Paolo Marconcini, "Transport simulation of armchair graphene ribbons with a generic potential in the presence of an orthogonal magnetic field", Proceedings of the 14th IEEE Conference on Nanotechnology, Toronto, Canada, 18-21 August 2014, IEEE Conference Proceedings, IEEE Catalog Number: CFP14NAN-USB, ISBN: 978-1-4799-4082-0, p. 543 (2014), DOI: 10.1109/NANO.2014.6968055.

Proceedings of the 14th IEEE International Conference on Nanotechnology Toronto, Canada, August 18-21, 2014

Transport simulation of armchair graphene ribbons with a generic potential in the presence of an orthogonal magnetic field Paolo Marconcini Abstract— The effect of an orthogonal magnetic field is introduced into a numerical simulator, based on the solution of the Dirac equation in the reciprocal space, for the study of transport in armchair graphene ribbons with a generic potential. Different approaches are proposed and their accuracy and efficiency is compared. The code is applied to simulate armchair ribbons with different potential profiles and to analyze the effect of an increasing magnetic field on their transport behavior.

I. INTRODUCTION In the last few years, graphene has attracted a significant interest in the scientific and industrial community, due to its outstanding electrical, optical, mechanical, thermal and chemical properties [1], [2]. Graphene exhibits also very uncommon physical properties: for example, the electric behavior of charge carriers is described by the Dirac equation [3] (i.e. the same relation that describes relativistic massless quantum particles) and thus in graphene relativistic effects (such as Klein tunneling [4]) appear at non relativistic speeds. Other anomalous phenomena include the unconventional integer quantum Hall effect in the presence of a magnetic field [5] and the presence of a negative refractive index in p-n junctions [6]. Therefore, many experimental, theoretical and numerical efforts have been devoted to the study and simulation of graphene and in particular of graphene-based devices [7]–[16]. Recently, we have presented a method [17]–[19] to investigate the transport properties of graphene structures with armchair edges in the presence of a generic electrostatic potential, using an envelope function model with an exact description of the boundary conditions. Following this approach, the device is subdivided into sections in which the potential is approximately longitudinally constant (a technique similar to the one that we have used for the simulation of 2DEG-based GaAs/AlGaAs devices [20]–[25]). In each of these sections the envelope function equation (which in the case of monolayer graphene corresponds to the Dirac equation [3]) with Dirichlet boundary conditions can be recast into an equivalent differential problem with periodic boundary conditions [26], which can be efficiently solved in the reciprocal space. Then, the continuity of the wave functions on the two sublattices of graphene across the interfaces between adjacent sections is enforced. Projecting the continuity equations onto a basis of sine functions and solving the resulting linear system, the scattering matrix Paolo Marconcini is with Dipartimento di Ingegneria dell’Informazione, Universit`a di Pisa, Via Girolamo Caruso 16, 56122 Pisa, Italy

[email protected]

978-1-4799-5622-7/$31.00 ©2014 IEEE

of the region extending across each interface is obtained. Finally, the scattering matrices are recursively combined and, exploiting the expression of the current corresponding to each transport mode, the transmission matrix (and, from it, the conductance) of the overall device is obtained. Here I discuss how it is possible to include the effects (non due to spin) of a magnetic field, orthogonal to the graphene plane, in this single-particle quantum simulator, and I show the results obtained for some simple structures using this extended code. II. NUMERICAL METHOD ~ can be taken into The presence of a magnetic field B ~ ~ ~ in account substituting the operator −i∇ with −i∇+(e/¯ h )A ~ ~ ~ the Dirac equation [3] (where B = ∇×A, e is the modulus of the electron charge, and ¯ h is the reduced Planck’s constant). The ribbon can be divided into sections (with effective ˜ ) where the potential U (x, y) and the vector powidth W ~ y) are approximately independent on x (x and tential A(x, y represent the coordinates in the transport and transverse direction, respectively) . In each section the four envelope functions Fβα~ (~r), corresponding to the 2 inequivalent Dirac ~ K ~ ′ and to the 2 graphene sublattices β = points α ~ = K, A, B, can be written as the product of a plane wave propagating along x and of a confined component in the y direction: ~ Fβα~ (~r) = eiκx x Φα β (y). Therefore the Dirac equation with Dirichlet boundary conditions becomes   e e d ~   + I Ax + σz i Ay + σx f (y) ϕ ~ K (y) = σz   dy h ¯ h ¯     ~   − κx ϕ ~ K (y)        d e e ~′ −σz + I Ax − σz i Ay + σx f (y) ϕ ~ K (y) = dy h ¯ h ¯    ~′ K   − κx ϕ ~ (y)     K ~′ ~   ~ K (0) ϕ ~ (0) = ϕ     K ~′ ˜ ~ ˜ ˜ ˜ K ) = ei2K W ϕ ~ (W ϕ ~ (W ) (1) where " # " # ~ ~′ K ′ (y) ΦK (y) Φ ~ ~ K K A A ϕ ~ (y) = , ϕ ~ (y) = i , ~ ~′ ΦK ΦK B (y) B (y) (2) f (y) =

