Quantum transport in nanowires with spin-orbit interaction: effect of quasi-bound states.

Quantum Transport in Nanowires with Spin-Orbit Interaction: effect of quasi-bound states Alba Pascual Gil Directed by Sebasti´an Bergeret and Vitaly Golovach Material Physics Department PhD Program of Physics of Nanostructures and Advanced Materials University of the Basque Country (UPV/EHU) December, 2019 (cc)2020 ALBA PASCUAL GIL (cc by-nc 4.0) A Pepa y Jos´e Luis Agradecimientos Primero quisiera agradecer a Sebasti´an Bergeret por su direcci´on y ayuda en estos cuatro a˜nos como supervisor de mi tesis. Tu perspectiva, optimismo y paciencia me han servido de gran ayuda e inspiraci´on para seguir este camino y crecer como persona. Gracias por acogerme en el grupo de Mesoscopic Physics durante esta etapa de mi vida. As´ı mismo, quiero agradecer a Vitaly Golovach todo el apoyo tanto acad´emico como personal que me ha permitido llevar a cabo esta tesis. Aspiro a tener al menos la mitad de tu generosidad como persona y tu amor por la ciencia. Quisiera tambi´en extender este agradecimiento al resto del grupo de Mesoscopic Physics del Centro de F´ısica de Materiales. Gracias Dario Bercioux por tus consejos al inicio de mi estancia y tambi´en gracias Tineke van den Berg por tu apoyo y amistad. Finalmente, gracias al resto de estudiantes y compaeros de despacho, Cristina, Mikel, Bogusz, Xian-Peng y Julie, por todos los ratos compartidos en este per´odo de mi vida. A nivel m´as personal, quiero dar las gracias a mis padres por transmitirme sus valores y apoyarme en todo momento para poder hacer siempre lo que yo quisiera. A mi familia por su amor incodicional y a mi hermana Clara por su paciencia infinita. A mi peque˜na familia en Donostia por todos esos momentos inolvidables y debates acalorados en las comidas y coffee breaks. En especial gracias a Moritz, Patri y Tom´aˇs, no ser´ıa hoy quien soy hoy en d´ıa sin vosotros. Gracias tambi´en a Chusa y Mateo por darme refugio espiritual, a Paloma por aportarle ese toque de humor a mi vida, a Yaiza por completarme, y al resto de amigos que me han acompa˜nado en este viaje. Finalmente, gracias a mi pareja Juan Carlos por estar ah´ı siempre. 2 Acknowledgements First of all, I would like to thank Sebasti´an Bergeret for all the guidance during the past four years as my PhD supervisor. Your vision, optimism and patience have been of great help and an inspiration to grow as a person. Thank you for welcoming me into the Mesoscopic Physics group during the duration of my stay. I would also like to thank Vitaly Golovach for all his help not only on an academic level but also on a personal one. I aspire to have your generosity as a person and your passion for science. I would like to extend this acknowledgement to the rest of the Mesoscopip Physics group from the Materials Physics Center in Donostia. Thank you Dario Bercioux for all your advice in the early stages of my stay here and Tineke van den Berg for all your support and friendship. Finally, thanks to the rest of students and office mates, Cristina, Mikel, Bogusz, Xian-Peng and Julie, for all this time spent together. On a more personal note, I would like to thank my parents for their upbringing and encouragement to pursue my dreams. To my family for their unconditional love and my sister Clara for her infinite patience. To my little family in Donostia for every unforgettable moment and every heated discussion during coffee breaks. And specially, thank you Moritz, Patri and Tom´aˇs, I wouldn’t be me today without you. Thank you Chusa and Mateo for offering me spiritual refuge, to Paloma for the spot of humor in my life, to Yaiza for completing me and to the rest of my friends wherever they are for accompanying me during this journey. Last but not least, thank you Juan Carlos for being there always. 3 Resumen Esta tesis tiene como objetivo el estudio de transporte en nanohilos semiconductores con interacci´on esp´ın-´orbita e impurezas. A d´ıa de hoy estos nanohilos son de los materiales m´as vers´atiles para el dise˜no de dispositivos cu´anticos. Ejemplo de ello, es el intenso estudio de los fermiones de Majorana, que pueden detectarse en los extremos de nanohilos, cuando ´estos est´an en contacto con un superconductor [1, 2, 3, 4, 5, 6, 7]. Crucial para la aparici´on de los fermiones de Majorana es la interacci´on esp´ın-´orbita y el bajo nivel de desorden en los nanohilos. De hecho, el desorden en nanohilos cu´anticos afecta fuertemente a la conductancia de los modos de Majorana [8, 9, 10, 11, 12, 13, 14]. Por otro lado, las estructuras semiconductoras con interacci´on esp´ın-´orbita nos llevan atr´as en el tiempo hasta la propuesta de Datta y Das de un transistor [15], que propone el control sobre la interacci´on esp´ın-´orbita por medio de un gate para rotar el esp´ın del electr´on y as´ı controlar el transporte de carga entre dos electrodos ferromagn´eticos. Los intentos de fabricar tal dispositivo han topado con varios problemas [16, 17, 18], inclu´ıda la baja eficiencia en la inyecci´on de esp´ın del ferromagneto en el semiconductor, pero tambi´en con la relajaci´on del esp´ın inducida por el scattering del electr´on debido al desorden en el dispositivo. De los ejemplos anteriores se desprende que el estudio del desorden requiere especial atenci´on. En particular en respuesta a la pregunta de c´omo afecta el desorden al transporte de carga y esp´ın en un nanohilo cuasi-unidimensional. Esta pregunta es absolutamente no trivial. En los nanohilos, el desplazamiento est´a confinado en una direcci´on y los portadores de carga solo pueden desplazarse en la direcci´on ortogonal a la del potencial de confinamiento. La combinaci´on del confinamiento con la presencia de la interacci´on esp´ın-´orbita de tipo Rashba induce el acoplamiento entre subbandas. Esto afecta fuertemente al transporte en el 4 hilo cu´antico, llegando por ejemplo a suprimir la modulaci´on de esp´ın para valores grandes de la interacci´on Rashba [19, 20, 21, 22]. Por otro lado, la presencia de una impureza puede llevar a la formaci´on de estados cuasi-ligados localizados entorno a la impureza. En trabajos previos se ha demostrado que la presencia de modos evanescentes lleva a fen´omenos poco usuales en el transporte electr´onico, como por ejemplo la perfecta transmisi´on en el umbral energ´etico en el que una nueva subbanda es accesible y comienza a propagar. Tambi´en se ha observado que cerca, pero por debajo de este umbral, la aparici´on de estados cuasi-ligados localizados en torno a una impureza atractiva es responsable del bloqueo total de canales de transmisi´on [23, 24]. La combinaci´on de ambos efectos nunca ha sido tratada. En esta tesis abordamos este tema y presentamos un estudio te´orico exhaustivo del transporte electr´onico en nanohilos cu´anticos semiconductores con interacci´on esp´ın-´orbita en la presencia de impurezas. Modelamos el nanohilo cu´antico como un sistema cuasi-unidimensional en el que el movimiento de los electrones est´a confinado en la direcci´on perpendicular a la de propagaci´on. La competici´on entre la interacci´on esp´ın-´orbita, el confinamiento lateral y la impureza hace que el problema sea altamente no-trivial. Para hacer frente a este problema usamos una combinaci´on de t´ecnicas y aproximaciones que nos permiten identificar novedosas propiedades del transporte de carga y del transporte de esp´ın. Espec´ıficamente, describimos la conductancia mediante el formalismo de Landauer-B¨uttiker, extendi´endolo para el caso de campos dependientes del esp´ın. Describimos el transporte a trav´es de este formalismo en funci´on de los coeficientes de scattering. Para calcular los coeficientes de la matriz de scattering usamos la ecuaci´on de Lippmann-Schwinger, un m´etodo ampliamente usado en el tratamiento del scattering en la mec´anica cu´antica. Una de las principales dificultades en el tratamiento de la interacci´on esp´ın-´orbita en un potencial de confinamiento es la hibridaci´on de las subbandas. Para superar este problema introducimos la transformaci´on de Schrieffer-Wolff, una transformaci´on de gauge que elimina esta hibridaci´on manteniendo la complicaci´on de los efectos de Rashba en la funci´on de onda transformada. Combinando las t´ecnicas mencionadas calculamos de forma anal´ıtica la coductancia para el transporte de carga y esp´ın en un nanohilo con interacci´on de tipo Rashba en presencia de una impureza puntual. 5 Encontramos que la impureza acopla estados propagantes y estados evanescentes en el nanohilo cu´antico, induciendo estados cuasi-ligados localizados entorno a la impureza. Por otra parte, la interacci´on esp´ın-´orbita de tipo Rashba permite distintos mecanismos de transporte electr´onico. Como consecuencia, la conductancia presenta transmisi´on bal´ıstica perfecta en la energ´ıa umbral donde el siguiente canal se vuelve transmisivo. Adem´as, por debajo de dicha energ´ıa umbral la conductancia presenta una reducci´on significativa. Demostramos que para las subbandas m´as bajas en esta energ´ıa resonante, todos los electrones inyectados en el nanohilo cu´antico en un estado preparado de tal manera que su esp´ın se al´ınea en cierta direcci´on preferente, solo tienen una forma de transmitir a trav´es de la impureza: mediante un proceso en el que su esp´ın salta a la orientaci´on opuesta. No solo es ´esta la ´unica forma de propagar hacia el otro lado de la impureza, sino que la probabilidad de que suceda este proceso de inversi´on del esp´ın aumenta notablemente respecto a la probabilidad fuera de la resonancia. Es m´as, en la energ´ıa exacta de la resonancia, mientras la probabilidad de transmisi´on manteniendo la misma orientaci´on en el esp´ın se reduce hasta cero, la probabilidad de transmisi´on con inversi´on del esp´ın es m´axima. M´as all´a del transporte de carga, tambi´en derivamos expresiones para las corrientes de esp´ın en el nanohilo y derivamos una expresi´on para el torque de esp´ın-´orbita inducido por la impureza. Demostramos que este torque depende completamente de los procesos de inversi´on del esp´ın en el scattering. Otro resultado clave de esta tesis es la relaci´on subyacente entre el campo de gauge SU(2) y la transmisi´on con inversi´on del esp´ın. La tesis est´a organizada de la siguiente manera: el Cap´ıtulo 1 es la introduccion a la tesis. En los Cap´ıtulos 2 y 3 extendemos esta introducci´on para hablar de conceptos generales que son usados a lo largo del resto de la tesis. En particular, en el Cap´ıtulo 2 proporcionamos un breve repaso sobre el tratamiento te´orico del scattering en nanohilos mediante la ecuaci´on de Lippmann-Schwinger. En el Cap´ıtulo 3 introducimos la interacci´on esp´ın-´orbita en sistemas de baja dimensionalidad como 2DEG y nanohilos. En este ´ultimo caso explicamos la dificultad de diagonalizar el Hamiltoniano correspondiente debido a la hibridaci´on de las subbandas. En el Cap´ıtulo 4 introducimos el formalismo de Landauer-B¨uttiker para la descripci´on del transporte cu´antico. Extendemos la derivaci´on habitual para incluir un sistema con interacci´on esp´ın-´orbita de tipo Rashba y como consecuencia el sistema mantiene no solo corrientes de carga sino tambi´en 6 CHAPTER 1. INTRODUCTION strongly affect conductance of the zero modes. [8, 9, 10, 11, 12, 13, 14]. On the other hand, the spin-orbit interaction couples efficiently the electron spin to its orbital degrees of freedom, making it possible to affect the spin by engineering the scalar potential along the path of the electron. One can envision designs in which the desired effect of the spin-orbit interaction is strongly enhanced, which can be used to improve device functionality. The idea of using the spin-orbit interaction to rotate the electron spin goes back to the Datta-Das transistor [15], in which the control over the spin-orbit interaction was proposed to be used to modulate the conductance of a ferromagnet-semiconductor-ferromagnet device. Attempts to implement this transistor [16, 17, 18] faced several problems, including the low spin injection efficiency from ferromagnet into semiconductor and the detrimental effect of the scattering of the electron on disorder, which leads to spin relaxation. The interplay between superconductivity and spin-dependent fields also plays a fundamental role in the emerging field of superconducting spintronics [32, 33, 34, 35]. Beside possible applications of semiconducting systems with spin-orbit interaction, there are still fundamental questions regarding the electronic transport in such structures that still require a theoretical analysis. In nanowires, an open question is how a defect may affect the spin and charge transport in a quasi one dimensional wire. The answer to this question is far from trivial in a quasi-one-dimensional quantum wire formed by applying a confining potential to a 2DEG . On the one hand the combination of the quantization of motion along the orthogonal axis and the presence of an intrinsic Rashba spin-orbit interaction gives rise to inter-subband mixing that can strongly affect transport properties of the nanowire, for instance suppressing spin-modulation for large values of Rashba coupling [19, 20, 21, 22]. On the other hand the presence of a scattering center, as for example an impurity, may lead to formation of quasi-bound states localized around the impurity. Previous studies have shown that the presence of evanescent modes leads to unusual properties in the transport such as perfect transparency when the Fermi energy approaches subband minima and the blocking of channels due to quasi-bound states localized around an attractive impurity [23, 24]. Combination of both spin-orbit interaction and impurity scattering remains almost unexplored . In this thesis we address this issue and present a thorough theoretical study of the electronic transport in semiconducting nanowires with Rashba 11 CHAPTER 1. INTRODUCTION spin-orbit coupling in the presence of impurities. We model the nanowire as a quasi-one-dimensional system where the motion of the electrons is confined in the direction perpendicular to the transport direction. The interplay between the spin-orbit coupling, confinement and impurity potential, makes the problem highly non-trivial. We tackle this issue through a combination of theoretical techniques and approximations, which allows us to identify striking novel properties of both the charge and spin transport. Specifically, we describe the conductance by using the well established Landauer-B¨uttiker formalism, which we extend for the case of spin-dependent fields. Within this formalism the transport is described in terms of the scattering coefficients. In order to calculate these coefficients we use the Lippmann-Schwinger equation, a widely used method to treat scattering in quantum mechanics [36, 37, 38, 20, 39]. One of the main difficulties when dealing with spin-orbit coupling in a confining potential is the intermixing of subbands. In order to overcome this problem we introduce the Schrieffer-Wolff transformation with which we gauge away this intermixing while still accounting for its effects. By the combination of the above techniques we compute the charge and spin conductances of the Rashba nanowire in the presence of a point-like impurity. We find that the impurity couples evanescent and propagating states in the nanowires, inducing quasi-bound states; while the Rashba spin-orbit interaction allows for different spin-dependent mechanisms for electronic transport. As a result, the charge conductance presents perfect ballistic transmission at the threshold energy for a channel that becomes propagating. In addition, below this threshold energy there appears a dip in the conductance as a consequence of the quasi-bound states strongly suppressing transmission. We prove that for the lowest subbands at this resonant energy all electrons injected in a prepared spin-up state scatter from the impurity to a spin-down state. Furthermore, this spin-flip mechanism is not only the only transmission allowed at resonant energy but it is also enhanced. We derive the expressions for the spin currents in the nanowire and find out and expression for the spin-orbit torque induced by the impurity and the spin-flip mechanisms for transport. While the effects of Rashba spin-orbit coupling in the charge conductance are quite relevant, our key result consists in finding the underlying relation between the spin-flip transmission and the SU(2) field. The thesis is organized as follows: In Chapters 2 and 3 we extend the introduction, by discussing general 12 CHAPTER 1. INTRODUCTION concepts used in the rest of the thesis. In particular, in Chapter 2 we provide a brief overview of scattering in semiconducting nanowires and the Lippmann-Schwinger equation, our theoretical tool to determine the scattering coefficients. In Chapter 3 we discuss the spin-orbit coupling in low dimensional systems as 2DEGs and quasi 1D nanowires. In the latter case we explain the difficulty of diagonalizing the Hamiltonian due to the subband intermixing. In Chapter 4 we introduce the Landauer-B¨uttiker formalism for the description of quantum transport. We extend the customary derivation to a system with Rashba spin-orbit coupling. This automatically extends the formalism to a spin-dependent situation. The system now supports both spin and charge currents and we introduce the concept of spin-bias in the leads connected to the nanowire. The non-conservation of the spin-current at both sides of the impurity is interpreted as a spin-orbit torque arising from the spin-flip transmission mechanisms induced in the impurity by the Rashba spin-orbit coupling. The main result of this chapter is the expression for the currents and torque in terms of the scattering coefficients. In Chapter 5 we calculate the scattering coefficients explicitly for the nanowire. In order to do this we perform a gauge transformation and derive an expression for the coefficients accurate up to second order of perturbation in the spin-orbit coupling strength. As a first step, we gauge away the intermixing of subbands due to Rashba spin-orbit interaction by performing a Schrieffer-Wolff transformation. This allows us to obtain the scattering states in the whole wire by means of the Lippmann-Schwinger equation following the discussion of Chapter 2. In a second step, we calculate the transmission coefficients, which are now spin-dependent due to the Rashba spin-orbit coupling. We identify two types of transmissions: one that preserves the spin of the scattered particle and one that flips it. In Chapter 6 we present the main results for the transport properties of the nanowire. We focus on both, charge and spin currents. We show that the conductance presents striking features related to the presence of quasibound states localized around the impurities. We show that a non-magnetic impurity can flip spin as a consequence of Rashba spin-orbit coupling, and that the spin-flip transmission reflects similar resonant behavior. As a result, we prove that at the resonant energy the only transmission allowed is through the spin-flip mechanism and we discuss how this measurable transmission SU(2) symmetry factors. This result paves the way for a sensitive interference technique to measure the SU(2) gauge field in nanowires. 