Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Conductance of non-interacting two-dimensional junctions of three quantum wires : general behaviors and… Shi, Zheng 2012

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2012_fall_shi_zheng.pdf [ 4.45MB ]
Metadata
JSON: 24-1.0085574.json
JSON-LD: 24-1.0085574-ld.json
RDF/XML (Pretty): 24-1.0085574-rdf.xml
RDF/JSON: 24-1.0085574-rdf.json
Turtle: 24-1.0085574-turtle.txt
N-Triples: 24-1.0085574-rdf-ntriples.txt
Original Record: 24-1.0085574-source.json
Full Text
24-1.0085574-fulltext.txt
Citation
24-1.0085574.ris

Full Text

Conductance of non-interacting two-dimensional junctions of three quantum wires General behaviors and Fano resonance  by Zheng Shi B. Sc., Peking University, 2010  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Master of Science in THE FACULTY OF GRADUATE STUDIES (Physics)  The University Of British Columbia (Vancouver) April 2012 c Zheng Shi, 2012  Abstract Junctions of three quantum wires, or ”Y junctions”, are among the basic building blocks of circuits of quantum wires. In most previous work the wires are modeled as one-dimensional objects. A step towards reality takes into account their finite width. In this thesis, we study non-interacting twodimensional Y junctions in the single-channel regime. The generic behaviors of zero-temperature conductance of two different types of Y junctions are discussed. Particular attention is given to the asymmetric line shapes, or Fano resonances, arising in the problem.  ii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Junctions of three Tomonaga-Luttinger liquid (TLL ) quantum wires . . . . . . . . .  2  1.1.1  g<1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.1.2  g=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.1.3  1<g<3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  1.1.4 1.1.5  g=3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3<g<9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5 5  1.1.6  g>9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6  2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  1.2  2.1  Quantum wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  2.2  Continuum model, Landauer formalism . . . . . . . . . . . . . . . . . . . . . . .  9  2.3  Fano resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10  2.4  Tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12  3 Conductance of two-dimensional Z3 symmetric Y junctions . . . . . . . . . . . . . .  20  3.1  Time-reversal invariant case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  20  3.2  Magnetic field: effect of Aharonov-Bohm phase, Fano resonance . . . . . . . . . .  21  3.3  Magnetic field: effect of Lorentz force . . . . . . . . . . . . . . . . . . . . . . . .  23  4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  39  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  40  iv  List of Figures Figure 1.1  Schematic of a junction of three quantum wires. . . . . . . . . . . . . . . . . .  Figure 1.2  Important fixed points of the interacting one-dimensional Y junction problem.  4  N and χ± behaviors are also found in Z3 symmetric non-interacting one-dimensional Y junctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Figure 1.3  RG flow of the three-wire-junction problem in the Γ-φ plane. . . . . . . . . . .  7  Figure 1.4  RG flow of the three-wire-junction problem in the Γ-φ plane (continued). . . .  7  Figure 2.1  Geometry of a two-dimensional quantum wire. . . . . . . . . . . . . . . . . .  15  Figure 2.2  Geometry of a two-dimensional Y junction of quantum wires. The junction area  Figure 2.3 Figure 2.4  is shaded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  16  Fano line shapes for various Fano parameters. . . . . . . . . . . . . . . . . . .  17  2  Transmission probability |t1 | for a quasi-one-dimensional (Q 1 D ) two-lead junction in the two-band approximation. . . . . . . . . . . . . . . . . . . . . . . .  Figure 2.5  18  Two-dimensional quantum wire modeled on a square lattice. Here the width of the ribbon N = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  18  Figure 2.6  Lattice models of Y junctions used in numerical simulation. . . . . . . . . . .  19  Figure 3.1  Total transmission probability Ttot versus energy in a Y junction without magnetic flux. The dotted line, Ttot = 8/9, shows the maximal Ttot in the Z3 symmetric and time-reversal invariant case. Fano resonances occurring in this figure will be explained in Sec. 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.2  25  Probability density and probability current distributions inside the junction area near a maximal transmission (E = 2.5, no magnetic flux applied, Ttot = 0.8889). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity. . . . . . . . . .  Figure 3.3  26  Probability density and probability current near a complete reflection (E = 10.25, no magnetic flux applied, Ttot = 0.0079). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity. . . . . . . . . . . . . . . . . . . . . . . . v  27  Figure 3.4  Total transmission probability Ttot versus energy and flux in Aharonov-Bohm (AB)-type Y junctions with different dimensions. . . . . . . . . . . . . . . . .  Figure 3.5  The difference of chiral transmission probabilities TR − TL versus energy and flux in AB-type Y junctions with different dimensions. . . . . . . . . . . . . .  Figure 3.6  Total transmission probability Ttot versus energy in a  AB -type  30  Total transmission probability Ttot and chiral transmission probabilities TL , TR versus energy in a AB-type Y junction with 1/4 flux quantum. . . . . . . . . .  Figure 3.8  29  Y junction with  different values of flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.7  28  30  Probability density and probability current near a perfectly chiral transmission (E = 25, 1/4 flux quantum, Ttot = 0.9967, TR = 0.9935, TL = 0.0032). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity. . . . . . . . . . . . . . .  Figure 3.9  31  Probability density and probability current distributions when the total transmission probability is independent of the magnetic flux (E = 6.71. There is no magnetic field in panels (a) and (b), Ttot = 0.5384; 1/4 flux quantum is applied in panels (c) and (d), Ttot = 0.5393, TR = 0.4283, TL = 0.1110). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity. . . . . . . . . . . . . . . . .  32  Figure 3.10 Probability density and probability current near a type I Fano resonance (1/4 flux quantum is applied. E = 4.7 in panels (a) and (b), Ttot = 0.8895, TR = 0.4933, TL = 0.3962; E = 4.73 in panels (c) and (d), Ttot = 0.0046, TR = 0.0023, TL = 0.0022). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity.  33  Figure 3.11 Probability density and probability current near a type II Fano resonance (1/4 flux quantum is applied. E = 8.8 in panels (a) and (b), Ttot = 0.8067, TR = 0.5844, TL = 0.2222; E = 8.81 in panels (c) and (d), Ttot = 0.0833, TR = 0.0460, TL = 0.0373). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity.  34  Figure 3.12 Total transmission probability Ttot versus energy in a Y junction without magnetic flux. Different single particle scattering potentials have been applied to the central region: V (r) = V0 sin(3θ ) and V (r) = V0 cos(3θ ), where V0 = 1 and  θ is the polar angle measured from the center of the junction, the polar axis pointing opposite to the 1st wire. . . . . . . . . . . . . . . . . . . . . . . . . .  vi  35  Figure 3.13 Total transmission probability Ttot,1 versus energy in an AB-type Y junction. Z3. symmetry is broken by a single particle scattering potential, V (r) = V0 sin(2θ ), in conjunction with a modification of geometry, namely replacing the straight wall connecting wires 1 and 3 with an arc that smoothly interpolates between the wires. V0 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  36  Figure 3.14 Total transmission probability Ttot versus energy and flux in a Lorentz-type Y junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  37  Figure 3.15 The difference of chiral transmission probabilities TL − TR versus energy and flux in a Lorentz-type Y junction. . . . . . . . . . . . . . . . . . . . . . . . .  