Paolo Marconcini, "Simplified method for the determination of the potential landscape in the transport simulation of graphene devices", 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. 318 (2014), DOI: 10.1109/NANO.2014.6968057.

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

Simplified method for the determination of the potential landscape in the transport simulation of graphene devices Paolo Marconcini ∆VG 2

Abstract— A method is described to approximately determine the potential in the channel of a graphene device as a function of the voltages applied to the gates, once the profile at a reference bias point is known. This technique, which avoids the selfconsistent solution of the electrostatic and transport equations, is adopted to perform a simplified but fast simulation of some interesting graphene structures.

C G 2 (r)

∆U(r)/(−e) r C G 1 (r)

I. INTRODUCTION

∆VG 1

Graphene, a two-dimensional honeycomb lattice of carbon atoms first isolated from bulk graphite by Geim and Novoselov in 2004 [1], represents a very interesting material due to its outstanding physical properties and has stimulated a large theoretical and experimental research in the last few years [2]–[10]. In particular, its high electrical mobility, planar nature and one-atom thickness, and thus the possibility to use standard lithographic techniques for the definition of devices with an improved electrostatic control on the channel (a very important issue in nanodevices [11]–[13]), make it a good candidate for electronic applications [13]–[18]. In order to understand, predict and optimize the behavior of graphene-based devices, it is useful to develop efficient numerical transport simulators. An accurate simulation requires the self-consistent solution of the electrostatic and transport equations for each set of voltages applied to the gates of the device. Indeed, in order to solve the transport equation one has to know the potential profile, which derives from the solution of the electrostatic equation; however, this last equation requires the knowledge of the mobile charge distribution and thus of the wave function, which is a solution of the transport equation. Therefore the two equations should be solved iteratively until a situation in which both the equations are satisfied within a given error is reached. This procedure can be numerically very demanding and thus in some cases, in which accuracy can be partially sacrificed in favor of a strong reduction of the simulation time [19], some approximation can be introduced in the electrostatic part of the problem (and in particular in the expression of the mobile charge density) in order to decouple the two equations. Once the approximate potential landscape is obtained, it can be directly used to solve the transport equation in the device, without any self-consistent procedure. Here a method is described that, generalizing approaches previously adopted in the literature [20], [21], makes it possible to estimate the effect on the potential landscape of the voltages applied to the gates, once the profile at a given 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

Fig. 1. The electrostatic coupling between the generic point ~ r of the graphene flake and the gates is described through the geometrical capacitances CGi , which in general depend on ~ r . The variations ∆VGi of the gates voltages determine a change ∆U (~ r )/(−e) of the potential energy at the point ~ r of the graphene sample.

bias point is known [22]. This approximate electrostatic potential is then passed to an envelope-function-based code for the quantum transport simulation of graphene armchair ribbons. As an example, the Fabry-Perot resonances which appear, as a function of the voltage applied to a backgate, in the presence of cavity-shaped potentials are simulated. II. NUMERICAL METHOD Once the graphene potential profile U0 for a given bias condition (with voltages VGi0 applied to the N gates) is known (for example from experimental measurements or theoretical calculations), the method makes it possible to approximately compute the variation ∆U (with respect to U0 ) induced on the graphene potential by changes ∆VGi of the gate voltages. In this model, the electrostatic effect of the gates is described with (in general position dependent) geometrical capacitances CGi (as schematically represented in Fig. 1). The variations ∆VGi of the gate voltages electrostatically induce a change ∆ρ in the charge density of the graphene layer (with respect to the charge density ρ0 corresponding to the profile U0 ). This change in the charge density is determined by a variation in the number of occupied states and thus is related (through the local density of states of graphene) to the shift ∆U of the local dispersion relations with respect to the Fermi energy EF (which is established by the potential of the contacts). In detail, the charge density ρ = ρ0 + ∆ρ is related to the potential energy U = U0 + ∆U through the local density of states. In general, the exact local density of states depends on the local value of the probability density and thus on the behavior of the wave function, which derives from the solution of the transport equation. However, here, in order

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to decouple the derivation of the potential profile from the solution of the transport problem, it is simply approximated (in the assumption of slowly varying potential) shifting the argument of the density of states by the local value of the potential energy U (as a consequence of the local shift by U of the energy dispersion relations). The charge density can be obtained summing the contribution of the holes in the valence band and of the electrons in the conduction band (with charge +e and −e, respectively): Z U DOS(E − U )[1 − f (E − EFD )] dE ρ =e Z −∞ ∞ (1) DOS(E − U )f (E − EFS ) dE −e U

