        ARTICLE_MAGNET = r"""it is well known that the classical magnetoresistance ( mr ) in metals or semiconductors with a closed free electron fermi surface increases quadratically with increasing magnetic field @xmath2 for @xmath3 and saturates when @xmath4 . here @xmath5 is the zero - magnetic - field mobility . hence , the extraordinarily high and linear mr ( lmr ) , which breaks this familiar rule , has been gaining much attention as soon as its discovery . in the past decade , this unexpected lmr has been reported in silver chalcogenide,@xcite indium antimonide,@xcite silicon,@xcite mnas - gaas composite material,@xcite and graphene.@xcite    kapitza s linear law@xcite indicates that the metal shows a magnetoresistance linear in perpendicular magnetic field when it has an open fermi surface and a mean free path longer than the electronic larmor radius . recently , another two models , irrespective of the open fermi surface , have been constructed to provide possible mechanisms for the lmr phenomenon . abrikosov suggested a quantum - limit origin of lmr for the homogenous system with a gapless linear energy spectrum.@xcite his model requires that landau levels are well formed and the carrier concentration is small that all electrons occupy only the lowest landau band . alternatively , parish and littlewood developed a classical model without involving linear spectrum.@xcite ignoring the concrete microscopic mechanism , they attributed this unusual mr to the mobility fluctuations in a strongly inhomogenous system . topological insulators@xcite ( tis ) are novel materials with a full energy gap in bulk , while there are gapless surface states . due to its unique band structure with only one helical dirac cone and linear energy dispersion,@xcite the surface states of the ti bi@xmath0se@xmath1 become an excellent platform for the study of quantum - limit lmr . the recent experiment in this flat surface system , however , reported that a large positive mr , which becomes very linear above a characteristic field of @xmath6@xmath7@xmath8 t , was observed even in an opposite situation where the carrier sheet density is high that electrons occupy more than one landau levels.@xcite moreover , they found that raising temperature to room temperature almost has no influence on the observed lmr . it is striking that this observation is in conflict with abrikosov s model and also with the classical parish - littlewood model . so far a reliable theoretical scheme capable of explaining this novel experiment has still been lacking .    in this paper , we generalize the balance - equation approach@xcite to a system modeling the surface states of a three - dimensional ti to investigate the two - dimensional magnetotransport in it . we find that a positive , nonsaturating and dominantly linear magnetoresistance can appear within quite wide magnetic - field range in the ti surface state having a positive and finite effective g - factor . this linear magnetoresistance shows up in the system of high carrier concentration and low mobility when electrons are in extended states and spread over many smeared landau levels , and persists up to room temperature , providing a possible mechanism for the recently observed linear magnetoresistance in topological insulator bi@xmath0se@xmath1 nanoribbons.@xcite we consider the surface state of a bi@xmath0se@xmath1-type large bulk gap ti in the @xmath9-@xmath10 plane under the influence of a uniform magnetic field @xmath11 applied along the @xmath12 direction.@xcite following the experimental observation,@xcite we assume that the fermi energy locates in the gap of the bulk band and above the dirac point , i.e. the surface carriers are electrons . further , the separations of the fermi energy from the bottom of bulk band and dirac point are much larger than the highest temperature ( @xmath13 ) considered in this work . hence , the contribution from the bulk band to the magnetotransport is negligible . these electrons , scattered by randomly distributed impurities and by phonons , are driven by a uniform in - plane electric field @xmath14 in the topological surface . the hamiltonian of this many - electron and phonon system consists of an electron part @xmath15 , a phonon part @xmath16 , and electron - impurity and electron - phonon interactions @xmath17 and @xmath18 : @xmath19 here , the electron hamiltonian is taken in the form @xmath20 , \ ] ] in which @xmath21 , @xmath22 , @xmath23 and @xmath24 , stand , respectively , for the canonical momentum , coordinate , momentum and spin operators of the @xmath25th electron having charge @xmath26 , @xmath27 is the vector potential of the perpendicular magnetic field @xmath28 in the landau gauge , @xmath29 is the fermi velocity , @xmath30 is the effective g - factor of the surface electron , and @xmath31 is the bohr magneton with @xmath32 the free electron mass . the sum index @xmath25 in eq.