Electron Transport in Graphene-Based Double-Barrier Structure under a Time Periodic Field

Transcription

1 Commun. Theor. Phys. 56 (2011) Vol. 56, No. 1, July 15, 2011 Electron Transport in Graphene-Based Double-Barrier Structure under a Time Periodic Field LU Wei-Tao ( å ) 1, and WANG Shun-Jin ( ) 2, 1 Department of Physics, Linyi University, Linyi , China 2 Department of Physics, Sichuan University, Chendu , China (Received November 8, 2010; revised manuscript received January 13, 2011) Abstract The transport property of electron through graphene-based double-barrier under a time periodic field is investigated. We study the influence of the system parameters and external field strength on the transmission probability. The results show that transmission exhibits various kinds of behavior with the change of parameters due to its angular anisotropy. One could control the values of transmission and conductivity as well as their distribution in each band by tuning the parameters. PACS numbers: Gk, Bd Key words: graphene-based double-barrier structure, Klein tunneling, external field 1 Introduction Great interest has been aroused in the investigation of the physical properties of graphene and graphene-based structures, since their experimental realization was carried out. [1] is a one-atom-thick sheet of carbon and consists of honeycomb lattice. The interaction between electrons and the honeycomb lattice causes the electrons to behave as if they have absolutely no mass. At low energy, the electrons in graphene could be described by an effective massless Dirac equation, having a linear energy dispersion E = v f k, where v f is the Fermi velocity. The linear dispersion and chirality of electrons make this material quite different from conventional solid-state materials, and lead to a number of unusual electronic properties such as unconventional quantum Hall effect, [2 5] strong electric-field effect, [6] finite minimal conductivity, [2] and Klein tunneling. [7] Graphene provides an opportunity for us to study the fundamental physics, and its notable properties also have the potential in technological applications. Klein tunneling predicts that a relativistic electron can penetrate and pass through a high potential barrier to approach perfect transmission. In contrast, for conventional non-relativistic tunneling, the transmission probability exponentially decays with increasing barrier height. Matching between electron and positron wavefunctions across the barrier leads to the high-probability tunneling described by Klein tunneling. Such a phenomenon in various graphene-based structures such as single-barrier and double-barrier structures, [7 10] quantum wells and dots, [11 13] n-p junctions, [14] and superlattice [15 16] has been investigated theoretically and experimentally. Recently, Sabeeh [8] and Trauzettel [17] researched electron transport in graphene under a time-oscillating field, respectively. The transport of conventional electron in various types of system with time-oscillating potential regions has been studied widely. In this paper, under the timeoscillating field, the transmission probability of electron through a monolayer graphene-based double-barrier is calculated numerically. The effect of external field on conductivity is studied with different values of system parameters. The relationship between transmission and conductivity is also discussed. The rest of the paper is organized as follow. In Sec. 2, we introduce the model. The numerical results and discussions are presented in Sec. 3. Section 4 is a brief summary. 