Conductance features of core-shell nanowires determined by the internal geometry
Miguel Urbaneja Torres, Anna Sitek, Sigurdur I. Erlingsson, Gunnar Thorgilsson, Vidar Gudmundsson, Andrei Manolescu
CConductance features of core-shell nanowires determined by the internal geometry
Miguel Urbaneja Torres, Anna Sitek,
1, 2
Sigurdur I. Erlingsson, Gunnar Thorgilsson, Vidar Gudmundsson, and Andrei Manolescu School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,Wroclaw University of Science and Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wroclaw, Poland Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
We consider electrons in tubular nanowires with prismatic geometry and infinite length. Sucha model corresponds to a core-shell nanowire with an insulating core and a conductive shell. Ina prismatic shell the lowest energy states are localized along the edges (corners) of the prism andare separated by a considerable energy gap from the states localized on the prism facets. Thecorner localization is robust in the presence of a magnetic field longitudinal to the wire. If themagnetic field is transversal to the wire the lowest states can be shifted to the lateral regions ofthe shell, relatively to the direction of the field. These localization effects should be observable intransport experiments on semiconductor core-shell nanowires, typically with hexagonal geometry.We show that the conductance of the prismatic structures considerably differs from the one of circularnanowires. The effects are observed for sufficiently thin hexagonal wires and become much morepronounced for square and triangular shells. To the best of our knowledge the internal geometryof such nanowires is not revealed in experimental studies. We show that with properly designednanowires these localization effects may become an important resource of interesting phenomenology.
I. INTRODUCTION
The fabrication of various types of nanostructures al-lows for the design of systems with controllable propertiesof the electronic states. Amongst such systems are core-shell nanowires, which are radial heterojunctions of two,or even more, different semiconductor materials. Typi-cally a central material (core) is surrounded by an outerlayer (shell). The length of such nanowires is usually ofthe order of microns, whereas the diameter is betweena few tens to a few hundreds of nanometers. An in-teresting aspect of this structure is the transversal ge-ometry. Semiconductor core-shell nanowires, most oftenbased on III-V materials, are almost always prismatic,and rarely cylindrical. The typical shape of the crosssection is hexagonal [1–6]. Interestingly, other prismaticgeometries can also be achieved, like square [7] or eventriangular [8–13]. The present art of manufacturing al-lows even for etching out the core and obtaining prismaticsemiconductor nanotubes with vacuum inside [1, 3].In general, the specific geometry of the core-shellnanowires has not been studied much experimentally. Itis either seen as a natural outcome, when the nanowiresare hexagonal, or as a curiosity, when they are of differ-ent shapes. The main interest is rather related to thematerials used, the quality of the nanowires (like de-fect free), a specific size of the core or the shell, spe-cific length, etc. Still, in particular, the triangular wiresshowed very interesting features such as a broad range ofemitted wavelengths at room temperature [9, 14] or diodecharacteristics[12], but such properties were not really as-sociated with the shape of the cross section and/or withparticle localization. Remarkably, triangular nanowirespossess intrinsic polarization effects that may break thethree-fold geometric symmetry of the cross-section [15].Theoretically, the prismatic geometry of the nanowire, and especially of the shell itself, is a unique, and a veryimportant feature, which can lead to very interesting andrich physics. If the materials, the geometry, the doping,and the shell thickness are properly adjusted, the shellbecomes a tubular conductor with edges, and each edgemay behave like a quasi-one-dimensional channel. A pio-neer theoretical work has shown that carriers in prismaticshells can form a set of quasi-one-dimensional quantumchannels localized at the prism edges [16]. More recentlyit has been predicted that such a prismatic shell can hostseveral Majorana states which may interact with eachother [17, 18]. Another recent prediction is that a mag-netic field transversal to a tubular nanowire, either madeof normal semiconductors, or of a topological insulatormaterial, may induce the sign reversal of the electric cur-rent generated by a temperature gradient [19–21].In this paper we consider prismatic tubes as models ofcore-shell nanowires with an insulating core and a con-ductive shell. Our models are also applicable to uni-form nanowires (i.e. not core-shell) which have conduc-tive surface states due to the Fermi level pinning [22].However, we are not addressing the nanowires built fromtopological materials, but we use in our calculations aSchr¨odinger Hamiltonian. In our models the cross sectionof the prismatic shell is a narrow polygonal ring (hexag-onal, square, or triangular), i.e., with lateral thicknessmuch smaller than the overall diameter of the nanowire.In this geometry the electrons with the lowest energiesare localized in the corners of the polygon and the elec-trons in the next layer of energy states are localized onthe sides [23–28]. The corner and side states are energet-ically separated by an interval that depends on the geom-etry and on the aspect ratio of the polygon, it increaseswith decreasing the shell thickness or the number of cor-ners, and it can become comparable or larger than theenergy corresponding to the room temperature [29, 