([helectron ] ) goes over all electrons of total number @xmath33 in the surface state of unit area .    in the frame work of balance equation approach,@xcite the two - dimensional center - of - mass ( c.m . ) momentum and coordinate @xmath34 and @xmath35 , and the relative - electron momenta and coordinates @xmath36 and @xmath37 are introduced to write the hamiltonian @xmath15 into the sum of a single - particle c.m . part @xmath38 and a many - particle relative - electron part @xmath39 : @xmath40 , with @xmath41.\end{aligned}\ ] ] in this , @xmath42 is the canonical momentum of the center - of - mass and @xmath43 is the canonical momentum for the @xmath25th relative electron . here we have also introduced c.m . spin operators @xmath44 and @xmath45 . the commutation relations between the c.m . spin operators @xmath46 and @xmath47 and the spin operators @xmath48 , @xmath49 and @xmath50 of the @xmath25th electron are of order of @xmath51 : @xmath52= n^{-1}2\,{\rm i}\,\varepsi lon_{\beta_1\beta_2\beta_3}\sigma_j^{\beta_3}$ ] with @xmath53 . therefore , for a macroscopic large @xmath33 system , the c.m . part @xmath38 actually commutes with the relative - electron part @xmath54 in the hamiltonian , i.e. the c.m . motion and the relative motion of electrons are truly separated from each other . the couplings between the two emerge only through the electron impurity and electron  phonon interactions . furthermore , the electric field @xmath55 shows up only in @xmath38 . and , in view of @xmath56={\rm i}\delta_{\alpha \beta}(\delta_{ij}-1/n)\simeq { \rm i}\delta_{\alpha\beta}\delta_{ij}$ ] , i.e. the relative - electron momenta and coordinates can be treated as canonical conjugate variables , the relative - motion part @xmath54 is just the hamiltonian of @xmath33 electrons in the surface state of ti in the magnetic field without the presence of the electric field .    in terms of the c.m . coordinate @xmath57 and the relative electron density operator @xmath58 , the electron impurity and electron  phonon interactions can be written as@xcite @xmath59 here @xmath60 and @xmath61 are respectively the impurity potential ( an impurity at randomly distributed position @xmath62 ) and electron  phonon coupling matrix element in the plane - wave representation , and @xmath63 with @xmath64 and @xmath65 being the creation and annihilation operators for a phonon of wavevector @xmath66 in branch @xmath67 having frequency @xmath68 . velocity ( operator ) @xmath69 is the time variation of its coordinate : @xmath70= v_{\rm f}(\sigma_{\rm c}^y\ , \hat{i}-\sigma_{\rm c}^x\ , \hat{j})$ ] . to derive a force - balance equation for steady state transport we consider the heisenberg equation for the rate of change of the c.m . canonical momentum @xmath71 : @xmath72= - n e({\bm v}\times { \bm b})- n e{\bm e}+{\bm { f}}_{\rm i}+{\bm { f}}_{\rm p},\ ] ] in which the frictional forces @xmath73 and @xmath74 share the same expressions as given in ref ..    the statistical average of the operator equation can be determined to linear order in the electron  impurity and electron phonon interactions @xmath17 and @xmath18 with the initial density matrix @xmath75 at temperature @xmath76 when the in - plane electric field @xmath77 is not strong . for steady - transport states we have @xmath78 , leading to a force - balance equation of the form @xmath79 here @xmath80 , the statistically averaged velocity of the moving center - of - mass , is identified as the average rate of change of its position , i.e. the drift velocity of the electron system driven by the electric field @xmath77 , and @xmath81 and @xmath82 are frictional forces experienced by the center - of - mass due to impurity and phonon scatterings : @xmath83,\label{fp}\end{aligned}\ ] ] in which @xmath84 is the bose distribution function , @xmath85 , and @xmath86 stands for the imaginary part of the fourier spectrum of the relative - electron density correlation function defined by @xmath87\big\rangle_{0},\ ] ] where @xmath88 and @xmath89 denotes the statistical averaging over the initial density matrix @xmath90.