Các tính chất truyền dẫn điện của một số cấu trúc nano graphene tt tiếng anh

MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY …… ….***………… NGUYEN THI THUY NHUNG ELECTRONIC TRANSPORT PROPERTIES OF SOME GRAPHENE NANOSTRUCTURES Major: Theoretical and Mathematical Physics Code: 44 01 03 DISSERTATION SUMMARY Ha Noi – 2020 Dissertation was completed at: Graduate University of Science and Technology - Vietnam Academy of Science and Technology Scientific Supervisor: Prof Dr NGUYEN VAN LIEN 1st Reviewer: … 2nd Reviewer: … 3rd Reviewer: … The dissertation will be defended at Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay District, Hanoi city At … hour … date … month … 2020 The dissertaion can be found in National Library of Vietnam and library of Graduate University of Science and Technology, Vietnam Academy of Science and Technology Introduction Carbon is a common element in nature and a fundamental element for life It is abundant in the crust of the Earth Diamond and graphite are 3D allotropes of carbon In 1985, the 0D allotrope of carbon, fullerence, was found by Kroto, Smalley and Curl In 1952, Radushkevich and Lukyanovich reported about carbon nanotube In 1991 Lijima and his colleagues successfully fabricated carbon nanotube Wallace was the pioneer who theoretically performed research about a single layer of carbon in 1947 The term “graphene” was first proposed by Boehm, Setton and Stumpp in 1994 to indicate a single layer of carbon in which carbon atoms are arranged at the nodes of a honeycomb lattice In 2004, Geim and Novoselov successfully separated graphene from graphite Graphene became the first ever 2D material created in the laboratory Other methods to fabricate graphene have been found gradually After being successfully created in the laboratory, graphene has become a hot subject of research Researchers expect graphene, with its superior conductivity and a good heat transfer property, to bring unique and important applications In electronics, graphene is an ideal material for ballistic transports to be realized Graphene has an advantage in fabricating p-n-p junctions, which are the basic components of bipolar devices Recently, scientists at the Massachusetts Institute of Technology (MIT) have created qubits in superconducting circuits using graphene From these facts, we chose the research topic “The electrical transport properties of some graphene nanostructures” The aim of this thesis is to study electrical transport characteristics of graphene nanostructures We focus on two research objects associated with two kinds of graphene nanostructures: graphene bipolar junctions (GBJs) and graphene quantum dots (GQDs) GBJs can be created by electrodes that are in contact with a graphene surface in a configuration that allows to control the transport in 1D The electrical transport characteristics of bipolar junctions (BJs) mainly depend on the potential at the transition region Previous theoretical studies assumed that this potential is rectangular or trapezoidal In this thesis, we propose to use a Gaussian potential barrier model for studying the transport properties of GBJs The advantage of Gaussian potential is that it better describes the potential profile in real BJ structures This allows to describe all modes of the carrier density as well as a smooth transition between the modes Our study focuses on calculating the characteristic quantities of electrical transport such as transmission probability, resistance, Volt-Ampere characteristics and shot noise depending on the model parameters, in order to understand clearly about the ballistic transport mechanism through GBJs Similar to p-n junctions, GQDs can be created by nanoelectrodes Thanks to Scanning Tunneling Microscope (STM), it is possible to fabricate nanosized electron confinement regions on graphene sheets In the GQD created by electrostatic potential, except for certain conditions that allow the existence of bound states (BSs), the carriers normally exist in quasi-bound states (QBSs) with a finite lifetime The determination of the lifetime of charge carriers in the GQD is paramount for the design of electronic devices based on GQD of desirable functions Therefore, in this thesis, we propose a theoretical model to study the lifetime and local density of states (LDOS) of the carriers in circular GQD (CGQD) created by electrostatic potentials The results that we achieve are compared to experiments We use the transfer matrix (T -matrix) method to solve the above problems