transport in curved nanoribbons in a magnetic...

PHYSICAL REVIEW B 84, 195419 (2011)

Transport in curved nanoribbons in a magnetic field

B. Novakovic,1,* R. Akis,2 and I. Knezevic1,†1Department of Electrical and Computer Engineering, University of Wisconsin—Madison, Madison, Wisconsin 53706, USA

2Department of Electrical Engineering, Fulton School of Engineering, Arizona State University, Tempe, Arizona 85287, USA(Received 4 August 2011; revised manuscript received 5 October 2011; published 4 November 2011)

We study the low-field and ballistic electronic transport in curved nanoribbons subject to static and uniformmagnetic fields, where the curved nanoribbon is placed between two leads that inject scattering states. The methodwe present is based on a tight-binding form of the time-independent two-dimensional Schrodinger equation incurvilinear coordinates, with electric scalar and magnetic vector potentials included, and it enables a numericaldescription of transport in arbitrarily shaped curved nanostructures. A description of transport in terms of thescattering-state basis in the presence of magnetic field, together with curvature and possibly misaligned contacts,requires the use of a local Landau gauge. Based on the use of a stabilized transfer matrix method, we compute theconductance and normalized electronic density at the Fermi level for several curved nanoribbons in cylindricaland toroidal geometry, with and without the magnetic field. The magnetic field determines the number of injectedpropagating modes by affecting the Landau-level energies in the contacts, while a complex interplay between themagnetic field and the nanoribbon shape (both its curvature and helicity) determines the transmission of injectedmodes.

DOI: 10.1103/PhysRevB.84.195419 PACS number(s): 73.63.−b, 73.23.Ad, 73.50.−h, 05.60.Gg

I. INTRODUCTION

Curved nanoribbons are narrow nonplanar strips fash-ioned from semiconductor heterostructures that support atwo-dimensional (2D) electron gas.1,2 They are among agrowing variety of curved nanostructures that can be fabricatedtoday.3–6 Curved nanostructures can be produced from planarheterostructures by selective underetching in combinationwith strain mismatch.3,7–10 Other methods involve transferfrom planar to curved substrates,11 growth of curved layersusing curved templates (nanowires, nanotubes, or nanor-ings) as substrates,12–15 or thermal transformation of curvedtemplates.16 Among the most prominent curved structuresare flexible electronic devices,17,18 used for applicationsranging from flexible integrated circuits19 to artificial retinas.20

Techniques such as dry transfer printing19,21 and electro-hydrodynamic jet printing22 have been developed to enabledeposition of electronic devices onto flexible substrates, suchas polymers.2,23

Curvature presents a novel, mechanical degree of free-dom available for tailoring these nanostructures’ physi-cal properties. Theoretical and experimental investigationsof optical properties of curved nanostructures for vari-ous resonators,24–29 photoluminescence tuning,30,31 coherentemission and lasing,32,33 and metamaterials34 have beenreported. There have also been a number of papers on mag-netic properties of curved nanostructures: magnetization,35–38

spin-orbit interaction,39,40 and spin-wave confinement andinterference.41,42

Theoretical work on the electronic properties of curvednanostructures has so far been largely focused on theeffects of the curvature-induced geometric potential andthe classification of states,43–49 as well as transport50–53 insimple structures with analytical shapes (e.g., cylindricalor with periodic corrugation). Similar features have beenaddressed experimentally.54–58 Considering that experimen-tally fabricated nanostructures typically have imperfections inshape,3 there is a need for a theoretical approach, necessarily

computationally based, that is capable of addressing transportin arbitrarily shaped curved nanostructures.

Within the envelope function and effective mass framework,electronic states in curved nanostructures can be obtainedby solving the 2D Schrodinger equation (SE) in curvilinearcoordinates, derived from the three-dimensional (3D) SE inthe limit of zero thickness.59–62 The problem of includingarbitrary magnetic fields into the 2D SE through the covariantPeierls substitution has been solved recently.63 It has beenshown that the dynamics of a particle on the surface canbe decoupled from the one along the normal to the surfaceby using a suitable gauge transformation in the derivationof the 2D SE. In the same work, analytic forms of the2D SE with magnetic field have been given for spherical,cylindrical, and toroidal coordinate systems. As the low-fieldand nearly ballistic quantum transport in nanostructures atlow temperatures is dictated by the transmission properties ofthe states in the vicinity of the Fermi level,64–67 a theoreticalaccount of electronic transport in curved nanostructures in thisregime can, therefore, be described by solving the 2D SE incurvilinear coordinates with open boundary conditions.

In this paper, we study the linear and ballistic transportproperties, specifically the electronic density and conductance,of curved nanoribbons subject to a static magnetic field. Thepresented theoretical and numerical framework is based on atight-binding form of the time-independent two-dimensionalSchrodinger equation in curvilinear coordinates and is entirelygeneral: it enables a computational description of ballis-tic transport in an arbitrarily shaped two-terminal curvednanostructure. A curved nanostructure is connected to largereservoirs via leads that are the source/drain for scatteringstates. We adopt a suitable orthogonal curvilinear coordinatesystem to parametrize a generic curved surface and discretizethe 2D SE, due to Ferrari and Cuoghi,63 by using a second-order finite-difference scheme. We explain the Hermiticityproperties of the discretized curvilinear Hamiltonian and dis-cuss the local Landau gauge used to include arbitrarily directed

195419-11098-0121/2011/84(19)/195419(18) ©2011 American Physical Society

http://dx.doi.org/10.1103/PhysRevB.84.195419

B. NOVAKOVIC, R. AKIS, AND I. KNEZEVIC PHYSICAL REVIEW B 84, 195419 (2011)

magnetic fields in the leads and in the curved nanoribbons.We also compare the Peierls phase approximation with theexact way of including the magnetic field in the curvilinear2D SE. The obtained tight-binding form of the 2D SE incurvilinear coordinates can be solved by well-establishednumerical methods for planar nanostructures, such as thetransfer matrix,68–70 scattering matrix,71,72 recursive Green’sfunction,73 and nonequilibrium Green’s function techniques.67

We use the stabilized transfer matrix approach due to Usukiet al.,68 since it provides both the transmission/reflectioncoefficients and a convenient way to reconstruct the wavefunction. By using the Landauer formula67 to calculate theconductance from the transmission coefficients, as well asby using the wave function to calculate the electron density,we show that coherent electron transport is affected by thecurvature and magnetic field through the number of injectedpropagating modes (modulated by the magnetic field flux inthe leads) and their transmission properties (determined bythe interplay between the magnetic field and the nanoribbon’scurvature and helicity).

This paper is organized as follows. In Sec. II, we lay out thetheoretical concepts and the corresponding numerical model.We explain how the surface divergence, which appears in thecurvilinear 2D SE, is related to the curvature properties andthe normal component of the magnetic vector potential. InSec. II A, we start from the 2D SE in curvilinear coordinates,with the electric scalar potential and the magnetic vectorpotential included, and focus on the curvilinear Laplacianby explaining its Hermitian properties and how they areaffected by its finite-difference version. We present severaldiscretization schemes that may be used. In Sec. II B, we derivea tight-binding form of the 2D SE in curvilinear coordinates byusing a second-order finite-difference scheme. We analyze theconsequences of the non-Hermitian nature of the tight-binding(discretized) curvilinear Hamiltonian matrix. In Sec. II C, weexplore the conditions for the validity of the Peierls phaseapproximation. In Sec. II D, we explain how the magneticfield is included in the lead/curved nanoribbon/lead system.We discuss the necessity of using the Landau gauge in theleads to ensure proper mode injection and discuss cases inwhich a local Landau gauge (amounting to a continuousgauge transformation) may need to be applied to the curvednanoribbon. In Sec. III, we apply the described numericaltechnique to calculate the electron density at the Fermi leveland conductance in several nanoribbons in cylindrical andtoroidal geometries, with and without the magnetic field. Asummary and concluding remarks are given in Sec. IV.

II. THEORETICAL AND NUMERICAL FRAMEWORK

Conductance in the linear, ballistic, steady-state quantumtransport regime in nanoribbons is determined by the trans-mission properties of the electronic states near the Fermilevel.65–67 In curved nanoribbons, we will calculate thetransmission and electron density near the Fermi level fromthe time-independent 2D SE in curvilinear coordinates in thescattering-state basis. As shown previously, by starting fromthe 3D SE and using the covariant Peierls substitution and alayer thinning procedure,63 one can arrive at its 2D version

completely decoupled from the direction normal to the thincurved surface. We state this equation in its time-independentform

1

2m

[ − h2∇2Sψ + ieh∇S · �ASψ

+ 2iehgijAi∂jψ + e2gijAiAjψ] + V ψ = Eψ, (1)

where the 2D curvilinear Laplacian is given by

∇2Sψ = 1√

g∂i(

√ggij ∂jψ) (2)

and the 2D curvilinear divergence by

∇S · �AS = 1√g

∂i(√

ggijAj ). (3)

Here, ∂i ≡ ∂/∂ui , �AS = (A1,A2,0), ui is a curvilinear co-ordinate on the surface S (here only i = 1,2 are includedsince the normal component i = 3 has been decoupled), Ai

are the covariant components of the magnetic vector potential,gij are the contravariant components of the metric tensor,g is the determinant of the metric tensor with covariantcomponents, V = e� + Vs , � is the electric scalar potential,and Vs is the geometric potential given by

Vs = − h2

8m[k1 − k2]2, (4)

where k1 and k2 are the principal curvatures of the surface S.74

It can be shown (see Appendix A) that there is a connectionbetween the surface divergence ∇S · �AS , the curvature prop-erties, and the normal component of the vector potential A3.Assuming an orthogonal coordinate system and ∇ · �A = 0, thefollowing relation holds

∇S · �AS = 1

gA3|u3=0(g11h22 + g22h11) − ∂A3

∂u3

∣∣∣∣u3=0

, (5)

where hij are the elements of the second fundamental formof the surface (covariant components of the metric tensor,gij , being the elements of the first fundamental form of thesurface)74 and A3 is the projection of �A onto the normal tothe curved surface (it simply corresponds to the component of�A along the normal to the surface). We can further introduce

effective potentials due to the magnetic vector potential

V3 = 1

2meh∇S · �AS , (6a)

VS = 1

2me2gijAiAj , (6b)

where V3 and VS define the potentials due to the normalcomponent A3 and the surface component �AS , respectively.We can now rewrite Eq. (1) in the form

1

2m

[− h2∇2Sψ + 2iehgijAi∂jψ

] + Veffψ = Eψ, (7)

where Veff = V + VS + iV3. Let us further rescale the previousequation, for convenience that will become apparent shortly,by multiplying it by

√g. We get

1

2m

[−h2∇2Sψ + 2ieh

√ggijAi∂jψ

]+ √gVeffψ = √

gEψ,

(8)

195419-2

TRANSPORT IN CURVED NANORIBBONS IN A MAGNETIC . . . PHYSICAL REVIEW B 84, 195419 (2011)

where ∇2S = ∂i(

√ggij ∂j ) ≡ √

g∇2S . At the end, it should be

mentioned that the Hamiltonian, as given by Eq. (1), isHermitian (see Appendix B 1).

