohmic contacts.

[0023] Also provided in other aspects, is an electromagnetic isolator comprising: a first port; a second port; a device structure described herein; and a magnetic field device configured to control a directional sense of electromagnetic isolation between the first port and the second port by controlling a magnetic field applied to the device structure.

[0024] Also provided in other aspects, is an electromagnetic circulator comprising: three or more ports; a device structure as described herein; and a magnetic field device configured to control a directional sense of electromagnetic circulation between the three or more ports by controlling a magnetic field applied to the device structure.

[0025] Also provided in other aspects, is a thermal isolator comprising: a first port;a second port; a device structure as described herein; and a magnetic field device configured to control a directional sense of thermal isolation between the first port and the second port by controlling a magnetic field applied to the device structure.

[0026] Also provided in other aspects, is a thermal circulator comprising: three or more ports; a device structure as described herein; and a magnetic field device configured to control a directional sense of thermal circulation between the three or more ports by controlling a magnetic field applied to the device structure.

[0027] Also provided in other aspects, is a thermoelectric isolator comprising: a first port; a second port; a device structure as described herein; an electric control device configured to electrically control a thermal current or temperature gradient associated with the thermoelectric isolator; and a magnetic field device configured to control a thermoelectric isolation between the first port and the second port by controlling a magnetic field applied to the device structure.

[0028] Also provided in other aspects, is a thermoelectric circulator comprising: three or more ports;a device structure as described herein; an electric control device configured to electrically control a thermal current or temperature gradient associated with the thermoelectric circulator; and a magnetic field device configured to control a thermoelectric circulation between the three or more ports by controlling a magnetic field applied to the device structure.

[0029] Also provided in other aspects, is a magnetic field rotation sensor comprising: a device structure as described herein, wherein the multiple ohmic contacts are arranged in a Hall effect geometrical configuration and the device structure further comprises a substrate layer contacting the bismuth film; and a sensor package comprising the substrate layer and multiple pins providing electrical connections to the multiple ohmic contacts, wherein the magnetic field rotation sensor is configured to provide sensor data indicating a magnetic field rotation angle, the sensor data being based on a relationship between the magnetic field rotation angle and a Hall-effect resistance measurement between a pair of the multiple ohmic contacts.

[0030] In some embodiments, the magnetic field rotation sensor is configured to provide the sensor data that is based on a linear relationship between the magnetic field rotation angle and the Hall-effect resistance measurement.

[0031] Also provided in other aspects, is a method of use of a magnetic field rotation sensor to determine a rotation angle of a magnetic field, the method comprising: positioning the magnetic field rotation sensor in the magnetic field; applying an excitation signal to the magnetic field rotation sensor; measuring a Hall-effect resistance signal generated by the magnetic field rotation sensor; anddetermining the rotation angle of the magnetic field based on a relationship between the rotation angle and the Hall-effect resistance, wherein the magnetic field rotation sensor comprises the device structure described herein.

[0032] Also provided in other aspects, is a method of use of a temperature sensor, the method comprising: applying an excitation signal to the temperature sensor; measuring a resistance signal generated by the temperature sensor; and determining the temperature based on a relationship between the temperature and the resistance, the relationship including a temperature calibration parameter while excluding magnetoresistance calibration parameters, wherein the temperature sensor comprises the device structure described herein.

[0033] Also provided in other aspects, is a method of use of a magnetic field sensor for temperature-independent magnetic field measurements in a 10mK to 300K temperature range, the method comprising: positioning the magnetic field sensor in a magnetic field; applying an excitation signal to the magnetic field sensor; measuring a Hall-effect resistance signal generated by the magnetic field sensor; and determining the magnetic field based on a relationship between the magnetic field and the Hall-effect resistance, wherein the magnetic field sensor comprises the device structure described herein, the multiple ohmic contacts being arranged in a Hall effect geometrical configuration.

[0034] Also provided in other aspects, is a method of use of a dual temperature and magnetic field sensor, the method comprising: positioning the dual sensor in a magnetic field; applying an excitation signal to the dual sensor; measuring a longitudinal resistance signal and a Hall-effect resistance signal generated by the dual sensor; determining the temperature based on a relationship between the temperature and the longitudinal resistance, the relationship including a temperature calibration parameter while excluding magnetoresistance calibration parameters, determining the magnetic field based on a relationship between the magnetic field and the Hall-effect resistance, wherein the dual sensor comprises the device structure described herein, the multiple ohmic contacts being arranged in a Hall effect geometrical configuration.BRIEF DESCRIPTION OF THE DRAWINGS

[0035] The drawings included herewith are for illustrating various examples of systems, tools, and processes of the present specification and are not intended to limit the scope of what is taught in any way. In the drawings:

[0036] FIG. 1A is a schematic diagram showing intrinsic AHE arises from time reversal symmetry breaking intrinsic to electronic structure;

[0037] FIG. 1 B is a schematic diagram showing extrinsic AHE arises from impurity scattering mechanisms that break time reversal symmetry;

[0038] FIG. 1 C is a schematic diagram showing a buckled honeycomb crystal structure of bismuthene, a 2-dimensional form of bismuth;

[0039] FIG. 2 is a schematic cross-section of a device structure, in accordance with an embodiment;

[0040] FIG. 3 is a schematic diagram of a temperature sensor, in accordance with an embodiment;

[0041] FIG. 4 is a schematic diagram of a sensor, in accordance with an embodiment;

[0042] FIG. 5 is a schematic diagram of an example electromagnetic isolator that includes the device structure of FIG. 2;

[0043] FIG. 6 is a schematic diagram of an operational principle of the electromagnetic isolator of FIG. 5;

[0044] FIG. 7 is a schematic diagram of another operational principle of the electromagnetic isolator of FIG. 5;

[0045] FIG. 8 is a schematic diagram of an example electromagnetic circulator that includes the device structure of FIG. 2;

[0046] FIG. 9 is a schematic diagram of an example thermal isolator that includes the device structure of FIG. 2;

[0047] FIG. 10 is a schematic diagram of an operational principle of the thermal isolator of FIG. 9;

[0048] FIG. 11 is a schematic diagram of an example thermal circulator that includes the device structure of FIG. 2;

[0049] FIG. 12 is a schematic diagram of an example thermoelectric isolator that includes the device structure of FIG. 2;

[0050] FIG. 13 is a schematic diagram of an operational principle of the thermoelectric isolator of FIG. 12;

[0051] FIG. 14 is a schematic diagram of another operational principle of the thermoelectric isolator of FIG. 12;

[0052] FIG. 15 is a schematic diagram of an example thermoelectric circulator that includes the device structure of FIG. 2;

[0053] FIG. 16a is a cartoon depiction of a process of mechanical exfoliation of bismuth by grating a bulk bismuth crystal against SiO2 micro-trench structures;

[0054] FIG. 16b is a schematic of a Bismuth device fabricated using a Bismuth flake generated by the process of FIG. 16a;

[0055] FIG. 16c is an optical microscope image of a fabricated Bismuth device in van der Pauw configuration where the dashed lines indicate the perimeter of the flake as confirmed by AFM;

[0056] FIG. 16d shows four-point resistance as a function of temperature of the Bismuth device of FIG. 16c;

[0057] FIG. 16e shows Raman spectroscopy of the Bismuth device of FIG. 16c compared to its bulk counterpart in an example study of AHE in thin bismuth (Note that 70 and 97 cm-1are Stokes shifts for pure bismuth whereas bismuth oxide β-Bi2O3has a Raman peak at 313 cm-1which was not observed for the device. Note that the Raman spectra were normalized to the bismuth peak of 70 cm-1);

[0058] FIGS. 17a-d show four-point resistances of different probing configurations versus magnetic field B (positive defined as pointed into the page) at 15 mK for the Bismuth device of FIG. 16c, FIGS. 17a and 17b showing four-point resistances that are in the XY configurations and form an Onsager pair, and FIGS. 17c and 17d showing four-point resistances that are in the XX configuration and are Onsager reciprocals, and the contacts configurations for FIGS. 17a-17d are shown in the insets;

[0059] FIG. 18a shows Rxyextracted from XX and XY Onsager pairs through anti- symmetrization, showing the Hall anomaly for the Bismuth device of FIG. 16c, the dashed lines show the slopes for the saturated region;

[0060] FIGS. 18b and 18c show the extracted Rxxfrom the same Onsager pairs as FIG. 18a, FIG. 18b showing Rxxsymmetrized from XX configuration and FIG. 18c showing Rxxsymmetrized from XY configuration;

[0061] FIG. 19a shows temperature dependence of the resistance for RxxOnsager pairs for the Bismuth device of FIG. 16c;

[0062] FIG. 19b shows temperature dependence of the resistance for RxyOnsager pairs for the Bismuth device of FIG. 16c;