2 Model The system under consideration is a monolayer graphene-based double-barrier structure driven by a timeoscillating field, which could be realized by two gate electrodes and an ac signal applied to the surface of graphene. In the vicinity of the Dirac points, the low-energy excitations are described by the two-dimensional massless Dirac equation, [v f (σ p) + V ]ψ = Eψ, (1) where v f m/s, the pseudospin matrix σ = (σ x, σ y ) is Pauli matrix, and p = (p x, p y ) is the momentum operator. V has a one-dimensional double-barrier and single well structure with external field V 1 cos(ωt) set in the well region, and its schematic profile is shown in Fig. 1. The static barrier is V 0. The widths of the barrier and well are presented by D and L, respectively. Based on perturbation theory, the external field will lead to new Supported in part by the National Natural Science Foundation of China under Grant Nos , , , , and , and by the Fund of Nuclear Theory Center of HIRFL of China twh4108@126.com sjwang@home.swjtu.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 164 Communications in Theoretical Physics Vol. 56 channels of electron transmission corresponding to the energies E ± n ω. Because the system is homogeneous along the y direction, and the y component of momentum is conserved. ψ 3 (x, y, t) = e ikyy ψ l (x, y, t) = e ikyy + m= + m= ( ) ( ) ] 1 [A 3m e ikmx B 3m e ikmx s m e iφm s m e iφm We assume that the incident electron propagates with an angle φ along the x axis. According to Tien-Gordon theory, [18] the solution of Eq. (1) in the five regions for a given incident energy E could be written as: n= ( V1 ) J n m e i(e+n ω)t/, (2) ω ( ) ( ) ] 1 [A lm e ikmx 1 + B lm e ikmx e i(e+m ω)t/, (3) s m e iφm s m e iφm where k = E/ v f, k y = k sin φ, φ m = arctan(k y /k m ), J n (V 1 / ω) is the n-th order Bessel function, and l = 1, 2, 4, 5. In region 1, A 1m = δ 0m, and the reflection coefficient r m = B 1m. In region 5, B 5m = 0, and the transmission coefficient t m = A 5m. In well regions 1, 3, and 5, k m = k mx = ((E + m ω)/ v f ) 2 ky 2 and s m = sgn(e + m ω). In barrier regions 2 and 4, k m = q mx = ((E + m ω V 0 )/ v f ) 2 ky 2 and s m = sgn (E + m ω V 0 ). The oscillating well results in infinite number of band states with energies E ± n ω. Reflected and transmitted waves distribute in central band and sidebands, satisfying + n= r n 2 + t n 2 = 1. Matching the wave functions at the boundaries of the five regions, we derive a set of infinite number of coupled equations. Actually, these equations can be truncated into finite number of equations starting from N to N, and N V 1 / ω, [8,19] which could be solved numerically. Then the transmission probability for the n-th sideband T n is obtained from T n = t n 2. Fig. 1 Schematic profile for monolayer graphene-based double-barrier structure driven by a time-oscillating field. The structure could be partitioned into five regions. After the transmission coefficients are obtained, the total conductivity can be calculated. According to Landauer Büttiker formula, [20] the angularly averaged conductivity at zero temperature is given by: π/2 G = g 0 T(E, sin φ)cosφdφ, (4) π/2 where g 0 = 2 e 2 mv f L y / 2, L y is the width of the graphene strip along y axis. 3 Numerical Results and Discussions In this section, we will study the transport and conductivity properties of the Dirac electron in graphene. The values of parameters are set temporarily as follow: the strength of the barrier is V 0 = 200 mev, the barrier width is D = 50 nm, and the well width is L = 50 nm. The Fermi wavelength of the incident electron is λ = 50 nm, and the corresponding Fermi energy is E 83 mev. A fixed modulation frequency is used with ω = Hz due to the limit of the numerical method. In all the following numerical calculations, the values of parameters are fixed as above, unless otherwise specified. Transmission probabilities T ±n as functions of the well width L and external strength V 1 for normal incidence φ = 0 and incidence with an angle φ = π/10 are shown in Figs. 2 and 3, respectively. For normal incidence (see Fig. 2), small V 1 leads to weak external strength, and small L leads to narrow driven zone, so result in little influence on transport. Therefore, the central band plays a dominant role to tunneling for small V 1 and L, and sidebands have no contribution nearly. With the increasing of V 1 and L, more and more sidebands will become important. From Figs. 2 and 3, one could find that all transmission probabilities T ±n present roughly a Bessel function-like oscillation, since t ±n is proportional