A. Curvilinear Laplacian in the tight-binding form

Unlike its Cartesian counterpart, the curvilinear Laplaciangiven by Eq. (2) can only be Hermitian up to the prefactor1/

√g once it is cast into a finite-difference form.75,76 In

other words, it is Hermitian only up to a truncation error ofthe Taylor expansion by which a particular finite-differencescheme is derived. A finite-difference approximation of theplanar Hamiltonian gives a Hermitian Hamiltonian matrix,as is well known, while the same approximation of thecurvilinear Hamiltonian in the curved nanoribbon can yield anon-Hermitian Hamiltonian matrix. The latter will be derivedin more detail in this section and in Sec. II B. By rescaling theLaplacian, as in Eq. (8), we remove this Hermiticity problemfrom the rescaled Laplacian, and, consequently, from therescaled Hamiltonian, by moving the prefactor that preventsachieving the Hermiticity to the main diagonal. The problemof finding appropriate discretization schemes for the rescaledHamiltonian in Eq. (8) is easier than for the original Hamilto-nian from Eq. (7). The non-Hermiticity can be minimized bychoosing a sufficiently small grid size, such that

√g does not

vary too much across one grid cell. In the limit where curvatureis constant (like in a cylinder), the curvilinear tight-bindingHamiltonian, obtained using the finite differences, is exactlyHermitian. Sufficiently smooth curvature variation on thefinite-difference grid is also important to be able to capturepotentially sharp curvature features.

In order to determine which conditions should be met tohave a small variation of

√g over one grid cell, we can imagine

an orthogonal coordinate system and refer to the definitionof the metric tensor (gii = ∂�r/∂ui∂�r/∂ui), which is diagonalin that case. If we assume that the curvilinear coordinateui is angular, the metric tensor component gii at any pointwill be equal to the squared radius of the curvature at thatpoint along ui . First, we need enough points on the surface tocapture all the curvature features. Therefore, the grid size (inradians) should satisfy �ui � 1, meaning that the radius of thecurvature should be much larger than the grid size (in meters)along the geodesic in that point. So, a grid size of 1 nm willgenerate a fine-enough mesh for curvature radii not smallerthen a few tens of nanometers. Second, the radii of curvaturein both surface directions should vary smoothly and slowlyover neighboring points; in other words, the relative change inthe grid size along the geodesic should be much smaller than1. The second requirement can be visualized by first defininga uniform (constant grid size) rectangular grid on a planarsurface and then introducing curvature, due to which the gridwill become nonuniform [Fig. 1(b)]. We can then assess themetric tensor variations by knowing the relative change in thegrid size along the geodesic over neighboring points.

Even after the rescaling, Eq. (8), we still cannot use arbitraryfinite-difference schemes, but only those that preserve the

Hermiticity of the rescaled Laplacian ∇2S . The reason for

possible non-Hermiticity in arbitrary finite-difference schemesare the off-diagonal terms of the metric tensor (g12 = g21

(- 12 ,- 1

2 ) ( 12 ,- 1

2 )

(- 12 , 1

2 ) ( 12 , 1

2 )

(a)

(b)ψn,m, gn,m

u1u2

gn± 12 ,m± 1

2

∂f0,0∂u1 , ∂f0,0

∂u2

FIG. 1. (Color online) (a) The four-point scheme for discretiza-tion of the first partial derivatives by using Eq. (9). (b) An exampleof a curved surface parametrized by curvilinear coordinates u1 andu2. Leads are planar continuations of the curved surface in bothdirections along the longitudinal coordinate u1. In order to calculatethe wave function value at the grid point denoted by the large solidcircle (central point) from the four-point scheme, we have to knowthe metric tensor values at all the surrounding points, denoted by thesmall solid circles, as well as at the central point.

or g12 = g21), which are nonzero if the coordinate system isnot orthogonal. Among possible finite-difference schemes to

discretize ∇2S , we will mention two: The four-point formula

(Fig. 1) given by

∂f0,0

∂u1= 1

2�1

(f 1

2 , 12− f− 1

2 , 12+ f 1

2 ,− 12− f− 1

2 ,− 12

) + O(�2

1

),

∂f0,0

∂u2= 1

2�2

(f 1

2 , 12+ f− 1

2 , 12− f 1

2 ,− 12− f− 1

2 ,− 12

) + O(�2

2

),

(9)

where �1 and �2 are the grid sizes along u1 and u2

axes, respectively, and the central difference scheme (bysimultaneously using two different grid sizes)75

∂f0,0

∂u1(2)=

⎧⎪⎨⎪⎩

f12 ,0

(0, 1

2

)−f− 1

2 ,0

(0,− 1

2

)�1(2)

, i = j in ∇2S

f1,0(0,1)−f−1,0(0,−1)

2�1(2), i �= j in ∇2

S .

(10)

Here, we will work only with orthogonal coordinate sys-tems, in which case there is no need for the more complicatedfinite-difference schemes given by Eqs. (9) and (10) and theordinary central-difference scheme is enough. [Actually, in thelimit gij = gij = 0, i �= j , the term in Eq. (10) for which i �= j

vanishes from ∇2S and we are left with the ordinary central-

difference scheme]. In the case a nonorthogonal coordinatesystem is needed, we can go back and use (9) or (10) andfollow the procedure outlined in the rest of this section. So,considering an orthogonal coordinate system and applying theordinary central-difference scheme we can come up with a

195419-3


tight-binding representation of ∇2S , discretized at some slice

n, as

− h2

2m

(∇2Sψ

)n

= Ln�n + Ln,n−1�n−1 + Ln,n+1�n+1,

(11)

where �n is a column vector containing all wave functionvalues on the nth slice (slice is a set of all points with thelongitudinal coordinate u1 constant) and the (M + 1) × (M +1) matrices L are given by

Ln =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

cn,0 −f 02 e

+ 12

n,0 0 · · · 0

−f 12 e

− 12

n,1 cn,1 −f 12 e

+ 12

n,1 · · · 0

0 −f 22 e

− 12

n,2 cn,2 · · · 0...

......

......

0 · · · 0 −f M2 e

− 12

n,M cn,M

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

(12a)

Ln,n±1 =

⎡⎢⎢⎢⎢⎢⎢⎣

−f n1 d

± 12

n,0 0 · · · 0

0 −f n1 d

± 12

n,1 · · · 0...

......

...

0 · · · 0 −f n1 d

± 12

n,M

⎤⎥⎥⎥⎥⎥⎥⎦

, (12b)

where

cn,m = f n1 d

+ 12

n,m + f n1 d

− 12

n,m + f m2 e

+ 12

n,m + f m2 e

− 12

n,m,

dln,m = tn+l

1 (√

gg11)n+l,m, (13)

eln,m = tm+l

2 (√

gg22)n,m+l ,

with t i+l1(2) = h2/2m�i

1(2) for 0 � l < 1, f i1(2) = 2/(�i

1(2) +�i−1

1(2)) and M + 1 the number of points in a slice. Theassumption is that M is constant for all slices. If the curvednanoribbon has varying width, we can always set a hardwallpotential at the extra points around the narrow regions.We assume a 2D nonuniform curvilinear grid, with �n

1 thelongitudinal grid size between slices n and n + 1 and �m

2the transversal grid size between points m and m + 1 alonga slice. Equation (11) reflects the lack of symmetry betweenthe two curvilinear surface directions in our problem: There isthe longitudinal direction with open boundaries, along whichthe current flows carried by complex traveling waves, andthe transversal direction along which the potential definesthe shape of real (confined) transverse modes. This formof Eq. (11) makes it suitable for use with the transfermatrix method, where the slice wave functions (containingall transverse points) are transferred along the longitudinaldirection.

The form of the matrix (12a) will depend on the boundaryconditions in the u2 direction. While the system is alwaysopen in the longitudinal (i.e., transport) direction along thecoordinate u1, it might be either closed (hardwall potential)or periodic (e.g., closing onto itself, like in a cylinder) in thetransverse direction u2. In the former case we use the Dirichletboundary conditions, which give the form (12a), while in the

latter the periodic boundary conditions, where we have thefollowing additional terms

Ln(1,M + 1) = −f 02 e

− 12

n,0 ,(14)

Ln(M + 1,1) = −f M2 e

+ 12

n,M.

If the coordinate system is not orthogonal (gij �= 0,i �= j ) andwe use either one of the schemes (9) or (10), then, besideshaving more complex matrix elements, Ln,n±1 would alsodepend on the boundary condition in the transverse direction(being a three-diagonal matrix in that case).

