Confined State and Electronic Transport in an Artificial Graphene-Based Tunnel Junction

Transcription

1 Commun. Theor. Phys Vol. 56, No. 6, December 5, 0 Confined State and Electronic Transport in an Artificial Graphene-Based Tunnel Junction YUAN Jian-Hui ï, ZHANG Jian-Jun, ZENG Qi-Jun É, ZHANG Jun-Pei, and CHENG Ze Ä School of Physics, Huazhong University of Science and Technology, Wuhan , China Received June 30, 0; revised manuscript received August 7, 0 Abstract Artificial graphene structures embedded in semiconductors could open novel routes for studies of electron interactions in low-dimensional systems. We propose a way to manipulate the transport properties of massless Dirac fermions in an artificial graphene-based tunnel junction. Velocity-modulation control of electron wave propagation in the different regions can be regarded as velocity barriers. Transmission probability of electron is affected profoundly by this velocity barrier. We find that there is no confinement for Dirac electron as the velocity ratio ξ is less than, but when the velocity ratio is larger than the confined state appears in the continuum band. These localized Dirac electrons may lead to the decreasing of transmission probability. PACS numbers: 73.3.Ad, Pm, 8.05.Uw Key words: ballistic transport, relativistic wave equations, Carbon diamond graphite Introduction Graphene can be described by a two-dimensional massless Dirac-fermion MDF model at low energies with chiral quasiparticles that are responsible for a number of unusual properties. This allows one to directly probe the physics of two-dimensional D Dirac Weyl fermions. [ 6] The presence of such Dirac-like quasiparticles is expected to induce some unusual electronic properties, which make much difference from the two dimensional electronic gas, such as Klein parodox, [3] the anomalous integer quantum Hall effect, [4 5] and observation of minimum conductivity. [6 8] Recently, Kim et al. [9] have developed a simple method to growth graphene films in large scale using chemical vapor deposition CVD on thin nickel layers. This is especially important means for the future design of graphene-based devices. The relativistic behavior of graphene electrons confined by electrostatic fields leads to Klein tunneling, [3] where a relativistic particle can tunnel through a high barrier by the process of pair production. But bound states do not occur when graphene is subjected to an external potential because there is no threshold for pair production. As a consequence, Dirac fermions are badly confined in a potential well barrier produced by electrostatic fields. To overcome this problem, several alternatives have been suggested. A possibility is to use an inhomogeneous magnetic field. [0 5] It was shown in numerous papers that an inhomogeneous magnetic field confines the usual electrons [] and the recent Dirac electrons as well. [0 ] It has been proved that a magnetic barrier can effectively block Klein tunneling and achieve confinement for such massless Dirac fermions in graphene. [3] In particular, magnetic confinement of Dirac electrons in graphene has reported in structures involving one or several magnetic barriers [0 ] as well as superlatices. [3 5] Recently, research has shown that modulating a two-dimensional electron gas with a long-wavelength periodic potential with honeycomb symmetry can lead to the creation of isolated massless Dirac points with tunable Fermi velocity. [6] This is a way to fabricate Artificial graphene AG in a twodimensional electron gas. An obvious difference from real graphene RG is that electrons in AG sheets behave like massless Dirac fermions with a tunable Fermi velocity. Researches have proved that an independent approach to the realization of artificial graphene in a nanopatterned DEG. [6 7] This way, which can date back to many decades ago, has been used to develop a similar condensedmatter analog of +-dimensional electrodynamics. [8] Such artificial graphene with a tuning Fermi velocity has been opened novel routes for studies of electron interactions in low-dimensional systems, [9 0] which is also realized in optical lattice, [] in cold atoms, [] and in trapped ion. [3] In this paper, we manipulate the transport properties of massless Dirac fermions in an AG/RG/AG tunnel junction. Velocity-modulation control of electron wave propagation in the different regions, can be regarded as velocity barriers. We find that transmission probability of electrons is affected profoundly by this velocity barrier. When the velocity ratio is larger than, these confined Supported by the National Natural Science Foundation of China under Grants Nos and Corresponding author, jianhui830@63.com c 0 Chinese Physical Society and IOP Publishing Ltd