[0063] FIG. 20a shows an original optical image of a Bismuth flake generated by the mechanical exfoliation of FIG. 16a;

[0064] FIG. 20b shows a corresponding AFM scan of the Bismuth flake of FIG. 20a;

[0065] FIG. 20c shows the height profile as selected in the AFM scan of FIG. 20b;

[0066] FIG. 21 a shows a 69 nm thin Bismuth device A patterned in the comb geometry using a Bismuth flake generated by the mechanical exfoliation of FIG. 16a (Note that the black dot is likely to be a particle of insulating SiO2 left on the surface after cleaving the substrate for packaging);

[0067] FIG. 21 b shows a 29 nm thin Bismuth device B patterned in the comb geometry using a Bismuth flake generated by the mechanical exfoliation of FIG. 16a;

[0068] FIG. 22 shows four-point resistance versus the magnetic field for the device A of FIG. 21 a;

[0069] FIG. 23 shows four-point resistance versus the magnetic field for device B of FIG. 21 b;

[0070] FIG. 24 shows a linear fit for the high and low field regions of the AHE data of the devices of FIGS. 21 a and 21 b;

[0071] FIG. 25a shows a cartoon depiction of a process of mechanical exfoliation of thin bismuth with micro-trench structures;

[0072] FIG. 25b shows a schematic of a sample mount that is rotated with respect of the magnetic field;

[0073] FIG. 25c shows an optical microscope image of a fabricated Bismuth device where the Bismuth flake is traced by the dashed line;

[0074] FIG. 25d shows a photograph of a header on which the Bismuth device of FIG. 25c is positioned, the header is further mounted on a sample holder;

[0075] FIG. 25e shows a rotating probe with the sample holder of FIG. 25d that is usable for studying anomalous hall effect in thin bismuth;

[0076] FIGS. 26a-d show four-point resistances of different probing configurations versus magnetic field 8 at T = 1.4 K for the Bismuth device of FIG. 25c, FIGS. 25a and 25b showing four-point resistances that are in the XY configurations and form an Onsager pair, and FIGS. 25c and 25d showing four-point resistances that are in the XX configuration and are Onsager reciprocals, and the contacts configurations for FIGS. 25a- d are shown in the insets;

[0077] FIG. 26e shows the anti-symmetrized data that extracts the true XY component from the XY configurations shown in FIGS. 26a and 26b;

[0078] FIG. 26f shows the symmetrized data that extracts the true XX component from the XX configurations shown in FIGS. 26c and 26d;

[0079] FIGS. 27a and 27b show the true Hall (XY) and true longitudinal resistances respectively, as a function of the magnetic field at various temperatures ranging from 1 .4 K to 300 K for the bismuth device of FIG. 25c, the inset in FIG. 27a is enlarged for data between -2 T ≤ B ≤ 2 T;

[0080] FIG. 27c shows a 3D plot version of the data shown in FIG. 27a;

[0081] FIG. 27d shows a 3D plot version of the data shown in FIG. 27b;

[0082] FIG. 28 shows angular dependence of the XY configuration 1 resistance as a function of the magnetic field for the Bismuth device of FIG. 25c, the inset cartoon illustrating the rotation angle of the magnetic field with respect to the fabricated device;

[0083] FIG. 29 shows angular dependence of the anti-symmetrized Hall resistance as a function of the magnetic field in another set of measurements for the Bismuth device of FIG. 25c, the inset cartoon illustrating the rotation angle of the magnetic field with respect to the fabricated device; and

[0084] FIG. 30 shows the transverse conductivity calculated from the transverse and longitudinal resistances for the Bismuth device of FIG. 25c.DETAILED DESCRIPTION

[0085] Numerous embodiments are described in this application and are presented for illustrative purposes only. The described embodiments are not intended to be limiting in any sense. The invention is widely applicable to numerous embodiments, as is readily apparent from the disclosure herein. Those skilled in the art will recognize that the present invention may be practiced with modification and alteration without departing from the teachings disclosed herein. Although particular features of the present invention may be described with reference to one or more particular embodiments or figures, it should be understood that such features are not limited to usage in the one or more particular embodiments or figures with reference to which they are described.

[0086] The terms “an embodiment,” “embodiment,” “embodiments,” “the embodiment,” “the embodiments,” “one or more embodiments,” “some embodiments,” and “one embodiment” mean “one or more (but not all) embodiments of the present invention(s),” unless expressly specified otherwise.

[0087] The terms “including,” “comprising” and variations thereof mean “including but not limited to,” unless expressly specified otherwise. A listing of items does not imply that any or all of the items are mutually exclusive, unless expressly specified otherwise. The terms “a,” “an” and “the” mean “one or more,” unless expressly specified otherwise.

[0088] In addition, as used herein, the wording “and / or” is intended to represent an inclusive-or. That is, “X and / or Y” is intended to mean X or Y or both, for example. As a further example, “X, Y, and / or Z” is intended to mean X or Y or Z or any combination thereof.

[0089] As used herein and in the claims, two or more parts are said to be “coupled”, “connected”, “attached”, “joined”, “affixed”, or “fastened” where the parts are joined or operate together either directly or indirectly (i.e. , through one or more intermediate parts), so long as a link occurs. As used herein and in the claims, two or more parts are said to be “directly coupled”, “directly connected”, “directly attached”, “directly joined”, “directly affixed”, or “directly fastened” where the parts are connected in physical contact with each other. None of the terms “coupled”, “connected”, “attached”, “joined”, “affixed”, and “fastened” distinguish the manner in which two or more parts are joined together.

[0090] Further, although method steps may be described (in the disclosure and I or in the claims) in a sequential order, such methods may be configured to work in alternate orders. In other words, any sequence or order of steps that may be described does not necessarily indicate a requirement that the steps be performed in that order. The steps of methods described herein may be performed in any order that is practical. Further, some steps may be performed simultaneously.

[0091] As used herein and in the claims, a group of elements are said to ‘collectively’ perform an act where that act is performed by any one of the elements in the group, or performed cooperatively by two or more (or all) elements in the group.

[0092] Some elements herein may be identified by a part number, which is composed of a base number followed by an alphabetical or subscript-numerical suffix (e.g., 112a, or 1121). Multiple elements herein may be identified by part numbers that share a base number in common and that differ by their suffixes (e.g., 1121, 1122, and 1123). All elements with a common base number may be referred to collectively or generically using the base number without a suffix (e.g. , 112).

[0093] It should be noted that terms of degree such as "substantially", "about" and "approximately" when used herein mean a reasonable amount of deviation of the modified term such that the end result is not significantly changed (e.g. + / - 10%). These terms of degree should be construed as including a deviation of the modified term if this deviation would not negate the meaning of the term it modifies.

[0094] The disclosed device structures include a thin, single-crystalline bismuth film. As described in the example studies included herein, the Hall (Rxy) resistance of the bismuth film show a Hall anomaly that is consistent with the AHE. The AHE results included in the example studies suggest that the existence of a broken time-reversal symmetry (TRS) mechanism in the bismuth film.

[0095] The disclosed device structure can be used in various sensing applications. For example, as described herein, the disclosed device structure can be used in thermal sensors and electromagnetic sensors.

[0096] The disclosed device structure can be used in various non-reciprocal electronic, thermal and thermoelectric device applications. For example, as described herein, the disclosed device structure can be used in electromagnetic isolators, electromagnetic circulators, thermal isolators, thermal circulators, thermoelectric isolators, and thermoelectric circulators.

[0097] Referring now to FIG. 2, shown therein is a schematic cross-section of a device structure 100. In the illustrated example, device structure 100 includes a thin single- crystalline bismuth film 110 having a first surface 120 and multiple ohmic contacts 130a, 130b (also collectively referred to herein as contacts 130) formed on first surface 120.

[0098] Bismuth film 110 can have any suitable thickness 140 that exhibits the AHE. For example, thickness 140 may be in a 25-120nm thickness range. In some embodiments, thickness 140 may be in a narrower 50- 100nm thickness range. Thickness 140 can vary across bismuth film 110, for example by about 10%.

[0099] Bismuth film 110 can be a high-purity bismuth film. For example, bismuth film 110 may have a purity greater than 98%. In some embodiments, bismuth film 110 may have a higher purity of greater than 99%.

[0100] Bismuth film 110 can operate in a semi-metallic regime and have a positive temperature coefficient of resistance in a temperature range between 10K and 300K. The resistance of bismuth film 110 may increase linearly as the temperature of the film is increased between 10K and 300K. For example, the resistance of a bismuth film of area 1 micron square, and thickness of approximately 100 nm , in the example study described herein, increased linearly from 3.2Ω at 1 K to 3.8Ω at 300K. The resistance of bismuth film 110 can remain stable after thermal cycling between 10K and 300K.

