The Aharonov-Bohm Effect and Persistent Currents in Quantum Rings
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The Aharonov-Bohm Effect and Persistent Currents in Quantum Rings Alexandre Miguel de Ara´ujoLopes Department of Physics University of Aveiro Advised by: Ricardo A. G. Dias 2007/2008 Preface When I first started this thesis I hadn’t had a solid state physics subject yet and therefore I had to acquire some basic knowledge before dedicating myself to this thesis. After some initial hickups, I think things worked out well and I certainly did learn a lot along the road. Throughout this thesis I tried to stick with S.I. units. Though most of the documentation that can be found on the subjects studied in this thesis are written in C.G.S units, and although I think that C.G.S units do make Maxwell’s equations look extremely beautiful and simple, I didn’t find any advantage in using them over the S.I. units in this case. The Mathematica notebooks used for the numerical calculations in this thesis can be found at http://www.quantumrings.vndv.com/. I would like to express my thanks to those persons who always stood there for me: my parents, my brother and Filipa - a big thanks to all of you. I would also like to thank Daniel Gouveia, my colleague and friend, for all the conversations that we had which have inspired me. To L´ıdiadel Rio, who always cheered up the time we spent at our advisor’s office by talking about really non sense stuff and fun comics. And finally I would like to thank my advisor, Ricardo Dias, for all the knowledge I have acquired from him and because without him this thesis would simply not exist. 1 Contents 1 Introduction 4 1.1 Objectives . 4 1.2 Persistent currents . 4 1.3 Aharonov-Bohm effect . 5 2 Ring models 6 2.1 Introducing ring models . 6 2.2 Free electron gas model . 6 2.3 Tight-binding model . 7 2.4 Interacting spinless fermions . 7 3 Free electrons in a ring 8 3.1 Flux dependence of the energy and magnetic moment levels . 8 3.2 Persistent currents . 10 4 Tight-binding perfect ring 13 4.1 Tight-binding Hamiltonian . 13 4.2 Persistent currents . 15 5 Disorder effects 16 5.1 One impurity ring . 16 5.2 Disordered ring . 18 6 Interacting spinless fermions 20 6.1 The strong coupling limit . 20 6.2 Currents from different subspaces . 20 6.3 t-V ring with an impurity . 21 A Electromagnetic Hamiltonian 24 B Second quantization Fermionic operators 25 C Electrical current operator 26 C.1 Free electron gas . 26 C.2 Tight-binding model . 26 D Wavefunctions for a free electron in a ring 28 2 Contents E Free electron persistent current calculation 29 F Tb Hamiltonian with external magnetic flux 31 3 Chapter 1 Introduction 1.1 Objectives This thesis objective is to study, theoretically, some properties of quantum rings enclosing an external magnetic flux, namely persistent currents, using different models. We will first study independent electrons via the free electron gas model and the tight-binding model, and finally we will address correlated electrons by studying the t-V model, which is basically an extension of the tight-binding model. To achieve the objectives of this thesis, analytical and numerical calculations were performed. The numerical calculations they were done using Mathematica and the notebooks can be found at http://www.quantumrings.vndv.com/. 1.2 Persistent currents Persistent currents are currents (i.e. charge movement) which are able to sustain themselves independently from an applied voltage difference. Persistent currents have been long known and studied in superconductors. In quantum rings, we can have same persistent currents too despite their origin being different. Theoretically, persistent currents in quantum rings have been known for some time [1, 2] and they have also been measured experimentally using SQUIDs [3]. Figure 1.1: Aharonov-Bohm effect general geometry. 4 1 - Introduction 1.3 Aharonov-Bohm effect The Aharonov-Bohm effect [4] is a quantum mechanical effect that results from gauge invari- ance. It states that if we have electrons travelling along two paths and if between those paths there is a magnetic flux which vanishes in the region where the electrons flow (i.e. there is no magnetic force acting on the travelling electrons), this same magnetic flux will produce a phase shift in the electrons’ wavefunction (Fig. 1.1). Assuming that the paths are equal, Aharonov-Bohm effect results in a phase shift given by, I q ∆ϕ = A.d~ ~l, (1.1) ~ C where A~ is the vector magnetic potential. 