Electron Spectrum and Tunneling Current of the Toroidal and Helical Graphene Nanoribbon-Quantum Dots Contact

Abstract

We study the electron spectrum and the density of states of long-wave electrons in the curved graphene nanoribbon based on the Dirac equation in a curved space-time. The current-voltage characteristics for the contact of nanoribbon-quantum dot have been revealed. We also analyze the dependence of the specimen properties on its geometry.

1. Introduction

The problem of modified graphene properties attracts a considerable attention of researchers [1, 2] because the “pure” graphene has no energy gap in the band structure and, therefore, the creation of different structures (e.g., analogs of transistors) is extremely difficult. However, the situation becomes more promising when various modifications of the specimen are introduced. As an example, we consider the modified graphene, for example, graphene nanoribbon, which have quantized electron energy spectrum due to the limited space in one dimension, which in turn can lead to the formation of the energy gap. Furthermore, we know that the graphene has a naturally wave-like curved surface due to the instability of the planar structure of its sheets [3, 4]. All of the above reasons have stimulated the study of different modifications of a curved graphene [5, 6]. The long-wave approximation, which is widely used to describe the properties of electrons in graphene, leads to an analog of the Dirac equation, which in turn makes it easy to produce generalization to the case when the graphene surface is curved. Note that in this case the degeneracy in the Dirac points is removed and therefore it becomes possible to create various structures with different band gaps. Consideration of the Dirac equation for curved graphene [5] also shows a change in the density of states of electrons and, therefore, it is possible to change the whole set of electrical characteristics of the graphene sample. Apparently, the easiest way to experimentally verify the changes in the density of states is to study the tunneling current [7], for example, through the contact with quantum dots. Quantum dots are still rather “young” objects of study, but their use in various fields of science and technology is obviously extremely promising (from the design of lasers and new generation displays to building qubits) [8].

2. Basic Equations and Spectrum of Electrons

We consider the graphene nanoribbon, which is curved along the toroidal and the helical surfaces, as presented in Figure 1.

(a)

(b)

Figure 1 

Geometry of a problem; (a) toroidal nanoribbon, (b) helical nanoribbon.

Properties of electrons in the graphene nanoribbons in the long-wave approximation in the vicinity of the Dirac point will be described on the basis of the Dirac equation generalized for the case of a curved space-time [9]: 𝛾^𝜇𝜕_𝜇−Ω_𝜇Ψ=0.(1) Here and below the repeated indices assume summation; 𝜕_𝜇 is the partial derivative with respect to coordinate 𝜇, Ω_𝜇 is the component of the spin connection, and Ψ=^𝜙𝜑 is the wave function (column vector) consisting of wave functions describing the electrons from different sublattices near the Dirac point.

As is known [9, 10], if we are given the metric tensor𝑑𝑠²𝑔=_𝛼𝛽𝑔_𝛼𝛽𝑑𝑥^𝛼𝑑𝑥^𝛼,𝑔^𝛽𝛾=𝛿^𝛾𝛼(2) (𝛿^𝛾𝛼: delta Kronecker symbol) then, we can define the field frames (tetrads): 𝑔𝑔_{𝛼𝛽𝛼𝛽}=𝑒^𝑎𝛼𝑒^𝑏𝛽𝜂_𝑎𝑏,𝜂_𝑎𝑏=𝑒^𝛼𝑎𝑒^𝛽𝑏𝜂^𝑎𝑏,𝜂^𝑏𝑐=𝛿^𝑐𝑎,(3) where, for the two-dimensional curved surface, we have 𝜂_𝑎𝑏=diag(1,−1,−1). Then,Ω_𝜇=14𝛾^𝑎Γ_𝜆𝜇𝜎𝛾_𝑏𝑒^𝑎𝜆𝑔^𝜆𝜎𝜕_𝜇𝑒^𝑏𝜎−Γ^𝜆𝜇𝜎𝑒^𝑏𝜆,=12𝑔^𝜆𝜈𝑔_{𝜎𝜈,𝜇}𝛾_𝜇+𝑔_{𝜈𝜇,𝜎}−𝑔_{𝜇𝜎,𝜈},=𝑒^𝜇𝑎𝛾_𝑎.(4) Using the torus and the helical parameterization𝑥𝑦==𝑅𝑅++𝑟𝑟ccooss𝑥𝑥₁₁cos𝑥₂,𝑧=𝑟sin𝑥₁sin𝑥₂,𝑥=𝑥₁,𝑦=𝑥₁cos𝑥₂,𝑧=ℎs⋅in𝑥₂𝑥₂,,(5) we find that the metrics on the torus surface and the helicoid is𝑑𝑠²𝑑=𝑠₂𝑑𝑥²⁰−𝑟²𝑑𝑥²¹−𝑅+𝑟cos𝑥₁²𝑑𝑥²²,=𝑑𝑥²⁰−𝑑𝑥²¹−ℎ²+𝑥²¹𝑑𝑥²².(6) Note that all the Christoffel symbols are equal to zero, except Γ²¹² and Γ¹²². For the torus, we have Ω₀=0, Ω₁=0, Ω₂=(1/2)𝛾₁𝛾₂𝑓^/𝑟(𝑓=𝑅+𝑟cos𝑥₁,and𝑓^=𝜕𝑓/𝜕𝑥₁), while in the case of helicoid Ω₀=0, Ω₁=0, and Ω₂=(1/2)𝛾₁𝛾₂(𝑥₁/(ℎ²+𝑥²¹)^1/2).