13 CHAPTER 1. INTRODUCTION Each chapter has its own conclusion section. Nevertheless we briefly summarize the whole thesis in Chapter 7. 14 Chapter 2 Theoretical description of transport in semiconducting nanowires Semiconductors are at the heart of modern electronics. In particular, they are important building blocks of nanostructures with versatile applications due to the accurate control of electronic transport via doping and external electric fields [40]. Furthermore, the advances in growth techniques such as molecular beam epitaxy (MBE) and patterning techniques, allows to create high-quatlity, meaning higher electron mobility, heterostructures that exhibit quantum confinement effects and a variety of quantum phenomena. The discovery of conductance quantization in low-dimensional systems (see Fig. 2.1)launches an intensive research of transport properties related to the charge of the electron [41]. In addition, the field of spintronics extended the research to spin-dependent transport phenomena, and the use of semicoductors for the design, and fabrication of novel spin-based electronic devices. The idea behind possible applications in this field relies on the control of the spin dynamics and relaxation by means of external fields. The cornerstone of many of the advances in these fields are two-dimentional electron gas (2DEG) typically formed at the interface of III-V semiconductor heterostructure which lead to the observation of new interesting phenomena, absent in bulk systems, such as Shubnikov-de Haas oscillations [42, 43], the integer [44] and fractional quantum Hall effect [45, 46, 46, 47] and the quantized conductance [48, 49]. In a two-dimensional electron gas electrons are confined to a narrow 15 CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN SEMICONDUCTING NANOWIRES Var UME 60, NUMBER 9PHYSICAL REVIEW LETTERS 29 FEaRU~RV 1988 15 1P LLj I— 5U3 LLJ CC -2 — 1.8—16-1.4-$.2-i -0.8-0 6 —14 GATE VOLTAGE IV) —12 GATE VOLTAGE (V) FIG. 1. Point-contact resistance as afunction of gate voltage at 0.6K. Inset: Point-contact layout. FIG. 2. Point-contact conductance as afunction of gate voltage, obtained from the data of Fig. 1after subtraction of the lead resistance. The conductance shows plateaus at multiples of e/xh. pinched off at Vg =— 2.2V. We measured the resistance of several point contacts as afunction of gate voltage. The measurements were performed in zero magnetic field, at 0.6K. An ac lockin technique was used, with voltages across the sample kept below kT/e, to prevent electron heating. In Fig. 1the measured resistance of apoint contact as afunction of gate voltage is shown. Unexpectedly, plateaus are found in the resistance. In total, sixteen plateaus are observed when the gate voltage is varied from — 0.6to — 2.2V. The measured resistance consists of the resistance of the point contact, which changes with gate voltage, and a constant series resistance from the 2DEG leads to the point contact. As demonstrated in Fig. 2, aplot of the conductance, calculated from the measured resistance after subtraction of alead resistance of 400 0, shows clear plateaus at integer multiples of e/&A. The above value for the lead resistance is consistent with an estimated value based on the lead geometry and the resistivity of the 2DEG. We do not know how accurate the quantization is. In this experiment the deviations from integer multiples of e/zh might be caused by the uncertainty in the resistance of the 2DEG leads. Inserting the point-contact resistance at V~= — 0.6V(750 0) into Eq. (1) we find for the width W,„=360nm, in reasonable agreement with the lithographically defined width between the gate electrodes. The average conductance increases almost linearly with gate voltage. This indicates that the relation between the width and the gate voltage is also almost linear. From the maximum width W,„(360nm) and the total number of observed steps (16) we estimate the increase in width between two consecutive steps to be 22 nm. We propose an explanation of the observed quantization of the conductance, based on the assumption of quantized transverse momentum in the contact constriction. In principle this assumption requires aconstriction much longer than wide, but presumably the quantization is conserved in the short and narrow constriction of the experiment. The point-contact conductance Gfor ballistic transport is given by " G=e NpW(It/2m)( [k„~ ). The brackets denote an average of the longitudinal wave vector k, over directions on the Fermi circle, Np =m/eh 2is the density of states in the two-dimensional electron gas, and Wis the width of the constriction. The Fermi-circle average is taken over discrete transverse wave vectors k» =~nz/W (n =1,2,...), so that we can write T &Ik. l&= Jd'krak, )&(k — kF) g6' k»— 7C F8',-)8' (3) Carrying out the integration and substituting into Eq. (2), one obtains the result N, (4) where the number of channels (or one-dimensional subbands) N, is the largest integer smaller than kFW/x. For 849 Figure 2.1: Quantized conductance for a point-contact as a function of gate voltage in a GaAs/InGaAs heterostructure. The conductance changes in quantized steps of 2e2/h. This figure taken from [48]. quantum well (QW) along the growth direction. Electrons can only move in the plane perpendicular to that direction. Transport properties, and band structure of 2DEGs can be modified by introducing dopants during growth, which contribute with electrons or holes to the QW, and by carefully choosing the materials in the quantum well and the barriers [47]. The two ways of realizing 2DEGs are by band inversion and heterostructure based systems. In Fig. 2.2 we can see the electrostatic potential Vz(z) (along the growth direction z) experienced by conduction band electrons in two situations: one shows a triangular quantum well, which forms, e.g., at the interface between n-doped AlGaAs and undoped GaAs and the other square quantum well, where a thin layer of the semiconducting material supporting the 2DEG, here GaAs, is sandwiched between layers of a different semiconducting material, here AlGaAs [50, 51]. In the first case, the Fermi energy of both will align at the interface of the semiconductors, where translational invariance is broken, with electrons coming out from the n-AlGaAs leaving behind an accumulation of holes which leads to a bending of the conduction and valence band [52]. For large enough hole concentration, the conduction band dips below the Fermi 16 CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN SEMICONDUCTING NANOWIRES   surface F  Figure 2.2: (a) Triangular quantum well in an inversion layer semiconductor heteroestructure. (b) Square quantum well in sandwich-like heteroestructure. energy at the surface. We can see then, that the electron density is sharply peaked near the GaAs-AlGaAs interface forming a thin conduction layer of thickness comparable to the Fermi wavelength, that is the two-dimensional electron gas, hence the name inversion layer. This layer is formed naturally, however, the mobility of layer-inversion 2DEGs is severely limited. Furthermore, since the electrons live at this interface, the quality of the 2DEG is highly dependent on details of fabrication. To achieve higher mobilities the quantum well has to be deeper [53, 54]. This can be done by introducing dopants in sandwiches of materials with differing bandgaps. When two such engineered materials with unequal bandgaps are brought into contact, the Fermi energy of the two materials will align and can form a quantum well. To align the chemical potential in the InGaAs/InAs/InGaAs sandwich structure, charge is transferred from remote dopants, introduced during growth, and into the quantum well. With the help of gate electrodes (external electric fields), or by clever sample fabrication, a large variety of potential energy structures can be achieved in the 2DEG. This way the motion of electrons can be further confined within the 2DEG semiconductor heterostructure plane leads to (quasi) one-dimensional quantum wires and zero dimensional quantum dots. Quantum confinement gives rise to new and fundamentally important physics phenomenon, it is therefore interesting to study quantum transport through these quantum confined mesoscopic systems [55]. The main focus in this thesis is the theoretical study of transport through a quantum nanowire in the presence of an impurity or defect. Therefore we summarize in this chapter the main theoretical tools for its description. 17 CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN SEMICONDUCTING NANOWIRES 2.1 Theoretical approach to scattering in nanowires Quantum wires have been proposed as basic elements in the design of many quantum devices. Because of their size the electronic transport rely on a full quantum mechanical approach rather than a classical one. The most important parameters (or length scales) that describe a semiconductor are the phase coherence length `φ, Fermi wavelength λF, and the mean free path `eof electrons. In a mesoscopic device the `φis much larger than the physical dimensions (length Land width W) of the device, while the λFis comparable to these dimensions. If `eis much larger than Land Wthe device is in the ballistic regime in which electrons propagate through the device without being scattered, either elastically or inelastically, by impurities or phonons respectively. In this thesis we are interested in transport through a nanowire in the presence of an impurity or defect. In particular we describe how the electronic transport is affected by the presence of the impurity, or in other words how the nanowire conductance depends on impurity and intrinsic properties of the wire. Because it is essential for our next analysis, we introduce here the Lippmann-Schwinger equation, which we will use for the description of quantum scattering . We start discussing this approach in a general 3D situation and then we focus on scattering on a delta-potential in a purely 1D system. We will discuss the limitations of the Born approximation. Finally we focus on a more realistic nanowire described by a transverse confining potential The main goal in scattering theory is to obtain the wave functions describing the scattering particle given a proper boundary conditions imposed, by the incoming particle. We can then begin by constructing the solution of the Schr¨odinger equation meeting these two criteria in formal terms. 2.1.1 The Lippmann-Schwinger Equation: theoretical description Following Ref.[56], we consider a system described by the following Hamiltonian H=H0+V , (2.1) 18 CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN SEMICONDUCTING NANOWIRES with H0=p2 2m∗ edescribes the free electrons with mass m∗ eand Vrepresenting the scattering potential. The goal is to find a solution for the states |ψiof the Schrdinger equation: H|ψi=E|ψi,(2.2) such that in the limiting of a vanishing potential V→0 the solution recovered would be that of the unperturbed system H0|φi=E|φi, i.e. |ψi→|φi. Formally one can write |ψi=V E−H0|ψi.(2.3) Then one can write the solution to Eq. (2.2) as a sum of the particular solution |φiand the homogeneous solution in Eq.(2.3) as follows, |ψi=1 E−H0 V|ψi+|φi.(2.4) Some complications arise from the singular nature of [E−H0]−1as the continuous spectrum of H0will include E. This problem can be circumvented by substituting E→E±ias a way to encode the boundary conditions for integration, so one may write the so-called Lippmann-Schwinger equation as follows, |ψ(±)i=|φi+1 E−H0±iV|ψ(±)i,(2.5) The physical meaning of (±) will be discussed later by evaluating |ψ(±)iat long distances. For the moment, we write the Lippmann-Schwinger equation in the coordinate basis, hr|ψ(±)i=hr|φi+Zdr0hr|1 E−H0±i|r0ihr0|V|ψ(±)i.(2.6) In order to solve this integral equation one must first evaluate the kernel defined by, G±(r,r0) = hr|1 E−H0±i|r0ihr0|,(2.7) which is nothing more than the Green’s function for the Helmholtz equation, ∇2+k2G±(r,r0) = 2m∗ e ~2δ(r,r0).(2.8) 19 CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN SEMICONDUCTING NANOWIRES The difficulty of most scattering problems lies in finding the proper Green’s function that solves Eq. (2.8) for the system. For the moment, in our derivation we may not write explicitly G±(r,r‘0) so that Eq. (2.6) reads, hr|ψ(±)i=hr|φi+Zdr0G±(r,r0)hr0|V|ψ(±)i.(2.9) Notice that the wavefunction hr|ψ(±)iin the presence of the scatterer is written as a sum of the incident wave hr|φiand a term that represents the scattering interaction. In most physical systems one works with the positive solution for the Green’s function G+(r,r0) as it satisfies the so called outgoing boundary conditions (as opposed to the negative solution G−(r,r0) corresponding to the less intuitive incoming boundary conditions). This means that G+(r,r0) garantees an outgoing flow from r0 to rchoosing E−H0+ iin Eq. (6.4), while G−(x, x0) on the other hand leads to an incoming current from rto r0choosing E−H0−i. From here on, we assume the positive case and drop the (±) sign reference in our description for notation simplicity. Now, in order to evaluate the specific behavior of hr|ψ(±)imore explicitly let us consider a local potential, that is a potential diagonal in the coordinate representation. The potential Vis considere to be local if it can be written as hr0|V|r00i=V(r0)δ(r0−r000),(2.10) and as a result, hr0|V|ψ(±)i=Zdr00hr0|V|r00ihr00|ψ(±)i =V(r0)hr0|ψ(±)i.(2.11) If we define the incident wavefunction to be a plane wave φ(r) = hr|φi,then the equation Eq. (2.9) simplifies to, ψ(r) = φ(r) + Zdr0G(r,r0)V(r0)ψ(r0),(2.12) giving the scattering states for an incoming particle evaluated at position x. For a finite range potential, the scattering state inside the support region will have a contribution limited to this space. So in effect the LippmannSchwinger equation provides a way to study scattering processes as a result 20 CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN SEMICONDUCTING NANOWIRES 41 EVANESCENT MODES AND SCATTERING IN QUASI-ONE-. .. 