U (y) − E hv F ¯

(where vF is the Fermi velocity of graphene, E is the ˜ = K − round (K W ˜ /π) · π/W ˜ , with injection energy, and K K = 4π/(3a) and a the graphene lattice constant).

543

If we introduce the function ( ~ ϕ ~ K (y) ϕ ~ (y) = ˜ ˜ K ~′ ˜ − y) ei2K W ϕ ~ (2W

˜] y ∈ [0, W ˜ , 2W ˜ ], y ∈ [W

(3)

˜] the system can be rewritten on the doubled domain [0, 2W in this form:     d σz +g(y)I +d(y)σz +h(y)σx ϕ ~ (y) = −κx ϕ ~ (y) dy (4)  ˜W ˜  i2K ˜ ϕ ~ (2W ) = e ϕ ~ (0) ,

where

e ˜ − |W ˜ − y|)), Ax (¯ x, W h ¯ e ˜ − y)Ay (¯ ˜ − |W ˜ − y|), x, W d(y) = i sign(W h ¯ ˜ − |W ˜ − y|), h(y) = f (W

g(y) =

(5)

x ¯ being the abscissa identifying the section. This represents a differential problem with periodic boundary conditions for ˜ the function e−iKy ϕ ~ (y), that can be solved in the reciprocal space. In the equation, we can substitute the functions ˜ e−iKy ϕ ~ (y), g(y), d(y), and h(y) (extended by periodicity ˜ ) with their Fourier expansions: with period 2W ˜

e−iKy ϕ ~ (y) =

∞ X

∞ X ˜ ˜ ~am eiπmy/W , g(y) = gℓ eiπℓy/W

m=−∞

∞ X ˜ d(y) = dℓ eiπℓy/W , ℓ=−∞

ℓ=−∞ ∞ X

h(y) =

(6)

˜

hℓ eiπℓy/W .

ℓ=−∞

Then, projecting the equation onto a basis of plane waves of ˜ ˜ the form ei((πn/W )+K)y , we obtain  +∞ h  X πn ˜ σz δn,m + +K i ˜ W (7) m=−∞ i (gn−m I + dn−m σz + hn−m σx ) ~am = −κx~an

for each n. This set of equations represents an (in principle, infinite-dimensional) linear problem in the Fourier compo˜ nents ~an of the function e−iKy ϕ ~ (y). Exploiting the decrease with frequency of the Fourier coefficients of g(y), d(y), and h(y), we can solve the problem considering in the system only the indexes |n|, |m| ≤ D, with an integer D such that πD ˜ ≫ max |gj |, max |dj |, max |hj |, (8) + K W ˜ j j j

and obtaining the Fourier coefficients ~an with |n| ≤ D. Then, exactly as in the absence of magnetic field [17], the Fourier coefficients ~an can be directly exploited to compute all the projections that are required for the calculation of the scattering matrices between adjacent sections. Since the relation between the current of each transport mode and the corresponding envelope functions does not change in the presence of a magnetic field, also these currents can be computed using the same procedure that is adopted in the absence of magnetic field [17].