5 Chapter 2 Ring models 2.1 Introducing ring models There are several models in solid state physics which can be used to treat the problem of electrons in a ring. Here we will address this problem using models of increasing complexity starting with non-correlated models and considering a perfect ring and finally we will address disorder and correlated electrons. We will address the problem of electrons in a ring using the free electron gas and the tight-binding models, for the non correlated case, and the t-V model for the correlated case. All these models are used for spinless fermions (although the tight binding model can be used for fermions with spin). The Hubbard model is a popular model which takes spin and Coulomb on-site interaction in account but we will not use it here. 2.2 Free electron gas model The free electron gas model was one of the first models used in solid state physics. It is an oversimplified model and that should not be forgotten when working with it. Nevertheless it is simple to solve analytically and it can prove extremely useful as a first approach to a problem. Electron gas model assumes that electrons behave much like a classical ideal gas. It assumes that electrons are independent, i.e., the Coulomb interaction between them is assumed to be zero, and free, i.e., they don’t feel a periodic potential from ions (only a global potential which prevents the electrons from leaving the solid). Using this approach we can treat every electron separately, using the free electron Hamiltonian1: ³ ´2 ~p − qA~ H = , (2.1) 2m where ~p is the conjugate momentum, A~ is the magnetic vector potential and q and m are the particle’s charge and mass respectively. 1See Appendix A for the Hamiltonian’s derivation. 6 2 - Ring Models 2.3 Tight-binding model The tight-binding model assumes that the Hamiltonian close to a lattice point can be approx- imated by the Hamiltonian of an isolated atom centered at that lattice point [5]. This way an electron’s wavefunction in a one-dimensional crystal can be written as a linear combination of the electron’s wavefunctions in the sites, that is,2 XN 1 i k j † |ψi = √ e cj |0i , (2.2) N j=1 † 3 where N is the number of sites, k the momentum and cj the fermionic creation operator. In second quantized form, the tight-binding Hamiltonian in a linear 1-D system can be written as a ”hopping” Hamiltonian, XN h i † H = −t cjcj+1 + h.c. , (2.3) j=1 where t is the one of the band theory overlap integrals [5]. Here we will call t the hopping factor. Figure 2.1: The tight-binding model (in second quantized form) assumes that an electron can ”hop” to its nearest neighbors if they are not occupied by an electron of the same spin. 2.4 Interacting spinless fermions In order to address correlations between electrons due to the Coulomb interactions we are going to study interacting spinless fermions using the t-V model. The t-V model is an extended version of the tight-binding model which takes in account the Coulomb interaction between nearest neighbors. Spin is not considered in this model as a simplification. Overall this is a decent model to apply in case we have a rather strong external magnetic field that aligns the system’s spins in which case, neglecting spin-spin interactions is the same as having no spin. The Hamiltonian of the t-V model in second quantized form is written as: XN h i XN † H = −t cjcj+1 + h.c. + V njnj+1. (2.4) j=1 j=1 † where V is the Coulomb energy between two nearest-neighbors electrons and nj = cjcj is the occupation number at site j. 2See Appendix B if you are unfamiliar with second quantization. 3Note that the lattice constant was chosen to be equal to 1. This is a usual convention in theoretical solid state physics and is used throughout this thesis. 7 Chapter 3 Free electrons in a ring In this chapter we will address a continuous ring using the free electron gas model. After calculating some electronic properties of the system we will introduce a weak potential and calculate the same properties again, comparing the differences between the system with and without the weak potential. 3.1 Flux dependence of the energy and magnetic moment lev- els Considering the Schr¨odinger’sequation for a free electron in a ring, ³ ´2 −i~∇~ polar + eA~ ψ = Eψ, (3.1) 2m 2 where ∇polar is the del operator in polar coordinates, we easily arrive at the electron’s wave- function,1 1 inθ ψn(θ) = √ e , (3.2) 2πR where R is the ring radius, θ the angular position on the ring (see Fig. 3.1) and n a quantum number. Operating with the free electron’s Hamiltonian (2.1), Hψn = Enψn, we arrive at an expression for En, µ ¶ ~2 RAe 2 E = n + , n = 0, ±1, ±2, ..