By obtaining the T -matrix, one can calculate electronic properties of the system such as transmission probability, conductance, current-voltage characteristics and shot noise From the components of the T-matrix, one can also determine the energies as well as the level widths of QBSs of electron in GQD The local density of states (LDOS) and scattering coefficients can be expressed accurately in terms of the T-matrix elements Apart from the T matrix method, we propose also a method for calculating LDOS directly from normalized wave functions for a CGQD with an arbitrary confined potential This thesis is divided into chapters, excluding the introduction, the conclusion and the references Chapter presents an overview of the electronic properties of graphene and the results of the previous research on n-p-n junction and GQD Chapter introduces the theoretical and numerical methods used in the thesis Chapter presents the results on electronic transport in n-p-n graphene junction Chapter describes the results of energy structure and related properties of CGQD created by electrostatic potentials as well as a theoretical development of the T-matrix method for CGQD in a perpendicular uniform magnetic field Chapter Electronic properties of graphene 1.1 Crystal structure and energy band structure of graphene Graphene is a monolayer of carbon atoms in which the atoms are on the vertices of a two-dimensional honeycomb lattice A graphene lattice can be considered as a hexagonal Bravais lattice with two atoms in an unit cell The energy band structure of graphene can be determined by the tight binding approximation When considering only the interaction between the nearest neighbors in the graphene lattice, the electronic band dispersion is given by √ E(k) = ±t cos(πkx a 3) cos(πky a) + cos2 (πky a) + , (1.1) ˆ where t = φ∗ (r − rA )Hφ(r − rB )d3 r is the hopping energy between the nearest neighbors For graphene, t ≈ 2.7 eV The energy band structure is given in terms of the formula (1.1) described in Fig.1.1 The minus sign on the right hand side of the formula (1.1) corresponds to the lower energy band, called the π band, whereas the plus sign corresponds to the higher energy band, called the π ∗ band We can see that these two bands are degenerated at the K and K points, called the Dirac points At zero Kelvin, the π band is fully filled while the π ∗ band is empty, and the Fermi energy is EF = Let k = K+q, where K is the momentum vector at the K or K point and q is the momentum relative to the momentum at Figure 1.1: (a) The energy band structure of graphene along the trajectory √ Γ → K → M → K with ky = kx / (b) The energy band structure in which E is expressed in terms of kx and ky the Dirac point By Taylor’s expansion of the energy E(k) around the Dirac point with the assumption |q| K we get: E(q) ≈ ± vF |q|, (1.2) where vF = 3ta0 /2 is the Fermi velocity In the vicinity of the K or K point, the velocity of electron is equal to the Fermi velocity and does not depend on the electron’s energy and momentum In graphene, the Fermi velocity is c with c is the speed of light about 106 m/s or 300 The existence of two sublattices A and B in graphene gives rise to a chirality in the kinetics of the carrier in graphene therefore two linear branches of the graphene’s carrier dispersion near the Dirac points become independent of each other According to the first quantization language, the carriers in graphene comply with the Dirac equation: vF σ · ∇Ψ(r) = EΨ(r) , (1.3) where σ = (σx , σy ) is the 2D vector formed by Pauli matrices σ is called quasi-spin The wave functions of the carriers in the vicinity of the Dirac points can be described in terms of two-component spinors that correspond to the probability of finding carriers in one of two corresponding sublattices 1.2 Graphene bipolar junction Several experimental studies have developed graphene heterosjunctions induced by local electrodes, also called local gates In general, to create a GBJ, one needs to create a design with two electrostatic gates By changing the gate voltage, Vb and Vt , independently of each other, one can create graphene bipolar heterosjunctions in all possible charge density regimes: p-n-p, n-p-n, p-p -p, or n-n -n ă Ozyilmaz and coworkers studied the quantum Hall transport in graphene n-p-n junctions and observed a series of fractional quantum Hall plateaus as the local charge density varies in the p and n regions at large magnetic field value Huard et al measured the resistance (R) across