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topological insulators ( ti ) are a new class of materials whose surfaces / edges host bound spin - polarized dirac fermions with high mobility . these states , in a time reversal invariant system are protected against perturbation and non - magnetic disorder . @xcite well - known examples of materials with such properties include bi@xmath0te@xmath1 , bi@xmath0se@xmath1 , sb@xmath0te@xmath1 , bi@xmath3sb@xmath4 alloy , and the cdte / hgte / cdte quantum well heterostructure . bi@xmath0te@xmath1 , bi@xmath0se@xmath1 , sb@xmath0te@xmath1 , and bi@xmath3sb@xmath4 alloy belong to the class of 3d topological insulators ( 3d ti )  @xcite and host bound states on their surface . these states are characterized by a linear hamiltonian and form a band - gap closing dirac cone on each surface at the @xmath5 point .    in this work , methods that alter the surface state dispersion through proximity induced ferromagnetism , superconductivity , and finite - size effects are described . alteration of the surface dispersion is driven in part by the need to utilize these materials as field effect transistors ( fets ) that uses the highly mobile zero - gap surface or edge states as the conducting channel . the zero band gap surface states though prohibit an easy turn - off mechanism and therefore inducing a band gap is essential for successful transistor operation . a key idea to open a band gap is destruction of time reversal symmetry  @xcite which is responsible for protecting the surface states . time reversal symmetry , which in principle , can be destroyed in the presence of an external perpendicular magnetic field  @xcite and appears to be a straightforward approach in a standard experimental set - up , may not be applied easily to a miniaturized device on account of electromagnetic interference with the surrounding circuitry . alternatively , the surface can be coated with a ferromagnet with an out - of - plane magnetization component to replicate this behaviour . it is now well established that doping with cu or pb can introduce superconducting states in bi@xmath0se@xmath1 . additionally , through the proximity effect at the interface between a superconductor ( sc ) and topological insulator , the superconductor s wave functions can penetrate the surface of a topological insulator and induce superconductivity . this induced superconductivity , by virtue of its intrinsic energy gap between the fermi - level and the superconducting ground state offers a possible way to open a band - gap in a topological insulator . a modified version of the bogoliubov - de gennes ( bdg ) hamiltonian for a 3d - ti and an _ s_-wave superconductor is used to compute the dispersion relationship of a ti - sc heterostructure . the paper is structured as follows : in section  [ meta ] , the 4-band k.p hamiltonian to compute the surface and edge state energy dispersion is presented . dispersion relations obtained through them are utilized later to investigate finite - size effects on surface and edge states of 3d and 2d tis respectively . the 4-band hamiltonian is modified via an exchange - interaction term to model proximity - induced ferromagnetism on the surface of a 3d ti . the ti - sc heterostructure is represented by a bdg hamiltonian and dispersion when the superconductor is an _ s_-wave or _ p_-wave type is worked out analytically . spin matrices using the basis states of the 4-band hamiltonian are set up to compute spin - polarization of surface states . numerical calculations based on the hamiltonians created for several ti systems in section  [ meta ] are collected in section  [ res ] . the paper concludes by summarizing key results in section  [ conc ] . surface and edge states in topological insulators are characterized by a linear dispersion and dirac cones . they further depend on dimensions and growth conditions of the structure that host them . low - energy continuum models for 3d and 2d tis used in deriving results included in this paper are described in this section . the dispersion relations of bi@xmath0te@xmath1 , bi@xmath0se@xmath1 , and sb@xmath0te@xmath1 films are computed using a 4-band k.p hamiltonian . the 4-band hamiltonian  @xcite is constructed ( eq .  [ eqn1 ] ) in terms of the four lowest low - lying states @xmath6 , @xmath7 , @xmath8 , and @xmath9 . additional warping effects  @xcite that involve the @xmath10 term are omitted in this low - energy effective hamiltonian . @xmath11    where @xmath12 , @xmath13 and @xmath14 . for bi@xmath0te@xmath1 and bi@xmath0se@xmath1 , the relevant parameters are summarized in table . [ table1 ] . [ table1 ]    lcc parameters & bi@xmath0te@xmath1 & bi@xmath0se@xmath1 + m@xmath15 ( ev ) & 0.30 & 0.28 + a@xmath16 ( ev  ) & 2.26 & 2.2 + a@xmath0 ( ev  ) & 2.87 & 4.1 + b@xmath16 ( ev @xmath17 ) & 10 & 10 + b@xmath0 ( ev @xmath17 ) & 57.38 & 56.6 + c ( ev ) & -0.18 & -0.0068 + d@xmath16 ( ev @xmath17 ) & 6.55 & 1.3 + d@xmath0 ( ev @xmath17 ) & 29.68 & 19.6 +    the dispersion for a 20.0 @xmath18 bi@xmath0se@xmath1 thick film which is approximately twenty quintuple - layers is shown in fig . the dirac cone is formed at an energy equal to 0.029 @xmath19 confirming that it is indeed a mid - gap state . the bulk band - gap of bi@xmath0se@xmath1 is approximately 0.32 @xmath19 at the @xmath20 point . in contrast to the thick - film dispersion , the band profile of a 3.0 @xmath18 ( approximately three quintuple - layers ) bi@xmath0se@xmath1 film has a finite band gap . in the case of a thin - film , the two surface states hybridize . the hybridization occurs because each state has a definite localization or penetration length . when the penetration length is comparable to film thickness , the opposite spin - resolved bands of the two surfaces will mix . since bands with identical quantum numbers can not cross , a gap(fig . [ fig1]b ) opens up at the @xmath20 point and the dispersion changes to dirac - hyperbolas . it must be stated here , that though a thin film on account of intrinsic hybridization , which is a function of dimension , is not a preferred choice to manipulate the band gap . the band gap control must be performed externally as described in the latter part of this paper . two variants of a topological insulator film are considered . the first is a free - standing symmetric thin - film of bi@xmath0se@xmath1 . two exactly degenerate dirac cones are formed for a symmetric film . the second thin film considered is assumed to grow on a substrate which induces an asymmetry between the two surfaces . the asymmetry can be explained by assuming that the top surface is exposed to vacuum while the bottom surface is fixed to a substrate . inversion symmetry therefore does not hold  @xcite for such a ti film . this asymmetry is simulated by adding a small symmetry - breaking potential to the hamiltonian along the confinement direction . it can also be viewed as a simple way of creating two chemically distinct surfaces . the dirac cones , one on each surface , are therefore no longer degenerate . they are now split as a function of the applied symmetry - breaking potential and positioned at distinct energies . this splitting is similar to a rashba - split  @xcite in presence of structural inversion asymmetry ( sia ) . the asymmetry of the surfaces , in this case , causes the sia . when the conditions of ultra - thin film and asymmetry are simultaneously fulfilled , it is observed that the two surface bands offer another instance of rashba - type splitting . to explain this phenomenon , first a low thickness film is considered . the two degenerate massless dirac cones in this case would mix and open a finite band - gap as explained above . each dirac cone can now be represented as a massive spin - degenerate dirac hyperbola . @xcite if an additional asymmetry is imposed through a potential , the spin degeneracy of each dirac hyperbola is broken and the bands split along @xmath21 in opposite directions . the degeneracy is only maintained at the @xmath20 point . the conventional rashba - type splitting follows exactly the same pattern , for example in an asymmetric gaas quantum well . @xcite    a hallmark of the surface bands of a 3d - topological insulator is their intrinsic complete spin - polarization and locking of the spin perpendicular to the momentum . within the framework of the 4-band k.p hamiltonian , expectation value of the three spin - polarized vectors is computed . the operators for the three spin - polarizations are given by eq . [ eqn2 ] and eq . [ eqn3 ] in the pauli representation @xmath22 @xmath23 . @xmath24    @xmath25    the above matrices are written under the basis set ordered as @xmath6 , @xmath7 , @xmath8 , and @xmath9 . the expectation value for each spin - polarization operator is calculated in the usual way in eq . (  [ eqn4 ] ) . @xmath26 where @xmath27 the spin - polarization shown in fig . [ fig2 ] is computed by choosing a @xmath28-vector of magnitude 