30]. a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Hence, such structures can contain a well-separated sub-space of corner states, with sharp localization peaks, andpotentially robust to many types of perturbations. Onthe contrary, if the shell is relatively thick with respect tothe diameter of the wire, the corner localization broadensand the polygonal structure has less effect on the electrondistribution [31].To the best of our knowledge, experimental resultswith features related to the internal geometry of thenanowire do not exist, or are very rare. One investi-gation, using inelastic light scattering, indicated the co-existence of one- and two-dimensional electron channels,along the edges and facets, respectively, of GaAs core-shell nanowires [32]. In transport experiments one canmention here the detection of flux periodic oscillations ofthe conductance in the presence of a magnetic field lon-gitudinal to the nanowire [33], which indicates the radiallocalization of the electrons in the shell. Or, flux peri-odic oscillations with the magnetic field perpendicular tothe nanowire, due to the formation of snaking states onthe sides of the tubular conductor [22, 34]. Nevertheless,experimental results indicating the presence of corner orside localized states, or the energy gap between them, orother details implied by the prismatic geometry of theshell, are not reported.The intention of this paper is to predict specific fea-tures of the conductance of core-shell nanowires whenthe electronic transport occurs within the shell, deter-mined by the prismatic geometry of the nanowire. Suchfeatures, which could be experimentally tested, can re-veal to what extent the electronic states are influencedby the polygonal geometry, and if not, to what extent thequality of the nanowire needs to be improved in order toachieve a robust corner localization.Next, in Section II we describe our model and method-ology, in Section III we discuss the transverse modes andthe expected conductance steps, in Section IV we con-sider a magnetic field longitudinal to the nanowire, inSection V a perpendicular magnetic field and finally inSection VI we summarize the conclusions.
II. MODEL AND METHODS
We analyze a system of non-interacting electrons,confined in a prismatic shell with polygonal cross sec-tion, in the presence of a uniform magnetic field B =( B x , B y , B z ). The axes x and y are chosen perpendicularto the shell and the axis z is longitudinal. The Hamilto-nian can be decomposed as H = H t + H l + H s , (1)where the terms H t , H l , and H s correspond to the trans-verse, the longitudinal, and the spin degrees of freedom,respectively.The transverse Hamiltonian is, H t = ( − i (cid:126) ∂ x + eA x ) + ( − i (cid:126) ∂ y + eA y ) m eff − e E · r , (2) -1-0.500.51 -1 -0.5 0 0.5 1(a) y [ un it s o f R e x t ] x [units of R ext ] R ext t-1-0.500.51 -1 -0.5 0 0.5 1(a) y [ un it s o f R e x t ] x [units of R ext ] R ext t-1-0.500.51 -1 -0.5 0 0.5 1(a) y [ un it s o f R e x t ] x [units of R ext ] R ext t-1-0.500.51 -1 -0.5 0 0.5 1(a) y [ un it s o f R e x t ] x [units of R ext ] R ext t -1 -0.5 0 0.5 1(b)x [units of R ext ]-1 -0.5 0 0.5 1(b)x [units of R ext ]-1 -0.5 0 0.5 1(b)x [units of R ext ]-1 -0.5 0 0.5 1(b)x [units of R ext ] -1 -0.5 0 0.5 1(c)x [units of R ext ]-1 -0.5 0 0.5 1(c)x [units of R ext ]-1 -0.5 0 0.5 1(c)x [units of R ext ]-1 -0.5 0 0.5 1(c)x [units of R ext ] FIG. 1. The cross sections of the prismatic shells are definedby applying boundaries on a circular ring discretized in polarcoordinates. Only the lattice points inside the polygonal shell(shown in yellow) are used in the transverse Hamiltonian (1),where we can include a chosen percentage of random-strengthimpurities (purple points). R ext and t indicate the nanowireradius and shell thickness, respectively. For clarity the figuresshow only a subset of lattice points, whereas the numericalcalculations are performed with 6000-10000 points, dependingon the polygon: (a) hexagon, (b) square, (c) triangle. where m eff is the effective electron mass in the shell mate-rial, A = ( A x , A y , A z ) = ( − yB z / , xB z / , yB x − xB y )is the vector potential, and E = ( E x , E y ,
0) is an externalelectric field perpendicular to the wire. The transverseHamiltonian depends only on the longitudinal magneticfield B z .Technically, the Hamiltonian (2) is restricted to a lat-tice of points that covers the cross section of the shell. Inorder to define the lattice we begin with a circular diskwhich is discretized in polar coordinates [35]. Next, weenclose the polygonal shell within this area and excludeall lattice points situated outside the shell, as shown inFig. 1. This method allows us to describe both symmet-ric and non-symmetric polygonal shells without the needof adapting the background grid geometry to the spe-cific polygon or redefining Hamiltonian matrix elements[29, 30]. As a second method we also used the Kwantsoftware [36] and obtained the same numerical results forthe matrix elements, this time using a triangular grid.The longitudinal Hamiltonian is H l = ( − i (cid:126) ∂ z + eA z ) m eff , (3)which depends on the magnetic field transverse to thenanowire B ⊥ = ( B x , B y ), which can be chosen at differ-ent angles relatively to the corners or sides of the shell.And finally the spin Hamiltonian is H s = − g eff µ B σ · B , (4)where g eff is the effective g factor, µ B is Bohr’s magneton, σ = ( σ x , σ y , σ z ) denote the Pauli matrices, and B is thetotal magnetic field.In our numerical calculations we first calculate theeigenstates of the transverse Hamiltonian, H t | a (cid:105) = (cid:15) a | a (cid:105) ( a = 1 , , , ... ), and obtain the corresponding eigenvec-tors in the position representation, | a (cid:105) = (cid:80) q ψ ( q, a ) | q (cid:105) ,where | ψ ( q, a ) | , is the localization probability on thelattice site q = ( x q , y q ). Next, we retain only a setof low-energy transverse modes, and together with theplane waves in the z direction | k (cid:105) = exp( ikz ) / √ L , k being the wave vector and L the (infinite) length ofthe nanowire, and with the spin states s = ±