10 363 with increasing strength of the scatterer. This is indeed true, although not shown in the figures. In Fig. 4the intersubband transmission is small because the potential is relatively weak. At the onset of the second subband in Fig. 4(a} only about 6% of the incident carriers are converted into the second normal mode through T,2, and 4— 5% are converted into the third normal mode via T,3 at the bottom of the third subband. Figure 4(b) gives only between 1%%A and 2%%uo conversion from the second to the third mode at the bottom of the third subband via T23~ We can understand some features of Fig. 4by arguing from the Fermi "golden-rule" scattering rate. To do this we do not consider the intrasubband transmission T&&, T~q, or T33 as they are simply the result of leftover particles which did not scatter and can be obtained from the requirements of current conservation. Consider first the intersubband transmission T&2, T&3, and T23. The intersubband transmission has amaximum near the onset of a subband and decays like the inverse square root of energy away from the maximum. This can be understood from a Fermi's "golden-rule" viewpoint, where the probability of scattering is proportional to the final density of states in the subband which decays like I/~E. In Appendix B we show that the dominant term in the intersubband scattering probability is indeed given by an expression similar to the golden rule. The intersubband transmission and reffection coefficient T,2=R,zin Fig. 4(a) also sho~s interesting behavior around the bottom of the third subband, staying zero on both sides of the subband minima. There is no scattering out of mode one into mode two at the bottom of the third subband. We have yet to find agood explanation for this lack of mode conversion or reflection at the subband minima. However, the overall shapes of the transmission and reflection coefficients are still well understood by golden-rule arguments. Given the golden-rule-like shapes of the intersubband transmission and reflection coefficients and the intrasubband reflection, we can argue for the shape of the intrasubband transmission. Let us do so for T». Because particles must be conserved so that 1=T» +T,2 +R,2+R», and since R» =0 on both sides of the subband minima, the drop in T» after reaching perfect transmission at the second subband must he equal to T,z+R» =2T&2, or just twice the intersubband transmission coefficient. This is shown in Fig. 4(a). Similarly, the discontinuity in T» in Fig. 4(a} at the minima of the third subband is just twice T,3. Next, let us examine the scattering coefficients for an attractive potential. Figure 5shows a5-function scatterer of comparable strength to the one in Fig. 4, but when IIII 020 40 60 80 100 Energy {meY} FIG. 6. Two-probe conductance through a5-function defect in the quasi-one-dimensional wire in units of 2e /h. The solid line corresponds to the repulsive scatterer from Fig. 4, while the dashed line gives the conductance of the attractive scatterer from Fig. S. When the electron energy aligns with asubband minimum, the conductance through the defect is equal to the ballistic conductance. At these special energies the wire is perfectly transparent as if no scatterer were present. There is only asmall difference between the conductance for the weak repulsive scatterer and the ideal ballistic conductance throughout the entire range of electron energies. For the attractive scatterer, the new dips in conductance correspond to quasi-bound-states developing in the wire. The distance in energy from these dips to the subband minimum is the quasi-bound-state energy. Note also that, even though the repulsive scatterer is stronger, the conductance of the attractive scatterer is much smaller due to the presence ofthe quasi-bound-state. I2.0Cd V o1.0V 020 I I 40 60 80 100 Energy {meY} FIG. 7. Two-probe conductance in units of 2e /h for an attractive scatterer having y=— 8feV cm (solid line), y=— 9 feV cm (dotted line), and y=— 20 feV cm (dashed line). Beginning with the dotted line from Fig. 6showing the weakest attractive scatterer having y= — 6feVcrn, the overall conductance level decreases and the new dips corresponding to the quasi-bound-states move lower in energy as the scatterer is made more attractive. As the scatterer becomes so attractive that the quasi-bound-states move below the bottom of the next lowest subband, the new dips first disappear and the conductance then increases as the scatterer is made stronger. This unusual effect occurs because the bound states have now moved below the energy range in which they can block conduction. Figure 2.3: Conductance through a delta-impurity in the quasi-1D nanowire in units of 2e2/h. For the attractive scatterer v0<0 (dashed line), the conductance at the threshold energy is the ballistic conductance and below these threshold energies conductance dips appear in the corresponding to quasi-bound states present in the nanowire. For the repusive scatterer v0>0 (solid line), the resonant dip is not present. This figure taken from [23]. 27 CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN SEMICONDUCTING NANOWIRES 63, 64, 65], double-delta scatterers[66], finite-size scatterers [67, 68, 69], and even in for magnetic impurities [70], in the presence of an eternal magnetic field [71] and for time-dependent potentials [72, 73, 68]. In all these works, the resonant characteristics of the transmission were discussed in relation to the presence of quasi-bound states in the system. However, to our knowledge the scattering from an impurity in the nanowire in the presence of Rashba spin-orbit coupling remains unexplored. 2.2 Conclusion In this chapter we introduce the Lippmann-Schwinger equation which we will use for the theoretical description of quantum scattering for a semiconducting nanowire. In particular we describe how the electronic transport is affected by the presence of the impurity in the nanowire. We first describe the approach in a general system and later focus on scattering on a delta-potential in a purely 1D system, discussing the limitations of the Born approximation. Finally we focus on a more realistic nanowire described by a transverse confining potential, discussing the emergence of resonant behaviour in the transmission as a consequence of quasi-bound states present in the nanowire. This effect arises from the localized impurity coupling the evanescent and propagating modes of the nanowire. Our interest is in the possible effects arising from the interplay between the Rashba interaction and quasi-bound states. For this reason, Chapter 3 introduces the key ingredient in our study, namely the Rashba spin-orbit interaction in quantum nanowires. 28 Chapter 3 Spin Orbit Coupling in semiconductors The field of spintronics aims to create devices that take advantages of both, the spin and charge degrees-of-freedom of electrons. In particular, semiconducting spintronics facilitate the study of the fundamental concepts in the field thanks to the easy integration with nowadays semiconductor electronics. One of the key ingredients garnering attention in this field is the spin-orbit coupling, specially some forms of symmetry-dependent spin-orbit coupling realized in semiconducting heterostructures hosting a two-dimensional electron gas (2DEG). Such is the case of the Rashba spin-orbit interaction. In this Chapter we provide a brief introduction to spin-orbit interaction of the Rashba type. We discuss the spectral properties of low dimensional strcutures with Rashba spin-orbit coupling and provide a brief overview of two possible applications of materials with Rashba spin-orbit coupling, namely the Datta-Das spin-transistor and the detection of Majorana Bound States. Our interest in Rashba spin-orbit coupling is related to spin-dependent transport in quantum nanowires. For this reason, we first introduce the Hamiltonian model for a 2DEG and discuss its spectrum and symmetries. By further confining the 2DEG, one can creates a quasi one dimensional guide, or quantum nanowire . The confinement potential add complexity to the problem, and the Hamiltonian is no longer analitically solvable without approximations. Indeed, the subbands are deformed and avoided crossings appear in the energy dispersion, as we describe in this chapter. Later, in Chapter 5, we will treat Rashba spin-orbit interaction in 29 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS a perturbative way. Therefore in the present chapter we present the exact solution for the momentum in the energy dispersion where the subbands cross (the only point that presents spin degeneracy) which we will use as the unperturbed solution. 3.1 Introduction to Spin-Orbit Coupling The spin-orbit coupling (SOC) is a widely studied effect that describes the interaction between the spin of a particle with its motion in the presence of an electric field. And it can be described by following Hamiltonian [56], HSO =~ 4m2c2ˆ ~σ (∇V×p),(3.1) where ˆ ~σ = (ˆσx,ˆσy,ˆσz) is the vector of Pauli matrices, mis the rest mass of the electron, V(r) is the electrostatic potential in which the electron propagates with momentum p. For example, in atomic physics V(r) is the Coulomb potential of the atomic core. In semiconductor physics V(r) is the potential of a crystalline lattice that arises from the hybridization of the electron orbitals of neighboring atoms. The spectral properties of these electrons are characterized by the band energy En(k) and affected by the spin-orbit coupling. The effects of spin-orbit coupling in InAs, GaAs, InSb or other materials that are commonly used in the realization of nanowires, where the energy of the top valence band is strongly splitted in subbands depending on spin. [74, 75, 76]. Furthermore, the lack of centro-symmetry in the zinc-blende structure of III-V crystals and the confinement of 2DEGs allows for significant inversion asymmetry spin-orbit coupling effects in the lattice potential, lifting the spin degeneracy by splitting the energy bands in the absence of a magnetic field. The effects of this type of spin-orbit coupling can be better understood by exploring the relation between symmetry and band splitting, specifically time-reversal symmetry (TRS) and spatial inversion symmetry (SIS). Spatial Inversion and Time Reversal Symmetries The relation between symmetry and the splitting of the bands is fundamental for the understanding of some types of spin-orbit coupling 30 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS Reversed Preserved p→ −p q →q B→ −B E →E σ→ −σ p2/2m→p2/2m Table 3.1: Observables preserved and reversed under the time reversal transformation. [77]. The first symmetry of relevance is the time reversal symmetry. When a system undergoes a time reversal transformation T:t→ −tcharacterized by the time reversal operator T, some observables are preserved while others are reversed. Some of these observables are presented in table 3.1. Then,under the reversal of time: because the angular momentum is reversed L → − L and the so is the spin σ→ −σ, the spin-orbit is preserved L ·σ→ L ·σand the momentum k → − k . If a system is symmetric under time reversal (and the spin is half-integer), the Kramers theorem implies En(σ, k) = En(−σ, −k) (3.2) for any band energy for a given spin σand for a given momentum k , corresponds a energy degenerate band with opposite spin −σand opposite momentum − k . The other important symmetry is the spatial inversion symmetry. Under space reversal R: r → − r , while L → L and σ→ −σand consequently the spin-orbit L ·σ→ − L ·σand momentum k → − k . Then, in the case of a system with spatial inversion symmetry, En(σ, k) = En(σ, −k) (3.3) meaning that for any band with given spin σand momentum k , there is another degenerate band with same spin σand opposite momentum − k . And if the system presents both time reversal and spatial inversion symmetries, then En(σ, k) = En(−σ, k).(3.4) Thus, in a system with two spin eigenstates ↑and ↓that presents both space inversion symmetry (SIS) and time reversal symmetry (TRS) the energy dispersion for the two subbands overlaps. However, for systems with TRS 31 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS E (a) k ↓ ↑ E (b) k ↓↑ E (c) k ↓↑ Figure 3.1: (a) Degenerate energy dispersion for a system with TRS and SIS.(b) Gaped spectrum for a system where TRS is broken.(c) Shifted energy dispersion for a system where SIS is broken and spin-degeneracy is lifted. but broken SIS the spin-↑and spin-↓subbands have different energy at a given momentum kfor the same spin so that, En(σ, k)6=En(−σ, k) (3.5) as is the case for systems with spin-orbit coupling in non-centrosymmetric materials (see Fig.3.1(c)). Furthermore, if time reversal symmetry is broken the Kramers degeneracy in Eq.(3.2) is lifted and En(σ, k)6=En(−σ, −k) (3.6) case when an external magnetic field is applied to the system (see Fig.3.1(b)). Thus a potential that breaks spatial inversion symmetry lifts spin-degeneracy as stated in Eq.(3.5) while a potential that breaks time reversal symmetry lifts spin-degeneracy and Kramers degeneracy as seen in Eq.(3.6). Symmetry dependent spin-orbit coupling As described by Eq.(3.1) the main sources of SOC are electric fields, originating from asymmetries of the crystalline potential through its gradient ∇V. Therefore, it is an intrinsic effect, strongly depending on the material and its structure. As we already discussed in the case of zincblende III-V heterostructures such as GaAs, AlGaAs, InAs, etc., these asymmetries break down the spatial inversion spin-splitting the spectrum with two possible origins, 1. The first one is bulk inversion asymmetry (BIA), i.e., contrary to other crystalline structures such as that of silicon, the zinblende structure 32 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS lacks an inversion center. This asymmetry is fixed for a given sample, is intrinsic of the system and it is not possible to manipulate it externally. The spin-orbit coupling caused by this inversion asymmetry is known as Dresselhaus interaction [78]. 2. The second one is only possible in low dimensional systems where the motion of electrons is confined to two dimension (2DEG), for example in quantum wells, where there is a lack of inversion symmetry in the growth direction. This is the structural inversion asymmetry (SIA), and the importance of this mechanism lies in the fact that the asymmetry in the confinement potential can be varied by electrostatic means, allowing to tune the SOC strength by an external gate voltage. The spin-orbit interaction corresponding to this asymmetry is called Rashba spin-orbit coupling (RSOC)[79]. The relative importance between both spin-orbit interactions, Dresselhaus and Rashba, varies depending on the band structure of the material, the electron density and the geometry of the sample under investigation. In narrow-gap III-V quantum wells, however, the Rashba SOC is generally much larger than the Dresselhaus,as well as being more interesting due to its tunability. As a consequence, in this thesis the focus will be on the Rashba interaction, neglecting the Dresselhaus term. 3.2 Applications of Rashba Spin-Orbit Coupling In 1990 the first application of RSOC was proposed as what is known as the Datta-Das transistor or spin-Field Effect Transistor (spin-FET)[15] but it was not realized until later [80, 81]. This toy-model was developed as an analog to the electro-optic modulator and is based on the spin precession induced by the Rashba effect. It follows from the general expression for the spin-orbit coupling in Eq.(3.1), that the Rashba SOC gives rise to an internal magnetic field BRSOC and it can be written as BRSOC =α(Ez) (k×z), i.e., the magnitude of the field is proportional to the momentum kand a voltage-dependent parameter α, and it is pointing in the direction 33 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS Semiconductor 2DEG Gate Voltage Figure 3.2: Skech of a Datta-Das spin transistor, with 2DEG sandwiched in between two ferromagnets. The injected spin can be controled by tuning the RSOC, which in turned is controlled by a gate voltage. If the alignment of the electron spins, as they reach the drain, is parallel to this ferromagnet, then the transistor will register a non-zero current. On the contrary, if like in this figure the magnetization of the drain ferromagnet is antiparallel to the electron spins, the transistor will register a zero-current. perpendicular to both kand z(with zbeing the growth direction of the quantum well). In the absence of an externally applied magnetic field, the spin will precess around this effective magnetic field BRSOC in a similar way as the Larmor-precession around an external magnetic field. The precession frequency depends on the magnitude of the internal magnetic field |BRSOC|, and hence can be tuned by applying a gate voltage [82, 83, 84, 85, 86]. This property has led to the proposal of a Datta-Das ”toy-model” [15], also known as the Datta-Das spin-transistor. Datta and Das consider a ballistic transport channel with Rashba SO coupling in-between ferromagnetic leads acting as spin polarizers(see Fig. 3.2). When a spin is injected from one of the leads, it precesses around the Rashba field BRSOC until the spin arrives at the other ferromagnetic lead (the drain). The electron transmission probability into the drain depends on the relative alignment of its spin with the magnetization of the drain (this being fixed). Since the frequency of the precession of the spin during the travel to the drain can be controlled via gate voltage, so can the sourceto-drain current (or conductance). The importance of this toy-model for the field of spintronics consists not on its physical realization but on the scientific discussion sparked around it about the role of Rashba SOC in the spin dynamics of 2DEG, its interplay with Dresselhaus SOC and Zeeman. In 34 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS return, this research has lead to the discovery of new spin-related phenomena and their applicability in new devices. NATURE PHYSICS DOI: 10.1038/NPHYS1915 ARTICLES s-wave superconductor Semiconducting wire B x y z E k μ B, 1 A, 1 B, 2 A, 2 A, 3 B, 3 B, N A, N a bc γγ γ γγγ γγ Figure 1 |Majorana fermions appear at the ends of a 1D ‘spinless’ p-wave superconductor, which can be experimentally realized in semiconducting wires21,22.a, Pictorial representation of the ground state of equation (1) in the limit µ=0, t=|1|. Each spinless fermion in the chain is decomposed in terms of two Majorana fermions γA,xand γB,x. Majoranas γB,xand γA,x+1 combine to form an ordinary, finite-energy fermion, leaving two zero-energy end Majoranas γA,1and γB,Nas shown23.b, A spin–orbit-coupled semiconducting wire deposited on an s-wave superconductor can be driven into a topological superconducting state exhibiting such end Majorana modes by applying an external magnetic field21,22.c, Band structure of the semiconducting wire when B=0 (dashed lines) and B6=0 (solid lines). When µlies in the band gap generated by the field, pairing inherited from the proximate superconductor drives the wire into the topological state. characteristics of Majorana fermions—they are their own antiparticle and constitute ‘half’ of an ordinary fermion. In this limit the Hamiltonian becomes H=−it N−1 X x=1 γB,xγA,x+1 Consequently, γB,xand γA,x+1combine to form an ordinary fermion dx=(γA,x+1+iγB,x)/2, which costs energy 2t, reflecting the wire’s bulk gap. Conspicuously absent from H, however, are γA,1and γB,N, which represent end-Majorana modes. These can be combined into an ordinary (although highly non-local) zero-energy fermion dend = (γA,1+iγB,N)/2. Thus there are two degenerate ground states which serve as topologically protected qubit states: |0iand |1i=dend†|0i, where dend|0i=0. Figure 1a illustrates this physics pictorially. Away from this limit the Majorana end states no longer retain this simple form, but survive provided the bulk gap remains finite23. This occurs when |µ|<2t, where a partially filled band pairs. The bulk gap closes when |µ|=2t. For larger |µ|, pairing occurs in a fully occupied or vacant band, and a trivial superconducting state without Majoranas emerges. Realizing Kitaev’s topological superconducting state experimentally requires a ‘spinless’ system (that is, with one pair of Fermi points) that p-wave pairs at the Fermi energy. Both criteria can be satisfied in a spin–orbit-coupled semiconducting wire deposited on an s-wave superconductor by applying a magnetic field21,22 (see Fig. 1b). The simplest Hamiltonian describing such a wire reads H=Zdxψx†−¯ h2∂x2 2m−µ−i¯ huˆ e·σ∂x −gµBBz 2σzψx+(|1|eiϕψ↓xψ↑x+h.c.)