vii  38  List of Abbreviations 2 DEG two-dimensional electron gas TLL  Tomonaga-Luttinger liquid  Q1D  quasi-one-dimensional  BW  Breit-Wigner  AB  Aharonov-Bohm  viii  Acknowledgments I am deeply grateful to my supervisor, Professor Ian Affleck, who has helped me through the preparation of this thesis with both deep physical insights and great patience.  ix  Dedication For my parents.  x  Chapter 1  Introduction Modern technologies have enabled study of systems which possess reduced dimensionality. An outstanding example comes from semiconductor structures [27]; when a physical dimension of the structure becomes comparable to the electron Fermi wavelength, the energy levels in that direction exhibit strong quantization. It is possible to fabricate heterostructures consisting of semiconductor layers whose thickness are of the order of 10 nm, and realize a two-dimensional electron gas (2 DEG ) with appropriate doping [23]. Further confining the 2DEG in one lateral direction, it is observed that only in the direction perpendicular to the direction of confinement can electrons move freely; in both other directions the energy levels are strongly quantized [10]. Such an effectively one-dimensional electronic waveguide is called a quantum wire. An alternative realization of quantum wires is found in carbon nanotubes [17]. The one-dimensional nature of quantum wires allows for non-Fermi-liquid behaviors in the presence of electron-electron interaction. It has been long established that the Tomonaga-Luttinger liquid (TLL ), instead of the Fermi liquid, describes the low-energy properties of such interacting one-dimensional models [12]. The strength of interaction in a  TLL  is characterized by the so-  called Luttinger parameter g. For spinless fermions, a repulsive interaction gives g < 1 while an attractive interaction produces g > 1; g is exactly 1 in a non-interacting model. The effect of reduced dimensionality is particularly crucial for an impurity embedded in a TLL . Examples include a single constriction in a quantum wire [18] or a weak link in a spin chain [11]; in a spinless fermion wire it has been verified that at zero temperature the conductance through the constriction becomes zero for g < 1 or the ideal value ge2 /h for g > 1, regardless of the strength of the constriction [18]. Molecular electronic devices can be built with quantum wires, and it is expected that junctions where three or more quantum wires meet will play an essential role in these devices. As  TLL  physics emerges in bulk quantum wires, intriguing behaviors are brought to many-wire junctions. Considerable theoretical effort has gone into electron transport in such nanostructures [3, 7, 13, 16, 19, 22, 24]. Before examining more closely the concept of one-dimensional wires, we review in the  1  following section existing results on junctions of three TLL quantum wires, or Y junctions [22].  1.1 Junctions of three TLL quantum wires The simplest model system is depicted in Fig. 1.1; three quantum wires are connected to the junction, represented by a ring that supports a magnetic flux. As a first approximation, the quantum wires are modeled as single-channel TLLs of spinless fermions (to which we simply refer as electrons). The Hamiltonian of the system is the sum of a bulk term Hbulk and a boundary term HB . Hbulk comprises three identical semi-infinite TLL wires with Luttinger parameter g, while 3  HB = − ∑ Γeiφ /3 ψ †j (0) ψ j−1 (0) + h.c. j=1  where j is a wire index, j = 1, 2, 3 ≡ j + 3; ψ j (x = 0) annihilates an electron at the end of the j-th wire connecting to the junction located at x = 0. Γ is the tunneling amplitude between wires, and φ is a phase determined by the magnetic flux in the junction. One may attach all three wires identically to the junction, which is the Z3 permutation symmetric case; alternatively, the Z3 symmetry may be broken by attaching the wires in different ways. The magnetic flux most generally will break time-reversal symmetry of the junction, but φ = 0 or φ = π leave time-reversal symmetry unbroken. We are interested in how such a junction conducts electricity. The characterizing quantity is the conductance tensor of the junction, which in the linear response regime is defined via I j = ∑ G jkVk k  where j, k are wire indices, I j is the current flowing into the junction from the j-th wire and Vk is the voltage applied to wire k. G jk is then the 3 × 3 conductance tensor characterizing electron transport of the junction. Simple physical considerations immediately lead to ∑ j G jk = ∑k G jk = 0. In the Z3 symmetric case the conductance tensor may be decomposed into a symmetric part and an antisymmetric one: G jk =  GS GA 3δ jk − 1 + ε jk 2 2  where ε jk is defined by ε j j = 0, ε j, j+1 = −ε j+1, j = 1. GA always vanishes in the presence of timereversal symmetry. We now list for different ranges of the Luttinger parameter g the stable RG fixed points that dominate the low-energy physics of the junction, and the corresponding conjectured RG flows in the parameter space of Γ − φ [22]. The reason that such a parameter space is usually sufficient is that Z3 symmetry is restored in most cases at low energies, even if it is initially broken in the  2  microscopic model.  1.1.1 g < 1 g < 1 in the bulk quantum wire implies a repulsive electron-electron interaction. The stable fixed point is no more than Γ = 0, namely completely decoupled wires. We call it the N fixed point (Fig. 1.2a). Obviously, it gives zero conductance G jk = 0. In the N fixed point, time-reversal symmetry is restored since the phase φ is of no importance. The Z3 asymmetry is irrelevant.  1.1.2 g = 1 g = 1 is the noninteracting case where the problem may be solved exactly as a single-particle one. The solution is represented by a 3 × 3 scattering matrix of the free electron, and a continuous manifold of fixed points exists. When time-reversal symmetry is imposed, the maximal conductance GS on the fixed manifold is 8/9e2 /h, as required by the unitarity of single-particle S-matrix. To prove this, parameterize the time-reversal invariant S-matrix as   Unitarity then leads to  |r| eiθr |t| eiθt |t| eiθt      S =  |t| eiθt |r| eiθr |t| eiθt  |t| eiθt |t| eiθt |r| eiθr  |r|2 + 2 |t|2 = 1 2 |r| |t| cos (θr − θt ) + |t|2 = 0 Eliminating r, |t|2 =  4 cos2 (θr − θt ) 8 cos2 (θr − θt ) + 1  Hence |t|2 ≤ 4/9. According to the Landauer formalism which will be further explained in Sec. 2.2, for non-interacting spinless electrons at zero temperature G jk = e2 /h δ jk − S jk  2  where the  S-matrix is evaluated at the Fermi energy; therefore GS ≤ 8/9 e2 /h. The N fixed point is a special point on this manifold; so are the chiral χ± fixed points (Fig. 1.2a), where electrons are perfectly transmitted from wire j to wire j ± 1. At the χ± fixed points, GA = ±GS and G jk =  e2 δ jk − δ j,k±1 h  3  Figure 1.1: Schematic of a junction of three quantum wires.  Figure 1.2: Important fixed points of the interacting one-dimensional Y junction problem. N and χ± behaviors are also found in Z3 symmetric non-interacting one-dimensional Y junctions.  4  Another simple fixed point, DA (Fig. 1.2b), emerges when Z3 symmetry breaking is allowed. It features maximally broken Z3 symmetry and connects two of the quantum wires perfectly leaving the third decoupled from the junction. When there is interaction in the junction, other unstable fixed points that are not described by single particle S-matrices do exist.  1.1.3 1 < g < 3 The N fixed point in this case becomes unstable against electron hopping at the junction. The system is renormalized, in the presence of time-reversal symmetry, into the M fixed point which is unstable against inclusion of magnetic flux. At the M fixed point, it is found in [24] that GS =  4gγ e2 2g + 3γ − 3gγ h  where γ ≈ 0.42. In the absence of time-reversal symmetry, the RG flows are directed to stable chiral fixed points  χ± , whose conductance tensor is given by GS =  4g e2 3 + g2 h  and GA = ±gGS . The DA fixed point is unstable.  