]> Coherent Spin Ratchets: Transport and Noise Properties

Coherent Spin Ratchets

Transport and Noise Properties

Diplomarbeit von Manuel Strehl aus Regensburg

durchgeführt am Institut für Theoretische Physik der Universität Regensburg unter Anleitung von Prof. Dr. Klaus Richter

Juli 2007

Preface

Throughout the 20th century computers started to change the way we work. In the last two decades they invaded private life as well, supporting us with completely new techniques for communication, design and time management. Just recently mobile phones and WLAN enabled notebooks add the possibility to use these features anywhere.

The key to those applications is the miniaturization of the underlying technology. Faster processors have to fit into smaller cases. Yet this procedure of minimizing the space of a component cannot be extended arbitrarily. Nowadays processors and circuits are mainly based on the properties of doped semiconductors. But inserting alien atoms in a structure, that is just a few atom widths broad, leads to regions, where dopants are only present by chance. Additionally thermodynamical and quantum mechanical effects start to play a significant role [1]. Moore’s law, the statement, that the complexity of integrated circuits will double about every 2 years, faces the end of its validity, if constricted to “classical” electronic processors. The effect finds its echo even in the planning of scientific projects. Researchers, who scheduled the processing of experimental output or numerical simulations in terms of computing power to be expected [2], could have to reformulate their assumptions.

Alternatively to the common production schemes additional components of a system could be used to transfer information. The spin is an unused intrinsic property of electrons in today’s semiconductor-based microelectronics. Therefore introducing spin-selective systems can allow transmission of more information than just the presence of a particle [3]. This idea has lead to the usage of the Giant Magneto-Resistance (GMR, [4]) to be used in ferromagnetic systems. With this combination the interplay of the spin with the applied magnetic fields allows for very small read heads for hard discs as well as non-volatile computer memory, the MRAM (Magnetoresistive Random Access Memory).

The usage of spin as information carrier in electronics has been introduced theoretically and experimentally under the term Spintronics [56]. These “spin-based electronics” have also a strong focus on semiconductors, which has the advantage of relying on the experience in manufacturing miniaturized structures, while on the other side theoretical models for electronic transport in these systems are well-founded.

The spin degree of freedom was first introduced in 1924 by Kronig and Pauli [7] in the context of the emission spectrum of alkali metals. In 1928 Paul Dirac consolidated the explanation for the observed effects with the introduction of special relativity to quantum mechanics. 60 years later the above mentioned GMR was discovered and found a rapid application in micro-electronics. The spin is used there by affecting the magnetic properties of solid state materials, ideal for implementation as memory devices. But the processing units of computers are still based solely on the charge transport of electrons.

Spin polarized current or even pure spin current inside semiconductors therefore attracts more and more attention. The possibility to process information with little or no charging effects but an accumulation of spin-polarized particles is a central target for many research projects. In this context Datta and Das suggested in 1990 a spin field-effect transistor (SFET, [8]). This spintronic device acts like a conventional transistor, that is, it tunes the carrier-concentration between a source and a drain contact by an applied gate voltage. But opposed to, e.g., MOSFETs an SFET utilizes the variation of the Rashba spin-orbit (SO) coupling in asymmetrically grown semiconductor heterostructures (see for example [9]) to achieve this effect. The electrons inside the device move in a two-dimensional electron gas (2DEG), where the Rashba effect is active. The effect can be understood in analogy to the electro-optic analyzer. By rotating polarized light inside an electro-optic material the following analyzer will measure a lower intensity. Because the tuning of the SO coupling with a gate voltage does not imply the charging of the quantum well [10], the SFET is predicted to work much faster than conventional field-effect transistors.

Many spintronic devices have been proposed since then, for example spin LEDs (light emitting diode), or spin RTDs (resonant tunneling device). Although none of them has reached industrial maturation so far, theoretical and experimental results are promissing to introduce new facilities to nowadays information procession. However, the generation or injection of spin current into semiconductor heterostructures is still a challenging task. The most obvious method, to inject a polarized current from a ferromagnetic contact, yields a very low efficiency due to the conductivity mismatch between the metal and the semiconductor at the interface. Alternatives involve tunnel injections or injections via ferromagnetic semiconductors [11].

Another attempt is the generation of the spin current inside the heterostructure. This could be accomplished by polarized laser pulses [121314], but a more direct approach would aim to create the polarization directly in the active region. The attention turns back to the proposed SFET and to the Rashba spin-orbit coupling inside semiconductors.

Following the line of thought for ratchets, that found recently high interest [1516], we focus our attention in this thesis on the generation of pure spin current directly inside a semiconductor heterostructure. The ratchet mechanism has theoretically proven to introduce spin polarization in systems with applied magnetic field [17] as well as with Rashba coupling [18]. We will concentrate on spin-orbit based devices and examine their properties in view of experimental realizations.

In chapter 1 an introduction into the state-of-the-art quantum ratchets is given. The possibility to extract useful work out of unbiased fluctuations and the intriguing application of the second law of thermodynamics are outlined and the idea of spin current ratchets is presented in more detail. The system in question could be realized by patterning a wire in a 2DEG and connecting it to two electron reservoirs. The transport in this area is assumed to be fully ballistic. Then the current through the device is related to the transmission probabilities, known as the Landauer-Büttiker theory [19]. We will introduce the current and the theory of scattering matrices in chapter 2.

In this chapter we also define the spin current as the property of interest. We will especially highlight the properties of this defined quantity, since it is not conserved as in the case of charge current. Finally, the spin-orbit coupling present in the 2DEG will be examined. The two contributions found by Rashba and Dresselhaus are considered and their interaction will be explained. The Rashba effect will be investigated closer, and the spin-dependent channel mixing between the discrete transmission modes of the constricted device will turn out to be a central requirement for the rectification.

Before we present results for the ratchet in chapter 4, the mechanism for the numerical simulation is presented in chapter 3. Inside the typical approximations the Lattice Green’s function method will be introduced and applied to the scattering scenery in the proposed device. The underlying approximation of a 2DEG in the tight-binding discretization approach has been used successfully in a wide range of physical problems [20].

We start the presentation of the results with the description of the system and an explanation of its properties using the Landau-Zener expression for a probability of channel transition. This ansatz is included in an intuitive picture describing the dynamics of particles passing the ratchet device. The results for the simple case of two open transverse modes are promissing. For future applications we extend the system by including more transmission channels and respecting the Dresselhaus spin-orbit coupling. We will point out the connection between the orientation of the underlying crystal lattice and the direction of motion of electrons in our device, leading to amplification or reduction of the observed spin polarization.

Chapter 5 will focus on spin-current fluctuations in the system. Going beyond the observation of average currents we expect to find new sources of information regarding the working of the ratchet device. The spin polarization of electrons finds a reproduction in the difference of the auto- and cross-correlation functions related to the two leads connecting to the ratchet.

Lastly we will outline the results of this work and give an overview about the direction of follow-up research.

In this last year of my studies I deeply appreciate the possibility to work at the chair of Prof. Dr. Klaus Richter. I want to thank him especially for the time he spent for me patiently despite his over-full schedule. It is the credit of Prof. Richter and Angela Reisser to establish such a harmonic environment, where the scientific exchange as well as the personal contact between the members of the chair can prosper.

A very special thank goes to Dr. Dario Bercioux, who was an invaluable teacher for me and ever optimistic supporter of my work. Although moving to Freiburg the successful completion of this thesis is due to his continued hints and advises. For the times, when I had to talk to a person face-to-face, I have to thank Matthias Scheid for his willingness to explain the physics and to bear with my problems.

I benefitted very much from the discussions with Andreas Laßl, Daniel Waltner, Michael Wimmer and Michael Hartung. The notes and comments of Andreas and Daniel to the results on shot noise were a very important source for my understanding in this area, and both Michaels spent a lot of time looking after my computation issues.

The inhabitants of the “Großraumbüro” have turned out to be every single one very pleasant people. I want to thank Marcus Bonança and the first generation of diplomates, Mirjam Schmid and Thomas Ernst, for their support in my first months as well as my successors Thilo Maurer, Martin Hetzenegger, Gabriel Niebler, Dominik Bauernfeind and Viktor Krückl for the nice time and the necessary distraction. Be it during hot summer days or deep in the night, I very much enjoyed having them around.

My parents supported me throughout my studies in every possible way. I want to express my gratitude to them for assuring me whenever I needed it most.

Finally I would not have been able to work with this enthusiasm and endurance, if not my fiancée Daniela Daum endorsed me in every situation. Thank you!

Regensburg, July 2007
Manuel Strehl

Contents

  1. Introduction to Ratchets
    1. Principles
      1. Brownian Motors
      2. Quantum Ratchets
    2. The Idea of Spin Ratchets
  2. Transport in Quantum Wires
    1. Transport and Currents
      1. The Scattering Matrix
      2. Definition of Currents
    2. Spin Currents
      1. Spin-Resolved Transmission
      2. Properties of the Spin Current
    3. Spin-Orbit Interaction
      1. Origins of Spin-Orbit Coupling
      2. The Weak Coupling Limit
      3. The Dresselhaus Effect
  3. Numerical Methods
    1. Composition of the Wire
      1. Description with Green’s Functions
    2. Tight-Binding Approach
      1. Splitting and Combining Green’s Functions
      2. Lattice Green’s Functions in a Clean Wire
      3. Group Velocities
    3. Scattering in the Wire
    4. Voltage Offset
  4. Results for a Multi-Mode Wire
    1. The Landau-Zener Ansatz
      1. Currents and Measurements in the Ratchet
      2. The Model System
      3. Level Crossings
      4. Numerical Results for the Simple Case
      5. The Transition Probability
      6. Spin Current Production
    2. Extending Landau-Zener to Multiple Modes
      1. Remaining Validity of the Landau-Zener Ansatz
      2. Higher Modes
    3. Introducing Dresselhaus Spin-Orbit Coupling
  5. Fluctuations and Shot Noise
    1. Introduction to Noise
    2. Shot Noise for Spin Currents
      1. Auto- and Cross-Correlation
      2. Properties of the Correlations
    3. Results for the Spin Ratchet
  6. Conclusions
    1. Summary
    2. Perspectives
  7. Bibliography

Introduction to Ratchets

The generation of spin-polarized current in semiconductor-based structures is a hot topic in mesoscopic physics. Especially the proposals and measurements of pure spin-current without underlying net particle motion arouse interest, since it focusses the advantages of this additional degree of freedom: increased data processing speed, decreased electric power consumption and increased integration densities compared with conventional semiconductor devices [621]. This can be achieved by optical generation and injection in the structure. However, a direct creation of spin current inside the semiconductor promisses, e.g., smaller devices.

Two processes get into the focus to carry out this task. Quantum pumping on one side creates a DC current by variation of two or more independent parameters, e.g., a gate voltage or an external magnetic field [22]. Most devices rely on the Coulomb blockade in quantum dots, small metal or semiconductor islands, that are connected to the outer system with point contacts. These set-ups showed in experimental realizations indeed pumping of DC charge as well as spin current [232425].

A different approach is characterized by the variation of only one parameter. To obtain again a situation, where current is generated, the system has to meet additional requirements, which we will point out in the following section. This mechanism, refered to as “ratchet”, is the basic pinciple, upon which we lay our interest in the course of this work.

Principles

T T 1 2 The “Smoluchowski-Feynman-Ratchet”. Particles, that move randomly in reservoir T1, should be the source of useful work, when controlled by a ratchet in T2.

Ratchets have found a broad interest in the last years [1516]. In general, a ratchet is a device that extracts usefull work out of unbiased fluctuations. Macroscopic ratchets belong to our everyday’s experience. From windmills to socket wrenches we use the effect in many applications.

In the first half of the 20th century it was considered, whether microscopic ratchets could bypass the second law of thermodynamics. In 1912 Smoluchowski [26] and later in the 1960s Feynman [27] examined a system, where gas particles in thermal motion hit a paddle. Connected to a pawl, this device should act as a rectifier for the randomly distributed momenta. Figure 1.1 shows a scheme of this system.

The figure reveals the first essential requirement for a ratchet mechanism. The spatial symmetry must be broken by an asymmetric potential to generate a directed motion. We can see this in the reservoir T2, where the saw tooth potential of the cogwheel provides this feature. Then the idea follows the guideline, that particles hitting the paddle from one direction move the axis, while the cogwheel blocks momenta from the opposite paddle site.

A closer inspection shows the flaw of this set-up. With increasing miniaturization all energy scales become comparable to the thermal fluctuations in the system. This includes the potential barriers of the ratchet saw teeth and deformation energies in the resetting spring. The downsized ratchet will not work. Indeed, this was proven experimentally [28] by using single triptycene[4]helycene molecules as ratchet and NMR techniques for detection.

This finding allows a differentiate view on Brownian motion. The thermal noise has an essentially different nature than the random non-equilibrium fluctuations rectified in macroscopic ratchets. It is related to the temperature, which is a quantity defined for an equilibrium situation. To achieve the effect of rectification we have to introduce an additional driving out of this equilibrium state.

Ratchets can be classified by the way, how this happens. Rocking or tilting ratchets are generated by a periodical offset of the ratchet potential, that “skips” the device. If on the other hand the potential is switched on and off, one speaks of flashing ratchets. The effect of a ratchet under these circumstances is, that a net motion occurs, even when the driving is unbiased and would otherwise not lead to an overall current.

We can therefore summarize the two main ingredients for a ratchet mechanism: broken spatial symmetry and unbiased external driving. However, the direction and amplitude of the created current depend strongly on the system’s variables. Therefore it is a sensitive task to determine these features, which might even reverse on changing parameters.

Brownian Motors

We examine dissipative systems, that are connected to a heat bath. In this connection classical ratchets are often called Brownian motors, and the mechanism can be found e.g. in biochemistry, where it governs intracellular transport [1529].

We use Newton’s equation of motion in one dimension to describe a particle in the ratchet,

mt + ηt = Vx + Ft + ξt ,

where the ratchet potential is modelled in Vx and Ft is the tilting force. The parameters η and ξt refer to the effects of the thermal environment, the friction (with coefficient η) and randomly fluctuating forces equivalent to thermal noise. The fluctuations are unbiased, i.e., ξ t = 0, and connected to the friction coefficient in terms of the fluctuation-dissipation theorem,

ξt ξ(t) = 2ηkB Tδ(tt) .

This typically very small system can be described as overdamped, that is, mt can be neglected. If the external driving Ft is missing, the net current for an ensemble of particles,

= lim t0 xt x(0) t ,

vanishes. This can be shown solving the so-called Fokker-Planck equation for the probability density of the ensemble above, which can take the form of a continuity equation for the probability. A detailed calculation can be found in ref. [15].