(3) The operator ψαxcorresponds to electrons with spin α, effective mass m, and chemical potential µ. (We suppress the spin indices except in the pairing term.) In the third term, udenotes the spin–orbit31,32 strength, and σ=(σx,σ y,σz) is a vector of Pauli matrices. This coupling favours aligning spins along or against the unit vector ˆ e, which we assume lies in the (x,y) plane. The fourth term represents the Zeeman coupling due to the magnetic field Bz<0. Note that spin–orbit enhancement can lead33 to g2. Finally, the last term reflects the spin-singlet pairing inherited from the superconductor by means of the proximity effect. To understand the physics of equation (3), consider first Bz=1=0. The dashed lines in Fig. 1c illustrate the band structure here—clearly no ‘spinless’ regime is possible. Introducing a magnetic field generates a band gap ∝|Bz|at zero momentum, as the solid line in Fig. 1c depicts. When µlies in this gap the system exhibits a single pair of Fermi points as desired. Turning on 1 weakly compared to the gap then effectively p-wave pairs fermions in the lower band with momentum kand −k, driving the wire into Kitaev’s topological phase21,22. (Singlet pairing in equation (3) generates p-wave pairing because spin–orbit coupling favours opposite spins for kand −kstates.) Quantitatively, realizing the topological phase requires21,22 |1|<gµB|Bz|/2, which we hereafter assume holds. The opposite limit |1|>gµB|Bz|/2 effectively violates the ‘spinless’ criterion because pairing strongly intermixes states from the upper band, producing an ordinary superconductor without Majorana modes. In the topological phase, the connection to equation (1) becomes more explicit when gµB|Bz|  mu2,|1|where the spins nearly polarize. One can then project equation (3) onto a simpler oneband problem by writing ψ↑x∼(u(ey+iex)/gµB|Bz|)∂x9xand ψ↓x∼9x, with 9xthe lower-band fermion operator. To leading order, one obtains Heff ∼Zdx9x†−¯ h2∂x2 2m−µeff9x +|1eff|eiϕeff 9x∂x9x+h.c.(4) whereµeff =µ+gµB|Bz|/2and the effective p-wavepair fieldreads |1eff|eiϕeff ≈u|1| gµB|Bz|eiϕ(ey+iex) (5) The dependence of ϕeff on ˆ ewill be important below when we consider networks of wires. Equation (4) constitutes an effective low-energy Hamiltonian for Kitaev’s model in equation (1) in the low-density limit. From this perspective, the existence of endMajoranas in the wire becomes manifest. We exploit this correspondence below when addressing universal properties such as braiding statistics, which must be shared by the topological phases described by equation (3) and the simpler lattice model, equation (1). We now seek a practical method to manipulate Majorana fermionsin the wire.Asmotivation, consider applyingagate voltage to adjust µuniformly across the wire. The excitation gap obtained from equation (3) at k=0 varies with µas Egap(k=0) = gµB|Bz| 2−p|1|2+µ2 For |µ|< µc=√(gµBBz/2)2−|1|2the topological phase with end Majoranas emerges, whereas for |µ|> µca topologically trivial phase appears. A uniform gate voltage thus allows the creation or removal of the Majorana fermions. However, when |µ|=µcthe bulk gap closes, and the excitation spectrum at small momentum behaves as Egap(k)≈¯ hv|k|, with velocity v=2u|1|/(gµB|Bz|). The gap closure is clearly undesirable, as we would like to manipulate Majorana fermions without generating further quasiparticles. This problem can be circumvented by employing a ‘keyboard’ of locally tunable gates as in Fig. 2, each impacting µover a finite NATURE PHYSICS |VOL 7 |MAY 2011 |www.nature.com/naturephysics 413 Figure 3.3: (a) Sketch of Majorana Fermions appearing at the ends of a nanowire. Each spinless fermion in the chain is formed by the overlap of two Majorana fermions γA,x and γB,x. Majoranas γB,x and γA,x+1 form an ordinary fermion with finite energy, leaving two uncombined Majorana Fermions at the ends of the nanowire. (b) Set-up for the observation of MF: a semiconducting nanowire with spin-orbit coupling sits on top of a swave superconductor while an external magnetic field Bis applied. (c) In the absence of a magnetic field (dashed lines), the energy spectrum is spinsplit, but if an external magnetic field is applied perpendicular to the HSO a helical gap opens (solid lines) and superconductivity by proximity can drive the nanowire to a topological state. Figure taken from [87]. More recently there has been a revival of interest in studying SOC in semiconducting hybrid structures due to the possibility of finding Majorana zero modes hosted in Rashba nanowires in contact with superconducting electrodes, which are possible candidates for topological quantum computation due to their non-Abelian statistics [88, 89, 90, 87]. The basic idea is that such a structure can become a topological superconductor 35 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS under the right circumstances and support two non-local Majorana Bound States at the ends of the nanowire (see Fig. 3.3(a)). The strong SOC present in the nanowire shifts the two parabolic bands depending on their spin polarization and applying an external magnetic field perpendicular to the SO field breaks the TRS of the system opening a gap at the crossing point of the parabolas (k= 0), as seen in Fig. 3.3(c). If the Fermi energy µ is inside the opened gap the degeneracy is two-fold instead of four-fold. The proximity of a s-wave superconductor induces pairing in the nanowire between electron states of opposite momentum and opposite spins and induces a superconducting gap, ∆. Combining this two-fold degeneracy with an induced gap creates a topological superconducting phase for BZ>p∆2+µ2lifting electron-hole symmetry and Majoranas arise as zero-energy (i.e. mid-gap) bound states, one at each end of the wire [1, 2, 3, 4, 5, 6]. A visualization of such a set-up can be seen in Fig. 3.3(b). 3.3 The Rashba model for 2DEG In this section we focus on how Rashba spin-orbit coupling affects the spectral properties of a free electron in a 2DEG, before going into a description of a quantum nanowire where further confinement is applied to the 2D system to obtain a quasi-1D system [91]. The effective Hamiltonian for an electron moving in a 2DEG system in the (x, y)−plane in the presence of the Rashba spin-orbit coupling and with effective electron mass meis given by, H0=p2 2m∗+α ~(σ×p)z,(3.7) with eigenvalues E±(k) = ~2k2 2m∗±αk =~2 2m∗(k±kR)2−∆R,(3.8) where k=pk2 x+k2 yis the momentum, kR=αm∗ ~2is the Rashba spin-orbit coupling constant with momentum dimensions and ∆R=αm∗ ~2. The last term of Eq.(3.8) results in a downward shift of the bands that renormalizes the chemical potential, altough it is often neglected as it is second order in α. 36 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS Each block can be solved by shifting pyas follows, ψ±(y) = e±iy/λSO Φn(y),(3.22) where Φn(y) can be shown to satisfy the equation for the quantum harmonic oscillator in Eq.(2.26) and λSO =~/m∗ eα. Both blocks have identical eigenvalues and their wave functions are related to each other by a gauge transform. The solution to the initial problem becomes ψn,±(y, s) = e±iy/λSO Φn(y)χ±(s),(3.23) with degenerate eigenvalues En,±≡En=~ω0n+1 2−m∗ eα2 2.(3.24) Note that the states ψnσ in Eq.(3.23) obey the orthonormalization condition hψn0σ0|ψnσi=δn0nδσ0σwhere the scalar product is taken in both the y-coordinate and the spin spaces, hψn0σ0|ψnσi:= X sZ+∞ −∞ dyψ∗ n0σ0(y, s)ψnσ(y, s).(3.25) However, without summation over the spin degree of freedom the states ψn0σ0 and ψnσ for n06=nare orthogonal only provided σ0=σ, Z+∞ −∞ dyψ∗ n0σ(y, s0)ψnσ(y, s)∝δn0n.(3.26) This is due to the phase factor e±iy/λSO dropping out only when same spin states are involved. To emphasize that the wave function in Eq.(3.23) does not separate into a product of a y-coordinate component and a spin component, we write the states as ψnσ(y, s) = eiˆσxy/λSO Φn(y)χσ(s).(3.27) A product of two states without summation over the spin indices reduces to the direct product of the operators eiˆσxy/λSO taken from each of the states 43 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS in Eq.(3.27). It is convenient to represent such a direct product simply by supplying an index to the Pauli matrix, e−iˆσxy/λSO ⊗eiˆσxy/λSO →ei(ˆσa−ˆσb)y/λSO ,(3.28) where ˆσa xand ˆσb xhave separate Hilbert spaces for the time being, until we contract the spin indices. The quantity of interest is, therefore, Z+∞ −∞ dyΦ∗ n0(y)Φn(y)ei(ˆσa x−ˆσb x)y/λSO ,(3.29) which reduces to the following Fourier transform Fn0n(q) = Z+∞ −∞ dyΦ∗ n0(y)Φn(y)eiqy.(3.30) Note that Fn0n(q)=[Fnn0(−q)]∗and also that Fn0n(0) = δn0n. Actually, we will need Fn0n(q) evaluated at q=±2/λSO. The form-factor Fn0n(q) can be calculated for the case of harmonic confinement with the functions Φn(y) as given above. Since Φn(y) are chosen to be real, we have Fn0n(q) = Fnn0(q), which subsequently leads to the relation Fn0n(−q) = [Fn0n(q)]∗. Then, without loss of generality, we take n0≥nand obtain Fn0n(q) = r2n0n! 2nn0!Ln0−n nq2λ2 y 2 ×iqλy 2n0−n exp −q2λ2 y 4,(3.31) where Lα n(ξ) is the Laguerre polynomial, Lα n(ξ) = 1 n!eξξ−α∂n ∂ξn(e−ξξn+α).(3.32) This limit case for kx= 0 serves as the unperturbed solution to build upon for the following subsection. 3.4.2 Perturbative solution around kx≈0 In order to build a solution around kx= 0 we take the unperturbed Hamiltonian to be that of Eq.(3.18) so that the perturbed system is given by H=H0+α~kx.(3.33) 44 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS where the terms proportional to αkxcan be treated by perturbation theory. Let us consider values of kxwhich are small enough, such that the following regime holds α~kxFnn0(q0)En−En0, n 6=n0.(3.34) This condition roughly refers to ”staying away from the avoided crossings” and is equivalent to λy/λSO 1. This small parameter is very important as it appears again in Chapter 5 in the context of perturbation theory but for the Schrieffer-Wolff transformation. In this case, we treat the term α~kxσyas perturbation, whereas the term −αpyσxis treated exactly. However, in Chapter 5 the opposite is true:α~kxσyis treated exactly, while the term −αpyσxis considered a perturbation. The interest in the calculation presented in the current subsection is to find possible contributions of order α2arising from the term −αpyσxalone. The reason behing this is because they may present corrections to the second order (α2) in our calculation in Chapter 5. One could expect that in order to determine this, it is sufficient to consider the point kx= 0, which is exactly solvable. However, that point is degenerate and we have to consider its vicinity to understand how the states propagate and what are their transport properties when scattering off an impurity. For this reason, we consider the zeroth-order of perturbation theory in the small parameter in Eq. (3.34). This corresponds to the degenerate perturbation theory around the point kx= 0 for each subband nseparately. While this approach is valid for a strong spin-orbit interaction and a very small kx, we are interested here in answering the question about the role of the second-order corrections due to −αpyσx. In matrix form, the diagonal (n0=n) part of the Hamiltonian of the system is given by, ˆ H1D n=En−iαn~kx iαn~kxEn,(3.35) where the basis is given as before by the states in Eq. (3.23). We denoted αn=αFn(q0) with q0= 2/λSO.The form-factor Fn(q)≡ Fnn(q) is real and simplifies to Fn(q) = Lnq2λ2 2exp −q2λ2 4.(3.36) Any correction arising from the form factor is is q2 0∝α2and by multiplying it by the α~kxof the perturbation theory in Eq.(3.35) it goes with α3and 45 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS hence is beyond the accuracy of any calculation done in this work. However, we care about any α2correction arising from the states and there is also an α2overall energy shift, see Eq. (3.24). The eigenstate of Eq. (3.35) corresponding to the energy En,+=En+~kxαn,(3.37) is constructed out of the states in Eq. (3.23) χ+=1 21 1eiy/λSO +i 2−1 1e−iy/λSO .(3.38) And the eigenstate corresponding to the energy En,−=En−~kxαn,(3.39) is constructed as χ+=1 21 1eiy/λSO −i 2−1 1e−iy/λSO .(3.40) They both are further multiplied by the same Φn(y) and by eikxx, since these orbital components of the wave function are in common for the subband n. As expected, to this zeroth order of perturbation theory in α~kx, the states are not affected at all by the parameter αn. It enters only in the energy and together with the constant term ~k2 x/2m∗ ewill determine the division into left and right movers. The eigenstates of Eq. (3.35) can also be written in a compact form. If we multiply both states by a phase factor eiπ/4, then we obtain χ+=  cos π 4+y λSO  isin π 4+y λSO  , χ−=  isin π 4+y λSO  cos π 4+y λSO  .(3.41) These states reduce at y= 0 to χ+=1 √21 i, χ−=1 √2i 1,(3.42) 46 CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS which are eigenstates of σy. However, at y6= 0, they are no longer eigenstates of σy. One can verify that the two states in Eq. (3.41) originate from the nonabelian gauge factor eiσxπ 4+y λSO ,(3.43) which multiplies the usual up and down states of the σzPauli matrix. As a result of this, the corrections to the states of order α2are not affecting the scattering potential, because the above gauge factor commutes with the scattering potential. This is a relevant result for all analytical calculations done in this thesis, as we now can ensure the accuracy of the calculation for the perturbative methods, up to the second order α2, use in subsequent chapters. 3.5 Conclusions In summary, in this chapter we introduce the Rashba spin-orbit coupling going over some of its applications in the field of spintronics and in the detection of Majorana Bound States. We briefly discuss the spectral properties of a 2DEG with Rashba spin-orbit coupling before reviewing the complexities involved in the analytical solution of the model Hamiltonian for a quantum nanowire where the 2DEG is further confined. As a result of this confinement, not only is the energy spectrum of the system strongly affected but also the polarization of the spin. The combined effect of RSOC and confinement gives raise to anti-crossings between branches of opposite spin, deforming the spectrum. Moreover, we discussed the subbband mixing at the origin of this phenomena which leads to coupling between propagating and evanescent states in the quantum nanowire. As a result, a proper calculation needs to take into account enough subbands. Finally, we present an exact solution for the kx= 0 that we use as the result for the unperturbed problem to obtain the solution of the problem in the vicinity of the point kx≈0. This calculation allows us to ensure the accuracy of our perturbative approach up to α2in Chapter 5. 