1.1.4 g = 3 The N fixed point is unstable. There is a manifold of fixed points characterized by the scattering matrix of a free fermion; but different from the case of g = 1, the free fermion is not the original electron. χ± are special points on the continuous manifold; another special point, DP , maximizes GS : GS = (4g/3)(e2 /h) while GA = 0. The enhanced conductance at DP may be interpreted as a result of pair tunneling. The M fixed point is not on the manifold and once again unstable against inclusion of magnetic flux.  1.1.5 3 < g < 9 DP becomes a stable fixed point, while M remains unstable against inclusion of magnetic flux.  1.1.6 g > 9 In this case, time-reversal invariant systems are renormalized into a stable fixed point DN whereas generic values of φ continue to drive the system into the stable fixed point DP . DN and DP have the same conductance tensor but different dimensions of the leading irrelevant operator.  5  1.2 This thesis Results in the previous section rest on the assumption that the quantum wires are strictly onedimensional. However, as previously explained, real wires despite exhibiting heavily one-dimensional nature are in fact higher-dimensional. The difference may be less important in bulk quantum wires, but could be striking in the vicinity of a quantum impurity such as a junction. To look into the consequences of abandoning the one-dimensional approximation, as a first step we present a study of two-dimensional non-interacting Y junctions of quantum wires which are quasi-one-dimensional (Q 1 D ), or have finite widths. Such junctions have also been objects of intense investigation [1, 5, 8, 14, 15], and are generally expected to host more complicated internal structures than their one-dimensional counterparts. Previous studies are focused on fabrication of devices that exhibit selective properties and perform specialized functions [5, 8, 15]; in contrast, we attach more importance to the universal aspects of the problem, which could serve as underpinnings of a full treatment of interaction in future work. The remainder of this thesis is organized as follows. In the second chapter, the models and formalisms of single-particle scattering in junctions of quantum wires are established. We will explain the concepts of open and closed channels through a simple model of a quantum wire, and the Landauer formalism that underlies the description of single-particle scattering in terms of the S-matrix. Then we show a common origin of Fano resonances which are frequently encountered in the Y junction problem. In Sec. 2.4 we discuss the numerical recipe for the problem based on a tight-binding description. Finally we present two types of models of two-dimensional Y junction to be studied. The two models mainly differ in the role played by the magnetic flux; a Lorentz force is exerted on electrons in one of the models, but in the other model the flux influences the phase of wave functions through purely quantum-mechanical interference or Aharonov-Bohm (AB) effects. The third chapter is devoted to presentation and analysis of the single-particle conductance tensors of two-dimensional Y junctions. First we study the general behaviors of conductance changing with energy and the magnetic flux; also analyzed are the corresponding probability density and current distributions. We then turn our attention to Fano resonances that arise in conductance of the purely-AB -type Y junction, and classify them according to their response to the magnetic flux and scattering potential. At the end of the chapter an explanation of the behaviors of conductance in the Lorentz-type Y junction is put forward.  6  (a) g < 1  (b) g = 1  (c) 1 < g < 3  Figure 1.3: RG flow of the three-wire-junction problem in the Γ-φ plane.  (a) g = 3  (b) 3 < g < 9  (c) g > 9  Figure 1.4: RG flow of the three-wire-junction problem in the Γ-φ plane (continued).  7  Chapter 2  Model 2.1 Quantum wires A more detailed model for higher-dimensional quantum wires will be needed to clarify a number of important concepts. For simplicity we begin with a model where the electrons move in a continuum and are confined by parallel hard walls (Fig. 2.1). The Hamiltonian takes the form H = Hkin + V , where Hkin = p2 /2m; a quadratic dispersion is expected to be accurate for semiconductor heterostructures where the Fermi energy is small compared to the bandwidth. V is zero between the parallel straight walls which model the boundaries of the wire; V goes to ∞ beyond the walls. The model is free of any scattering, with the intention of addressing the ideal electron waveguide behavior of the wires [26]. It is now a textbook result that the dispersion relation of the electron is given by En (k) = h¯ 2 k2 /2m + π 2 n2 h¯ 2 /2ma2 where a is the width of the wire. Each positive integer n represents a band or channel in the scattering theory glossary. We will be working at a given energy E which later turns out to be the Fermi energy. For a given E there may be several channels for which π 2 n2 h¯ 2 /2ma2 < E, and these bands are called open channels; other n’s then correspond to closed channels. For a given E, there is a real wavevector k in an open channel, and therefore a charge carrier with wavevector k and energy E is able to propagate along the wire; in contrast, in a closed channel the wavevector k acquires an imaginary part, signifying an evanescent wave or an exponential decay of the wave amplitude: hence the names open and closed channels. For this particular model of quantum wire there are always a finite number of open channels and infinitely many closed ones. The definitions of open and closed channels extend to other models of quantum wires - the only requirement is that the model is translational invariant along one direction and spatially confined in other direction(s).  8  2.2 Continuum model, Landauer formalism The picture of higher-dimensional quantum wires having been established, a description of the junction of matching accuracy is now called for. Our model is defined on a two dimensional domain, where a junction area is connected to three long straight wave guides that represent the quantum wires (Fig. 2.2) [5, 8, 14, 15]. Each waveguide is connected to a particle reservoir which is far away from the junction area. There may be a scattering potential and/or a magnetic flux applied to the junction area. In general the system does not have to be Z3 symmetric. Neglecting interactions, the Hamiltonian inside the domain may be written as the sum of a kinetic energy term and a potential energy term H = Hkin + V , where Hkin = p + eA/c  2  /2m. A  stands for the vector potential of the applied magnetic flux; note that it does not have to vanish everywhere outside of the junction area. e is the absolute value of electron charge. An immediate insight is that the wave function inside the wires far from the center is asymptotically none other than a linear combination of open channel wave functions for the wires [5, 8, 15]:  ψ∼  ∑  A jn φ jn e−ik jn x j + B jn φ jn eik jn x j  for x j → ∞  j,n∈open  where j = 1, 2, 3 is the wire index and n the channel index (summed over open channels n ≤ n j , n j can be different for different wires). φ jn stands for the transverse part of the wave function and is in the simple model above proportional to sin  nπ y j a  where the y j axis is perpendicular to the x j  axis. k jn the wavevector is given in the same model by k jn =  2mE/¯h2 − π 2 n2 /a2j . The x j axis is  parallel to the jth wire; larger x j corresponds to positions farther away from the junction. One may define a ”gluing interface” where the wire is connected to the central region of the junction and set x j = 0 on this interface. [See Fig. 2.2.] An j is the incident part of the wave function and Bn j the outgoing part. Further defining the S matrix which relates the column vector B to the column vector A:  (B11 , · · · , B1n1 , B21 , · · · , B2n2 , B31 , · · · , B3n3 )T = S (A11 , · · · , A1n1 , A21 , · · · , A2n2 , A31 , · · · , A3n3 )T What makes such a definition possible is the linearity of the Schroedinger equation. S must be unitary, as follows from probability current conservation. We pause to remark