GBJs and reported a noticeable asymmetry of the R versus Vt curves with respect to the maxima Using a suspended ‘air-bridge’ top gate to avoid a decrease of the carrier mobility in the region under this gate, Gorbachev et al were able to fabricate ballistic graphene p-n-p junctions The Coulomb blockade in graphene nanoribbon based bipolar junctions is found out in the single electron transistors 1.3 Graphene quantum dot Experimentally, the methods of fabricating QD using electrostatic potentials to confine carriers have been studied Zhao and coworkers created nanoscale confinement areas in graphene in the form of the circular p-n junction using a scaling tunneling microscope (STM) In the research of Lee et al., GQDs are created using a technique related to the generation of local charge defects inside the insulating substrate that is beneath the graphene monolayer film In the experiment that was carried out by Gutierrez et al., the CGQD was created by a piece of copper substrate of the size of several tens of nanometers causing a potential difference between the inside and the outside of the QD Theoretically, CGQDs are usually modeled by a radial confinement potential depending on the distance in the form of a step or an exponential form For pristine graphene with no energy gap, when there is no magnetic field, the states of the carriers in the GQDs induced by an electrostatic potential are generally quasi-bound states (QBS) with a finite trapping time instead of bound states (BS), except for a few special cases in which BSs are observed In the case of graphene with an energy gap, researchers have shown that the energy gap makes the life-time of the carriers longer The work of Chen and coworkers suggests that the trapping time of the carriers in GQD increases with the smoothness of the exponential confinement potential The research by Lee et al shows that the charge density of a GQD measured experimentally varies smoothly at the boundary of the p-n junction This suggests that QD is induced by a smooth potential One of the objectives of this thesis is studying the GQD with the different shapes of the confinement potentials such as trapezoidal, Gaussian potential in order to better understand the dependence of states and the trapping time of electrons in GQDs on the shape of the confinement potential 1.4 Trapping carriers in graphene by a magnetic field The effect of a magnetic field on graphene sheets can also give rise to BS because of eliminating the Klein tunneling De Martino and coworkers theoretically pointed out that a perpendicular heterogeneous magnetic field can be used to create GQDs with the existence of BSs Another way to create a GQD, overcoming the obstacles caused by the Klein’s paradox, is the use of the combination of the electric and magnetic fields to confine the carriers in a certain confinement area Giavaras and coworkers examined the states of the massless Dirac particles in a GQD created simultaneously by electric and magnetic fields The other solution is to create a GQD having the confinement states that are in the gap between the Landau levels when QD is put in a magnetic field Maksym et al pointed out that a CGQD can be induced by an electrostatic confinement potential along with a plane asymptotic potential outside the QD which is placed in an uniform perpendicular magnetic field Chapter Calculation methods 2.1 Transfer matrix method Transfer matrix (T -matrix) is widely used in problems in which electrons follow the Schră odinger equation In the case electrons obey the Dirac equation and are affected by a smooth barrier, one can still apply effectively the T matrix method The reason of this is, in principle, any smooth 1D potential can be approximately considered as a series of the step potentials in which the potential at each step can be considered constant By multiplying T matrix elements obtained from the solution for this step potential one can find an overall T -matrix On the other hand, for each step potential, T matrix elements can be obtained from the solutions of the Hamiltonian on its left and right sides (at that point, the potential is considered constant) by satisfying an appropriate condition of continuity at the step interface In the next chapters, we will use and develop the T -matrix method for both the graphene bipolar junctions and graphene circular quantum dots For the problem of electronic transport through a potential barrier in graphene, using the obtained overall T -matrix, the transmission