1, weform a basis in the Hilbert space of the total Hamil-tonian, | aks (cid:105) . In the absence of the magnetic fieldtransverse to the wire ( B ⊥ = 0), the kets | aks (cid:105) areeigenvectors of the total Hamiltonian, with eigenvalues E aks = (cid:15) a + (cid:126) k / m eff − g eff µ B sB z . If B ⊥ (cid:54) = 0 the trans-verse motion becomes dependent on k , and then we di-agonalize the total Hamiltonian for a discretized series of k values, to obtain its eigenvalues E mks ( m = 1 , , , ... ),and its eigenvectors | mks (cid:105) expanded in the basis | aks (cid:105) .The next step of our calculations is to evaluate theelectric current along the nanowire in the presence of avoltage bias. The operator describing the contributionof an electron which is localized at point r , to the totalcharge-current density observed in a spatial point r , isdefined as j ( r , r ) = e δ ( r − r ) v + v δ ( r − r )] , where δ ( r − r ) is the particle-density operator and v ( r ) = i (cid:126) [ H, r ] the velocity operator [37]. In ourcase we need only the component along the nanowire, v z = ( p z + eA z ) /m eff . We can define the expected valueof the total charge current flowing in the positive or nega-tive direction along the nanowire (i.e. in the z direction)as I ± = (cid:90) (cid:34)(cid:88) mks F (cid:18) E mks − µ ± k B T (cid:19) (cid:104) mks | j ( r , r ) | mks (cid:105) (cid:35) d r . (5)The integration is performed over the cross section ofthe shell, practically as a summation over all lattice sites r q = ( x q , y q ), whereas the scalar product included in thesquare brackets is an integration over the electrons po-sition r . F ( w ) = 1 / [exp( w ) + 1] represents the Fermifunction, T is the temperature, and k B Boltzmann’s con-stant. Here µ ± are the chemical potentials associatedwith electrons having positive or negative velocity alongthe nanowire. Obviously, in equilibrium µ + = µ − thecorresponding currents compensate each other and thetotal current I = I + − I − is zero.To generate a current along the nanowire we create animbalance between the states with positive velocity, i.e., ∂E mks /∂k >
0, and negative velocity, i.e., ∂E mks /∂k <
0, by considering in Eq. (5) different chemical potentials, µ + and µ − , respectively. The current is thus driven alongthe nanowire by the potential bias eV = µ + − µ − . Thisprocedure, well established in ballistic transport theory[38], allows us to calculate the I − V characteristic andthe conductance G = I/V in the small bias limit.To include the effect of disorder on the conductance weconsider a nanowire with a scattering region of a finitelength containing a random distribution of impurities,and we assume elastic electron-impurity collisions. In this case the impurities can be represented by an extrarandom term in the total Hamiltonian (1). The currentand the conductance can be calculated using the wellknown concept of transmission function. Here we com-pute the transmission function by using the Fisher-Leeformula [39] and the recursive Green’s function method[40]. A summary of the method for cylindrical geometrycan be found in Ref. [19] (Supplemental Material).We also address in our paper the case of a longnanowire, much longer than the scattering length, whenthe transport is far from ballistic. In principle thiscase can be treated with the recursive Green’s functionsmethod, by including a large number of impurities, butit becomes computationally very expensive. As an alter-native approach we will consider in this case a nanowireof infinite length, and we will use the traditional Kuboformula, which allows one to calculate the conductivityby performing a statistical average over impurity config-urations [41]: σ zz = he V (cid:90) dE (cid:20) − ∂ F ∂E (cid:21) × (cid:88) m k s m k s | (cid:104) m k s | v z | m k s (cid:105) | A m k s ( E ) A m k s ( E ) , (6)where V is the volume of the conductor and A mks ( E ) isthe spectral function which represents the broadening ofthe energy level E mks due to the disorder. In this studywe model the spectral function for each energy level E nks as a Gaussian function, A mks ( E ) = 1 √ π Γ e − ( E − Emks )22Γ2 , (7)whose width parameter Γ represents the disorder energy.This model of spectral function was used in the past fordescribing the impurity effects in the two-dimensionalelectron gas [42], but also more recently to incorporatethe effect of the electrodes on the density of states inmolecular nanowires [43].In the next sections we will show results for the con-ductance of core-shell nanowires, related to their internalgeometry, in different situations, using the computationalmethods mentioned above. III. TRANSVERSE MODES FOR ZEROMAGNETIC FIELD
In all the following examples the external radius of thepolygonal shell, which is the distance from the centerto one corner, as indicated in Fig. 1(a), is R ext = 30nm, whereas the side thickness t varies between 2-8 nm.The numerical calculations were performed for InAs bulkparameters, which are m eff = 0 . m e ( m e being theelectron mass of a free electron), and g eff = − . FIG. 2. Single-particle probability distributions associatedwith the low-energy transverse modes of (a) hexagonal, (b)square, and (c) triangular shells. The top figures illustrate thecorner states, which include the ground state, and the bot-tom figures illustrate the next, higher-energy group of modes,localized on sides. The energy separation between corner andside modes is indicated in Fig. 3.