to J n (γv 1 / ω), where γ is determined by system parameters, [21] indicating that the parameters determine electron transport. Due to the symmetry of this driven system, the electron distributes in sidebands ϕ n and ϕ n with equal probability for φ = 0. Thus, T n and T n display uniform change along V 1 and L. However, the symmetry does not hold for the incidence other than normal one (see Fig. 3). Thus, T n and T n display kindred-but not conform change along V 1 and L. Compared with Fig. 2, the phase diagram (L V 1 ) with regular tori is destroyed and exhibits many islands in Fig. 3, implying that the symmetry is destroyed. These islands will become smaller and finally disappear with the increase of incident angle, similar to that in chaotic dynamics. Therefore, one could control the transmission of different bands by changing external strength and well width. Figure 4 shows the angular dependence of transmission probabilities at different values of L and V 1. Solid line and dashed line correspond to transmissions in static single barrier and double barriers, respectively, and the result is consistent with previous work. [7,15] For special incident angle, for instance, φ satisfies q 0x D = nπ for static single barrier system, it may lead to resonant tunneling, namely, Klein tunneling, where the backscattering

3 No. 1 Communications in Theoretical Physics 165 process is suppressed. The resonant condition should be given by a function f(q 0x, D, L) = nπ for static double barriers. [16] In the presence of external field (V 1 0), the probability T is spread among the central band and sidebands, due to the appearance of band states. The probabilities T ±n exhibit complicated character for large value of φ. It is difficult to determine the positions of resonant tunneling of sidebands, but the resonant condition should relate to system parameters q mx, D, L, and V 1, that is f(q mx, D, L, V 1 ) = nπ. many new properties. Figure 5 shows the effect of barrier width D on transport for different values of V 1 and φ. It can be seen that, for normal incidence, T and T n (= T n ) are independent of D, and transport is Klein tunneling at V 1 = 0. However, T and T ±n oscillate with the increase of D at φ = π/10, and the oscillation frequencies increase with increasing static barrier V 0. One may get Klein tunneling at the peaks satisfying resonant condition. Indeed, without the field in the well region, the transport behavior depending on L is similar to that here. Fig. 2 Transmission probabilities as functions of L and V 1 for φ = 0. Fig. 5 Transmission probability as a function of D for different values of V 1 and φ. Straight dashed lines correspond to T and T ±n for φ = 0, and oscillatory curves below them are T and T ±n for φ = π/10, respectively. Fig. 3 Transmission probabilities as functions of L and V 1 for φ = π/10. Fig. 4 Transmission probability as a function of φ for different values of V 1 and L. Solid line and dashed line correspond to static single barrier (L = 0, V 1 = 0) and static double barriers (L = 50 nm, V 1 = 0), respectively. Dotted line, dash-dotted line, and dash-dot-dotted line correspond to T 1, T 0, and T 1 of double barriers driven by external field (L = 50 nm, V 1 = 10 mev), respectively. The above discussed property of electron transport is analogous to that in periodic driven single barrier. [8] Nevertheless, electron in double-barrier system also manifests The energy E dependence of transmission probabilities and variable mod (q 0x D, π) at different values of φ and V 1 is shown in Fig. 6. From Figs. 6(b) and 6(c), it can be seen that, unlike conventional transport, there exists a well in T and T ±n curves. The width of this well increases with increasing incident angle, and the well disappears for normal incidence. In the absence of external field, for normal incidence, one may get t = e 2iD(k0x+q0x) from Eqs. (2) and (3), and T = t 2 = 1, that is Klein tunneling, independent of energy and other parameters, as shown in Fig. 6(b). At φ = π/20, the transmission displays disordered oscillation, since T is