@xcite    the force - balance equation describes the steady - state two - dimensional magnetotransport in the surface state of a ti . note that the frictional forces @xmath81 and @xmath82 are in the opposite direction of the drift velocity @xmath91 and their magnitudes are functions of @xmath92 only . with the drift velocity @xmath93 in the @xmath9 direction , the force - balance equation eq . yields a transverse resistivity @xmath94 , and a longitudinal resistivity @xmath95 . the linear one is in the form @xmath96 for calculating the electron density correlation function @xmath97 we proceed in the landau representation.@xcite the landau levels of the single - particle hamiltonian @xmath98 of the relative - electron system in the absence of electric field are composed of a positive `` @xmath99 '' and a negative `` @xmath100 '' branch@xcite @xmath101 with @xmath102 and @xmath103 , and a zero ( @xmath104 ) level @xmath105 the corresponding landau wave functions are @xmath106 and @xmath107 for @xmath108 ; and @xmath109 for @xmath104 . here @xmath110 is the wavevector of the system along @xmath9 direction ; @xmath111 with @xmath112 ; and @xmath113 is the harmonic oscillator eigenfunction with @xmath114 being the hermite polynomial , @xmath115 , and @xmath116 . each landau level contains @xmath117 electron states for system of unit surface area . the positive branch @xmath118 and the @xmath104 level @xmath119 of the above energy spectra are indeed quite close to those of the surface states in the bulk gap of bi@xmath0se@xmath1-family materials derived from microscopic band calculation.@xcite    the landau levels are broadened due to impurity , phonon and electron - electron scatterings . we model the imaginary part of the retarded green s function , or the density - of - states , of the broadened landau level @xmath120 ( written for `` + ' ' -branch and @xmath104 levels ) , using a gaussian - type form:@xcite @xmath121,\ ] ] with a half - width @xmath122 of the form:@xcite @xmath123^{1/2}$ ] . here @xmath124 is the single - particle lifetime and @xmath125 is the cyclotron frequency of linear - energy - dispersion system with @xmath126 being the zero - temperature fermi level . using a semi - empirical parameter @xmath127 to relate @xmath124 with the transport scattering time @xmath128 , and expressing @xmath129 with the zero - field mobility @xmath5 at finite temperature,@xcite we can write the landau - level broadening as @xmath130^{1/2}.\ ] ]    in the present study we consider the case of @xmath120-doping , i.e. the fermi level is high enough above the energy zero of the dirac cone in the range of `` + ' ' -branch levels and the states of `` @xmath100''-branch levels are completely filled , that they are irrelevant to electron transport . special attention has to be paid to the @xmath104 level , since , depending on the direction of exchange potential the effective g - factor of a ti surface state , @xmath30 , can be positive , zero or negative.@xcite the sign and magnitude of the effective g - factor determines how many states of the zero level should be included in or excluded from the available states for electron occupation in the case of @xmath120-doping at a magnetic field . ( i ) if @xmath131 , the @xmath104 level center is exactly at @xmath132 and the system is electron - hole symmetric . the total number of negative energy states ( including the states of the lower half of the @xmath104 level and states of the @xmath100"-branch levels ) and that of positive energy states ( including the states of the upper half of the @xmath104 level and states of the @xmath99"-branch levels ) do not change when changing magnetic field . therefore , the lower - half negative energy states of this level are always filled and the upper - half positive - energy states of it are available for the occupation of particles which are counted as electrons participating in transport in the case of @xmath120-doping . ( ii ) for a finite positive @xmath133 , the @xmath104 level @xmath134 moves downward to negative energy and its distance to the nearest  @xmath100"-branch level is @xmath135 closer than to the nearest  + " -branch level at finite magnetic field strength @xmath2 . this is equivalent to the opening of an increasingly enlarged ( with increasing @xmath2 ) energy gap between the  + " -branch states and the states of the zero - level and the  @xmath100"-branch levels . the opening of a sufficient energy gap implies that with increasing magnetic field the states in the  + " -branch levels would no longer shrink into the zero - level , and thus the @xmath104 level should be completely excluded from the conduction band , i.e. only particles occupying the  + " -branch states are counted as electrons participating in transport in the case of @xmath120-doping , when the magnetic field @xmath2 gets larger than a certain value ( depending on the magnitude of @xmath30 ) . ( iii ) for a finite negative @xmath136 , the @xmath104 level @xmath134 moves upward to positive energy and an increasingly enlarged energy gap will be opened between the states of the zero - level and the  + " -branch and the states of  @xmath100"-branch levels , and particles occupying the @xmath104 level and  + " -branch states are electrons participating in transport when the magnetic field @xmath2 gets larger than a certain value .    