The transport simulation can then be concluded combining all the single scattering matrices and obtaining the total transmission matrix, and thus (using the Landauer-B¨uttiker approach) the conductance of the device. ~ exist for each Infinite choices for the vector potential A ~ fixed magnetic field B [27]. Let’s consider a magnetic field orthogonal to the graphene ~ = B zˆ (where zˆ is the unit vector orthogonal to structure B the ribbon) and let’s assume B constant. A possible choice for the vector potential is the gauge ~ = Bx A ¯yˆ. With this gauge g(y) = 0, while d(y) is an odd imaginary function represented by a square wave, the Fourier coefficients of which are dn = 0 for even n, and dn = (e/¯ h)(2B x ¯/(πn)) for odd n. The fact that these coefficients decrease very slowly (as 1/n) clearly limits the possibility to reduce the size of the numerical problem represented by the reciprocal space solution of the Dirac equation in each section. However, using this Landau gauge an alternative approach is possible. It is easy to prove that, with this gauge, the Dirac equation with Dirichlet boundary conditions in each section ~ (Eq. (1)) is satisfied, with longitudinally constant U and A ~ by the in the presence of an orthogonal magnetic field B, same solutions as in the absence of magnetic field, apart from a multiplication of the functions Φβα~ (y) by a Peierls factor e−i(e/¯h)Ay y = e−i(e/¯h)(B x¯)y . In particular, the longitudinal wave vectors κx remain the same and thus preserve the Z2 × Z2 symmetry which characterizes them in the absence of magnetic field [17], [26]. Following this alternative approach, we can solve the differential system in each section in the absence of magnetic field, and then we can include the effect of the Peierls factors in the other computations. While the Peierls factors do not affect the evaluation of the currents (where each Peierls factor is multiplied by its complex conjugate), during the scattering matrix calculation they slightly increase the numerical complexity of the projections of the envelope functions onto the basis functions. While the two approaches we have described give the same results, the first one is less efficient than the second one, since it requires the solution of a linear system with larger size in each section. The use of this gauge can be generalized also to the case ~ = B(x, y)ˆ of a non constant magneticRfield B z , considering x ¯ ~ a vector potential A = [ 0 B(x, y)dx] yˆ. If B does not depend on y both the approaches we R x¯have described can still be adopted, substituting B x ¯ with 0 Bdx in the Fourier coefficients of d(y) and in the Peierls factors. Instead, if B depends on y only the first method can be generalized, numerically computing the Fourier coefficients of the odd function d(y), which in that case differs from a simple square wave. However, the choice of this gauge has some drawbacks. ~ Since using this gauge the magnetic vector potential A depends on x, in order to consider sections with nearly ~ the device has to be longitudinally constant U and A

544

U (eV)

0.1

0.1

0.05

0.05

0

U (eV)

0

−0.05

−0.05

−0.1

−0.1

0 100 x (nm)

200

100 y (nm)

0

200

Fig. 1. Nonzero potential, consisting in the sum of two Gaussian functions, that has been considered in the 200 nm wide and 250 nm long semiconducting armchair graphene ribbon. 12 B =0 T B =0.25 T B =0.5 T B =0.75 T B =1 T

10 8 G/G0

subdivided into a large number of thin sections. In each of them the magnetic flux has to be much less than the flux quantum h/e (where h is Planck’s constant), in such a way to reduce the discretization errors which derive from ~ as longitudinally constant inside each section. considering A The presence of a large number of sections strongly increases the computational burden of the transport simulation. This gauge has another serious drawback. As we have seen, if we adopt this choice, in each section the solutions in the presence of magnetic field are equal to those in the absence of magnetic field, apart from the multiplication of the envelope functions by a Perierls factor, which does not alter the current associated with each mode. Therefore, the number of propagating modes in each thin section remains identical with and without magnetic field. The correct wave function and transport behavior in the presence of magnetic field then is obtained when, through the scattering matrix calculation, the mode-mixing introduced by the longitudinal variation of the vector potential is taken in account and the contributions of all the sections are composed. However, the accuracy of the method reduces when the number of propagating modes entering and leaving the structure is low. In particular, adopting this gauge, if for a given energy the geometry and the potential of the entrance or exit leads preclude the propagation of transport modes in those sections in the absence of magnetic field, the same thing happens also in the presence of magnetic field. Therefore, for that energy the Landauer-B¨uttiker formula gives a null conductance, a result that can differ from the real value. In order to avoid this kind of problems, we have tested another possible gauge. In particular, another vector potential ~ = B zˆ is A ~ = −Byˆ corresponding to B x. In this case, when we solve the Dirac equation in each ~ we have section with longitudinally constant U and A that d(y) = 0, while g(y) is a real even function represented by a triangle wave, with Fourier coefficients g0 = ˜ /2), gn = 0 for nonzero even n, and gn = −(e/¯ h)(B W ˜ (e/¯ h)2B W /(nπ)2 for odd n. The solutions inside each section lose the close similarity (that we have observed using the previous gauge) with those in the absence of magnetic field, and in particular the Z2 × Z2 symmetry of the longitudinal wave vectors κx disappears. However, since in this case the vector potential does not depend on x, it does not represent any more a constraint for the length of the sections, which now is determined only by the variations along x of the potential energy U . In general, this strongly reduces the number of sections that has to be considered in the calculation and thus the time required for the simulation. Moreover, while the previous gauge, reproducing the effect of the constant magnetic field through the dependence of the vector potential on x, was able to give the correct wave function and conductance behavior only when the contributions of all the sections were composed, using this alternative gauge the magnetic field reveals its effects, and in particular modifies the number of propagating modes, already inside the single section. Therefore, using this gauge

