Patent Application: US-201113019779-A

Abstract:
partially or fully saturated doped graphene materials are found to be superconducting . the saturation is with hydrogen or halogen . doping is performed by substitution of carbon atoms or by applying an electric field . diamond nano - rods are also found to be superconducting . these materials can be used in electronic devices having a gate .

Description:
description herein is offered by reference to particular embodiments of the technology . it is to be understood that the nature and scope of the technology sought to be claimed is not limited to those specific embodiments but admits of variations that can be practiced by those of skill in the art . a first example is graphane . this is where all the carbon atoms in a sheet of hexagons are each bonded to a respective hydrogen atom . hence the formula is ( ch ) n where n is the number of carbon atoms in the sheet . ( this formula may be inaccurate when taking into account the carbon atoms at the boundary of the sheet since carbons at the edge do not have three neighbouring carbons to bond to , which carbon atoms may have more than one bonded atom other than carbon .) thus when considering this substance relative to graphene it may be said to be fully hydrogenated , fully saturated with hydrogen or fully substituted with hydrogen . even if the material were to be prepared by routes other than from graphene this terminology would still apply to the final structure . a second example is where only a proportion of the carbon atoms of the sheet has a hydrogen atom . hence the formula is c n h xn , where n is the number of carbon atoms in the sheet and x is the proportion ( up to but less than 1 ) of carbon atoms that have a hydrogen atom bonded to them . ( this formula may be inaccurate when taking into account the carbon atoms at the boundary of the sheet since carbons at the edge do not have three neighbouring carbons to bond to , which carbon atoms may have more than one bonded atom other than carbon .) thus when considering the structure of this substance relative to graphene it may be said to be partially hydrogenated , partially saturated with hydrogen or partially substituted with hydrogen ( although of course no carbon is replaced ). even if the material were to be prepared by routes other than from graphene this terminology would still apply to the final structure . these materials may be rendered superconducting by the application of an electric field , for example by an electrode , which would commonly termed as a gate . the material is of course also cooled to below its superconducting critical temperature t c . fig6 shows an example of a suitable structure , which is analogous to a mosfet transistor in semiconductor technology . a silicon substrate has a 11 has layer of silicon dioxide 12 thereon . the active layer 13 made of the hydrogenated graphene is on the oxide layer . the electrode 14 , or gate , is a layer of metal thereon and separated and insulated from the active superconducting material by an insulating layer 15 . contacts 16 to pass a current along the active layer 13 of a metal such as gold or aluminium are provided . alternatively ( or additionally ) the materials are rendered superconducting by chemical doping , for example by substituting carbon atoms with boron . a third example is fluorinated . this is where all the carbon atoms in a sheet of hexagons are each bonded to a respective fluorine atom . hence the formula is ( cf ) n where n is the number of carbon atoms in the sheet . ( this formula may be inaccurate when taking into account the carbon atoms at the boundary of the sheet since carbons at the edge do not have three neighbouring carbons to bond to , which carbon atoms may have more than one bonded atom other than carbon .) thus when considering this substance relative to graphene it may be said to be fully fluorinated , fully saturated with fluorine or fully substituted with fluorine . even if the material were to be prepared by routes other than from graphene this terminology would still apply to the final structure . a fourth example is where only a proportion of the carbon atoms of the sheet has a fluorine atom . hence the formula is c n f xn , where n is the number of carbon atoms in the sheet and x is the proportion ( up to but less than 1 ) of carbon atoms that have a fluorine atom bonded to them . ( this formula may be inaccurate when taking into account the carbon atoms at the boundary of the sheet since carbons at the edge do not have three neighbouring carbons to bond to , which carbon atoms may have more than