2 36 Communications in Theoretical Physics Vol. 56 states appear in the continuum band, which leads to the decreasing of transmission probability and a characteristic of analogous photons in an optical fiber. Nevertheless there is no confinement for Dirac electron when the velocity ratio ξ is less than, but a critical incident angle exists for transport. In Sec., we introduce the model and method for our calculation. In Sec. 3, the numerical analysis to our important analytical issues are reported. Finally, a brief summary is given in Sec. 4. Model and Method Let us consider the Hamiltonian properties of massless Dirac fermions in an AG/RG/AG tunnel junction shown in Fig.. As reproted in Ref. [9], this tunnel junction Fig. Schematic representation of the model of an AG/RG/AG tunnel junction. This tunnel junction leads to different Fermi velocities in the two parts of the material. In the region of AG sheet Fermi velocity can be controlled to greater or less than that of graphene. leads to different velocities in the two parts of the material. This situation calls for a model where v F is position dependent: v F = vr. This is quite conceivable in condensed matter physics, since the value of v F is determined by the material under consideration different materials can have different Fermi velocities. Taking into account wave propagation control by spatial modulation of velocity in the different regions, we write [,9,5] i υr σ r υrψr = Eψr, where σ is Pauli matrices with σ = σ x, σ y. In using the Dirac equation we are assuming that velocity variations are slow on a lattice constant scale. In this limit spin and valley degrees of freedom play a passive role. Note that the derivative will act on the product vrψr. It is nevertheless convenient to introduce the auxiliary spinor Ψr = vrψr which satisfies: i υ eff σ r Ψr = EΨr. Now we consider an AG/RG/AG tunnel junction with a simple velocity, which changes only along the x direction as follows υ eff = { υ AG F, if x < 0 and x > L, υf RG, otherwise, The system presented can be separated into three regions by the effective Fermi velocity v eff in an AG/RG/AG tunnel junction as I x < 0, II 0 < x < L, and III x > L. The wavefunction in the different velocity regions can be written in terms of incident and reflected wave, as is the case of the photons. Because of the translational invariance of the system along y direction, the equation HΨx, y = EΨx, y admits solutions of the form Ψx, y = Ψ A x, Ψ B x T expik y y with Ψ A x, Ψ B x obeying the coupled equations i υ eff [ x +k y ]Ψ B = EΨ A and υ eff [ x k y ]Ψ A = EΨ B, respectively. In what follows we put = υf RG =. In region I, we have [] with k = we have with q = Ψx = + r k y Ψx = a + b k + ik y k ik y e ikxx and ξ = υag F q + ik y E q ik y E e iqxx e ikxx, /υrg F e iqxx,. In region II, E ky. Finally in region III, we have a transmitted wave only, Ψx = t k + ik y e ikxx. The coefficients r, a, b, and t are determined by the correct boundary conditions at x = 0 and x = L: Continuity of the wave functions Ψ at x = 0 and x = L are Ψ0 = Ψ0 + and ΨL = ΨL +, respectively. These conditions imply that the physical ψ satisfies the following matching conditions: [9 0] ξψ0 = ψ0 +, ψl = ξψl +. 3 These discontinuities in ψ guarantee that the divergence of the local current Jr = vrψ r σψr vanishes. Under these matching conditions we are able to obtain the form of transmission probability 4k q ξ T = 4k q ξ cos ql+q +ky ξ +k ξ sin ql.4 This expression has the advantage that can well describe the propagating states and evanescent states in a unified form and can avoid the discussion of the chiral features of Dirac electron. As reported in Refs. [,0,4 5], we can also define the incident and refractive angles for electrons in propagating states: φ = arctanky/k and θ = arctanky/q. In order to discuss the transmission probability dependance of the incident angle, we only make replacement ky = E sinφ/ξ in Eq. 4.