This result is quite counterintuitive, since the breaking of the spatial symmetry does not lead to a preferred current direction, as it is in macroscopic devices. Then, introducing a small, still microscopic tilting Ft, the ratchet effect shows up. The second law of thermodynamics is no longer applicable, because here the system is out of the thermal equilibrium.

We can formulate this finding in Curie’s principle, namely that if a certain phenomenon is not ruled out by symmetries, then it will occur [30]. In the case of the microscopic ratchet in equilibrium, the macroscopically observable current rectification is suppressed by the so-called detailed ballance symmetry, the condition for a system to be in thermal equilibrium.

Quantum Ratchets

We have introduced small ratchets, even minimized to a single molecule. The next consequent step is to take into account quantum effects, that arise in this regime. Here we have to distinguish two situations. The one, which could be called quantum Brownian motor, is in principle a reformulation of the ratchet above in terms of quantum mechanics. The motion will take part in a dissipative environment [31].

The second system is sized in a way, that quantum effects become relevant, e.g., the particle propagation is truly ballistic [323334]. These coherent ratchets are set up in an environment without thermal noise, which is an essential part of the dissipative ratchet’s system. We will first take a look at the generalization of classical ratchets to dissipative quantum ratchets and then come back to coherent systems, which build the basis for the device proposed in this work.

We consider the Hamiltonian

Ĥt = p̂2 2m + V(x̂) x̂Ft + ĤB(x̂,q) ,

from which equation (1.1) follows as exact Heisenberg equation for the coordinate operator x̂t [35]. The friction and thermal noise turn into ĤB(x̂,q), where the heat bath is modelled as an ensemble of harmonic oscillators q. This model can be adapted, since it is not possible to tell the difference between a set of harmonic oscillators and the actual heat bath, viewing from the reduced dynamics of the small system xt.

Classical ratchets rely on the effect of the saw-tooth potential. Entering quantum mechanical realms, additionally tunneling probabilities have to be considered. The competition between these two influences leads to a characteristic crossover temperature TC, under which the tunneling takes over, while transport is dominated by noice-induced diffusion for temperatures above TC.

The tunneling itself can introduce a ratchet effect. By rocking the asymmetric potential the barrier can become narrower and wider, respectively, in the two situations, leading to favourized or suppressed tunneling probabilities. Thus the width of a ratchet barrier enters the ratchet mehanism additionally to the still present classical parameter of the barrier height. This effect has however a different dependency on the external parameters, so around TC even a reversal of the net current can be expected. Also for T0 the ratchet still produces current. This result can not be understood in terms of classical dissipative systems.

PIC Scanning electron micrograph of an array of triangular quantum dots. The array, consisting of 10 dots etched from a GaAs/AlGaAs semiconductor heterostructure, was used in ref. [36] as ratchet. Picture taken from [15].

Most of the predicted properties of quantum ratchets could be observed in experiments [373839]. A key experiment was published in 1999 by H. Linke et al. [36]. This ratchet works fully in the coherent regime and shows very nicely the predicted behaviour.

The setup includes a central region, where ballistic transport takes place, and a connection to two external electronic reservoirs. The driving is realized by applying an unbiased, periodically varying voltage between those two reservoirs.

The ballistic region is patterned like depicted in fig. 1.2 introducing the breaking of spatial symmetry in the direction of motion for the particles. The electron motion was restricted to a two-dimensional electron gas (2DEG) in a GaAs/AlGaAs heterostructure. The lateral confinement creates an effective ratchet potential for the coherent particle dynamics.

PIC Theoretical model for the experiment related to fig. 1.2. Δt (left axis, bold curve) is the difference between the transmission functions for positive and negative voltages. The Fermi window Δf defines the region for electrons contributing to the current. The net current is given by the integrated product ΔtE ΔfE. Taken from [36].

Upon driving this system with a square wave AC voltage of amplitude V0 (typically 1mV), using a switching period, that is long compared to any adjusting times in the system, the net DC current is given by

IV0 = 1 2 I+V0+IV0 .

Indeed it was found, that at a temperature TC current reversal appears. This finding aligns with the theoretical results, namely, that there is a non-zero current even for T0 and the existence of an inversion point for the net current. An intuitive theoretical approach to explain this behaviour inside the Landauer-Büttiker transport theory is shown in fig. 1.3. The particles, that can contribute to the current, have to be inside the Fermi window, that is opened up by the two Fermi distributions of the source and the drain, respectively. Then the net DC current is gained by integrating the product of the transmission probability tE and ΔfE.

The rocking ratchet introduced here acts essentially as a non-linear rectifier. To explain its behaviour one has to go beyond the linear response of the system, that is characterized by the linear part of the conductance G0=limV0IVV. For this case, the currents in the two rocking situations would cancel out, I+V=IV, so that the average current can only be different from zero, if higher order terms are respected,

IV = G0V + G1V2 + G2V3 +

A nice usage of the ratchet 1.2 involves the interpretation, that here “hot” and “cold” particles move in different directions. So the ratchet, operated at TC, can be used for “cooling”, that is, separating particles depending on their kinetic energy [40].

The Idea of Spin Ratchets

As mentioned above the generation of spin current in semiconductor structures yields still a challenge in the development of spintronic devices. The idea to apply a ratchet mechanism based on the above considerations for spin polarization is therefore an obvious aim. One possible set-up can be realized by applying asymmetric magnetic fields [1741]. The Zeeman splitting leads for spin-up and -down electrons to different effective potentials, which can be tailored to achieve net spin currents without corresponding charge currents.

The alternative principle, that guides the course of this work, is based on researches into spin-transport properties in quantum wires [4243], that show Rashba spin-orbit coupling. The combination of rocking the system by means of an external applied voltage and the spin selection from the SO coupling has already shown its potential to generate spin-resolved current [18].

In chapter 4 we will further study this system and concentrate on the properties of single conducting channels inside the coherent central region. Thermal noise will diminish the effect of the ratchet, but shot noise can be a valuable source of information about the mechanism. We will investigate this in chapter 5.

Both setups, the Zeeman ratchet based on magnetic fields as well as the system with Rashba spin-orbit coupling, can be tuned in a way, that charge current vanishes and the rectification effect is only present for spin current.

Transport in Quantum Wires

The term “quantum wires” describes a system, where particle motion is restricted in two dimensions. Most commonly, quantum wires are modelled of a two-dimensional electron gas (2DEG) by a constricting potential. 2DEGs on the other hand appear at the transition between different semiconductors, where the bandstructures are bent to fit the Fermi energy inside the materials. Typical examples for the appearance of 2DEGs are the inversion layers in Silicium MOSFETs (Metal-Oxide-Semiconductor FieldEffect Transistors) or at the interface in GaAs-AlGaAs heterostructures [44].

The constriction for a quantum wire is realized, e.g., by epitactical growth on non-planar substrates so that the wire is formed inside a lithographically created V-shaped trench [45]. Here, additional quantum effects come into play, such as the 1D density of states. If scattering from phonons can be neglected at low temperatures, the phase coherence length of the electrons has the order of the system length or even outreaches it. So, scattering of the charge carriers in semiconductor heterostructures can be controlled mainly by manipulable barriers introduced for example by gate voltages. Due to scattering at impurities this is not true for metals.

Today, the possibilities to create ultra-pure and small semiconductors are highly developed. Indeed the engineering of structures based on doped semiconductors enters a regime, where even materials with a high concentration of impurities contain a dopant only by chance. To further develop electronic circuits based on this technique one has to control individual electrons.

Additionally, one could start using other channels to transport information. Investigating features of a quantum system to serve this task, one immediately finds the spin of the electrons, that could be manipulated. This is the basic idea in the physical field that lately was called “Spintronics” [65].

In this chapter we will introduce in transport theory in quantum wires and the concept of scattering matrices and generalise the results for spin resolved measurements. The current as important quantity will be derived and peculiarities of spin current will be outlined. Finally, transport is modified by various properties of the wire. We will concentrate on the spin-orbit like Rashba and Dresselhaus effects.

Transport and Currents

For the considerations in this chapter we will look at a two-dimensional electron gas (2DEG), that is confined in one direction to form a wire. The 2DEG will be oriented in the xz-plane and the constriction will be towards the z-direction, leaving the x-direction for free particle motion. In the simplest case, this wire will be connected to some outer reservoirs by two leads, which will be assumed as semi-infinite and reflectionless. A more detailed view of this set-up will be given in chapter 3. For the moment, we will use a generic Hamiltonian of the form

Ĥ0 = px2 2m + pz2 2m + U(x,z) ,

where U(x,z) contains the constriction in z-direction and an arbitrary potential landscape in x-direction.

The Scattering Matrix

When considering transport through a quantum system, one has to look at the wave packets moving in the system. In the sketched system we can simply distinguish them in ingoing, that is, stemming from one of the leads, and outgoing, pointing towards the leads, waves. The amplitudes will be denoted as a for incoming and b for outgoing direction. In this representation they are vectors with two entries per transverse mode in each lead, since the motion in z-direction is restricted by the confinement and we respect the spin orientation. The z-depending part of a single wave can be expressed as a transverse wave function χn based on the shape of the constriction, that adds quantized terms to the particle energy. Relying on the principle of superposition we can then compose a resulting wave packet in one of the leads:

ΨLx = nσL anσ χnσ ψnσ+ + nσL bnσ χnσ ψnσ ,

where ψLnσ± is the wave function in ±x-direction and σ denotes a spin state. In section 2.1.2 ψLnσ± will be introduced accurately. We will interpret the result above in terms of second quantization. Then equation (2.2) does not represent a wave packet with complex amplitudes aαn but an operator Ψ̂ acting on a Fock space [46]. The Fock space is a direct sum over all possible N-particle-Hilbert spaces

Fock = (0) (1) N ,

with N a particle number. For this task the amplitudes aαn have to be replaced by operators âαn, that annihilate carriers in the incoming lead. The same applies to the amplitudes bαn. For both the corresponding creation operators âαn and b̂αn can be defined:

âLnE b̂LnE annihilation operators
âLnE b̂LnE creation operators

These operators obey the anticommutation rule

âαmE,âβnE+ = δαβδmnδEE

For both, amplitudes and operators, one can define a relation between the incoming and the outgoing states in the form of a matrix, the scattering or S-matrix:

b̂ = Sâ

or, expressed in coefficients for the single contributing channels, for an operator in the left lead

b̂mς =nσLrmς,nσânσ +nσRtmς,nσânσ

where r and t refer to submatrices

S = rttr .

For convenience, we will assign t to transmission from left to right, r to reflection in the left lead and the primed values correspondingly for the right lead. The letter s will denote a generic element of the S-matrix. The sub-blocks of the scattering matrix are related to the reflection and transmission probabilities of the system. So, the appearance of the S-matrix characterizes the conductor.

From the elements we can extract certain transmission and reflection probabilities via

Tmn,ςσ = |tmςnσ|2 ,

where a state nσ passes into a state mς.

Unitarity

To assure the conservation of particle current, the S-matrix has to be unitary. To show this, we assume, that the square amplitudes of incoming and outgoing states are related to the current, so that we can state

ââ = b̂b̂ â1 SSâ = 0 .

Here we used the relation (2.5), that connects â and b̂ via the scattering matrix. The second term is true only for S =S1 and hence for a unitary matrix.

From unitarity we can derive sum rules for certain elements. Obviously a particle injected into the device has to end up in some resulting state. So we can state

mςLR|smς,nσ|2 =mςL|rmς,nσ|2 +mςR|tmς,nσ|2 = 1 .

Analogous a rule for the incoming amplitudes exists:

nσLR|smς,nσ|2 =nσL|rmς,nσ|2 +nσR|tmς,nσ|2 = 1 .

We define the transmission function TE as

TE =mς,nσ|tmςR,nσLE|2

and can relate this to functions RE and NLE =nσL1, the number of modes in the lead L, via eqn. (2.8) and eqn. (2.9):

TE = NLERE

and

TE = NRERE .

Comparing these with the corresponding relations for TE, the transmission function for the right lead, and assuming the leads to have the same number of open channels, NLE = NRE, the two have to be equal

TE = TE .

This result is an expression of current conservation in the system.

Time-Reversal Symmetry

We will present another symmetry of the device, when we can assume, that the Hamiltonian (2.1) possesses time-reversal symmetry. This assumption is well supported, as long as the potential Uxz is real and no magnetic field is considered. Then the non-relativistic time-reversal operator for a Pauli-spinor reads

K̂ iσyĈ ,

where the operator Ĉ introduces complex conjugation of all components. The action of K̂ is identical to a reflection of the coordinate system at the y-axis, in which the complex conjugation inverts the direction of motion

K̂p̂xzK̂1 = p̂xz

and the spin of the state is flipped,

K̂σ̂xzK̂1 = σ̂xz .

We can thus state, that incoming and outgoing states are exchanged, which can be sketched as

K̂bn,σ = σan,σ .

Now, using equation (2.5) to involve the scattering matrix, we obtain

σan,σ =mLRς=±1σs1nm,σςbm,ς .

We use again the time-reversal transformation and find for the transformed states

K̂bn,σ =mLRς=±1σςs1nm,σςK̂am,ς .

The application of K̂ does however not change the scattering matrix itself, since it is derived from the Hamiltonian, which is invariant under time-reversal. Therefore a second relation between the transformed states can be derived from the definition of the S-matrix:

K̂bn,σ =mLRς=±1snm,σςK̂am,ς .

We compare these two results and use the unitarity of the scattering matrix. This leads to the important symmetry relation for entries of the S-matrix

snm,σςE = σςsmn,σ,ςE ,

from which, after summation over the square norm, we could retrieve equation (2.12).

Definition of Currents

The quantity to be measured is the current, that is produced by the model system. The particle current will be determined at some position x in one of the leads. Using the current operator [47]

Ĵx =2mi x x

an expression for the particle current, e.g., in the left lead, can be defined [48]:

ÎPxL,t=dzΨLx,z;tĴxΨLx,z;t ,

where the wave functions ΨLx,z;t have the structure, that we introduced for eqn. (2.2) and are assembled according to the following considerations. In z-direction hard walls are assumed for the constriction that builds the quantum wire from the 2DEG. The transverse part of the wave functions can then be written as

χnz = 2Wsin πnzW .

In x-direction plain waves contribute to the particle motion. A wave function looks therefore like

ψLnσ,E±x = 1knEe±iknEx σ,xL ,

with the sign in the exponent determining the direction of motion and σ the spin state. As we outlined above, we have to take into account, that there are waves to and from the scatterer. So a resulting wave function in the left lead is a linear combination of ψLnσ,E±, weighted with the annihilation operators âαn and b̂αn for incoming and outgoing states. We also will change terminology in this step and look at Ψ̂ as an operator in Fock space rather than as wave packet. The resulting operator takes the form

Ψ̂Lx,z L;t = dEeiEtχnz nσL âLnσEΨ̂Lnσ,E+xz +b̂LnσEΨ̂Lnσ,Exz .