6 4 2 0 −0.06

−0.04

−0.02

0

0.02

0.04

0.06

Energy (eV)

Fig. 2. Normalized conductance (as a function of the injection energy) that has been obtained, in the presence of an orthogonal magnetic field of 0 T, 0.25 T, 0.5 T, 0.75 T and 1 T, for a null potential energy, including two 5 nm long sections with a potential of -0.5 eV at the entrance and exit of the ribbon. In this case, the results obtained with the three presented methods coincide.

it is possible to correctly simulate also structures which in the absence of magnetic field have no (or a low number of) propagating modes in their entrance and exit sections. Also this gauge can be generalized to the case of a ~ = B(x, y)ˆ generic, spatially-varying magnetic field B z. ~ = In Rthis case we have to consider a vector potential A y ˆ (that now can depend, in general, also [− 0 B(x, γ)dγ] x on x) and the length of the sections has to be chosen in such a way that inside each of them both the potential ~ are approximately energy U and the vector potential A independent of x. In each section the function g(y) = RW ˜ −|W ˜ −y| B(¯ x, γ)dγ is not simply a triangle wave −(e/¯ h) 0 but a real even function for which the Fourier coefficients gn have to be numerically computed. However, with this small change, the method can be used to simulate the transport behavior of an armchair graphene structure in the presence of a generic orthogonal magnetic field. III. NUMERICAL RESULTS Here the results that have been obtained, using the discussed methods, for a 200 nm wide and 250 nm long semiconducting armchair ribbon in the presence of a magnetic field are shown. The simulations have been performed both in the absence of potential, and in the presence of a potential landscape given by the sum of two Gaussian functions with standard deviation 10 nm and amplitude 0.1 eV and −0.1 eV (see Fig. 1).

545

9

8 B =0 T B =0.25 T B =0.5 T B =0.75 T B =1 T

7

G/G0

6 5

7

A=(Bx)y

6

A=(−By)x

5 G/G0

8

4

4 3

3 2

2

1

1

0 −0.06

−0.04

−0.02

0

0.02

0.04

0 −0.06

0.06

−0.04

−0.02

0

0.02

0.04

0.06

Energy (eV)

Energy (eV)

Fig. 3. Normalized conductance (as a function of the injection energy) that has been obtained, in the presence of an orthogonal magnetic field of 0 T, 0.25 T, 0.5 T, 0.75 T and 1 T, for the nonzero potential profile shown in Fig. 1, including two 5 nm long sections with a potential of -0.5 eV at the entrance and exit of the ribbon. In this case, the results obtained with the three presented methods coincide.

Fig. 4. Normalized conductance (as a function of the injection energy) that has been obtained, in the presence of an orthogonal magnetic field of 1 T, for a null potential energy, without low-potential sections at the entrance and exit of the ribbon, using the first gauge (dashed line) and the second gauge (solid line). 14 B =0 T B =0.25 T B =0.5 T B =0.75 T B =1 T

12 10 G/G0

In the first simulations two 5 nm long sections with a potential of -0.5 eV have been included at the entrance and exit of the structure. These low-potential regions, which represent a possible model for the contacts [28], increase the number of propagating modes at the entrance and exit, and require the inclusion in the simulation of a large number of modes. In Fig. 2 and Fig. 3 the behavior of the conductance (normalized to the conductance quantum G0 = 2e2 /h, where h is Planck’s constant) as a function of the energy of the charge carriers in the presence of a magnetic field of 0 T, 0.25 T, 0.5 T, 0.75 T, and 1 T are shown (for the case of null and non null potential, respectively). In these cases, the three different techniques that have been described (the first two, which exploit the gauge with non null Ay , and the third one, which uses the gauge with non null Ax ) give the same results. However, the third method is more efficient than the first two, especially when the potential is null and thus the only factor limiting the length of the slices is the variation of the vector potential along the transport direction. In particular, the first method that has been presented, requiring the solution of large-size eigenproblems, is the most time-consuming one. Then we have performed the simulations without including the low-potential sections at the entrance and exit. In Fig. 4 we report the normalized conductance for a null potential in the presence of a magnetic field of 1 T, obtained with the first two approaches (which give the same results) and with the third one. In this case, the first two give less accurate results, especially for the energies for which in the absence of magnetic field no propagating mode exists in the input and output sections of the structure. Indeed, the choice of the first gauge forces at zero the conductance for these energies, leaving unchanged the energy gap with respect to the case of no magnetic field (contrary to the correct result [29], [30]). Repeating the simulations of Fig. 2 and Fig. 3, this time without the sections at -0.5 eV and using only the second gauge, the conductance behavior represented in Fig. 5 and Fig. 6 has been obtained (with a magnetic field of 0 T, 0.25 T,