one bonded atom other than carbon .) thus when considering this substance relative to graphene it may be said to be partially fluorinated , partially saturated with fluorine or partially substituted with fluorine ( although of course no carbon is replaced ). even if the material were to be prepared by routes other than from graphene this terminology would still apply to the final structure . these materials may be rendered superconducting by the application of an electric field , for example by an electrode , which would commonly termed as a gate . the material is of course also cooled to below its superconducting critical temperature t c . fig6 shows an example of a suitable structure , which is analogous to a mosfet transistor in semiconductor technology . a silicon substrate has a 11 has layer of silicon dioxide 12 thereon . the active layer 13 made of the fluorinated graphene is on the oxide layer . the electrode 14 , or gate , is a layer of metal thereon and separated and insulated from the active superconducting material by an insulating layer 15 . contacts 16 to pass a current along the active layer 13 of a metal such as gold or aluminium are provided . alternatively ( or additionally ) the materials are rendered superconducting by chemical doping , for example b substituting carbon atoms with boron . a diamond nano - rod doped with boron of 2 nm in diameter is prepared by cvd doped with boron . an undoped diamond nano - rod of 2 nm in diameter is provided with an electrode to which connected to a power supply to produce an electric field in the rod . in all these embodiments the material is cooled with either liquid nitrogen or liquid helium to obtain superconductivity . other forms of cryogenic apparatus could also be used . these materials can be used to form conductors . these may be used to convey power via electric currents , or to convey electric signals . these conductors can also be used to form superconducting magnets by , for example , being formed into coils and passing a current through them . superconducting magnets find application in imaging systems , manipulation of ferrous objects and levitation devices ( for example for trains and other vehicles ). the inventors have modelled fully hydrogenated and fully fluorinated graphene . the following discussion of their investigations is taken from their paper of 3 feb . 2010 [ arxiv : 1002 . 0653v1 [ cond - mat . mtrl - sci ]] which deals with fully hydrogenated graphene or “ graphane ”. that paper , and also “ first - principles prediction of doped graphane as a high - temperature electron - phonon superconductor ” from physical review letters [ prl 105 , 037002 ( 2010 )] of 14 jul . 2010 , are incorporated herein by reference . numbered citations refer to the list of references appended to the description , before the claims . results obtained from similar investigations of fully fluorinated graphene are as follows immediately below . the markings * 1 and * 2 added in the text of the paper later mark the comparable results for graphane . the electronic structure for fully fluorinated graphene is 2 - dimensional as in graphane — the density of states at the top of the valence band is & gt ; 0 . 4 states / ev / cell , which is about a factor 2 larger than in graphane . however the states at the top of the valence band in fully fluorinated graphene are mainly associated with fluorine atoms , therefore the electron - phonon coupling is smaller than in graphane and the superconducting pairing is correspondingly weaker . the superconducting transition temperature is significantly smaller than in graphane , but can be just above the boiling point of liquid helium ( 4 k ). the discovery of superconductors such as magnesium diboride [ 1 ] and iron pnictides [ 2 , 3 ] opened new horizons in the landscape of superconductivity research , fueling renewed interest in the quest for high - temperature superconductivity in materials other than the copper oxides [ 4 , 5 ]. the critical temperature , t c , reflects the energy scale of the quantum - mechanical interactions driving the electron condensation into the superconducting state [ 6 ]. in high - t c copper oxides [ 7 ] the nature of the interaction leading to superconductivity is still under debate [ 8 ], yet it is generally accepted that coulomb exchange and correlation effects , with energy scales around few hundred mevs , play an important role [ 9 , 10 ]. in contrast , in conventional superconductors the pairing is known to be driven by the interaction between electrons and lattice vibrations , with an associated energy scale of only a few