Inserting Ψ̂ in eqn. (2.15) the derivatives with respect to the longitudinal coordinate x have to be evaluated. To simplify the resulting expression we can make the assumption, that the energies E and E, stemming from the operators Ψ̂ and Ψ̂, respectively, either coincide or are close to each other for all quantities, that are of interest in the course of this work. When we also state, that knE varies slowly with the energy, we arrive at the expression

knE knE ,

that allows us to derive the expression

ÎPx L,t = ehdEdEeiEEt nσL âLnσEâLnσE b̂LnσEb̂LnσE .

This equation can be rewritten to

ÎPx L,t = ehdEdEeiEEt αβmς,kςâαmςE𝒜αmς;βkςL;E,EâβkςE

by introducing a matrix 𝒜 with

𝒜αmς;βkςL;E,E = δmkδςςδαLδβL nσLsLnσ;αmςEsLnσ;βkςE

and α and β denoting leads.

In chapter 5 we will look at the noise part of the current. The thermal average over the current on the other hand is determined by the average of the annihilation operators

âαmςEâβkςE = δαβδmkδςςδEEfαE ,

with fαE the Fermi-Dirac distribution in the lead α

fαE =11 +expEμαkBT .

We find for the average charge current1 in the left lead

1
The charge current is related to the particle current via ÎC = e h ÎP
ÎCx L,t = ehdE NLEREfLE TEfRE .

Using eqn. (2.11), the expression can be simplified finally to

ÎCx L,t = ehdETEfLEfRE ,

Since TE = TE, it does not matter, in which lead we evaluate the charge current. The number of particles transmitted and reflected is the same, independent from the location, where we measure, that is, ÎCx L,t = ÎCx R,t. This situation will be different for the spin current.

We will see in chapter 3, how TE and thus the average current can be calculated with the help of Green’s Functions theory from the Hamiltonian (2.1).

Spin Currents

In this section the previous results will be expanded to respect the spin state of the system. To determine, to which degree current from the device is spin-polarized, we define the spin current by introducing the spin current operator [4950]

ĴS,x = 2σzĴx = 2 4miσz x x .

We assumed for this definition, that the quantization axis of the spin is parallel to the z-direction of the system. The matrix σz is the Pauli matrix

σz = 1 0 01 .

With this definition the spin current takes the form

ÎSx L,t =dzΨ̂L(x,z,t)Ĵ S,xΨ̂L(x,z,t) .

The wave functions or rather the operators defined in eqn. (18) are inserted. With the same assumptions, that we did for the particle current, the spin current reads

ÎSx L,t = 1 4πdEdEeiEEt nσL σ âLnσEâ LnσE b̂ LnσEb̂ LnσE .

Again a matrix 𝒜 is defined to compactify the expression. With

𝒜αmς;βkς(L,σ;E,E) = σ δ mkδςςδαLδβL nLsLnσ;αmςEs Lnσ;βkςE

we arrive at

ÎSx L,t = 1 4πdEdEeiEEt αβmς,kςσâαmςE𝒜 αmς;βkς(L,σ;E,E)â βkςE .

In a last step we take again the thermal average:

ÎSxL = 1 4πdEαmςσσ 1 nLsLnσ;αmςs Lnσ;αmς fαE = 1 4πdEσ σmςL 1 nLrLnσ;αmςr Lnσ;αmς fLE+ σmςR nLt Lnσ;αmςt Lnσ;αmςfRE

The expression for the particle current could be simplified by applying sum rules for the transmission and reflection functions. This gives the motivation to re-define these quantities in a spin-resolved way. In the following section we will investigate these functions and try to find properties, that allow to simplify equation (2.27).

Spin-Resolved Transmission

Spin-resolved transmission and reflection functions distinguish between the the incident and the outgoing spin state of the particle. We define the quantities corresponding to the spin-degenerate case

TςσE =m,n|tmς,nσE|2

and

RςσE =m,n|rmς,nσE|2 .

The full transmission function is still the sum over all contributions, that means

TE =ςσTςσE .

For the task to simplify the expression for the average spin current we further define the spin transmission

TSE =ςσςTςσE = T++E + T+ET+ETE .

We apply the sum rules (2.8) and (2.9) to show relations between the spin-resolved quantities

Rς+E + RςE = 1 2NLETς+ET ςE ,

and

Rς+E + R ςE = 1 2NLETς+ETςE .

So we arrive at

TSE = RSE .

This term yields the property, that is needed to reduce the average spin current

ÎSxL = 1 8πdE fLEfRETSE .

We can also see a very important property of the spin current. Since in general TSETSE it does matter, in which lead the spin current is evaluated. This result is expected, because due to spin flip possibilities in the device the spin current is not conserved, as it is in the case of particle current.

Properties of the Spin Current

For charge current the continuity equation

ρt + j = 0

holds. Due to the possibility of spin flips inside the system we cannot state this anymore in the case of the spin current, that was defined in the last section. An additional “torque density” τS appears at the right-hand side:

ρSt + jS = τS .

This torque density takes the form [50]

τS = ReΨ̂ 1iσ̂zĤΨ̂ .

In the model system, that we investigate in chapter 4, spin-orbit coupling based on the Rashba effect will be present. In this case the commutator does not vanish and the spin current is not conserved. This result assures, what we have stated in the last section: The position, where we evaluate spin current, is important. We cannot easily relate the values for the left and the right lead, as we did for the charge current. Later on we will introduce two rocking situations for our system, that set the two leads to differing electrochemical potentials, and evaluate the average over the two resulting spin currents. The finite value for the torque density then makes it inavoidable to choose explicitly, which leads we will use in either of the two situations to evaluate the spin current.

Basically we distinguish between two possibilities and hence two definitions for the average spin current produced by our system. In the first case, the spin transmissions will always be evaluated at the side with the lower electrochemical potential. We will assume, that this is the right lead for V0 > 0 and the left lead for the opposite case. The average of the produced spin current is then

S = 12 ÎSxR,+V0 + ÎSx L,V0 =18πdEΔfEV0 TSEV0TSEV0 ,

with ΔfEV0 = fEεF+V02fEεFV02 the difference between the Fermi distributions in the two reservoirs.

In the limit of vanishing potential differences V0 0 the average charge current of the ratchet will vanish, which we will show in more detail in section 4.1.1. However, equation (2.30) still contains the difference TSE0TSE0, that might be finite. We are thus faced with the interesting fact, that the ratchet can produce spin current, according to this definition, even in the absence of any directed particle motion. In this case the quantity (2.30) must be interpreted as the possibility to generate polarized spin current.

Furthermore definition (2.30) requires the possibility to measure the spin polarization in both leads, that are connected to the device. If we are interested only in the spin rectification in one lead and hence in one reservoir, it is straight forward to introduce a second measure for spin current, that will be evaluated in one fixed lead, e.g., the left one,

ISxL = 12 ÎSxL,+V0 + ÎSx L,V0 =18πdEΔfEV0 TSEV0TSEV0 .

When we examine the behaviour of this definition in the case of linear response, that is, V0 0, we find that it vanishes in the same way the particle current does. But when we rock the system, as proposed in chapter 4, the difference can be larger than 0 and therefore spin rectification can occur.

In the following part of this work we will mainly investigate the properties of the second definition and show, that a model system with Rashba spin-orbit coupling and a scattering barrier will produce spin current upon driving by external voltages.

Spin-Orbit Interaction

Since 1986, when the first experimental realisation of a quasi-one-dimensional electron system was achieved, a number of techniques have been developed to create such a set-up on top of a 2DEG. Especially the split-gate technique, wet and dry etching and cleaved edge overgrowth have to be mentioned [5152]. Experimental structures allow effective widths over ranges from 10 nm to 10 μm and therefore access to ballistic as well as diffusive transport regimes. Inside a 2DEG several properties lead to a connection between the spin state of a particle and its motion through the gas. This fact, the spin-orbit (SO) coupling, will turn out to be a crucial ingredient for the ratchet effect proposed in chapter 4. We will investigate in this section two reasons for spin-orbit coupling in quantum wires [53].

The Dresselhaus effect stems from electrostatic potential gradients rising from the bulk-semiconductor crystal lattice and the microscopic features of the heterostructure interfaces. One speaks in this context of “bulk inversion asymmetry” (BIA) and “interface inversion asymmetry” (IIA) [54]. We will discuss this source of spin-orbit coupling in the last part of the section.

Opposed to that, the so-called Rashba effect stems from the asymmetry of the band structure in growth direction of the quantum well. This “structure inversion asymmetry” (SIA) also leads to potential gradients and thus to spin-orbit interaction [515556].

Origins of Spin-Orbit Coupling

An electron in the electrostatic field of a proton moves with a velocity v = pme [57]. This gives rise to a magnetic field B in the resting system of the electron. This magnetic field is given in first order by

B = 1 c2v ×E(r) ,

where E(r) Φ(r). The Hamiltonian of an electron, that shows fine structure, therefore contains a term

HSO Φ(r) σ ×p .

This form was derived by quadratic vc expansion of the Dirac equation. The presence of an electrostatic potential gradient leads to a coupling between the spin state and the motion of the particle. This statement is the starting point for evaluating SO-coupling terms in bulk material. As stated above, the Rashba effect arises due to a gradient of the electric field of the confining potential creating the 2DEG. The general term added to the overall Hamiltonian has therefore the shape

Ĥα = α σ ×pz .

The factor α denotes the Rashba coupling constant, that is proportional to the confining field. This causes the SO coupling to be tunable by application of gate voltages over an order of magnitude [58]. In chapter 4 we will use the coupling strength as a parameter for the ratchet device.

The spin quantization axis is chosen in the z-direction. The coupling constant α will be replaced in the following course by a wave number kα, so that the Hamiltonian takes the form

ĤR = kα m (σxpzσzpx) ,

with α = kα m . This term is responsible for transition probabilities between different conduction channels, that lead to a spin flip at certain crossing points.

We will use now a perturbative ansatz for the spin-mixing to derive properties, that we will use further in chapter 4.

The Weak Coupling Limit

For the perturbative ansatz the Hamiltonian of our system is separated in an exactly solvable part Ĥ0 and the spin-mixing part Ĥp [59],

Ĥ = Ĥ0 +Ĥp Ĥ0 = p̂x2 2m kα m σ̂zp̂x + p̂z2 2m + V cz Ĥp = kα m σ̂xp̂z

We divided here the Rashba interaction into two contributions. The part remaining in Ĥ0, proportional to σ̂zp̂x, raises the spin degeneracy by shifting the dispersion relations for the two spin directions differently. The contribution of Ĥp is in contrast to introduce the mixing of different transverse channels. Using σz2 = 1 the Hamiltonian of the unperturbed part can be rewritten into

Ĥ0 = 1 2m(p̂xkασ̂z)2 2kα2 2m + p̂z2 2m + V cz .

V cz contains the confining potential in z-direction. We will assume hard walls, which leads to transverse bound states

p̂z2 2m χnz + V czχnz = ɛnχnz

with transverse wave functions χnz = 2πsin(nπzW), that depend on the width W of the confined sink. Using this relation and the vanishing commutators Ĥ0p̂x = 0 and Ĥ0σ̂z = 0 we easily find the solutions Ψ of the corresponding Schrödinger equation Ĥ0Ψ = EΨ:

Ψn,k,σxz = 1 2πeikxχnz σ .

The solutions are distinguished by the transverse quantum number n, the longitudinal wavenumber k (,) and the spin state. With this the eigen-energies take the form

Enσk = 2m(kσkα)2 2kα2 2m + ɛn .
PIC Properties of the Rashba energy spectrum in a 2DEG. Panel a) Portion of the energy spectrum of a Hamiltonian containing a Rashba interaction (2.32). Panel b) The Fermi contours relative to a Hamiltonian with term (2.32), the spin states are indicated. Panel c) Section of the energy spectrum for a free electron. Panel d) Section of the energy spectrum for an electron in presence of a magnetic field (Zeeman splitting). Panel e) Section of the energy spectrum for an electron in presence of Rashba spin-orbit interaction. Taken from [53].

From the dispersion relation (2.34) we see a basic effect of the Rashba term in the Hamiltonian Ĥ0: The branches of each spin sort are shifted in different directions along the k-axis and down to lower energies proportional to kα2. This situation is depicted in fig. 2.1, panel e. This alone does not yet lead to transitions between the branches. Respecting the perturbation part Ĥp we look now at matrix elements of this Hamiltonian between unperturbed states

n,k,σĤp n,k,σ = kα m δkkδσ,σ np̂z n .

We will evaluate np̂z n in chapter 4, equation (4.30). For now we point out, that the expectation value np̂z n vanishes and the equation above assures, that a particle switching between transverse modes also changes its spin state. So a perturbative coupling between different subbands nn arises. The effect is pronounced for states with the same longitudinal momentum k. This is exactly the case at the crossing points that were arranged by the first part of the Hamiltonian, where states of opposite spin are hybridized due to the action of the perturbation.

The dispersion relation as well as the solutions presented above are valid only if inter-subband mixing can be neglected, defining the regime, where the splitting of the Hamiltonian is reasonable [4260]. But for large Rashba coupling constants this does not hold true anymore and we would have to include higher terms. We evaluate the following at the branch minima kmin = ±kα and compare this to the wave number of a crossing point

En,(kn=m)=!Em,+(kn=m)kn=m = 2m2 ɛmɛn4kα .

The weak coupling regime is identified with the condition kn=m kα. We rewrite the level energies as ɛn = 2kn2 2m and introduce the spin-precession length Lp = πkα [42]. In a one-dimensional quantum waveguide with hard wall confinement kn nπW holds. Then the condition takes the form

kα2 = πLp 2 14mn2 πW2 .

We end up at the intuitive result, that the spin-precession length has to be much larger than the width of the wire

WLp 1 .

The Dresselhaus Effect

Unlike the Rashba SO coupling, that was examined in the previous part, the Dresselhaus effect stems from the crystal-lattice structure and interface effects at semiconductor heterostructures [61]. We have outlined above, that spin-orbit coupling emerges from potential gradients, that affect the electron. The Dresselhaus term is based on the details of crystallographic elementary cells of the investigated structure and therefore an effect of the material itself more than of an external parameter like the gate voltage.