probability T through the studied junction can be calculated as a function of the incident energy E and the incident angle θ 2.2 Methods of calculating the current, conductance, shot noise and Fano factor Consider a simple potential barrier shown in Fig 2.1 One can calculate the transmission probability T (E, θ) by T -matrix approach At equilibrium condition (V = 0), electrons obey the Fermi distribution, the left and the right hand sides of the barrier have the same Fermi energy µ0 When the bias V is applied, each side of the barrier has its own Fermi energy, µL and µR corresponding to the left and the right, respectively The difference between them is |µL − µR | = eV Figure 2.1: The barrier and a Fermi sea of electrons in an ideal case when the 1D potential bias V is small Consider a graphene system under the effect of a small bias The current at the linear region I–V is as follows: I= geW |(µ0 − UL )eV | h2 vF π/2 dθT (θ) cos θ (2.1) −π/2 Then one can calculate the conductance G = I/V A quantity that is usually considered in the ballistic transport is the Fano factor F It is defined as ratio of the actual shot noise power S and the Poissonian noise SP , S S F= = (2.2) SP 2eI The shot noise is a consequence of the charge quantization The Poissonian noise is a noise produced when the charge carriers are uncorrelated The general formula of the bias-dependent shot noise power was derived by Buttiker for a 2D system in the case of a continuous spectral near K: S = ge2 W h2 vF µL π/2 dE |E − UL | µR dθT (E, θ)[1 − T (E, θ)] cos θ −π/2 (2.3) 90o o 90 60o 1.0 90o o 60 60o 0.8 0.6 30o 30o o 30 0.4 0.2 0o 0o 0o 0.2 0.4 −30o 0.6 −30o −30o 0.8 1.0 −90o −60o (a) o 90 1.0 −90o −60o (b) (c) o o 90 60o −60o −90o 90 o 60 60o 0.8 0.6 o o 30 o 30 30 0.4 0.2 0o o 0o 0 0.2 0.4 −30o 0.6 o −30o −30 0.8 1.0 −90o −60o (d) −90o −60o (e) −90o o −60 (f) Figure 3.2: Polar graphs depicting T (θ) for GBJs in rectangular potential model (dashed blue curves) and Gaussian-type potential model (solid red curves): the outmost semicircle corresponds to T = and the center to T = T (θ)-graphs are shown for various values of parameters of [L (nm), E (meV), Vb (V), Vt (V)]: (a) [25, 0, 60, −12]; (b) [25, 50, 60, −12]; (c) [50, 50, 60, −12]; (d) [25, 0, 40, −6]; (e) [25, 50, 40, −6]; v (f) [50, 0, 40, −6] (c) maximum at Vt ≈ Vt , the junction enters the n-n -n regime, where the structure becomes much more transparent and consequently the resistance (c) experiences a sharp reduction at Vt > Vt In the range of Vb under study Fig 3.3(a) also shows that an increase of Vb leads to a decrease of not only (c) the transition voltage Vt , but also the average resistance in both regions, (c) (c) Vt < Vt v Vt > Vt Overall, the calculated R(Vt )- dependence shown in Fig 3.3 describe quite well the experimental data reported Observed oscillations of R versus Vt can arise from the oscillations of the transmission probability caused by the inside-barrier interference of chiral waves 11 ĐIỆN TRỞ R HỆ SỐ FANO F ĐIỆN THẾ CỦA TOP GATE Vt (V) Figure 3.3: Resistances R (a), odd resistance 2Rodd (b), and Zero-bias Fano factors F (c) are plotted versus Vt for three cases with Vb = 40 V (dash-dotted red lines), 60 V (solid blue lines), and 80 V (dashed green lines) Arrows (c) indicate the transition top gate voltages Vt where the transition between (c) n-p-n and n-n -n regimes occurs (Vt = −2.59 V, −5.39 V , and −8.19 V for Vb = 40 V, 60 V , and 80 V , respectively) 3.4 Current shot noise In order to study the currentvoltage (IV) characteristics, it is assumed that a symmetric bias [+eVsd /2, −eVsd /2] is applied to the two leads (source and drain), linked to the structure under measurement Figure 3.4(a) shows the IV curves for three different GBJs In general, by gradually rising the bias voltage Vsd , starting from Vsd = 0, the current I first increases progressively, then experiences a slowing down at some bias voltage, which mainly depends on the back gate voltage Vb Crossing this bias voltage, currents weakly fluctuate and even go through a slightly negative differential resistance (NDR) region Calculations reveal a close association between the observed NDR and the bias-dependence of the transmission probability In any case, examining the IV curves for a number of GBJs with different values of parameters, including Vb , Vt , L, and W also, shows that within the model of interest the NDR effect is always rather weak Figure 3.3(c) represents the Fano factor as a function of top gate Vt when 12 DÒNG