ten mevs [ 11 ]. due to the order - of - magnitude difference between such energy scales , it is generally assumed that conventional superconductors cannot exhibit t c as high as copper oxides [ 4 , 5 ]. here , we report first - principles calculations showing that p - doped graphane would make a conventional superconductor with t c well above the boiling point of liquid nitrogen . the bardeen - cooper - schrieffer ( bcs ) theory [ 11 ] defines the basic theoretical framework to understand conventional superconductivity . its generalization , known as the migdal - eliashberg theory [ 12 ], incorporating the lattice dynamics , provides a predictive computational tool . within bcs , t c is given by [ 11 ]: k b t c = 1 . 14 hω 0 exp (− 1 / n f v ) ( 1 ) where k b is the boltzmann constant , hω 0 a characteristic phonon energy , n f the electronic density of states ( edos ) at the fermi energy , e f , v an effective pairing potential resulting from the net balance between the attractive electron - phonon coupling ( epc ) and the repulsive electron - electron interaction [ 11 ]. even though the original bcs formula for t c is now replaced by more refined expressions such as , e . g ., the modified mcmillan equation [ 13 ], eq . ( 1 ) still proves useful for discussing trends . eq . ( 1 ) indicates that one could maximize t c by increasing the materials parameters ω 0 , n f , v . however , these are strongly intertwined , making such optimization complex [ 13 , 14 ]. here , we propose a simple procedure , based on eq . t c to design a high - t c superconductor . fig1 ( a ) is as follows . edos per carbon atom of 1d ( nanotube ; diamond nanowire ), 2d ( graphene ; graphane ) and 3d ( diamond ) systems . with the exception of graphene , with linear dispersions , the edos is proportional to e − 1 / 2 in 1d , a step - like function in 2d , and e 1 / 2 in 3d . the step - like edos in graphane implies that nf is large even at low doping . fig1 ( b ) is as follows . edos of pristine ( solid black line ) and 12 . 5 % p - doped graphane ( dashed red line ). the top of the valence band is set as zero , and e f =− 0 . 96 ev ( green line ). the edos at e f is similar in the two models ( 0 . 26 states / ev / cell in rigid - band and 0 . 27 states / ev / cell in supercell ). fig1 ( c ) is as follows . band structure of pristine ( solid black line ) and 12 . 5 % p - doped graphane ( dashed red line ). ( inset ) ball - and - stick 2 × 2 supercell with one substitutional b ( top and side views ). let us first consider the conventional superconductor with the highest t c , mgb 2 ( t c = 39 k )[ 1 ]. for simplicity , we neglect multi - band and anisotropy effects , which were the object of detailed investigations [ 15 , 16 , 17 , 18 , 19 ]. in mgb 2 the epc contribution to v is large (˜ 1 . 4 ev , from λ = n f v , using n f = 0 . 7 states / cell / ev , and λ ˜ 1 [ 16 ]) because the states with energy close to e f ( those which condensate in the superconducting state [ 11 ]) are of σ character , i . e ., derive from bonding combinations of planar b sp 2 hybrids localised around the middle of b — b bonds [ 15 , 16 , 17 , 18 , 19 ]. these electronic states are significantly affected by the b — b bond length variation associated with bond - stretching e 2g phonons [ 20 , 16 ], resulting into a large epc contribution to v . at the same time , the e 2g phonon energy is large (˜ 60 mev [ 16 ]), due to the small b mass , leading to a large ω 0 in eq . ( 1 ). furthermore , mgb 2 is a metal with a significant edos at e f (˜ 0 . 7 states / cell / ev [ 16 ]). these three factors cooperate in eq . ( 1 ) to establish a superconducting state with t c = 39 k [ 15 , 16 , 17 , 18 , 19 ]. however , many attempts to improve upon mgb 2 , by investigating related materials , only met limited success [ 21 ], with the experimental t c never higher than mgb 2 . we thus search for an alternative material having at least some of the desirable features of mgb 2 , i . e ., ( i ) σ electrons at the fermi surface ,( ii ) large bond - stretching phonon frequencies , and ( iii ) large edos at e f . we note that the first two requirements are both met by b - doped diamond , a conventional bcs superconductor with t c = 4 k [ 22 ], where a small hole - like fermi surface appears around the top of the valence band [ 23 ]. the electronic states at e f have σ character deriving from the bonding combination of tetrahedral c sp3 hybrids , bearing some analogy to mgb 2 . as these σ states are localized in the middle of the c — c bonds , they couple considerably to bond - stretching phonons [ 23 , 24 ], resulting in a large epc contribution to v , even superior to mgb 2 (˜ 3 ev , from λ = n f v , using n f = 0 . 