0.01 @xmath29 at various polar angles.for each such vector , the spin - polarization obtained was perpendicular to momentum with a zero out - of - plane component . the red - arrows in fig . [ fig2 ] denote the direction of spin - polarization and is always tangential to the circular contour traced out by the @xmath28-vector . point for a 20.0 @xmath30 thick bi@xmath0se@xmath1 film . the spin is locked to momentum ( which is a radial vector on the circle ) shown by the tangential lines on the plot . ] 2d - tis which exhibit edge states are also highly mobile and suitable for nanoscale applications . @xcite one example of edge states in a 2d - ti is a cdte / hgte / cdte nanoribbon obtained by quantizing one of the edges of the corresponding quantum well . it would be useful to compare length scales up to which an edge and surface state can be sustained in a 3d and 2d topological insulator system . while an eight - band k.p hamiltonian describes the full set of six valence ( including spin split - off ) and two conduction bands and their mutual interaction through the off - diagonal terms , it is sufficient to focus on bands that exclusively take part in the inversion process  @xcite . this interaction of bands is governed by the coupling of conduction and valence states , represented through a linear term as shown in eq .  .    @xmath31    where @xmath32 and @xmath33 @xmath34 describes band bending and 2__m@xmath15 _ _ = -__e__@xmath35 corresponds to energy gap between bands and is negative in the inverted order bands . this hamiltonian is written in the basis of the lowest quantum well sub - bands @xmath36 , @xmath37 , @xmath38 , and @xmath39 . here @xmath40 stands for the two kramers partners . the sign of the gap parameter _ m _ determines if it is a trivial insulator ( _ m _ @xmath41 0 ) or a topological insulator ( _ m _ @xmath42 0 ) . experimentally , _ m@xmath15 _ is tuned by changing the quantum well width . the parameters _ a , b , c , d , and m@xmath15 _ are geometry dependent . for numerical calculations performed in this paper , the parameters were set to : @xmath43 , @xmath44 , @xmath45 , @xmath46 , and @xmath47 .      before a complete hamiltonian for a ti - superconductor heterostructure can be written , the conventional bcs description  @xcite of an _ s_-wave superconductor must be examined . the bcs hamiltonian in its simplest form can be written by starting with a hamiltonian ( eq .  [ bcs3 ] ) that describes a many - fermion system with a spin - independent interaction potential . such a hamiltonian in general form is expressed as : @xmath48 by restricting to paired fermions with zero total momentum and opposite spin , the bcs hamiltonian can be written as @xmath49    the spectrum of this hamiltonian when solved using the bogoliubov transformation  @xcite yields a band structure  @xcite with a gap in the spectrum . for studying proximity effect between a superconductor  @xcite and a topological insulator , the 4-band k.p model and the bcs hamiltonian is used in conjunction . the fundamental assumption of the bcs hamiltonian is the formation of cooper pairs which are paired electrons with zero momentum and spin . superconductivity which is induced on the ti side of the ti - sc heterostructure must therefore agree to this principle . at this point it is worth mentioning again that the four orbitals participating in the electronic bonding process are @xmath6 , @xmath7 , @xmath8 , and @xmath9 . the composite hamiltonian for the ti - sc structure similar to the bdg hamiltonian can now be written as @xmath50    in eq . [ bdgprimer ] , @xmath51 denotes the chemical potential and _ t _ is the time reversal operator . h@xmath52 is the composite hamiltonian and h@xmath53 represents the intrinsic 4-band k.p hamiltonian . in the composite hamiltonian , eq . [ bdgprimer ] , @xmath54 is the pair potential where @xmath55 is the order parameter and @xmath56 characterizes the strength of the attractive interaction potential as established in the bcs formulation . for the _ s_-wave superconductor considered here , the pair - potential , @xmath57 is isotropic and just a number . the analytic representation of pair - potential changes to a @xmath58 dependent quantity if _ p _ or _ d_-type superconductors are considered . the pair potential @xmath59 denotes a correlated pair of electrons with opposite momentum and spin . for the case of a ti , which is turned in to a superconductor , the orbitals with opposite spin and momentum are paired . the two sets of orbitals in the 4-band ti hamiltonian are therefore coupled by two pair potentials . the full ti - sc hamiltonian _ h@xmath52 _ in the basis set @xmath6 , @xmath7 , @xmath8 , @xmath9 , @xmath60 , @xmath61 , @xmath62 , and @xmath63 can be now written as    @xmath64    where @xmath12 , @xmath13 and @xmath14 .    