3 No. 6 Communications in Theoretical Physics 37 3 Results and Discussions In what follows we present some numerical examples of the calculated transmission probability: = υf RG =. In Fig., we show transmission probability of electrons through an AG/RG/AG tunnel junction as a function of the incident angle with the different ξ for fixed length L and E F shown in a L = 50 nm and E F = 0 mev, b L = 50 nm and E F = 80 mev, c L = 0 nm and E F = 0 mev, d L = 0 nm and E F = 80 mev respectively. Firstly, we note that Tφ = T φ, and the value of q x L satisfying the relation q x L = nπ where n is an integer, the velocity barriers become completely transparent independent of the value of k y or φ [see in Eq. 4]. This feature is obvious for the larger Fermi energy E F and the bigger size of graphene [see in Fig. d]. The result shows that there are obvious difference from the value ξ < and the value ξ >. We find that there are a gap of transmission probability corresponding to the value ξ < with increasing of the incident angle; that is to say, when the angle of incidence exceeds the critical angle the electron does not penetrate through the AG/RG/AG tunnel junction. This critical angle is analogous to the called Brewster angle in optics. From the definition of the refracted angle, this so called Brewster-like angle is φ b = arcsinξ. [6] In fact, this Brewster-like angle φ b is not equal to critical angle φ c, but it can distinguish between propagating wave mode and evanescent wave mode. The critical angle φ c obviously depends on the Fermi energy and the length of sample [see in Figs. a d]. We find that the transmission declines sharply for a fixed incident energy or a fixed size of graphene sheet when the incident angle φ exceeds the Brewster-like angle φ b. The reason is that the wavevector q x in the graphene region becomes imaginary, denoting the appearance of evanescent modes, and then is blocked corresponding to the critical angle φ c. When the propagating wave mode dominates, we note that the transmission probabilities as the function of the incident angle have multi-resonance peaks. Factually, as the energy of the incident Dirac fermions is increased or the length of the sample of graphene is larger, more ballistic channels peaks will be opened. [0] The transmission can also be tuned by the Fermi velocity [see in Figs. e f]. Fig. Transmission probability of electrons through an AG/RG/AG tunnel junction as a function of the incident angle with the different velocity ratio ξ for fixed length L and E F shown in a L = 50 nm and E F = 0 mev, b L = 50 nm and E F = 80 mev, c L = 0 nm and E F = 0 mev, d L = 0 nm and E F = 80 mev respectively. Transmission probability of electrons through an AG/RG/AG tunnel junction as a function of the ratio ξ with the different incident angle for fixed length L nm and E F shown in e L = 0 nm and E F = 0 mev, f L = 0 nm and E F = 80 mev respectively.

4 38 Communications in Theoretical Physics Vol. 56 q = E F /ξ k y and There is a threshold value ξ for a fixed incident angle φ; that is to say, the Dirac electron can be penetrated through the AG/RG/AG tunnel junction when the velocity ratio ξ exceeds the threshold value ξ T. We find that the threshold value ξ T is increasing with the increasing of the incident angle. Puzzled us here is that the transmission declines sharply for a fixed Fermi energy when the velocity ratio ξ exceeds a certain value ξ 3 in Fig. e as an example, especially for the bigger size of graphene sheet [see in Fig. f]. These tunneling features are well appreciated by inspecting the phase diagram, as shown in Fig. 3. It is obvious that the Fermi wavevector is different in AG and RG regions as a consequence of the appearance of different wave mode. We can see that k = E F ky. As the value k y exceeds the value ky c where k c y = E F for ξ <, the value q becomes imaginary, which denotes the appearance of evanescent modes. However q is always real, which corresponds to the propagating modes for the velocity ratio ξ >. It is