1 states / cell / ev , and λ ˜ 0 . 3 [ 25 ]) [ 26 , 27 ]. in addition , the light c atoms have high energy optical phonons (˜ 130 mev , even after softening induced by the large epc [ 25 , 24 ]). however in b - doped diamond the edos at e f is rather small (˜ 0 . 1 states / cell / ev for 2 % doping [ 23 , 28 ]). this compromises t c . thus , while b - doped diamond shares some of the desirable features of mgb 2 , its 3 - dimensional ( 3d ) nature implies that the edos in proximity of the valence band scales as ˜ e 1 / 2 ( with e measured from the valence band edge ) [ 29 ], fig1 ( a ). then , the number of carriers available for the superconducting state remains relatively small even for large doping . superconducting diamond is thus a 3d analogue of mgb 2 [ 23 , 24 ]. this leads to the question of what would happen in a hypothetical b - doped diamond structure with reduced dimensionality , such as a thin film or a nanowire , where the edos can be significantly enhanced by quantum confinement . indeed , the edos of a two - dimensional semiconductor goes as ˜ θ ( e ) ( θ being the step function ) [ 29 ], hence the number of available carriers can be large , even at low doping . in order to estimate the expected edos increase in a diamond thin film it is helpful to consider a simple parabolic band model . for 2 % b - doping , bulk diamond has n f = 0 . 1 states / ev / cell at e f . a 0 . 5 nm thick diamond film with the same doping would have n f ˜ 0 . 5 states / ev / cell . such an edos increase would significantly enhance t c . using the electron - phonon potential and the phonon frequency of bulk diamond , eq . ( 1 ) gives that a 0 . 5 nm film would superconduct at t c ˜ 80 k . however the question remains whether it is possible to synthesize an atomically thin diamond film . recent work on graphene and its derivatives points to a positive answer . soon after the discovery of graphene [ 30 ] several works considered how to functionalise and chemically modify this novel 2d material [ 31 , 35 , 36 , 32 , 33 , 34 ]. in particular , it was proposed that fully hydrogenated graphene ( graphane ) would be stable [ 37 ]. the main difference between graphene and graphane is that , while the former is fully sp 2 bonded , the latter is sp 3 , as diamond [ 37 ]. recently , some experimental evidence of graphane was reported [ 38 ]. since graphane is the 2d counterpart of diamond , our scaling arguments immediately point to doped graphane as a potential high - t c superconductor . doping could be achieved by gating , including using an electrolyte gate , or by charge - transfer , as done in graphene [ 31 - 34 , 39 , 40 ]. substitutional doping of graphene was also reported , up to ˜ 10 14 cm − 2 [ 35 , 41 ]. we thus perform density functional perturbation theory ( dfpt ) calculations of epc and superconductivity in doped graphane within the framework of the migdal - eliashberg theory [ 12 ] and the local density approximation ( lda ) [ 42 , 43 ]. by analogy with b doped diamond , we consider p - doping . this is simulated using the rigid - band approximation [ 44 ]. fig1 ( b ) shows that the calculated edos in p - doped graphane close to the valence band maximum follows a step - like behavior , as expected for a 2d system . at 3 % doping the edos is 0 . 22 states / ev / cell , compared to 0 . 13 states / ev / cell in bulk diamond , with a factor 1 . 7 enhancement (* 1 — similar results for the fluorinated equivalent were given earlier hereinabove ). fig1 ( c ) indicates that the dispersions close to e f are essentially identical for a supercell containing b and for a rigid - band model of doped graphane . we expect this to hold also for lower doping , where the perturbation to the pristine dispersions is smaller . the similarity between these two models justifies our use of the rigid - band approximation . a supercell calculation with the b dopant explicitly included does not show impurity states inside the gap . fig2 ( a ) is as follows . phonon dispersion of pristine ( solid black line ) and 1 % p - doped graphane ( dashed blue lines ). the c — h stretching modes have higher frequencies ( 2655 - 2711 cm − 1 ) and are not shown . fig2 ( b ) is as follows . optical modes around the zone centre , showing the kohn anomalies . the horizontal ( green ) arrows indicate the average fermi surface diameter . fig3 ( a ) is as follows . pdos of pristine and doped graphane . fig3 ( b ) is as follows . eliashberg function in p - doped graphane for increasing doping . the largest