in the above hamiltonian , the fermi - level @xmath51 has been set to zero . also , the pair potential @xmath65 is assumed for the zero - momentum fermion pair @xmath66 . for the corresponding , @xmath67 , the pair potential is @xmath68 . @xcite    a more accurate order parameter can be obtained through a self - consistent calculation . for instance the order parameter @xmath65 , is given as @xmath69 in eq . [ delta1 ] , @xmath70 and @xmath71 refer to the wave function components in the 8 @xmath72 1 column vector that correspond to @xmath6 and @xmath62 . details of the self - consistent calculation of the order parameter is discussed in section  [ res ] .    as a concrete demonstration of band - gap alteration , an _ s_-wave superconductor which is characterized by an isotropic energy - gap and conveniently represented as @xmath73 is considered . the surface states can be modeled through a dirac hamiltonian written as @xmath74 . adding a pair potential such that fermions with opposite spins and momentum are coupled , the hamiltonian takes a simple form @xmath75 for an s - type superconductor , the pair potential is independent of the @xmath76 , the hamiltonian given in eq . (  [ red_ham ] ) can now be solved to obtain a dispersion relationship of the form @xmath77 for a p - type superconductor , the pair potential is generally expressed as @xmath78 . the corresponding dispersion gives the following eigenvalues @xmath79 it is easy to see that at the @xmath5 point the pair potential is equal to zero and band gap closing is preserved . this is qualitatively different from the _ s_-wave case discussed earlier . @xcite      applied magnetic field represents a significant way to open a band gap in the spectrum near @xmath80 . the @xmath80 degeneracy is protected by time reversal symmetry which can be destroyed by a magnetic field . the applied magnetic field is incorporated in the hamiltonian through the standard peierls substitution @xmath81 where @xmath82 is the magnetic vector potential . the simplest way to add a magnetic field is layer the surface of a ti with a ferromagnet . the ferromagnetic proximity exchange field polarized along an arbitrary direction @xcite can be written in the most general way as @xmath83 . to illustrate the effect of a magnetic field , the spectrum of a ti slab under a @xmath84 field directed parallel and perpendicular to the surface is computed . for a parallel orientation , the states remain gap - less but are shifted along the in - plane direction perpendicular to the field . the surface states around each dirac cone are approximately described as @xmath85 this gives a linear dispersion relationship of the form @xmath86 . a @xmath87 field perpendicular to the surface ( 0,0,@xmath84 ) can be represented using a landau gauge @xmath88 . using the standard peierls transformation , the simple dirac hamiltonian is transformed to @xmath89 the eigen values are landau levels given as @xmath90 . a simpler dirac hamiltonian which captures the phenomenlogical behaviour of surface electrons in presence of the exchange field is shown in eq . [ dferro ] . @xmath91    however , for a surface parallel to the magnetic field , the same gauge can now be written as @xmath92 . using the same peierls transformation , the @xmath93 vector is now transformed to @xmath94 . the @xmath95 vector remains unchanged . for an x - y surface , _ z _ is a constant which means that @xmath96 is a constant too and commutes with @xmath93 . the new set of eigen values are therefore written as @xmath97 the only difference is that dirac cones now appear at @xmath98 . in other words , the eigen energies of the system are independent of the choice of gauge . two structures are considered to understand the geometric dependence of surface and edge states . 3d topological insulators with surface states are modeled as thin films of finite thickness along the _ z _ direction while 2d topological insulators exhibiting edge states are nano - ribbons with two dimensional confinement along _ y _ and _ z _ directions . it must be pointed out here that a nano - ribbon is constructed out of an inverted quantum well  @xcite to allow topological edge states . 