(the energy level which separates the valence and the conduction bands being equal to U ). In this equation f is the Fermi-Dirac function and, assuming no inelastic scattering, f (E − EFS ) and 1 − f (E − EFD ) represent the occupation factors of the electrons and of the holes (injected by the source and drain contacts with Fermi energies EFS and EFD , respectively). In the hypotheses of quasi-equilibrium (EFD ≈ EFS ) and low temperature (Fermi-Dirac distribution approximated with a unit step function), Eq. (1) becomes Z U DOS(E − U ) dE , (2) ρ=e EF

which represents a positive quantity if EF < U (and thus the charge is due to the holes in the valence band) and a negative quantity if EF > U (and thus the charge is due to the electrons in the conductance band). In particular, the value of ρ0 can be obtained substituting in the previous equations U0 to the potential energy U . Instead, the electrostatic relation between the change of the graphene charge density and the variations of the gate voltages is approximately expressed through the presence of the geometrical capacitances CGi (per unit area) between the gates and the graphene layer. Taking into consideration the variations of the voltages on the plates of these capacitors, i.e. ∆VGi = VGi −VGi0 on the gates and ∆U/(−e) on graphene (see Fig. 1), and summing the resulting charge variations on graphene, we have that the change of the graphene charge density is equal to   X ∆U ∆ρ = CGi (3) − ∆VGi . −e i Therefore, the variation ∆U (and thus ∆ρ) deriving from the bias changes ∆VGi can be computed solving the system of Eq. (2) and Eq. (3), i.e. (substituting U and ρ with U0 + ∆U and ρ0 + ∆ρ, respectively) the equation   X ∆U ρ0 + CGi − ∆VGi −e i (4) Z U +∆U 0

DOS(E − U0 − ∆U ) dE ,

=e

EF

where U0 is known and ρ0 is given by Z U0 DOS(E − U0 ) dE . ρ0 = e EF

(5)

Since the quantities U , U0 , ρ, ρ0 and CGi in general depend on the specific point ~r of the graphene sample, this equation has to be solved for each of these points, in order to find the spatial distribution of the potential energy U (and of the charge density ρ) in the sheet. Eq. (2) connects the total quantities ρ and U through the density of states, rather than exploiting the quantum capacitance [23] (which is a differential parameter). Therefore, this method can be applied also in the case of large variations. With respect to previously adopted approaches [20], [21], this procedure allows also the inclusion in the potential profile of contributions not deriving from the electrostatic action of the biasing gates, such as those deriving from the presence of doping or of charged impurities. This is obtained through U0 , the potential profile for the gate voltages VGi0 , which is supposed known and in which all the different effects, evaluated through experimental measurements or analytical approximations, can be considered. For large graphene samples and low energies, we can consider the following expression for the density of states [4]: DOS(E) =

2|E| π(¯ hv F )2

(6)

(which derives from the ~k · p~ dispersion relations of unconfined monolayer graphene [24], [25]). Substituting this expression into Eq. (2), the charge in the graphene sample becomes Z U 2e |E − U | dE ρ= π(¯ hv F )2 E F (7) e 2 = sign(U − EF ) (U − E ) . F π(¯ hv F )2 Therefore the relation (4) can be rewritten as:   X ∆U − ∆VGi ρ0 + CGi −e i (8) e 2 = sign(U0 +∆U −EF ) (U +∆U −E ) , 0 F π(¯ hv F )2 where ρ0 is given by ρ0 = sign(U0 − EF )

e (U0 − EF )2 . π(¯ hv F )2

(9)

Since U0 is known, ρ0 can be immediately obtained from Eq. (9). Then, in order to compute ∆U , we can solve Eq. (8) twice, under the two possible hypotheses on the sign of U0 + ∆U − EF . In each of these two cases Eq. (8) represents a second-order equation a(∆U )2 + b(∆U ) + c = 0, with X CGi e 2e a= , b= (U0 − EF ) ± , 2 2 π(¯ hv F ) π(¯ hv F ) e i X e 2 (U − E ) ∓ ρ ± c= CGi ∆VGi (10) 0 F 0 π(¯ hvF )2 i (where the upper and lower signs refer to U0 +∆U −EF ≥ 0 and U0 + ∆U − EF < 0, respectively). In both cases, this equation can be analytically solved in the unknown ∆U . If the discriminant is negative no real solution exists, while if it is positive we have to verify if one of the two possible