ĐIỆN I HỆ SỐ FANO F ĐIỆN THẾ BIAS Vsd (mV) Figure 3.4: (a) Current-voltage and (b) Fano factor-voltage characteristics for three junctions with [L(nm), Vb (V ), Vt (V )] = [25, 35, −6] (solid blue lines), [25, 40, −6] (dashed red lines), and [50, 40, −3.5] (dash-dotted green lines) The bias voltage Vsd is symmetrically applied to the source and the drain the bias is equal Within our potential model, the zero-bias Fano factor is depicted in Fig 3.3(c) as a function of the top gate voltage Vt While the observed harmony of oscillations in resistance R (Fig 3.3(a)) and in Fano factor F (Fig 3.3(c)) of the same GBJ is quite well understood, the fact that all three curves for various GBJs in Fig 3.3(c) exhibit practically the same maxima of 0.36 and the same minima of 0.08 causes a little surprise In any case, this value of F = 0.36 is rather close to the experimental value of 0.38 Thus, our model provides the shot noise with F ≈ 0.36 for n-p-n GBJs in the linear regime A question might then arise of whether the bias voltage which modifies the potential barrier and changes the transmission probability can enhance the noise or even cause a super-Poissonian noise as it did in conventional semiconductor/metal nanostructures To shed light on this question, we show in Fig 3.4(b) the F(Vsd ) characteristics for the same GBJs with I(Vsd ) characteristics presented in Fig 3.4(a) Actually, Fig 3.4(b) demonstrates that for a given junction, in accordance with the current fluctuation in Fig 3.4(a), the Fano factor, starting from the value F(Vsd = 0), fluctuates against the bias between the values ∼ 0.18 and ∼ 0.25 13 Chapter Circular graphene quantum dot induced by electrostatic potential 4.1 The T -matrix method for circular graphene quantum dot We consider a single-layer CGQD defined by the radial confinement potential U (r) that is assumed to be smooth on the scale of the graphene lattice spacing Using the units such that = and the Fermi velocity = 1, the low-energy electron dynamics in this structure can be described by the 2D Dirac-like Hamiltonian H = σ · p + ν∆σz + U (r), (4.1) where σ = (σx , σy ) are the Pauli matrices, p = −i(∂x , ∂y ) is the 2D momentum operator, ν = ±1 is the valley index for the valleys K and K respectively, and ∆σz is a constant mass term This term is due to the interaction between the graphene sheet and the substrate Assume that Ψ(r, φ) is an eigen-function corresponding to the energy E Ψ(r, φ) = eijφ e−iφ/2 χA (r) e+iφ/2 χB (r) , (4.2) where the total angular momentum j takes half-integer values and the radial 14 spinor χ = (χA , χB )t satisfies the following equation: U (r) − E + ν∆ −i(∂r − j− 12 r ) −i(∂r + j+ 12 r ) U (r) − E − ν∆ χA (r) χB (r) = (4.3) Consider a particular case of GQD defined by an electrostatic potential of the form:   Ui , r ≤ ri , Uf , r ≥ rf , (4.4) U (r) =  bt k, r cn li We consider some region < r < rb , where the potential U (r) is constant, ¯ For E = U ¯ ± ν∆, the general solution to equation (4.3) in this U (r) = U region can be written in terms of two independent integral constants C = (C (1) , C (2) )t ¯ , r)C , χ(r) = W (U (4.5) ¯ , r) = W (U Yj− 12 (qr) Jj− 21 (qr) iτ Jj+ 21 (qr) iτ Yj+ 12 (qr) , (4.6) ¯ )2 − ∆2 and τ = q/(E−U ¯ +ν∆) In which Jj± and Yj± where q = (E − U 2 ¯ ± ν∆, are the Bessel function of the first and the second kind In case E → U the basic solutions of the equation (4.6) become divergent For simplicity, we ¯ ± ν∆ and consider the case E = U ¯ ± ν∆ assume that E = U The spinors at r1 and r2 should be linearly related by a matrix G: χ(r2 ) = G(r2 , r1 )χ(r1 ) (4.7) When we represent the spinors at r1 ≤ ri and r2 ≥ rf by the basic coefficients Ci and Cf , these coefficients are related: Cf = T Ci , (4.8) T = W −1 (Uf , r2 )G(r2 , r1 )W (Ui , r1 ) (4.9) By replacing the equation (4.7) into the equation (4.3) we get i ∂G(r2 , r1 ) = H(r2 )G(r2 , r1 ) , ∂r2 (4.10) where the Hamiltonian is assumed by H(r) = i j− 12 r U (r) − E − ν∆ U (r) − E + ν∆ 15 −i j+ 12 r (4.11) This equation needs to be solved for G(r2 , r1 ) with the initial condition such that G(r1 , r1 ) is the (2 × 2) identity matrix In fact G(r2 , r1 ) can be solved by a relevant numerical method such as Runge-Kutta method After getting G(r2 , r1 ), the equation (4.9) allows us to calculate the T -matrix 4.1.1 Bound states The bound states are the states of the carriers affected by a potential but these carriers tend to localize in a given region The general equation to determine the energy spectrum of all the bound