Because of the polygonal cross section of the shell thelowest transverse modes are localized in the corners ofthe polygons while the probability distributions corre-sponding to higher modes have maxima on the sides [29],as illustrated in Fig. 2. In three dimensions the cornerstates form 1D conductive channels along the edges ofthe prismatic shell [28]. For each polygon there are 2 N corner states, where N = 6 , , t = 6 nm shell thick-ness these states fit into the intervals of 15.3 meV for thethe hexagon, 3.1 meV for the square, and 0.2 meV for thetriangle (measured from the ground state). For each ge-ometry, above the group of corner states, there is anothergroup of 2 N states, localized on the the sides of the poly-gons (the lower part of Fig. 2). For the present radius(30 nm) and side thickness (6 nm) the energy dispersionwithin each group of corner and side states is exceededby the energy interval (or gap) between these groups,which are ∆ h = 21 . s = 34 . t = 87 . t (cid:28) R ext , one expects the transverse energies ε m = (cid:126) m / m eff R , with m = 0 , ± , ± , ... the quantumnumber of the angular momentum, i.e., with energy inter-vals uniformly increasing as m . In addition, in the cylin-drical case (not shown in Fig. 2), at zero magnetic field,the transverse modes are four fold degenerate, both spinand orbital, except the ground state which is only twofold, spin degenerate. For symmetric polygons, becauseof the reduced, discrete symmetry, the orbital degener-acy is lifted between consecutive groups of 2 N transversestates. In particular, for symmetric shells, the sequenceof degeneracy orders (2-fold or 4-fold) of the corner/side ∆ t ∆ s ∆ h G [ G ] µ -E [meV]CircleHexagonSquareTriangle FIG. 3. Conductance steps expected for tubular nanowires ofcircular and polygonal cross sections, versus the chemical po-tential µ adjusted by subtracting the the ground state energy E . ∆ h , t , s indicate the energy separation between the cornerand side modes for hexagon, square, and triangle, respectively.The conductance is represented in units of G = e /h . Here t = 0 . R ext = 6 nm states, with increasing the energy, is 2442/2442 for thehexagon, 242/242 for the square, and 24/42 for the tri-angle.In Fig. 3 we compare the conductance steps expectedin the cylindrical and in the three prismatic geometries,when the chemical potential µ is varied. Here we assumeballistic transport and symmetric shells. The conduc-tance is derived with Eq. (5), using a small bias andtemperature T = 1 K. The steps obtained for circularand hexagonal structures are visibly different, with a ex-tra plateau at 12 G for the hexagon, ( G = e /h ) cor-responding to the gap between corner and side states.These plateaus increase dramatically for the triangularand square geometries, at 8 G and at 6 G , respectively.For the hexagonal shell the corner states are visible assteps at 2 G , 6 G , and 10 G , similar to the circular case.For the square, instead, we see only some shoulders in-dicating the corner states at 2 G and 6 G , and for thetriangular case they are not resolved.In order to obtain the conductance of a nanowire withimpurities we consider scattering centers, with randomcharacteristic energy between 0 and 0.5 meV, at randomlocations within the tubular nanowire. We calculate theconductance, this time with the recursive Green’s func-tion approach, for the example of the triangular geome-try, at zero temperature. The results are shown in Fig.4 for a nanowire of 54.2 nm length with different im-purity concentrations corresponding to a mean distancebetween the nearest neighbors of 2.7 nm, 1.6 nm and 1.2nm. The impurity configurations are fixed in these cases,such that this situation corresponds to a specific meso-scopic sample with individual randomness. The impurityconcentration is increased until the largest plateau, cor-responding to the gap between the corner and side states,becomes nearly indistinguishable. This impurity concen-tration can offer a hint on how dirty a nanowire can be G [ G ] Energy [meV]Clean λ = 2.6 nm λ = 1.7 nm λ = 1.2 nm FIG. 4. Conductance steps expected for the triangularnanowire for different impurity concentrations characterizedby the mean distance λ between the nearest neighbors. Theassociated potentials are repulsive and of a random strengthbetween 0 and 0.5 meV. G [ G ] µ -E [meV]t = 2 nmt = 4 nmt = 6 nmt = 8 nm 01020304050600 2 4 6 8 10 ∆ h [ m e V ] t [nm] FIG. 5. Dependence of the conductance plateaus on the sidethickness t for the hexagonal shells. Decreasing the side thick-ness leads to a significant increase of the plateau between thecorner and side states, at 12 G , and to a decrease of all theother plateaus. such that the conductance steps cannot be detected inthe experiments.We now return to the clean nanowire to discuss theeffect of the side thickness on the hexagonal geometry.With the parameters used in Fig. 3 the plateaus corre-sponding to side modes, i.e. above 12 G , are comparableor larger than ∆ h . However, the relative magnitude ofthe energy intervals, i.e., of ∆ h relatively to the disper-sion of the corner and side states, increases with reducingthe aspect ratio of the polygon, i.e., t/R ext [30]. As a re-sult, with reducing the thickness parameter t , while R ext is kept constant, the plateau at the transition betweencorner and side states becomes much more prominent.At the same time, the ratio between ∆ h and the otherenergy intervals rapidly increases such that the plateausin the corner domain become negligible relatively to themain one of width ∆ h , as shown in Fig. 5.In these examples we assumed the geometric symme- G [ G ] E = 0 meV/RE = 0.25 meV/RE = 0.5 meV/R G [ G ] µ -E [meV]E = 0 meV/RE = 0.25 meV/RE = 0.5 meV/R FIG. 6. Conductance steps in the presence of an asymmetryinduced by an electric field perpendicular to the nanowire,along the x -axis, for (a) the hexagonal case and (b) the cylin-drical one for comparison. The orbital gaps are split and thegap between corner and side states for the hexagonal geom-etry is reduced, yielding to shorter plateaus at 12 G . Here t = 0 . R ext = 6 nm try of the nanowires. Nevertheless, shells with perfectpolygonal symmetry cannot be fabricated, such that thesymmetric case should be considered only for reference.Geometric imperfections, random impurities, or externalelectric fields (gates) remove the orbital degeneracies. Inorder to account for asymmetry we consider an electricfield transversal to the wire, included in Eq. 2. In prac-tice, this is a way to model also the effect of a lateralgate attached to the nanowire, or of the substrate wherethe nanowire is situated, typically used to control thecarrier concentration with a voltage [29]. Assuming thisvoltage is not very large, such that the electrons are stilldistributed over the entire shell (and not simply crowdednear the gate), new plateaus can be obtained between thesplit orbital states, as shown in Fig. 6, where we comparethe results obtained for a cylindrical and a hexagonalnanowire.The electric field changes the electron density distri-bution by favoring some corner or sides over the others,depending on its orientation. This allows for the con-trollability of the electron localization within the shell.Although the first plateaus are similar for both the cylin-drical and the hexagonal case the main difference arisesfrom the orbital degeneracy pattern of the hexagonalnanowire, which is split at 12 G , separating corner fromside states (2442/2442). It is also interesting to observethat, in the cylindrical nanowire, the electric field ap-plied needs to be progressively stronger in order to splitthe orbital degeneracy of higher energy levels, as shownin Fig. 6(b). Additionally, for the hexagonal wire, rotat-ing the direction of the electric field leads to a change inthe secondary plateaus but leaves the main one (at 12 G )unaffected (not shown). IV. EFFECTS OF A LONGITUDINALMAGNETIC FIELD
We consider now a magnetic field in Eq. (2), longitu-dinal to the nanowire and its effects on the transversemodes. For a circular nanowire with t (cid:28) R ext a periodicenergy spectrum vs. the magnetic field is expected, withperiod of one flux quantum in the cross sectional area.Such flux-periodic oscillations, related to the Aharonov-Bohm interference, have been observed experimentally onInAs/GaAs hexagonal core-shell nanowires with t = 25nm and R ext ≈
75 nm [33]. The oscillations had ad-ditional modulations that were attributed to impurities,or subsequently to the spin splitting, by using a circularnanowire model of zero thickness [44].Indeed, the presence of the spin disturbs or breaks theflux periodicity of the transverse modes when the mag-netic field increases, as seen in Fig. 7(a) for a circularshell. In addition, the flux periodicity is also disturbedby the shell thickness. Flux periodic energy spectra forthin polygonal shells, without spin effects, have alreadybeen obtained by other authors [16, 45, 46].In Fig. 7(b) we show the energies of the transversemodes for our hexagonal shell. The energy gap betweenthe corner and side states decreases when the magneticfield is increased, but, remarkably, still a large value sur-vives at high magnetic fields, in our case above 10 T. Thereason is that the orbital energy of the states localizedin corners is significantly reduced compared to states ofhigher energies, an effect that becomes more pronouncedin the square geometry, Fig. 7(c). In fact, the differencesreveal the symmetry reduction from circular to hexagonalor square shapes. In these examples the orbital degree offreedom of the corner states is provided by their mutualcoupling via tunneling across the polygon sides. Instead,in the triangular case, where the corner localization isthe strongest, the tunneling is suppressed, and the or-bital motion of the corner states is completely frozen,such that only the spin Zeeman energy is observable forthe corner states in Fig. 7(d). The same can also happenfor thinner square and hexagonal shells.The conductance steps expected in the presence of alongitudinal magnetic field would develop according tothe evolution of the transverse modes when increasingthe field, by lifting both spin and orbital degeneracies.It is worth noting that, up to 10 T, the gap separatingcorner and side states prevails for the three polygonalshells, with the geometric parameters used, and that anew gap separating the corner states with different spinis created as a consequence of the Zeeman splitting forthe triangular and square nanowires, where the orbital E - E [ m e V ] B [T] -20020406080 0 2 4 6 8 10(b) ∆ h B [T]-50510 0 2 4 6 8 10(c) ∆ s E - E [ m e V ] B [T]354555 -50510 0 2 4 6 8 10(d) ∆ t B [T]8090100