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III. NUMERICAL RESULTS For example, here the method has been applied to study the Fabry-Perot resonances in a large graphene ribbon with cavity-shaped potential profiles. The ribbon is biased through a backgate separated from the flake by a tox = 300 nm thick silicon oxide layer, and thus a geometrical capacitance CG = ǫ0 ǫr /tox per unit area is assumed between the backgate and the graphene ribbon (ǫ0 and ǫr = 3.9 are the vacuum permittivity and the relative permittivity, respectively). Three different profiles, varying only along the transport direction x, have been considered for U0 (the potential energy at VG = 10 V, that we take as the reference backgate voltage VG0 ): a Lorentzian shape with half-width at half-maximum 150 nm, a parabolic cavity with half-width 250 nm, and a square cavity with half-width 150 nm. In all three cases, the depth of the cavity represented by U0 is equal to 100 meV; in order to consider more realistic profiles, in the parabolic and square cases the regions containing discontinuities in the first derivative have been smoothed substituting them with parabolic connections. These profiles are reported in Fig. 2. The method that has been presented has been used to compute the potential U corresponding to different values of VG , assuming a Fermi energy EF = 0. For example,

0.02 0

U (eV)

−0.02 −0.04 −0.06 −0.08 V G =10 V

−0.1 −0.12 0

100

200

300

400

500

600

700

800

900

1000

x (nm)

Fig. 2. Dependence on x (the coordinate in the transport direction) of the three different potential profiles U0 (for VG = 10 V): Lorentzian (solid curve), parabolic (dashed curve) and square (dotted curve).

0.1 V G =−5 V 0.05 V G =5 V U (eV)

solutions ∆U that we obtain satisfies the hypothesis that has been done on the sign of U0 + ∆U − EF . At the end, the Eq. (8) has only one solution ∆U , which corresponds to the actual variation of the graphene potential. The sum of this solution ∆U and of U0 gives the potential energy U at the graphene level for the considered gate voltages. The approximate potential profile on the sample is obtained performing this calculation in each point ~r of the graphene flake. Then this profile can be passed to a transport simulator, in order to make an approximate, non self-consistent study of the electrical properties of the graphene device. For the following calculations, the numerical method described in Ref. [26] has been adopted, where an envelopefunction approximation is used to study the transport properties of an armchair graphene structure in the presence of a generic potential profile and considering the exact boundary conditions (see also Refs. [27], [28]). In analogy with the procedure that has been previously adopted for the study of devices based on semiconductor heterostructures [29]– [34], the graphene sample is divided into a series of sections in each of which the potential is approximately constant in the transport direction. In each section the envelopefunction equation (i.e. the Dirac equation [24], [25]) with Dirichlet boundary conditions is solved. In particular, with a proper analytical manipulation, the problem is recast into a differential equation with periodic boundary conditions [35], which is solved in the reciprocal space. Then the obtained solutions are used to compute the scattering matrices which connect adjacent sections. Finally, all the single scattering matrices are recursively composed and the transmission of the overall sample is obtained.

V G =0 V

0 V G =10 V −0.05

−0.1 0

200

400

600

800

1000

x (nm)

Fig. 3. Potential profile along the transport direction x for a backgate voltage VG = −5 V, 0 V, 5 V and 10 V. The profile for VG = 10 V is assumed as known, while those for VG = −5 V, 0 V and 5 V are obtained with the described approximate method.