on the general form of S when the system has time-reversal symmetry. The time-reversal transformation can be effectively regarded as a complex conjugate operation on the wave function. Specializing to the wave function in question, the incident and outgoing parts are exchanged; also, the amplitudes are transformed into their complex conjugate. In other words, B → A∗ and A → B∗ ; consequently S → S−1 reversal symmetric system S =  ∗  ∗ S−1 ; recalling  under the time-reversal transformation. For a time∗  that S is always unitary, ST = S−1 , we are able to 9  identify the symmetry of the S matrix, S = ST , as the only additional requirement that time-reversal symmetry imposes on it. At this stage within the Landauer formalism the conductance of the Y junction may be obtained from the S matrix. This is done by noticing the total current in wire j is the incoming current in the j-th wire minus the reflected current in the same wire minus the transmitted currents from the other two wires. For spinless electrons,  Ij = e∑ n  dk 2π  v jn (k) nF (ε jn (k) − µ − eV j ) − ∑ v j′ n′ (k) S jn, j′ n′ (k) nF ε j′ n′ (k) − µ − eV j′ 2  j′ n′  where nF stands for the Fermi distribution, µ is the chemical potential, v jn is the group velocity for the n-th channel of the j-th wire and V j is the voltage applied to the j-th wire. The assumption of a thermal equilibrium distribution for the charge carriers has been made, as is standard in linear response theory. Expanding to linear order in voltage, at zero temperature Ij =  e2 2 V j − ∑ S jn, j′ n′ (EF ) V j′ h∑ n j′ n′  Comparing this to the definition of linear conductance, one immediately sees that the 3 × 3 conductance tensor is given by [9] G j j′ =  e2 δ j j′ − ∑ S jn, j′ n′ (EF ) h∑ n n′  2  It is then observed that ∑ j′ G j j′ = 0 as is consistent with current conservation. Now it becomes clear that in order to study the conductance tensor it is necessary to find the S matrix at the Fermi energy.  2.3 Fano resonance To show the prevalence of Fano resonances in the Y junction problem at hand, we retreat in this section to the general scattering theory in restricted geometries. In Q 1 D scattering, the symmetric Breit-Wigner (BW) line shape is not the only type of resonance encountered. Instead, the most general resonant line shape of the conductance takes the form of an asymmetric Fano function [21, 25]: G ∝ σE =  (qΓres /2 + E − Eres )2 (Γres /2)2 + (E − Eres )2  10  Γres is the half-width of resonance, and q is called the asymmetry parameter or Fano parameter. q → ∞ leads us back to the symmetric  BW  resonance while q → 0 is the symmetric  BW  anti-resonance.  Intermediate values of q give rise to Fano resonances (Fig. 2.3). A Fano resonance often results from interference between two scattering amplitudes, one of them due to a continuum background and the other due to the excitation of a discrete state. In elastic Q1D  scattering such discrete states are usually bound states which are embedded in open channels  but are nevertheless stable due to the presence of closed channels. The interference mechanism can be illustrated with a minimal two-band model following [25]. Consider a Q 1 D two-lead junction whose dynamics is governed by the non-interacting Schroedinger equation −¯h2 ∇2 /2m +Vc (y) +Vsc (x, y) ψ (x, y) = E ψ (x, y) Here the wire is infinite in the x direction; Vc (y) is the lateral confining potential of the quantum wire and Vsc (x, y) is a scattering potential that models the two-lead junction. The equation is usually difficult to solve, which motivates the assumption of a simplified scattering potential: Vsc (x, y) = v (y) δ (x) where v (y) is a real function of y. In addition, for sake of simplicity we always work in the single channel regime, so that the only open channel is the one with the lowest energy range; in the simplest continuum model, this requirement corresponds to E ∈ π 2 h¯ 2 /2ma2 , 4π 2 h¯ 2 /2ma2 where the only open channel is n = 1. Assuming all electrons are injected into the system at x = −∞, one can expand the wave function in φm , the transverse modes or eigenstates of Vc :  ψ (x, y) =  ∑m φm (y) δ1m eikm x + rm e−ikm x (x < 0) ∑m φm (y)tm eikm x (x > 0)  The wave function being continuous at x = 0,  ∑ φm (y) (δ1m + rm ) = ∑ φm (y)tm m  m  The Schroedinger equation integrated from x = 0− to x = 0+ yields −  h¯ 2 φm (y) [ikm tm − ikm (δ1m − rm )] + v (y) ∑ φm (y)tm = 0 2m ∑ m m  Making use of the orthogonality of φm , we eliminate rm : h¯ 2 ikm (tm − δ1m ) = ∑ vmn tn m n where vmn = φm |v| φn . The simplest approximation is to neglect all channels other than the lowest two, reducing the problem to a 2 × 2 one:  11  2  − h¯m ik1 + v11 v21  2  v12 h¯ 2 m κ2 + v22  t1 − h¯m ik1 = 0 t2  where κ2 = −ik2 is the magnitude of the imaginary wavevector. The solution reads 2  t1 =  − h¯m ik1  h¯ 2 m κ2 + v22  2  h¯ 2 m κ2 + v22  − h¯m ik1 + v11  − |v12 |2  For small |v12 |2 , |t1 |2 has an asymmetric line shape, as is shown in Fig. 2.4. In fact, we can find explicitly the condition for complete reflection h¯ 2 κ2 + v22 = 0 m and perfect transmission |v12 |2 h¯ 2 κ2 + v22 = m v11 when |v12 |2 is small compared to other energies in the problem, the difference between the energy at which complete reflection occurs and the energy at which perfect transmission occurs is also small, as is characteristic of a Fano resonance. It is shown above that in a two-lead junction Fano resonances appear when a continuum level from an open channel has the same energy as a bound state from a closed one. Fano resonances are indeed quite common and may be found in other similar situations, including the Y junction of three one-dimensional wires with a two-dimensional junction area [20]. In the following chapter we will demonstrate that they are also present in the fully two-dimensional Y junction. It is worth noticing, however, that Fano resonances do not exist exclusively in higher dimensions; in principle they can be found in strictly one-dimensional systems which combine resonant scattering and background scattering.  2.4 Tight-binding model We return to the question of how to find the S matrix. The problem is defined on a semi-infinite geometry, but the semi-infinite parts are described by the column vectors A and B and in numerical simulations one should be able to eliminate them. Their elimination is achieved by continuity of the wave function and its normal derivative at the gluing interfaces:  ψ (x j = 0) = ∑ (A jn + B jn ) φ jn j,n  12  dψ dx j  x j =0  = ∑ (−ik jn A jn + ik jn B jn ) φ jn j,n  Here n is summed over all channels i.e. including closed ones. Only by including closed channels can completeness of the Hilbert space be ensured. It in fact brings in an obstacle for numerical simulations: there are infinitely many closed channels to sum over in the continuum model of quantum wires introduced earlier, and numerically one has to cut off the number of channels considered to proceed. There are two resolutions to the problem. The first one involves introducing a buffer zone [5, 8, 15] for each wire 0 < x j < Λ j , including those buffer zones in the junction area, and taking only a limited number of closed channels, with the assumption that other closed channels with larger magnitude of wavevector have all decayed away for Λ sufficiently large. The second method, which is the one we use, is based on the observation that in a lattice model there are a finite number of channels to start with, so the whole model can be reconstructed on a lattice[2]. As an example of the tight-binding approach, we reconstruct a square lattice model of a quantum wire (Fig. 2.5), in the hope it will reduce to the continuum model in certain limits. The dispersion relation of a square lattice ribbon is given by En (k) = −2t cos kaT B − 2t cos  nπ N+1  where t is the hopping amplitude, aT B is the lattice constant, n is the band index and N the width of the ribbon. The continuum limit is recovered by taking a sufficiently large N and a sufficiently small aT B : En (k) ≈ −4t + tk2 a2T B + t  n2 π 2 N2  By identifying it with the continuum dispersion, we obtain ta2T B =  h¯ 2 