2d tis are modeled using the bhz hamiltonian described in section  [ meta ] . for a bi@xmath0se@xmath1 film , it is observed that surface states are preserved for thickness as low as 9.0 @xmath30 while the corresponding edge states in a cdte / hgte / cdte 2d - ti nanoribbon requires a thickness of at least 100 @xmath2 . a shows a nanoribbon 100.0 @xmath18 wide and 8.0 @xmath18 thick . this structure shows a dirac crossing and topological edge state . [ fig3]b is 50.0 @xmath18 wide and 8.0 @xmath18 thick ribbon with a finite band gap . ] this suggests that much smaller samples that retain their conducting surface states can be fabricated with 3d tis than in the case of a cdte / hgte / cdte 2d - ti . the physical interpretation of these length scales can be sought in the delocalization length or penetration depth of these states . the penetration depth is usually expressed as @xmath99 , where _ m _ is the bulk band gap and @xmath100 is the fermi - velocity . by just noting the charge excitation gap , it is obvious that large difference in penetration depth can be easily explained . the bulk excitation gap of bi@xmath0se@xmath1 is much higher than that of hgte , which is a zero band gap compound . another result that supports this observation is the film thickness at which a gap opens up in a sb@xmath0te@xmath1 compared to bi@xmath0se@xmath1 . as a numerical example , the band gap opening in a 5.0 @xmath18 thin - film is 0.0084 @xmath19 and 0.0629 @xmath19 in bi@xmath0se@xmath1 and sb@xmath0te@xmath1 respectively . assuming comparable fermi velocities , sb@xmath0te@xmath1 which has a smaller bulk band - gap at @xmath20 , the surface states would be more delocalized than bi@xmath0se@xmath1 . as a result of greater delocalization , complete hybridization occurs and gives a larger band gap opening . the dependence of band - gap for various bi@xmath0se@xmath1 film - thickness is plotted in fig . [ fig4 ] . se@xmath1 film thickness . band gap opens because the two surfaces hybridize . ]      for a symmetric thin film with identical surfaces , the two dirac cones representing the surface states are degenerate . asymmetry due to two different surfaces though can exist on account of inequivalent surface termination or presence of a substrate . this naturally occurring asymmetry  @xcite can be reproduced by applying an electric field along the confinement direction . an example of a real arrangement of atoms in a bi@xmath0te@xmath1 thin film with bi and tellurium surface termination is shown in fig . [ fig5 ] . under ideal conditions ( ignoring impurity effects ) , a dipole is formed between the two surfaces which then host a single dirac cone as shown in fig . the separation of the dirac cones on each surface depends on the asymmetry induced by the potential . te@xmath1 thin film with two different surfaces . the two surfaces have bi and te termination thus making them chemically inequivalent ]    se@xmath1 thin film with built - in asymmetry . asymmetry in this film was artificially introduced by using a small potential along the confinement direction . the two dirac cones are now separated in energy ]    a more interesting dispersion relationship arises in the presence of an asymmetric potential in an ultra - thin film . this situation can be broken in to two parts and results obtained above can then be combined . for an ultra - thin film whose surfaces have hybridized , two massive dirac hyperbolas are created . each hyperbola is spin - degenerate as shown in fig . if film asymmetry impresses an electrostatic potential breaking inversion symmetry , the two dirac hyperbolas will now split in to four sets of dirac hyperbolas ( fig . [ fig7 ] ) . the spin degeneracy of the dirac - hyperbolas is only maintained at the @xmath20 because it happens to be one of the time - reversal - invariant - momenta ( trim ) points . @xcite    se@xmath1 film with asymmetry . the two dirac hyperbolas from the ultra - thin film in presence of asymmetry are now spin - split . they form four copies , two from conduction and valence band and maintain degeneracy only at the @xmath20 point ]    for a symmetric free - standing film , the spin - up and spin - down components of the in - plane spin - polarization are exactly anti - parallel . interestingly , if spin - polarization is measured in an asymmetric 3d - ti film , the two anti - parallel spin components are no longer of same strength . the amplitude of each spin - component is now a function of the asymmetry expressed as an electric potential . the two surfaces of a 3d - ti therefore have unequal spin - amplitudes . [ fig8 ] was obtained by computing the spin polarization for @xmath101 . the _ k_-vector is chosen close to the @xmath5 point to satisfactorily represent the dispersion in the dirac regime .          