Effects of localized trap-states and corrugation on charge transport in graphene nanoribbons

We investigate the role played by electron traps on adiabatic charge transport for graphene nanoribbons in the presence of an acoustically induced longitudinal surface acoustic wave (SAW) potential. Due to the weak longitudinal SAW-induced potential as well as the strong transverse confinement by a nanoribbon, minibandsof sliding tunnel-coupled quantum dots are formed so that by varying the chemical potential to pass through the minigaps, quantized adiabatic charge transport may be obtained. We analyze the way that the minigaps may be closed, thereby destroying the likelihood of current quantization in a nanoribbon. We present numerical calculations showing the effects due to electron traps which lead to localized-trap energy levels within the minigaps. Additionally, for comparison, we present results for the minibands of a corrugated nanoribbon in the absence of a SAW.


I. INTRODUCTION
A considerable amount of research work has been carried out so far on the design and improvement of electronic devices which are based on the use of quantized adiabatic charge transport. [1][2][3][4][5][6][7][8][9][10] Moreover, under a surface-acoustic wave (SAW), the inelastic capture and tunneling escape effects on the non-adiabatic transport of photo-excited charges in quantum wells was also investigated. [11] The underlying challenge is to produce a device with an accuracy for the quantized current of one part in 10 8 on the plateaus. When this goal is achieved, one application of this device would be in metrology for standardizing the unit of current.
At the present time, a SAW is launched on a piezoelectric heterostructure, such as GaAs/AlGaAs, and GHz single/few-electron pumps have been gaining close scrutiny due to the fact that the measured currents lie within the nanoamp range, high enough for the measured current to be suitable as a current standard. However, these pumps have so far been capable of delivering electrons/holes in each cycle of a sliding dynamic quantum dot (QD), giving rise to a quantized current with an accuracy of one part in 10 6 as reported in Refs. [3][4][5][6][7][8]. Interestingly, in Ref. [12] a measurement was carried out of the noise accompanying a 3-GHz SAW pump. It was observed in this experiment that the current near the lowest plateau, corresponding to the transfer of one electron per SAW cycle, is dominated by shot noise. However, away from the plateau, the noise is attributed to electron traps in the material. There have been some attempts to increase the flatness of the plateaus by applying magnetic fields. [13,14] Some time ago, a proposal was put forward by Thouless [10] which would make use of quantized adiabatic charge transport. This adiabatic approach involves the use of a one-dimensional (1D) electron system subjected to a slowlysliding periodic potential. Relatively simple analysis indicates that in such a 1D system minigaps are generated in instantaneous electronic spectra as a function of the SAW amplitude. With the use of a gate, the chemical potential can be varied by applying a voltage to the gate. Consequently, when the chemical potential lies within a minigap, there will be an integral multiple of electron charge transported across the system during a single time period. [1] In other words, by combining with the strong transverse confinement of a nanoribbon, the weal longitudinal SAW potential has induced a series of dynamic (sliding) tunnel-coupled QDs whose impenetrable"wall is constructed through destructive interference of the electronic wave functions around a minimum of the SAW potential. In principle, such an adiabatictransport device could provide an important application, like a current standard. Talyanskii, et al. [1] investigated the physical mechanisms of quantized adiabatic charge transport in carbon nanotubes for a SAW to produce a periodic potential required for miniband/minigap formation.
In the presence of a SAW, the scattering effects from impurities embedded in a 1D electronic system are expected to play an important role on the flatness of a current plateau. The current quantization should be completely smeared out when the level broadening from impurity scattering becomes comparable to the minigaps of dynamic tunnel-coupled QDs. On the other hand, we can also simulate localized electron traps by superposing a series of negative δ-potentials onto a SAW potential within each spatial period. Consequently, we expect a set of localized trap states occurring within the minigaps of dynamic QDs. This provides an escape channel for the QD-confined electrons being carried by the SAW. This trap mechanism is quite different from the impurity one [1] where a spatial average with respect to the distribution of impurities within a dynamic QD is inevitable due to a SAW.
In this paper, we consider a 1D Dirac-like electron gas in a graphene nanoribbon in the presence of a SAW. We will introduce two mechanisms for miniband formation. First, the nanoribbon is modulated by a longitudinal potential from a SAW. Secondly, the nanoribbon is periodically corrugated. We notice that the second mechanism does not lead to a quantized current but instead produces traps for Dirac electrons, thereby limiting electron mobility. Our numerical calculations reveal that localized electron trap states are an effective mechanism to adversely affect the adiabatic transport because the localized-trap levels lying within the minigaps are very sensitive to the phase of either the SAW or the corrugation-induced potential. Varying the weight or the position of the δ-potential leads to different positions of localized trap levels within the minigaps of the nanoribbon. Therefore, these inevitable fluctuations of the trap potential in a realistic system would most likely impede the current quantization.
The rest of the paper is organized as follows. In Sec. II, we present the formalism for calculating band structure with localized trap states for nanoribbons in the presence of a SAW. In Sec. III, numerical results for nanoribbons in the absence/presence of a SAW and those for corrugated nanoribbons in the absence of a SAW are presented to demonstrate and explain the localized trap states within the minigaps. The conclusions drawn from these results are briefly summarized in Sec. IV.