t π 2 π 2 h¯ 2 , 2 = 2m N 2ma2  Their ratio simply gives NaT B = a, which suggests one may interpret the tight-binding lattice as a mesh laid down on the continuum model. As N is increased, the error due to discretization decreases. As a summary, the continuum model can be studied with a tight binding model as a numerical tool. Before ending this chapter we present the lattice models used in actual simulations. The junction is modeled by a hexagon that connects the three wires on its three edges (but does not have to be regular). In the center of the hexagon there may or may not be a ”hole”, or a packet of space that cannot be reached by the electron. It is quite convenient to add a finite single-particle scattering potential in the junction area. These junctions have been mostly chosen to be Z3 symmetric, partly because Z3 asymmetry has been reported to be typically irrelevant in 13  the RG sense in the one-dimensional case [22]. The other reason is that a Z3 symmetric model simplifies the tight-binding meshing process: it is convenient to use a honeycomb lattice to carry out the discretization, while general junctions without Z3 symmetry usually have to be tackled with the relatively computationally costly finite element method [14]. We have, however, also studied a Z3 asymmetric junction at the end of Sec. 3.2 to ensure that no generality is lost. There are two straightforward schemes to include the magnetic field (Fig. 2.6): we could focus on its  AB -like  effect, or influence on the phase of the wave function, and to this end a magnetic  flux is injected in the center of the hole. Alternatively, we could investigate the Lorentz force on electrons induced by the magnetic field, and spread the field uniformly within the junction area; the magnetic field vanishes inside the wires, and is for simplicity modeled to change abruptly across the interfaces to which the wires are attached [2]. We admit that probably neither configuration is realistic in experiments, and they represent no more than a first attempt on the effects of  AB  phase and Lorentz force, respectively. Nevertheless, in both cases the magnetic field is conveniently incorporated through Peirels substitution into the lattice model: t → t exp −  e h¯  A · dl hopping path  Also, we once again restrict ourselves to the single-channel regime. This is plausible because in many applications the system should be as close to one-dimensional as possible, which means only the presence of one species of conducting fermions is desired. Due to Z3 symmetry, the S-matrix is now reduced to   r  t2 t1   S =  t1  r  t2 t1     t2  r  In other words, the transmission amplitude from wire 1 to wire 2 is identical to that from wire 2 to wire 3 (both are equal to t1 ), and the transmission amplitude from wire 2 to wire 1 is identical to that from wire 3 to wire 2 (both are equal to t2 ). However, t1 does not have to be equal to t2 unless the system is time-reversal invariant. The conductance tensor at zero temperature takes the form  G=    e2   h  |t1 |2 + |t2 |2  − |t2 |2  − |t1 |2  − |t1 |2  |t1 |2 + |t2 |2  − |t2 |2  − |t2 |2  − |t1 |2  |t1 |2 + |t2 |2  14      Figure 2.1: Geometry of a two-dimensional quantum wire.  15  Figure 2.2: Geometry of a two-dimensional Y junction of quantum wires. The junction area is shaded.  16  q=0 q=1  5  q=2  4  E  3  2  1  0 -4  -2  0  (E-E )/ 0  2  4  res  Figure 2.3: Fano line shapes for various Fano parameters.  17  1.0  v v 0.8  v v  =0  12  =0.05  12  =0.25  12  =0.5  12  1  |t |  2  0.6  0.4  0.2  0.0 0.0  0.2  0.4  0.6  0.8  1.0  E  Figure 2.4: Transmission probability |t1 |2 for a proximation.  Q1D  two-lead junction in the two-band ap-  Figure 2.5: Two-dimensional quantum wire modeled on a square lattice. Here the width of the ribbon N = 10. 18  Figure 2.6: Lattice models of Y junctions used in numerical simulation. 19  Chapter 3  Conductance of two-dimensional Z3 symmetric Y junctions In this chapter, we present and analyze various numerical results extracted from the models, namely the dependence of zero temperature conductance on energy and magnetic field. Since the conductance is conveniently expressed in terms of the transmission probabilities TL = |t1 |2 and TR = |t2 |2 , these probabilities will be the main objects of interest. Natural units of h¯ = 2m = e = c = 1 are always chosen to remove the dimensions of various quantities.  3.1 Time-reversal invariant case The simplest possible situation is found when no magnetic field is turned on, in which case one necessarily finds |t1 |2 = |t2 |2 . Consider the Y junction with a hole of radius rh in the middle; Z3 symmetric leads with width a are connected by straight hard walls of length b, as in the upper panel of Fig. 2.6. Unless specifically stated, we set a = 1, b = 3, rh = 0.1 and do not apply a single particle scattering potential; width of the honeycomb ribbons which model the quantum wires is w = 100. In Fig. 3.1, the energy dependence of total transmission probability Ttot = |t1 |2 + |t2 |2 is plotted in the single channel regime E ∈ 0, 3π 2 h¯ 2 /2ma2 (the energy is measured relative to the lower threshold of the single channel regime). In a gallium arsenide wire (effective mass m = 0.067me , where me is the mass of an electron) with a = 100nm, one energy unit h¯ 2 / 2ma2 is equal to 5.687 × 10−5 eV. Motivated by the fixed points that governs the low-energy physics in one-dimensional Y junctions, we first concentrate on special values of transmission probabilities. The maximal transmission probability again turns out to be 8/9 in time-reversal symmetric situations. We are also interested in the two-dimensional counterpart of the N fixed point, namely complete reflections. Apart from the trivial complete reflection just above the bottom of the band (or k = 0), complete reflection may occur at certain nonzero energies (E ∼ 10.4, 21.2, 24.8 and 25.4 in Fig. 3.1). Complete reflection at energies other than the band bottom is a very special feature which is not found in strictly one20  dimensional noninteracting systems [4], and can be attributed to the presence of evanescent waves in closed channels. We have plotted the probability density and probability current distribution inside the junction area for a maximal transmission (Fig. 3.2) and a complete reflection (Fig. 3.3) to demonstrate their typical behaviors. Vortices are seen to form due to the spatial confinements; the higher the energy of incident electron, the more vortices there are. Intuitively, whenever strong reflections occur, the vortices are found to occupy much space immediately next to the openings to the other two wires, thus hampering the flow of electrons.  3.2 Magnetic field: effect of Aharonov-Bohm phase, Fano resonance We proceed to examine the influence of a  AB  magnetic flux on the system, i.e. the flux penetrates  through the hole but nowhere else. Figs. 3.4 and 3.5 feature total transmission probabilities and difference of chiral transmission probabilities TL − TR plotted against both energy and the flux, where the flux dependence is plotted in [0, π ], or zero up to half a flux quantum only; the rest is covered by symmetry. For a hole radius of rh = 10nm, one flux quantum means a magnetic field of 6.582T. Fig. 3.6 compares the energy dependence of total transmission probability at several different values of flux, and Fig. 3.7 depicts typical trends of chiral transmission probabilities. A very straightforward observation from Figs. 3.4 and 3.5 is that the larger the size of the junction, the more complicated the energy dependence of transmission probabilities. For b close to zero (i.e. each pair of wires are very close to each other), the energy dependence has considerably less structure than larger values of b. A special value of transmission probability when time-reversal symmetry becomes broken is found in perfectly chiral transmissions (e.g. E ∼ 25 in Fig. 3.7 and Fig. 3.4(c)(d)). They are less often found in two-dimensional models than in one-dimensional ones [22]; a heuristic explanation is that electrons have more freedom in choosing their ”paths” or wave function distributions