as a result , the experimentally accessible sheet density @xmath33 of electrons participating in transport is related to the fermi energy @xmath137 by the following equation valid at finite @xmath30 for the magnetic field @xmath2 larger than a certain value : @xmath138 in which @xmath139 + 1\}^{-1}$ ] is the fermi distribution function at temperature @xmath76 and the summation index @xmath120 goes over @xmath140 for @xmath133 , or @xmath141 for @xmath136 . in the case of @xmath131 , @xmath142\ ] ] valid for arbitrary magnetic field , in which @xmath143 . the imaginary part of relative - electron density correlation function in the presence of a magnetic field , @xmath86 , can be expressed in the landau representation as@xcite @xmath144 in which the transform factor @xmath145 ^ 2,\end{aligned}\ ] ] with @xmath146 , @xmath147 , @xmath148 , and @xmath149 being associated laguerre polynomials . the landau - representation correlation function @xmath150 in eq.([piqw ] ) can be constructed with the imaginary part of the retarded green s function @xmath151 , or the density - of - states , of the @xmath120th landau level as@xcite @xmath152\nonumber\\ & \hspace{1.2cm}\times{\rm im}g_n(\epsilon+\omega){\rm im}g_{n'}(\epsilon).\end{aligned}\ ] ] the summation indices @xmath120 and @xmath153 in eq.([piqw ] ) are taken over @xmath140 for @xmath133 , or @xmath154 for @xmath136 . in the case of @xmath131 , eq.([piqw ] ) still works and the summation indices @xmath120 and @xmath153 go over @xmath154 but with @xmath155 replaced by @xmath156 in eq.([p2nn ] ) . numerical calculations are performed for the magnetoresistivity @xmath157 of surface state in a uniform ti bi@xmath0se@xmath1 . at zero temperature the elastic scattering contributing to the resistivity is modeled by a coulomb potential due to charged impurities:@xcite @xmath158 with @xmath159 being the impurity density , which is determined by the zero - magnetic - field mobility @xmath5 . at temperatures higher than @xmath160,@xcite phonon scatterings play increasingly important role and the dominant inelastic contribution comes from optical phonons . for this polar material , the scattering by optical phonons via the deformation potential can be neglected . hence , we take account of inelastic scattering from optical phonons via frhlich coupling : @xmath161 . in the numerical calculation we use the following parameters:@xcite fermi velocity @xmath162 , static dielectric constant @xmath163 , optical dielectric constant @xmath164 , and phonon energy @xmath165 . the broadening parameter is taken to be @xmath166 . as a function of the magnetic field @xmath2 having different effective g - factors : @xmath167 and @xmath168 for a ti surface system with electron sheet density @xmath169 in the cases of zero - magnetic - field mobility @xmath170 ( a ) and @xmath171 ( b ) . several integer - number positions of filling factor @xmath172 are marked in ( b).,scaledwidth=40.0% ]    fig.[diffg ] shows the calculated magnetoresistivity @xmath157 versus the magnetic field strength @xmath2 for a ti surface system with electron sheet density @xmath169 but having different effective g - factors : @xmath167 and @xmath168 for two values of zero - magnetic - field mobility @xmath170 and @xmath171 , representing different degree of landau - level broadening . in the case without zeeman splitting ( @xmath131 ) the resistivity @xmath157 exhibits almost no change with changing magnetic field up to 10 t , except the shubnikov - de haas ( sdh ) oscillation showing up in the case of @xmath171 . this kind of magnetoresistance behavior was indeed seen experimentally in the electron - hole symmetrical massless system of single - layer graphene.@xcite in the case of a positive g - factor , @xmath173 , the magnetoresistivity increases linearly with increasing magnetic field ; while for a negative g - factor , @xmath174 , the magnetoresistivity decreases linearly with increasing magnetic field . is shown as a function of the magnetic field @xmath2 for different values of zero - magnetic - field mobility : ( a ) @xmath175 , ( b ) @xmath176 , ( c ) @xmath177 , ( d ) @xmath178 , ( e ) @xmath179 , and ( f ) @xmath180 . the inset of ( a ) illustrates the same for a larger magnetic - field range @xmath181 . the filling factor @xmath182 is plotted versus the magnetic field in ( f ) ; and several integer - number positions of @xmath182 are also marked in ( d ) and ( e ) . here the surface electron density @xmath169 and the lattice temperature @xmath183.,scaledwidth=47.0% ]    in the following we will give more detailed examination on the linearly increasing magnetoresistance in the positive @xmath30 case . fig.