B. The tight-binding 2D Schrodinger equation

We will now discretize the rest of Eq. (8) to obtain thecurvilinear 2D SE in the tight-binding form. To avoid havingimaginary potential on the main diagonal of the Hamiltonianmatrix, we will first rewrite the imaginary terms containingthe magnetic vector potential in Eq. (8) as

2ieh√

ggijAi∂jψ + ieh√

g∇S · �ASψ

= ieh√

ggijAi∂jψ + ieh∂i(√

ggijAjψ). (15)

Since these terms contain the first derivative of the wavefunction, we will use the central-difference scheme like for thecase i �= j in Eq. (10), in order to avoid having wave-functionvalues at half-integer grid points. We obtain the followingtight-binding form at some slice n

ieh

2m[√

ggijAi∂jψ + ∂i(√

ggijAjψ)]n

= Pn�n + Pn,n−1�n−1 + Pn,n+1�n+1, (16)

where

Pn(m,m + 1) = ie

hf m−1

2

1

2

[e0n,m−1A2(n,m − 1)�m−1

2

+ e+1n,m−1A2(n,m)�m

2

],

Pn(m,m − 1) = − ie

hf m−1

2

1

2

[e0n,m−1A2(n,m − 1)�m−1

2

+ e−1n,m−1A2(n,m − 2)�m−2

2

], (17)

Pn,n±1(m,m) = ± ie

hf n

11

2

[d0

n,m−1A1(n,m − 1)�n1

+ d±1n,m−1A1(n ± 1,m − 1)�n±1

1

].

(The matrix indices in the above equations are m ∈ [1,M],m ∈ [2,M + 1], and m ∈ [1,M + 1], respectively). Note thatwhen m is used as a matrix index on the left-hand side ofEqs. (17) its range is [1,M + 1], while the correspondingtransverse grid position on the right-hand side is in the range[0,M], hence m − 1 is used on the right-hand side. Themagnetic vector potential components in Eqs. (17) are thecovariant components, given by �A = Ai

�ei , where �ei = ∇ui .The relations of the covariant components to the contravariantcomponents �A = Ai �ei , where �ei = ∂�r/∂ �ui , are A1 = g11A

1

and A2 = g22A2, if the coordinate system is orthogonal. The

relations to the magnetic vector components expressed with

respect to the unit curvilinear vectors �ai , �A = Ai �ai , which is

often done if one starts with calculating �A in the Cartesiancoordinate system and then performs either a coordinate

195419-4


system transformation to the desired curvilinear system ora vector projection in the case of arbitrary curvature, are

A1 = √g11A

1and A2 = √

g22A2. In the case of using A

1(2)we

see that it should be multiplied by the actual grid size along thegeodesic (in meters), (

√g11(22))n,m�

n(m)1(2) , at the grid position

(n,m).As with the L matrices in Eqs. (12), if the transverse

boundary condition is periodic, we have additional terms ofthe following form in Eqs. (17)

Pn(1,M + 1) = − ie

hf 0

21

2

[e0n,0A2(n,0)�0

2

+ e−1n,0A2(n,M)�M

2

],

(18)

Pn(M + 1,1) = ie

hf M

21

2

[e0n,MA2(n,M)�M

2

+ e+1n,MA2(n,0)�0

2

].

Finally, the tight-binding 2D SE can be written as

Hn,n+1�n+1 + Hn,n−1�n−1 + Hn�n = EGn�n, (19)

with

Gn =

⎡⎢⎢⎢⎢⎣

(√

g)n,0 0 · · · 0

0 (√

g)n,1 · · · 0...

......

...

0 · · · 0 (√

g)n,M

⎤⎥⎥⎥⎥⎦ , (20)

and

Hn = Ln + Pn + GnV n,(21)

Hn,n±1 = Ln,n±1 + Pn,n±1,

where V n is a column vector containing the potentialVeff − iV3 = V + VS of the nth slice (since iV3 is alreadyincorporated into the Pn and Pn,n±1 matrices).

Equation (19) is the central equation of this paper. Itsform is similar to that of the planar case,68,73 but with morecomplicated matrix elements. In order to solve it, we makeuse of the stabilized transfer matrix technique due to Usukiet al.,68 because it provides both the reflection/transmissioncoefficients and a convenient way to calculate the wave func-tions needed for electron density calculation. The modifiedstabilized transfer matrix technique, as used in this work, isdescribed in detail in Appendix C.

Although Eq. (19) is derived from the exact curvilinearSchrodinger equation [Eq. (1)], the usual discretization error,associated with the second-order finite-difference scheme,here may lead to a non-Hermitian tight-binding Hamiltonian.The non-Hermiticity stems from the discretization of the

curvilinear Laplacian in the case when curvature is spatiallynonuniform, as explained in Sec. II A, and/or from the factthat we allow the curvilinear grid to be nonuniform in general.The Hermiticity of a discretized Hamiltonian matrix is adesirable property, because it makes the numerical solution ofthe eigenvalue problem easier. Even though our computationalmethod requires the solution of the eigenvalue problem onlyin the leads—which are typically, but not necessarily, flat,so the Hamiltonian in them is Hermitian—we employeddiscretization schemes in Eq. (19) that enable the constructionof a discretized non-Hermitian curvilinear Hamiltonian whoseeigenvalue problem can be mapped onto that of a Hermitianmatrix with the help of certain matrix transformations. InAppendix B, we give a detailed account of the Hermitianproperties of both the exact and the tight-binding Hamilto-nians, along with the aforementioned matrix transformations.After transforming the original non-Hermitian eigenvalueproblem, we obtain a Hermitian eigenvalue problem withthe same eigenvalues and transformed wave functions. Fromthis we draw two conclusions: First, the eigenvalue problembecomes more manageable computationally, albeit with theadditional reverse transformation necessary to reconstruct theoriginal wave functions, and, second, although the transformedwave functions do form a complete and orthonormal basis,being eigenvectors of a Hermitian matrix, our original wavefunctions may not form such a basis. However, the latterconsequence is unavoidable, since it is a side effect of thediscretization error of the finite-difference approximation inthe case of spatially nonuniform curvature and/or nonuniformcurvilinear grid. As explained in Sec. II A, this side effect canbe minimized by choosing a sufficiently small grid size thatprevents sharp curvature variations across one grid cell. If inaddition the curvilinear grid is nonuniform, one should ensuresufficiently smooth grid size variations over neighboring gridcells.

C. The Peierls phase approximation

It is instructive to see how the matrix elements look in thelimit of small curvature variations (smooth and slow transitionsbetween different curvatures) and a weak magnetic field. Forthe matrix element Hn (m,m + 1) the first condition allows usto write

e+ 1

2n,m ≈ e0

n,m, e+ 1

2n,m ≈ e+1

n,m, (22)

since the metric tensor varies very slowly over one unit cellin this limit. Further taking a weak magnetic field limit suchthat terms of the form Ai�i are small, we obtain the followingPeierls phase form:

Hn(m,m + 1) ≈ −f m−12 e

+ 12

n,m−1

{1 − ie

h

1

2

[A2(n,m − 1)�m−1

2 + A2(n,m)�m2

]}

≈ −f m−12 e

+ 12

n,m−1 exp

{− ie

h

1

2

[A2(n,m − 1)�m−1

2 + A2(n,m)�m2

]}. (23)

195419-5


Similarly,

Hn(m,m − 1) ≈ −f m−12 e

− 12

n,m−1 exp

{ie

h

1

2

[A2(n,m − 1)�m−1

2 + A2(n,m − 2)�m−22

]},

(24)

Hn,n±1(m,m) ≈ −f n1 d

± 12

n,m−1 exp

{∓ ie

h

1

2

[A1(n,m − 1)�n

1 + A1(n ± 1,m − 1)�n±11

]}.

Of course, there will be additional terms in Eq. (23) and the firstof Eqs. (24) if we use the periodic boundary conditions in the u2

direction, obtained by summing Eqs. (14) and (18) in a similarway. If we further neglect the remaining field-dependent termVS from Eq. (6b), since it is second-order in magnetic vectorcomponents, we have represented the effect of the magneticfield in a curved nanoribbon through the Peierls phase, as isoften done in planar nanostructures.68,73

In planar nanostructures subject to a perpendicular mag-netic field, the Landau gauge with zero transverse component(A2 = 0) is typically chosen to ensure plane-wave injectionfrom the leads (more on this issue in the next section).Therefore, the phase in Hn(m,m ± 1) is typically zero inplanar structures. In contrast, in curved nanostructures subjectto a magnetic field, the curvature may make it impossible toeliminate A2 through gauge transformations, so there may benontrivial phase factors in Hn(m,m ± 1). We will discuss theissue of gauge transformations in the next section.

Let us estimate the magnitude of the magnetic field forwhich the Peierls phase approximation is justified. We can say| �A| ≈ | �B|D, where D is the largest transverse dimension ofthe device. For the Peierls phase to hold we need

e

h| �A|� � 1, | �B| � h

e�D= 6.6 × 10−16Tm2

�D. (25)

In a typical nanostructure, � ≈ 1 nm and D ≈ 100 nm, so wehave | �B| � 6.6 T. To estimate VS, which depends on boththe curvature and the magnetic field, we introduce R as theradius of curvature in our device and notice that gii ∼ R2.By again using | �A| ≈ | �B|D and Eq. (6b), we can easily showthat VS � 1 eV for | �B| � √

m/|e|/D. We see that for VS thecondition is inversely proportional to D, which is reasonablebecause VS is of the second order in �A and, therefore, smallfor small device size. In GaAs and with D ≈ 100 nm we haveVS � 1 eV for | �B| � 6 T, which is a similar condition as theone given by Eq. (25).

The Peierls phase approach is often used for simulating theeffect of magnetic field in planar nanostructures.68,73 Equation(19), modified such that matrix elements that contain themagnetic vector potential are given by Eqs. (23) and (24),along with setting VS = 0, represent the generalization of thatapproach for curved nanostructures. Equations (23) and (24)reduce to the form used earlier for planar nanostructures in thelimit of flat metric and a uniform grid, as do the other matrixelements in Eq. (19). Having in mind the approximations usedto derive Eqs. (23) and (24) and to be able to neglect VS , thePeierls phase approach for curved nanostructures has limitedapplicability for weak magnetic fields and small curvaturevariations. Since planar nanostructures are the limit of a flat

metric (unit metric tensor, no curvature variations), the onlycondition for the Peierls phase approximation validity in themis a sufficiently weak magnetic field.