8 6 4 2 0 −0.06

−0.04

−0.02

0

0.02

0.04

0.06

Energy (eV)

Fig. 5. Normalized conductance (as a function of the injection energy) that has been obtained, in the presence of an orthogonal magnetic field of 0 T, 0.25 T, 0.5 T, 0.75 T and 1 T, for a null potential energy, without including low-potential sections at the entrance and exit of the ribbon, using the second gauge.

0.5 T, 0.75 T, and 1 T) for the null and non null potential, respectively. In particular, considering the case of null potential (Fig. 5) we see that in the absence of magnetic field the conductance of the ribbon exhibits steplike increases by a single conductance quantum each time the threshold energy for the propagation of a further conduction mode is reached. Instead, when a large magnetic field is applied the conductance increases by steps with energy assuming values equal to 2n + 1 conductance quanta, with n = 0, 1, 2, . . ., in agreement with the unconventional integer quantum Hall effect theoretically predicted and experimentally observed in monolayer graphene [5]. These conductance steps (which are present for high magnetic fields also in all the other results shown here) correspond to the presence of edge states propagating along the borders of the ribbon, and appear more clearly as the magnetic flux across the overall device increases. Comparing these results with the dispersion relation obtained through a tight-binding calculation (which include the effect of the nonzero magnetic field through the multiplication by proper Peierls factors of the transfer integrals between nearest-neighbor atoms [31]), we have

546

12

8 G/G0

model. This is particularly useful for the simulation of quite large graphene samples, for which an atomistic analysis would be too computationally expensive.

B =0 T B =0.25 T B =0.5 T B =0.75 T B =1 T

10

6

R EFERENCES

4 2 0 −0.06

−0.04

−0.02

0

0.02

0.04

0.06

Energy (eV)

Fig. 6. Normalized conductance (as a function of the injection energy) that has been obtained, in the presence of an orthogonal magnetic field of 0 T, 0.25 T, 0.5 T, 0.75 T and 1 T, for the nonzero potential profile shown in Fig. 1, without including low-potential sections at the entrance and exit of the ribbon, using the second gauge.

verified that the energy threshold of these conduction steps correspond to the edges of the energy bands of the ribbon in the presence of magnetic field. In particular, the energy gap of the semiconducting ribbon decreases with the application of the magnetic field. This energy-gap modulation, which has been previously observed in the literature through tightbinding simulations and experimental measurements, could be exploited for the fabrication of graphene devices with a large magnetoresistance effect [29], [30], [32]–[35]. Analogous considerations are valid in the presence of a non null potential landscape (Fig. 6). IV. CONCLUSIONS Three possible methods have been presented to include the effect of an orthogonal magnetic field into a numerical code for the simulation of the transport properties of armchair graphene structures, based on a scattering matrix approach and on the solution of the envelope function equation in the reciprocal space. In particular, two different gauges have been considered for the vector potential: the first one with a tranversally oriented vector potential, the second one with a longitudinally oriented vector potential. For the first gauge, two alternative methods have been discussed: one based on the numerical solution of the transport equation in the presence of magnetic field, and another one based on the use of Peierls phase factors. Comparing the accuracy and the efficiency of the three approaches, the second gauge has turned out to be the most convenient. The described methods have been applied to the simulation of armchair ribbons with different potential profiles, and the transport behavior characteristic of the unconventional integer quantum Hall effect has been obtained for high values of the magnetic field. The presented simulators can be used (by coupling them with a Poisson solver in order to perform a complete selfconsistent calculation, or by exploiting an approximate calculation of the potential profile [36]) to simulate the transport characteristics of graphene-based devices in the presence of an orthogonal magnetic field using an envelope-function

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