To measure the single contributions of Rashba and Dresselhaus coupling the spin-galvanic effect [1254] has proven to work. Measurements indicate, that the strength of both effects can be of comparable size. In this section we will see, that the presence of a Dresselhaus term can have significant impact on the proposed ratchet mechanism.

Details on the Effect

In materials with zinc-blende or wurtzite lattice structure [61] the inversion symmetry for the point groups is not present. This causes unbalanced crystal fields, that act as source for the SO coupling. This can be expressed by a Hamiltonian

Ĥk3 = γ σ̂xp̂x(K̂y2 K̂ z2) + σ̂ yp̂y(K̂z2 K̂x2) + σ̂ zp̂z(K̂x2 K̂ z2) .

The bulk Dresselhaus coupling constant γ depends on the characteristics of the elementary cell. We assume the coordinate axis to be parallel to the crystallographic cubic axis, that is, ex 100 and so on. In section 4.3 we will show the results for rotating the crystal lattice against the direction of the electron motion.

Now a quantum well with width dy is built on this semiconductor, which leads to electrons moving freely in the xz-plane, while the wave vector in y-direction is quantized:

ky π dy 2 .

We can recast the Hamiltonian (2.36) then into two parts, representing the “bulk” and the “interfacial” interaction

Ĥk3 γ K̂y2 (σ̂xp̂x σ̂zp̂z) γ (σ̂zp̂zK̂x2 σ̂xp̂xK̂z2) .

For a quantum well thin enough, which we will assume in favour for a 2DEG, the kinetic energy of the in-plane motion is much smaller than the energy of the quantised degree of freedom. Therefore the term quadratic in kxz can be neglected. A good approximation for the Dresselhaus Hamiltonian reads then

ĤD = kβ m (σ̂xp̂x σ̂zp̂z) .

In this equation we introduced a factor kβ = m 2 γ K̂y2 similar to the Rashba coupling strength kα. To reemphasize the above, kβ depends on the material-specific constant γ as well as on the effective width of the 2DEG. We state, that a Hamiltonian with a parity symmetry P̂, defined in section 4.1.1, and time reversal symmetry keeps these symmetries, when a Dresselhaus term is added:

Ĥ =Ĥwire +Ĥα +Ĥβ P̂Ĥp̂1 = Ĥ .
Interaction with Rashba Coupling

Now we will sketch a perturbative treatment of the Dresselhaus effect in a quantum wire in analogy to our considerations in section 2.3.2. The confinement in z-direction is again assumed to consist of hard walls and is modelled by a potential V cz. When we split the Hamiltonian again in an exactly solvable and a perturbative part, we find

Ĥ =Ĥwire +Ĥα +Ĥβ=!Ĥ0 +Ĥp ,
Ĥ0 = p̂x2 2m kα m σ̂zp̂x + p̂z2 2m kβ m σ̂zp̂z + V cz, Ĥp = kα m σ̂xp̂z + kβ m σ̂xp̂x .

The solutions of Ĥ0Ψ(0) = E(0)Ψ(0) are simply

Ψn,k,σ(0)xz = 1 2πeikxψ n,σz σ ,

where ψn,σz is the transverse wave function, multiplied with a spin-dependent spatial phase-factor [62]

ψn,σz = eiσkβzχnz .

With this solution we find the dispersion relation corresponding to the unperturbed Ĥ0

En,σ(0)k = 2 2m(kσkα)2 + ɛn2 2m(kα2 + k β2) .

The only difference to the dispersion relation (2.34) is an additional energy shift quadratic in kβ. The matrix element of Ĥp coupling two eigenstates of Ĥ0 however shows a new contribution, that competes with the one of the Rashba term,

n,k,σĤp n,k,σ = kα m δ(kk)δ σ,σ np̂z n + kβ m kδ(kk)δ n,nδσ,σ .

There is still a coupling between neighbouring subbands mediated by the Rashba effect. But a second term connects the two spin states of the same transverse mode at a fixed momentum k, being proportional to this factor.

We will look at the specific case kα = ±kβ. As shown in ref. [62], the effects on observables will cancel each other. An explanation for this behaviour introduces a new conserved spin quantity Σ̂. Setting kα = kβ = kSO the Hamiltonian takes the form

Ĥ = p̂x2 2m + p̂z2 2m + V cz + kSO m p̂x(σ̂x σ̂z) + p̂z(σ̂x σ̂z) = p̂x2 2m + p̂z2 2m + V cz + kSO m p̂x2Σ̂ + p̂z2Σ̂ .

The new spin operator

Σ̂ := 1 2(σ̂x σ̂z)

is hermitian, unitary and has the eigenvalues Σ = ±1. Evaluating the commutators

Ĥp̂x = ĤΣ̂ = 0

we find, that there exists a set of exact eigenstates

Ψn,k,Σ(α=β)xz = 1 2eikxψ n,Σz ± ,
ψn,Σz = ei2ΣkSOzχnz ,

that yields a dispersion relation, that is exactly given by the two k-shifted subband parabolas. Since Σ̂ is a conserved quantity, we cannot observe spin-flips between two branches in the dispersion relation.

We make a note, that the ratchet effect should vanish in the case, where kα = kβ. We will see in chapter 4, that this is indeed the case. Furthermore, the strength and the direction of the generated spin current can be modified by tuning the two parameters independently.

Numerical Methods

The prediction of spin current generation according to chapter 2 will be confirmed by data, that was achieved by numerical calculations. For this reason a scheme has to be developed to compute the properties of the Hamiltonian (2.1) with a sufficient accuracy. In this chapter we will introduce the ideas of the lattice Green’s function method, that will allow us to extract fundamental features from the system by using a grid to model the 2DEG [20] and with this knowledge to determine the scattering matrix.

This grid must not be confused with the crystal lattice of the system. The latter is respected in the effective mass approach, where electrons are handled as freely moving particles with a modified mass

m = 2 d2ɛ dk2 1 .

In principle, the effective mass is a direction-depending tensor. But since it is real and symmetric, one can find a direction, where it takes on a diagonal form, so that one ends up with a value for the effective mass in each direction. We will simplify this assumption further by setting the two relevant masses mx and mz to the value m. This step is justified, since we concentrate on the conduction band at its minimum, where we can assume a quadratic dispersion relation. The effective mass is usually much smaller than the electron mass me [63]. A value for m for a certain material or combination of materials can be derived from theoretical calculation of the band structure or from experimental results.

As stated in the previous chapter the 2DEG will also be confined to create a wire with a certain width W. This will be modeled by a potential Vcz. Additionally many-particle interactions will be neglected for the following considerations. Since the number of electrons, that are excited into the conduction band, is comparably small for semiconductors, this simplification holds.

Composition of the Wire

left reservoir right reservoir right lead left lead central region z x Scheme of the system. A central region is attached to two reservoirs by “clean” leads.

We look at the system in more detail. In chapter 2 the quantum wire was introduced. We will model it by assuming, that there is a central region, where effects like the Rashba SO coupling or additional scattering barriers are active.This region is connected via clean leads to two reservoirs. The leads are supposed to show neither SO coupling nor any other effects, that would disturb the propagation of a particle within.

The reservoirs act as an electron source. They themselves are in equilibrium, but may differ in the electrochemical potential1. This will lead to a particle current from the reservoir with the higher potential through the wire towards the one with the lower potential. The connection between leads and reservoirs is “reflectionless”, that is, the probability for particles to be reflected back in the lead upon exiting into the reservoir is negligible [19].

1
In fact, we will present in chapter 4 a mechanism for generation of spin current, where this feature is required.

Description with Green’s Functions

We aim to find a possibility to describe the quantum wire in terms of a grid, that can be computed. To tackle this task, we introduce Green’s functions. These are completely determined by the Hamiltonian of the system and can be connected to the reflection and transmission probabilities in the scattering matrix.

We start from the time-independent Schrödinger equation

EĤΨ = 0 .

Ĥ is the single-particle Hamiltonian describing our system with the eigenvalues En. The corresponding eigenstates are denoted as Φn. The Green’s function for this problem is defined as

EĤGE = 1 .

Technically, a solution for the Green’s function takes the form

GE = 1E Ĥ ,

but it is easy to see, that the expression is not defined at points E = En. A common way to avoid this behaviour is to add a small, imaginary term ±η to the energy:

G± =limη0+G(E ± iη) .

When we now use the completeness relation n Φn Φn = 𝟙 and expand the denominator into a power series, the Green’s function looks like

G±E =nΦnΦnEEn ± iη .
Green’s Functions and Propagators

What is the physical motivation for introducing Green’s functions? We take a look at the Fourier transform of (3.2), where we execute the integral through a closed contour integration in the complex plane. For the following we focus on the case +η, where the constant is added to the Green’s function. Then we obtain a time-dependent Green’s function, the Green’s operator

iG+t = i2πdEG+EeiEt = θteiĤteηt ,

which has the form of the definition of a “propagator” K, that describes the motion of waves with the correct causality [57]. Using the time-evolution operator U(t2,t1) the state of a particle at time t2 can be expressed via

Φt2 = Ut2,t1 Φt1 .

For the spatial representation of the state this yields

Φr2,t2 =d3r1 r2 Ut2,t1 r1 r1 Φt1 ,

where r2 Ut2,t1 r1 determines the time-propagation of the state Φt1. To assure the correct causal behaviour an additional θ function is introduced, so that the propagator is defined as

Kr2,t2;r1,t1 = r2 Ut2,t1 r1 θt2t1 .

This form of the propagator is the introduction of Huygen’s principle in quantum mechanics, where the θ function only allows states with t2 > t1 to be influenced by the state of the system at time t1. The physical interpretation of K is simply the probability amplitude to find the particle with initial position r1 at time t1 after some time, at t2, at the location r2.

From this equivalence we can derive two important statements. The first is, that, since the propagator connects the state of an incoming particle with the state, when the particle leaves the central region, this motivates to look for a connection between the Green’s function defined above and the scattering matrix of the system. We will introduce such a relation in section 3.3. Then we see, why the choice of + η makes sense. This leads to the so-called retarded solution for the Green’s function and therefore to a result, that we can use to describe the dynamics in the system in the right time order.

Tight-Binding Approach

In order to find a representation of the Green’s function, that can be computed effectively, we introduce the tight-binding Hamiltonian

Ĥ =r rɛr r +r,r rV r,r r ,

where ɛr is the on-site energy at the point r and V r,r is the hopping energy between the lattice points r and r. This equation assumes the continuous motion of a particle to be replaced by localized δ-shaped “orbitals” at the gridpoints r. In this form a wavefunction looks like

Ψ(x) =rδ(x r)Ψ(r) .

In order to respect the spin state, Ψ is a Pauli spinor Ψ = Ψ+xzΨxz. Introducing the creation and annihilation operators âr and âr we state for these the fermionic commutator rules

âr,âr+ = δr,r âr,âr+ = 0 = âr,âr+

and can with this definition rewrite the tight-binding Hamiltonian in second quantization

Ĥ =rɛrârâr +r,r V r,rârâr + h.c. .

For the representation of the grid coordinates we will choose the notation

x la z ta,

with l and t integer numbers labeling the position on the lattice and a the lattice constant describing the acuteness of the discretization. For the following we assume the grid points to be arranged in a square shape.

A common assumption is to restrict the hopping between lattice sites to nearest neighbours only. This means, that V r,r = 0 for all positions r, that are not next to r.

In chapter 4 we will describe the ratchet system with the Hamiltonian (4.20)

Ĥ = p̂x2 + p̂z2 2m +Uxz+kα m (σ̂xp̂zσ̂zp̂x)+kβ m (σ̂xp̂xσ̂xp̂x) .

We will reshape this Hamiltonian to its tight-binding representation. Therefore we have to find expressions for the operators at each lattice site (x0,z0) = (la,ta). The results are, each for a certain spin state σ,

x0,z0;σp̂x Ψ = i xx0Ψσ(x,z 0) = i Ψl+1,tσΨl1,tσ 2a + 𝒪(a2) x0,z0;σp̂z Ψ = i zz0Ψσ(x 0,z) = i Ψl,t+1σΨl,t1σ 2a + 𝒪(a2) x0,z0;σp̂x2 Ψ = 2 2 x2 x0Ψσ(x,z 0) = 2Ψl+1,tσ2Ψl,tσ + Ψl1,tσ 2a + 𝒪(a4) x0,z0;σp̂z2 Ψ = 2 2 z2 z0Ψσ(x 0,z) = 2Ψl,t+1σ2Ψl,tσ + Ψl,t1σ 2a + 𝒪(a4).

We will neglect higher-order terms, so that we can express the Hamiltonian as

x0,z0 Ĥ Ψ =2 2m Ψl+1,t+2Ψl,t+ + Ψl1,t+ Ψl+1,t 2Ψl,t + Ψl1,t 2 2m Ψl,t+1+2Ψl,t+ + Ψl,t1+ Ψl,t+1 2Ψl,t + Ψl,t1 + U(la,ta) Ψl,t+ Ψl,t + kα m 2iaσ̂x Ψl,t+1+Ψl,t1+ Ψl,t+1 Ψl,t1 kα m 2iaσ̂z Ψl+1,t+Ψl1,t+ Ψl+1,t Ψl1,t + kβ m 2iaσ̂x Ψl+1,t+Ψl1,t+ Ψl+1,t Ψl1,t kβ m 2iaσ̂z Ψl,t+1+Ψl,t1+ Ψl,t+1 Ψl,t1 .

This expression can be simplified by collecting terms. To receive the Hamiltonian dimensionless, we additionally divide by 2 2ma2. With ɛ0 = 4 the on-site energy and Ũ = U 2 2ma2 the dimensionless potential we arrive at

H̃a = Ĥ 2 2ma2 =l,t ɛ0 + Ũlta lt lt +l,t 1 + ikαa ikβa ikβa 1ikαa l,t l + 1,t + h.c. +l,t 1 + ikβa ikαa ikαa 1ikβa l,t l,t + 1 + h.c.

In section 4.3 we will meet again this expression, where it will be generalized for rotations of the crystal lattice against the discretization grid.

We want to point out, that the accuracy of the discretization procedure can be increased by manipulating the grid constant a. Then the following quantities have to be rescaled according to this scheme:

Splitting and Combining Green’s Functions

The basic idea is, that Green’s functions for different regions of a system can be “attached”. If this is possible, analytic solutions can be evaluated for parts of the device, while the remaining smaller parts can be computed more effectively. This is especially important, when we introduce the model for the clean leads. There we will assume them as semi-infinite, which would leave us with an infinite matrix to invert.