for the case of the Lorentzian U0 the profiles obtained for VG = −5 V, 0 V and 5 V are reported in Fig. 3. Then, for each value of VG , the potential profile has been passed to the envelope-function-based transport code described in Ref. [26]. In Fig. 4 the resulting behavior of the conductance (normalized to the conductance quantum G0 = 2e2 /h, where h is Planck’s constant), as a function of the voltage VG applied to the backgate, is shown with a thick line for the case of the Lorentzian U0 . While for low and high values of the gate bias (which correspond to a profile of the potential energy U completely over or under the Fermi energy, respectively) the conductance is large, for intermediate values of VG FabryPerot oscillations, deriving from the interference between transmitted and reflected wave functions, clearly appear. Then the calculation has been repeated superimposing on the Lorentzian potential at VG = 10 V a disorder, modeled through a sum of randomly distributed Gaussian functions with a concentration 1012 cm−2 , half-width at half-maximum 4 nm, and an amplitude randomly distributed between −δ and +δ. As shown in Fig. 4 (where the results for δ = 10, 20, 30, 40 meV are reported with a dashed, dotted, dashdotted, and thin solid line, respectively), the Fabry-Perot resonances gradually disappear as the strength of the disorder is increased.

320

180

180 δ =0 δ =10 meV δ =20 meV δ =30 meV δ =40 meV

160 140

140 120 G/G 0

G/G 0

120 100 80

100 80

60

60

40

40

20

20

0

−5

0

5

δ =0 δ =10 meV δ =20 meV δ =30 meV δ =40 meV

160

10

0

15

−5

0

5

V G (V)

Fig. 4. Normalized conductance, as a function of the backgate voltage, obtained with the Lorentzian profile U0 with δ = 0 (no disorder), 10 meV, 20 meV, 30 meV and 40 meV.

15

Fig. 6. Normalized conductance, as a function of the backgate voltage, obtained with the square profile U0 with δ = 0 (no disorder), 10 meV, 20 meV, 30 meV and 40 meV.

180

10 δ =0 δ =10 meV δ =20 meV δ =30 meV δ =40 meV

160 140 120

8 6

100

V G (V)

G/G 0

10

V G (V)

80

4 2

60

Cavity with parabolic profile −2

20 0

Cavity with Lorentzian profile

0

40

−5

0

5

10

−4

15

V G (V)

Cavity with square profile 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18

Peak index

Fig. 5. Normalized conductance, as a function of the backgate voltage, obtained with the parabolic profile U0 with δ = 0 (no disorder), 10 meV, 20 meV, 30 meV and 40 meV.

Then the simulation has been repeated for the parabolic and the square cavities, both in the absence and in the presence of potential disorder. The results are reported in Fig. 5 (parabolic U0 ) and in Fig. 6 (square U0 ). Also in these cases, Fabry-Perot oscillations are clearly visible in the transmission, at least in the presence of a sufficiently weak potential disorder. In Fig. 7 the values of the backgate voltage for which the maxima of the Fabry-Perot oscillations in the conductance appear are reported (as a function of an index identifying the peaks) for the three different potential profiles that have been considered in the simulations. Finally, in order to test the validity of the method, a comparison has been performed with the results achieved with NanoTCAD ViDES, a transport code (available online [36]) which self-consistently solves the electrostatic and transport equations, using a tight-binding model and the NEGF formalism for the solution of the quantum transport equation. Since this solver relies on an atomistic description of the structure, it has been necessary to consider just a very narrow graphene ribbon (in order to limit the requirements in terms of memory and of CPU time). In detail, a 10 nm long and 4 nm wide ribbon (with 32 dimer lines across its width) has been simulated, with a backgate underneath the ribbon, insulated by a 1 nm thick silicon oxide layer. Above the nanoribbon a 1 nm thick and 1 nm wide transversal stripe of silicon oxide,

Fig. 7. Values of the gate voltage for which the maxima of the Fabry-Perot oscillations in the conductance appear, reported as a function of an index identifying the peaks, for the three different potential profiles: Lorentzian (dots), parabolic (triangles) and square (squares). The curves represent only a guide for the eyes.

containing a uniformly distributed positive fixed charge with a concentration of 1026 m−3 , has been considered. By means of the self-consistent simulator the potential profile at the graphene level for a backgate voltage VG = 0.5 V has been computed, and this result has been used as the potential U0 in the approximate calculation. In Fig. 8, the profiles obtained in the middle of the ribbon with the described approximate calculation for VG = −0.5 V and VG = −3 V, along with those obtained for the same backgate voltages with the selfconsistent solver, are reported. The agreement is surprisingly good, since the approximate calculation is rigorously valid only for unconfined graphene (the density of states of 2D graphene has been used), while the comparison has been performed for a narrow ribbon. However, it can be noted that this approximate technique can be extended to the analysis of narrow ribbons substituting in the calculation the expression (6) of the density of states with that reported in Ref. [23] for graphene nanoribbons. Moreover, in order to overcome the hypotheses of quasiequilibrium and low temperature, it is also possible to substitute Eq. (2) with Eq. (1). These generalizations, however, require to substitute the purely algebraic solution that we have presented with a numerical solution procedure that increases the simulation times.