states in the considered energy regions for a GQD is given by T11 + iT21 = 4.1.2 (4.12) Quasi-bound states Electron can be confined temporarily at the quasi-bound states (QBSs) with a finite lifetime Each QBS is characterized by a complex energy E = Re(E) + i Im(E) with the imaginary part Im(E) < relatively small This part plays a role of a disturbance Re(E) defines the position of the QBS (i.e the resonant level) |Im(E)| is a measure of the resonant level width and its inverse is a measure of the carrier lifetime at the QBS, τ0 ∝ 1/(2|Im(E)|) The general equation to determine the energy spectrum of the QBSs in CGQD is as follows T11 + isT21 = 4.1.3 (4.13) Density of states By using T -matrix we can easily get LDOS With each angular momentum j, LDOS can be calculated as ρ(j) (E) ∝ |E| π (j) T11 (j) (4.14) + T21 By summing Eq (4.14) over j, we obtain the total LDOS: +∞ ρ(j) (E) ρ(E) = j=−∞ 16 (4.15) 4.2 Graphene quantum dot defined by a radial trapezoidal potential Consider a CGQD induced by the radial trapezoidal potential Ui = U0 , ri = (1 − α)L, Uf = 0, rf = (1 + α)L, i v U (r) = Ui + rr−r (Uf − Ui ) vi ri < r < rf f −ri This potential becomes the rectangular potential in the limiting case of α = 0.50 ● ● −Im[E] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (a) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 20 Re[E] (b) LDOS α = 0.3 α = 0.5 α = 0.7 10 20 E Figure 4.1: QBS spectra (a) and LDOS (b) of a GQD induced by the trapezoidal radial potential of L = and U0 = 20 are presented for ν = +, j = 23 and various α In (a): curves correspond to QBS levels, each describing how the QBS energy (Im(E) and Re(E)) changes as α varying regularly from 0.3 (top) to 0.7 (bottom) from larger point-sizes to smaller point-sizes Figure 4.1(a) shows how the complex energies of five different QBSs change as the smoothness α varies from 0.3 to 0.7 With increasing potential smoothness α, while the real part of the energy just changes slightly, the imaginary part decreases substantially This result proposes an approach of creating the states with a long lifetime and can be manipulated such that they can satisfy the requirements of manufacturing electronic devices Figure 4.1(b) represents LDOS (in arbitrary unit) for three spectra with the α values examined in figure 4.1(a) to compare the correlation between QBSs and LDOS The positions of QBSs in (a) and the resonant peaks of LDOS in (b) are in a good 17 agreement Moreover, the imaginary parts of the QBS energies represent the widths of the corresponding LDOS peaks quite well 4.3 CGQD induced by an arbitrary axially symmetric electrostatic potential 4.3.1 Method for calculating LDOS from normalized wave functions The eigen function Ψ(E) (r, φ) has the form which is defined by the equation (E,j) (E,j) (4.2) The wave function χ(E,j) (r) = (χA (r), χB (r))T obeys the equation: ∂χ(E,j) (r) i = H(r)χ(E,j) (r), (4.16) ∂r j− 12 r U (r) − E Because of the axij+ U (r) − E −i r ally symmetric potential, LDOS only depends on the radial coordinate r: i where the Hamiltonian H(r) = +∞ ρ(j) (E, r), ρ(E, r) = ρ(j) (E, r) ∝ j=−∞ ρ(j) (E, r) ∝ χ(E,j) (r) , ∆E (4.17) χ(E,j) (r) , ∆E (4.18) where ∆E is the level spacing at the energy E and χ(E,j) (r) is the normalized wave function For r ≥ rf , the wave function can be represented in term of two integral (1) (2) constants Cf = (Cf , Cf )T : χ(E,j) (r) = Wf (r)Cf , where Wf (r) = Yj− 12 (qf r) Jj− 21 (qf r) , iτf Jj+ 21 (qf r) iτf Yj+ 12 (qf r) (4.19) Because the wave function is equal to at r = L, we can find the level spacing: π ∆E = L 4L C The normalization condition can be found in the form: |E−Uff | = The eigenfunction of the equation (4.16) near the center of QD has the form χ(E,j) (r) = N Jj− 21 (qi r) , iτi Jj+ 21 (qi r) 18 (4.20) Năng lượng Năng lượng Figure 4.2: (a, b) LDOS of CGQD with R0 = 5.93 nm, V0 = 0.43 eV (a background correction leads to a shift of ED = −0.347eV in the energy position.): (a) Experimental data; (b) Calculated results using the present approach; (c) Two TDOSs calculated from the data in (a) (dashed) and (b) (solid line) [log scale, arbitrary unit] The resonances are labelled by their angular momentum Panels (d−f ) compare the partial LDOS for the state of j = 21 : (d) from the experimental paper; (e) Eq (4.18) without normalization; (f ) Eq (4.18) with normalization with qi = |E − Ui |, τi = sign(E − Ui ) and N is the normalization coefficient Then we take the solutions given in the equation (4.20) at ri as the initial values and solve the