D. Handling the magnetic field: The local Landau gauge

In order to properly inject transverse modes and computetheir velocities (the latter being critical for the calculationof the transmission matrix, as described in Appendix C 1),we have to include the leads in the transport calculation. Acurved nanoribbon is placed between two leads that connectit to macroscopic reservoirs of charge. The leads act as thesource/sink for the scattering states, whose plane-wave formin the leads allows us to calculate the conductance usingthe Landauer-Buttiker formalism.67 So, we will include anecessary portion of the leads in the simulation region; inthe case of planar leads, we will in general have to connecta planar rectangular grid of the leads to a curvilinear grid ofa nanoribbon via a transition grid. The magnetic field will bepresent in the entire simulation region, including the leads. Thebasis that describes the lead/curved nanoribbon/lead systemconsists of scattering states, whose form in the injecting leadis a sum of the incident and reflected waves and in the outgoinglead a single transmitted wave.

In order to keep the scattering-state basis for the purposesof transport calculation in the presence of magnetic field, onehas to use the Landau gauge in the planar leads, such that thereis no transverse component of the magnetic vector potentialin the leads and no explicit dependence on the longitudinalcoordinate.67 The Landau gauge ensures that the Hamiltonianeigenstates are plane waves along the longitudinal direction.In Fig. 2, we have an example of a lead injecting scatteringstates into a nanoribbon in toroidal geometry. The leadconnects to the structure at the bottom left corner. The lead’sCartesian coordinate system, shown at the bottom left (z is thelongitudinal coordinate in the lead, y is the transverse, and x isthe normal coordinate) serves as the reference. We can write anarbitrary magnetic field, present in the whole structure, in termsof the lead’s Cartesian coordinate system, �B = (Bx,By,Bz),and define a Landau gauge for each component separately;namely we can define �Ax = (0,0,Bxy), �Ay = (0,0, − Byx),and �Az = (−Bzy,0,0), so for an arbitrarily directed magneticfield �B, we can employ the superposition principle andwrite �A = �Ax + �Ay + �Az. That is, the total magnetic vectorpotential will then be given by

�A = (−Bzy,0,Bxy − Byx). (26)

195419-6


x

y

z z

z

y y x

x

x

zy

y

x

z

Bz GT

Bx GT

By GT

GT - gauge transformation

FIG. 2. (Color online) An example of a curved nanoribbon forcalculating the magnetic vector potential: the nanoribbon (dark blue)on top of a half-torus (light blue), with the leads extending downwardat both ends. The Cartesian coordinate system in the left lead(shown in the lower left corner) is taken as a reference system,according to which the magnetic field components Bx , By , and Bz

are specified. Gauge transformations (GT), equivalent to coordinatesystem translations and rotations in the plane perpendicular to eachcomponent, are also shown.

With �A in place for the injecting lead, we can calculate theinjected scattering states by solving the eigenvalue problemof the transfer matrix between the first two slices that belongto the lead.68 Explanation of one technical point is neededhere. In order to obtain the correct number of forward-and backward-propagating states by numerically solving theeigenvalue problem in the lead, as in Appendix C, it isimportant to properly choose the reference level for �A. Thiscan be done by centering the reference coordinate system tofollow the midline that “cuts” the lead in half in the transversedirection: In other words, we can choose the u2 = 0 gridline such that it extends into the lead and splits it in twohalves symmetrically. On such a choice, the terms in �A inEq. (26) due to the different �B components will always beeither even or odd functions of the transverse position, whichgives the correct number and distribution of forward- andbackward-propagating states. We will refer to this u2 = 0 linesimply as the midline or the bisector. More details on thecalculation and sorting of injected modes can be found inAppendix C.

Now, our nonplanar structure is placed between two leads,which could, in principle, be misaligned, and we need to beable to inject scattering states from both of them. It was shown(see Appendix E in Baranger and Stone77) that, in planarstructures with misaligned leads, in order to have the Landaugauge and properly inject plane waves from each lead, onehas to use suitable gauge transformations in order to properlyaccount for the magnetic field in all leads. These gaugetransformations ensure that the magnetic vector potential iscalculated in the Landau gauge locally for each lead. Theproblem of including magnetic field in the presence of multiplemisaligned leads is addressed in a similar way elsewhere.78

In order to smoothly connect the magnetic vector potential

in different leads, gauge transformations will be required inthe device region as well. Since there is no unique analyticalexpression for gauge transformations for different geometries(like cylindrical or toroidal), we illustrate the approach withthe aid of Fig. 2. We generalize the requirement to havea local Landau gauge in each lead to the requirement tohave a local Landau gauge in every slice of the lead/curvednanoribbon/lead system.

The idea is the following: For a given magnetic fieldorientation, any coordinate system translation as well ascoordinate system rotations in the plane perpendicular to thefield can be shown to be equivalent to gauge transformationsthat make the gauge locally Landau.77 In other words, ifwe define �A associated with a given �B in some coordinatesystem, then rotate and/or translate that system as specifiedabove, after which we define �A′ in the transformed coordinatesystem in the same way as in the initial coordinate system(i.e., give it the same functional dependence on the newcoordinates as we had in the old), then it can be shownthat there exists a scalar function f such that (in the sameset of coordinates, for instance, the old ones) �A′ = �A + ∇f .When employing the transfer matrix method, which usesslice-specific dynamical matrices as described in Sec. II B,the relevant matrices in each slice contain the magnetic vectorpotential according to the local Landau gauge (however, itis written in the curvilinear coordinates that parametrize thesurface). Gauge-independent quantities, such as the electrondensity or conductance, are, of course, insensitive to thesegauge transformations. In Appendix D 1, we show that themultitude of gauge transformations required to obtain the localLandau gauge slice by slice can be represented as a singlegauge transformation.

In Fig. 2, we apply this approach to the magnetic vector po-tential components ( �Ax , �Ay , or �Az) generated by each magneticfield component (Bx , By , or Bz) independently, starting fromEq. (26) and having in mind the superposition principle. Asalready mentioned, we can always identify a midline throughour device on which the reference coordinate system originis situated in the reference (injecting) contact. The gaugetransformations are then performed for each magnetic vectorpotential component ( �Ax , �Ay , or �Az) independently such thatthe coordinate system origin moves along the bisector fromone lead to the other, while the local z axis is tangentialto the projection of the bisector onto the plane normal tothe respective magnetic field component. In the example ofFig. 2, nontrivial gauge transformations are needed only forthe magnetic vector potential associated with the By fieldcomponent, because it is the only field component requiringrotations; the projection of the bisector onto the plane normal toBy is a semicircle (more details can be found in Appendix D 2).Projections of the bisector onto the planes normal to Bx andBz are straight lines, and, thus, require only translations alongthe longitudinal axis (the z axis of the reference system for Bx

and the x axis for Bz), while the values of the components of�A in Eq. (26) related to these magnetic field components vary

along the y axis, which is normal to the respective longitudinalaxes and, therefore, neither directions nor values of thesemagnetic vector components are changed going from slice toslice.

195419-7


III. EXAMPLES

We apply the method of Sec. II to calculate the electrondensity and conductance for various nanoribbons. To definecurved nanoribbons of various shapes, we use a virtualsubstrate specified by a curvilinear coordinate system, likecylindrical, spherical, and torodial [in other words, we mimicplacing the ribbon on top (or wrapping it around) a cylinder,sphere, or a torus] and then we apply a hardwall potentialto define the ribbon edges. This method allows us to userelatively simple coordinate systems to work with complicatedshapes. For each curvilinear coordinate system there areseveral quantities, related to the nonzero curvature, that haveto be precalculated: the first fundamental form (or the metrictensor) g and the two principal curvatures, k1 and k2. Sufficientinformation for this task can be found in many textbooks andhandbooks74,79 and online mathematical resources.

A. Cylindrical geometry

The cylindrical geometry is relatively simple and veryuseful for modeling various curved nanoribbons that are beingproduced today.4,7,23,80 We assume the ribbon will be placedon a cylinder of radius r , with (φ,x) being the natural surfacecoordinates and x being the direction of cylinder axis. Byapplying a hardwall potential in (φ,x) we can define the edgesof the curved nanoribbon that is placed on top of or wrappedaround the cylinder and proceed to calculate its conductanceand electronic density by the previously described numericaltechnique. The leads are assumed cylindrical to avoid theneed for transitional regions, which is a good approximationif the nanoribbon is narrow compared to the circumferenceof the cylinder. The leads extend away from both ends ofthe nanoribbon and along the cylinder axis, as depicted inFig. 3(b).

In Figs. 3 and 4, we present calculation results for acylindrical nanoribbon with helicity (wrapped around thecylinder) for two different magnetic field directions. The radiusof the cylinder is r = 40 nm and its length is 4πr , while thenanoribbon width corresponds to one-third of the circumfer-ence. The helix occupies one-half of the length. The Fermilevel is at 8 meV, and the grid contains 500 × 250 points.The curved nanoribbon has no scattering centers; therefore,any difference between the actual normalized conductanceand the number of injected propagating modes (formed inthe injecting lead subject to the magnetic field), which canbe seen in Figs. 3(a) and 4(a), is due to the curvature andhelicity and their coupling with the magnetic field. For thedirection of magnetic field as in Fig. 3(b), there is a very weakdependence of the Landau level energies in the injecting leadon the magnetic field, due to the fact that the field has a smallflux through the narrow cylindrical lead, but the dependence isnot strong enough to affect the number of injected propagatingmodes. (The flux would be zero if the leads were perfectlyplanar.) In Fig. 3(a), there are three modes injected throughoutthe magnetic field range, but even at B = 0 only two aretransmitted; the reason for the reflection is the ribbon’s helicity(we will discuss helicity in more detail later). From Fig. 3(a),we can see that, for higher magnetic fields, the transmissiongradually decreases, until it is completely suppressed at around

(a)

(b)

FIG. 3. (Color online) (a) Normalized conductance vs. magneticfield for a nanoribbon with helicity in cylindrical geometry. The leadsare assumed cylindrical to avoid the need for transitional regionsand, as a consequence, have the Landau levels energies affectedby the magnetic field flux in the leads (Landau level quantization),albeit weakly. The lead on the left-hand side is the injecting lead.Magnetic field direction with respect to the structure is shown in (b)and would be parallel to the leads if they were exactly planar. Evenat zero field, not all of the three injected modes propagate through,due to the ribbon helicity. (b) Electron density at the Fermi levelshown for By = 2.663 T on the log10 scale, where there is resonanttransmission, and for By = 4.5 T, where the normalized conductanceis zero (transmission suppressed). A quasibound state [right panel of(b)] is associated with the resonant transmission feature from (a).