The solution is, to split the representation of the Green’s function into

GL GLS GLS GS = (E + iη)𝟙ĤL τLS τLS (E + iη)𝟙ĤS 1 ,

where we divided the Hamiltonian into a part, that describes a central scattering region, ĤS, a part ĤL, where analytic solutions for the Green’s function are known, e.g., the leads, and the coupling term between those two regions, τLS [6419]. The coupling matrix has non-vanishing entries only for adjacent points at the interface between the regions S and L.

Then we can relate GLS with GS using

(E + iη)𝟙ĤL GLS + τLS GS = 0

and find

GLS = gLτLSGS

with gL = (E + iη)𝟙ĤL 1 the Green’s function of the isolated semi-infinite lead. We can insert this in the second relation, that can be derived from (3.7), E𝟙ĤS GS + τLSGLS = 𝟙 , and find finally

GS = E𝟙ĤSτLSgLτLS 1 .

Now we have reduced the Green’s function of the scattering region to an expression of finite dimension. When we find a way to calculate the value of ΣE := τLSgLτLS, which we will call the “proper self-energy” of the leads, the relevant Green’s function can be calculated by inverting a matrix of size C × C, where C is the number of discretization points in the scattering region.

We remark, that equation (3.8) can be recast using the property

(E + iη)GSHSGS = 𝟙

into an expression in the form of the Dyson equation

GS = GL + GL(τLSgLτLS)GS ,

which can be solved recursively [65]. This gives the mechanism the name recursive Green’s function method. It has been successfully used to determine properties of a quantum wire with Rashba-SO active center [66].

Lattice Green’s Functions in a Clean Wire

We are left with the problem to find an expression for the self-energy of a clean, semi-infinite lead. Clearly, this cannot be computed by brute force, since it involves inversion of an infinitely extended matrix. The self-energy, generally a not hermitian matrix, simulates the interaction between the central region and the lead. The effect is a shift of the eigen-energies of the Hamiltonian in the central region and a level-broadening leading to finite life-times of the eigenstates. These life-times describe the duration, while a particle is in a certain state, before it escapes through the lead.

For clean leads we mention again, that they are assumed not to contain any effect, that would disturb a moving particle, hence also no SO coupling. The width of the lead is expressed in units of the lattice constant W = (M + 1)a. Then the Hamiltonian of a lead is simplified from (3.4)

H̃a =l=1t=1Mɛ 0 lt lt l=1t=1M l,t l + 1,t + h.c. l=1t=1M l,t l,t + 1 + h.c.

The transverse profile of the lead is assumed to have a hard-wall shape. Then it can be shown [19], that the self-energy in the range we are interested in, that is, at the surface towards the scattering region, takes the form

ΣSE =n=1MχntχntFE

with FE = exp(iknEa) and χn the transverse profile of the lead.

We compare the eigen-energies of the Lattice Green’s Function ansatz with the ones of a continuous system to get a possibility for describing the error produced by the discretization.

For a continuous system the wave function would be

Ψn,kxz = 2 L0sin(kx) 2 Wsin(πn Wz) = Φkxχnz .

The eigen-energies are simply

Enk = ɛn + 2 2mk2 = 2 2m n π W2 + 2 2mk2 ,

where n labels transverse modes and k > 0 is the wave-number of a standing wave vanishing at x = 0. This assures the correct boundary behaviour.

When we now switch to the grid representation, the normalized wave function changes and we find

Ψn,k(l,t) = 2a L0sin(kla) 2 M + 1sin( πn M + 1t) = Φklχnt .

If we let act the Hamiltonian above on the state represented by this wave function, the dispersion relation for the lattice states reads

Enk = ɛ02cos(ka)2cos( nπ M + 1)with ka 0π .

The two dispersion relations (3.9) and (3.10) will be compared in section 4.1.4 to find an estimation for the accuracy of the presented results.

Group Velocities

When we introduce the connection between the Green’s functions and the scattering matrix, we will need the group velocities of different modes in the leads. These will in principle act as normalization to ensure the unitarity of the resulting S-matrix.

The group velocity is the expectation value of the operator =iĤx. If we use now the dispersion relation (3.10), that we found above, we arrive at the expression

vnk = nknk = 1 kEnk = 2asin(knEa) .

Scattering in the Wire

We are now able to derive the Green’s function of the system from the Hamiltonian. The last step to evaluate current through the device is the connection between this matrix and the scattering coefficients, that determine the current through the Landauer-Büttiker formalism. The scattering matrix has the form (2.6)

S = rt tr

with r denoting reflection back in the same lead and t transmission to another lead. We will assume two connections L and R to a central region, where scattering is present.

Then the Green’s function of the central region can be evaluated by the mechanism described above. We end up with a matrix characterizing the propagation of a spin state (n,σ) to a state (m,ς). These states are simple, spin-degenerate plain waves, since this is the appearance they take deep inside the semi-infinite leads,

Ψn,k,σ(l,t) = a 2πeikla 2 M + 1sin(n π M + 1l) σ .

The S-matrix is now simply derived from the projection of the Green’s function onto these states. But since it has to be unitary, which we explained in section 2.1, to assure current conservation, and since the group velocities differ in different transverse channels n, this has to be taken into account. The result are the Fisher-Lee relations for the reflection entries [6768]

Rmς,nσ = δmnδςσ+i avmς Evnσ EjAiAΨmς,E(0,j)G jς,iσEΨnσ,E(0,i)

and for the transmission

tmς,nσ = i avmς Evnσ EjBiAΨmς,E(x B,j)Gjς,iσEΨnσ,E(0,i) .

These are the desired connections that allow us to determine the current from the knowledge of Green’s functions.

Voltage Offset

The last point to be respected for the further proceeding is the application of a voltage offset, setting the two reservoirs to different electrochemical potentials. We will use this to rock the system, which is a necessary process to find spin current according to definition (2.31).

When this offset is applied, the voltage will drop along the device in a certain way. Inside clean wires there is no resistance, hence no voltage drop. The scattering region contributes with a certain amount, that depends on the structure of this region. And lastly there is the contact resistance between the leads and the “real” reservoirs. This resistance GC1 arises, because the electrons from the “continuous” reservoirs have to be redistributed into the transverse open channels inside the leads. It can be determined to

GC1=4πe21N(ɛF) ,

where N(ɛF) is the number of open transverse modes inside the leads.

The Landauer-Büttiker formula delivers us furthermore an expression for the total resistance

G1=4πe21T(ɛF)

with T(ɛF) the total transmission probability (2.10). With this knowledge we can determine the resistance GS1 of the central region in the device:

GS1=G1GC1=4πe21N(ɛF)R(ɛF)T(ɛF) .

Results for a Multi-Mode Wire

In the previous chapters we introduced the properties of a quantum wire with constrictions. Now we consider this system with an offset in the injection energy. Eventually, we introduce a scatterer, i.e., a voltage offset inside the device, that could be created by a gate. The results show a spin rectification, that is, a non-vanishing average spin current upon driving the system. We will explain this based on the Landau-Zener theory for time-dependent level crossings.

Realistic wires will contain multiple open transmission channels. Therefore the findings must be extendable for this case. We will show, that the Landau-Zener ansatz is appropriate for three modes, and present results for broader wires.

Finally, the Dresselhaus spin-orbit coupling has to be taken into account in some materials. But with this effect the orientation of the crystal lattice becomes important. We will address that issue in the last section of this chapter.

The Landau-Zener Ansatz

Currents and Measurements in the Ratchet

To quantify the spin rectification in the device, we evaluate the average spin current of the system. We have shown in chapter 2, that therefore we have to fix the lead, where the spin current will be evaluated. For the following, we will choose the right one and evaluate spin current IS only in this location.

The system will be rocked, so that there are two situations to evaluate. They differ in the chemical potential of the leads and therefore in the direction of the particle motion. In the next section we will argue, that effects, that stem from the switching, can be neglected due to the high difference between rocking intervals, that is, AC frequencies, and the transition time for a given particle. Therefore we can calculate the average spin current simply by taking the arithmetic average over the spin currents in the two rocking situations +V0 and V0:

ISR=12(IS(R,+V0)+IS(R,V0)) .

We insert eqn. (2.29) and get for the average spin current of our system:

ISR=18πdEΔf(E,V0)TS(E;+V0)TS(E;V0)

with TS(E;V) = T++(E;V) + T+(E;V)T+(E;V)T(E;V). The minus sign for the spin transmission function stems from fLV0= fRV0. The central factor for the ratchet is thus the difference of the spin transmission functions in the two rocking situations. The difference has to be unequal to 0 for the system to produce a net spin current upon average. This is not automatically implied, as we will show in the case of the charge current.

Net Charge Current of the Ratchet

For the particle motion, we received in chapter 2 as expression of the average charge current

ICx R,t = e hdETE(fREfLE) .

Following the argumentation above, we introduce for the net charge current of the system the expression

IC R = 1 2(IC(R,+V0) + IC(R,V0))

and insert eqn. (4.2):

IC = e 2hdEΔfEV0 T(E;+V0)T(E;V0) .

We will now use symmetry considerations to argue, that the sum T(E;+V0)T(E;V0) vanishes. Assume an operator Â, that commutes with the Hamiltonian of the system, i.e.,

ĤÂ = 0 .

With Ψ being a solution of the corresponding Schrödinger equation, we also find

ÂĤΨ = ÂEΨ Ĥ(ÂΨ) = E(ÂΨ) ,

so that ÂΨ is also an eigenfunction of the Hamiltonian Ĥ. We keep in mind, that application of  also does not change the scattering matrix, since it is calculated from the Hamiltonian and does not depend on the states of the system.

The system is assumed to be invariant under mirroring at the y-axis, if one changes at the same time the voltage at the leads. This transformation exchanges the terminals. Therefore, also changing all properties of the leads will leave the system in the same state as before. We will test this by applying the operator

P̂ = iσyp̂0xz

to our Hamiltonian. p̂0xz denotes here the simultaneous inversion of x- and z-direction, so that p̂0xzΨ(x,y,z) = Ψ(x,y,z). The effect to the parts of the Hamiltonian (4.20), that we will introduce in the next section, is as follows:

P̂Usxp̂1 = US(x) = Usx P̂V czp̂1 = V c(z) = V cz P̂p̂xp̂1 = p̂x P̂p̂zp̂1 = p̂ z P̂σ̂xp̂1 = σ̂x P̂σ̂zp̂1 = σ̂ z

The function g(x,z;t) for the voltage drop along the device is assumed to be invariant under the transformation. This is true, if Usx and V cz share the same kind of spatial symmetry, because the voltage drop along the device, and with it g(x,z;t), depends on the total potential in that area. Therefore the model Hamiltonian is invariant under application of P̂ with simultaneous changing of the voltage sign:

P̂Ĥ(V )p̂1 = Ĥ(V )

Here the voltage is treated as a tuning parameter of the Hamiltonian. The result is intuitive, because inversion of x-direction changes leads. Therefore properties of the leads have to be exchanged, too.

Next we look at the effect of P̂ on the eigenstates of the Hamiltonian. Due to the symmetric transversal confinement we can state, that the eigenfunctions in z-direction have a defined parity, namely

χn(z) = pnχnzwith pn = ±1 ,

where pn is depending on the channel n. Because the system is connected to two identical leads, we also note, that there is always a corresponding channel n R to a channel n L, described by the same function χnz. The transformation P̂ flips the spin of a state. This can be derived easily from the definition (4.4) of P̂. A transformed state will have an additional factor σ compared to the original state.

In chapter 2 we found for the connection between the amplitudes of incoming and outgoing waves

b̂mςL =nσLrmς,nσânσ +nσRtmς,nσâ nσ .

A given state, e.g., in the left lead, can always be expressed as a linear combination of ânσ and b̂nσ with left and right moving states ψ± as defined in section 2.1:

ΨL(x,z,t) =dEeiEtnσL ânσEψnσ,E+xz +b̂ nσEψnσ,Exz

Now we use these formulae to express the action of P̂ on states in terms of scattering amplitudes:

P̂b̂nL,σ = σpnbnR,σ = mςLσpnsn,σ;mς(V )âmς +mςRσpnsn,σ,mς(V )âmς nR

We repeat this step for the incoming amplitudes to receive a connection between transformed states similar to the one above for states before the transformation:

P̂ânL,σ = σpnânR,σ .

Inserting in the expression for b̂ leaves us with

P̂b̂nL,σ = mςLσςpnpmsn,σ;mς(V )(P̂âmς)+ +mςRσςpnpmsn,σ,mς(V )(P̂âmς)|nR

A second relation is derived from the invariance of the Hamiltonian under P̂. With changing the voltage sign in both leads one finds:

P̂b̂nL,σ = mςLsn,σ;mς(V )(P̂âmς)+ +mςRsn,σ,mς(V )(P̂âmς)|nR.

Comparing these two results, the elements of the scattering matrix are related like

snmL;σς(V ) = σςpnpmsnmR;σ,ς(V ) .

For the transmission function, that enters the net current, we evaluate the square norm of the scattering matrix elements. Since pn = ±1 we find

snmL;σς(V )2 = s nmR;σ,ς(V )2 .

which, according to eqn. (2.10), yields an expression for the transmission function

T(E;V ) = T(E;V )

or, with respect to equation (2.13), that was derived from the time reversal symmetry,

T(E;V ) = T(E;V ) .

Taking a look at eqn. (4.3) we see, that the particle current vanishes

IC = e 2hdE fREfLE T(E;+V0)T(E;V0) 0
Considerations for Spin Current

The previous result assures, that the device does not act as charge rectifier, as long as it possesses inversion symmetry. In both rocking situations the transmitted number of charge carriers is equal, so that the resulting average charge current vanishes. For spin current, however, we have to look at the spin states of the transmitted and reflected particles. Here we will find a different picture. We outline it in the case of vanishing voltage drop to show the principle effect for the spin transmission.

In the case V0 = 0 eqn. (4.9) gives

snmL;σς 2 = s nmR;σ,ς 2 .

Taking a look at the spin mixing terms we find

snmL;+2 = s nmR;+ 2 s nmL;+ 2 = s nmR;+2

Additionally the time-reversal symmetry (2.13) gives

snmL;+2 = s mnL;+2 s nmL;+ 2 = s mnL;+ 2

Therefore we can show

T+E=parityT+E=timerev.T +E ,

and thus that the transmissions with spin flip are equal and do not contribute to the spin transmission TS. This result might be a bit surprising at the first glance, since spin flip is a convenient method to create spin currents [697071], e.g., in ferromagnetic structures. T++E and TE on the other hand do not need to be equal, since there is no similar relation between them. The effect of the spin ratchet relies on this difference.

With this explanation in mind we will argue in the next section, that the shape of the voltage drop along the device has no influence on the basic mechanism of spin rectification. The functionality of spin polarization can be explained by finding different transmission probabilities T++E and TE.