equation (4.16) for χ(E,j) (r) With the normalized wave function, we can calculate LDOS using the Eq (4.17) 4.3.2 Comparison with experiment To illustrate the method of calculating LDOS that we introduce in the above contents we calculate LDOS in two cases of the step and smooth potential Besides we also compare our results with experimental ones Figure 4.2 compares the experimental data (a) and the LDOS that we calculated (b) We point out that the widths of the resonant peaks that are extracted from the cal19 culated LDOS are almost compatible with the corresponding values obtained by T -matrix method and describe quantitatively the experimental data Especially, our result shows that the QBS of the lowest angular momentum, in accordance with our expectation, is located at a higher energy level compared to the theoretical calculations present in the experimental paper of Gutierrez et al 4.4 CGQD in the magnetic field In this section we develop the T -matrix approach for CGQD induced by an axially symmetric electrostatic potential and subjected to a perpendicular uniform magnetic field 4.4.1 Wave function form When an uniform magnetic field is applied on the QD, the Hamiltonian of the Dirac fermion has the form: e (4.21) H = vF σ · p + A + ν∆σz + U (r), c where A is a vector potential, B = ∇ × A Assume that the magnetic field is perpendicular, B = (0, 0, B), one can choose A = B2 (−y, x, 0) The Schră odinger equation with the operator H is as follows HΨ(r, φ) = EΨ(r, φ) , (4.22) where Ψ is the eigenfunction corresponding to the energy E Because U (r) is axially symmetric, in polar coordinates this eigenfunction is represented by Ψ(r, φ) = eijφ e−iφ/2 χA (r) e+iφ/2 χB (r) , (4.23) in which the total angular momentum j takes half-integral values Let b = 2lB vi lB = c eB the magnetic characteristic length ¯ , let aσ = 2b j + For a constant potential value U (r) = U − (E−U¯ ) −∆2 v2 F and nσ = j − σ2 with σ = ±1 for the lattice A/B The general solutions are given in term of the Kummer hypergeometric function: ασ M (qσ , + nσ , br2 ), r ≤ R χσ (r) = 2(1+nσ ) e−br /2 rnσ , (4.24) βσ U (qσ , + nσ , br2 ), r > R where R is the effective radius of a GQD and qσ = 20 aσ b + (1 + nσ ) 4.4.2 T -matrix formula Using the hypothetical solution given in Eq (4.23) to the eigenvalue problem of the Hamiltonian in the Eq (4.21), we receive the equation for the radial spinor χ(r) = (χA (r), χB (r))t as follows   U (r) − E + ν∆ −i vF ∂r − j− 12 r −i vF ∂r + − br j+ 12 r + br U (r) − E − ν∆   χA (r) χB (r) =0 (4.25) Consider the case of a CGQD defined by an axially symmetric electrostatic potential:  r ≤ ri ,  Ui , Uf , r ≥ rf , (4.26) U (r) =  arbitrary, ri < r < rf , We consider the equation (4.25) in some region < r < rb , where the ¯ , and the spinor components have potential has a constant value, U (r) = U the form: χσ (r) = e−br /2 nσ r C (1) ασ M(qσ , + nσ , br2 ) + C (2) βσ U(qσ , + nσ , br2 ) , (4.27) where qσ = aσ = 2b j + aσ + 2(1 + nσ ) , b σ ¯ )2 − ∆2 ]/( vF )2 , − [(E − U 2 and nσ = j − σ2 (σ = A/B is identified with σ = ±1), b = 1/2lB The coefficients ασ and βσ are defined only based on the ratio between them The solution of the equation (4.25) can be written in the form: χ(r) = ¯ , r)C , where C = (C (1) , C (2) )t are basic coefficients and the matrix W W(U is given by the equation: α+ rn+ M(q+ , + n+ , br2 ) β+ rn+ U(q+ , + n+ , br2 ) α− rn− M(q− , + n− , br2 ) β− rn− U(q− , + n− , br2 ) (4.28) ¯ ± ν∆ The spinors at For simplicity, we just consider the case of E = U r1 and r2 are mutually related by a matrix G: χ(r2 ) = G(r2 , r1 )χ(r1 ) The spinor at r1 ≤ ri and r2 ≥ rf are represented by basic coefficients Ci and Cf which are linearly related, Cf = T Ci ¯ , r) = e− W(U br 2 21 The equation (4.4.2) is simply a basic transformation of the equation (4.4.2) and we have the following relation T = W −1 (Uf , r2 )G(r2 , r1 )W (Ui , r1 ) (4.29) This equation is correct for all r1 ≤ ri and r2 ≥ rf , including r1 = ri and r2 = rf Replace the equation (4.4.2) into the equation (4.25) we get the differential equation for G(r2 , r1 ) in term of the form of an equation of motion in the r direction ∂G(r2 , r1 ) i = H(r2 )G(r2 , r1 ) , (4.30) ∂r2 Note that in case of the presence of the magnetic field the hypothetical Hamiltonian H(r) in the differential equation for G(r2 , r1 ) is defined by   j− 21 i + b r U (r) − E − ν∆ r  (4.31) H(r) =  j+ 12 + b r U (r) − E + ν∆ −i r The equation (4.30) needs to be solved to find G(r2 , r1 ) One can use a relevant numerical