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0.8 VG =−3 V

0.6

U (eV)

0.4 VG =−0.5 V 0.2 0 VG =0.5 V

−0.2 −0.4 0

2

4

6

8

10

x (nm)

Fig. 8. Comparison between the results obtained with the described approximate method (solid red curve) and with a self-consistent solver (dashed blue curve) in the case of a narrow graphene ribbon. The profile for VG = 0.5 V, obtained with the self-consistent solver, has been used as the potential U0 in the approximate calculation.

IV. CONCLUSIONS A method has been presented to approximately calculate the potential profile in a graphene structure under the electrostatic action of biased gates, once the profile for a reference bias point is known. This approach, which is numerically very efficient and which allows the inclusion in the potential landscape also of effects not deriving from the gate bias, has been used to study the variation with the voltage applied on a backgate of a cavity-shaped potential profile in a graphene flake. The obtained results have been passed to an envelopefunction-based transport code, in order to study the FabryPerot oscillations in cavities with different shapes and in the presence of different levels of potential disorder. R EFERENCES [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science, vol. 306, p. 666, 2004. [2] A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nature Materials, vol. 6, p. 183, 2007. [3] A. K. Geim, “Graphene: Status and Prospects,” Science, vol. 324, p. 1530, 2009. [4] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys., vol. 81, p. 109, 2009. [5] T. Enoki, T. Ando, Physics and Chemistry of Graphene: Graphene to Nanographene, Singapore: Pan Stanford Publishing, 2013. [6] M. I. Katsnelson, Graphene: Carbon in Two Dimensions, Cambridge: Cambridge University Press, 2012. [7] D. R. Cooper, B. D’Anjou, N. Ghattamaneni, B. Harack, M. Hilke, A. Horth, N. Majlis, M. Massicotte, L. Vandsburger, E. Whiteway, V. Yu, “Experimental Review of Graphene”, ISRN Condensed Matter Physics, vol. 2012, p. 501686, 2012. [8] S. Mikhailov, Physics and Applications of Graphene, Rijeka: InTech, 2011. [9] M. R. Connolly, R. K. Puddy, D. Logoteta, P. Marconcini, M. Roy, J. P. Griffiths, G. A. C. Jones, P. A. Maksym, M. Macucci, C. G. Smith, “Unraveling Quantum Hall Breakdown in Bilayer Graphene with Scanning Gate Microscopy”, Nano Lett., vol. 12, p. 5448, 2012. [10] P. Marconcini, M. Macucci, “A novel choice of the graphene unit vectors, useful in zone-folding computations,” Carbon, vol. 45, p. 1018, 2007. [11] J. Dura, S. Martinie, D. Munteanu, F. Triozon, S. Barraud, Y.M. Niquet, A. Jean-Luc, “Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices,” in Electrostatics, H. Canbolat, Ed., Rijeka: InTech, 2012, pp. 113–136.