[0101] Any suitable substrate may be used to mechanically support bismuth film 110. In the illustrated example, device structure 100 includes a Si substrate 10 and a SiO2 layer 20 that contacts bismuth film 110. In other examples, the substrate may be a different material (e.g., other semiconductors, metals, insulators). In some embodiments, layer 20 may be absent. For example, substrate 10 can be an insulating substrate that directly contacts bismuth film 110 and layer 20 may be absent.

[0102] In some embodiments, device structure 100 may be connected to other electrical components to provide electrical interconnections and / or additional functionality. For example, substrate 10 can be a semiconductor substrate (e.g., Si) and device structure 100 can be connected to metal interconnections and semiconductor devices (e.g., transistors) fabricated on substrate 10.

[0103] In some embodiments, device structure 100 can include a capping layer 160 on first surface 120. Any suitable material and / or thickness may be used for capping layer 160. For example, capping layer 160 may be a polymethyl methacrylate (PMMA) layer having thickness in a 150-300nm thickness range. Capping layer 160 may protect bismuth film 110 from surface oxidation and / or degradation. In other embodiments, device structure 100 may not include capping layer 160.

[0104] Any suitable method may be used to form bismuth film 110. In some embodiments, bismuth film 110 may be formed by mechanically exfoliating thin bismuth flakes. In other embodiments, bismuth film 110 may be deposited as a thin film on substrate 10 using any suitable thin film deposition methods. The thin film deposition methods can include physical vapor deposition methods or chemical vapor deposition methods. For example, bismuth film 110 may be deposited using molecular beam epitaxy, metal organic chemical vapor deposition, etc.

[0105] Single-crystal line bismuth film 110 can have any suitable orientation that exhibits the AHE. In some embodiments, first surface 120 of single-crystalline bismuth film 110 may have a (1 , 1 , 1 ) orientation.

[0106] In the illustrated example, multiple ohmic contacts 130 includes two ohmic contacts 130a and 130b. In other examples, multiple ohmic contacts 130 may include a greater number of ohmic contacts (e.g., three or more ohmic contacts).

[0107] In some embodiments, multiple ohmic contacts 130 may be arranged in a Hall effect geometrical configuration. The Hall effect geometrical configuration can be any configuration that enables Hall effect-related electrical measurements (e.g., voltage, current and / or resistance measurements) to be performed. Each ohmic contact 130 can be connected to an electrical lead to enable electrical measurements to be performed. In some embodiments, the Hall effect geometrical configuration may be a van der Pauw configuration, a comb configuration, a Hall bar configuration, or a Corbino configuration. In some instances, Hall effect-related electrical measurements may be performed to characterize properties of bismuth film 110. In some instances, Hall effect-related electrical measurements may be performed to utilize bismuth film 110 in sensor applications, as described in further detail herein.

[0108] Any suitable method may be used to form multiple ohmic contacts 130. For example, any suitable thin film deposition method (e.g., physical vapor deposition, chemical vapor deposition, electro-plating etc.) may be used to deposit one or more contact metals on contact areas of first surface 120. In the illustrated example, ohmiccontact 130a is formed on contact area 150a and ohmic contact 130b is formed on contact area 150b.

[0109] The deposited contact metals may have any suitable number of layers. For example, the deposited contact metal film may be a single layer or multiple layers. The deposited contact metal film may have any suitable thickness. In some embodiments, the total thickness of the deposited contact metal film may be in a 50-200nm thickness range. In other embodiments, the total thickness of the deposited contact metal film may be smaller than 50nm or greater than 200nm.

[0110] Any suitable contact metal can be used to form ohmic contacts 130. In some embodiments, the contact metal used may be selected based on compatibility of the contact metal deposition tool with other tools used for forming the device structure. For example, an electron-beam evaporator system may be used to deposit Au and / or Ti contact metals. In other examples, different contact metals and / or contact metal deposition tools may be used.

[0111] As a first example, ohmic contact 130 may be formed using a single 100nm thick layer of Au. As another example, ohmic contact 130 may be formed using a combination of two layers - a 20nm thick layer of Ti and a 10Onm thick layer of Au. In other examples, any other suitable combination of contact metals, thicknesses and number of layers may be used.

[0112] In some embodiments the same contact metals, thicknesses and number of layers may be used for all ohmic contacts 130. In other embodiments, at least one of the contact metal, thicknesses, or number of layers may be different between two ohmic contacts of multiple ohmic contacts 130. For example, ohmic contact 130a may be formed using a single layer of Au and ohmic contact 130b may be formed using a combination of Ti and Au layers.

[0113] The as-deposited contact metal films on bismuth film 110 may form a non-ohmic or Schottky contact. The as-deposited contact metal films can be annealed to form an ohmic contact to bismuth film 110. A thermal annealing process may damage cappinglayer 160 (if present) and / or the crystalline structure of bismuth film 110. In some embodiments, a current annealing process may be used to anneal the as-deposited contact metal film. Any suitable process parameters may be used for the current annealing process. For example, the current annealing process may include applying a current bias of 1 μA or greater between a pair of as-deposited non-ohmic contacts. The current annealing process can convert the as-deposited non-ohmic contacts into ohmic contacts. In some embodiments, a threshold contact resistance may be used to control the duration of the current bias. For example, as-deposited contacts may have a high contact resistance > 1 MΩ and the current bias may be applied until the contact resistance < 1 kΩ. In other examples, other values of the threshold contact resistance may be used.

[0114] In some embodiments, a contact resistance of each ohmic contact 130 may be in a range from 1 -1000ohms. In other embodiments, the contact resistance may be smaller than 1 ohm or greater than 1000 ohms.Sensors

[0115] The Hall-effect related properties of the disclosed thin Bismuth films may be used to provide one or more sensors. Any suitable method may be utilized to fabricate the sensors. For example, any suitable electronic packaging technique may be used for packaging a disclosed Bismuth device structure (e.g., device structure 100 shown in FIG. 2 that includes bismuth film 110, multiple ohmic contacts 130 and substrate 10) to form a packaged sensor. The packaged sensor may include multiple pins or terminals that provide electrical connections to the multiple ohmic contacts 130 for voltage / current signals and / or resistance measurements.

[0116] Reference is now made to FIG. 3 showing a schematic diagram of an example temperature sensor 300 operating in magnetic field environment 30. In the illustrated example, temperature sensor 300 includes bismuth film 110 and ohmic contacts 130c, 130d formed on first surface 120 of bismuth film 110.

[0117] Temperature sensor 300 can be configured to provide temperature data based on a resistance between ohmic contacts 130c and 130d, and a calibration parameter. Forexample, a relationship between the resistance and temperature is illustrated in FIGS. 19a and 19b. The calibration parameter can be determined in an initial temperature calibration process. For example, temperature data generated by temperature sensor 300 for measurements of known / controlled temperatures can be used to determine the calibration parameter.

[0118] Temperature sensor 300 can provide temperature sensing operation in a wide temperature range from 10K to 300K. During operation, a voltage signal can be applied between ohmic contacts 130c and 130d. A resistance measurement between ohmic contacts 130c and 130d can provide an indication of the sensed temperature. The calibration parameter can be used to convert the resistance measurements to temperature data.

[0119] Temperature sensor 300 can provide an advantage compared with other temperature sensors when operating in magnetic field environments. Other temperature sensors can have an unknown magnetoresistance thereby requiring a magnetoresistance calibration (in addition to a temperature calibration) to perform temperature sensing operations in magnetic field environments.

[0120] In contrast, a magnetoresistance calibration may not be required for temperature sensor 300. The resistance measurements for bismuth film 110 are not affected by magnetic fields between -30 to 30 Tesla. This can enable conversion of resistance measurement to temperature data based on the temperature calibration and without requiring a magnetoresistance calibration.

[0121] Reference is now made to FIG. 4 showing a schematic diagram of an example sensor 400 operating in magnetic field environment 30. In the illustrated example, sensor 400 includes bismuth film 110 and ohmic contacts 130c-130f formed on first surface 120 of bismuth film 110.

[0122] Ohmic contacts 130c-130f can be arranged in any suitable Hall effect geometrical configuration. In the illustrated examples, ohmic contacts 130c-130f are arranged in a van der Pauw configuration. In other examples, a different Hall effect geometricalconfiguration may be used. The exhibition of the AHE by bismuth film 110 is not dependent on the specific Hall effect geometrical configuration that is used for sensor 400.

[0123] In a first embodiment, sensor 400 can be configured to operate as a magnetic field sensor. Sensor 400 can be configured to provide magnetic field data based on a Hall resistance (Rxy) between ohmic contacts 130e and 130f. For example, a relationship between the Hall resistance and magnetic field is illustrated in FIGS. 18a and 27a.