II. MINIBAND STRUCTURE WITH LOCALIZED TRAP STATES
The work done by Talyanskii, et al. [1] on quantum adiabatic charge transport focused on the coupling between a semimetallic carbon nanotube and a SAW. The electron backscattering from the SAW potential is used to induce a miniband spectrum. The electron interactions enhance the minigaps thereby improving current quantization. The effect due to impurities in the carbon nanotube is averaged by a SAW potential.
For the cases of a semimetallic carbon nanotube, semiconducting carbon nanoribbon with applied SAW potential and corrugated nanoribbon, the energy levels are given by the spectra of discretized 1D Dirac Hamiltonian (see the Appendix A for detailed derivations) The eigenvalue problem is defined within the spatial interval 0 < x < 2π/k and assumes periodic boundary conditions. In this notation, k stands for either the wave number k SAW of the SAW potential or the wave number k c of an effective potential induced by the corrugation. The discretization of the Hamiltonian is provided by the mesh x n = nδx with n ∈ 0 . . . N − 1 and δx = 2π/kN .
For either the carbon nanotube or graphene nanoribbon, the parameters for the Hamiltonian matrix in Eq. (1) are given by a n = 0 , where we have introduced the SAW and impurities combined phase with λ = 2A/( ṽ F k SAW ) the normalized SAW amplitude,ṽ F is the Fermi velocity of graphene. Additionally, ) and x 0 denote the normalized trap-potential amplitude and position of the short-range dynamic trap for electrons, respectively, and the trap is sliding together with the SAW potential. The mass term involving ∆ is the original energy gap for the system in the absence of a SAW. In case of a nanotube, ∆ may be generated by a magnetic field. For a nanoribbon, the gap is structural for the semiconducting nanoribbon ṽ F k being the transverse electron wave number due to finite size across the ribbon. For nanoribbons, the explicit form for the phase introduced in Eq. (3) can be found from the Appendix A.
As far as the minigaps are concerned, the effect due to the SAW potential on the nanoribbon may be compared with corrugation. A sinusoidal corrugated semiconducting ribbon can be mapped on to a flat ribbon. The mapping introduces an additionalσ 1 term into the Dirac equation, as described in Appendix A. This yields where the corrugation-induced gap is given by ∆ c (x n ) = 1 + C 2 k 2 c cos 2 (k c x n ), with C being the normalized amplitude of the corrugation, and k (m) y is the quantized wave number across the nanoribbon.