in a two-dimensional junction, and are therefore less likely to follow extreme patterns that would lead to perfectly chiral transmission. However, based on calculations for Z3 asymmetric models, it is reasonable to believe that perfectly chiral transmissions do exist in many junctions. Examples of probability density and probability current inside the junction area at a perfectly chiral transmission are given in Fig. 3.8. One more interesting feature is flux-independent nonzero total transmission probabilities at a certain energy while transmission remains chiral, i.e. TL = TR ; see E ∼ 6.7 in Fig. 3.6. It is a novel phenomenon which does not appear in one-dimensional models, and is remarkably robust even when Z3 symmetry is broken. Fig. 3.9 show the associated probability densities and probability currents. The probability density distribution pattern does not change considerably as the flux is adjusted. The Fano resonances are now readily identified in Fig. 3.6, as well as Figs. 3.4 and 3.5. When there is no scattering potential, there are two basic types of Fano resonance: 21  Type I - Zero line width in time reversal symmetric cases (zero flux / half a flux quantum); gradually broadened and distorted by additional magnetic field, completely dissolved when timereversal symmetry is reached from the other side (half a flux quantum / zero flux). In Fig. 3.4 they manifest themselves as groove-shaped regions of low transmission probability (bright red). Most Fano resonances assume this form; examples include E ∼ 4, 10.5, 17 and 28 in Fig. 3.6 and Fig. 3.4(c), E ∼ 27 in Fig. 3.4(b), E ∼ 4, 10.5, 17 and 28 in Fig. 3.4(d). It is also worth mentioning that in some type I Fano resonances, a flux-independent complete reflection is found at a certain energy (e.g. E ∼ 28 in Fig. 3.4(a), E ∼ 13.5 in Fig. 3.4(b), E ∼ 10.5 in Fig. 3.6 and Fig. 3.4(c)). The chirality of scattering is different below and above this value of energy (i.e. TL − TR changes sign), signaling the formation of a new pattern of the wave function. Type II - Finite line width in time reversal symmetric cases, but persistently sharp at all fields. They appear as thin color stripes in Fig. 3.4. See E ∼ 8.8 in Fig. 3.6 and Fig. 3.4(c); E ∼ 26 in Fig. 3.4(d). When the system obeys Z3 symmetry, parity symmetry and time-reversal symmetry, the two types of Fano resonances are distinguished by their parity. As one can see most clearly from the probability current plots (Fig. 3.10), a type I resonance is caused by interference with a discrete state which is parity-odd with respect to all three axes of symmetry of the Y junction (i.e. the central axes of the wires). In the single channel regime, any traveling wave state in a wire is parityeven with respect to the central axis of that wire. These parity-odd discrete states are therefore true bound states which cannot decay into traveling waves; thus the zero line widths of the corresponding Fano resonances.1 Upon the inclusion of a flux these bound states become unstable, and the resonances acquire finite widths. On the other hand, Fig. 3.11 suggests that the discrete states associated with type II resonances are parity-even with respect to the axis of symmetry along the direction of incidence. Consequently, they are allowed to decay into traveling waves (i.e. they are by definition quasi-bound states rather than bound states), which explains the finite line width of type II resonances in the absence of a magnetic field. In Fig. 3.12 we demonstrate the effect of scattering potentials which preserve the Z3 symmetry. The potentials are chosen to be not too strong; a strong potential barrier necessarily destroys any Fano resonance by suppressing transmission, whereas a deep potential well distorts the profile of energy dependence wildly and does not facilitate comparison with potential-free results. Fig. 3.12 shows that in the time-reversal symmetric case, type I resonances are broadened by a parity-odd potential but not a parity-even one; this is because the latter preserves parity so that the parityodd bound states continue to exist, while the former breaks the parity symmetry and mixes up the parity-odd bound states with the parity-even quasi-bound ones. The type II resonance found at E ∼ 8 retains its shape in both cases. 1 In the time-reversal symmetric case, the line widths seen in simulation despite being small are not strictly zero, as they should be due to this symmetry argument. It is likely an unimportant numerical error which traces back to the slight asymmetry of the underlying lattice model.  22  Before ending the section we briefly discuss the consequences of breaking Z3 symmetry. Fig. 3.13 shows the influence of a weak Z3 asymmetric scattering potential, combined with modifying one of the straight walls of the junction area to an arc; note that both modifications combined break not only the Z3 symmetry but also any parity symmetry. In the present situation the Fano resonances are no longer classified by the parity of their related bound and quasi-bound states. Instead, as the parameters which quantify the asymmetry of the Y junction are continuously adjusted, it is natural to assume that the bound and quasi-bound states in the Z3 symmetric case evolve continuously into quasi-bound states which, respectively, lead to type I and type II Fano resonances in the Z3 asymmetric case. For all three wires, type I resonances at E ∼ 7.5 and 27 are broadened even with the time-reversal symmetry unbroken; in contrast, type I resonances at E ∼ 10.5, 17 and the type II resonance at E ∼ 8.8 in the Z3 symmetric case do not correspond to any Fano resonances in the Z3 asymmetric case. Also note the flux-independent non-zero total transmission probability; for each wire these probabilities are generally different and occur at different energies. In addition, perfectly chiral transmissions again emerge, but there are no non-trivial complete reflections observed. This may indicate that the presence of complete reflections in the single channel regime is more the exception than the rule in two dimensions and requires additional symmetries such as parity. The special conductance tensor for the DA fixed point, where two of the wires are perfectly connected and the third is decoupled, is likewise rarely encountered in two dimensions.  3.3 Magnetic field: effect of Lorentz force Finally we explain the characteristics of the transmission probabilities of Y junctions where an electron feels a Lorentz force in the junction; the magnetic field is uniform inside the central region, and falls abruptly to zero across the gluing interface to which the wires are attached (see Fig. 2.6, lower panel). Z3 symmetric leads with width a = 1 are smoothly connected by arcs with radius b. Highlighted in Figs. 3.14 and 3.15 are total transmission probabilities and difference of chiral transmission probabilities plotted against both energy and the flux. When a = 100nm, one magnetic field unit equals 3.291 × 10−2 T. The most striking difference with the AB flux scenario is the abrupt boundary lying in the middle of the plots. This boundary separates the almost perfect chiral transmission region to its left from the perfect reflection region to its right, and is almost a straight line if the ratio b/a is not too small. This may be interpreted as the combination of two results: that the electron needs to overcome an energy barrier directly proportional to the magnetic field before being accepted into the junction area, and that once inside the junction area it usually propagates in an edge state - known as a skipping orbit in literature [6] - before falling into the opening to another wire to be transmitted. For the combination of parameters to the right of this boundary line, electrons cannot overcome the barrier and therefore are almost completely reflected. To the left of it, as long as the magnetic field is not too weak, electrons are driven by the Lorentz force onto the skipping orbits near one of the walls, and scatter 23  predominantly into the wire at the end of that wall. However, in case of too weak a magnetic field, quantum effects again dominate, and the complicated zero flux behavior is recovered. Such a heuristic picture appears to be consistent with the observation that adding a small hole in the center of the junction does not greatly impact these results. More evidence comes from the fact that using straight walls in place of curved ones enhances chiral transmission. The electrons may get backscattered along a curved wall, but not a straight one.  24  T  0  2  4  6  8  10  12  14  16  18  20  22  24  26  28  1.0  1.0  0.9  0.9  0.8  0.8  0.7  0.7  0.6  0.6  0.5  0.5  0.4  0.4  0.3  0.3  0.2  0.2  0.1  0.1  0.0  0.0 0  2  4  6  8  10  12  14  16  18  20  22  24  26  28  E  Figure 3.1: Total transmission probability Ttot versus energy in a Y junction without magnetic flux. The dotted line, Ttot = 8/9, shows the maximal Ttot in the Z3 symmetric and timereversal invariant case. Fano resonances occurring in this figure will be explained in Sec. 3.2.  