FIG. 7. Evolution of the transverse energy modes with mag-netic field for the circular shape (a) and for the three polygo-nal shapes: (b) hexagonal, (c) square and (d) triangular. Thered color stands for the spin down levels and the blue for thespin up ones. Here t = 0 . R ext . Note that, in order to includeboth corner and side states in the graphs, the vertical axes inpanels (c) and (d) are broken. degree of freedom is most restricted. V. EFFECTS OF A MAGNETIC FIELDPERPENDICULAR TO THE NANOWIREA. Charge and current distributions
A magnetic field orthogonal to the nanowire axis,which can be incorporated in the longitudinal Hamilto-nian, Eq. (3), creates a second localization mechanism,in addition to the localization imposed by the polygonalgeometry. This type of magnetic localization has beenstudied in recent years for the cylindrical shell geometryby several authors [47–51]. If the electrons are confinedon a cylindrical surface, their orbital motion is governedby the radial component of the magnetic field. Assuminga magnetic field uniform in space, its radial componentvanishes and changes sign along the two parallel lines onthe cylinder situated at ± ◦ angles relatively to the di-rection of the field, and if the field is strong enough theelectrons have snaking trajectories along these lines. Atthe same time, at angles 0 ◦ and 180 ◦ the electrons per-form closed cyclotron loops. This localization mechanismleads to an accumulation of electrons on the sides of thecylinder, where the snaking states are formed, and to thedepletion the regions hosting the cyclotron orbits [49].In a thin prismatic shell the geometric and magneticlocalization of the electrons coexist. Therefore, the dis-tribution of electrons is expected to depend on the orien-tation of the magnetic field relatively to the prism edges (a)-2×10 -5 -1.5×10 -5 -1×10 -5 -5×10 -6
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0 5×10 -6 -5 -5 -5 -2×10 -5 -1.5×10 -5 -1×10 -5 -5×10 -6
0 5×10 -6 -5 -5 -5 -2×10 -5 -1.5×10 -5 -1×10 -5 -5×10 -6
0 5×10 -6 -5 -5 -5 FIG. 8. The color scale shows the change of the electrondensity, in units of nm − , to the orientation of the magneticfield, of magnitude B = 2 T, indicated by the arrows, rela-tively to the three polygonal shells: (a) hexagonal, (b) square,(c) triangular. The magnetic field points to the corners inthe left column, it is perpendicular to the sides in the rightcolumn, and half way in between these two orientations inthe middle column. In each case the electron density for B = 0 is subtracted. The average electron density is fixedto 1 . × − nm − . The geometry parameters are R ext = 50nm and t = 10 nm. (a)-4 -3 -2 -1 0 1 2 3 4-4 -3 -2 -1 0 1 2 3 4-4 -3 -2 -1 0 1 2 3 4(b)-4 -3 -2 -1 0 1 2 3 4-4 -3 -2 -1 0 1 2 3 4-4 -3 -2 -1 0 1 2 3 4(c)-4 -3 -2 -1 0 1 2 3 4-4 -3 -2 -1 0 1 2 3 4-4 -3 -2 -1 0 1 2 3 4 FIG. 9. The current density in equilibrium (no potential bias)for the same cases as in Fig. 8 above. The color scale is inunits of nA/nm or facets, as in the examples shown in Fig. 8. Here weconsider a carrier concentration of 1 . × cm − asreported in the recent literature [22]. In each case thecolor scale represents the difference between the carrierconcentration at B = 2 T and at B = 0 for the three ge-ometries. With the present parameters these differencesare somewhere up to 10%. Furthermore, for each geom-etry, the local changes of the density when the angle ofthe magnetic field is varied, are up to 1% only. Still, aswe shall see, this variation should be sufficient to haveimplications on the transport data.Next, in Fig. 9 we show the current distributions inthe equilibrium states, i.e. in the absence of a biasedchemical potential ( µ + = µ − ). In the absence of themagnetic field the current distribution is zero everywherein the shell, i.e. the positive currents and negative cur-rents compensate in each point. In the presence of themagnetic field, as expected, the Lorentz force may createlocal currents, as shown in the figure, which no longercompensate locally, but indeed the integrated current iszero. The local currents are in fact loops along the z axisof the nanowire, closing up at infinity. Still, it is inter-esting to observe the compensation of these loops. In thecases where the magnetic field points along the directionof one of the symmetry axes of the cross section the loopsare compensated, with the same current flowing in bothdirections, and the channels are paired with the ones onthe opposite side of the geometric symmetry axis relativeto the magnetic field direction. The pairing can happenwithin the same corner or side or on opposite ones. Inthe cases where the magnetic field does not point alongone of the symmetry axes the loops are no longer com-pensated and the current traveling in both directions isnot the same. Instead, the compensation occurs whenadding the loop on the opposite side of the shell. B. Energy spectra and conductivity