contribution comes from the optical modes , similar to diamond [ 24 ], but also the acoustic phonons couple to holes at the fermi surface , similar to sic [ 44 ]. fig3 ( c ) is as follows . contributions to the eliashberg function arising from the to stretching modes ( hashed region ). ( insets ) ball - and - stick representations of two to modes . the arrows indicate the in - plane c — c stretching motions ( carbons are shown in grey , hydrogens in white ). fig2 ( a ) and 2 ( b ) report the phonon dispersions of pristine and p - doped graphane and fig3 ( a ) the corresponding phonon density of states ( pdos ). upon doping , the optical zone - centre modes with in - plane c — c stretching soften as a result of the inception of kohn anomalies [ 45 ]. the two degenerate to modes , having planar c — c stretching and h atoms moving in - phase with the c atoms , downshift from 1185 to 715 cm − 1 ( 147 to 89 mev ). this is due to the large epc of planar c — c stretching , which significantly affects the sp 3 - like electronic states at the fermi surface . the two degenerate zone - centre modes , having in - plane c — c stretching and h atoms moving out - of - phase with respect to the c atoms , downshift from 1348 to 1257 cm − 1 ( 167 to 156 mev ). the lo mode , with out - of - plane c — c stretching , does not couple to the electrons due to the different parity of potential and wavefunctions , resulting into a vanishing epc . the two degenerate optical modes corresponding to the shear motion of the c and h planes ( at ˜ 1133 cm − 1 ) and the c — h stretching modes ( 2 modes at 2663 and 2711 cm − 1 ) do not undergo softening upon doping . this is consistent with the electronic states associated with the c — h bonds having little weight at e f , hence a small epc . fig4 ( a ) is as follows . epc of graphane as a function of doping , calculated using the standard dfpt formalism [ 51 ]: the brillouin zone is sampled with an electron grid up to 300 × 300 × 1 , smearing from 50 to 270 mev , and phonon grid of 100 × 100 × 1 . for comparison , we plot literature values for mgb 2 ( solid line [ 24 ]), cac 6 ( dashed line [ 52 ]), and diamond ( solid line [ 24 ]; triangles [ 25 ]). more sophisticated calculations taking explicitly into account a substitutional dopant , such as b , could slightly change the epcs [ 28 , 25 ]. however , in b - doped diamond a rigid - band model provides a lower epc and a lower bound to t c [ 25 ]. fig4 ( b ) is as follows . t c calculated using the modified mcmillan formula and a coulomb pseudopotential μ *= 0 . 13 [ 49 ]. the left - side hashed region indicates doping below the estimated mit , where our formalism applies only to charge - transfer or gate - induced doping . above mit , it applies to substitutional doping as well . we use the isotropic eliashberg formalism [ 49 ]. a more sophisticated description based on the fully anisotropic eliashberg theory is expected to increase t c [ 15 , 53 ]. for comparison we also show t c of mgb 2 ( solid line , t c = 39 k [ 1 ]), cac 6 ( dashed line , t c = 11 . 5 k [ 54 ]), and diamond ( solid black line , t c = 4 k at ˜ 3 % b [ 22 ]; 11 k at ˜ 7 % b [ 55 ] the softening of modes with a large c — c stretching component is similar to that reported in b - doped diamond [ 24 , 25 ]. in particular , the region of reciprocal space where the phonon softening is observed matches the diameter , 2k f , of the hole fermi surface around the γ point , this being a typical signature of the kohn effect [ 45 ]. the calculated phonon softening of the to c — c stretching modes (˜ 58 mev or ˜ 470 cm − 1 ) is significantly larger than in other materials , as typical kohn anomalies range from ˜ 5 mev ( graphite and graphene [ 46 ]) to ˜ 10 mev ( tac [ 47 ]). in the case of b - doped diamond the phonon softening takes place through the creation of a non - dispersive defect branch associated with the b dopant [ 25 ]. a similar effect could happen in b - doped graphane , but we expect the magnitude of the doping - induced softening to be reasonably well described within our rigid - band model . also , more sophisticated calculations , taking b explicitly into account [ 28 , 25 ] or with non - adiabatic corrections [ 48 ], may slightly revise the softening . nevertheless , such a large softening stands out as a qualitative effect . fig3 plots the eliashberg spectral function [ 49 ], which shows the relative contribution of different modes to the superconducting pairing [ 49 ]. we get that the to in - plane c — c bond - stretching phonons with c and h atoms moving in - phase ( see fig3 ( c )) have the largest epc , due