47 Chapter 4 The Landauer-Buttiker description of Transport The Landauer-B¨uttiker formalism is widely used as a method to study charge transport in mesoscopic systems [93, 94, 95]. It provides a very intuitive description of macroscopic effects in terms of scattering properties, that may be related to microscopic details of a system. In this chapter we generalize the Landauer-B¨uttiker formalism to include the effects of Rashba spin-orbit coupling. Besides the importance of such generalization, our results will help us to derive our results on transport properties of a nanowire with Rashba spin-orbit coupling and a single point-like impurity in the next chapters. In the next sections we present the Landauer-B¨uttiker formalism to describe both the charge and spin conductance. In this way we provide a simple description of spin-dependent transport that allows us to separate the spin-bias and voltage-bias contributions to the spin current. The Landauer-B¨uttiker formalism approach to deal with spin currents has been treated numerically or discussed in some models in the presence of Rashba spin-orbit coupling for either the Sin Hall Effect or three-terminal spin-filters [96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106]. Very recently an extension of the formalism was developed to better understand the origin and symmetries involved in spin currents in magnetic multi-layered systems [107]. In this chapter we derive an analytical expression for the spin current in terms of the spin-dependent transmission coefficients and discuss further the implications of scattering at an impurity on the transport. We identify a spin-torque as a consequence of spin-flip transmission mechanism mediated by such an impurity (see subsection 48 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT (a) scatterer 𝒂 𝑳↑ 𝒂 𝑹↑ 𝒃 𝑳↑ 𝒃 𝑹↑ 𝒂 𝑳↓ 𝒂 𝑹↓ 𝒃 𝑳↓ 𝒃 𝑹↓ (b) scatterer 𝑎 𝑎 𝑎 𝑎 𝑏  𝑏  𝑏  𝑏  Figure 4.1: (a) Two-terminal spin-dependent geometry. (b) Four-terminal geometry. 4.3.1). In addition, we find a conection between our results and the concept of the spin-mixing conductance introduced in Ref.[108]. 4.1 The Landauer-B¨uttiker approach for quantum transport The goal of the Landauer-B¨uttiker approach is to write expressions for the current in terms of transmission probabilities between different terminals connected by a scattering region. Here we follow we Reference [109], and generalize the derivation of the formalism for the case of spin-dependent transport. Indeed, we are considering wires with Rashba spin-orbit interaction and therefore the scattering amplitudes depends on the spin. In Fig.4.1 (a) we show the typical two-terminal system. The scattering region, in our case a nanowire, is connected to two ideal leads that we refer as left (L) and right (R). We treat each spin species as independent channels labeled by the index σ=↑,↓, see sketch in Fig.4.1 (a). The leads are characterized by temperature Tασ and chemical potential µασ, with α=L, R. It is worth noticing that the two-terminal spin-dependent geometry is equivalent to the four-terminal geometry sketched in 4.1 (b), where the temperatures for the terminals are T1,2=TL↑,L↓and chemical potentials 49 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT µ1,2=µL↑,L↓, and similarly T3,4=TR↑,R↓and µ3,4=µR↑,R↓. Thus, our analysis of 2-terminal setup for spin-dependent channels can be mapped to a 4-terminal situation with independent channels. It is assumed that the leads are at local equilibrium and therefore the electronic distribution functions in each lead is given by the Fermi distribution function: fα,σ (E) = e(E−µα)/kBTα+ 1−1, α =L, R ;σ=↑,↓(4.1) (see Fig.4.1). It is important to note that we are considering the contact leads to be wide compared to the crosssection of the quasi-1D nanowire, so that as far as the reservoirs are concerned, the nanowire represents only a small perturbation, and thus it is valid to describe the local properties in terms of an equilibrium state. Even though the dynamics of the scattering problem are described in terms of a Hamiltonian, the problem considered is irreversible[109]. This means that the processes for a particle entering or exiting the nanowire are uncorrrelated events; the reservoirs are fully determined by their respective Fermi distributions, and act as perfect sources and sinks for the particles independently of the energy of the particle entering or leaving the nanowire. Between the leads we consider a ballistic nanowire with Rashba spin-orbit coupling and a local impurity potential that will act as a source of scattering inside the nanowire. Far from the impurity we assume that transverse motion longitudinal motion of particles are separable. As described in Section 2.1.3 the motion from left to right contact (longitudinal motion) is not-confined and the system is characterized by the conserved wave-vector kn, where ndenotes the index number of transverse channels introduced by the quantization across the leads in the transverse direction corresponding to transverse energies EL,R;n,σ, which can be different for the left and right leads. We denote with NL,R (E) the number of incoming channels in the left and right lead, respectively. We now introduce the creation and annihilation operators denoted by α, nand σ.The operators ˆa† αnσ (E) and ˆaαnσ (E) create and annihilate electrons respectively, with total energy Ein the transverse channel nin the αlead, which are incident upon the sample. Similarly, the creation ˆ b† αnσ (E) and annihilation ˆ bαnσ (E) operators refer to electrons in the outgoing states. They 50 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT obey anticommutation relations: ˆa† Lnσ (E) ˆaαn0σ0(E0) + ˆaαn0σ0(E0) ˆa† αnσ (E) = δαβδnn0δσσ0δ(E−E0),(4.2a) ˆaαnσ (E) ˆaαn0σ0(E0) + ˆaαn0σ0(E0) ˆaαnσ (E) = 0 ,(4.2b) ˆa† αnσ (E) ˆa† αn0σ0(E0) + ˆa† αn0σ0(E0) ˆa† αnσ (E) = 0 .(4.2c) We introduce creation and annihilation operators, ˆ b† αnσ (E) and ˆ b† αnσ (E), and their anticommutation relations in outgoing states in the same way as incoming states in eqs. (4.2a) to (4.2c). The operators ˆaαnσ (E) and ˆ bαnσ (E) are related trough the scattering matrix Sas follows,                  ˆ bL1↑ ˆ bL1↓ ... ˆ bLN↑ ˆ bLN↓ ˆ bR1↑ ˆ bR1↓ ... ˆ bRN↑ ˆ bRN↓                  =S                 ˆaL1↑ ˆaL1↓ ... ˆaLN↑ ˆaLN↓ ˆaR1↑ ˆaR1↓ ... ˆaRN↑ ˆaRN↓                 .(4.3) We can write a similar equation to Eq. (4.3) for the hermitian conjugated matrix S†relating the creation operators ˆa† αnσ (E) and ˆ b† αnσ (E). The matrix Shas dimensions (NL+NR)×(NL+NR). Its elements are energy-dependent, and it has the following block structure S=rσσ0t0 σσ0 tσσ0r0 σσ0.(4.4) Here the diagonal blocks rσσ0and r0 σσ0describe electron reflection to the left and to the right reservoir, respectively. The off-diagonal blocks tσσ0and t0 σσ0 correspond to the electron transmission through the sample from the left to the right reservoir and from the right to the left reservoir, respectively. As discussed in Section 2.1.3, the flux conservation in the scattering process implies the unitarity of matrix S. In addition, in the presence of time-reversal symmetry as discussed in Section ?? the scattering matrix is also symmetric. Our goal in the following sections is to describe the transport through a nanowire with an impurity. Specifically, we derive an expression for the total 51 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT current operator ˆ IL(r, t) in the left lead far away from the localized impurity, and obtain an expression for the charge and spin-dependent conductances. In the Landauer-B¨utiker formalism the current operator is expressed in terms of the creation and annihilation operators. 4.2 Charge conductance The conservation of charge implies the continuity equation in quantum mechanics for the charge density ρ= eˆ Ψ†ˆ Ψ dρ dt +∇·j= 0 .(4.5) From this expression and teh Schr¨odinger equation we can derive an expression for the charge current density j. To do so we start by differentiating with respect to time the expression for the charge density dρ dt = e "dˆ Ψ† dt ˆ Ψ + ˆ Ψ†dˆ Ψ dt #,(4.6) where ˆ Ψ = ˆ Ψ1,ˆ Ψ2Tis a two-component spinor and ˆ Ψ†its hermitian conjugate. We now make use of the Shr¨odinger equation i~dˆ Ψ dt =Hˆ Ψ and its adjoint, where the Hamiltonian of the system is given by, H=−~2∇2 2m∗ e−i~α(∂xˆσy−∂yˆσx) + Vconf (y).(4.7) On the one hand, the kinetic term and the confinement potential in Eq.4.7 leads us to write the kinetic contribution for the evolution of the charge density in Eq.(4.6) as dρK dt =e i~ˆ Ψ†−−~2 2m∇2ˆ Ψ−−−~2 2m∇2ˆ Ψ†ˆ Ψ + ˆ Ψ†Vconf ˆ Ψ−Vconf ˆ Ψ†ˆ Ψ =−e~ 2mi∇hˆ Ψ†(∇ˆ Ψ) −(∇ˆ Ψ†)ˆ Ψi,(4.8) whereas the contribution from the spin-orbit coupling of the Hamiltonian to Eq.(4.6), usually referred as the anomalous term of the current, leads on the 52 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT conductance as G=I/V with current Igiven by Eq.(4.29), resulting in G=1 V e 2π~ZdE Tr t†t[f(E−µL)−f(E−µR)] =1 V e 2π~ZdE Tr t†t[−(µL−µR)f0(E)] =e2 2π~ZdE Tr t†tδ(E−EF). In the last step in Eq.(??) we assume in the zero-temperature limit the Fermi distribution function is a step function whose derivative becomes a delta in energy −f0(E) = δ(E−EF). Finally we obtain: G=e2 2π~Tr t†(EF)t(EF).(4.30) Eq. (4.30) is the well-known Landauer-B¨uttiker expression. It establishes the connection between the scattering matrix and the conductance of the system. We must notice that independently of the choice of basis, the conductance can be expressed in terms of transmission probabilities Tnfor each channel, as the expression t†(EF)t(EF) is diagonalizable and hence Tr t†t=PnTn. Furthermore, another version of Eq. (4.30) allows us to write the conductance in terms of the transmission probabilities for electrons leaving the left lead Lfrom a channel nand with spin σto arrive to the mchannel in the right lead Rwith spin σ0, G=e2 2π~X mσ0,nσ |tmσ0,nσ|2.(4.31) Once we have reviewed the way to write the charge conductance in the Landauer-B¨uttiker formalism we will generalize it in the next section for the case of spin-dependent observables such as the spin-current and spin-polarized conductance. 4.3 Spin current along the nanowire As we mention in the previous section, in Chapter 5 we obtain the solutions for the Hamiltonian described by Eq.(4.7) via a gauge transfromation that preserves the dynamics of the system. In this transformed system, effectively described by the Hamiltonian in Eq. (4.14) we can define a spin-density (not 59 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT equivalent to the ”true spin”) the y-component. Since we expect that away from the impurity of the spin to be conserved we can use the continuity equation. This way, we are able to derive expressions for the spin currents in both leads. In analogy to previous subsection, we start first deriving the corresponding conservation equation from the spin density Sy= eˆ Ψ†ˆσyˆ Ψ by differentiating with respect to time, dSy dt = e "dˆ Ψ† dt ˆσyˆ Ψ + ˆ Ψ†ˆσy dˆ Ψ dt #.(4.32) For the kinetic term and confinement described by our problem Hamiltonian in Eq.(4.14) we can write (dSy)K dt =e i~ˆ Ψ†ˆσy−−~2 2m∇2ˆ Ψ−−−~2 2m∇2ˆ Ψ†ˆσyˆ Ψ +ˆ Ψ†ˆσyVconf ˆ Ψ−Vconf ˆ Ψ†ˆσyˆ Ψi =−e~ 2mi∇hˆ Ψ†ˆσy(∇ˆ Ψ) −(∇ˆ Ψ†)ˆσyˆ Ψi.(4.33) And for the additional spin-orbit or anomalous term the contribution is given by, d(Sy)SO dt =e i~ˆ Ψ†ˆσyHSO ˆ Ψ−HSO ˆ Ψ†ˆσyˆ Ψ =e i~(ˆ Ψ∗ 1,ˆ Ψ∗ 2)ˆσyαx~−∂xˆ Ψ2 ∂xˆ Ψ1−αx~(−∂xˆ Ψ∗ 2, ∂xˆ Ψ∗ 1)ˆσyˆ Ψ1 ˆ Ψ2 =eαx i(ˆ Ψ∗ 1,ˆ Ψ∗ 2)−i∂xˆ Ψ1 −i∂xˆ Ψ2−(−∂xˆ Ψ∗ 2, ∂xˆ Ψ∗ 1)−iˆ Ψ2 iˆ Ψ1 =−αe∂xhˆ Ψ†ˆ Ψi.(4.34) Taking into account both kinetic and spin-orbit contributions in Eq.(4.33) and Eq.(4.34) respectively, we can write the total evolution for the spindensity polarized along y-axis as follows dSy dt =−e~ 2mi∂xhˆ Ψ†ˆσy(∂xˆ Ψ) −(∂xˆ Ψ†)ˆσyˆ Ψi−eαx∂xˆ Ψ†ˆ Ψ.(4.35) 60 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT All the terms in Eq.(4.35) can be gathered under the same partial derivative ∂xand therefore written in the form of dSy dt +∂xjy x= 0, that is to say we can define a conserved spin current along x with spin-polarization along ysuch that, jy x= (jy x)K+ (jy x)SO =e~ 2mihˆ Ψ†ˆσy(∂xˆ Ψ) −(∂xˆσyˆ Ψ†)ˆ Ψi−eαxˆ Ψ†ˆ Ψ.(4.36) Then, in the framework of the second quantization one can write the current operator as an integral of Eq.(4.36) in terms of the field operators ˆ Ψ with a kinetic contribution given by (ˆ Iy L)K(x, t) = e~ 2mZdEdE0X nσ ei(E−E0)t/~ 2π~√vLn0σ0vLnσ ×nˆa† Lnσ(E) (kn(E)σ+σkn0(E0)−2kR) ˆaLnσ(E0)e−i(kn(E)−kn0(E0))x −ˆ b† Lnσ(E) (kn(E)σ+σkn0(E0)+2kR)ˆ bLnσ(E0)ei(kn(E)−kn0(E0))x +ˆa† Lnσ(E) (kn(E)σ−σkn0(E0)−2kR)ˆ bLnσ(E0)e−i(kn(E)+kn0(E0))x −ˆ b† Lnσ(E) (kn(E)σ−σkn0(E0)+2kR) ˆaLnσ(E0)ei(kn(E)+kn0(E0))xo. Again, the expression in Eq.(4.37) can be significantly simplified by taking into account that values of Eand E0are close to each other and that the wave vectors kn(E) and velocities vary slowly with energy around the Fermi energy. This way, (ˆ Iy L)K(x, t) = e 4πm ZdEdE0X nσ ei(E−E0)t/~ vLn0σ0 ×2nˆa† Lnσ(E) (kn(E)σ−kR) ˆaLnσ(E0)−ˆ b† Lnσ(E) (kn(E)σ+kR)ˆ bLnσ(E0) −2kRˆa† Lnσ(E)ˆ bLnσ(E0)e−2ikn(E)x+ˆ b† Lnσ(E)ˆaLnσ(E0)e2ikn(E)xo. (4.37) 61 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT And for the spin-orbit contribution Eq.(4.36), (ˆ Iy L)SO(x, t) = eαx 2π~ZdEdE0X nσ ei(E−E0)t/~ vLn0σ0 ×nˆa† Lnσ(E)ˆaLnσ(E0) + ˆ b† Lnσ(E)ˆ bLnσ(E0) +ˆa† Lnσ(E)ˆ bLnσ(E0)e−2ikn(E)x+ˆ b† Lnσ(E)ˆaLnσ(E0)e2ikn(E)xo. (4.38) Finally the expression for the spin current polarized along the y-axis in the left lead is given by the sum of both kinetic Eq. (??) and spin-orbit contributions Eq. (4.38), so that ˆ Iy L(t) = ˆ IK L+ˆ ISO L =e 2π~ZdEdE0ei(E−E0)t/~X nσ nˆa† Lnσ(E)σˆaLnσ(E0)−ˆ b† Lnσ(E)σˆ bLnσ(E0)o. (4.39) Now we will focus on the term in between the brackets where the sum over the σindex is going to influence the final result as we will see. Reorganizing a bit the different sums it is possible to rewrite the expression for the spinpolarized current in Eq. (4.39) as given by the following expression only in terms of the creation and annihilation operators of the incoming basis, ˆ Iy L(t) = e 2π~X αβ X m0mX s0sZdEdE0ei(E−E0)t/~ˆa† αnσ(E)Bm0s0,ms αβ (L;E, E0)ˆaβnσ(E0), (4.40) here again, αand βtake the reservoir values Lor Rand Bm0s0,ms αβ (L;E, E0) = δm0mδs0sδαLδβL s−Pnσ S† αm0s0,Lnσ(E)σSLnσ,βms(E0). In order to derive the average spin-polarized current, we need to know that the product of the creation and annihilation operators of a electron Fermi gas at thermal equilibrium is Dˆa† αm0s0(E)ˆaβms(E0)E=δm0mδs0sδαβδ(E−E0)fαs0(E).(4.41) We have added a spin index in the Fermi distribution, fαs0(E) = e(E−µαs0)/kBTαs0+ 1−1, to describe eventually different spin 62 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT chemical potential and spin temperature in each lead. This means we have an additional index with respect to Eq.(4.25). Similarly, in the averaging process we notice that only α=βterms are going to survive and that SLnσ,Lms =rnσ,ms and SLnσ,Rms =t0 nσ,ms . Then, using the previously derived unitary identities and this other one Psr† s,σrσs +t† s,σtσs= 1 we can rewrite the integrand in Eq. (4.39) using the Eq.(4.41),as follows X αβ X m0mX s0sDˆa† αms(E)Bm0s0,ms αβ (L;E, E0)ˆaβms(E0)E =X m0σ0 σ0−X mσ r† m0σ0,mσσrmσ,m0s0!fLσ0(E)δ(E−E0) +X m0σ0X mσ t0† m0σ0,mσσt0 mσ,m0σ0fRσ0(E)δ(E−E0).(4.42) Substituing Eq. (4.42) into Eq. (4.39) we obtain the expression for the average spin-current : Dˆ Iy L(t)E=e 2π~ZdE "X m0σ0 σ0−X mσ r† m0σ0,mσσrmσ,m0s0!fLσ0(E) +X m0σ0X mσ t0† m0σ0,mσσt0 mσ,m0σ0fRσ0(E)#.(4.43) Due to the spin-dependence of the Fermi distribution we can write explicitly the spin current of Eq. (4.43) as follows Dˆ Iy L(t)E=e 2π~ZdE T↑fL↑(E)−T↓fL↓(E)−T0 ↑fR↑(E) + T0 ↓fR↓(E) +r† ↓↑r↑↓ +r† ↑↓r↓↑(fL↑(E)−fL↓(E)) +t0† ↓↑t0 ↑↓ +t0† ↑↓t0 ↓↑(fR↑(E)−fR↓(E))i.(4.44) where T↑=t† ↑↑t↑↑ +t† ↓↑t↑↓ is the transmission amplitude with spin-up on the left lead. Similarly, T↓=t† ↓↓t↓↓ +t† ↑↓t↓↑ for the transmission amplitude with spin-down. In order to keep track of the origin of each transmission amplitudes we work with T0 ↑,↓=t0† ↑↑t0 ↑↑+t0† ↓↑t0 ↑↓, that is to say, the transmission amplitudein the left lead for electrons incident from the right, although due 63 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT to the unitarity of the scattering matrix they are equivalent to T↑,↓. In the absence of spin bias, i.e.(fL,R↑=fL,R↓), Eq. (4.44) is reduced to: Dˆ Iy L(t)E=e 2π~ZdE [T↑−T↓] (fL(E)−fR(E)) , where we used T↑,↓=T0 ↑,↓. In a general case, when no assumption is made regarding the chemical potentials, we can write in linear response ZdET↑(E) (fL↑(E)−fR↑(E)) = T↑µ↑ L−µ↑ R, ZdET↓(E) (fL↓(E)−fR↓(E)) = T↓µ↓ L−µ↓ R. (4.45) and similarly, ZdE r† ↓↑r↑↓ +r† ↑↓r↓↑(fL↑(E)−fL↓(E)) = r† ↓↑r↑↓ +r† ↑↓r↓↑µ↑ L−µ↓ L, ZdE t0† ↓↑t0 ↑↓ +t0† ↑↓t0 ↓↑(fR↑(E)−fR↓(E)) = t0† ↓↑t0 ↑↓ +t0† ↑↓t0 ↓↑µ↑ R−µ↓ R. (4.46) These differences in chemical potential are µ↑,↓ L−µR↑,↓=eV ↑,↓ L→Rand for the spin-biases µ↑ L,R −µ↓ L,R= eVy L,R as we polarize the spin along the y-direction. Substituting Eqs.(??)-(4.46) into Eq.(4.44) that finally we can express the spin current polarized alon y-direction in the left lead as hIs Li=e2 2π~hT↑V↑ L→R−T↓V↓ L→R+r† ↓↑r↑↓ +r† ↑↓r↓↑Vy L+t0† ↓↑t0 ↑↓ +t0† ↑↓t0 ↓↑Vy Ri. (4.47) With this expression we close the subsection giving a way to probe the spin current in the left lead in order to find the spin-dependent conductance for different spin and charge biases applied. The same calculation for the right lead is straightforward and is used in the next subsection as we discuss the appearance of the spin torque related to the spin-flip processes in transport calculations. 64 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT 4.3.1 Spin Torque in the nanowire In the previous section we derived the spin current on the left lead, far away from the impurity, due to the presence of Rashba spin-orbit coupling. However, close to the impurity we can no longer assume continuity of the spin-density and Eq.