[rhob ] shows the calculated resistivity @xmath157 versus the magnetic field strength @xmath2 at lattice temperature @xmath183 for system of carrier sheet density @xmath169 and @xmath173 , having different zero - field mobility @xmath184 and @xmath180 . all resistivity curves for mobility @xmath185 exhibit clear linearity in the magnetic - field range and appear no tendency of saturation at the highest field shown in the figure . especially , for the case @xmath170 , the linear behavior extends even up to the magnetic field of @xmath186 , as illustrated in the inset of fig.[rhob](a ) . this feature contradicts the classical mr which saturates at sufficiently large magnetic field @xmath187 . note that here we only present the calculated @xmath157 for magnetic field @xmath2 larger than @xmath188 t , for which a sufficient energy gap @xmath135 is assumed to open that with further increase of the magnetic field the states in the `` + ' ' -branch levels no longer shrink into the zero level and thus it should be excluded from the conduction band . this is of course not true for very weak magnetic field . when @xmath189 the energy gap @xmath190 , the situation becomes similar to the case of @xmath131 : the whole upper half of the zero - level states are available to electron occupation and we should have a flat resistivity @xmath157 when changing magnetic field . with increasing @xmath2 the portion of the zero - level states available to conduction electrons decreases until the magnetic field reaches @xmath191 . as a result the resistivity @xmath157 should exhibit a crossover from a flat changing at small @xmath2 to positively linear increasing at @xmath192 . this is just the behavior observed in the ti bi@xmath0se@xmath1.@xcite    note that in the case of @xmath170 , the broadened landau - level widths are always larger than the neighboring level interval : @xmath193 , which requires @xmath194 ^ 2 $ ] , even for the lowest landau level @xmath195 , i.e. the whole landau - level spectrum is smeared . with increasing the zero - field mobility the magnitude of resistivity @xmath157 decreases , and when the broadened landau - level width becomes smaller than the neighboring level interval , @xmath196 , a weak sdh oscillation begin to occur around the linearly - dependent average value of @xmath157 at higher portion of the magnetic field range , as seen in fig.[rhob](c ) , ( d ) and ( e ) for @xmath197 and @xmath198 . on the other hand , in the case of large mobility , e.g. @xmath199 , where the broadened landau - level widths @xmath200 are much smaller than the neighboring level interval even for level index @xmath120 as large as @xmath201 , the magnetoresistivity shows pronounced sdh oscillation and the linear - dependent behavior disappears , before the appearance of quantum hall effect,@xcite as shown in fig.[rhob](f ) . abrikosov s model for the lmr requires the applied magnetic field large enough to reach the quantum limit at which all the carriers are within the lowest landau level,@xcite while it is obvious that more than one landau levels are occupied in the experimental samples in the field range in which the linear and non - saturating magnetoresistivity was observed.@xcite for the given electron surface density @xmath202 , the number of occupied landau levels , or the filling factor @xmath172 , at different magnetic fields is shown in fig.[rhob](f ) , as well as in the fig.[rhob](d ) and ( e ) , where the integer - number positions of @xmath203 , i.e. filling up to entire @xmath182 landau levels , coincide with the minima of the density - of - states or the dips of sdh oscillation . this is in contrast with @xmath131 case , where the integer number of @xmath203 , which implies a filling up to the center position of the @xmath182th landau levels , locates at a peak of sdh oscillation , as shown in fig.[diffg]b . the observed sdh oscillations in the bi@xmath0se@xmath1 nanoribbon exhibiting nonsaturating surface lmr in the experiment@xcite favor the former case : a finite positive effective @xmath133 .     