Choosing 𝛾₀=𝜎₃, 𝛾₁=−𝑖𝜎₂, and 𝛾₂=−𝑖𝜎₁, where 𝜎 are the Pauli matrices, we obtain the following system of equations:𝑉^𝐹−1𝜕_𝑡𝜑=−1𝑟²𝜕_𝑥1Ψ−𝑖𝑓²𝜕_𝑥2Ψ+𝑓′𝑉2𝑓²𝑟Ψ,_𝐹−1𝜕_𝑡𝜑=−1𝑟²𝜕_𝑥1𝜑+𝑖𝑓²𝜕_𝑥2𝜑+𝑓′𝑉2𝑓²𝑟𝜑,_𝐹−1𝜕_𝑡𝜑+𝜕_𝑥1Ψ+𝑖ℎ²+𝑥²¹𝜕_𝑥2Ψ−𝑥¹2ℎ²+𝑥²¹^3/2−𝑉_𝐹−1Ψ=0,𝜕_𝑡Ψ−𝜕_𝑥1𝜑+𝑖ℎ²+𝑥²¹𝜕_𝑥2𝜑−𝑥¹2ℎ²+𝑥²¹^3/2𝜑=0.(7) Here, 𝑉_𝐹 is the Fermi velocity for planar graphene, 𝜕₀=𝑉^𝐹−1𝜕_𝑡. Note that since the metric (6) admits two Killing vectors corresponding to translations along 𝑥₀,𝑥₂, the solution (7) can be found in the formΨ𝜑⟶𝜑𝑥¹Ψ𝑥₁𝑒^{𝑖𝐸𝑡−𝑖𝑘𝑥2},(8) which finally gives Ψ^=−𝐸²𝑟⁴𝑉^2𝑓+𝑘^2𝑛𝑟⁴𝑓⁴Ψ+2𝑟𝑓𝑓^2Ψ^+2𝑘^𝑛𝑟²𝑓^𝑓³+𝑟𝑓^2𝑓²−𝑟𝑓^2𝑓³−𝑟²𝑓^2Ψ4𝑓⁴Ψ,_=⎛⎜⎜⎝−𝜀²𝑉^2𝑓+𝑘²ℎ²+𝑥²¹²+⎛⎜⎜⎝−𝑘𝑥₁⎞⎟⎟⎠Ψℎ²+𝑥²¹^5/2+𝑥²¹4ℎ²+𝑥²¹³⎞⎟⎟⎠Ψ.(9) Note that the wave vector 𝑘 is found from the boundary conditions at the ends of the nanoribbon. In our particular case, we have chosen the armchair-type ribbon [6] and𝑘_𝑛=2𝜋3𝑎₀2𝑀2𝑀++1+1𝑛,(10) where 𝑎₀ is the distance between the atoms in the carbon lattice, 𝑀 is the number of atoms along nanoribbon axis, and 𝑛 is the quantum number. We can consider (9) as a Schrodinger equation with perturbation𝑉_Torus=2𝑘^𝑛𝑟²𝑓′𝑓³+𝑟𝑓^2𝑓²−𝑟𝑓′²𝑓³−𝑟²𝑓′²4𝑓⁴+2𝑟𝑓𝑓′²𝜕_𝑥𝑉_Helicoid,=⎛⎜⎜⎝−𝑘𝑥¹ℎ²+𝑥²¹^5/2+𝑥²¹4ℎ²+𝑥²¹³⎞⎟⎟⎠.(11) In this case, the spectrum of perturbation looks like𝐸=±𝑘^2𝑛+𝑘^2𝑦.(12) Expanding the functions in the denominator in Taylor series to the second order, we calculate the first perturbation correction to the spectrum, 𝑉_Torus and 𝑉_Helicoid, as follows:𝐸₁=Ψ^∗𝑉Ψ𝑑𝑥,Ψ_𝑛=𝐴⋅Sin𝑘_𝑛𝑥₁.(13) The integration is performed from 0 to 𝐿=(3𝑀+1)𝑎₀, and the corrections are as follows:𝐸₁=2𝐿−4𝑟𝑅²²Sin(𝐿)+𝑟²8𝑅²1−𝑘1_𝑛Sin2𝑘^𝑛−1𝐿+𝑟2𝑘_𝑛−1₂8𝑅²1+𝑘1_𝑛Sin2𝑘^𝑛+1𝐿𝐸2𝑘_𝑛+1₁,=2𝐿−𝑘4𝐿ℎ²⁵+𝑘4ℎ⁵𝑘^2𝑛+ℎ^−2/3𝐿³+Sin2𝑘_𝑛24𝐿𝑘𝐿4ℎ⁵𝑘_𝑛+ℎ^−2/3321𝑘^3𝑛−𝐿²+Cos2𝑘_𝑛16𝑘_𝑛𝐿−𝑘4ℎ⁵𝑘^4𝑛+ℎ^−2/3𝐿16𝑘^2𝑛.(14) The dependence of the perturbations on the atom numbers along the nanoribbons 𝑀 is presented in Figure 2.