(4.32) presents a source term, ∂Sy ∂t +∂xjy x=T(x).(4.48) the right hand side is a torque term T(x) = T0δ(x−ximp) due to the presence of the impurity. By integrating the expression for the spin-current for the static problem (∂Sy ∂t = 0) around the position of the impurity, T0=Zximp+ ximp− (∂xjy x)dx =−hIy Ri−hIy Li.(4.49) the expression for the spin current on the right side of the impurity ximp + corresponds to a current traveling from left to right, implying a negative sign. Then, we need to calculate the expression for the average of the spincurrent on the right lead from Eq.(4.40) where the indexes for the Llead have been substituted with Lindexes. This implies a change of the scattering coefficients involved (tinstead of t0and r0instead of r), hIs Ri=e2 2π~hT↑V↑ R→L−T↓V↓ R→L+r0† ↓↑r0 ↑↓ +r0† ↑↓r0 ↓↑Vy R+t† ↓↑t↑↓ +t† ↑↓t↓↑Vy Li, (4.50) where V↑,↓ R→L=−V↑,↓ L→R. So that the Eq.(4.49) for the torque in combination with the results of Eq.(4.47) and Eq.(4.50) becomes, T0=−e2 2π~ht† ↓↑t↑↓ +t† ↑↓t↓↑ +r† ↓↑r↑↓ +r† ↑↓r↓↑Vy L +t0† ↓↑t0 ↑↓ +t0† ↑↓t0 ↓↑ +r0† ↓↑r0 ↑↓ +r0† ↑↓r0 ↓↑Vy Ri.(4.51) As a result evidence by Eq.(4.51), applying a spin-bias on either the right or left lead generates a spin-orbit torque expressed here as a function of the scattering coefficients related to spin-flip processes mediated by the impurity. In the result section of Chapter 6 we discuss fully the importance of this result. 65 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT 4.3.2 Relation the spin-mixing conductance The mixing conductance is a concept of great importance for transport between noncollinear ferromagnets and is responsible for the spin rotation around the magnetization axis of the ferromagnet [108]. This quantity is itnroduced as a way to highlight the fact that scattering mixes the two components in a spinor and one must not see spin species as independent. In order to show this, they propose a toy model where an electron incides in the scatterer from the right, ψie−iikxxand is reflected onto the right side of the scatterer as ψfe−iikxx. Both the incident and the final state are related through the scattering matrix and contribute to the spin current as shown in the Landauer-B¨utiker expression of Eq.(4.39), j(S) α=~ 2vxψ∗ iˆσαψi−ψ∗ fˆσαψf,(4.52) with final states given by ψf=r0 0r⊗ψi,(4.53) substituting the spinors polarized in the x, y-directions and integrating over energy we obtain the spin currents: IS x≈(Re G↑↓Vx R+ Im G↑↓Vy R) (4.54) IS y≈(Re G↑↓Vy R+ Im G↑↓Vx R),(4.55) where the complex conductance G↑↓ is given by, G↑↓ =G01−rr∗ ⊗.(4.56) We want to establish a comparison between the y-component of the spin current in Eq.(4.55) to our previous result for the spin current on the right side of the scatterer as described by Eq.(4.50). First, in this toy model there appears to be spin bias only on the right side of the scatterer, hence Vy L= 0 in Eq.(4.50). In the toy model the only process considered is the reflection, resultin in T↑=T↓= 0, one must remember that these transmission amplitudes include both spin-conserved and spin-flip transmission processes. By this account, the expression in Eq.(4.50) 66 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT becomes for the toy-model where a spin bias is applied on the right side of the scatterer hIs Ri=e2 2π~r0† ↓↑r0 ↑↓ +r0† ↑↓r0 ↓↑Vy R.(4.57) Nazaroz writes his scattering matrix in the basis of the ˆσzmatrix, ˆ S=|↑zih↑z|ˆ S+|↓zih↓z|ˆ S⊗,(4.58) but when we calculate our S-matrix in Chapter 5 it will be in the basis of ˆσy. So, by writing the scattering matrix in Eq.(4.58) in our basis, ˆ S=1 2h(ˆ S+ˆ S⊗)(|↑yih↑y|+|↓yih↓y|)+(ˆ S−ˆ S⊗)(|↑yih↓y|+|↓yih↑y|)i. (4.59) From Eq.4.59 we can write the reflexion matrix as, ˆr0=1 2r+r⊗r−r⊗ r−r⊗r+r⊗.(4.60) The out-of-diagonal elements in Eq.(4.60) are the r0 ↑↓ and r0 ↓↑ in Eq. (4.57). This equation written in the language of rand r⊗becomes, hIs Ri=e2 2π~ 1 42(r−r⊗)†(r−r⊗)Vy R =e2 2π~ 1 2hr† r+r† ⊗r⊗−r† r⊗−r† ⊗riVy R =e2 2π~ 1 4h2−2 Re(r† ⊗r)iVy R =e2 2π~Re(1 −r† ⊗r)Vy R.(4.61) If we consider no spin-bias for the polarization along x-direction, Vx R= 0, it is pretty clear that we recover Eq.(4.55) in Eq. (4.61). We see then that even in the absence of the traditional conductance T↑=T↓= 0, we recover some mixing conductance on the right side of the impurity. This means that the spin current flows even in the absence of charge transport. 4.4 Conclusions In summary, in this chapter we extend the Landauer-B¨uttiker formalis to include the effect of the spin-orbit coupling for the description of the 67 CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF TRANSPORT transport properties of a nanowire. We derive an expression for the spin current polarized along y-direction in the left lead to be far away from the impurity. This expression, Eq.(4.47), is the highlight of this chapter. It allows us to express such a spin current as a function of the scattering coefficients of the S-matrix making it easy to track the contributions to the spin current from the different voltage and spin-biases. In combination with the expression for the spin current on the right lead, Eq.(4.50), allows us to describe a torque that arises at the impurity position, as a consequence of the spin-flip transport mechanisms resulting from the Rashba spin-orbit coupling. This mechanism will have important consequence on the conductance as it is discussed in subsequent chapters. Furthermore, we are able to make a connection between our expressions and those obtained in the context of the spin-mixing conductance in magnetic hybrid structures. So far we have expressed all transport properties in terms of the scattering coefficients. In the next chapter we determine such coefficients for the scattering from an impurity in a nanowire in the presence of Rashba spin-orbit coupling. 68 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE term. At the same time, the sine term drops out, since it has only offdiagonal matrix elements, i.e. it admixes excited subbands. We expand that latter Hamiltonian up to order α2, Hwire =p2 y 2m∗ e +m∗ eω2 0 2y2+αx~kxσz −2m∗ eαxαykxσyy+~2k2 x 2m∗ e−m∗ eα2 y 2,(5.25) and re-write it as follows Hwire =p2 y 2m∗ e +m∗ eω2 0 2y−2αxαy ω2 0 kxσy2 +αx~kxσz +~2k2 x 2m∗ e−m∗ eα2 y 2−2m∗ eα2 xα2 y ω2 0 k2 x.(5.26) The shift in the harmonic oscillator center can be gauged away in a similar way as the linear in momentum terms of the spin-orbit interaction, which also could be interpreted as a shift of the kinetic energy central position as a function of momentum. The next transformation has the form ˜ ψkxn(y) = e−iσypyy0/~¯ ψkxn(y),(5.27) where py=−i~∂yand y0=2αxαy ω2 0 kx.(5.28) After this transformation the Hamiltonian reads Hwire =p2 y 2m∗ e +m∗ eω2 0 2y2 +αx~kxσzcos 2pyy0 ~−σxsin 2pyy0 ~ +~2k2 x 2m∗ e−m∗ eα2 y 2−2m∗ eα2 xα2 y ω2 0 k2 x.(5.29) The last term is of fourth order in αand can be omitted, because we are accurate only to the second order. Similarly the cosine and sine contain y0 75 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE which is proportional to α2and there is another αin front of the whole term. As a result, we obtain, up to order α2, the final effective Hamiltonian Hwire =p2 y 2m∗ e +m∗ eω2 0 2y2+αx~kxσz +~2k2 x 2m∗ e−m∗ eα2 y 2.(5.30) This Hamiltonian is rather simple. It can be solved analytically and the corresponding Green’s function can be written straightaway, as we will see in the next sections. The eigenfunctions can be the written by summarizing up the above transformations: Ψkxn(x, y) = eiσxm∗ eαy ~y+π 4e−iσy2αxαy ~2ω2 0 pxpy1 √LeikxxΦn(y),(5.31) where we restored ~kx→pxin the second factor. We can also restore ~kx→pxin Eq. (5.30), since the Hamiltonian is diagonal in the quantum number kx. The lateral wavefunctions Φn(y) for a harmonic confinement are given by Eq.(2.30) as discussed in Chapter 2. 5.2 Scattering states We now follow the procedure described in Chapter 2 to obtain the scattering states for the effective system of a multiband quasi-1D nanowire in the presence of Rashba interaction. We focus first on the Green’s functions corresponding to the effective Hamiltonian Eq.(5.30). The Green’s function is a sum of Green’s functions of the independent sub-bands, G(r,r0) = X n Φn(y)Φ∗ n(y0)Gn(x, x0).(5.32) This expression implies separation of variables for the channel without impurity. In our case, we take out the spin degree of freedom into a matrix structure, ˆ G(r,r0) = X nσ Φn(y)Φ∗ n(y0)|σihσ|Gnσ(x, x0).(5.33) The Green’s function for each individual channel is given by Gnσy(x, x0) = 2m∗ e ~2 i 2kn eikn|x−x0|e−iσy m∗ eαx ~(x−x0),(5.34) 76 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE where kn=1 ~v u u t2m∗ e"E−n+m∗ eα2 x+α2 y 2#.(5.35) For what follows it is convenient to write these Green’s functions as follows, Gnσy(x, x0) = gnσyunσy(x)u∗ nσy(x0), x > x0, vnσy(x)v∗ nσy(x0), x < x0,(5.36) where unσy(x) = rm∗ e ~kn eiknxe−iσy m∗ eαx ~x,(5.37) vnσy(x) = rm∗ e ~kn e−iknxe−iσy m∗ eαx ~x.(5.38) These functions are normalized to carry unit flux density. With such a normalization, we have the common factor to be gnσy=i ~.(5.39) The state unσy(x) is the outgoing state on the right side of the impurity(x>x0 or x→+∞). It is a right mover and we shall choose this state also as an incident state from the left, ΦL(x) = unσy(x). Similarly, the state vnσy(x) is the outgoing state on the left side of the source (x<x0or x→ −∞). It is a left mover and we shall choose this state also as an incident state from the right, ΦR(x) = vnσy(x). The Lippmann-Schwinger equation after the Schrieffer-Wolff transformation reads: Ψ(r) = Φ(r)−Zˆ G(r,r0)eMVimp(r0)e−MΨ(r0)d2r0,(5.40) where we explicitly write the transformed ˆ Vimp(r0). Notice that in the basis Eq. (5.36), the incoming state Φ(r) can be written as ΦL nσ(x, y) = unσ(x)Φn(y)|σi,(5.41) for incident from the left, and similarly ΦR nσ(x, y) = vnσ(x)Φn(y)|σi,(5.42) 77 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE for incident from the right. Because of the small support of Vimp(r0), the exponents get projected on a state with r0=r0, and effectively behaves as a delta-like function. After introducing the dimensionless projector |r0ihr0|,(5.43) we obtain from Eq. (5.40) in the limit of a point-like scatterer, Ψ(r) = Φ(r)−v0eM(r0)ˆ G(r,r0)|r0ihr0|e−M(r0)Ψ(r0).(5.44) By expanding the exponential functions and using Eq. (5.14) we obtain: Ψ(r) = Φ(r)−v0 ×ˆ G(r,r0) + 2iαxαy ω2 0h∂x0∂y0ˆ G(r,r0)iˆσz ×Ψ(r0)−2iαxαy ω2 0 ˆσz[∂x0∂y0Ψ(r0)].(5.45) In order to determine the wave function e−SΨ(r) at position r=r0, we act with e−Son the Lippmann-Schwinger equation e−SΨ(r) = e−SΦ(r)−Ze−Sˆ G(r,r0)eSVimp(r0)e−SΨ(r0)d2r0,(5.46) and set r=r0:  1 +Ze−Sˆ G(r0,r0)eMVimp(r0)d2r0e−SΨ(r0) = e−MΦ(r0).(5.47) The contribution of evanescent states to the sum over nin the Green’s function diverges if we simply set r0=r0for a δ-like Vimp(r0). However, we can do that for the propagating states and this is the reason why one can take the term e−MΨ out of the integrand. The bound states are determined by the condition: det  1 +Ze−Sˆ G(r0,r0)eSVimp(r0)d2r0= 0 .(5.48) The expression e−Sˆ G(r0,r0)eSrepresents the exact Green’s function of the channel (without impurity). However our treatment is perturbative in the 78 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE spin-orbit interaction, and thefore: e−Sˆ G(r,r0)eS≈ˆ G(r,r0) +2iαxαy ω2 0h∂x0∂y0ˆ G(r,r0)iˆσz −2iαxαy ω2 0 ˆσzh∂x∂yˆ G(r,r0)i+. . . . (5.49) To attain unitarity of the scattering matrix, additional (overcounting) terms might be necessary, as explained in the next section. Notice that the Green’s function is diagonal in the basis of σy, |+i=1 √21 i, |−i =1 √21 −i,(5.50) and therefore σzreads. ˆσz=|+ih−|+|−ih+|.(5.51) In this basis the Green’s function is a diagonal 2 ×2 matrix ˆ G(r,r0) = G++ 0 0G−− ,(5.52) where G±± is the Green’s function projected onto the state with σy=±1. We construct symmetric and antisymmetric combinations with respect to the change of sign of x−x0, Gs=G++ +G−− 2=X nσ Φn(y)Φ∗ n(y0)i m∗ e ~2kn eikn|x−x0|cos m∗ eαx ~(x−x0), (5.53) Ga=G++ −G−− 2=X nσ Φn(y)Φ∗ n(y0)m∗ e ~2kn eikn|x−x0|sin m∗ eαx ~(x−x0).(5.54) Because of the translational invariance over x, the single band Green’s function obeys ∂xGnσ(x, x0) = −∂x0Gnσ(x, x0).(5.55) 79 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE This is also valid to the full Green’s function ˆ G(r,r0), but only with respect to xand x0. Therefore, we can write Eq. 5.49 as e−Sˆ G(r,r0)eS≈ˆ G(r,r0) −2iαxαy ω2 0h∂x∂y0ˆ G(r,r0)iˆσz −2iαxαy ω2 0 ˆσzh∂x∂yˆ G(r,r0)i,(5.56) or in a matrix form e−Sˆ G(r,r0)eS≈ G++ −2iαxαy ω2 0∂x(∂y0G++ +∂yG−−) −2iαxαy ω2 0∂x(∂y0G−− +∂yG++)G−− !. (5.57) The scatterer potential is assumed to be symmetric. In particular Vimp(r) has mirror symmetry with respect to x→ −x, where xis measured for this purpose with respect to x0. Therefore, in Eq. (5.48) only the symmetric part of Gcontributes to integrals of the form ZG++(r0,r0)Vimp(r0)d2r0.(5.58) Notice that because Ghas a block structure in spin space, we can consider each block contribution to Eq. (5.48) independently. On the other hand he anti-symmetric part of Gcontributes to the integrals of the form Z[∂xG++(r,r0)]|r=r0Vimp(r0)d2r0,(5.59) only the anti-symmetric part of Genters. As a result, the matrix in Eq. (5.57) which enters in Eq. (5.48) can be written as Gs−2iαxαy ω2 0∂x(∂y0−∂y)Ga 2iαxαy ω2 0∂x(∂y0−∂y)GaGs!.(5.60) We introduce the short notations: ⟪Gs⟫=1 v0ZGs(r0,r0)Vimp(r0)d2r0, ⟪∂2Ga⟫=1 v0Z[∂x(∂y0−∂y)Ga(r,r0)]|r=r0Vimp(r0)d2r0. (5.61) 80 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE and rewrite Eq. (5.48) as det h 1 +v0⟪ˆ G⟫i= 0 ,(5.62) where ⟪ˆ G⟫= ⟪Gs⟫−2iαxαy ω2 0⟪∂2Ga⟫ 2iαxαy ω2 0⟪∂2Ga⟫ ⟪Gs⟫!.(5.63) Or explicitly (1 + v0⟪Gs⟫)2−2αxαyv0 ω2 02 ⟪∂2Ga⟫2= 0 .(5.64) This is the condition for bound states, which can be written also as 1 + v0⟪Gs⟫±2αxαy ω2 0 ⟪∂2Ga⟫= 0 .(5.65) According to the Kramers theorem, the two solutions obtained from this equation (for the ±sign) must be degenerate and therefore, ⟪∂2Ga⟫has to vanish at the leading order of our approximation (a rigorous proof is presented in Appendix A. Such that Eq. (5.47) results in e−SΨ(r0) = 1 1 + v0⟪Gs⟫e−SΦ(r0).(5.66) Inserting this result into Eq. (5.45) we finally obtain Ψ(r) = Φ(r)−v0 1 + v0⟪Gs⟫ ×ˆ G(r,r0) + 2iαxαy ω2 0h∂x0∂y0ˆ G(r,r0)iˆσz ×Φ(r0)−2iαxαy ω2 0 ˆσz[∂x0∂y0Φ(r0)].(5.67) The scattering matrix we can now calculate the scattering matrix. For this, we send a sate ΦL nσ(r) incident from the left and look at x→+∞. The Green’s function at x→+∞ is ˆ G(r,r0) = X nσ Φn(y)Φ∗ n(y0)|σihσ|gnσunσ(x)u∗ nσ(x0).(5.68) 81 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE The transmission amplitude to scatter from left to right is found from tRL mσ0,nσ =δΨ(r) δΦL mσ0(r)Φ(r)→ΦL nσ(r) .(5.69) The final expression for the transmission is the obtained from Eqs(5.67-5.69). In the absence of scatterer, the transmission amplitude is unity tRL mσ0,nσ =δmnδσ0σ.(5.70) In the presence of a scatterer, an additional term appears tRL mσ0,nσ =δmσ0,nσ +ARL mσ0,nσ ,(5.71) where ARL mσ0,nσ ≡Amσ0,nσ is the forward scattering amplitude. It is convenient to write Amσ0,nσ as a 2 ×2 block-matrix in the spin space. In fact we can write ˆ Am,n =−iˆ Amˆ Bn,(5.72) with ˆ Amand ˆ Bngiven by ˆ Am=−iv0ˆgmΦ∗ mˆu† m+2iαxαy ω2 0 Φ0∗ mˆu0† mˆσz, ˆ Bn=1 1 + v0⟪Gs⟫Φnˆun−2iαxαy ω2 0 ˆσzΦ0 nˆu0 n.(5.73) Here, all functions are evaluated at the position of the scatterer. We have also introduced such matrices: ˆun(x) = rm∗ e ~kn eiknxe−iˆσy m∗ eαx ~x, ˆu† n(x) = rm∗ e ~kn e−iknxeiˆσy m∗ eαx ~x, ˆvn(x) = rm∗ e ~kn e−iknxe−iˆσy m∗ eαx ~x, ˆv† n(x) = rm∗ e ~kn eiknxeiˆσy m∗ eαx ~x.(5.74) Similarly, to obtain the reflection amplitude we calculate rRL mσ0,nσ =δΨ(r) δΦR mσ0(r)Φ(r)→ΦL nσ(r) ,(5.75) 82 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE where we adopted the basis for the outgoing states on the left to be ΦR mσ0(r), i.e. to be the incident states from the right. It is important that the Green’s function is now taken for x→ −∞, which reads ˆ G(r,r0) = X nσ Φn(y)Φ∗ n(y0)|σihσ|gnσvnσ(x)v∗ nσ(x0).(5.76) In a similar way we obtain ˆrm,n =−iˆ Cmˆ Bn,(5.77) where ˆ Cm=−iv0ˆgmΦ∗ mˆv† m+2iαxαy ω2 0 Φ0∗ mˆv0† mˆσz,(5.78) and ˆ Bnis given by Eq. (5.73). 5.3 Brief discussion of the unitary of the Smatrix The problem with our approach to obtain the scattering coefficients is that being a perturbative method the unitarity of the scattering matrix is affected and ˆ S†ˆ S6= 1 but some other hermitian matrix ˆ A, such that ˆ S†ˆ S=ˆ A. One can device a method to recover unitarity. Being Hermitian, the matrix ˆ Ahas real eigenvalues and is diagonalizable via a unitary transformation, ˆ A=ˆ U†ˆ Adiag ˆ U , (5.79) where the eigenvalues of ˆ Aare the diagonal elements of ˆ Adiag and the eigenvectors of ˆ Aare the colums of ˆ U. Now this means ˆ S†ˆ S=ˆ U†ˆ Adiag ˆ U=⇒ˆ Uˆ S†ˆ U†ˆ Uˆ Sˆ U†=ˆ Adiag .(5.80) As the matrix ˆ Adiag is diagonal with real values, it is possible to write it as the square of qˆ Adiag in order to invert them on the left hand side of Eq. (5.80) and rewrite the new unitary scattering matrix, ˆ S0=ˆ Uˆ Sˆ U†ˆ Adiag−1/2.(5.81) 83 CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A NANOWIRE This implies renormalization of the scattering coefficients which can be recalculated from the above expression. In the next chapter when we present the main results we impose unitarity for the calculation of all observables. 