25  T  tot  200  150  100  0.000 0.2094  50  0.4187  Y  0.6281  0  0.8375 1.047  -50  1.256 1.466  -100  1.675  -150  -200 0  50  100  150  200  250  300  X (a) 200  150  100  Y  50  30x  0  -50  -100  -150  -200 0  50  100  150  200  250  300  X (b)  Figure 3.2: Probability density and probability current distributions inside the junction area near a maximal transmission (E = 2.5, no magnetic flux applied, Ttot = 0.8889). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity. 26  200  150  100 0.000 0.2219  50  0.4437  Y  0.6656  0  0.8875 1.109  -50  1.331 1.553  -100  1.775  -150  -200 0  50  100  150  200  250  300  X (a) 200  150  100  Y  50  3000x  0  -50  -100  -150  -200 0  50  100  150  200  250  300  X (b)  Figure 3.3: Probability density and probability current near a complete reflection (E = 10.25, no magnetic flux applied, Ttot = 0.0079). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity. 27  Total transmission probability b/a=0.01,r =0.1 h  b/a=1,r =0.1 h  b/a=3,r =0.2  b/a=3,r =0.1  h  h  b/a=3,r =0.5 h  0.000 0.05000  25  25  25  0.06250  25  25  0.1250 0.1875 0.2500 0.3125 0.3750  20  20  20  0.4375  20  20  0.5000 0.5625 0.6250 0.6875  Energy  0.7500  15  15  15  0.8125  15  15  0.8750 0.9375 0.9950 1.000  10  10  10  10  10  5  5  5  5  5  0  0  0  0  0  0  1  2  (a)  3  0  1  2  (b)  3  0  1  2  3  0  1  2  3  0  (d)  (c)  1  2  3  (e)  Magnetic flux  Figure 3.4: Total transmission probability Ttot versus energy and flux in with different dimensions.  28  AB -type  Y junctions  Difference in chiral transmission probabilities T -T R  b/a=0.01,r =0.1 h  25  b/a=1,r =0.1 h  h  h  25  25  b/a=3,r =0.2  b/a=3,r =0.1  L  b/a=3,r =0.5 h  25  25  -1.000 -0.9900 -0.7500 -0.5000  20  20  20  20  20  -0.2500 -0.1250 0.000  Energy  0.1250  15  15  15  15  15  0.2500 0.5000 0.7500 0.9900  10  10  10  10  10  5  5  5  5  5  0  0  0  0  0  0  1  2  (a)  3  0  1  2  (b)  3  0  1  2  3  0  1  2  (d)  (c)  3  1.000  0  1  2  3  (e)  Magnetic flux  Figure 3.5: The difference of chiral transmission probabilities TR − TL versus energy and flux in AB-type Y junctions with different dimensions.  29  0  2  4  6  8  10  12  14  16  18  20  22  24  26  28  1.0  1.0  a=1, b=3, r =0.1, w=100 h  0.9  0.9  T  0.8  T  0.8  0.7  0.7  0.6  0.6  0.5  0.5  0.4  0.4  0.3  0.3  0.2  0.2  0.1  0.1  0.0  T T T T  tot  tot  tot  tot  tot  ( =0) ( = /4) ( = /2) ( = ( = )  0.0 0  2  4  6  8  10  12  14  16  18  20  22  24  26  28  E  Figure 3.6: Total transmission probability Ttot versus energy in a different values of flux.  0  2  4  6  8  10  12  14  16  18  20  AB -type  22  Y junction with  24  26  28 1.0  1.0 T 0.9  T T  T  0.8  a=1, b=3, r =0.1, w=100,1/4 flux quantum h  tot  0.9  R  0.8  L  0.7  0.7  0.6  0.6  0.5  0.5  0.4  0.4  0.3  0.3  0.2  0.2  0.1  0.1  0.0  0.0 0  2  4  6  8  10  12  14  16  18  20  22  24  26  28  E  Figure 3.7: Total transmission probability Ttot and chiral transmission probabilities TL , TR versus energy in a AB-type Y junction with 1/4 flux quantum.  30  /4)  200  R  150  100  0.000 0.3388  50  0.6775  Y  1.016  0  1.355 1.694  -50  2.033 2.371  -100  2.710  -150  L -200 0  50  100  150  200  250  300  X (a) 200  R 150  100  Y  50  10x  0  -50  -100  -150  L -200 0  50  100  150  200  250  300  X (b)  Figure 3.8: Probability density and probability current near a perfectly chiral transmission (E = 25, 1/4 flux quantum, Ttot = 0.9967, TR = 0.9935, TL = 0.0032). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity. 31  200  200  150  150  100  100  0.000 0.1035  50  50  0.2070  0.4140  0  0.5175  Y  Y  0.3105  100x  0  0.6210  -50  -50  0.7245 0.8280  -100  -100  -150  -150  -200  -200 0  50  100  150  200  250  0  300  50  100  150  X  X  (a)  (b)  200  200  150  150  100  200  250  300  100  0.000 0.1163  50  50  0.2325  0.4650  0  0.5813 0.6975  -50  Y  Y  0.3487  100x  0  -50  0.8138 0.9300  -100  -100  -150  -150  -200  -200 0  50  100  150  200  250  300  0  50  100  150  X  X  (c)  (d)  200  250  300  Figure 3.9: Probability density and probability current distributions when the total transmission probability is independent of the magnetic flux (E = 6.71. There is no magnetic field in panels (a) and (b), Ttot = 0.5384; 1/4 flux quantum is applied in panels (c) and (d), Ttot = 0.5393, TR = 0.4283, TL = 0.1110). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity.  32  200  200  150  150  100  100  0.000 1.681  50  50  3.363  6.725  0  8.406  Y  Y  5.044  30x  0  10.09  -50  -50  11.77 13.45  -100  -100  -150  -150  -200  -200 0  50  100  150  200  250  300  0  50  100  150  X  X  (a)  (b)  200  200  150  150  100  200  250  300  100  0.000 3.750  50  50  7.500  15.00 18.75 22.50  -50  Y  Y  11.25  0  100x  0  -50  26.25 30.00  -100  -100  -150  -150  -200  -200 0  50  100  150  200  250  0  300  50  100  150  X  X  (c)  (d)  200  250  300  Figure 3.10: Probability density and probability current near a type I Fano resonance (1/4 flux quantum is applied. E = 4.7 in panels (a) and (b), Ttot = 0.8895, TR = 0.4933, TL = 0.3962; E = 4.73 in panels (c) and (d), Ttot = 0.0046, TR = 0.0023, TL = 0.0022). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity.  33  200  200  150  150  100  100  0.000 18.38  50  50  36.75  0  73.50 91.88  Y  Y  55.12  110.3  -50  30x  0  -50  128.6 147.0  -100  -100  -150  -150  -200  -200 0  50  100  150  200  250  300  0  50  100  150  X  X  (a)  (b)  200  200  150  150  100  200  250  300  100  0.000 4.538  50  50  9.075  18.15  0  22.69 27.23  -50  Y  Y  13.61  300x  0  -50  31.76 36.30  -100  -100  -150  -150  -200  -200 0  50  100  150  200  250  0  300  50  100  150  X  X  (c)  (d)  200  250  300  Figure 3.11: Probability density and probability current near a type II Fano resonance (1/4 flux quantum is applied. E = 8.8 in panels (a) and (b), Ttot = 0.8067, TR = 0.5844, TL = 0.2222; E = 8.81 in panels (c) and (d), Ttot = 0.0833, TR = 0.0460, TL = 0.0373). Electrons are injected from the wire connected to the leftmost edge of the junction, and the flux of incoming electrons is normalized to unity.  34  1.0  T T  (V=V sin3 )  tot  0  (V=V cos3 )  tot  0  0.8  T  0.6  0.4  0.2  0.0 0  5  10  15  20  25  E  Figure 3.12: Total transmission probability Ttot versus energy in a Y junction without magnetic flux. Different single particle scattering potentials have been applied to the central region: V (r) = V0 sin(3θ ) and V (r) = V0 cos(3θ ), where V0 = 1 and θ is the polar angle measured from the center of the junction, the polar axis pointing opposite to the 1st wire.  35  1.0  T T 0.8  T T T  T  0.6  0.4  0.2  0.0 0  5  10  15  20  25  E  Figure 3.13: Total transmission probability Ttot,1 versus energy in an AB-type Y junction. Z3. symmetry is broken by a single particle scattering potential, V (r) = V0 sin(2θ ), in conjunction with a modification of geometry, namely replacing the straight wall connecting wires 1 and 3 with an arc that smoothly interpolates between the wires. V0 = 1.  36  tot  tot  tot  tot  tot  ( = ) ( =  )  ( =  )  ( =3 ( = )  )  Total transmission probability, b/a=3  0.000  25  5.000E-04 0.1250 0.2500  20  0.3750 0.5000  Energy  0.6250 0.7500  15  0.8750 0.9995 1.000  10  5  0 0  10  20  30  40  Magnetic field  Figure 3.14: Total transmission probability Ttot versus energy and flux in a Lorentz-type Y junction.  