[0124] Sensor 400 can provide magnetic field sensing operation in a wide temperature range from 10mK to 300K. During operation, a voltage signal can be applied between ohmic contacts 130c and 130d. Hall resistance (Rxy) measurement between ohmic contacts 130e and 130f can provide magnetic field data for magnetic field 30. The Hall resistance (Rxy) measurement exhibits the AHE in a wide temperature range from 10mK to 300K.

[0125] Sensor 400 can provide high sensitivity of the Hall resistance (Rxy) measurements to magnetic field 30 in a range of ± 1 Tesla. This can enable magnetic field sensing operation with a high signal to noise ratio in the ± 1 Tesla range. In some embodiments, sensor 400 may provide sensitivity of the Hall resistance (Rxy) measurements to magnetic field 30 in a range of ± 30 Tesla.

[0126] Sensor 400 can provide an advantage compared with other magnetic field sensors made using semiconductor materials (e.g., Si, GaAs, GaN, InN and graphene) in which the magnitude of the “ordinary” Hall effect is dependent on temperature. In contrast, the magnitude of the AHE in the resistance measurements between ohmic contacts 130e and 130f is substantially independent of temperature. This can enable accurate magnetic field sensing for a wide temperature range from 10mK to 300K.

[0127] In a second embodiment, sensor 400 can be configured to operate as a dual temperature and magnetic field sensor. As described above, sensor 400 can be configured to provide magnetic field data based on a Hall resistance (Rxy) measurement between ohmic contacts 130e and 130f . Additionally, as described above with referenceto temperature sensor 300, sensor 400 can be configured to provide temperature data based on a longitudinal resistance (Rxx) between ohmic contacts 130c and 130d, and a calibration parameter.Electromagnetic isolators and circulators

[0128] The AHE observed in Hall resistance (Rxy) of bismuth film 110 implies that electromagnetic waves over a broad frequency range will exhibit non-reciprocal scattering properties, including Faraday rotation of transmitted waves and Kerr rotation of reflected waves. The Faraday rotation and / or Kerr rotation can be used to create electromagnetic devices with non-reciprocal response, including isolators and circulators with directional steering. The application of a magnetic field can be used to tune the sign of Rxy, and thus the orientation of the Faraday I Kerr rotation, and thus the directional sense of isolation and circulation. In some embodiments, a magnetic field of approximately 2T is used to tune the sign of Rxyand thereby control the directional sense of isolation and circulation.

[0129] Reference is now made to FIG. 5 showing a schematic diagram of an example electromagnetic isolator 500 that includes device structure 100. Any suitable configuration may be used for isolator 500. In the illustrated example, isolator 500 includes a first port 510, a second port 520, device structure 100, and a magnetic field device 530.

[0130] Magnetic field device 530 can control the directional sense of isolation between first port 510 and second port 520 by controlling the direction of a generated magnetic field. For a first direction of the magnetic field, isolator 500 can permit propagation of electromagnetic (EM) waves from first port 510 to second port 520, and block propagation of EM waves from second port 520 to first port 510. For a second direction of the magnetic field, isolator 500 can permit propagation of EM waves from second port 520 to first port 510, and block propagation of EM waves from first port 510 to second port 520.

[0131] Reference is now additionally made to FIG. 6 showing a schematic diagram 600 of an operational principle of isolator 500 for an incident EM wave 610. Bismuth film 110 of device structure 100 creates Kerr rotation of reflected EM wave 620. The orientation of the Kerr rotation of reflected EM wave 620 (and thereby directional sense of isolationbetween first port 510 and second port 520) can be controlled by the direction of magnetic field 30 generated by magnetic field device 530.

[0132] Reference is now additionally made to FIG. 7 showing a schematic diagram 700 of another operational principle of isolator 500 for an incident EM wave 710. Bismuth film 110 of device structure 100 creates Faraday rotation of transmitted EM wave 720. The orientation of the Faraday rotation of transmitted EM wave 720 (and thereby directional sense of isolation between first port 510 and second port 520) can be controlled by the direction of magnetic field 30 generated by magnetic field device 530.

[0133] Reference is now made to FIG. 8 showing a schematic diagram of an example electromagnetic circulator 800 that includes device structure 100. Any suitable configuration may be used for circulator 800. In the illustrated example, circulator 800 includes a first port 810, a second port 820, a third port 830, device structure 100, and a magnetic field device 840. In other examples, circulator 800 may have a greater number of ports.

[0134] Magnetic field device 840 can control the directional sense of circulation between adjacent ports by controlling the direction of a generated magnetic field. For a first direction of the magnetic field, circulator 800 can permit a clockwise propagation of EM waves (e.g., from first port 810 to second port 820, from second port 820 to third port 830, from third port 830 to first port 810), and block a counter-clockwise propagation of EM waves (e.g., from first port 810 to third port 830, from third port 830 to second port 820, from second port 820 to first port 810). For a second direction of the magnetic field, circulator 800 can permit clockwise propagation of EM waves and block counter- clockwise propagation of EM waves.

[0135] The operational principle of circulator 800 can be based on the Kerr rotation of reflected EM waves by bismuth film 110 (e.g., as described herein above with reference to FIG. 6) or on the Faraday rotation of transmitted EM waves by bismuth film 110 (e.g., as described herein above with reference to FIG. 7).Thermal isolators and circulators

[0136] The Wiedemann-Franz law implies that thermal transport coefficients (i.e. , thermal resistivity) associated with electronic degrees of freedom are proportional through a constant to electronic transport coefficients (i.e., electronic resistivity). Consequently, AHE in Hall resistance (Rxy) will manifest itself in an AHE in pxy (transverse thermal resistance). Heat transport will thus exhibit analogous behaviour to electronic transport, exhibiting non-reciprocal characteristics that can be switched with a magnetic field. Thermal isolators and circulators can be implemented by applying the non-reciprocal thermal transport coefficient pxy.

[0137] Reference is now made to FIG. 9 showing a schematic diagram of an example thermal isolator 900 that includes device structure 100. Any suitable configuration may be used for isolator 900. In the illustrated example, isolator 900 includes a first port 910, a second port 920, device structure 100, and a magnetic field device 930.

[0138] Magnetic field device 930 can control the directional sense of isolation between first port 910 and second port 920 by controlling the direction of a generated magnetic field. For a first direction of the magnetic field, isolator 900 can permit propagation of heat current from first port 910 to second port 920, and block propagation of heat current from second port 920 to first port 910. For a second direction of the magnetic field, isolator 900 can permit propagation of heat current from second port 920 to first port 910, and block propagation of heat current from first port 910 to second port 920.

[0139] Reference is now additionally made to FIG. 10 showing a schematic diagram 1010 of an operational principle of isolator 900 for a heat current 1020. The AHE in ρxycreates a temperature gradient 1030. The directionality of temperature gradient 1030 (and thereby directional sense of isolation between first port 910 and second port 920) can be controlled by the direction of magnetic field 30 generated by magnetic field device 930.

[0140] Reference is now made to FIG. 11 showing a schematic diagram of an example electromagnetic circulator 1110 that includes device structure 100. Any suitable configuration may be used for circulator 1110. In the illustrated example, circulator 1110 includes a first port 1120, a second port 1130, a third port 1140, device structure 100, anda magnetic field device 1150. In other examples, circulator 1110 may have a greater number of ports.

[0141] Magnetic field device 1150 can control the directional sense of circulation between adjacent ports by controlling the direction of a generated magnetic field. For a first direction of the magnetic field, circulator 1110 can permit a clockwise propagation of heat currents (e.g., from first port 1120 to second port 1130, from second port 1130 to third port 1140, from third port 1140 to first port 1120), and block a counter-clockwise propagation of heat currents (e.g., from first port 1120 to third port 1140, from third port 1140 to second port 1130, from second port 1130 to first port 1120). For a second direction of the magnetic field, circulator 1110 can permit clockwise propagation of heat currents and block counter-clockwise propagation of heat currents.

[0142] The operational principle of circulator 1110 can be based on the AHE in ρxyof bismuth film 110 (e.g., as described herein above with reference to FIG. 10).Thermoelectric isolators and circulators

[0143] Bismuth is a thermoelectric material, with non-zero Seebeck and Peltier coefficients for correlated transport of heat and charge. As both electric and thermal coefficients exhibit AHE, the thermoelectric coefficients will exhibit AHE. For example, the transverse Seebeck coefficient Sxyand transverse Peltier coefficient Πxywill be non-zero. Consequently, magnetic field can be used to tune and to digitally switch via AHE the transverse thermoelectric coefficients. Magnetic field can digitally switch electrical current control of transverse thermal current, and / or electrical potential control of transverse temperature gradient.

[0144] Reference is now made to FIG. 12 showing a schematic diagram of an example thermoelectric isolator 1210 that includes device structure 100. Any suitable configuration may be used for isolator 1210. In the illustrated example, isolator 1210 includes a first port 1220, a second port 1230, device structure 100, an electric control device 1240, and a magnetic field device 1250.