The energy dispersion with respect to the wave vector k , corresponding to the situations shown in Figs. 8 and 9,can be seen in Fig. 10. The dashed horizontal lines indi-cate the chemical potential corresponding to the selectedcarrier concentration. In all cases, in the absence of themagnetic field the position of the chemical potential issomewhere at the level of the side states. The presenceof the magnetic field mixes the corner and the side statesand leads to quite complex changes in the spectra and inthe charge or current distributions. And, as we can see,the energy dispersions are also sensitive to the orientationof the field relatively to the prismatic shell.Note that, in the triangular case, the spectra may notbe symmetric (even) functions of the wave vector k ifthe magnetic field is not aligned with a symmetry axis ofthe shell, like in the second example of Fig. 10(c), cor-responding to the second example of Figs. 8(c) and 9(c).In this case the magnetic field is parallel to the side ofthe triangle. The sign reversal of the magnetic field leads (a) E n e r gy ( m e V ) (b) E n e r gy ( m e V ) (c) E n e r gy ( m e V ) k (nm -1 ) 0 10 20 30 40 50 60 -0.2 -0.1 0 0.1 0.2k (nm -1 ) 0 10 20 30 40 50 60 -0.2 -0.1 0 0.1 0.2k (nm -1 ) FIG. 10. Energy spectra obtained by varying the angle of theperpendicular magnetic field, as in Fig. 8: (a) hexagon, (b)square, (c) triangle. In each case the energy of the groundstate is considered zero. The temperature is 1 K. to the sign reversal of the wave vector, i.e., to the sameenergy spectrum. The reason for the asymmetric spectrais that the triangular geometry is somehow special, com-pared to the hexagonal or square, because of the absenceof an inversion center.According to these results we can predict that due tothe internal geometry of the nanowire the conductanceshould depend on the orientation of the magnetic fieldin a manner that indicates the polygonal cross sectionof the shell. We demonstrate this in Fig. 11 where weshow the results obtained with the Kubo formula (6).In these calculations we assume infinitely long nanowiresand we include a disorder broadening of the energy spec-tra described by the parameter Γ = 1 meV in Eq. (7),i.e., we are far from the ballistic regime. In addition weconsider temperatures up to 50 K, to emphasize that theconductance anisotropy should be also robust to thermalperturbations.For each geometry the zero angle is considered whenthe field is oriented towards a prism edge, as shown inthe first column of Fig. 8. The angular period of the con-ductivity, when the magnetic field rotates, is obtainedwhen the magnetic field points to the next corner forthe hexagon and square, i.e. 60 ◦ and 90 ◦ , respectively,whereas for the triangle it corresponds to a half of it, i.e.60 ◦ . Obviously, all figures can be continued by periodic-ity, up to a complete rotation of the magnetic field.The conductivity described by the Kubo formula is, inthis case, an example of band conductivity, i.e., it is di-rectly related to the velocity of carriers along the trans-port direction, like in the ballistic case, and decreaseswhen the disorder parameter Γ increases [52]. The de- C ondu c ti v it y ( S / µ m ) Angle (deg)T = 1 KT=10 KT=50 K 0.250.300.35 0 30 60 90(b) Angle (deg) 0.400.41 0 20 40 60(c) Angle (deg)
FIG. 11. Conductivity as a function of the angle of the mag-netic field, for temperatures 1 K, 10 K, and 50 K, within onefull period for each geometry (a) hexagonal, (b) square, and(c) triangular shells. As before, the magnetic field is B = 2T and the carrier concentration 1 . × − nm − . The re-sults are obtained with Kubo formula including a disorderbroadening of 1 meV. pendence on the temperature is more complicated. Inthis calculations we completely ignored the dependenceon the temperature of the collisional broadening param-eter Γ, which is a separate problem. In our model themain effect of increasing the temperature is the popula-tion/depopulation of the states above/below the chemicalpotential, respectively, and to a lesser extent the varia-tion of the chemical potential itself. Because the energycurves are nonlinear functions of the wave vector, Fig.10, the velocity distribution of the carriers changes ina nonuniform manner, such that our conductivity mayeither increase or decrease with increasing