5.4 Conclusions In this chapter we present a detail derivation of the scattering coefficients for a short-range, deltalike impurity in a nanowire with Rashba spin-orbit coupling where electrons motion is confined in the ydirection. We do so by probing the scattering states at the extremes of the nanowire via the Lippmann-Schwinger equation. The intersubband mixing arising from the interplay between Rashba spin-orbit coupling and the harmonic confinement complicates the analytical solution of the Lippmann-Schwinger equation. Our way to deal with these difficulties is by performing a Schrieffer-Wolff transformation. In this manner, we gauge away the intersubband mixing, up to second order in perturbation theory. As a result, the Green’s function in the Lippmann-Schwinger can be obtained straightforwardly. On the other hand the complexity is absorbed into the impurity potential or the eigenfunctions of the system. Consequently, the impurity, that is assumed from the beginning to be a scalar, acquires a spin-structure. From the form of the scattering coefficients we already understand that by scattering at the the impurity the electron spin may flip. Such spin-flip processes will have important consequence on the transport properties of the wire, as discussed in the next chapter, where we will use the results of the preset and previous chapters. We close the chapter describing a method to recover unitarity of the S-matrix, after the perturbative approach used in the calculation. 84 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING (a) 10 5 0 5 10 ² 0.0 0.2 0.4 0.6 0.8 1.0 f ( ² ) (b) 10 5 0 5 10 ² 0.0 0.2 0.4 0.6 0.8 1.0 f ( ² ) (c) 10 5 0 5 10 ² 0.0 0.2 0.4 0.6 0.8 1.0 f ( ² ) Figure 6.3: Fano line-shape f() as a function of the dimensionless energy parameter for: (a) q→ ∞, where the transmission occurs through the discrete state, (b) q= 1 and the transition through the discrete the discrete and the continuum of states is of equal strength with minimum at Emin = ER−Γ/2qand maximum at Emax =ER+ Γ/2q, and (c) q= 0 for the resonant symmetric line-shape. quasi-bound states was first introduced by Bagwell [23]. This discussion was later picked up by [24] and related to the sum over evanescent modes throught the Green’s function. As the evanescent mode is associated with a decaying length(corresponding to the evanescent κn), these states are not properly bound as opposed to the stable bound-state of a delta-scatterer in Eq.(2.23)), as discussed by other authors [57, 58, 59]. At this point, it is worth emphasizing the advantages of using the Lipmann-Schwinger approach. As noted by Ref. [65], many other approach the system by solving the Schr¨odinger equation through matching the wavefunctions for the modes on both sides of the impurity potential and obtaining an infinite set of coupled equations[23, 117, 118]. In order to solve such a problem one needs to truncate the system of equations, and correspondingly rescale the coupling constants. In contrast, by using the Lippmann-Schwinger equation the system is analytically solvable, as all the information for the coupling is encoded in the Green’s function of the bare nanowire. Quasi-bound states as Fano resonances As pointed out before, the presence of quasi-bound states is due to the coupling between a discrete state and the continuum of subbands available in the nanowire. This is nothing but a Fano resonance, described by the asymmetric Fano line-shape [119], f() = (+q)2 1 + 2,(6.9) 91 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING where = (E−ER)/Γ is the dimenssionless energy measured from the resonance, Γ is the resonance width and qis the asymmetry parameter introduced by Fano in his original paper[120]. By equation Eq.(6.9) we see that the minimum fmin = 0 occurs at =−qand the maximum fmax = 1 + q2at = 1/q. In the limit |q| → ∞, the transition occurs through a discrete state as the transition trough the continuum becomes very weak. This results in a Lorentzian peak of the form f()→1/(1 + 2) (see Fig. 6.3 (a)). If the asymmetry parameter is close to unity q→1, transition through both the discrete and the continuum and Eq.(6.9) leads to curves of the type shown in in 6.3 (b). Finally, for the case q→0 the Fano resonance becomes f()→q2/(1 + q2) forming a dip at E=ERwith a symmetrical lineshape (see Fig. 6.3 (c)). This last case is unique to Fano resonance and is sometimes referred in the literature as anti-resonance [119]. We can now fit our result for the conductance as expressed in Eq. (6.3) to Eq. (6.9) and obtain, the following expressions for the Fano parameters, ER=En−v2 0Im2(G)−1 v2 0Im2(G)+12v2 0Φ4 n 2,(6.10) q=±2v0Im (G) |v2 0Im2(G)−1|,(6.11) Γ = ±v3 0Im (G) Φ4 n|v2 0Im2(G)−1| v2 0Im2(G)+1 .(6.12) The resonance energy in Eq. (6.10) coincides, up to a correction factor, with Eq. (6.7). In the limit of weak impurity such factor equals unity and we recover that result. This is equivalent to a Lorentzian when q→ ∞. The possibility of destructive interference leading to asymmetric lineshapes due to disorder has been widely studied in quasi-one-dimensional waveguides [72, 58, 67, 121, 122]. But the inclusion of RSOC as a source of Fano resonances is even more interesting, either in nanowires [92, 123, 124]. 6.1.2 Effect of Rashba spin-orbit coupling in the conductance So far we have described the effect of delta-like impurity in a simple nanowire. In this section we study how the Rashba spin-orbit coupling 92 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING 0.0 0.5 1.0 1.5 2.0 E / ω 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 G / G 0 (a) 1.375 1.380 1.385 1.390 1.395 1.400 1.405 1.410 E / ω 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 G / G 0 (b) Figure 6.4: (a) Conductance in the presence of a scatterer of strenght v0=−0.9 and RSOC αx=αy= 0.2 (solid red line). The ballistic conductance is shown in black dashed lines. At the threshold energy below 1.5~ω, the transmission is perfect. Close and below the threshold the conductance exhibits a dip related to the quasi-bound state forming in the nanowire as explained in the main text. (b) A zoom into the quasi-bound state resonance. One clearly see that the transmission is not fully suppressed. 93 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING affects the nanowire conductance. The expression for the conductance is no longer as simple as the one describe in Eq. (6.3). As mentioned in the introduction to this section, in the presence Rashba spin-orbit coupling transport properties are spin-dependent. The combination of the Rashba spin-orbit coupling with the delta-like impurity leads to striking transport properties, both in the charge and spin conductances, due to spin-flip events discussed below. At first glance, the charge conductance shown in Fig.6.4(a) shows similar resonant features as those in a wire without Rashba spin-orbit coupling (see previous section and Fig.6.1): a perfect ballistic transmission at the threshold and the dips just below the threshold. However, a closer look shows important differences. One is the shift to lower energies of threshold energy given by n=~ω0(n−1/2) −m∗ e(α2 x+α2 y)/2) as a consequence of the sinking in the subband energy dispersion. As mentioned in the introduction of this chapter, the presence of Rashba spin-orbit coupling allows for different transmission mechanisms mediated by the impurity embedded in the nanowire. As a result, tranmission of an incoming electron can occur conserving the spin-alignment with probability |t↑↑|2for states prepared with spin-up (|t↓↓|2for states prepared with spin-down) or with a flip in the alignment of the spin with probability |t↑↓|2for states prepared with spin-up(|t↓↑|2). Although spin-flip transmission is much smaller than spin-conserved transmission as it depends on 2αxαy/ω2 0as shown in Eq. (6.1), it is not negligible. The relevance of this factor will become clear in Section 6.2, where we study the spin transport and its relation with a SU(2) field. As in the case where Rashba spin-orbit coupling is absent, the resonant behaviour of the conductance is dictated by the denominator in Eq. (6.1). When the energy approaches the threshold for the opening of the next subband, the dominant nature of Re (G)→ ∞ in the denominator results in a zero-probability of the spin-flip transmission. This is the same characteristic that responsible for the perfect transmission of the spin-conserved component. Consequently, the conductance at the threshold energy is 2Ndue to the lifting of spin-degeneracy (see Fig.6.4(a)). The main effect of Rashba spin-orbit coupling is the absence of the full supression of the conductance at the resonant energy, as shown in detail in Fig.6.4(b). To clarify the origin of this, we refer again to Eq. (6.1) in order 94 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING 0.4 0.6 0.8 1.0 1.2 1.4 1.6 E / ω 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 G/G 0 ballistic conductance 0 . 0 λy 0 . 5 λy 1 . 0 λy 1 . 5 λy (a) 1.30 1.35 1.40 1.45 1.50 E / ω 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 G/G 0 ballistic conductance 0 . 0 λy 0 . 5 λy 1 . 0 λy 1 . 5 λy (b) Figure 6.5: (a) Conductance for a scatterer of strenght v0=−0.9. The different colour correspond to different position of the impurity yimp. The position is given in units of the confinement length λy=p~/m∗ eω0. The chosen RSOC is αx=αy= 0.2. (b) Zoom in of the resonance for the same plots. to write the following expression for the transmission of a spin-up state, T0↑∼N−Im2(G) Im2(G)+2αxαy ω2 02(2Φ0Φ0 0)2/k0 Im2(G),(6.13) 95 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING It becomes clear from Eq. (6.13) that while the spin-conserved contribution to the transmission will be fully suppressed when there is only one propagating band, there is a finite contribution to the spin-flip transmission. We can conclude then, that the effect of the quasi-bound state in the charge conductance is strongly spin-dependent in the presence of RSOC interaction. Specifically , at the resonant energy only spin-flip transmission is allowed while spin-conserved transmission is completely suppressed. In a next section we discuss this in more detail and the consequences on the spin-dependent transport. But before that we present a systematic study of the charge conductance as a function of the impurity strength and lateral position. Dependence on the impurity position and strength As briefly mentioned above, due to the translational symmetry along x-axis, the impurity position ximp is irrelevant. However, the lateral position yimp is important. As the even lateral modes Φn(yimp) vanish at the center of the harmonic oscillator, resulting in the decoupling of the subbands as the imaginary part of the Green’s function in the denominator of Eq. (6.1) vanishes Re (G)=Φ2 n(0)/κ0= 0. This is a consequence of the symmetric states of the quantum harmonic oscillator for even modes, n= 0,2,4, .... As a consequence there is no propagating weight for the impurity potential to compensate. The lack of coupling between subbands suppresses the dip resonance. In most of the plots, when not explicitly said, it is assumed that the impurity is located at yimp 6= 0, in order to study the dip. In Fig.6.5(a) we present the conductance as a function of different lateral positions for the impurity. One can see the suppression of the resonant dip for yimp = 0, which only appears when the transverse position of the impurity breaks the mirror symmetry. We define the binding energy of the quasibound state as the energy difference between the threshold and the resonant energy EQBS =Eth −ER. Focusing into energies close to the resonance (see Fig. 6.5(b)), we can appreciate that first EQBS increases with the distance from the centre up to a certain maximum value at around yimp = 1.0λy, and then decreases for larger values. It is also worth noticing that the further away the impurity is from the center, the higher conductance minima is, implying there is a higher spin-flip transmission probability. This becomes clear from Eq. (6.13), as the spin-flip transmission depends on the derivative of the wave-function at the impurity 96 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING position Φ0 0(yimp). Similarly, in Fig. 6.6(a) we study the dependence of the quasi-bound state resonant energy with the impurity strength for a fixed position, yimp =λy. We can see in Fig. 6.6 (b) that while the binding energy of the quasi-bound state, EQBS, strongly increases with the strength of the impurity potential v0, the conductance minima weakly depends on v0. This can be understood from Eq. (6.13) since both spin-conserved and spin-flip transmission probabilities depend on the impurity strength in the same way. Effective 1D potential We have also derived an effective one-dimensional model (see Appendix C for details). Within such effective model, the effect of all higher subbands is projected onto the a single band. This results in the following transmission coefficient, tRL 00,σ0σ=δσ0σ−i m∗ e ~2v0 1+iv0m∗ e ~2 Φ2 0 k0hΦ∗ 0Φ0 k0+2αxαy ω2 0(2k0Φ∗ 0Φ0 0)ei2(k0+kR)ximp k0ˆσzi.(6.14) Comparison between Eq. (6.14) and Eq. (6.1) reveals similarities. However, in the strict 1D situation the sum over all the evanescent modes is not taken into account. This sum, which appear in the denominator of Eq. (6.1) for the quasi 1D case, is absence in Eq. (6.14). As discussed in Section 6.1.1, evanescent modes are required to form a quasi-bound state. As for the numerator in Eq. (6.1), we recover the previous result of Eq. (6.14), but of course for a single propagating band per spin species (m=n= 0) and energy E0. The absence of the quasi-bound state in the 1D model prevent the resonant behaviour to be observed, blue curve in Fig.6.7. Thus the result for the conductance obtained from Eq. (6.1) is a good approximation for the energies close to the bottom of the propagating band. However, there is a way to recover the results obtained in the quasi 1D case from the pure 1D, by adding ”by hand” in the denominator of the second term in Eq.(6.14), the contribution of the evanescent modes. By doing this one obtains an excellent agreement between the full solution and the one obtained from the 1D model, as shown in Fig.6.8. 97 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING 0.4 0.6 0.8 1.0 1.2 1.4 1.6 E / ω 0.0 0.5 1.0 1.5 2.0 2.5 G/G 0 ballistic conductance − 0 . 9 − 0 . 7 − 0 . 5 − 0 . 3 − 0 . (a) 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 E / ω 0.0 0.1 0.2 0.3 0.4 0.5 G/G 0 ballistic conductance − 0 . 9 − 0 . 7 − 0 . 5 − 0 . 3 − 0 . (b) Figure 6.6: The dependence of the conductance on the impurity strength for a nanowire with RSOC: (a) Conductance for different strength v0of the attractive scatter. We have chosen RSOC αx=αy= 0.2, and yimp = 1.0λy. (b) Zoom of the resonance for the same values. 6.2 Spin-dependent transport properties We understand from previous sections the relevant role of Rashba spin-orbit coupling on the charge transport properties of a nanowire with an impurity. In previous sections, we identified two different scattering mechanisms 98 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING 0.4 0.6 0.8 1.0 1.2 1.4 1.6 E ( ω ) 0.0 0.5 1.0 1.5 2.0 Conductance Figure 6.7: Conductance for an effective 1D model (blue) vs conductance for the full quasi-1D model (red) in units of G0for an impurity potential of strength v0=−0.9 at position yimp = 1.0λyrespect to the center of the nanowire with Rashba parameters αx=αy= 0.2. 0.4 0.6 0.8 1.0 1.2 1.4 1.6 E ( ω ) 0.0 0.5 1.0 1.5 2.0 0.6 0.8 1.0 1.2 1.4 1.6 E ( ω ) 0.00 0.02 0.04 0.06 0.08 0.10 (a) Conductance (b) T ↑ ↓ Figure 6.8: (a)Conductance and (b) spin-flip transmission for the quasi-1D model (solid red line) and for the effective 1D model (dashed blue line) after adding by hand the contribution from the evanescent modes. 99 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING mediated via the impurity potential, which are distinguished by whether the spin is conserved or flipped after scattering. The interplay between these two mechanism manifest on a modification of the quasi-bound states resonance in the charge conductance. Specifically, as shown in Fig. 6.4 and Eq. (6.13), at the resonant energy the dip in the conductance does not reach zero as was the case in for transport in the absence of Rashba spin-orbit interaction. Because this effect is due to spin-dependent processes, one expects that it has consequences on the spin transport itself. Therefore in this section we focus on the spin-dependent transport. In Fig. 6.9(a) we plot the spin-flip transmission as a function of the injection (Fermi) energy. Even though it is small in comparison to the total conductance, spin-flip transport is not negligible. We can observe two features that are closely related to the resonant characteristics discussed in Section 6.1.2. To begin with, the spin-flip transmission is exactly zero at the threshold energy where the next propagating subband in opened. This is in agreement with our conclusions in Section from Eq. (6.1), namely: the dominant nature in the denominator of the real part of the Green’s function Re (G)→ ∞ at the threshold energy results in perfect transport for the spin-conserved transmission while the spin-flip transport is completely suppressed. On the other hand, below the threshold energy the spin-flip transmission presents a significant enhancement. In Fig.6.9(b) we plot the ratio between spin-flip transmission probability and total transmission probability T↑↓/(T↑↑ +T↑↓) for an incoming spin-up state. We can observe that at certain energy below the threshold the only transport allowed is spin-flip transport. 