[0145] In some embodiments, electric control device 1240 can control a thermal current between first port 1220 and second port 1230 by controlling an electric current through the bismuth film of device structure 100. In other embodiments, electric control device 1240 can control a temperature gradient between first port 1220 and second port 1230 by controlling an electric potential across the bismuth film of device structure 100.

[0146] Magnetic field device 1250 can control the directional sense of isolation between first port 1220 and second port 1230 by controlling the direction of a generated magnetic field. As an example, for a first direction of the magnetic field, isolator 1210 can permit propagation of heat current from first port 1220 to second port 1230, and block propagation of heat current from second port 1230 to first port 1220. For a second direction of the magnetic field, isolator 1210 can permit propagation of heat current from second port 1230 to first port 1220, and block propagation of heat current from first port 1220 to second port 1230.

[0147] Reference is now additionally made to FIG. 13 showing a schematic diagram 1310 of an operational principle of isolator 1210 for a heat current 1330 that is controlled by an electrical current 1320. Magnetic field 30 generated by magnetic field device 1250 can be used to tune and to switch the non-zero transverse Peltier coefficient flxy of bismuth film 110, thereby enabling control of isolation between first port 1220 and second port 1230.

[0148] Reference is now additionally made to FIG. 14 showing a schematic diagram 1410 of an operational principle of isolator 1210 for a temperature gradient 1430 that is controlled by an electrical potential 1420. Magnetic field 30 generated by magnetic field device 1250 can be used to tune and to switch the non-zero transverse Seebeck coefficient Sxy of bismuth film 110, thereby enabling control of isolation between first port 1220 and second port 1230.

[0149] Reference is now made to FIG. 15 showing a schematic diagram of an example thermoelectric circulator 1510 that includes device structure 100. Any suitable configuration may be used for circulator 1510. In the illustrated example, circulator 1510 includes a first port 1520, a second port 1530, a third port 1540, device structure 100,electric control device 1550 and a magnetic field device 1560. In other examples, circulator 1510 may have a greater number of ports.

[0150] In some embodiments, electric control device 1550 can control a thermal current between adjacent ports by controlling an electric current through the bismuth film of device structure 100. In other embodiments, electric control device 1550 can control a temperature gradient between adjacent ports by controlling an electric potential across the bismuth film of device structure 100.

[0151] Magnetic field device 1560 can control the directional sense of circulation between adjacent ports by controlling the direction of a generated magnetic field. As an example, for a first direction of the magnetic field, circulator 1510 can permit a clockwise propagation of heat currents (e.g., from first port 1520 to second port 1530, from second port 1530 to third port 1540, from third port 1540 to first port 1520), and block a counter- clockwise propagation of heat currents (e.g., from first port 1520 to third port 1540, from third port 1540 to second port 1530, from second port 1530 to first port 1520). For a second direction of the magnetic field, circulator 1510 can permit clockwise propagation of heat currents and block counter-clockwise propagation of heat currents.

[0152] The operational principle of circulator 1510 can be based on the AHE exhibited by the transverse thermoelectric coefficients of bismuth film 110 (e.g., as described herein above with reference to FIGS. 13 and 14).An Example Study of AHE in Thin Bismuth

[0153] The example study was conducted in sub-100nm bismuth films whereby an unambiguous signature of the AHE was observed in electronic transport measurements. Bismuth is known to be a diamagnetic material and as such the manifestation of AHE indicates the breaking of TRS.Method

[0154] Traditional mechanical exfoliation methods have shown to be challenging to obtain thin bismuth flakes as the atomic layers have relatively strong inter-layer bonds comparedto other group V elements. To overcome such challenge, a novel technique was developed using micro-trench structure to mechanically exfoliate thin bismuth flakes [6], The micro-trench structure was prepared by etching a SiO2 thermal oxide layer above a degenerately doped silicon substrate, effectively turning it to a mechanical file whereas the latter can be used as a backgate. As shown in FIG. 16a, bulk bismuth crystal, with its orientation carefully chosen to be the (1 ,1 ,1 ) surface, was attached to the tip of a metal pen. By grating the bismuth crystal against the micro-trench file, thin flakes of bismuth were obtained and were found to be as thin as ~ 10 nanometers [6],

[0155] Using this technique, 60 to 100 nm thin bismuth flakes were obtained and characterized by atomic force microscopy (AFM) (see e.g., Supplemental Information section below). Ti / Au contacts were deposited via electron beam lithography and electron beam vapor deposition. Note that the environment was carefully controlled with all major fabrication steps performed in a vacuum, or in a nitrogen-filled glovebox. Lastly, a polymethyl methacrylate (PMMA) capping layer was spincoated for protecting the bismuth flake against oxidization. As shown in the Results and Discussions section below, the small resistance of the fabricated device as well as Raman spectroscopy suggest that negligible bismuth oxide had formed on the flake’s surface during the fabrication process. An optical image of the device fabricated in the van der Pauw (vdP) geometry is shown in FIG. 16c, with the schematic shown in FIG. 16b. The average thickness of the Bismuth flake used for the vdP device fabrication was 68nm with a ±47nm thickness range. Two other devices fabricated in a comb geometry are shown in the Supplemental Information section. Note that due to the small size of the flake (~ 1 x 1 μm), will be discussed below, the ohmic contacts are subject to misalignments and hence mixing of electronic transport components is to be expected.

[0156] The chemical nature of bismuth flakes was confirmed via Raman spectroscopy with a Broker Senterra confocal Raman equipped with a 785 nm laser. The flake was compared to its bulk counterpart of the same crystal as a benchmark. The small Raman signal of the flake is due to its small size, however it was still possible to observe the 70 cm-1and 97 cm-1Raman shift peaks associated with pure bismuth [7,8], see FIG. 16e (lighter grey). Additionally, the most common type of of bismuth oxide typically formed atlower temperatures (≤ 300°C), β-Bi2O3, was observed in the bulk crystal (darker grey) at 125 cm-1and 313 cm-1[9-11 ] but not in the mechanically exfoliated flake.

[0157] Finally, the resistance of the thin bismuth device was measured with a voltage pre- amplifier using a quasi-DC technique at a frequency of 17.777 Hz and with an excitation current of 100 nA. The low-temperature measurements were performed in a Bluefors BF- LD250 dilution refrigerator fitted with a 9 T magnet and at the National High Magnetic Field Laboratory in a variable temperature inset with temperature ranging from 1 to 300K, and magnetic fields + / - 31 T. The temperature dependence of the resistance of the van der Pauw device is plotted in FIG. 16d from 3 to 250 K, and is consistent with what were previously reported in the literature for bulk semi-metallic bismuth [12,13] as well as thin films down to 500 nm

[0014] ,Results and Discussions

[0158] As mentioned earlier, the small size and the limitation imposed by electron beam lithography resulted in the van der Pauw contacts to be deposited relatively close to each other, as can be seen from the optical image shown in FIG. 16c. Since the contacts are not exactly at the comers as depicted in FIG. 16b, the longitudinal (XX) and Hall (XY) resistances are expected to be mixed to some degree in every probe configuration. To overcome this mixing, the Onsager symmetrization was used to reconstruct the true longitudinal Rxxand Hall Rxyresistances. According to the Onsager’s reciprocity theorem

[0015] , inverting the current and voltage contacts in a linear system allows measurement of the transpose of the resistance tensor given bywhere B is the applied magnetic field. Consequently, by measuring the resistance in one configuration as well as its Onsager reciprocal, the true longitudinal Rxxand Hall Rxyresistances can be obtained by respectively symmetrizing and anti-symmetrizing the two configurations withandwhere R and R’ form an Onsager pair. For example, a four-point measurement configuration labeled ABCD (corresponding to probes I+ / V+ / V- / I-) would have its Onsager reciprocal with contact configuration BADC.

[0159] The resistances versus magnetic fields (with positive field defined as pointed into the page, see FIG. 17 insets) were measured at 15 mK and are shown in FIG. 17. It may be stressed that mixing of the Rxxand Rxywas inevitably observed, however FIG. 17a and FIG. 17b are XY configurations that maximize the Hall signal and that are also Onsager reciprocals to one another. Similarly, FIG. 17c and FIG. 17d are the XX configurations that optimize the XX signal while also capturing a mixed Hall signal. The corresponding ideal probe configurations in the van der Pauw geometry are shown in the insets of FIG. 17).

[0160] FIG. 18 summarizes the main result: the extracted true Hall and longitudinal resistances as a function of the magnetic field B. FIG. 18a shows the anti-symmetrized signal from the XY configurations, and as expected it is greater than the anti-symmetrized signal from the XX configurations. Similarly, the true longitudinal signal symmetrized in a similar fashion is shown in FIG. 18b and FIG. 18c for the XX and XY configuration pairs, respectively. In particular, both true Rxxare constant as a function of the magnetic field up to ±9 T. The resistance values are 3.2 Ω (extracted from XX) and 1 .7 Ω (extracted from XY). The difference in the extracted resistances is attributed to the contacts that are not equidistant from one another, as the ratio of the Rxxresistances is approximately equal to the ratio of the distances between the voltage probes.