Now we will take a look at a concrete system and introduce the Hamiltonian that we mentioned to derive the properties of the P̂-transformation. Then the rise of a difference between T++E and TE in this system will be explained based on transition probabilities at level crossings.

The Model System

The region of interest in our system is a constricted area in a 2DEG. It is attached to two clean, semi-infinite leads that act as ideal quantum waveguides to reservoirs. The interfaces between the leads and the reservoirs are treated as reflection-less. So, every particle leaving through a lead will certainly arrive in the reservoir. This assumption allows wave packets to be handled as a composition of waves incoming from the left and the right lead, respectively. The reservoirs set the electrochemical potential and the temperature of the system.

left lead left region centre region right region scatterer right lead spin-orbit coupling potential offset x z x kaa x (kFa)2 The scatterer. The 2DEG is confined to create a quantum wire. Connected to two leads are regions, where spin-orbit coupling increases. In the central region the coupling strength is assumed constant. Here the scattering potential is applied.

The dimensions of our device are chosen to be in a range, where we can assume coherent transport. Length-scales of the system have therefore to be smaller than the phase coherence length in the material. In semiconductor heterostructures at low temperature this is easily met by structures up to several μm in each direction of the 2DEG [58]. The spin quantization axis is fixed in the z-direction. Fig. 4.1 shows the principle geometry of the following considerations.

The confinement in z-direction is modelled by a potential Vcz. Usually the two scenarios of parabolic vs. hard wall confinement are chosen for the shape of this potential. In this chapter we will consider the constriction to consist of hard walls. In any case, the potential is assumed to be symmetric in z-direction.

In x-direction a ballistic scatterer Usx of arbitrary shape is introduced. Mostly it will be handled as one adiabatic barrier.The potential Usx will always be set to be symmetric regarding the x-direction. This ensures, as we have shown in the previous section, that the net particle current through our device vanishes.

So far, we can introduce a Hamiltonian that describes these features:

Ĥwire = p̂x22m + Usx + p̂z22m + Vcx .

Here, m denotes the effective mass of the electrons moving through the scatterer. This approximation covers the periodic structure of the crystal lattice of our system. The neglect of electron-electron interaction and the description as free motion with a typically reduced effective mass has been proven to be a good approximation for semiconductor heterostructures.

As outlined in chapter 2 we have to consider an additional term in the form of a SO coupling. This Rashba effect is modelled by a Hamiltonian

Ĥα = kαm σxp̂zσzp̂x ,

where σi denotes the Pauli matrix in i-direction:

σx = 0 1 1 0 , σz = 1 0 0 1 .

This is a crucial point for the functionality of the suggested spin ratchet mechanism. The lifting of the spin degeneracy occurs without spatial asymmetry in the xz-plane. Following this term, as we have seen in chapter 2, spin currents appear without a net charge flowing.

Finally, a voltage-drop along the device is applied, that adds a term

Ĥt = eV tg(x,z;t)

to the Hamiltonian of the system. Here Vt is the time-dependent voltage offset in one lead and g(x,z;t) describes the distribution of the voltage-drop along the scattering region. For simplicity we assume an AC square wave voltage

Vt = +V0 for 0 t < T2 V0 for T2 t < T

with a period T, that is much larger than transition times for particles through the system. With a length of about 500nm and a Fermi velocity of vF = m2πn 3 107cm/s one expects usual transmission times of about LvF 1012s. Here a carrier density n = 5 1011cm2 is assumed [72]. Compared to the usual timescale of AC driving in the region of 50Hz up to 1GHz we can neglect effects of the switching of the voltage. Thus we can evaluate the net average current of the system by considering the arithmetic average over the currents in the two rocking situations.

Through the device the voltage will drop according to g(x,z;t). The exact shape of this function must be calculated selfconsistently, since the Poisson and the Schrödinger equation for the leads have to be calculated simultaneously. It depends, as described in chapter 3, on the ratio RT. Actually, this is not necessary to show the effect of the ratchet. We have seen in section 4.1.1, that the important quantity is the difference between the non-flipping spin transmission functions. Since this difference will be present even for a vanishing voltage drop, we can approximate g(x,z;t). Relying on calculations for transport through nanoscale wires [737475] we will assume, that the voltage is constant through the device for one adiabatic barrier. In the case of an array of short barriers a linear voltage drop along the central region would seem appropriate.

In any case, g(x,z;t) has to show similar symmetric properties as Usx and Vcz, since it depends on their form. Therefore it follows, that reflexion on the y-axis translates Ĥ(t,V) in Ĥ(t,V ).

The full Hamiltonian of our system, that we will study during this chapter, reads at last

Ĥ = Ĥwire + Ĥα + Ĥt .

Level Crossings

The Rashba spin-orbit coupling shifts the branches of different spin states in the dispersion relation in different directions along the k-axis. This shift is positive for spin-up particles and negative for spin-down. Therefore crossing points between different levels arise. In this section, particles are assumed to be either in the first or in the second channel. Higher modes are closed. The appearance of level crossing points will be the central starting point for the explanation of spin polarization.

We split the Hamiltonian (4.20) of our system in an exactly solvable part Ĥ0 = Ĥ kαm σ̂zp̂x and a perturbation part Ĥp = kαm(σ̂zp̂x)  [76]. The level crossings show up in the dispersion relation for Ĥ0, while Ĥp introduces possibilities for mixing branches at the crossings. We assume the potential in x-direction to vary adiabatically, that is, dUsxdx λF, where λF is the Fermi wavelength of the particle. This assures, that good quantum numbers are conserved during the transition of the wave packet.

U(x) E x EF(x) kx E 1+ 1- 2- 2+ Motion of a particle through the scattering region. Due to the barrier the local Fermi energy of the particle changes. The level crossing is marked with a circle.

We follow the path of the particle through the device. While Usx increases, the subbands move upwards. Alternatively the local Fermi energy decreases. We can describe this as motion of the electron along its branch in the dispersion relation. It may happen, that the Fermi energy drops below the band edge of the channel. In this case the particle will be reflected. Furthermore, if the height of the barrier and the injection energy of the particle are in a certain range, a level crossing point will be passed on the way. At this point the electron has a certain probability to change the branch. This is induced by the perturbation part of the Hamiltonian, that we introduced above, and will be explained in more detail in section 4.1.5. By changing the channel the particle flips the spin simultaneously.

If the barrier height is chosen in a way, that completely closes the second band, and if this barrier is long enough to prevent tunneling, particles in the second mode, that have already passed the crossing, will be reflected. At this point, the second subband will be completely depopulated and a remaining particle is surely in the first mode. After passing the top of the barrier the Fermi energy of this particle starts to rise again. The degeneracy point will be traversed a second time, so we have to respect these transition probabilities, too. With two open channels and considering propagation in + x-direction there is exactly one level crossing, that has to be considered: The crossing between the (1-) and the (2+) branch. The probabilities for the level switching are directly related to the transmission of the system. Therefore we can set up a table that summarizes this result for the transmission of a particle from the left lead:

Transmission probability between channels according to the course along branches in the dispersion relation.
Tmς,nσ 1+ 1 2+ 2
1+ 1 0 0 0
1 0 1P122 1P12P12 0
2+ 0 P121P12 P12P12 0
2 0 0 0 0

Particles in the (2-) branch will be reflected with probability 1. We also see, that the (1+) mode will be fully transmitted. From the values above we can then calculate TS straight forward by summing over the relevant entries:

T++ = 1 + P122
T = 1P122
T+ = 1P12P12
T+ = P121P12

We have stated above, that for the system T+ = T+. This result is recovered here. A second consequence of time reversal symmetry is the symmetry of the table with respect to the diagonal. For the spin transmission we find

TS =σσσTσσ = 1 + P122 + 1P122 = 2P12 .

This expression gets maximal for P12 = 1, that is, every passing of the degeneracy point leads to a branch crossing.

In chapter 2 two definitions for the net spin current (2.30) and (2.31) upon rocking the system were introduced. When we measure the spin at two different leads for each rocking situation, eqn. (4.25) is sufficient to get spin current from the device. Since we can relate TS(V ) and TS(V ),

TSVTSV=4.92TSV ,

the average spin current from the device considering both rocking situations reads

S = 18πdEΔfE,V0TSE;+V=14πdEΔfE,V0P12 .

The above relation has no counterpart in the case, where spin current is measured in one lead for the two situations. In order to produce spin current, there has to be a voltage drop along the scatterer. Then, assuming P12 = P12V, spin current is generated, if

P12VP12V ,

and therefore if the transmission probabilities depend on the applied voltage.

We will try to verify basic statements of this theory by calculating the system with appropriate parameters. Afterwards we will introduce an expression for the transition probability, that has the required feature, namely to depend on the barrier height. The last point will be to generalize the relation found between the transmission functions and the probability for systems with more conducting channels.

Numerical Results for the Simple Case

As we have seen above, the spin current depends on the transmission functions that are determined by the scattering matrix

S = r t t r .
PIC Squared norm of elements of the scattering matrix (submatrix T), calculated for the system introduced in section 4.1.4. The probabilities are plotted against the Rashba coupling constant for parameters from fig. 4.7. One can see, that the transmission probabilities between channels vanish exactly according to table 4.1. The transmission probability T1+,1+ is almost constantly 1, thin lines correspond to probabilities equal 0.

We will therefore show the dependence of the transmission on various parameters. In several plots we will show channel-resolved results. In these cases the output channels

Tm,ςσ =nTmn,ςσ

are given. This is equivalent to sums over the columns of table 4.1 with respect to the spin state. In all of these sums one of the two summands is equal to 0, so that there should be no loss of information. Indeed, when taking a look at the scattering matrix, that our system delivers, the appropriate parts vanish. Fig. 4.3 shows the squared norm of the full resolved t-submatrix of the sample system, that we will investigate in the next section.

Units

The injection energy of the particle determines its mode and velocity. We use dimensionless units

F=kFa2=EF22ma2

to describe the energy. An important parameter is the width of the wire. It will be expressed in units of the lattice discretisation constant a. As mentioned in chapter 3 a is not the lattice constant of the underlying crystal but a tool to evaluate continuous motion in the wire. It has to be chosen small enough to fulfill this task. To get a magnitude for the deviation from the continuous limit, we can compare the two dispersion relations, that were found in section 3.2.2. The resulting equation from dividing both,

fx = 12x21 cosx 12x2 ,

is evaluated for the wave number ka of an electron in the wire. Requiring less than 5% deviation we find, that this is given for |ka| < 0.75. We will therefore restrict the injection energy of particles to a maximal value of 0.5. Additionally we will fix the width to W = 12a. With this value, we assure two open transmission channels in the considered energy interval. To show this, we use

n = kna2 = n πWa2

and find

1 =π122 =0.069 2 =2π122 =0.274 3 =3π122 =0.617

We see, that in the energy interval < 0.5 the third mode will not open for transmission.

The last ingredient is the scattering barrier. The height can be chosen from the above considerations. In the last section we wanted the second mode to be fully closed. Therefore by adjusting the height to 0.5 2 0.2222ma2 the second band should be depopulated completely in the observed region. The barrier is also required to vary adiabatically along the x-direction. We will mimic this by using a cosine shaped potential

Usx =U021 cos 2πxLwith x 0,L ,

where L is the length of the scattering region inside the wire. This shape was used successfully in references [7677] for barriers in quantum point contacts. For adiabatic behaviour L will be set to 60a.

Results

Figure 4.4 shows the basic results for a clean wire without Rashba SO coupling. In the case of a missing barrier the transmission shows the expected step-like structure. When a barrier with height U0 = 0.22 2 2ma2 is introduced, the second subband is suppressed. This is clearly shown by the contributions of the single channels.

PIC Transmission probability through a clean wire (dotted line) and a wire with barrier U0 = 0.22 2 2ma2 (solid line). The y-axis values are the transmission neglecting the spin degree of freedom. The single channels (dashed for ch. 1, dash-dotted for ch. 2) show the expected behaviour. The second subband is almost completely suppressed.
PIC Dispersion relations for the cases kαa = 0.08 (left) and kαa = 0.16. The dashed lines show the position of the crossing point. Dotted lines refer to the branches with spin down, solid lines to spin up.
PIC Spin resolved transmission probabilities for a spin orbit coupling strength kαa = 0.08. The barrier is set to completely cover the second band. There is almost no effect for the two different spin sorts. The reason is the too small Rashba effect, that has not a sufficient impact on the dispersion relation.
PIC Channel resolved spin transmissions for a spin orbit coupling strength kαa = 0.16. The plot shows only transmissions without resulting spin flip, e.g., (1+)in + (2+)in contributing to (1+)out . All other parameters are equal to fig. 4.6. In contrast to the case without Rashba effect here is a non-vanishing probability for particles in the second mode to be transmitted. This is due to the mechanism introduced in the last section.
PIC Same system as fig. 4.7. The plot shows the spin resolved transmission probabilities. In contrast to fig. 4.6 we find a significant difference between the two spin states, while the spin polarization based on the spin-flip transmissions T+ and T+ does not play an important role.

Introducing now spin-orbit interaction it is important to state, that the Rashba coupling strength has to be large enough to create significant spin current. This can be understood by imagining the effect of the coupling strength as “pulling apart” the branches of the dispersion relation. Looking at the crossing point between (1-) and (2+), it will move simultaneously to lower energies. Finally, it gets in a regime, where particles with the above determined injection energy pass it on their way through the scatterer. This is a required condition, as we lined out in the previous section.

Figure 4.5 shows the dispersion relations for the two cases, where the spin-orbit coupling is large enough (right) and where the crossing point lies above 0.5, which is the maximal injection energy (left). In the latter case we expect to find no relevant spin transmission in the observed region. Figure 4.6 shows only very weak differences in the two spin directions. Since the crossing is not a sharp point but merely a region with level switching probability increasing towards it, the result is still in agreement with the dispersion relations above. It has to be pointed out, that this is not covered by the lowest-order perturbation theory, that we introduced in the last section, but it is an explicable result.

Raising the Rashba coupling strength the crossing point gets accessible for electrons passing the device. In fig. 4.7 we find for the channel resolved spin transmissions, that indeed the possibility for transmissions in the second channel is finite. It can be shown [78], that the barrier is an essential ingredient for the system to work as a spin ratchet. A vanishing barrier leads to reflection of electrons only in a part of the observed second channel, namely of the ones with energies Ê2 E Ê2 + U0.