(a)

(b)

Figure 2 

Dependence of the correction to the energy 𝑉 caused by the perturbation 𝑉 on atoms number along nanoribbon axis 𝑀: (a) for torus (𝑟/𝑅=0.1,𝑛=1); (b) for helicoid (ℎ=1.5): (a)𝑛=1,(b)𝑛=2,(c)𝑛=3.

The dependence shown in Figure 2(a) is rather complex, which is associated with the quantization of the electron spectrum in graphene nanoribbons in accordance with (10). It should be noted that the dependence of the energy gap in carbon nanotubes of zigzag type is pretty much similar, which also arises from the quantization of the electron spectrum in the direction along the circumference of the nanotube. The calculations (Figure 2(b)) show that the value of the helicoids parameterization ℎ influences most strongly the correction to the energy (as well as its sign).

3. Tunneling Characteristics

The Hamiltonian of the system of electrons can be written in the following form:𝐻=_𝑝𝐸^𝐴𝑝𝑎^+𝑝𝑎_𝑝+_𝑞𝐸^𝐵𝑞𝑏^+𝑞𝑏_𝑞+_𝑝𝑞𝑇_𝑝𝑞𝑎^+𝑝𝑏_𝑞+𝑏^+𝑞𝑎_𝑞,(15) where 𝑎^+𝑝,𝑎_𝑝 are the electron creation and annihilation operators with momentum 𝑝 in the carbon nanoribbons, 𝐸^𝐴𝑝 is the electron spectrum of the carbon nanoribbons (12) with taking into account (14), 𝑇_𝑝𝑞 is the matrix element of tunneling operator between 𝑝 and 𝑞 states, 𝑏^+𝑞,𝑏_𝑞 are the electron creation and annihilation operators with momentum 𝑞 in a substance which is in a contact with a carbon nanoribbon, and 𝐸^𝐵𝑝 is the electron spectrum of another substance. It should be noted that 𝑝 and 𝑞 are multi-indices in (15). So, for graphene nanoribbon (further, we consider an arm-chair nanoribbon only), 𝑝=(𝑝_𝑦,𝑛), 𝑛=0,1,…,𝑀−1. Multi-index 𝑞 is determined by the substance which contacts with carbon nanoribbon and, for example, for quantum dots 𝑞=(𝑝_𝑥,𝑝_𝑦,𝑝_𝑧), whereas for grapheme 𝑞=(𝑝_𝑥,𝑝_𝑦). A consideration of the external electric field 𝐸⃗ (and choosing 𝐸⃗=−(1/𝑐)(𝜕𝐴/⃗𝜕𝑡)) can be carried out by the replacement 𝑝→𝑝−𝑒𝐴/𝑐.