the surface band dispersion of a 3d ti is first presented followed by a self - consistent calculation of the order parameter . a 50.0 @xmath2 bi@xmath0se@xmath1 film was layered with an _ s_-wave superconductor . the superconductor is assumed to extend up to 25.0 @xmath2 . the remaining half of the slab is pristine bi@xmath0se@xmath1 and possesses regular 3d ti properties . s_-wave superconductor is assumed to have the material properties of aluminium  @xcite whose order parameter(@xmath102 in eq . [ bdg_full ] ) at _ t _ = 0 k is roughly equal to 0.34 @xmath103 . band dispersion of the surface states for the bi@xmath0se@xmath1 film coupled to the superconductor is shown in fig . [ fig9]a . since the superconductor extends only until half of the structure , the second surface still shows a dirac cone while the top surface has an open band gap . this is shown in the zoomed out fig . additionally , in this calculation , the chemical potential @xmath51 and the second order parameters @xmath104 has been set to zero . bi@xmath0se@xmath1 film when coated with an _ s_-wave superconductor . [ fig9]a shows the overall band dispersion while fig . [ fig9]b displays the energy dispersion around the dirac cones . the surface with no superconductor penetration has a ti surface state . ]    to compute the spatial profile of the order parameter self - consistently , @xmath105 is initially set to the bulk superconducting gap value for aluminium . with this value , the ti - sc hamiltonian(@xmath106)is diagonalized and a new order parameter @xmath107 using eq . [ delta1]is produced . in eq . [ delta1 ] , the function @xmath108 is set equal to the bulk superconducting energy gap . @xmath105 in the ti - sc hamiltonian(@xmath106 ) is set to @xmath107 and the process is iterated until convergence is achieved . the spatial profile shows that @xmath105 is significantly suppressed at 25.0 @xmath2 mark which is the ti - sc interface and drops to zero beyond that as expected since the superconductor extends only until the halfway mark . the interface therefore strongly distorts the superconductor wave function as demonstrated through this self - consistent calculation . the oscillations seen in the order parameter is due to finite discretization along the quantized _ z_-direction . for calculations presented here , a matrix of the size 2008 @xmath72 2008 was diagonalized . two spatial profiles are included for @xmath51 = 0 and @xmath51 = 0.2 @xmath19 . as evident , the chemical potential does not significantly impact the order parameter . calculated self - consistently at two different chemical potentials is plotted as a function of the quantized _ z_-axis . the bi@xmath0se@xmath1 film is 50.0 @xmath2 in thickness and superconductor is assumed to extend until 25.0 @xmath18 . ] an exchange field operational in a ferromagnet and with a component along the normal ( the normal is chosen to be along the _ z_-axis)to the ti film surface opens a band gap . a relatively thick slab of bi@xmath0se@xmath1 ( 10.0 @xmath2 in present calculation ) which ensures a decoupled top and bottom surface in presence of a ferromagnetic exchange field ( @xmath109 ) shows a band gap opening . a gap equal to twice the exchange energy appears and the bands acquire a parabolic character . the linear dispersion , though is preserved away from the @xmath5 point and the perpendicular spin - momentum locking , is observed again . for thin slabs which allow hybridization of the two surfaces , the dirac bands change to dirac hyperbolas as mentioned elsewhere in this paper . in presence of an exchange field , which breaks time reversal symmetry , and near _ k _ = 0 the bands split in two states with spin up energies @xmath110 and two spin down states @xmath111 . for an exchange energy @xmath112 , the gap induced in a thick ti slab as shown in fig . [ fig11]a is equal to 0.0585 @xmath19 . this split is roughly twice the chosen exchange energy . the corresponding split ( fig . [ fig11]b ) for a thin slab ( 3.0 @xmath2 ) is computed to be 0.0510 @xmath19 .     _ z_-directed exchange field . the exchange field gives different band gap openings for a 3.0 @xmath2 film ( left ) with hybridized surfaces in contrast to decoupled surfaces of a thicker 10.0 @xmath2 film . ]    finally , as a demonstration of the assertion made and analytically derived in eq . (  [ bfield_par ] ) , the axis of magnetization for the ferromagnet is assumed to lie along the _ y_-axis . the magnetic field equal to 10.0 tesla , in landau gauge is represented as @xmath113 where _ z _ is the set of discretized coordinates along the confined axis . this choice of magnetic vector potential can be easily verified through the standard @xmath87 = @xmath114 relation to represent a @xmath87 field along the _ y_-axis . further , it was also assumed that the ferromagnetic proximity effect extends only for the first three layers of the film . two dirac cones are formed as shown in fig . [ fig12 ] . one of the dirac cones which corresponds to the surface unaffected by the ferromagnet is at the @xmath115 point . ti slabs with an in - plane magnetic field ( along _ y_-axis ) of 10.0 tesla . the dirac cone shifted along _ k@xmath3_-axis comes from the ferromagnet coated surface . ] the second dirac cone which is from the top surface coated with a ferromagnet has shifted along the _ k@xmath3_-axis to -0.01403 nm@xmath116 . the energies of the two dirac cones are equal to 0.02914 @xmath117 . the valence and conduction band component are clearly identified in blue and red colour respectively in fig . [ fig12 ] . it is therefore imperative to have an out - of - plane magnetization for a finite band gap and destruction of topological states . the results presented above demonstrate the finite - sized effects on 2d and 3d topological insulator states . 3d ti films of thickness as low as 6.0 @xmath2 were found to support surface states . in contrast , a 2d ti cdte / hgte / cdte nanoribbon must be atleast 100.0 @xmath2 in thickness to prevent hybridization of edge states and open a band gap . the role of asymmetry in a 3d - ti was found to gproduce dirac hyperbolas which for a hybridized thin film is split similar to a rashba splitting . it was also shown that influence of an external magnetic field can be emulated by layering the surface of a 3d ti with a ferromagnet . the important condition to note is that the axis of magnetization must have a component perpendicular to the surface . a ferromagnet whose magnetic axis lies in the plane of the surface will shift the dirac cone and no band gap opening is observed . finally , coating a 3d - ti with an _ s_-wave superconductor leads to a band gap on account of the well - known proximity effect . the band - gap opening depends on the strength of the order parameter which was computed self - consistently in the calculations shown . one of us ( ps ) wishes to thank chandan setty from the dept . of physics at purdue university for several illuminating discussions on superconductors . we also acknowledge support from intel corp . during the early stages of this work .
bi@xmath0te@xmath1 and bi@xmath0se@xmath1 are well known 3d - topological insulators . 
 films made of these materials exhibit metal - like surface states with a dirac dispersion and possess high mobility . 
 the high mobility metal - like surface states can serve as channel material for ti - based field effect transistors . 
 while such a transistor offers superior terminal characteristics , they suffer from an inherent zero band gap problem . 
 the absence of a band gap for the surface states prevents an easy turn - off mechanism . in this work , techniques that can be employed to easily open a band gap for the ti surface states 
 is introduced . 
 two approaches are described : 1 ) coating the surface states with a ferromagnet which has a controllable magnetization axis . 
 the magnetization strength of the ferromagnet is incorporated as an exchange interaction term in the hamiltonian . 
 2 ) an _ s_-wave superconductor , because of the proximity effect , when coupled to a 3d - ti opens a band gap on the surface . 
 this ti - superconductor heterostructure is modeled using the bogoliubov - de gennes hamiltonian . 
 a comparison demonstrating the finite size effects on surface states of a 3d - ti and edge states of a cdte / hgte / cdte - based 2d - ti is also presented . 
 3d - ti nanostructures can be reduced to dimensions as low as 10.0 @xmath2 in contrast to 2d - ti structures which require a thickness of at least 100.0 @xmath2 . 
 all calculations are performed using the continuum four - band k.p hamiltonian .