PIC Dependence of the spin-resolved transmissions from the Rashba coupling strength. The electrons are injected in the scatterer in the second band, that will be completely closed by the barrier. Therefore the ratchet condition is satisfied.
PIC Same system as fig. 4.9. The plot shows the channel and spin resolved transmissions for the spin-up and spin-down case without spin-flip. (2-) stays constantly zero, which assures, that particles in the second band would be reflected totally without SO coupling. Nevertheless, the contributions from (2+) rise continuously, until the peak value 0.8 is achieved. At the same time, (1-) loses weight.

The general dependence of the device from the spin-orbit coupling is clear so far. But the coupling strength can be tuned over a large interval, e.g., using appropriate gate voltages [5879]. Therefore kαa = 1 2α 2 2ma2 can be used as tuning parameter. Fig. 4.9 shows the connection between the spin resolved transmissions and the Rashba parameter. Afterwards we will compare this result to fits for the transmission probability (4.29), that we will introduce then.

We expect, that transmission for the second channel should be highly Rashba-strength-dependent, since this value determines the existence and accessibility of the level crossings. In fig. 4.10 we find, that for vanishing SO coupling there is no transmission in the second band. With increasing kαa the contributions of the (2+) channel start to rise. At some value for kαa TS will be maximal. Then, with further increasing Rashba strength, the effectiveness starts to sink again, since the level crossing gets again out of reach for the particles. In this regime we are also leaving the weak coupling limit (2.35).

The Transition Probability

In eqn. (4.25) TS depends on the transition probability P12 between the (1-) and the (2+) conducting channel. We are still missing an expression for this factor. Moreover, P12 has to depend on the applied voltage, so that the ratchet generates spin current IS. We have seen, that TS is connected to the Rashba strength, so this must hold true for P12 as well.

Clearly P12 has to be related to the perturbation part

Ĥp = kαm σ̂zp̂x

of the Hamiltonian and to the coupling matrix element mςĤp nσ respectively. This gives the dependence on the Rashba strength as well as on the particle energy and on the shape of the confining potential in z-direction. For a more accurate expression we follow the argumentation in ref. [76]. The Landau-Zener theory yields an expression of the form

P12 = 1 exp 2πλ12

to describe the transition probability. We mark, that the original definition of the Landau-Zener probability was the one for staying in a certain branch, P̃12 = 1P12, but since we are interested in the changing of channels, the above definition is more appropriate. λ12 is a measure for the adiabaticity of the system and can be expressed in terms of the coupling matrix element and the change velocity of the level spacing: λ12 = mςĤp nσ2Q, where Q = t E2+E1, evaluated at the crossing point. We can recast Q in a product of the potential change and the group velocity of a particle in either of the two branches:

Q = dUsxdxv2,+Ev1,E ,

so that P12 takes the form

P12 = 1 exp 2πmςĤpnσ2Usx v2,+Ev1,E .

Next, we have to evaluate the coupling matrix element between the two unperturbed states (1+) and (2-). From the form of the perturbation Hamiltonian (4.27) we can state

mςĤp nσ = kαm δς,σ mp̂z n .
PIC Spin resolved transmission probabilities fitted and plotted against the Rashba coupling strength. The values are derived from the Landau-Zener ansatz (4.29). Main frame: Channel and spin resolved transmission like fig. 4.10. Inset: Spin resolved transmission like fig. 4.9. The fitting parameter c was fixed to 42 in both cases.

Here we can see nicely, that the perturbation Hamiltonian Hp introduces spin flip, when particles cross branches. For the remaining part we have to consider the boundary conditions. As mentioned above we will take hard walls, that leave us with the perpendicular wave functions χnz = 2πsin nπzW. Evaluating the matrix element then provides

mp̂z n =0Wdzχnzi zχn = 2iWnnn2n2 cosnπcosnπ1 ,

which was derived using partial integration from inserting the wave functions. For usage in section 4.2 we state, that eqn. (4.30) assures vanishing transition probabilities between even non-neighbouring states like P13. But at the same time, odd non-neighbouring states like P14 will again contribute to the level switching. We will use this result to explain the spin transmission functions for broader wires, where one has to take more open modes into account.

Inserting (4.30) eqn. (4.29) shows a dependence from the Rashba strength, that goes like 1 exp cα2. We will now use this formula to verify, that the Landau-Zener ansatz is adequate to describe the ratchet system. The spin resolved transmissions were already presented in fig. 4.9 and fig. 4.10. With a fitting parameter c = 42 we can reconstruct a similar result. This is presented in fig. 4.11.

Spin Current Production

PIC Spin current produced by the system introduced in the last section. The spin transmissions in the two rocking situations differ in the way implied by eqn. (4.31).

So far only systems in equilibrium were considered. We can see from fig. 4.9, that there will be a non vanishing spin current if it is measured always at the lead with the lower chemical potential, since in this case the relevant factor is just 2TS+V0. But by defining spin current like (4.1) this will hold no more and the ratchet ceases to work. Therefore in the Hamiltonian (4.20) a driving with a voltage offset was introduced. But this driving only leads to a spin current, if the transition probability between modes is somehow related to it.

In eqn. (4.29) we did not include the voltage drop. For this calculation we will assume it to sink linear along the device. Then the generalisation of Usx containing the additional potential due to the applied voltage takes the form:

UfullxV0=Usx +12xLV0,

with L the overall length of the scatterer. In the Landau-Zener probability this potential will be derived with respect to x. The above then yields

dUfullxdx=Usx+V0L

which in turn takes care of the voltage dependence of P12:

P12V0= 1 exp2πmςĤpnσ2Usx+V0Lv2,+Ev1,E .

We expect therefore, that the level crossing probability decreases with increasing voltage. This means, that the spin transmission at the lead in question will be higher in the case where it experiences the negative voltage and vice versa. Indeed, the spin transmissions shown in fig. 4.12 show exactly this behaviour.

The derivation above does not fit smoothly into the picture of a stepwise voltage drop, i.e., a voltage distribution, that is locally constant along the scatterer. But this is a very rough model, that states at a closer look, that the voltage completely drops (to a factor 12) at the very start of the scatterer, thus leading to two delta functions in the denominator in the exponent. We can assume in any case, that the voltage will sink somehow along the scatterer, which leads again to an x-dependence of the potential along the device. In the simplest case, this would just be the above linear decay, which yielded a reproducible result.

Extending Landau-Zener to Multiple Modes

PIC The dispersion relation for three open modes at a Rashba coupling strength of kαa = 0.16. We mark, that particles on their way “up” the (1-) branch, that are scattered into the second branch, cannot reach the crossing point between ch.s 2 and 3, that would allow them to enter the third mode.

We have seen in the previous section, that the Landau-Zener ansatz is appropriate to explain the spin rectification mechanism in the case of two open transmission channels. For realistic devices, the theory must hold for more modes, that is, for broader wires.

For the examination of the transmission functions we use again our model system from the last section. Expanding the system to a width of 15a yields a third open mode. The Rashba strength will not be altered. Now the barrier height must be chosen. Opposed to the 2-channel situation two cases have to be considered:

  1. The scattering barrier closes the second and the third mode. Like in the last section, only particles in the first mode will be transmitted.
  2. The barrier closes only the third mode, second and first mode will be transmitted.

The difference will be clear, when we look at the resulting probabilities for spin rectification in both cases. Therefore we take a look at the tables summing up the different transmission probabilities.

Transmission probabilities: only channel 1 stays open.
Tmς,nσ 1+ 1 2+ 2 3+ 3
1+ 1 0 0 0 0 0
1 0 1P1321P122 1P131P12P12 0 P131P131P132 0
2+ 0 1P131P12P12 P122 0 P13P121P12 0
2 0 0 0 0 0 0
3+ 0 P131P131P132 P13P121P12 0 P1321P122 0
3 0 0 0 0 0 0
Transmission probabilities: channels 1 and 2 stay open.
Tmς,nσ 1+ 1 2+ 2 3+ 3
1+ 1 0 0 0 0 0
1 0 1P1321P122 + 1P132P122 + P132P232 21P131P12P12 P13P231P23 P131P13 P122 + P232 + 1P122 0
2+ 0 2 1P131P12P12 P122 + 1P122 0 2 P13P121P12 0
2 0 P13P231P23 0 1P232 P231P231P13 0
3+ 0 P131P13 P122 + P232 + 1P122 2 P13P121P12 P231P231P13 P1321P122 + P132P122 + 1P132P232 0
3 0 0 0 0 0 0

Table 4.2 shows the probabilities for case 1. One can see, that the sub-table for the first two modes is equal to table 4.1, when the transition probability P13 vanishes. Additionally there are transition probabilities to and from the third mode. When we calculate the spin resolved total transmissions

T++ = 1 + P122 + 2 P12P131P12 + P1321P122 T = 1P1321P122 T+ = P121P121P13 + P131P1221P13 T+ = P121P121P13 + P131P1221P13

we get an expression for the spin transmission

TS = T++T = 2P12 + 2P132P12P13 .

We have already seen in eqn. (4.30), that the transition probabilities between even non-neighbouring states vanish. That means, we can set P13 = 0 and achieve a simple expression for the spin resolved transmission:

TS = 2P12 .

It is not surprising, that this result is exactly the same as in the case of two open channels, since there are no transitions back into the third mode. The crossing point between second and third mode cannot be reached from particles coming from the first channel, and a direct transition is prohibited. Because the second subband will be depopulated completely by the barrier, too, the third mode shows no contribution to the spin rectification effect.

When we repeat the calculation for the second case and consider table 4.3, where first and second mode stay open during the transit of a particle through the device, we find

T++ = 1 + P122 + 1P122 + 4 P121P12P13+ +P1321P122 + P132P122 + 1P132P232 T = 1P1221P132 + 1P132P122 + P132P122+ + P13P231P23 + 1P23P23P13 + 1P232 T+ = 2 1P131P12P12 + P131P13 P122 + P232 + 1P122 + + 1P23P231P13 T+ = 2 1P131P12P12 + P131P13 P122 + P232 + 1P122 + + 1P23P231P13

and for the spin transmission

TS = T++T = 2P232P13P23 + 2P13 + P232P132P132P122 .

If we again neglect P13, we find

TS = 2P23 ,

so the spin current depends only on the transmission probability between the second and the third mode. This result was also expected. A particle injected in the first open mode with spin down will in fact have a non-vanishing probability to end up in the second mode (2+), but since the probability for the reversed case is exactly the same, and the second subband is populated in this case, the totalled contribution to the spin transmission vanishes. The only possibility to break this symmetry would have been a transition of the outgoing (1-) particle into the (3+) branch. To conclude, the only relevant crossing to be considered is the one between second and third mode.

Remaining Validity of the Landau-Zener Ansatz

PIC Calculation of the model system: Channel 3 is closed by the barrier, channels 1 and 2 are transmitted.
PIC Fitting of the transition probabilities in fig. 4.14. P13 = 0 and the fitting parameters c12 = 60 and c23 = 30 were used.
PIC Calculation of the model system: Channels 2 and 3 are closed by the barrier, while channel 1 will be transmitted.
PIC Fitting of the transition probabilities in fig. 4.16. The fitting parameters are the same as in fig. 4.15.

We have to prove, that in the above case the assumption of a transition probability in the form of eqn. (4.29) is still correct. Furthermore the ratio between, e.g., P12 and P23 is of interest, likewise the prediction, that P13 vanishes. We can address these questions with the attempt to fit the numerical results in the same way that we introduced in section 4.1.5. The Landau-Zener probabilities of the form

Pnm = 1 expcnmkα2

will therefore be plotted against the Rashba strength. The fitting constants cnm determine the size of a certain probability. We compare these fittings to calculations of our model system. To simulate the two cases, we adjust the height of the barrier to close either only the third subband or both the second and the third.

Fig. 4.15 shows the results of the fitting for the case, where only the first mode will stay open. Comparing it to the calculations shown in fig. 4.14 we find good agreement in the single channel resolved contributions, as long as the Rashba strength is in a reasonable range. The fitting parameters are determined to

c12 =60 c13 =0 c23 =30

Going to a system, where both first and second mode stay open, we can fit the transition probabilities with the same set of parameters. The difference lies in the adaption of table 4.3 for the calculated values. The comparison between the calculated and the fitted transmission functions Tn,σς shows, that the qualitative behaviour is approximated good at least for small Rashba coupling strengths.

We see, that the transition probability between first and second channel is 0, as we expected. c12 is larger than c23. In fig. 4.17 we find, that this is the reason for the peaks of T2,+ and T3,+ to be shifted in the same way, as we can observe in fig. 4.16. One would in the first place argue, that c12 < c23. Eqn. (4.30) implies, that cnm nmn2m2, and thus c12 23, while c23 65. But since the crossing point between (2-) and (3+) is still beyond the injection energy in the selected system, its effect is reduced, just as we have seen in fig. 4.6.

Higher Modes

PIC Contributions of the single channels to TS plotted against the number of open transverse channels in the device. The channels contribute depending on the accessibility of the level crossing with the next higher channel. If this crossing drops below the barrier height of 0.22, particles will not be scattered in the higher mode, hence no spin-resolved transmission occurs.
PIC Spin transmission TS plotted against the number of open transverse channels in the quantum wire. The black solid curve is the sum of the contributions of fig. 4.18. The red dashed curve is derived from the assumption, that the crossing point with the next higher mode has to be achievable to produce spin current. Is this the case, the spin transmission is set to a fixed value of 0.9, which is in loose agreement to the contributions, that are found from the single channels (see fig. 4.18).

The next step after explaining the spin current generation for three modes is the examination of wires with arbitrary width. We take a look at the channel openings for an injection energy of kFa2 = 0.36. The number of channels as a function of the width of the wire is

nW = Waπ arccosɛ00.362γ1 0.194Wa

with ɛ0 = 4 and γ = 1 fixed by the discretisation procedure in chapter 3. We will take Wa as parameter and look at the spin transmission of the system. It is important to state, that we will fix the height of the barrier to 0.222 2ma2. This means, that, while the width of the wire increases, more and more modes will stay above the barrier height and will not be depopulated. After some steps, the first crossing point of a mode with its next higher neighbour will also remain above the barrier and will therefore be not achievable for the particle. If this is the case, neglecting crossings between non neighbours, the mode will not contribute to the resulting spin current.

On the other hand, while the mode is open and reachable crossings lead to channel mixing, the mode acts in the way, that we have outlined in the previous sections. We have therefore to compare different factors to determine, if a mode produces spin current. The channel must be open for transmission and a crossing point must lie in the energy span between the injection energy and the barrier maximum.

For charge current the expected result is a linear increase [80]. Since more channels will survive the barrier, more particles can be transmitted. Eqn. (4.34) assures, that the number of channels rises linearly, which holds true for the number of modes beneath the barrier. Spin resolved quantities show at the first glance no such obvious reason, why the spin transmission should rise linearly. We will show, that this is nevertheless the case, resulting from the wandering of the crossing points in the dispersion relation for increasing width.