37  Difference in chiral transmission probabilities T_R-T_L, b/a=3  -1.000  25  -0.9995 -0.7500 -0.5000 -0.2500  20  -1.000E-03 1.000E-03  Energy  0.2500 0.5000  15  0.7500 0.9995 1.000  10  5  0 0  10  20  30  40  Magnetic field  Figure 3.15: The difference of chiral transmission probabilities TL − TR versus energy and flux in a Lorentz-type Y junction.  38  Chapter 4  Summary In this thesis, we have performed calculations of zero-temperature conductance of two-dimensional non-interacting Y junctions in Landauer formalism. Two types of Y junctions are considered: one where a magnetic flux is injected through a hole in the junction area (”AB -type”), and one where the flux is applied uniformly to the junction (”Lorentz type”). In the time-reversal symmetric case, corresponding to the N fixed point in one dimension, complete reflections are sometimes found. For AB -type Y junctions perfectly chiral transmission exists in many cases; also frequently seen is a total  transmission probability independent of the magnetic flux, and both behaviors are reasonably robust against breaking of Z3 symmetry in the Y junction geometry. Another particularly outstanding phenomenon is the Fano resonance. We are able to categorize the Fano resonances into two groups based on their response to magnetic flux and single particle scattering potential, among which type II Fano resonances are robust against both potentials and magnetic fields. Finally, for Lorentz-type Y junctions, there are two regimes on the Fermi energy-magnetic field plane, one of almost perfectly chiral transmission and one of almost complete reflection, separated by a largely straight boundary. Such a separation of regimes is explainable on classical grounds.  39  Bibliography [1] H. U. Baranger. Multiprobe electron waveguides: Filtering and bend resistances. Phys. Rev. B, 42:11479–11495, Dec 1990. doi:10.1103/PhysRevB.42.11479. URL http://link.aps.org/doi/10.1103/PhysRevB.42.11479. → pages 6 [2] H. U. Baranger, D. P. DiVincenzo, R. A. Jalabert, and A. D. Stone. Classical and quantum ballistic-transport anomalies in microjunctions. Phys. Rev. B, 44:10637–10675, Nov 1991. doi:10.1103/PhysRevB.44.10637. URL http://link.aps.org/doi/10.1103/PhysRevB.44.10637. → pages 13, 14 [3] X. Barnab´e-Th´eriault, A. Sedeki, V. Meden, and K. Sch¨onhammer. Junctions of one-dimensional quantum wires: Correlation effects in transport. Phys. Rev. B, 71:205327, May 2005. doi:10.1103/PhysRevB.71.205327. URL http://link.aps.org/doi/10.1103/PhysRevB.71.205327. → pages 1 [4] G. Barton. Levinson’s theorem in one dimension: heuristics. Journal of Physics A: Mathematical and General, 18(3):479, 1985. URL http://stacks.iop.org/0305-4470/18/i=3/a=023. → pages 21 [5] L. Baskin, V. Grikurov, P. Neittaanm¨aki, and B. Plamenevskii. Quantum effects controlling electron beams. Technical Physics Letters, 30:650–653, 2004. ISSN 1063-7850. URL http://dx.doi.org/10.1134/1.1792302. 10.1134/1.1792302. → pages 6, 9, 13 [6] C. Beenakker, H. van Houten, and B. van Wees. Skipping orbits, traversing trajectories, and quantum ballistic transport in microstructures. Superlattices and Microstructures, 5(1):127 – 132, 1989. ISSN 0749-6036. doi:10.1016/0749-6036(89)90081-5. URL http://www.sciencedirect.com/science/article/pii/0749603689900815. → pages 23 [7] B. Bellazzini, M. Mintchev, and P. Sorba. Quantum wire junctions breaking time-reversal invariance. Phys. Rev. B, 80:245441, Dec 2009. doi:10.1103/PhysRevB.80.245441. URL http://link.aps.org/doi/10.1103/PhysRevB.80.245441. → pages 1 [8] J. Br¨uning and V. Grikurov. Numerical simulation of electron scattering by nanotube junctions. Russian Journal of Mathematical Physics, 15:17–24, 2008. ISSN 1061-9208. URL http://dx.doi.org/10.1134/S1061920808010020. 10.1134/S1061920808010020. → pages 6, 9, 13 [9] M. B¨uttiker. Four-terminal phase-coherent conductance. Phys. Rev. Lett., 57:1761–1764, Oct 1986. doi:10.1103/PhysRevLett.57.1761. URL http://link.aps.org/doi/10.1103/PhysRevLett.57.1761. → pages 10 40  [10] L. T. Canham. Silicon quantum wire array fabrication by electrochemical and chemical dissolution of wafers. Applied Physics Letters, 57(10):1046 –1048, sep 1990. ISSN 0003-6951. doi:10.1063/1.103561. → pages 1 [11] S. Eggert and I. Affleck. Magnetic impurities in half-integer-spin heisenberg antiferromagnetic chains. Phys. Rev. B, 46:10866–10883, Nov 1992. doi:10.1103/PhysRevB.46.10866. URL http://link.aps.org/doi/10.1103/PhysRevB.46.10866. → pages 1 [12] T. Giamarchi. Quantum physics in one dimension. International series of monographs on physics. Clarendon, 2004. ISBN 9780198525004. URL http://books.google.ca/books?id=GVeuKZLGMZ0C. → pages 1 [13] D. Giuliano and P. Sodano. Y-junction of superconducting josephson chains. Nuclear Physics B, 811(3):395 – 419, 2009. ISSN 0550-3213. doi:10.1016/j.nuclphysb.2008.11.011. URL http://www.sciencedirect.com/science/article/pii/S0550321308006457. → pages 1 [14] C. I. Goldstein. A finite element method for solving helmholtz type equations in waveguides and other unbounded domains. Mathematics of Computation, 39(160):pp. 309–324, 1982. ISSN 00255718. URL http://www.jstor.org/stable/2007317. → pages 6, 9, 14 [15] V. E. Grikurov. Scattering, trapped modes and guided waves in waveguides and diffraction gratings. eprint arXiv:quant-ph/0406019, June 2004. → pages 6, 9, 13 [16] C.-Y. Hou and C. Chamon. Junctions of three quantum wires for spin- 12 electrons. Phys. Rev. B, 77:155422, Apr 2008. doi:10.1103/PhysRevB.77.155422. URL http://link.aps.org/doi/10.1103/PhysRevB.77.155422. → pages 1 [17] S. Iijima and T. Ichihashi. Single-shell carbon nanotubes of 1-nm diameter. Nature, 363: 603–605, Jun 1993. doi:doi:10.1038/363603a0. → pages 1 [18] C. L. Kane and M. P. A. Fisher. Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas. Phys. Rev. B, 46:15233–15262, Dec 1992. doi:10.1103/PhysRevB.46.15233. URL http://link.aps.org/doi/10.1103/PhysRevB.46.15233. → pages 1 [19] S. Lal, S. Rao, and D. Sen. Junction of several weakly interacting quantum wires: a renormalization group study. Phys. Rev. B, 66:165327, Oct 2002. doi:10.1103/PhysRevB.66.165327. URL http://link.aps.org/doi/10.1103/PhysRevB.66.165327. → pages 1 [20] V. A. Margulis and M. A. Pyataev. Fano resonances in a three-terminal nanodevice. Journal of Physics: Condensed Matter, 16(24):4315, 2004. URL http://stacks.iop.org/0953-8984/16/i=24/a=013. → pages 12 [21] J. U. N¨ockel and A. D. Stone. Resonance line shapes in quasi-one-dimensional scattering. Phys. Rev. B, 50:17415–17432, Dec 1994. doi:10.1103/PhysRevB.50.17415. URL http://link.aps.org/doi/10.1103/PhysRevB.50.17415. → pages 10  41  [22] M. Oshikawa, C. Chamon, and I. Affleck. Junctions of three quantum wires. Journal of Statistical Mechanics: Theory and Experiment, 2006(02):P02008, 2006. URL http://stacks.iop.org/1742-5468/2006/i=02/a=P02008. → pages 1, 2, 14, 21 [23] E. H. C. Parker. The Technology and Physics of Molecular Beam Epitaxy. Plenum Press, New York, 1992. → pages 1 [24] A. Rahmani, C.-Y. Hou, A. Feiguin, M. Oshikawa, C. Chamon, and I. Affleck. General method for calculating the universal conductance of strongly correlated junctions of multiple quantum wires. Phys. Rev. B, 85:045120, Jan 2012. doi:10.1103/PhysRevB.85.045120. URL http://link.aps.org/doi/10.1103/PhysRevB.85.045120. → pages 1, 5 [25] E. Tekman and P. F. Bagwell. Fano resonances in quasi-one-dimensional electron waveguides. Phys. Rev. B, 48:2553–2559, Jul 1993. doi:10.1103/PhysRevB.48.2553. URL http://link.aps.org/doi/10.1103/PhysRevB.48.2553. → pages 10, 11 [26] G. Timp, A. M. Chang, P. Mankiewich, R. Behringer, J. E. Cunningham, T. Y. Chang, and R. E. Howard. Quantum transport in an electron-wave guide. Phys. Rev. Lett., 59:732–735, Aug 1987. doi:10.1103/PhysRevLett.59.732. URL http://link.aps.org/doi/10.1103/PhysRevLett.59.732. → pages 8 [27] C. Weisbuch and B. Vinter. Quantum Semiconductor Structures. Academic Press, San Diego, 1991. → pages 1  42  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0085574/manifest

Comment

Related Items