is plotted as a function of the surface electron density @xmath33 at magnetic field @xmath204 : ( a ) at different values of zero - field mobility @xmath5 , and ( b ) at different values of zero - field conductivity @xmath205.,scaledwidth=40.0% ]     at various lattice temperatures . here the zero - magnetic - field mobility at zero temperature is @xmath206.,scaledwidth=35.0% ]    next , we examine the density - dependence of the linear magnetoresistivity . to compare with abrikosov s quantum magnetoresistance which suggests a @xmath207 behavior,@xcite we show the calculated @xmath208 for above lmr versus the carrier sheet density @xmath33 in fig.[rhon ] at fixed magnetic field @xmath209 t . the mobility is taken respectively to be @xmath210 and @xmath211m@xmath212/vs to make the resistivity in the lmr regime . a clearly linear dependence of @xmath213 on the surface density @xmath33 is seen in all cases , indicating that this non - saturating linear resistivity is almost inversely proportional to the carrier density . in the figure we also show @xmath208 versus @xmath33 under the condition of different given conductivity @xmath214 and @xmath215 . in this case the half - width @xmath216 is independent of surface density . the linear dependence still holds , indicating that this linear behavior is not sensitive to the modest @xmath33-dependence of landau level broadening @xmath216 as long as the system is in the overlapped landau level regime . from the above discussion , it is obvious that lmr shows up in the system having overlapped landau levels and the separation of landau levels makes the mr departure from the linear increase . at high temperature , the thermal energy would smear the level separation and phonon scatterings further broaden landau levels . hence , it is believed that this lmr will be robust against raising temperature . this is indeed the case as seen in fig.[rhot ] , where we plot the calculated magnetoresistivity @xmath157 for the above system with zero - temperature linear mobility @xmath217m@xmath212/vs versus the magnetic field at different lattice temperatures . we can see that raising temperature to room temperature has little effect on the linearity of mr . due to the decreased mobility at higher temperature from phonon scattering , the weak sdh oscillation on the linear background tends to vanish . these features are in good agreement with the experimental report.@xcite in summary , we have studied the two - dimensional magnetotransport in the flat surface of a three - dimensional ti , which arises from the surface states with a wavevector - linear energy dispersion and a finite , positive zeeman splitting within the bulk energy gap . when the level broadening is comparable to or larger than the landau - level separation and the conduction electrons spread over many landau levels , a positive , dominantly linear and non - saturating magnetoresistance appears within a quite wide range of magnetic field and persists up to room temperature . this remarkable lmr provides a possible mechanism for the recently observed linear magnetoresistance in topological insulator bi@xmath0se@xmath1 nanoribbons.@xcite    in contrast to quantum hall effect which appears in the case of well formed landau levels and to abrikosov s quantum magnetotransport,@xcite which is limited to the extreme quantum limit that all electrons coalesce into the lowest landau level , the discussed lmr is a phenomena of pure classical two - dimensional magnetotransport in a system having linear - energy - dispersion , appearing in the regime of overlapped landau levels , irrespective of its showing up in relatively high magnetic field range . furthermore , the present scheme deals with spatially uniform case without invoking the mobility fluctuation in a strongly inhomogeneous system , which is required in the classical parish and littlewood model to produce a lmr.@xcite    the appearance of this significant positive - increasing linear magnetoresistance depends on the existence of a positive and sizable effective g - factor . if the zeeman energy splitting is quite small the resistivity @xmath157 would exhibit little change with changing magnetic field . in the case of a negative and sizable effective g - factor the magnetoresistivity would decrease linearly with increasing magnetic field . therefore , the behavior of the longitudinal resistivity versus magnetic field may provide a useful way for judging the direction and the size of the effective zeeman energy splitting in ti surface states . this work was supported by the national science foundation of china ( grant no . 11104002 ) , the national basic research program of china ( grant no . 2012cb927403 ) and by the program for science&technology innovation talents in universities of henan province ( grant no . 2012hastit029 ) ."""