approach for a ordinary differential equation such as Runge-Kutta The equation (4.29) allow us to calculate T -matrix after getting G(r2 , r1 ) 22 Conclusions The major results presented in this thesis can be summarized as follows: We have suggested a model of Gaussian-type confinement potential to describe the transport properties of the locally gated GBJs The advantages of this potential are: (i) practicality, it better reflects the electrostatic potential profile created by the gates with the parameters determined exactly like in the experiment and (ii) simplicity, thanks to a continuous variation of the Gaussian-type potential, the Dirac equation can be solved effectively and easily by the T -matrix approach Besides, this type of potential allows us to examine GBJ in all possible charge density regimes (n-p-n, p-n-p, n-n -n, and p-p -p), including a smooth transition between the regimes We have studied systematically the ballistic transport characteristics of the GBJs at zero temperature The achieved results are new They prove the presence of the Klein tunneling effect as well as describe quantitatively the picture of the radiation and interference of electron waves inside the barrier The calculation results of the resistance and shot noise are in a good agreement with the corresponding experimental results We haved developed T -matrix formalism to study the electronic properties of CGQDs induced by an axially symmetric confinement potential In the case that the confinement potential has a flat form near the center of the QD, we have derived exactly: (1) the equation that determines BS, (2) the equation that determines QBS, (3) LDOS and (4) the resonant scattering coefficients All of them are represented simply in term of the T -matrix components These results are the general, completely new and effective for studying the characteristics of the QBS spectrum of CGQDs in which the positions and the lifetime of QBSs can be determined exactly We have proposed a method for calculating LDOS of CGQDs created by an axially symmetric confinement potential in an arbitrary form We can find the QBSs spectrum from LDOS Besides, LDOS tightly relates to the tunneling through the CGQD: Measurement of the tunneling differential conductance gives a picture of LDOS Hence, the calculation of LDOS is so meaningful to study the QBSs spectrum in comparison to experiment In fact, our calculation method gives the results that is 23 consistent with Gutierrez’s experiment data and fits better than their theoretical results Our method of calculating LDOS is totally new and very general in the sense that it is not limited by the electrostatic potential profile If the potential has a flat form near the center of QD, the QBS spectrum obtained from LDOS and from T -matrix are the same We have rudimentally developed the T -matrix approach to study the energy structure of CGQDs induced by electrostatic potential in the presence of an perpendicular uniform magnetic field All of the above results are new The models and calculation methods suggested in this thesis are universal and can be easily applied or extended to other structures or materials 24 List of publications Nhung T T Nguyen, D Quang To and V Lien Nguyen, “A model for ballistic transport across locally gated graphene bipolar junctions,” Journal of Physics: Condensed Matter, vol 26, p 015301, 2014 H Chau Nguyen, Nhung T T Nguyen, V Lien Nguyen, “The transfer matrix approach to circular graphene quantum dots,” Journal of Physics: Condensed Matter, vol 28, p 275301, 2016 H Chau Nguyen, Nhung T T Nguyen, V Lien Nguyen, “On the density of states of circular graphene quantum dots,” Journal of Physics: Condensed Matter, vol 29, p 405301, 2017 Nhung T T Nguyen, “Trapping electrons in a circular graphene quantum dot with Gaussian potential,” Communications in Physics, vol 28, no 1, pp 51–60, 2018 25 ... graphene nanostructures” The aim of this thesis is to study electrical transport characteristics of graphene nanostructures We focus on two research objects associated with two kinds of graphene nanostructures:... of graphene 1.1 Crystal structure and energy band structure of graphene Graphene is a monolayer of carbon atoms in which the atoms are on the vertices of a two-dimensional honeycomb lattice A graphene. .. interference of chiral waves 11 ĐIỆN TRỞ R HỆ SỐ FANO F ĐIỆN THẾ CỦA TOP GATE Vt (V) Figure 3.3: Resistances R (a), odd resistance 2Rodd (b), and Zero-bias Fano factors F (c) are plotted versus Vt for three