4 T. The suppression can also be observed in the plot of theelectron density [Fig. 3(b)], where at 4.5 T the wave functionis confined to the region close to the injecting lead. In contrast,at 2.663 T, a quasibound state is responsible for the resonanttransmission, visible as a sharp peak in Fig. 3(a).

The variation in the Landau level energies in the leads withvarying magnetic field is much more pronounced in Fig. 4,where the magnetic field is nearly perpendicular to the leadand produces maximal flux. Consequently, the number ofinjected propagating modes varies appreciably in the magneticfield range of interest. We use this example to illustrate theonset of discrepancies between a calculation based on theexact equation (19) and the Peierls phase approximation.The discrepancy between the curves calculated using theexact equation (19) and the Peierls phase approximationbecomes apparent at fields higher than about 2.3 T, Fig. 4(a).

195419-8


(a)

(b)

FIG. 4. (Color online) (a) Normalized conductance vs. magneticfield obtained with the exact equation (19) (thick solid red curve)and within the Peierls phase approach (thin solid black curve) fora nanoribbon with helicity in cylindrical geometry. The leads areassumed cylindrical to avoid the need for transitional regions. Thelead on the left-hand side is the injecting lead. The magnetic fielddirection with respect to the structure is shown in (b) and would benormal to the leads if they were exactly flat. The number of injectedpropagating modes is shown with a thin dashed black curve andis influenced by the magnetic field flux in the leads (Landau levelquantization). There is a discrepancy in the conductance between theexact and Peierls phase approaches starting at around 2.3 T. Resonantreflection is present at B = 4.582 T in the exact (thick red) curve.(b) Electron density at the Fermi level, represented through color onthe log10 scale, corresponding to the resonant reflection at B = 4.582T. A quasibound state is formed, primarily confined to the right-handside of the helical part, looking from the left, injecting, lead.

The appearance of the resonant reflection (the sharp dipin transmission), often observed in 2D coherent electronsystems of nonuniform width,81–83 is present in the exact(thick red) curve at 4.582 T. In these nanostructures, althoughthe nanoribbon has constant width (there are no cavitiesalong the nanoribbon), the interplay between helicity and themagnetic field might lead to the formation of quasiboundstates responsible for resonant reflections. These reflectionsare very sensitive to the amplitude and direction of magneticfield. The interplay between the helicity and magnetic field isalso responsible for the incomplete transmission from about2 T to 3.3 T.

B. Toroidal geometry

Toroidal geometry is more complicated than the cylindricalone and enables us to demonstrate the technique on nanorib-bons with spatially nonuniform curvature. We use a standardparametrization of a torus of radii R and r , where R is theradius of the large circle going through the middle of the torusand r is the radius of the small circle, whose center remainson the larger circle as its rotation around the large circle’s axismaps the torus surface. The surface coordinates are the twoangles, (θ,φ), where θ is related to r and φ to R. To avoidtransitional regions, we adopt cylindrical leads here as well,because of their natural compatibility with a “straightened”torus. Having a curved nanoribbon that consists partially ofa cylindrical shape and partially of a toroidal shape causesproblems in constructing the tight-binding Hamiltonian, sincethe adopted longitudinal curvilinear coordinates have differentunits: length in the cylindrical part and angle in the toroidalpart. To avoid this we can rescale the longitudinal coordinatein the torus to get the units of length there as well:

θ → θ = cθ, �1,tor → �1,tor = c�1,tor, (27)

where �1,tor is the longitudinal grid size in the toroidal part.The constant c has units of length and if we choose c =�1,cyl/�1,tor, we can, in addition to equating the units, obtaina uniform grid along the cylindrical lead-toroidal nanoribbon-cylindrical lead system. The corresponding change in themetric tensor is

g11 = ∂�r∂θ

∂�r∂θ

= 1

c2

∂�r∂θ

∂�r∂θ

= g11

c2. (28)

Similarly, g11 = c2g11 and g = g/c2.In Figs. 5–7 we present calculation results for several

different nanoribbons in toroidal geometry, with and without amagnetic field. The radii of the torus are R = 200 nm andr = 40 nm. The Fermi level is at 8 meV, while the gridsize is 575 × 230 for the toroidal part and 50 × 230 foreach of cylindrical leads. The leads extend downward fromthe nanoribbon as shown in the electron density plots. Thisparticular geometry, in combination with the magnetic field inthe x direction [see the configuration in Fig. 5(b)] leads to oneimportant point. Even though the leads can be geometricallyequivalent, due to the combination of curvature and magneticfield, the calculated transverse mode profiles in the leads mightnot be the same. This fact requires a slight modification of thetransfer matrix method by Usuki et al.68 [see their Eq. (2.17)]to account for different mode-to-site conversion matrices inthe incoming and outgoing leads. In Appendix C, we presentthe details of these necessary changes in case the two contactscannot accommodate the same mode profile.

In Fig. 5, for the magnetic field along the x axis (the largestflux through the leads), we see that the actual conductancefollows the number of propagating modes for most of the mag-netic field magnitudes applied (there is no nanoribbon helicity)and then undergoes oscillating behavior when the magneticfield is sufficiently strong so there is only one propagatingmode. This is a much more visible consequence of the resonantreflection, mentioned in connection to Fig. 4. In Fig. 5(b)electron density plots are shown for two characteristic points:zero conductance at Bz = 4.456 T (resonant reflection) and

195419-9


(a)

(b)

FIG. 5. (Color online) (a) Normalized conductance vs. magneticfield for a nanoribbon in toroidal geometry. The leads are assumedcylindrical to avoid the need for transitional regions. The numberof propagating modes, given by the dashed line, is influenced bythe magnetic field in the leads (Landau level quantization). There arestrong resonant reflections at high magnetic fields, where there is onlyone injected mode. (b) Electron density at the Fermi level, representedthrough color, for unit conductance (complete transmission) at Bx =4.646 T and zero conductance (resonant reflection) at Bx = 4.456 T.For complete transmission at Bx = 4.646 T, there is an edge channelformed that changes sides according to what can be expected fromthe classical Lorentz force. An extended quasibound state, responsiblefor the resonant reflection, is visible on the log10 scale in the centralregion of the nanoribbon at Bx = 4.456 T.

unit conductance (complete transmission) at Bz = 4.646 T. Anextended quasibound state in the middle part of the nanoribbonis visible in the electron density plot in the log10 scale atBz = 4.456 T, while for the case of complete transmissionthere is an edge state that changes sides according to what canbe expected from the classical Lorentz force.

Figure 6 illustrates a situation where it is necessary toinclude gauge transformations, as explained in Sec. II D andAppendix D 2. Since the magnetic field is in the y direction(perpendicular to the plane of the torus and leads), theprojection of the bisector (the middle longitudinal grid lineof the nanoribbon) onto the plane normal to the magnetic fieldis a half-circle. In each slice perpendicular to the current flow,a local coordinate system (as depicted in Fig. 2) is rotatedwith respect to the reference system; each such rotation in theplane perpendicular to the magnetic field results in a gaugetransformation. Therefore, we have a series of slice-dependentlocal Landau gauges as we move along the nanoribbon, whichin this specific case means that A2 = 0.

FIG. 6. (Color online) Sample electron density plot at the Fermilevel for a toroidal nanoribbon subject to a magnetic field in the y

direction. Electron density is represented through color (light denoteshigh and dark denotes low). This situation illustrates the use of thegauge transformations, as explained in Sec. II D and Appendix D 2,since the projection of the middle longitudinal grid line on the planenormal to the field is a half-circle, which requires the use of rotationsin every slice of the toroidal part of the nanoribbon. The formed edgestate reflects the symmetry of the problem and agrees qualitatively(the side at which it is formed) with the result obtained by applyingthe classical Lorentz force.

The edge state visible in the plot of electron densitycorresponds qualitatively (the side at which it is formed) towhat can be expected from the classical Lorentz force andreflects well the symmetry of the problem. The details on howone performs the gauge transformations in this case in orderto connect the local (slice-by-slice) Landau gauges are givenin Appendix D 2.

In Fig. 7, we can see the results for a torodial nanoribbon,with and without helicity, in the absence of magnetic field. Thenormalized conductance for the nanoribbon without helicityagrees exactly with the number of injected propagating modes,meaning that there is complete transmission. Comparing theresult of the conductance vs. Fermi level for the nanoribbonswith and without helicity confirms that the observed differencebetween the normalized conductance and the number ofpropagating modes is due to helicity. As we also observed inFigs. 3 and 4, helicity plays a role similar to a quantum pointcontact in a flat wire (i.e., has the ability to quench propagationof certain modes). As shown earlier,84 even without magneticfield one might expect resonant conductance features. Thereason we do not see them here is because, due to the relativelylarge radius of the curvature of the nanoribbons consideredhere, the geometric potential is on the order of 0.1 meV, whichis much smaller then the Fermi energy (8 meV). Anotherreason is that there are no sharp changes in curvature and,therefore, no sharp variations in the geometric vector potential,which can cause quantum interference effects.

IV. SUMMARY AND CONCLUDING REMARKS

We presented a theoretical and numerical technique forinvestigating the linear, ballistic transport properties of curvednanoribbons in a static, uniform, and arbitrarily directed

195419-10


(a)

(b)

FIG. 7. (Color online) (a) Normalized conductance vs. Fermilevel in the leads for a nanoribbon in toroidal geometry with andwithout helicity, in the absence of magnetic field. The leads areassumed cylindrical to avoid the need for transitional regions. Thenumber of injected modes is changing due to the change in the Fermienergy. Comparison of the conductance for these two nanoribbonsshows that helicity is the reason for the observed difference betweenthe actual normalized conductance and the number of propagatingmodes. (b) Normalized electron density at the Fermi level, representedthrough color (white denotes high and black denotes low) for thenanoribbons with (right panel) and without helicity (left panel) whenthe Fermi level energy is equal to 13.5 meV.

magnetic field. Having started from the two-dimensionalcurvilinear Schrodinger equation with magnetic vector andelectric scalar potentials included, after a rearrangement andrescaling of the Schrodinger equation we identified an effectivepotential, which is in general a complex quantity and dependson the magnetic field. The resulting equation was cast into atight-binding form by using a second-order finite-differencescheme, which can be conveniently solved by a stabilizedtransfer matrix method. We analyzed the constraints on theHermiticity of the discretized curvilinear Hamiltonian andproposed several suitable finite-difference schemes. While weadopted orthogonal coordinate systems throughout this paper,the framework is general enough that it can be extended tononorthogonal coordinate systems if required.