to the σ character of the electronic states at ef and the large c displacements associated with these modes . this is similar to b - doped bulk diamond [ 23 , 24 , 25 , 28 ] and validates our hypothesis that p - doped graphane can be regarded as an atomically thin diamond film , exhibiting similar epc and vibrational frequencies , but larger edos at ef . we note that the in - plane c — c bond - stretching phonons , with c and h atoms moving out - of - phase , do not contribute to the epc . this happens because , upon softening , the four c — c planar stretching modes hybridize in such a way that those at 715 cm − 1 carry an increased weight on the c atoms , while the opposite happens for the two modes at 1257 cm − 1 . fig4 ( a ) plots the epc as a function of doping , and fig4 ( b ) the corresponding t c . we find that t c exceeds the boiling point of liquid nitrogen , and falls within the same t c range as copper oxides [ 50 ] (* 2 — similar results for the fluorinated equivalent were given earlier above ). due to the relatively constant edos below the top of the valence band [ cf . fig1 ( b )], t c is rather insensitive to doping . this is important for the practical realization of superconducting graphane . our results should be valid throughout the entire doping range considered here in the case of gate - or charge transfer - induced doping , since in these cases the holes are delocalized and doped graphane is in the metallic regime . on the other hand , for substitutional doping we expect our results to be valid only beyond the mott metal - to - insulator transition ( mit ). in absence of experimental mit measurements in graphane , we estimate the critical doping concentration , n , using the following argument . in 3d the mit occurs when the impurity wavefunctions are close enough that their overlap is significant [ 56 ]. for many materials a h n c 1 / 3 ˜ 0 . 26 , a h being the radius of the ground - state wavefunction of an hydrogenic donor [ 56 ]. the radius can be calculated as a h = εm * a 0 / 2 , a 0 being the bohr radius , ε the dielectric constant , and m * the effective mass [ 56 ]. in diamond a h ˜ 4 å and n c ˜ 4 * 10 20 cm − 3 [ 56 ], therefore the average separation between nearest neighbor b atoms is ˜ 15 å . in the case of graphane we use a similar criterion , replacing the 3d hydrogenic impurity with a 2d one . the ground - state hydrogenic wavefuction in 2d has a radius a h ( 2d )= εm * a 0 / 2 [ 57 ]. using the dielectric constant and hole effective mass of diamond ( ε = 5 . 7 $; m *= 0 . 74 ) we find a h ( 2d )= a h / 2 ˜ 2 å . thus , the average separation between nearest neighbor b atoms at the mit is ˜ 7 . 5 å , and the corresponding doping can be estimated as 5 % b ( 1 b every 20 c atoms ) or 2 * 10 14 holes cm − 2 . this could be feasible , considering that substitutional doping in graphene was reported up to 5 % [ 41 ]. the calculated high - t c for p - doped graphane bears consequences both for fundamental science and applications . one could envision hybrid superconducting - semiconducting circuits directly patterned through lithographic techniques , graphane - based josephson junctions for nanoscale magnetic sensing , and ultimately an ideal workbench for exploring the physics of the superconducting state in two dimensions [ 58 ]. the superconducting phase transition in graphane could also be controlled by gating [ 34 , 59 ]. a high - t c superconductor with gate - controllable t could lead to novel switching mechanisms in nanoscale field - effect transistors . furthermore , the discovery of an electron - phonon superconductor with t c above liquid nitrogen would mean that ( i ) there are no fundamental reasons to believe that bcs superconductors cannot have t c & gt ; 40 k ( mgb 2 ), and ( ii ) high - t c superconductivity does not take place exclusively in the copper oxides . in particular , our calculations indicate that at least one material could exist where a very strong epc leads to t c in the copper oxide range without triggering a lattice instability . the superconducting phase transition in systems with reduced dimensionality has been the subject of numerous theoretical studies [ 60 , 61 ]. quantum fluctuations could destroy the superconducting order in 2d [ 62 ]. however , recent experimental evidence suggests that this is not necessarily the case [ 58 , 59 , 63 ]. in particular , for thin pb it was reported that the superconducting state is robust down to two atomic layers [ 58 ]. since our proposed mechanism of superconductivity in doped graphane is bcs - 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