an oscillating function of q 0x, and q 0x is determined by E, q 0x = ((E V 0 )/ v f ) 2 ky. 2 Specifically, in the region of E V 0 < v f k y, where mod (q 0x D, π) is zero in Fig. 6(a), the evanescent states appear inside the barrier since q 0x becomes imaginary. Then the transmission returns to ordinary tunneling, and the electrons act as Schrödinger electrons. Thus, quantum resonance occurs within the electron continuum in the electrodes and the well, [10] which is shown as the line-type resonant peaks appeared in Fig. 6(b). However, in the region of E V 0 > v f k y, propagating states appear in the barrier. For energy at which electronic states outside the barrier match the hole states inside it, transmission is governed by Klein tunneling. By comparing Figs. 6(a) and 6(b), one may find that the peaks of transmission correspond to the resonance condition q 0x D = nπ. In the

4 166 Communications in Theoretical Physics Vol. 56 presence of external field, the transmissions are independent of energy for normal incidence, and T n = T n. At φ = π/20, all the transmissions display disordered oscillation. In the forbidden region E + n ω V 0 < v f k y, the number of resonant peaks is increased. While, in the continuum region E+n ω V 0 > v f k y, the number of resonant peaks is decreased. This implies different influences of Schrödinger and Dirac dynamics on transmission. From Figs. 6(a) and 6(c), it can be seen that the peaks of T 0 correspond to the resonance condition. The number of resonant peaks of T and T ±n increases with increasing barrier width. The feature here is similar to that of transmissions T and T ±n as functions of V 0, because of k m = ((E + m ω V 0 )/ v f ) 2 ky 2. the appearance of quantum resonance tunneling, the conductivity is nonzero. In the region E > V 0, conductivity increases when energy increases. Furthermore, with the decrease of barrier width, the transmission trends to transparency, and the conductivity becomes independent of energy and approaches constant. Fig. 7 Conductivity as a function of energy E for different values of D, at V 1 = 0. Solid line, dashed line, and dotted line correspond to D = 50 nm, 20 nm, and 5 nm, respectively. Fig. 6 Transmission probability and the variable mod (q 0xD, π) as functions of E for different values of φ and V 1. (a) The variable mod (q 0xD, π) as a function of E. (b) At V 1 = 0, solid line and dashed line depict T for φ = π/20 and 0, respectively. (c) At V 1 = 10 mev, solid line, dashed line, and dotted line depict T 1, T 0, and T 1 for φ = π/20, respectively. Straight dash-dotted line and dash-dot-dotted line depict T 0 and T 1 = T 1 for φ = 0, respectively. Next, the effect of system parameters on the conductivity is studied. Figure 7 displays the conductivity as a function of energy E for various widths of barrier without external field. It can be seen that, in the lower energy region E < V 0, the conductivity exhibits irregularly oscillatory decline with increasing energy. This phenomenon could be understood in the light of transmission behavior in Fig. 6(b). The amplitude of transmission increases with increasing energy, so the averaged transmission exhibits decreasing tendency. Thus, conductivity oscillates and decreases, due to Eq. (4). At E V 0, owing to Fig. 8 Conductivity as a function of system parameters (a) E, (b) V 0, (c) L, and (d) D for different values of V 1. Solid line, dashed line, dash-dotted line, and dotted line correspond to G(V 1 = 0), G 1, G 0, and G 1 in all figures, respectively. V 1 = 10 mev in (a), (b), and (d), V 1 = 33 mev in (c). Figure 8 shows the contribution of sidebands to conductivity in the presence of external field. Figure 8(a) displays the conductivity as a function of energy E. One may find that the conductivity of central band G 0 exhibits chaotically oscillatory decline with increasing energy in this energy region. The behavior of G 0 corresponds to that of T 0 in Fig. 6(c), and the same as G ±n. For weak external field, G 0 consists with G(V 1 = 0), and the central band