III. NUMERICAL SIMULATION AND DISCUSSIONS
In our numerical calculations, all the energies in Figs. 1 and 2, such as ε, ∆ and E g , are normalized to ṽ F k SAW . The SAW potential amplitude A is also normalized to ṽ F k SAW . Additionally, all the energies in Fig. 3, such as ε and ∆, are measured in units of ṽ F k c , and the corrugation amplitude C is normalized to 1/k c . Besides, the transverse wave number k (m) y in all the plots is scaled by 2π/3a 0 . In this way, we are able to draw some universal conclusions concerning the effects due to minigaps.
In Fig. 1, we compare the energy band structure of nanoribbons for two values of ∆ in the absence of electron traps. Two values of k (m) y were chosen (light and dark colors) to describe the two lowest energy levels (see the discussions in the Appendix A). The minigaps are generated by a sliding dynamic QD and they oscillate as a function of the SAW amplitude A, as may be verified using perturbation theory, vanishing at values close, but generally not equal, to the roots of Bessel functions. Increasing or decreasing the value of ∆ results in a shift of the nodes on the graph as evidenced by comparing our results in Fig. 1. Therefore, ∆ determines not only the magnitude of the original gap in the absence of a SAW but also the size of the minigaps in the presence of a SAW. Higher energy minigaps are partially closed by the energy levels corresponding to a larger value of k We now introduce electron traps into our nanoribbon by superposing a negative δ-potential onto the SAW potential so as to simulate a short-range Coulomb interaction. In this case, the position of the trap is fixed in the moving SAW frame of reference, which is quite different from embedded impurities in a nanostructure. In the moving SAW frame, the embedded impurities are moving against the dynamic QDs created by both the transverse dimension of the nanostructure and the longitudinal SAW potential. This results in an average of the impurity effects with respect to these dynamic QDs in the longitudinal direction. As seen in the results presented in Fig. 2, localized trap states occur within the minigaps once the weight of the trap potential V 0 becomes larger than ∆/( ṽ F k SAW ). Relative energy value of these trap states in the presence of SAW is sensitive to the position x 0 of the trap within a dynamic QD. If we set λ = V 0 , then the contribution to α(x) from the trap located in the nodes of the SAW potential is fully compensated by the cosine term in Eq. (3). As a result of this compensation, the localized trap states will disappear from the gap and minigap regions. If we extend the single-trap model employed in this paper to a uniform distribution of traps, the fluctuations in the phase term α(x) [see Eq. (3)] would fill up the entire minigap region with a delocalized trap band. Consequently, the adiabatic approximation may not be applicable. In other words, to satisfy the adiabatic assumption, one must have dominance of the SAW potential, i.e., λ V 0 must be satisfied.
We compare the results for SAW-based dynamic QDs in Figs. 1 and 2 with those for static QDs created by corrugation on a graphene nanoribbon in the absence of a SAW and electron traps. This we do by displaying in Fig. 3 the minigaps induced by the corrugation. We find from the figure that minigaps only exist for finite but small values of the corrugation amplitude C. This means that these minigaps are generally much less than those induced by a SAW. As a matter of fact, the existence of non-vanishing diagonal terms a n given in Eq. (4), effectively mitigates the phase fluctuations in the off-diagonal terms b n . This keeps the minigaps open and the energy spectra robust even after traps have been introduced to cause a fluctuation in the phase term α c (x).
Finally, let us assume that a narrow channel is formed within a two-dimensional electron-gas layer lying in the xyplane. We will neglect the finite thickness of the quantum well for the heterostructure in the z-direction and consider the electron motion as strictly two dimensional. We will employ one of the simplest models for the gate-induced or etched [15] confining electrostatic potential. In this way, a 1D channel is formed on the two-dimensional electron-gas layer and the dynamics of massive electrons can be modeled by a discretized 1D Schrodinger equation. However, from numerical results (not shown here), we find no evidence of the minigaps for this model, i.e., the minigaps are the characteristics of Dirac fermions.