Special attention was paid to the way in which the magneticfield should be included in the model. The requirementof working in the scattering-state basis, coupled with thestructure’s curvature, led us to a model for the inclusion of the

magnetic field through a local Landau gauge. The use of thelocal Landau gauge is necessary to properly inject modes fromthe leads and avoid artificial numerical suppression of modepropagation, which can happen in finite-difference approachesto solving the Schrodinger equation in magnetic field, becausean inadequate gauge in the effective complex potential defacto acts as a barrier to transmission. We have explained howslice-by-slice gauge transformations can be generated basedon the structure’s shape and the magnetic field direction.

Since the Peierls phase is often used to account forthe magnetic field in planar nanostructures, we have alsoinvestigated the requirements for the validity of the Peierlsphase approximation generalized to curved structures, wherethe magnetic field influence is captured through complexexponential terms. We find that there exists an upper limit onthe magnetic field strength and curvature variations, connectedto the structure’s dimensions as well as the grid size, beyondwhich the Peierls phase approach is no longer valid.

We illustrated the method by numerically calculating theelectron density at the Fermi level and conductance forseveral cylindrical and toroidal nanoribbons, with and withouthelicity, and subject to magnetic fields of different orientations.Interesting features that demonstrate the interplay betweengeometry and the magnetic field can be observed in thesestructures: For example, the prominent resonant reflectionconductance features that occur due to a formation of boundstates can be manipulated by curvature, helicity, and magneticfield.

ACKNOWLEDGMENT

The authors thank R. Joynt for helpful discussions. Thiswork was supported by the NSF, Grant No. ECCS-0547415.

APPENDIX A: THE SURFACE DIVERGENCE OF THEMAGNETIC VECTOR POTENTIAL

By using the fact that ∇ · �A = 0 for the Landau gaugein the Cartesian coordinate system and that the curvilinearcoordinate system we are using is orthogonal, we derive arelationship between the surface curvilinear divergence ∇S ·�AS and the normal component A3 and curvature properties,

given by Eq. (5).Since ∇ · �A = 0 in the rectangular coordinate system, it will

remain zero in any other coordinate system. So if we assumea general curvilinear coordinate system suitable for defining acurved surface in the limit where the thickness goes to zero,59

we get

∇ · �A = 1√G

∂i(√

GGijAj ) = 1√G

∂a(√

GGabAb)

+ 1√G

∂3(√

GA3) = 0. (A1)

Here, a and b are the surface curvilinear coordinates and

G =

⎡⎢⎣

G11 G12 0

G21 G22 0

0 0 1

⎤⎥⎦ , (A2)

195419-11


where G is the determinant of the matrix (A2) and Gij are thecontravariant components. If we evaluate Eq. (A1) at u3 = 0,we get

1√g

∂a(√

ggabAb

∣∣u3=0) = − 1√

G∂3(

√GA3)

∣∣∣∣u3=0

, (A3)

since

Gab = gab + [αg + (αg)T

]ab

u3 + (αgαT

)ab

(u3)2 (A4)

and α is a matrix with elements satisfying Weingartenequations59

α11 = 1

g(g12h21 − g22h11), α12 = 1

g(h11g21 − h21g11),

α21 = 1

g(h22g12 − h12g22), α22 = 1

g(h21g12 − h22g11).

(A5)

Using the fact that the coordinate system is orthogonal (g12 =g21 = 0) and that in our notation �AS = (A1|u3=0,A2|u3=0,0),where A1,2,3 are the covariant components of the magneticvector potential, after some algebra we arrive at the finalequation (5) from (A3).

APPENDIX B: HERMITICITY OF THE EXACT ANDTIGHT-BINDING HAMILTONIANS

1. Exact Hamiltonian

Here, we will investigate the Hermiticity property of theexact Hamiltonian, given by Eq. (1). The first term on theleft-hand side of Eq. (1) is the curvilinear Laplacian and it iswell known that it is Hermitian. Therefore, we will not give theformal proof, but the method is similar to the one we will usefor other terms. The fourth term on the left-hand side is realand does not contain any derivatives, so it is Hermitian, too.The terms that are not obviously Hermitian are the imaginaryterms containing the magnetic vector potential (the secondand third terms on the left-hand side). If we introduce h =(ieh/m)gijAi∂j + iV3, we need to prove∫

φ∗ (hψ) dS =∫

(hφ)∗ ψdS, (B1)

where dS is an infinitesimal surface element on the curvedsurface S. It is known that the infinitesimal volume elementcan be expressed in a curvilinear coordinate system as79 dV =√

Gdu1du2du3, where the three-dimensional metric tensorGij is defined in Appendix A. We can now write

(dV )u3=0 = √gdu1du2du3 ⇒ dS = √

gdu1du2. (B2)

The first term in h consists of a sum of four terms (two ifthe curvilinear coordinate system is orthogonal). For example,for (i,j ) = (1,1), we can use the integration by parts and thefact that the wave function vanishes at infinity:

ieh

m

∫du2

∫du1φ∗g11A1

√g∂1ψ

={u=φ∗g11A1

√g

dv=∂1ψdu1

}= ieh

m

∫du2

[ψφ∗g11A1

√g∣∣u1=+∞u1=−∞

−∫

ψ∂1(φ∗g11A1√

g)du1

]

= − ieh

m

∫ψ∂1(φ∗g11A1

√g)du1du2

= − ieh

m

[ ∫ψg11A1∂1φ

∗dS

+∫

ψφ∗ 1√g

∂1(√

gg11A1)dS

]. (B3)

Similar derivations can be carried out for the other three terms,so we have in total for the first term in h

ieh

m

∫φ∗gijAi∂jψdS

= − ieh

m

[∫ψgijAi∂jφ

∗dS

+∫

ψφ∗ 1√g

∂i(√

ggijAj )dS

]. (B4)

The second term on the right-hand side in Eq. (B4) is thematrix element of −2iV3. After the partial cancellation withiV3 in h we then have∫

φ∗ (hψ) dS = − ieh

2m

∫ψφ∗ 1√

g∂i(

√ggijAj )dS

− ieh

m

∫ψgijAi∂jφ

∗dS =∫

(hφ)∗ ψdS,

(B5)

which proves the Hermiticity of the exact Hamiltonian.

2. Tight-binding Hamiltonian

The tight-binding Hamiltonian derived by using finite-difference schemes does not have to be Hermitian, regardlessof the fact the exact Hamiltonian is Hermitian. However, itis computationally advantageous to have Hermitian matriceswhen one wants to solve the eigenvalue problem. Here, we willshow that the discretization schemes employed in Secs. II Aand II B, along with some matrix transformations, allow for thedefinition of Hermitian matrices. To obtain the tight-bindingHamiltonian, we can rewrite Eq. (19) in the form of aneigenvalue problem. We have⎡

⎢⎢⎢⎢⎣H0 H0,1 0 · · · 0

H1,0 H1 H1,2 · · · 00 H2,1 H2 · · · 0...

......

......

0 · · · 0 HN,N−1 HN

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

�0

�1

�2...

�N

⎤⎥⎥⎥⎥⎦

= E

⎡⎢⎢⎢⎢⎣

G0 0 · · · · · · 00 G1 0 · · · 00 0 G2 · · · 0...

......

......

0 · · · · · · 0 GN

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

�0

�1

�2...

�N

⎤⎥⎥⎥⎥⎦ −

⎡⎢⎢⎢⎢⎣

A0...0B

⎤⎥⎥⎥⎥⎦ ,

(B6)

where A = H0,−1�−1 and B = HN,N+1�N+1 are the openboundary conditions. If we extend all the matrices to includeslices sufficiently far away from the curved region, we will

195419-12


eventually be able to use the Dirichlet boundary conditions(analytically the wave function vanishes at infinity), so A = 0and B = 0. We then can write the eigenvalue problem as

H� = EG�, (B7)

where the matrix definitions follow from Eq. (B6) with A = 0and B = 0.

As mentioned in Sec. (II A), one of the consequences ofusing the second-order discretization scheme is that the finaltight-binding Hamiltonian matrix G−1H will not be Hermitiandue to the existence of the curvilinear Laplacian. By rescalingthe Schrodinger equation [Eq. (8)] we eventually arrived ata generalized eigenvalue problem given by Eq. (B7), whereH is the rescaled tight-binding Hamiltonian matrix. Using asuitable matrix transformation we can obtain the eigenvalueproblem in a standard form

HG−1

G� = EGG�, (G−1

HG−1

)G� = EG�, (B8)

where the definition of G is obvious, since G = GG andG is diagonal. In this transformed eigenvalue problem, theHamiltonian is given by G

−1HG

−1and the wave function

by G�. Since the matrix G is diagonal, the condition forHermiticity of this transformed Hamiltonian reduces to H =H†, where † represents the conjugate transpose. Furthermore,this condition reduces to

{Hn = H†

n

Hn,n±1 = H†n±1,n

}⇒

⎧⎪⎪⎨⎪⎪⎩

Ln = L†n

Ln,n±1 = L†n±1,n

Pn = P†n

Pn,n±1 = P†n±1,n

⎫⎪⎪⎬⎪⎪⎭ , (B9)

where we used definitions from Eq. (21).If our curvilinear grid is uniform, such that �n

1 = const =�1 and �m

2 = const = �2, the conditions in Eq. (B9) aresatisfied, as can be seen from Eqs. (12a)–(14) and Eqs. (17) and(18). Therefore, the eigenvalue problem in Eq. (B8) contains aHermitian matrix. Since, in general, we prefer the possibilityof using a nonuniform curvilinear grid, with �n

1 varying inthe longitudinal direction and �m

2 in the transversal direction,the rescaled Hamiltonian matrix H will not be Hermitian. Inthat case we can introduce further matrix transformations,85–87

defined by a diagonal block matrix F, with diagonal matricesFn on its main diagonal given by

Fn(m,m) = 2√(�n

1 + �n−11

) (�m−1

2 + �m−22

) , (B10)

where, as before, the index m on the left-hand side designatesmatrix elements with limits m ∈ [1,M + 1], while on the right-hand side all the quantities are defined on the grid having limits[0,M] and, therefore, we use m − 1 instead of m. We can nowconstruct a Hermitian eigenvalue problem by the followingmatrix transformation in the second of Eqs. (B8):

(F−1G−1

HFF−1G−1

)G� = EF−1G�,

(G−1

F−1HFG−1

)GF−1� = EGF−1�,

which can be rewritten as

H� = E�, (B11)

where H = G−1

F−1HFG−1

is a Hermitian matrix and � =GF−1�.