[0161] From the Hall response shown in FIG. 18a, a careful inspection of the data shows the presence of a very small slope in the saturated regime of the Hall signal. In particular, for the XY configuration’s saturated high-field region (|B| ≥ 2 T), a fit to the linear slopeyields the value -0.0010(4) Ω / T, and similarly a linear fit to the XY configuration’s low field region (|B| ≤ 0.5 T) yields a slope of -0.174(3) Ω / T, please see the Supplemental Information section. While the negative values found for both linear Hall signals support the electronic transport being hole-dominated, it is unfortunately not possible to reliably extract carrier densities and mobilities. Bismuth is known to host both electron and hole pockets in the bulk [16, 17], as well as in thin films [18-20], Furthermore, in contrast to bulk bismuth where quantum oscillations can be observed well under 1 T

[0021] , in this case SdH oscillations in the longitudinal resistance were not observed. As discussed below, this unexpected behavior was nevertheless observed previously in a 100 nm thick film grown by molecular beam epitaxy

[0014] ,

[0162] Using the device geometry and the longitudinal resistance measured in the absence of a magnetic field, the conductivity is calculated to be σxx~ 105(Ωcm)-1which places the bismuth device in a good-metal regime dominated by scattering-independent mechanisms

[0001] , Bismuth has a high spin orbit coupling (SOC). However, time-reversal symmetry breaking is a necessary condition for both intrinsic and extrinsic AHE, and as such, SOC alone is not sufficient to explain the anomalous Hall signal observed here. Usually, intrinsic time-reversal symmetry breaking is achieved by ferro- or antiferromagnetism, but bulk bismuth is known to be the most diamagnetic element with a magnetic susceptibility value of -1.66 x 10-4at room temperature

[0022] , Consequently, unless a magnetic transition would occur as the thickness of bismuth is reduced, TRS breaking must originate from elsewhere.

[0163] More recently, B. C. Camargo et al. claimed to have observed the AHE in bulk bismuth but were unable to find the source of the TRS breaking either

[0023] , A portion of their work was dedicated to eliminate magnetic contamination and superconductivity, and it was concluded that the AHE may arise from the topologically non-trivial surface or hinge states rather than in the bulk. Such arguments align with the observation here of the AHE in thin devices with flake thicknesses of 68 nm (main text), 29 nm and 69 nm (see Supplemental Information section). Strikingly, these two other devices were fabricated in a comb geometry, yet the four-probe resistances demonstrated a magnetic field response similar to that of AHE measured in the vdP device presented in the main section of thisexample study. While the longitudinal resistances of the vdP and comb devices were different (2 to 32 Ω), the anomalous Hall responses were unexpectedly similar (0.1 to 0.4 Ω), hinting that the observed AHE does not arise from the bulk, but instead from the surface. The resistance’s temperature dependence which is consistent with that of the bulk bismuth down to 500 nm also supports this observation as it suggests that the bulk is measured simultaneously with the surface or hinge states.

[0164] Finally, in addition to a non-trivial topology in bismuth [4, 24], there are also other prospect origins for spontaneous time-reversal symmetry breaking. These include band- flattening as in the case of twisted bi-layer graphene, orbital magnetism of Dirac electrons that could arise in bismuth, and strain-induced band distortion [25-27] that could all lead to a broken TRS.Conclusion

[0165] Using the inventors’ micro-trench technique, sub-100nm bismuth flakes were mechanically exfoliated which were then fabricated into devices in the van der Pauw and comb configurations for electronic transport measurements. With the help of Onsager symmetrization, the reconstructed Hall resistance shows a clear anomaly consistent with the intrinsic Anomalous Hall Effect, even if bismuth is known to be a diamagnetic material and thus a priori does not break time-reversal symmetry. Altogether, the results provide support for the existence of a broken TRS mechanism.Supplemental InformationTemperature Dependence

[0166] Given that there are four different configurations used in the Onsager symmetrization, resistance versus temperature traces for all of them are shown for completeness. The Rxxand Rxyare shown in FIGS. 19 (a) and (b), respectively. Note that the trace for XX2 was shown herein above.Optical Images and AFM Scans

[0167] The optical image before contact patterning is shown in FIG. 20(a). The corresponding AFM scan is shown in FIG. 20(b). A profile is selected across the flake where the height versus position graph is shown in FIG. 20(c).Additional Comb Devices

[0168] The Anomalous Hall Effect was observed in two other devices which are labeled device A and device B. These two devices were fabricated in a comb geometry rather than in a van der Pauw geometry as for the device presented herein above. The AHE was observed in these two comb devices because of the mixing of transport components due to imperfections in the contacts. The optical images of the devices are shown in FIG. 21 . The average Bi flake thickness for device A was 69nm with a ± 20nm thickness range. The average Bi flake thickness for device B was 29nm with a ± 19nm thickness range. The resistances are 19Ω and 32Ω for devices A and B, respectively.

[0169] For device A, the raw measured four-point resistance versus the magnetic field is presented, shown in FIG. 22.

[0170] The conventional configuration is the usual I+ / V+ / V- / I- configuration where the current contacts are on the outside of the comb geometry and the voltage probes are in the center. In contrast, the unconventional configuration is obtained by inverting l+ with V+ and I- with V-. This effectively becomes the Onsager reciprocal of the conventional setup. Although the AHE is clearly observed, the two curves do not cross at the origin which is attributed tentatively to the non-ideal configuration for Hall response used here.

[0171] In FIG. 23, the resistance dependence on the magnetic field for device B is shown. Similar to device A, the AHE is clearly observed despite a vertical offset most likely due to the non-ideal configuration for measuring a Hall response.

[0172] Linear fits were performed on the anti-symmetrized data shown herein above. The high-field region is chosen to be |B| > 2 T and the low field region is chosen as |B| ≤ 0.5 T. The corresponding slopes are shown in FIG. 24.An Example Study of Anomalous Hall Effect in Thin Bismuth at 300K

[0173] In this example study, the anomalous Hall effect was unambiguously observed in sub-100 nm bismuth flakes. Most notably, it was found that the anomalous Hall resistance RAHE is nearly temperature independent for a wide range of temperatures from 1 .4 K to 300 K. Such temperature independence strongly points to an intrinsic nature. It is worthwhile to note that, to the inventors’ knowledge, the intrinsic AHE has never been identified for such a large temperature range since the extrinsic mechanisms typically dominate at high temperatures.Methods

[0174] Thin bismuth was exfoliated from a bulk crystal through a mechanical exfoliation method using micro-trench structures. As shown in FIG. 25a, a bulk single crystalline bismuth was grated against a SiO2micro-trenched surface to facilitate breakage of interlayer bonds in bismuth which produced thin flakes with thicknesses down to the 10 nm range. Note that during the entire fabrication process, including this exfoliation step, oxidation was carefully controlled with a nitrogen-only glovebox (02 and water concentration both less than 1 ppm) and a PMMA protection layer.

[0175] Three flakes of thicknesses 29, 68 and 69 nm were fabricated into electronic transport devices and all demonstrated the anomalous Hall effect discussed in this example study. The 68 nm device was fabricated in the van der Pauw geometry and its optical microscope image is shown in FIG. 25c. FIG. 25d shows the device's substrate mounted on a 16-pin header. Gold wires were indium soldered and connected the device contacts to the pins of the header. In particular, this 68 nm device was used to measure all the temperature and angular dependences which are presented in the Results section.

[0176] FIG. 25e shows a drawing of the rotating probe configuration used with the variable temperature insert (VTI), and the magnetic field orientation with respect to the header socket is shown in FIG. 25b. The temperature was recorded with a Cernox thermometer (model CX-1010-CU-0.1 L) that is calibrated between 0.1 K and 325 K. A resistive magnet of 31 T at National High Magnetic Field Laboratory (NHMFL) in Tallahassee, Florida was used to measure all the field response shown in this example study. Finally, the four-point resistance was measured using a quasi-DC technique with a Stanford Research SR560low noise voltage preamplifier and Stanford Research SR830 lock-in amplifier. The frequency used was 17.671 Hz and the excitation current was 500 nA.ResultsHall and Longitudinal Resistances

[0177] The 68 nm flake used to fabricate the four point van der Pauw device has a size of the order 1 x 1 μm. As shown in FIG. 25c, the small size of the flake results in the contacts, deposited by electron beam lithography, to be located nearby one another. In contrast, an ideal van der Pauw geometry would have the contacts at the corners as depicted in the insets of FIG. 26 a-d. As a consequence, the Hall Rxyand longitudinal Rxxresistances are mixed and are measured simultaneously regardless of the probe arrangements.