5 No. 1 Communications in Theoretical Physics 167 dominates the conductivity. With the increase of strength V 1, sidebands become significant, and the corresponding conductivities G ±n increase as well. Figure 8(b) shows the conductivity as a function of V 0. In the region V 0 > E, the averaged transmissions increase with increasing barrier. Thus, the conductivities are also oscillating functions of V 0, although the transmissions are Klein tunnelings for normal incidence. One could see that G(V 1 = 0) and G ±n exhibit oscillatory and ascending behavior when V 0 increases, and the oscillations become stronger with increase of D. Figure 8(c) shows the conductivity as a function of L. We can see that G(V 1 = 0) exhibits oscillatory behavior, but its amplitude decreases with increased well width L. In the presence of field, G 0 decreases rapidly with increasing L. Compared with Figs. 2 and 3, it implies that various angular dependence of transmission at different well widths is the origin of this interesting phenomenon. Thus, one can control the conductivity through a particular band or not by adjusting the value of L. Changing the barrier width D can also change the conductivity, as depicted in Fig. 8(d). It demonstrates that the conductivities G(V 1 = 0) and G ±n vary periodically, and their amplitudes decay to zero with increasing D. From the above results on conductivity, it is clear that conductivity manifests many significant behaviors with the change of system parameters. The primary reason is the angular anisotropy of transmission, even though the transmissions are independent of parameters E, V 0, and D for normal incidence. 4 Conclusion In conclusion, we have studied numerically the property of transmission probability of electron in graphenebased double-barrier structure with a periodical modulation in potential well. The transmissions of central band and sidebands as functions of external strength, well and barrier widths, as well as energy and angle of electron are discussed. The results show that electron transport behavior depends strongly on those parameters. Moreover, the contribution of each band to conductivity is also studied with different values of system parameters. The results obtained in this paper indicate that by adjusting the external strength and system parameters, one could control the electron transport and conductivity as well as their distribution in each band. References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, and A.A. Firsov, Science 306 (2004) 666. [2] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov, Nature (London) 438 (2005) 197. [3] Y. Zhang, Y.W. Tan, H.L. Stormer, and P. Kim, Nature (London) 438 (2005) 201. [4] V.P. Gusynin and S.G. Sharapov, Phys. Rev. Lett. 95 (2005) [5] K.S. Novoselov, E. Mccann, S.V. Morozov, V.I. Falko, M.I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A.K. Geim, Nat. Phys. 2 (2006) 177. [6] Y. Zhang, J.P. Small, M.E.S. Amori, and P. Kim, Phys. Rev. Lett. 94 (2005) [7] M.I. Katsnelson, K.S. Novoselov, and A.K. Geim, Nat. Phys. 2 (2006) 620. [8] M. Ahsan Zeb, K. Sabeeh, and M. Tahir, Phys. Rev. B 78 (2008) [9] J.M. Pereira, P. Vasilopoulos, and F.M. Peeters, Appl. Phys. Lett. 90 (2007) [10] R. Zhu and Y. Guo, Appl. Phys. Lett. 91 (2007) [11] K. Wakabayashi, Y. Takane, and M. Sigrist, Phys. Rev. Lett. 99 (2007) [12] M.I. Katsnelson, Eur. Phys. J. B 51 (2006) 157. [13] P.G. Silvestrov and K.B. Efetov, Phys. Rev. Lett. 98 (2007) [14] V.V. Cheianov and V.I. Falko, Phys. Rev. B 74 (2006) (R). [15] C. Bai and X. Zhang, Phys. Rev. B 76 (2007) [16] N. Abedpour, A. Esmailpour, R. Asgari, and M.R.R. Tabar, Phys. Rev. B 79 (2009) [17] B. Trauzettel, Ya. M. Blanter, and A.F. Morpurgo, Phys. Rev. B 75 (2007) [18] P.K. Tien and J.P. Gordon, Phys. Rev. 129 (1963) 647. [19] E.N. Bulgakov and A.F. Sadreev, J. Phys.: Cond. Matt. 8 (1996) 8869; G.P. Berman, E.N. Bulgakov, D.K. Campbell, and A.F. Sadreev, Physica B 225 (1996) 1. [20] M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31 (1985) 6207; M. Büttiker, Phys. Rev. Lett. 57 (1986) [21] M. Wagner, Phys. Rev. B 49 (1994) 16544; Phys. Rev. A 51 (1995) 798.