Tunneling current is considered to be given by𝐽=𝑖𝑒_𝑝𝑞𝑎^+𝑝𝑏_𝑞−𝑏^+𝑞𝑎_𝑝.(16) With a gauge transformation [11],𝑆=e𝑎x_𝑝p⟶𝑖𝑒𝑉𝑆𝑡⁻¹_𝑝𝑎_𝑝𝑆,𝑎^+𝑝𝑎_𝑝,(17) where 𝑉is the applied voltage and 𝑒 is the electron charge. Formally, it is possible to reduce a problem of calculation of the current-voltage characteristics to the calculation of the operator response𝐽_𝑡=𝑖𝑒_𝑝𝑞𝑎^+𝑝𝑏_𝑞𝑒^{𝑖𝑒𝑉𝑡}−𝑏^+𝑞𝑎_𝑞𝑒^{−𝑖𝑒𝑉𝑡}(18) on the external influence [11]𝐻_𝑡=_𝑝𝑞𝑇_𝑝𝑞𝑎^+𝑝𝑏_𝑞𝑒^{𝑖𝑒𝑉𝑡}+𝑏^+𝑞𝑎_𝑞𝑒^{−𝑖𝑒𝑉𝑡}.(19) The solution was obtained within the framework of the Kubo theory:𝐽=4𝜋𝑒||𝑇||²^∞−∞𝜈𝑑𝐸𝜈_𝐴(𝐸+𝑒𝑉)𝜈_𝐵(𝐸)𝑛_𝑓(𝐸)−𝑛_𝑓(𝐸+𝑒𝑉),_𝐴(𝐸)=_𝑝𝛿𝐸−𝐸^𝐴𝑝;𝜈_𝐵(𝐸)=_𝑞𝛿𝐸−𝐸^𝐵𝑞,(20) where 𝛿(𝑥) is the Dirac delta function, 𝑣_𝐴(𝐵)(𝐸) is the tunneling density of states, and 𝑛_𝑓(𝐸) is the equilibrium number of fermions with the energy 𝐸. The approximation of a “rough” contact is hereinafter used, so that 𝑇_𝑝𝑞=𝑇 (this imposes certain restrictions on the contact geometry, i.e., for the case discussed below means that nanoribbon should be perpendicular to the contact material surface). For definiteness, we choose the dispersion law for the graphene nanoribbons given by (12) and (14) and the dispersion law for the quantum dots as the contact material being𝐸^𝐴𝑞=𝜀₀−Δcos(𝑝),(21) where 𝜀₀ is the electron energy of a quantum well, Δ is the tunneling integral determined by the overlap of electron wave functions in the adjacent wells, and the momentum 𝑝 is directed along the axis 𝑍.

Equation (20) under study has been solved numerically. The current-voltage characteristic of the contact is presented in Figure 3.

Figure 3 

The current-voltage characteristic of the contact: curved grapheme nanoribbon-quantum dots: (a) for torus; (b) for helicoid.

Figure 3 shows the asymmetric behavior of current versus voltage applied to the contact. This is due to both the peculiarities of the electronic structure (density of states) of the metal and graphene nanoribbon and the processes of carrier recombination in the transition contact, which dominate over the thermal processes when 𝑉>0. The resulting dependence may have important practical applications in the study of nanocontacts and the design of tunnel diodes based on graphene nanoribbons. Also, the region with negative differential resistance was observed for some values of 𝑉. The presence of such region allows the use of a tunnel diode as a high-speed switch.

4. Conclusion

(1)An effective equation, which describes the electrons in a curved graphene nanoribbon and the tunneling contact nanoribbon-quantum dot has been obtained.(2)The dependence of the electron spectrum on the geometry has been revealed. In general, the first perturbation correction to the spectrum is defined by the atoms number along the nanoribbon axis 𝑀 and by the parametric coefficient, which describes the “curvature” of nanoribbon, 𝑟/𝑅 (in the torus nanoribbon case), and the step for helical nanoribbon ℎ. (3)The current-voltage characteristics for the contact graphene nanoribbon-quantum dots was constructed; the regions with negative differential conductivity were found.

Acknowledgments

This work was supported by the Russian Foundation for Basic Research under project no. 08-02-00663 and by the Federal Target Program “Scientific and pedagogical manpower” for 2010–2013. This work was partially supported by NSERC and CRC Program, Canada, and by the Russian Foundation for Basic Research under project no. 11-02-97054-r_povolzhè_a.