[0178] To isolate the two signals, a symmetrization based on Onsager's reciprocal relations was used. Specifically, Onsager symmetrization states that in a linear system, the resistance tensoris transposed upon reversal of the current and voltage probes. By measuring the same contacts twice while reversing the current and voltages probes would therefore allow the decoupling of the true longitudinal Rxxand the true Hall Rxyresistances by symmetrizing and anti-symmetrization the obtained data via andwhere R and R’ form an Onsager reciprocal pair whose voltage and current leads are reversed through V+I+and V- I-.

[0179] The resistances as a function of the magnetic field were measured in two different Onsager pairs: one that maximizes the Hall signal (XY configurations 1 and 2) as shown in FIG. 26 a and c, and one that maximizes the longitudinal signal (XX configurations 1 and 2) as shown in FIG. 26 b and d. The ideal probe configurations are shown in the insets along with the positive direction of the magnetic field pointing out of the page. Equation (1 ) was then used to symmetrize XX Onsager pairs to extract the longitudinal resistance, shown in FIG. 26 f. Similarly, Eq. (2) was used to anti-symmetrize XY Onsager pairs for the Hall resistance in FIG. 26 e.Temperature Dependence

[0180] With a variable temperature insert (VTI), the resistances versus magnetic field for all van der Pauw configurations shown in FIG. 26 a-d were measured at temperatures of 1.4 K, 40 K, 80 K, 160 K, 240 K and 300 K. The true Hall responses Rxywere then extracted via anti-symmetrization of the XY On-sager pairs and are shown in full scale in FIG. 27 a. The inset focuses on the jump near zero field where the Hall responses are plotted for -2 T ≤ B ≤ 2 T. The same Hall resistances Rxyare plotted in 3-dimensions and are shown in FIG. 27 b. Similarly, the longitudinal resistances are extracted from the XX Onsager pairs and are plotted in FIG. 27 c, d. In FIG. 27 d, the zero field temperature dependence Rxx(T) for the XX configuration 1 is shown in the background for comparison.Angular Dependence

[0181] Reference is now made to FIG. 28 showing angular dependence of the XY configuration 1 resistance (measured using the configuration shown in FIG. 26a) as a function of the magnetic field. The angle of the magnetic field with respect to the device is varied between 0° and 90° by changing the orientation of the probe on the VTI (see FIG. 25 e). 0° is defined as the magnetic field being entirely perpendicular to the flake's plane that is in the (111 ) direction. As this angle θ is varied, the magnetic field B becomes more aligned with the plane of the flake, and upon reaching 90°, the magnetic field is entirely in the plane with the substrate. The angle θ and a schematic of the magnetic field rotation is shown in the inset of FIG. 28.

[0182] In the XY configuration 1 which is the XY configuration with the least noise, Hall signal is measured between 0 T to 30 T for angles 0° ≤ θ ≤ 90°. The obtained curves are plotted in FIG. 28. Note that the signal is not anti-symmetrized to extract the true Hall response, because the noise in the required XY configuration 2 became too large at 300K, The anti-symmetrization was therefore omitted. However, given that the temperature is fixed at 300 K and the true longitudinal resistance Rxxis field-independent (see FIG. 27 c), the XY configuration 1 alone should capture the essence of the Hall signal despite the mixing between Rxyand Rxx.

[0183] It is also worthwhile to note that for θ = 90°, one would expect the effective magnetic field to be Bzcos 90° = 0. Consequently, no Hall response should be observed. Conversely, in the curve for θ = 90° shown in FIG. 28, there is clearly a small yet non zero dependence. This is due to a small angle offset which is estimated to be 7°. However, this offset cannot be avoided, because the rotating probe of the VTI is prone to mechanical backlash and differential contraction during the cooling or warming process.

[0184] The angular dependence of the measured resistance may be utilized to provide a magnetic field rotation sensor 2810 using the configuration illustrated in the inset of FIG. 28. As illustrated in FIG. 25c, the ohmic contacts may be formed in a four point van der Pauw device configuration. In some embodiments, any other suitable Hall effect geometrical configuration may be used.

[0185] As illustrated in FIG. 25d, the substrate of the magnetic field rotation sensor may be mounted on a 16-pin header. Further, the device contacts may be connected to the pins of the header. In some embodiments, any other suitable configuration may be used to make the electrical connections.

[0186] As illustrated in FIG. 25e, a rotating probe configuration may be used with a variable temperature insert (VTI) for the magnetic field rotation sensor. In some embodiments, a stationary probe configuration may be used that measures the angle of a magnetic field that rotates with respect to the device.

[0187] As illustrated in FIG. 28 and described herein above, the measured XY configuration 1 resistance shows a dependence on the magnetic field rotation angle in a range from 0 T to 30 T for angles 0° ≤ θ ≤ 90°. Magnetic field rotation sensor 2810 may generate sensor data indicating the rotation angle of the magnetic field based on the measured XY configuration 1 resistance. In some embodiments, magnetic field rotation sensor 2810 may generate sensor data based on other resistance measurements. For example, magnetic field rotation sensor 2810 may generate the sensor data based on the measured XY configuration 2 resistance (e.g., FIG. 26c) or the anti-symmetrized resistance RXY(e.g., FIG. 26e).

[0188] Reference is now made to FIG. 29 showing angular dependence of the anti- symmetrized Hall resistance RXYof an example thin bismuth device (illustrated in FIG. 25c) measured at a base temperature of 15mK in a dilution refrigerator equipped with a 3-axis magnet. The magnetic field may be tilted to change the magnetic field rotation angle by adjusting the current in the coils. The measurements were performed at various magnetic field magnitudes ranging from 0.05 T to 1 T.

[0189] The angular dependence of the measured resistance may be utilized to provide a magnetic field rotation sensor 2910 using the configuration shown in the inset of FIG. 29. The cartoon shown in the inset of FIG. 29 illustrates the angle θ of the magnetic field with respect to the bismuth device.

[0190] For the data measurements shown in FIG. 29, the angle θ was swept between 0 and 360 degrees. A four-terminal quasi-DC (100 nA of excitation current @ 17.781 Hz) measurement technique was used, and the anti-symmetrized data using Onsager reciprocity theorem is plotted in FIG. 29. It is noted that given the lack of angular dependence and magnetic field dependence of the longitudinal resistance (Rxx), plotting the anti-symmetrized data effectively only subtracted a constant from the measured resistance data so as to put in light the angular dependence of RXY as a function of the magnetic field. As illustrated in FIG. 29, RXYdisplays a linear dependence on θ for magnetic field magnitudes ranging from 0.05 T to 1 T. Further, the high-resolution 1 T data measured at 15mK and illustrated in FIG. 29 overlaps with the 1 T data measuredat 300K (shown in FIG. 26e) indicating no substantial temperature dependence for this large temperature range.

[0191] The data plotted in FIGS. 28 and 29 illustrate that the disclosed magnetic field rotation sensors (e.g., sensors 2810, 2910) can provide an advantage compared with other magnetic field rotation sensors (e.g., other sensors made using semiconductor materials like Si, GaAs, GaN, InN and graphene) in which the magnitude of the “ordinary” Hall effect is dependent on temperature. In contrast, the disclosed magnetic field rotation sensors can provide sensing operation that is substantially independent of the temperature in a wide temperature range from 10mK to 300K.

[0192] The angular dependence of the measured resistance may be linear or non-linear. For example, magnetic field rotation sensor 2810 may provide sensor data that displays a sine / cosine relationship on the rotation angle for magnetic fields above ~1T. As another example, magnetic field rotation sensor 2910 may provide sensor data that displays a linear relationship on the rotation angle for magnetic fields below ~1T.DiscussionLongitudinal Resistance

[0193] The true longitudinal resistance as a function of the magnetic field and of the temperature is shown in 3D in FIG. 27 d. The temperature dependence of Rxxis consistent with the four-point resistance measured for the XX1 configuration between 3K and 260K (shown in gray in FIG. 27 d). The increasing resistance with increasing temperature is also consistent with the semi-metallic nature of bismuth. Notably in FIGS. 27 c, d, it is observed that the true longitudinal resistance is independent of the magnetic field between -30 T and 30 T. This result is unexpected given that bulk bismuth shows clear signs of the Shubnikov-de Haas (SdH) oscillations for fields as low as 1 T. It is noted that the ordinary Hall effect is also nearly absent up to ±30 T, which is in agreement with the absence of SdH oscillations since both would require exceptionally high carrier concentration.Intrinsic Anomalous Hall Effect

[0194] It is equally surprising to observe the AHE in bismuth, since it is only expected for a system with broken time-reversal symmetry to manifest AHE. Nonetheless, it is emphasized that the strong spin-orbit coupling found in bismuth is characteristic of the intrinsic AHE. The current transport data presented in this example study provides support that the origin of AHE in bismuth is in fact intrinsic.