APPENDIX C: STABILIZED TRANSFER MATRIXSOLUTION TO EQ. (19)

To solve Eq. (19) for the transmission/reflection coefficientsand electron density, we first rearrange it in the transfer matrixform: [

�n

�n+1

]= Tn

[�n−1

�n

], (C1)

where the transfer matrix Tn is given by

Tn =[

0 1−H−1

n,n+1Hn,n−1 H−1n,n+1(E · Gn − Hn)

]. (C2)

For our case of linear (small temperature and bias) and ballistictransport we solve this equation only for E = EF , where EF

is the Fermi level energy. Writing equations of the form givenby Eq. (C1) for every slice in our curved nanoribbon we canconnect the two slices at the interface with the left lead to thetwo slices at the interface with the right lead[

�N

�N+1

]= TN · · · T0

[�−1

�0

], (C3)

where the domain bounded by slices 0 and N belongs to thecurved nanoribbon and the rest belongs to the left (smallerthan 0) and right (greater than N ) leads.

We now express the wave functions on the left- andright-hand sides of Eq. (C3) in terms of the reflection andtransmission coefficients multiplying forward- and backward-propagating eigenfunctions of the leads. In the Landau gauge,these eigenfunctions are proportional (up to the normalizationconstant) to the product of plane waves along the lead andtransverse modes.67 The decomposition of the wave functionsin Eq. (C3) is possible because each �n (“slice wave function”)is a superposition of forward and backward traveling waves,with amplitudes determined by specific injection conditionsas well as any reflection and transmission that the waveundergoes. For the wave incident at the left lead (with unitamplitude) we can write[

�−1

�0

]= TL

[1r

],

[�N

�N+1

]= TR

[t0

], (C4)

where

TL/R =[

�+L/R �−

L/R

�+L/Rλ+

L/R �−L/Rλ−

L/R

], (C5)

with

�±L/R = [

ψ±0,L/R, . . . ,ψ±

M,L/R

],

(C6)λ±

L/R = diag[λ±

0,L/R, . . . ,λ±M,L/R

],

where ψ±l,L/R is the eigenfunction of the l-th transverse mode at

the (−1)-th (for L) and N -th (for R) slice, where + stands forforward and − for backward propagation, while λ±

l,L/R (similarnotation as for eigenfunctions) is a phase factor that representsthe difference in phase between eigenfunction values at twoneighboring slices (eigenfunctions are plane waves along thelead). It should be noted that �±

L/R is a matrix of dimension

195419-13


(M + 1) × (M + 1), while �n is a column vector of dimension(M + 1) × 1.

We can construct the matrices TL/R by using the transfermatrix form [Eq. (C1)] within the leads (planar geometry) andknowing that wave function values at two neighboring slices inthe leads differ only by a phase factor. We obtain the followingeigenvalue problems for the left and right leads[

0 1

−H−10,1H0,−1 H−1

0,1 (EF · 1 − H0)

][ψ±

l,L

λ±l,Lψ±

l,L

]

= λ±l,L

[ψ±

l,L

λ±l,Lψ±

l,L

], (C7)

[0 1

−H−1N+1,N+2HN+1,N H−1

N+1,N+2 (EF · 1 − HN+1)

]

×[

ψ±l,R

λ±l,Rψ±

l,R

]= λ±

l,R

[ψ±

l,R

λ±l,Rψ±

l,R

]. (C8)

Here, for the purpose of keeping the same notation as inEq. (C1), slices 0 and 1 appear, but it should be consideredthat they belong to the leads. In fact, since the leads are planarand uniform and the magnetic field is also uniform, we couldhave chosen any other combination of three consecutive slicesthat belong to the same lead in Eqs. (C7) and (C8). MatricesH0, H0,±1 and HN+1, HN+1,N+1±1 follow from the matrices inEq. (21) in the zero-curvature limit and are given in earlierpapers on coherent transport in planar nanostructures.68,73

Finally, we obtain the following matrix equation to be solved[t0

]= T−1

R TN · · · T0TL

[1r

]. (C9)

Straightforward solution of Eq. (C9) is not stable nu-merically because of the existence of evanescent waves(exponentially growing or decaying), whose numerical valuecan become extremely small or large after only a few matrixmultiplications. We employ a stabilized iterative technique dueto Usuki et al.68 with two minor modifications: in the wavefunction calculation and in allowing the left and right leadHamiltonians to differ (either through different geometry ormagnetic field flux). The fact that the lead Hamiltonians candiffer is made explicit in the above equations. The stabilizediterative version of Eq. (C9) for the calculation of transmissioncoefficients in the column vector t is[

C(j+1)1 C(j+1)

2

0 1

]= Tj

[C(j )

1 C(j )2

0 1

]P(j ),

− 1 � j � N + 1, (C10)

with T−1 = TL and TN+1 = T−1R and

P(j ) =[

1 0

P(j )1 P(j )

2

]. (C11)

By using the block matrix inversion to calculate T−1R and

setting T12R = 1, T22

R = 0, where the position of the block

matrix in TR is denoted by the superscript [their value is notimportant because they are multiplied by 0 in Eq. (C4)] we get

TN+1 =[

0[�+

Rλ+R

]−1

1 −�+R

[�+

Rλ+R

]−1

]. (C12)

The form of P(j ) and its block matrices is determined from thecondition that the form of the matrix on the left-hand side ofEq. (C10) is satisfied. We obtain

P(j )1 = −P(j )

2 T21j C(j )

1 , P(j )2 = [

T21j C(j )

2 + T22j

]−1, (C13)

where different block matrices in Tj are specified with therow and column numbers in the superscript. Starting from theinitial conditions C(0)

1 = 1 and C(0)2 = 0 and solving Eq. (C10)

iteratively we obtain the transmission coefficients from

t = C(N+2)1 . (C14)

In a similar way for the reflection coefficients we have[D(j+1)

1 D(j+1)2

] = [D(j )

1 D(j )2

]P(j ), r = D(N+2)

1 , (C15)

with the initial conditions D(0)1 = 0 and D(0)

2 = 1.For the calculation of electron density we define a new

operator �(j ), using the already calculated operator P(j ),

�(j−1) = P(j−1)1 + P(j−1)

2 �(j ), N + 1 � j � 1, (C16)

with the initial condition �(N+1) = P(N+1)1 . Now the electron

density originating from a single transverse mode l in theinjecting lead will be

nl (n,m) = ∣∣�(n) (m,l)∣∣2 , n ∈ [0,N ] , (C17)

where the indices (n,m) represent longitudinal and transversegrid positions, respectively.

1. Velocities of propagating states in the presenceof magnetic field

With �A in place as in Eq. (26), we can calculate theinjected (evanescent and propagating) scattering states fromeach lead by solving the eigenvalue problem of the transfermatrix between the first two slices (n = −1,0) that belong tothe leads [here we make use of the fact that the scattering statesin the leads are plane waves in the longitudinal (transport)direction]68

exp (ik1�1)

[�−1

�0

]= T 0

[�−1

�0

],

(C18)

T 0 =[

0 1

−H−10,1 H0,−1 H−1

0,1 (E · 1 − H0)

],

where k1 denotes the longitudinal component of the wavevector. Leads are assumed planar, so the H matrices satisfythe tight-binding 2D SE for a planar grid,68 into whichEq. (19) reduces in the limit of zero curvature (gii = gii = 1).Explanation of one technical point is needed here. In orderto obtain the correct forward- and backward-propagatingstates from a numerical solution to Eq. (C18), it is importantto properly choose the reference level for �A. This can bedone by centering the reference coordinate system to followthe midline that “cuts” the lead in half in the transverse

195419-14


direction: In other words, we can choose the u2 = 0 grid linesuch that it extends into the lead and splits it in two halvessymmetrically. On such a choice, the terms in �A in Eq. (26), dueto the different �B components, will always be odd (linear in y)or even (independent of y) functions of the transverse position,which gives correct number and distribution of forward- andbackward-propagating states.73

A correct evaluation of velocities is important since oneof our goals is to calculate the transmission through a curvednanoribbon,

Tij = |tij |2v(i)/v(j ), (C19)

where i is the outgoing mode index, j is the incoming modeindex, with vi and vj being the corresponding velocities, whiletij is the matrix of transmission amplitudes, Eq. (C14).

It turns out that the relation v(l) = hk(l)1 /m∗, where k

(l)1

for a propagating mode of index l is calculated as explainedabove, does not hold in the presence of a magnetic field.77,88

To calculate the appropriate relationship we can use a tight-binding Hamiltonian on a planar grid in a magnetic field. Byworking in the |n,m〉 basis (n denotes slice index and m denotesthe transverse index in a slice)

x|n,m〉 = n�1|n,m〉, |ψ〉 =∑n,m

ψn,m|n,m〉 (C20)

and defining the velocity operator in the x direction (which isassumed longitudinal in the leads) as usual,

υ = 1

ih[x,H ], (C21)

we can define the velocity at the slice n as

υ(l)n = [〈ψ |υ|ψ〉](l)

n = 2t1�1

h

∑m

|ψn,m|2

×[sin

(k

(l)1 �1

) − e

hcos

(k

(l)1 �1

)A1(n,m)�1

].