IV. CONCLUDING REMARKS
In conclusion, we have calculated in this paper the energy band structure for graphene nanoribbons, embedded with a single electron trap, upon which a SAW is launched. Our results show that localized trap states appear in the minigaps. More importantly, the location of the trap state-energy level is determined by the positions of the trap with respect to the phase of the sinusoidal SAW. Consequently, the adiabatic approximation might not be appropriate whenever the minigap is less or comparable with the weight of a short-range δ-potential for the trap (see Fig. 2). On the other hand, in Fig. 1, where there are no electron traps, a larger value of ∆ in the energy spectrum leads to a substantial increase in the number of minigaps as well as the ballistic current quantization. Periodic corrugation of the nanoribbon may be used instead of a SAW as a mechanism for inducing minigaps. Those are expected to be less sensitive to the presence of charged impurities or electron trap potentials.
Appendix A: Energy Band Calculations in Absence of Electron Traps

Carbon Nanotubes
The electron eigenstates in a semi-metallic nanotube are described by a 1D Dirac equation. For simplicity, a noninteracting system is considered here. Under the stationary approximation, the single particle energy spectrum ε(k) is obtained from the following perturbed 1D Dirac equation In this notation, k represents the electron wave number along the nanotube,ṽ F is the Fermi velocity of Dirac electrons, A is the SAW amplitude, ∆ is the energy gap of the system in the absence of a SAW, α and β label the two sublattices of graphene from which the nanotube is rolled. In addition, we require k = k SAW to satisfy momentum conservation, where k SAW is the wave number of a SAW propagating along the nanotube. To explore the miniband structure due to quantum confinement in the radial direction, a gauge transformation is implemented and is defined by where λ = 2A/( ṽ F k) is the normalized SAW amplitude. Substituting Eq. (A3) into Eqs. (A1) and (A2), we obtain +∆ e (i/2)λ cos(kx) ψ β (x) + A sin(kx) e (−i/2)λ cos(kx) ψ α (x) .

By introducing the following identities
Eqs. (A5) and (A 1) may be simplified as Furthermore, by employing the basis setΨ (x) ≡ ψ α (x), ψ β (x) , the above equations can be rewritten into a compact matrix form, given by We will solve the eigenvalue problem within the spatial interval 0 < x < 2π/k and introduce the N -point mesh x n = n δx, where n ∈ 0 . . . N − 1 and δx = 2π/kN . In this way, the derivative on the mesh can be approximated by . Especially, on this spatial mesh, the Hamiltonian in Eq. (A11) may be projected into the matrix given in Eq. (1).

Graphene Nanoribbons
Here, we consider an armchair graphene nanoribbon lying along the x-direction. The total number of carbon atoms (in both sublattices) across the ribbon is assumed to be M . The armchair edges mix up the graphene K and K valleys so that the wave function becomeŝ where the transverse wave number is given by m = 0, ±1, ±2, · · · is an integer, L is the nanoribbon width, and a 0 = √ 3a/2 (a ≈ 1.42Å) is the size of the unit cell in graphene. Nanoribbons with width L/a 0 = 3M + 1 give rise to the following relation and it is clear that the minimum energy occurs at k Those nanoribbons are semiconducting with the energy gap determined by The x-component of the wave function in Eq. (A12) may be determined by where the ± signs correspond to K and K valleys. Additionally, the Hamiltonian in Eq. (A18) can be transformed into the form in Eq. (A11) after applying the following unitary transformation where α(x) = −A cos(k SAW x)/( ṽ F k SAW ). As a result, the transformed Hamiltonian takes the form Formally, this Hamiltonian is equivalent to that in Eq. (A11) after we applying the following substitutions The valley sign ± does not change anything due to the mirror symmetry in the energy dispersion relation ε(k) with respect to ±k.

Corrugated Nanoribbons
By making use of the finite-difference method for calculating ∂/∂x along with the following basis set the transformed Hamiltonian matrix in Eq. (A23) may be projected aŝ The above Hamiltonian matrix has the same form as Eq. (1). . The graph at the top comes from the lowest quantized energy subband, whereas the graph at the bottom is due to the first-excited subband.