[0195] Despite the noise for |B| > 2 T, FIG. 27 a, b show an anomalous Hall effect that is almost perfectly independent of temperature, especially in the low field region of |B| < 2 T where the Rxy(B) are completely overlapped for temperatures ranging between 1 .4 K to 300 K. From the high field regions where AHE is saturated, the anomalous Hall resistance RAHE can be extracted via the zero field intercept of a linear fit. The extracted RAHE has a very weak and nearly absent dependence on the temperature. At first sight, this lack of temperature dependence may seem surprising, but an important feature of the intrinsic AHE is that the anomalous Hall conductivity σintAHE is temperature independent. The conductivity σxy(B) is calculated from the true RXX(B) and Rxy(B) for temperatures ranging from 1 .4 K to 300 K and shown in FIG. 30. The change in Rxxinduces a weak temperature dependence of σAHE.Conclusion

[0196] The anomalous Hall effect has been found in thin flakes of bismuth. Using Onsager's reciprocity theorem, the transverse Hall resistance was extracted as a function of the magnetic field, and the anomalous Hall response was found to be nearly independent of the temperature.

[0197] While the above description provides examples of the embodiments, it will be appreciated that some features and / or functions of the described embodiments are susceptible to modification without departing from the spirit and principles of operation of the described embodiments. Accordingly, what has been described above has been intended to be illustrative of the invention and non-limiting and it will be understood by persons skilled in the art that other variants and modifications may be made without departing from the scope of the invention as defined in the claims appended hereto. Thescope of the claims should not be limited by the preferred embodiments and examples, but should be given the broadest interpretation consistent with the description as a whole.References

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Claims

AMENDED CLAIMS received by the International Bureau on May 20, 2025 (20.05.2025)WE CLAIM:

1. A device structure to utilize anomalous Hall effect (AHE), the device structure comprising: a thin single-crystalline bismuth film having a first surface, a thickness of the bismuth film being between 25-120nm; and multiple ohmic contacts formed on the first surface, the multiple ohmic contacts being arranged in a Hall effect geometrical configuration.

2. The device structure of claim 1 , wherein a purity of the bismuth film is greater than 98% or 99%.

3. The device structure of claim 1 , wherein the Hall effect geometrical configuration is a van der Pauw configuration, a comb configuration, a Hall bar configuration, or a Corbino configuration.

4. The device structure of any one of claims 1 to 3, further comprising a capping layer on the first surface.

5. The device structure of claim 4, wherein the capping layer is a polymethyl methacrylate (PMMA) layer of thickness between 150-300nm.

6. The device structure of any one of claims 1 to 5, wherein the thickness of the bismuth film is between 50- 100nm.

7. The device structure of any one of claims 1 to 6, wherein the bismuth film is formed by mechanically exfoliating thin bismuth flakes.

8. The device structure of any one of claims 1 to 7, wherein the multiple ohmic contacts are formed by: depositing one or more metals on contact areas of the first surface; andannealing the deposited one or more metals, the annealing being a thermal annealing or a current annealing.

9. The device structure of claim 8, wherein the one or more metals includes Ti andAu.

10. The device structure of any one of claims 1 to 9, wherein the bismuth film has a positive temperature coefficient of resistance in a temperature range between 10K and 300K.

11. The device structure of any one of claims 1 to 10, wherein the first surface has a (1 ,1 ,1 ) orientation.

12. The device structure of any one of claims 1 to 11 , further comprising a substrate layer contacting the bismuth film.

13. The device structure of claim 12, wherein the substrate layer is a Si / SiO2 substrate.

14. A method of forming a device structure, wherein the device structure is according to any one of claims 1 to 13, the method comprising: forming a thin single-crystalline bismuth film having a thickness between 25- 120nm; and forming multiple ohmic contacts arranged in a Hall effect geometrical configuration on a first surface of the bismuth film.

15. A temperature sensor comprising the device structure according to any one of claims 1 to 13, wherein the temperature sensor is configured to provide temperature data based on a resistance between a pair of the multiple ohmic contacts and a calibration parameter.

16. A magnetic field sensor comprising the device structure according to any one of claims 1 to 13, wherein the magnetic field sensor is configured to provide magnetic field data based on a resistance between a pair of the multiple ohmic contacts.

17. A dual temperature and magnetic field sensor comprising the device structure according to any one of claims 1 to 13, wherein the dual temperature and magnetic field sensor is configured to: provide temperature data based on a calibration parameter and a first resistance between a first pair of the multiple ohmic contacts; and provide magnetic field data based on a second resistance between a second pair of the multiple ohmic contacts.

18. An electromagnetic isolator comprising: a first port; a second port; the device structure according to any one of claims 1 to 13; and a magnetic field device configured to control a directional sense of electromagnetic isolation between the first port and the second port by controlling a magnetic field applied to the device structure.

19. An electromagnetic circulator comprising: three or more ports; the device structure according to any one of claims 1 to 13; and a magnetic field device configured to control a directional sense of electromagnetic circulation between the three or more ports by controlling a magnetic field applied to the device structure.

20. A thermal isolator comprising:a first port; a second port; the device structure according to any one of claims 1 to 13; and a magnetic field device configured to control a directional sense of thermal isolation between the first port and the second port by controlling a magnetic field applied to the device structure.

21. A thermal circulator comprising: three or more ports; the device structure according to any one of claims 1 to 13; and a magnetic field device configured to control a directional sense of thermal circulation between the three or more ports by controlling a magnetic field applied to the device structure.

22. A thermoelectric isolator comprising: a first port; a second port; the device structure according to any one of claims 1 to 13; an electric control device configured to electrically control a thermal current or temperature gradient associated with the thermoelectric isolator; and a magnetic field device configured to control a thermoelectric isolation between the first port and the second port by controlling a magnetic field applied to the device structure.

23. A thermoelectric circulator comprising:three or more ports; the device structure according to any one of claims 1 to 13; an electric control device configured to electrically control a thermal current or temperature gradient associated with the thermoelectric circulator; and a magnetic field device configured to control a thermoelectric circulation between the three or more ports by controlling a magnetic field applied to the device structure.

24. A magnetic field rotation sensor comprising: the device structure according to claim 1 , wherein the device structure further comprises a substrate layer contacting the bismuth film; and a sensor package comprising the substrate layer and multiple pins providing electrical connections to the multiple ohmic contacts, wherein the magnetic field rotation sensor is configured to provide sensor data indicating a magnetic field rotation angle, the sensor data being based on a relationship between the magnetic field rotation angle and a Hall-effect resistance measurement between a pair of the multiple ohmic contacts.

25. The magnetic field rotation sensor of claim 24, wherein the magnetic field rotation sensor is configured to provide the sensor data that is based on a linear relationship between the magnetic field rotation angle and the Hall-effect resistance measurement.

26. A method of use of a magnetic field rotation sensor to determine a rotation angle of a magnetic field, the method comprising: positioning the magnetic field rotation sensor in the magnetic field; applying an excitation signal to the magnetic field rotation sensor;measuring a Hall-effect resistance signal generated by the magnetic field rotation sensor; and determining the rotation angle of the magnetic field based on a relationship between the rotation angle and the Hall-effect resistance, wherein the magnetic field rotation sensor comprises the device structure according to any one of claims 1 to 13.

27. A method of use of a temperature sensor, the method comprising: applying an excitation signal to the temperature sensor; measuring a resistance signal generated by the temperature sensor; and determining the temperature based on a relationship between the temperature and the resistance, the relationship including a temperature calibration parameter while excluding magnetoresistance calibration parameters, wherein the temperature sensor comprises the device structure according to any one of claims 1 to 13.

28. A method of use of a magnetic field sensor for temperature-independent magnetic field measurements in a 10mK to 300K temperature range, the method comprising: positioning the magnetic field sensor in a magnetic field; applying an excitation signal to the magnetic field sensor; measuring a Hall-effect resistance signal generated by the magnetic field sensor; and determining the magnetic field based on a relationship between the magnetic field and the Hall-effect resistance, wherein the magnetic field sensor comprises the device structure according to any one of claims 1 to 13.

29. A method of use of a dual temperature and magnetic field sensor, the method comprising: positioning the dual sensor in a magnetic field; applying an excitation signal to the dual sensor; measuring a longitudinal resistance signal and a Hall-effect resistance signal generated by the dual sensor; determining the temperature based on a relationship between the temperature and the longitudinal resistance, the relationship including a temperature calibration parameter while excluding magnetoresistance calibration parameters, determining the magnetic field based on a relationship between the magnetic field and the Hall-effect resistance, wherein the dual sensor comprises the device structure according to any one of claims 1 to 13.

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