implied that there are more ballistic channels peaks, which will be opened for ξ >. However the result shows that transmission probability declines with the increasing of the velocity ratio ξ [see in Figs. e f]. The reason may be the result of the Dirac electron confined by the velocity barrier. ξ = 0.5, b ξ =.0, c ξ =., d ξ =.0. The branches of the spectrum for localized states are shown by black curves between the straight solid line and the straight dashed line. Confinement of electrons in the x- direction gives rise to nondispersive transporting in the y-direction and the group velocities along the y-direction are indicated by the slope of the dispersion relation. The necessary condition for the existence of bound states is k y E F ξ k y, which implies the existence of bound states only for ξ >. We find that the spectrum is symmetric with respect to the transverse wavevector ky = 0, guaranteed by the time reversal symmetry. That is to say the transport properties are isotropic for each incident terminal. When the velocity ratio ξ =.0, the energy spectrum of electron denotes the sea of delocalized states [see in Fig. 4b] because of no anyway confinement for electron. As we all known, Dirac fermions are badly confined in a potential barrier. The appearance of the localized state mainly exists in forbidden band of Dirac electron confined by a potential barrier can enhance the electron density of zero Fermi energy level with increasing of the potential barrier. [8] As a consequence, there is a minimum in conductivity. [6 8] However, we find that the appearance of the localized state mainly exists in continuum band of Dirac electron confined by a velocity barrier for the velocity ratio ξ > [see in Figs. 4c 4d] and no confined state exists in the AG/RG/AG tunnel junction for the velocity ratio ξ < [see in Fig. 4a]. It sounds rather reasonable based on the fact that when waves travel across different media: the velocity ratio ξ < is corresponding to the wave passing from a denser to a rarer Fig. 3 The phase diagram in momentum space. The outer circle indicates the wavevector in the incident AG region and the inner circle indicates the wavevector in the transmitted RG region for the velocity ratio ξ < and vice versa for the velocity ratio ξ >. In order to observe the interesting confined electronic state in the AG/RG/AG tunnel junction, we firstly assume the parameters = υf RG = L =. We calculate the energy dispersion relation of the bound states when the Dirac electrons penetrate through an AG/RG/AG tunnel junction. Matching of propagating waves to evanescent waves at x = 0 and x = L, the energy spectrum of the bound states can be obtained from the zero point of the determinant of the coefficients, which can be reduced to a transcendental equation, tanql = kqξ/q + kyξ k ξ. In Fig. 4, we show energy spectrum of the bound states as a function of k y for the different velocity ratio ξ shown in a Fig. 4 Energy spectrum of the bound states as a function of k y for the different velocity ratio ξ shown in a ξ = 0.5, b ξ =.0, c ξ =., d ξ =.0, respectively. The branches of the spectrum for localized states are shown by black curves between the straight dashed line and the straight solid line. The energy unit is equal to E 0 = υf AG /L.

5 No. 6 Communications in Theoretical Physics 39 medium but the velocity ratio ξ > is corresponding to the wave passing from a rarer to a denser medium. [6] With the increasing of the velocity ratio ξ, the electron in continuum band will confine and more delocalized electron will change into localized electron. The number of modes is roughly determined by the integer n satisfying that 0 n < ξ k y L/π. Noting that electron is confined easily for the bigger k y or φ, this is why transmission probability declines with the increasing of the velocity ratio ξ [see in Figs. e f]. 