(C22)

Here, H is the Hamiltonian with the matrix representationgiven by Eq. (B6) in flat metric and with a uniform gridassumed in the leads, so t1 is the longitudinal hoppingenergy. We also make use of the fact that for a sufficientlysmall grid size one can employ the translational operator ofthe form ψn±1,m = ψn,m exp(±ik1�1).89 In the Peierls phaseapproximation (weak magnetic field so a small �A) Eq. (C22)equals

υ(l)n ≈ 2t1�1

h

∑m

|ψn,m|2 sin{[

k(l)1 − e

hA1(n,m)

]�1

}.

(C23)

As A1 depends only on the transverse coordinate [a conse-quence of the Landau gauge, see (26)], the only dependenceof the current on the slice index n would occur through theterm |ψn,m|2. However, since a mode’s transversal profile isconstant inside a lead of constant width, and is modulated by aplane wave in the longitudinal direction, its square amplitudedoes not depend on n for any m and velocity (and current)per mode is conserved between slices in the lead. Therefore,

it is sufficient to evaluate the mode velocity at (n = −1) inEqs. (C22) and (C23).

APPENDIX D: LOCAL LANDAU GAUGE

1. Local Landau gauge: Gauge transformations

The local Landau gauge transformation can be defined in aformal mathematical language. In analogy with a misalignedmultiterminal structure,77 here we can derive very similarequations with a slightly different interpretation: A localLandau gauge in every lead becomes a local Landau gaugein every slice, while the planar device region between theleads, used to smoothly connect local Landau gauges in theleads, becomes the curved regions between slices used tosmoothly connect local Landau gauges in slices. The sliceat the position of the reference coordinate system is numbered0 (slice −1 belongs to the left lead), so we can write for thegauge transformation between slices 0 and n

�An (�x0) = �A0 (�x0) + ∇fn (�x0) , (D1)

where �x0 is a point in the reference coordinate system, whichis positioned at the slice 0. It is straightforward to derive thefunction fn for an arbitrary rotation77 and a translation neededfor the gauge transformation between the magnetic vectorpotential at the 0th slice, �A0, and the magnetic vector potentialat nth slice, �An. The gauge transformation then generated byfn in Eq. (D1) allows us to express �An in the same form as�A0 but in a different coordinate system, translated and rotated

with respect to the reference coordinate system in which �A0

is calculated (see Fig. 2). For example, if �A0 = −By0 �x0, then�An = −Byn �xn.

The gauge transformation given by fn should be turned ononly at the position of nth slice and turned off at the positionsof all other slices, whereas in the rest of the domain (curvednanoribbon’s surface), it should be a smooth function. Thiscan be achieved by multiplying fn by the following auxiliaryfunction:

ξn =⎧⎨⎩

1, on slice nsmooth, between slices n − 1 and n + 10, everywhere else .

(D2)

Equation (D1) now becomes

�An = �A0 + ∇ (ξnfn) = �A0 + ∇ξnfn + ξn∇fn. (D3)

At the slice n, ∇ξn = 0 and ξn = 1 and we have a correctgauge transformation, given by Eq. (D1) evaluated at the slicen. In the domain bounded by slices 0 to n − 1 and n + 1to N both the auxiliary function and its gradient are zero,so there is no gauge transformation of the form generatedby fn. In the domain bounded by slices n − 1 to n + 1 boththe auxiliary function and its gradient are nonzero and wehave some unavoidable gauge transformation, but we are notinterested in its exact form since we work on the discretegrid only (space between discrete slices is not included in ourcalculation). To generate the total gauge transformation wecalculate functions fi and ξi for each slice i and sum them up

f =N∑

i=1

ξifi . (D4)

195419-15


For a magnetic field of general direction we can write thetotal magnetic vector potential in the following form:

�A0 = �Ax0 + �Ay

0 + �Az0, (D5)

where the magnetic vector potential components are each dueto a magnetic field component denoted by the superscript.Therefore, for Eq. (26) we have that �Ax

0 = Bx0y0z0, �Ay

0 =−By0x0z0, and �Az

0 = −Bz0y0x0. Since the curl operator islinear we can use the principle of superposition and applythe necessary gauge transformations independently to thesemagnetic vector potential components. By applying the semi-graphical and formal mathematical methods explained above,we can decide which component needs a gauge transformationand calculate it.

In deriving Eq. (5) we used the fact that ∇ · �A = 0 in theLandau gauge. If we use the separation into components ex-plained in the previous paragraph, we see that the componentsof �A not requiring gauge transformations will not alter the factthat ∇ · �A = 0. However, in general, we might have a fieldcomponent that requires a gauge transformation generated bythe function in Eq. (D4). If we designate the field componentrequiring a gauge transformation by q we have

�Aq = �Aq

0 + ∇f = �Aq

0 +N∑

i=0

∇ (ξifi)

= �Aq

0 +N∑

i=0

[fi∇ξi + ξi∇fi] . (D6)

Therefore,

∇ · �Aq = ∇ · �Aq

0 +N∑

i=0

[2∇fi · ∇ξi + fi∇2ξi + ξi∇2fi]

=N∑

i=0

[2∇fi · ∇ξi + fi∇2ξi + ξi∇2fi], (D7)

since ∇ · �Aq

0 = 0, �Aq

0 (as well as any other �Aq

i ) being in theLandau gauge. This is clearly nonzero for arbitrary point onthe surface of the curved nanoribbon, but since we work on adiscrete grid let us evaluate Eq. (D7) at the points belongingto some slice n:

(∇ · �Aq)n = (∇2fn)n = (∇ · ∇fn)n = [∇ · ( �Aqn − �Aq

0

)]n

= (∇ · �Aqn − ∇ · �Aq

0

)n

= 0, (D8)

where we used the properties of the function ξi and the factthat all �Aq

i ’s are in the Landau gauge. So, at the points on thegrid we still have zero divergence, meaning that we can stilluse Eq. (5) to calculate V3 on the grid.

2. Gauge transformation for Fig. 6

From Eq. (26) we see that

�A0 = (0,0, − By0x0). (D9)

From our discussion in Sec. II D we know that the slice 0does not need a gauge transformation: We can directly useEq. (D9) evaluated at the slice 0, because the longitudinal axis(the z0 axis for the By0 component) is already tangential to

the bisector at the point of its intersection with the slice 0,and the coordinate system is positioned at that point. Whenwe transform the coordinates in Eq. (D9) to our toroidalcoordinates (θ,φ,r) and evaluate the result at the position ofthe slice 0 (φ0 = 0) we get

�A0(�x(0)

0

) = φBy0r (cosθ − 1) , (D10)

where �x(0)0 are points for which �x0 ∈ {slice 0}.

Angle φ is measured from the slice 0, which is at theinterface with the left lead in Fig. 2, while angle θ is measuredfrom the bisector toward the negative y0 axis (where the y0 axisbelongs to the reference coordinate system in Fig. 2). If theCartesian coordinate system is like the reference coordinatesystem in Fig. 2, but positioned in the center of the torus (fromwhere R is measured), then the formulas relating the Cartesiancoordinates of the torus surface (x,y,z) to the curvilinearcoordinates (θ,φ) are given by

x = − (R + rcosθ ) cosφ, y = −rsinθ,(D11)

z = (R + rcosθ ) sinφ,

where φ is measured from the negative x axis and θ in theclockwise direction looking along the current flow (from leftto right in Fig. 2).

The transformation to the toroidal coordinates can be doneby using Eq. (D11) and the following relations:

x0 = x + R + r, y0 = y, z0 = z, (D12)

x0 = φsinφ − rcosθcosφ + θsinθcosφ,

y0 = −θcosθ − rsinθ, (D13)

z0 = φcosφ + rcosθsinφ − θsinθsinφ.

Since we know that after the gauge transformation the formof �A is the same, but expressed in a new coordinate system,which is translated and rotated according to the slice position(see Fig. 2 for the By component), we conclude that

�Ai

(�x(i)i

) = φBy0r (cosθ − 1) , 0 � i � N. (D14)

Equation (D14) can be derived in a formal way by usingEq. (D1). If we assume that for some slice n the coordinatesystem (as shown in Fig. 2) is rotated by the angle αn

(clockwise here) and translated by the lengths a along thex0 axis and b along the z0 axis we obtain for fn

fn (�x0) = By0

[(x0z0 − bx0) sin2αn + az0cos2αn

+ 1

2

(1

2z2

0 − 1

2x2

0 − bz0 + ax0

)sin2αn

]. (D15)

Now, since �A0 = −By0x0z0, by using Eq. (D1) we obtain

�An(�x0) = −By0

{z0[(x0 − a) cos2αn − 1

2 (z0 − b) sin2αn

]+ x0

[12 (x0 − a) sin2αn − (z0 − b) sin2αn

]}.

(D16)

Using the transformations between coordinates (x0,y0,z0) and(xn,yn,zn)

x0 = a + znsinαn + xncosαn,

z0 = b + zncosαn − xnsinαn, (D17)

195419-16


x0 = znsinαn + xncosαn,

z0 = zncosαn − xnsinαn, (D18)

it can be shown that, from Eq. (D16), �An = −By0xnzn. On theother hand, by using the transformations between coordinates(x0,y0,z0) and (θ,φ,r) given by Eqs. (D11) and (D12), alongwith the transformations between unit vectors in Eq. (D13),we obtain

�Ai

(�x(i)0

) = φBy0r (cosθ − 1) , 0 � i � N, (D19)

where we used the facts that αn = φn, sinφn = b/ (R + r), andcosφn = (R + r − a) / (R + r). Equation (D19) is the sameas Eq. (D14), previously obtained in an easier way, with theformal distinction in the argument of �Ai : the transformation tothe toroidal coordinates in Eq. (D14) is done from the rotatedand translated coordinate system (xi,yi,zi), since we know�Ai (�xi) and �A0 (�x0) have the same form, while in Eq. (D19) it

is from the reference coordinate system by formally derivingthe gauge transformation according to Eq. (D1).

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