100 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING (2αxαyσz) that appears in Eq. (6.21). The physical interpretation of this displacement is that depending on the orientation of the spin-orbit field FSO ∼(2αxαyσz) the particle will ”see” the center of the confinement in one direction or the opposite. In Fig.6.13 we illustrate the behaviour of the SU(2) field in the nanowire as an electron propagates along it. One see that an electron describing a cyclic motion on a closed path in the 2DEG (xy−plane) gives rise to a spinorbital SU(2) field FSO, while in (b) the field FSO changes sign when the path is traversed in the opposite direction. Fig.6.13(c) sketches an electron moving uniformly in the nanowire with speed vn(E) along x. The trajectory undergoes an oscillatory motion along ywith frequency ω0, as governed by size quantization due to the potential V(y). The area swept by the radiusvector of the electron during its motion is oscillating about zero average value, resulting in a consistently positive (negative) FSO for the lower (upper) half of the wire. The fluctuating field FSO, despite being zero on average, couples the electron spin to the central position y0of the electron wave function that is displaced with respect to to the center of the wire according to the orientation of the FSO field. We can see the emergence of the ∼(2αxαy) with origin in the SU(2) symmetry in Eq. (6.1). This equation directly relates the spin-flip transmission probability to the effect of SU(2) gauge. In Fig.6.14 we sketch the different transport processes that may occur for an electron traveling in the x-direction. If the electron is prepared with spinup, and its trajectory is in the lower half of the nanowire plane that results in a positive SU(2) field FSO, then the electron will ”see” the impurity potential as if it were displaced from its position by a quantity y0≈2αxαykx/ω2 0. On the other hand, if the electron is describing a trajectory in the upper half of the nanowire plane that results in a negative SU(2) field FSO, then the electron will ”see” the impurity potential as if it were displaced from its position by a quantity −y0. Upon interference of both possible tranmission paths, we pick up a phase ϕresulting in a tilt of the spin. In other words, at the resonant energy where the only allowed tranmission is through spin-flip a measurement of the conductance would serve to probe the SU(2) gauge field. 107 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING (f) ±y0 S S φ R e-iφ/2 e iφ/2 x y Powered by TCPDF (www.tcpdf.org)Powered by TCPDF (www.tcpdf.org)Powered by TCPDF (www.tcpdf.org) Figure 6.14: Transport mechanisms for an incoming spin-up state. Due to the SU(2) there is a shift in the position of the impurity that depends on spin, as a consequence the spin picks up a phase translating in a tilt with respect to the quantization axis. 6.3 Conclusions To summarize, in this the chapter we use the approaches developed in previous chapters of the thesis to study the transport properties of a nanowire with RSOC and a delta-like impurity. We first discuss the dip in the conductance that appears in the absence of Rashba spin-orbit interaction and its relation with the Fano resonance. In Subsection 6.1.2 we discuss the effect of RSOC on the transport and found a spin-flip transmission contribution as a consequence of the inter-band mixing mediated via an attractive impurity. As a consequence, we show that the presence of a quasi-bound state blocks the transmission channel that preserves spin, while it enhances the spin-flip transmission. In addition, we make a systematic study of the quasi-bound state resonant energy with respect to the lateral position and the strength of the impurity and recovering a result from the Chapter 4 that depends entirely on the spin-flip transmission, namely the torque. While there are effects of Rashba spin-orbit interaction of relevance in the charge conductance, our key result consists in finding the underlying relation between the spin-flip transmission and the SU(2) field. At the resonant energy, the only 108 CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS OF THE RASHBA SPIN-ORBIT COUPLING transmission allowed is through spin-flip permitting us to connect a physical quantity with the corresponding SU(2) symmetry. 109 Chapter 7 Conclusions The goal of this thesis is to formulate a theoretical model to study analytically electronic transport in quantum nanowires in the presence os Rashba spin-orbit interaction. The topic may have direct impact on the field of semiconducting and superconducting spintronics, and Majorana fermions. In general terms we have shown that the Rashba interaction in combination with an impurity and confinement potential affects drastically the charge and spin transport. At certain energies we found a hitherto unknown spin-flip transmission which is related to the appearance of a spin-orbit torque. Our theoretical results can be used in multiple ways to further studies on transport properties of confined systems in the presence of spin-dependent fields and impurities. In the introductory Chapters 1 and 2, we describe the motivation behind our work and our goal of studying the properties of scattering from an impurity in a quasi-1D semiconducting nanowire with intrinsic spin-orbit coupling of Rashba type. We introduce the Lippmann-Schwinger equation, a useful method for the theoretical description of quantum scattering. In particular, we use this method to describe how the electronic transport is affected by the presence of the impurity in the nanowire. We first describe the approach in a general 3D system and later focus on scattering on a delta-potential in a purely 1D system, discussing the limitations of the Born approximation. Finally we focus on a more realistic nanowire described by a transverse confining potential, discussing the emergence of resonant behavior in the transmission as a consequence of quasi-bound states present in the nanowire. This effect arises from the localized impurity coupling the evanescent and propagating modes of the 110 CHAPTER 7. CONCLUSIONS nanowire. We focus the possible effects arising from the interplay between the Rashba interaction and the quasi-bound states. Because the Rashba spin-orbit interaction plays a central role in this thesis, in Chapter 3 we provide an introduction in this topic. Specifically, we briefly discuss the spectral properties of a 2DEG with Rashba spin-orbit coupling before reviewing the complexities involved in the analytical solution of the model Hamiltonian for a quantum nanowire where the 2DEG is further confined. As a result of this confinement, the energy spectrum is strongly affected,as well as the spin polarization. The combined effect of RSOC and confinement gives raise to anti-crossings between branches of opposite spin and different band index, deforming the spectrum.These are a consequence of the subband mixing that couples propagating and evanescent states. We conclude that to address this problem enough subbands have to been taken into account. At the end of the chapter, we present an exact solution for the kx= 0 that we use later as the zeroth order solution in our perturbation around the point kx= 0. This calculation allows us to ensure the accuracy of our perturbative approach up to α2in Chapter 5. In Chapter 4 we extend the widely used Landauer-B¨uttiker formalism to include the effect of the spin-orbit coupling for the description of the transport properties of the nanowire. We derive an expression for the spin current polarized along y-direction in the left lead considered to be far away from the impurity. This expression allow us to express such spin current as a function of the scattering coefficients of the S-matrix. Within this representation it is easy to track the contributions from the voltage and spin-biases. This result in combination with the expression for the spin current allowed us to describe the torque that arises at the impurity position, as a consequence of the spin-flip transport mechanisms resulting from the Rashba spin-orbit coupling. These mechanisms have important consequences on the conductance as discussed in subsequent chapters. Furthermore, we make a connection between our expressions and those obtained in the context of the spin-mixing conductance in magnetic hybrid structures. In Chapter 5 we present a detailed derivation of the scattering coefficients for a short-range, deltalike impurity in a nanowire with Rashba spin-orbit coupling where electrons motion is confined in the y direction. We do this with the help of the Lippmann-Schwinger equation introduced in Chapter 2. The intersubband mixing, arising from the interplay between Rashba spin-orbit coupling and the harmonic 111 CHAPTER 7. CONCLUSIONS confinement, complicates the analytical solution of the equation. However, we deal with these difficulties by performing a Schrieffer-Wolff transformation. In this manner, we gauge away the intersubband mixing, up to second order in perturbation theory. As a result, the Green’s function in the Lippmann-Schwinger can be obtained straightforwardly. On the other hand, the complexity is absorbed into the impurity potential or the eigenfunctions of the system. Consequently, the impurity, that is assumed from the beginning to be a scalar, acquires a spin-structure. From the form of the scattering coefficients we already understand that by scattering at the the impurity the electron spin may flip. Such spin-flip processes have important consequences on the transport properties of the wire, as discussed in the next chapter, where we use the results of Chapter 5 and previous chapters. We close the Chapter by describing a method to recover unitarity of the S-matrix within the perturbative approach used in the calculation. Finally, we present the transport results in Chapter 6. Specifically, we use the approaches developed in previous chapters to study the transport properties of a nanowire with RSOC and a delta-like impurity. We first discussed the dip in the conductance that appears by scattering at a delta-like impurity in a nanowire in the absence of Rashba spin-orbit interaction and its relation with the Fano resonance. In Subsection 6.1.2 we discuss the effect of RSOC on the transport and found a spin-flip transmission contribution as a consequence of the inter-band mixing mediated via an attractive impurity. We show that the presence of a quasi-bound state blocks the transmission channel that preserves spin, while it enhances the spin-flip transmission. In addition, we make a systematic study of the quasi-bound state resonant energy with respect to the lateral position and the strength of the impurity. Another key result is the underlying relation between the spin-flip transmission and the SU(2) field. At the resonant energy, the only transmission allowed is through spin-flip permitting us to connect a physical quantity with the corresponding SU(2) symmetry. Besides the effects discussed in this thesis, the methods developed here can be used in future research. We envision a possible application of the results from Chapter 6, namely the enhancement of spin-flip transmission, in the spirit of the Datta-Das spin-transistor introduced in Chapter 3. Indeed, one can design a device based on a nanowire in which one can externally tune the strength of the impurity potential and so, control the 112 CHAPTER 7. CONCLUSIONS spin-flip probability of injected electrons from a ferromagnetic lead. By tuning the chemical potential at the resonant energy, we can ensure that the spin of transmitted electrons is flipped. This application is in the spirit of previous works that extend the spin filter model to take advantage of Fano resonances in quantum dots [130] or side rings coupled to nanowires [131]. Moreover, the use these resonances has been already proposed in spin inversion devices based in semiconducting lattices with spin orbit-interaction and magnetic fields [132]. Furthermore, we can think of other ways to extend the study of disorder in nanowires. For example, placing a second defect and studying the possible spin-dependent transmission. In principle our methods, based on the scattering matrix, can be extended straightforwardly to two and more impurities. One further perspective of the present work is its extension to include superconductivity and a Zeeman field and see what are the effects in the context of Majorana physics. Taking these two ingredients into account will be the next step in the theoretical approach to the scattering problem proposed in this thesis. Adding Zeeman to the Hamiltonian described in Eq.(5.1) would imply further work since the Greens function will be spin-dependent even in the absence of the impurity and additional terms in the Schriffer-Wolff transformation will appear. Moreover, introducing superconductivity require enlargement of the space to include the Nambu structure. Another possible extension of our results, is the study of the Josephson current in a nanowire attached to two superconducting reservoirs. The Josephson current is an equilibrium current that can be determined from the knowledge of the subgap spectrum, Andreev bound sates . One can address the question how the quasi-bound states affects such spectrum and hence the Josephson current. According to Beennakker theory[133], the transport properties of such junction can be fully determine by the knowledge of the scattering matrix scattering matrix, that we know from our analysis. 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Giazotto, “The ω-squipt as a tool to phase-engineer josephson topological materials,” Nature nanotechnology, vol. 11, no. 12, p. 1055, 2016. 128 Appendix A 1D Green’s function For the resolution of the Lippmann-Schwinger equation for a 1D system, Eq.(2.16), we will need the Green’s function for the Helmholtz equation, ∇2+k2G±(x,x0) = 2m∗ e ~2δ(x,x0).(A.1) This requires the evaluation of, G±(x,x0) = hx|1 E−H0±i|x0i.(A.2) Then, G±(x,x0) = Z∞ −∞ dp 2π~hx|1 E−H0±i|pihp|x0i =Z∞ −∞ dK 2π eipx/~ p2 2m∗ e−(E±i)e−ipx0/~ =1 2π 2m∗ e ~2Z∞ −∞ dKeiK(x−x0) K2−2m∗ e ~2(E±i),(A.3) with poles given by K=±kr1±i2m ~2k2≃ ±k1±im∗ e ~k.(A.4) The problem then can be solved using Cauchy’s integral formula, IC f(z)dz = 2πi×(sum of residues enclosed by C) ,(A.5) 129 APPENDIX A. 1D GREEN’S FUNCTION where Cis the contour defining the path of integration, taken counter-clockwise for the upper half-plane and clockwise for the lower half-plane. For E≥0 and taking into account Eq.(A.5) for the poles in Eq. (A.4) in Eq.(A.3), then G±(x,x0) = 2m∗ e ~2 i 2k{∓e±ik(x−x0)|x>x0∓e∓ik(x−x0)|x<x0} =∓2m∗ e ~2 i 2ke±ik|x−x0|.(A.6) The ±in the Green’s function G±(x,x0) corresponds to the incoming (−) or outgoing (+) boundary conditions, which means taht for the positive exponent we close in the upper half-plane and include the pole (+k+ i) and for the negative exponent we close in the lower half-pane and include (−k−i). 130 Appendix B Ensuring Kramers reversivility The introduction of the notations ⟪Gs⟫and ⟪∂2Ga⟫in Chapter 5, led us to the following expression for the bound-states, 1 + v0⟪Gs⟫±2αxαy ω2 0 ⟪∂2Ga⟫= 0 .(B.1) According to the Kramers theorem, the two solutions obtained from this equation (for the ±sign) must be degenerate. Therefore, we strongly suspect that ⟪∂2Ga⟫has to vanish. In Chapter 5 we introduce the short notations: ⟪Gs⟫=1 v0ZGs(r0,r0)Vimp(r0)d2r0,(B.2) ⟪∂2Ga⟫=1 v0Z[∂x(∂y0−∂y)Ga(r,r0)]|r=r0Vimp(r0)d2r0.(B.3) (B.4) Let us model the scatterer potential as Vimp(x, y) = V0e−(x−x0)2 2σ2e−(y−y0)2 2σ2 =v0δσ(x−x0)δσ(y−y0),(B.5) where v0= 2πσ2V0is the strenght of the impurity, V0is the height and we define the δ-like potential as δσ(x) := 1 √2πσe−x2 2σ2.(B.6) 131 APPENDIX B. ENSURING KRAMERS REVERSIVILITY The support of the integral, Eq.(B.2), is centered around r0=r0within a circle of radius a∼σ. The following integral is useful Z+∞ −∞ dx0eikn|x0−x0|δσ(x0−x0) = e−1 2k2 nσ21 + erf iknσ √2.(B.7) Since erf(0) = 0 and the exponential tends to 1, we have this integral to be 1 in the limit σ→0. Another useful integral is Z+∞ −∞ dy0Φ∗ n(y0)δσ(y0−y0) = 1 √2nn!√πλ qλ2 λ2+σ2λ2−σ2 λ2+σ2n/2 ×Hny0λ √λ4−σ4exp h−y2 0 2(λ2+σ2)i.(B.8) This integral in the limit σ→0 is equal to the transversal wavefunction at the impurity position Φn(y0). As a result, from Eqs.(B.7) and (B.8) in the point-like limit, Eq. (B.2) becomes, ⟪Gs⟫= Φ∗ n(y0) Φn(y0).(B.9) For the calculation of Eq.(B.3) it is sufficient to focus on the y0integral. Taking into account that the partial derivative of Gaas defined in (5.54) is, ∂y0Ga=1 p2nn!√piλ 2n λHn−1(y0/λ)−y0 λ2Hn(y0/λ)e−y02/λ2Φn(y), (B.10) we can write the following integral for the y0dependence of Eq.(B.3), Z∞ −∞ dy0(∂y0−∂y)Gaδσ(y−y0) = e−y2 0/2(λ2+σ2)Φn(y0) p2nn!√piλ ×(2n λ2+σ2λ2−σ2 λ2+σ2n−1/2 Hn−1λy0 √λ4−σ4 −σ2 λ2+σ2λ2−σ2 λ2+σ2n−1/2 Hn−1λy0 √λ4−σ4 −y0 1 λ2+σ2rλ2 λ2+σ2λ2−σ2 λ2+σ2n/2 Hnλy0 √λ4−σ4) (B.11) 132