4 Conclusion In this paper, we propose a way to manipulate the transport properties of massless Dirac fermions in an AG/RG/AG tunnel junction. Artificial graphene AG structures with a tunable Fermi energy embedded in semiconductors could open novel routes for studies of electron interactions in low-dimensional systems. It is possible, in principle, to manipulate the transmission properties of a system described as a Dirac equation by controlling the Fermi velocity. We calculate the angular-dependent transmission probability to investigate the effect of resonant tunneling and to study the confined state of Dirac electron in this tunnel junction. Transmission probability of electrons is affected profoundly by this velocity barrier. We find that there is no confinement for Dirac electron as the velocity ratio ξ is less than, but when the velocity ratio is larger than the confined state appears in the continuum band. These interesting features will be more helpful for developing new type of devices. References [] K.S. Novoselov, et al., Science ; C. Berger, et al., Science ; M.I. Katsnelson, Mater. Today [] A.H.C. Neto, et al., Rev. Mod. Phys [3] O. Klein, Z. Phys ; F. Sauter, Z. Phys ; A. Hansen and F. Ravndal, Phys. Scripta ; Stefano De Leo and Pietro P. Rotelli, Phys. Rev. A ; N. Stander, B. Huard, and D.G. Gordon, Phys. Rev. Lett ; C.W.J. Beenakker, Rev. Mod. Phys ; J.H. Yuan, Z. Cheng, M. Yin, Q.J. Zeng, and J.P. Zhang, Commun. Theor. Phys ; C.X. Bai and X.D. Zhang, Phys. Rev. B [4] K. Nomura and A.H. MacDonald, Phys. Rev. Lett ; V.P. Gusynin and S.G. Sharapov, Phys. Rev. Lett ; K.S. Novoselov, et al., Nat. Phys [5] K.S. Novoselov, et al., Nature London ; Y.B. Zhang, et al., Nature London [6] K. Ziegler, Phys. Rev. B ; N.M.R. Peres, F. Guinea, and A.H. Castro Neto, Phys. Rev. B ; P.M. Ostrovsky, I.V. Gornyi, and A.D. Mirlin, Phys. Rev. B ; S. Ryu, C. Mudry, A. Furusaki, and A.W.W. Ludwig, Phys. Rev. B ; K. Ziegler, Phys. Rev. Lett [7] M.I. Katsnelson, Eur. Phys. J. B ; M.I. Katsnelsona, Eur. Phys. J. B [8] J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz, and C.W.J. Beenakker, Phys. Rev. Lett ; J.J. Palacios, Phys. Rev. B [9] K.S. Kim, et al., Nat. Lett [0] F. Zhai and K. Chang, Phys. Rev. B ; H. Xu, T. Heinzel, M. Evaldsson, and I.V. Zozoulenko, Phys. Rev. B ; M.R. Masir, et al., Appl. Phys. Lett ; N.M.G. Ihm, Phys. E ; Y.Y. Guo and Y. Guo, J. Appl. Phys [] M.R. Masir, P. Vasilopoulos, and F.M. Peeters, Phys. Rev. B ; M.R. Masir, P. Vasilopoulos, A. Matulis, and F.M. Peeters, Phys. Rev. B [] F.M. Peeters and A. Matulis, Phys. Rev. B ; J. Reijniers, F.M. Peeters, and A. Matulis, ibid ; H.Z. Xu and Y. Okado, Appl. Phys. Lett [3] M.R. Masir, P. Vasilopoulos, and F.M. Peeters, New J. Phys [4] L.Z. Tan, C.H. Park, and Steven G. Louie, Phys. Rev. B ; C.H. Park, L. Yang, Y.W. Son, M. L. Cohen, and S.G. Louie, Nat. Phys ; M. Gibertini, A. Singha, V. Pellegrini, M. Polini, G. Vignale, A. Pinczuk, L.N. Pfeiffer, and K.W. West, Phys. Rev. B R. [5] R. Biswas, A. Biswas, N. Hui, and C. Sinha, J. Appl. Phys ; L. Dell, Anna, and A.D. Martino, Phys. Rev. B [6] C.H. Park and S.G. Louie, Nano Lett [7] M. Gibertini, A. Singha, V. Pellegrini, M. Polini, G. Vignale, and A. Pinczuk, Phys. Rev. B R. [8] S. Deser, R. Jackiw, and S. Templeton, Phys. Rev. Lett ; G.W. Semenoff, Phys. Rev. Lett ; A.N. Redlich, Phys. Rev. Lett ; A.N. Redlich, Phys. Rev. D ; R. Jackiw, Phys. Rev. D [9] N.M.R. Peres, J. Phys.: Condens. Matter [0] A. Concha and Z. Te sanović, Phys. Rev. B ; A. Raoux, M. Polini, R. Asgari, A.R. Hamilton, R. Fazio, and A.H. MacDonald, Phys. Rev. B ; D. Juan F, A. Cortijo, and M.A.H. Vozmediano, Phys. Rev. B [] J. Wu and S.D. Sarma, Phys. Rev. B ; S.L. Zhu, B.G. Wang, and L.M. Duan, Phys. Rev. Lett ; S.L. Zhu, D.W. Zhang, and Z.D. Wang, Phys. Rev. Lett ; B. Wunsch, F. Guinea, and F. Sols, N. J. Phys C. [] G. Juzeliunas, J. Ruseckas, M. Lindberg, L. Santos, and P. Ohberg, Phys. Rev. A R. [3] L. Lamata, J. Leon, T. Schatz, and E. Solano, Phys. Rev. Lett [4] Z.H. Wu, Appl. Phys. Lett [5] V.A. Yampolskii, S. Savelev, and F. Nori, N. J. Phys [6] S. Ghosh and M. Sharma, J. Phys.: Condens. Matter ; S. Ghosh and M. Sharma, J. Phys.: Condens. Matter