[12] P. D’Amico, P. Marconcini, G. Fiori, G. Iannaccone, “Engineering Interband Tunneling in Nanowires With Diamond Cubic or Zincblende Crystalline Structure Based on Atomistic Modeling,” IEEE Trans. Nanotechnol., vol. 12, p. 839, 2013. [13] F. Schwierz, “Graphene transistors,” Nature Nanotech., vol. 5, p. 487, 2010. [14] H. Raza, Graphene Nanoelectronics: Metrology, Synthesis, Properties and Applications, Heidelberg: Springer, 2012. [15] L. E. F. Foa Torres, S. Roche, J.-C. Charlier, Introduction to GrapheneBased Nanomaterials: From Electronic Structure to Quantum Transport, Cambridge: Cambridge University Press, 2014. [16] P. Avouris, “Graphene: Electronic and Photonic Properties and Devices,” Nano Lett., vol. 10, p. 4285, 2010. [17] G. Iannaccone, G. Fiori, M. Macucci, P. Michetti, M. Cheli, A. Betti, P. Marconcini, “Perspectives of graphene nanoelectronics: probing technological options with modeling,” Proceedings of IEDM 2009, IEEE Conference Proceedings, p. 245, 2009, DOI: 10.1109/IEDM.2009.5424376. [18] P. Marconcini, A. Cresti, F. Triozon, G. Fiori, B. Biel, Y.-M. Niquet, M. Macucci, S. Roche, “Atomistic Boron-Doped Graphene FieldEffect Transistors: A Route toward Unipolar Characteristics,” ACS Nano, vol. 6, p. 7942, 2012. [19] P. Marconcini, “The role of the choice of the physical model in the optimization of nanoelectronic device simulators,” International Journal of Circuits, Systems and Signal Processing, vol. 7, p. 173, 2013. [20] A. Das, B. Chakraborty, S. Piscanec, S. Pisana, A. K. Sood, A. C. Ferrari, “Phonon renormalization in doped bilayer graphene,” Phys. Rev. B, vol. 79, p. 155417, 2009. [21] A. Das, S. Pisana, B. Chakraborty, et al.: “Monitoring dopants by Raman scattering in an electrochemically top-gated graphene transistor,” Nature Nanotech., vol. 3, (4), p. 210, 2008. [22] P. Marconcini, M. Macucci, “Approximate calculation of the potential profile in a graphene-based device,” IET Circuits Devices Syst., DOI: 10.1049/iet-cds.2014.0003. [23] T. Fang, A. Konar, H. Xing, D. Jena, “Carrier statistics and quantum capacitance of graphene sheets and ribbons,” Appl. Phys. Lett., vol. 91, p. 092109, 2007. [24] P. Marconcini, M. Macucci, “The k·p method and its application to graphene, carbon nanotubes and graphene nanoribbons: the Dirac equation,” La Rivista del Nuovo Cimento, vol. 34, no. 8-9, p. 489, 2011. [25] T. Ando, “Theory of Electronic States and Transport in Carbon Nanotubes,” J. Phys. Soc. Jpn., vol. 74, p. 777, 2005. [26] D. Logoteta, P. Marconcini, C. Bonati, M. Fagotti, M. Macucci, “Highperformance solution of the transport problem in a graphene armchair structure with a generic potential,” arXiv:1401.1178. [27] P. Marconcini, M. Macucci, “Symmetry-dependent transport behavior of graphene double dots,” J. Appl. Phys., vol. 114, p. 163708, 2013. [28] P. Marconcini, D. Logoteta, M. Macucci, “Sinc-based method for an efficient solution in the direct space of quantum wave equations with periodic boundary conditions,” J. Appl. Phys., vol. 114, p. 173707, 2013. [29] P. Marconcini, M. Macucci, G. Iannaccone, B. Pellegrini, G. Marola, “Analysis of shot noise suppression in mesoscopic cavities in a magnetic field,” Europhys. Lett., vol. 73, p. 574, 2006. [30] P. Marconcini, M. Macucci, G. Iannaccone, B. Pellegrini, “Quantum analysis of shot noise suppression in a series of tunnel barriers,” Phys. Rev. B, vol. 79, p. 241307(R), 2009. [31] R. S. Whitney, P. Marconcini, M. Macucci, “Huge conductance peak caused by symmetry in double quantum dots,” Phys. Rev. Lett., vol. 102, p. 186802, 2009. [32] M. Totaro, P. Marconcini, D. Logoteta, M. Macucci, R. S. Whitney, “Effect of imperfections on the tunneling enhancement phenomenon in symmetric double quantum dots,” J. Appl. Phys., vol. 107, p. 043708, 2010. [33] P. Marconcini, M. Macucci, D. Logoteta, M. Totaro, “Is the regime with shot noise suppression by a factor 1/3 achievable in semiconductor devices with mesoscopic dimensions?,” Fluct. Noise Lett., vol. 11, p. 1240012, 2012. [34] P. Marconcini, M. Totaro, G. Basso, M. Macucci, “Effect of potential fluctuations on shot noise suppression in mesoscopic cavities,” AIP Advances, vol. 3, p. 062131, 2013. [35] M. Fagotti, C. Bonati, D. Logoteta, P. Marconcini, M. Macucci, “Arm-

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[36] Code is available at “NanoTCAD ViDES”, DOI: 10254/nanohub-r5116.5; http://nanohub.org/resources/vides/

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