the temper-ature. The relative variation of the conductivity withthe angle is also not simple, meaning that it can increaseor decrease, depending on how the energy spectra be-have around the chemical potential when the magneticfield is changing orientation. In our examples we can seethat the variation is the weakest for the triangular case,which is because the corner localization is stronger andtheir mixing with the side states is relatively weak.The conductance anisotropy in hexagonal core-shellnanowires was already predicted for the case with elec-trons localized in the core [24], but only in the ballisticregime, at much higher magnetic fields, and much lesspronounced. In our case the results obtained in the bal-listic regime are qualitatively similar to those shown inFig. 11, except at low temperatures, when the conduc-tance depends critically on the number of intersectionsof the Fermi energy with the energy bands, which mayvary discontinuously. But, since it is difficult to obtainthe ballistic regime in realistic experimental samples, ourresults show that the anisotropy of the shell could be eas-ier resolved, in well attainable experimental conditions,and possibly not only at low temperatures. C. Nonlinear I-V characteristics
Another indication of the internal geometry of the pris-matic shell can be found in the I − V characteristic, as weshow in Fig. 12. The I − V characteristics are obtainedfollowing the same method used for the conductance, i.e.Eq. (5). This time we create an imbalance between stateswith positive and negative velocity by increasing the po-tential bias far from the linear regime, starting with achemical potential close to the lowest band bottom. I ( µ A ) V bias (meV) B=0 TB=2 TB=4 T bias (meV) 0 6 12 18 24 0 10 20 30 40 50 60(c) V bias (meV)
FIG. 12. I − V characteristic for different values of a magneticfield perpendicular to an edge of the nanowire, for the threegeometries: (a) the hexagonal, (b) square, and (c) triangular.Here the temperature is 25 K. The geometry parameters are R ext = 50 nm and t = 10 nm. We first notice the bias threshold where the linearregime breaks. That is the lowest for the hexagonal ge-ometry, below 5 meV, whereas for the square and triangu-lar cases it increases to 15 meV and 35 meV, respectively.The change of slope of the curves, more clearly seen forsquare and triangular cases, indicate the transition fromcorner states to side states, as more channels are crossed,and could offer a possibility to observe experimentally theexistence of the gap separating these groups of states. Insome cases the magnetic field yields a reduction in energyand a shift in k-space of two mixed-spin bands, leadingto an avoided crossing at k = 0. In these cases an in-flection point is observed, more clearly for the hexagonaland square cases, as a consequence of the reduction inthe number of levels crossed by the potential bias.In the results shown in Fig. 12 we used a tempera-ture of 25 K in order to increase the population of thestates above the chemical potential. The I − V curveswere obtained assuming an infinite wire without impuri-ties. To a first approximation their averaged effect can be incorporated in Eq. (5) by the substitution F (cid:18) E mks − µk B T (cid:19) → (cid:90) A mks ( E ) F (cid:18) E − µk B T (cid:19) dE . In this way we could obtain results similar to those shownin Fig. 12 with a lower temperature, T = 1 K, but witha disorder energy Γ = 1 meV. VI. CONCLUSIONS
We presented several features of the conductance ofcore-shell nanowires with a conductive shell and an in-sulating core, which are consequences of the internal ge-ometry of such nanowires, and can be probed in wellachievable experimental conditions. The motivation ofour work is to stimulate the interest of the experimen-tal groups to do such investigations and to achieve thecorresponding quality of the samples with a clear mani-festation of the internal geometry.Most of the presented results are basically determinedby the energy spectra and by the geometric localizationof the electrons. In the ballistic cases, with no impurities,there is no real need for transport calculations, the con-ductance can be obtained by simply counting the trans-verse modes. The transport calculations that we per-formed were intended to support the predictions from thespectra, in a qualitative manner. We used the transmis-sion function for short, ballistic or quasi-ballistic wires,and Kubo formula for long (infinite) non-ballistic wires.One of the most interesting aspects is that the stateslocalized along the edges and those localized on thefacets can be separated by a large energy gap, robustto many kinds of perturbations, such that a single core-shell nanowire may possibly function like a collection ofthinner nanowires. Another interesting result is that thelocalization, and thus the conductance features, can bemodified or controlled by external electric or magneticfields.
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