Fig. 4.18 shows the contributions to the spin resolved transmission TS for each channel. We can see, that channels open and close again as a function of the width of the system. The resulting spin resolved transmission TS is shown in fig. 4.19. The second curve is derived by comparing the energy of the crossing point of a mode

Ekn=ma = π2W2 π216W2kα2 m2n22 + m2+n22

with the injection and the potential energy. If it is inside a certain area, that is chosen 0.3 higher than the maximal injection energy and 0.1 lower than the barrier height, the channel is marked as “open for spin transmission”. We assigned an arbitrary constant value of 0.9 to each contributing channel and 0 otherwise. The energy area is extended, because, as we stated in section 4.1.4, the crossing is not a sharp point and will be active even if it is not in the assigned energy span.

Though this is a very rough estimate, that could be improved by remodelling the shape of the contributions, that can be derived from fig. 4.18, we see a very good agreement between the numerical data and the estimated result.

Introducing Dresselhaus Spin-Orbit Coupling

We have introduced the Dresselhaus effect in section 2.3.3. This spin-orbit coupling based on the inversion asymmetry of the underlying crystal lattice becomes important, e.g., in InAs quantum wells [54]. In this section we want to outline the relevance of the orientation of the crystal relative to the quantum wire. Therefore we rotate the system around an angle Φ along the crystalographic axis by applying the operator

R̂yΦ =expiJyΦ= =cosΦsinΦsinΦcosΦspin cosΦsinΦsinΦcosΦreal

to the Hamiltonian, that includes a Dresselhaus part

Ĥcompl. = Ĥ +ĤD = Ĥ + kβmσ̂xp̂x σ̂zp̂z .

Here, Ĥ is the Hamiltonian (4.20). One finds, that the part without Dresselhaus term is invariant under the rotation

R̂yΦĤR̂y1Φ = Ĥ ,

whereas the Dresselhaus part becomes

R̂yΦĤDR̂y1Φ = kβm cos2Φσxpxσzpz+sin2Φσxpz + σzpx .

We have to update the tight binding Hamiltonian (3.4) with this result. This gives finally

H̃a = l,tɛ0 + Ũalt1l,t l,t +l,t1+ikαaikβasin2Φikβacos2Φikβacos2Φ1ikαa+ikβasin2Φ l,t l + 1,t+h.c. +l,t1+ikβacos2Φikαaikβasin2Φikαaikβasin2Φ1ikβacos2Φl,tl,t+1+h.c.

The Hamiltonian depends thus on twice the angle Φ between the [100]-direction of the crystal lattice and the x-direction of the quantum wire. Now we can recalculate properties of our model system from section 4.1. Again, a width of W = 12a leads, at injection energies up to 0.5, to two open modes at a reasonable accuracy.

PIC T++ against the angle Φ between the [100]-direction of the crystal lattice and the x-direction. The different curves relate to different values of the Dresselhaus coupling strength. For vanishing kβ the influence of Φ is trivial. At finite coupling strengths the dependence on 2Φ can be seen clearly from the axial symmetry for angles 45° and 135°.
PIC TS+V, the spin resolved transmission function in the rocking situation, where the voltage is positive at the left lead, against Φ. It is clearly visible, that the spin-resolved transmission vanishes completely, where kα = 0.1>6 = kβ.
PIC Resulting essential factor for the spin current IS in the two rocking situations TS+VTSV. The spin current vanishes as expected in the case, where kα = kβ. Additionally, it takes a maximum at Φ = 135, so that an experimentally realised qunatum wire for usage as spin ratchet should be grown in the 1̄01-direction for maximal efficiency.

The barrier height assures, that electrons in the second channels would be reflected completely. Particles will be injected at energies kFa2 = 0.36, which yields an open second transport mode. The Rashba strength of kαa = 0.16 has appeared in the plots above to produce reasonable results for spin current. We introduce now the new parameter kβa representing the Dresselhaus coupling strength. The variation of this parameter reveals basic differences in the figures 4.20, 4.21 and 4.22. The most obvious result is, that the spin current vanishes completely, if kα = kβ. This behaviour is explained in section 2.3.3 and ref. [81].

Fluctuations and Shot Noise

In this work we have so far only considered average current through the device. But these considerations miss an important source of information about the system, that was more and more brought to mind in mesoscopic physics through the last two decades. The current fluctuations produced in the output of a mesoscopic device can be used to gather additional properties of the system, e.g., detecting open transmission channels or distinguishing regimes of particle against wave propagation [82].

This chapter addresses spin shot noise in quantum ratchets. The first section will give a brief overview over shot noise in general. Then we will turn our focus on spin shot noise and finally we present results for the system, that we introduced in chapter 4.

Introduction to Noise

Reference [82] cites Rolf Landauer with the words “The noise is the signal”. Indeed, the investigation of noise in a system becomes more important, as it tends to scale down. Since the carrier of information in Spintronic devices is the spin current, the knowledge of correlations in this factor is a valuable source to determine system properties.

We will divide noise in general in two contributions [83]. The one is the thermal or Nyquist-Johnson noise, stemming from the thermal motion of particles in the conductor. As stated in chapter 1 thermal noise is not an ingredient to the ratchet mechanism in coherent systems. On the other hand shot noise is introduced by the discreteness of the current carriers. This can however carry information about the effectivity of the device. We introduce a simple argumentation, that leads us to expressions for the fluctuation spectra correlating reflection and transmission. In the next section we will use a scattering approach to derive the corresponding equations for the spin shot noise [84].

A particle in a waveguide is incident on a barrier. The probability to transmit is T, while reflection happens with probability R = 1T. When we consider the occupation number nin of the incident state, that should be identical 1, we find for the average occupation numbers of the outgoing states nT = T and nR = R. When we check the mean squared fluctuations in the two outgoing beams and their correlation, using the definition ΔnTR = nTR nTR, we readily find

ΔnT2 = ΔnT2 = ΔnTΔnT = TR .

This partition noise, stemming from the division of a beam into two streams, is maximal for T = 12 and vanishes for situations with certain result T = 1 and T = 0.

If we now let the incoming state be populated with the Fermi distribution, nin = f, the fluctuations take the form

ΔnT2 = Tf1Tf, ΔnT2 = Rf1Rf, ΔnTΔnT = TRf2.

The next consequent step is to relate this result with the charge current ITR through the device. This happens, as we have seen in chapter 2, by applying the Landauer-Büttiker ansatz, that relates S-matrix probabilities with the current. We then end up with the fluctuation spectra for transmitted and reflected current

SITIT = 2e2h dETf1Tf, SIRIR = 2e2h dERf1Rf

and with the correlation between these two

SITIR = 2e2h dETRf2 .

We point out, that the autocorrelation of the transmitted beam yields in the case of very small T or small f

SITIT = 2e I

the result of Schottky for shot noise of electrons in a vacuum tube [85]. In most systems, shot noise will be sub-Poissonian, that is, smaller than the Schottky value.

Shot noise was measured with success in various mesoscopic systems [48], e.g., tunnel barriers [86], resonant quantum wells with parallel magnetic field exhibiting super-Poissonian noise [87] or quantum Hall bars [88].

Shot Noise for Spin Currents

For application in Spintronics the spin current is selected as information carrier. It is therefore straightforward to investigate the shot noise of spin current in a corresponding system. We will use a scattering approach [4684] to derive an expression for this quantity. Therefore we recall from chapter 2 the expression for the spin current (2.26)

ÎSαxα,t=14πdEdEeiEEt βγmς,kςσâβmςE𝒜βmς;γkςα,σ;E,EâγkςE

with 𝒜 defined as

𝒜αmς;βkςL,σ;E,E=σ δmkδςςδαLδβLnLsLnσ;αmςEsLnσ;βkςE .

We are interested in the correlation function

Sαβtt = 12 ΔÎSαtΔÎSβt + ΔÎSβtΔÎSαt

between the leads α and β, which we will Fourier-transform to receive an expression for the noise power

2πδω+ωSαβω = ΔÎSαωΔÎSβω + ΔÎSβωΔÎSαω

with ΔÎS = ÎS ÎS. We hence Fourier-transform the spin current

ÎSαω =dtexpiωtÎSαt = 14πdEdEσβγâβEAβγασEEâγEdtexpiωt + iEEt = 12dEdEσβγâβEAβγασEEâγEδEE+ω

and arrive after evaluation of the integral over E at

ÎSαω = 12dEσβγâβEAβγασEE+ωâγE+ω .

The crucial factor for the correlation function is

ΔÎSαωΔÎSβω = ÎSαωÎSβω ÎSαω ÎSβω ,

where we will neglect the second part, since it is a constant value [89]. Inserting (5.2) two times we are left with

ΔÎSαωΔÎSβω = 14dEdEσσγδεζ âγEAγδασEE+ωâδE+ω âεEAεζβσEE+ωâζE+ω

For a Fermi gas the expectation value of the product of four operators â is

âαkE1âβlE2âγmE3âδnE4 âαkE1âβlE2 âγmE3âδnE4 = δαδδβγδknδlmδE1E4δE2E3fαE1 1fβE2 ,

which allows us to evaluate the above average to

ΔÎSαωΔÎSβω = 14dEσσγδδω+ωfγE1fδE+ω TrAγδασEE+ωAδγβσE + ωE .

We combine (3) with the expression for ΔÎSβωΔÎSαω and find finally for the noise power

2πSαβω = ΔÎSαωΔÎSβω + ΔÎSβωΔÎSαω = 14dEσσγδδω+ωfγE1fδE+ω + fδE+ω1fγE TrAγδασEE+ωAδγβσE + ωE .

In the following we go to the zero-frequency limit, since we are interested in low-frequency noise. Thermal noise vanishes with decreasing temperature, so that we can cut its contribution for the limit T 0. Since fα1fα = kBTdfαdE all terms with γδ disappear in this case. Therefore we find for the spectrum

Sαβ=1 4πdEσσγδTrσsαγsαδσsβδsβγ fγE1fδE .

We note, that the trace in this equation cannot be expressed in terms of transmission and reflection probabilities [48]. This is a sign of interference possibilities between particles from different carriers and shows, that they must remain indistinguishable.

We have to prove, that the trace will give a real result, since the single entries in the scattering matrix are in general complex. This is easy to see, when we write an entry in the S-matrix as sαβ = rαβexpiϕαβ. Then building the trace yields due to the conjugation always corresponding pairs of phases expiϕαβ +expiϕαβ = 2cosϕαβ.

Auto- and Cross-Correlation

We will now consider a two-terminal device with a left and a right lead. Then the auto-correlation of the outgoing current in the left lead reads

SLL = 1 4πdEσσγδTrσsLγsLδσsLδsLγ fγE1fδE = =14πdEσσ TrσsLLsLRσsLRsLL fLE1fRE+ TrσsLRsLLσsLLsLR fRE1fLE .

We simplify this expression by cyclic rotation under the trace and quadratic expansion of the last term:

fLfLfR + fRfLfR = fLfR2 + fL1fL+fR1fR

But since fγ1fγ = 0 in the zero temperature limit the expression for the auto-correlation in the left lead finally reads

SLL=14πdEσσTrσsLLsLLσsLRsLRfLEfRE2 .

Replacing the trace by summation over single indices we receive an expression that can be evaluated using the Lattice Green’s function method of chapter 3

SLL=14πdEnσLmςLL,n,σR,m,ς σsLnσ,LnσsLnσ,RmςςsLmς,RmςsLmς,LnσfLfR2 .

For the derivation of the cross-correlation SLR we have to make use of the unitarity of the scattering matrix. We find for S-matrices sαβ in the case of a two-terminal device

δsαδsβδ = 0 sαLsβL = sαRsβR .

Then the cross-correlation reads

SLR = 1 4π dEσσTrσsLLsLLσsRLsRL fLEfRE2 ,

and expressed in terms of single S-matrix entries

SLR=14πdEnσLmςRγ=L,n,σδ=L,m,ς σsLnσ,LnσsLnσ,LmςςsRmς,LmςsRmς,LnσfLfR2 .

Properties of the Correlations

Spin Shot Noise at Vanishing Current

Shot noise is connected to the current through the system. For vanishing particle current it holds true, that both correlations of the corresponding charge shot noise will also go to 0. This, however, is different in the case of spin current. Even in the case, when the output of the device is not spin polarized, there are still particles transmitted and reflected. Therefore one expects contributions to the spin shot noise even for ÎS = 0.

In the system of chapter 4 spin current surely does not occur, when the Rashba SO coupling is missing. We will evaluate the auto-correlation in this case. The crucial quantity has the form

nσLmςLnσLmςR σrnσ,nσtnσ,mςςtmς,mςrmς,nσ .

For kα = 0 there is no channel mixing and thus no off-diagonal elements in r and t with respect to the channel number. We can therefore restrict the sums to the case n = n = m = m. We remain with four sums over the spin state and two summands with n = 1 and n = 2, respectively:

σLςLσLςR σr11,σσt11,σςςt11,ςςr11,ςσ+ +σr22,σσt22,σςςt22,ςςr22,ςσ .

The vanishing Rashba coupling also prohibits spin flip. We can therefore introduce delta functions δσσδςσ and δσςδςς, that yield finally for the auto-correlation

SLL=14πdE r11,++t11,++t11,++r11,++ + r11,t11,t11,r11,+ r22,++t22,++t22,++r22,++ + r22,t22,t22,r22,fLfR2 .

This expression is in general non-vanishing. We find therefore contributions to shot noise even without spin current. Furthermore evaluating the cross-correlation we arrive at a similar result

SLR =14πdE r11,++r11,++t11,++t11,++ + r11,r11,t11,t11,+ r22,++r22,++t22,++t22,++ + r22,r22,t22,t22,fLfR2 .

But in this case, we can relate t and t via equation (2.13) and find, that the absolute values of auto- and cross-correlation are identical. This is surely the case, when the system does not produce spin current.

The Sign of the Correlations

For charge shot noise it has been shown [46], that auto-correlation in the zero temperature limit is positive defined, while the cross-correlation is negative. Due to current conservation we find for a two-terminal device additionally

SLLC = SLRC .

More generally, cross-correlation for fermions is negative at any temperature, while it can, under certain circumstances, become positive for bosons.

We have seen in section 2.2.2, that spin current is in general not conserved. Therefore equation (5.8) remains only valid for spin shot noise, when spin flip processes are suppressed. In this case we also find, that the trace of the auto-correlation is positively defined, so that SLR has to become negative.

But as soon as spin-flip processes occur, the elements under the trace can become negative. In certain cases the spin cross-correlation can have a positive value and the sum SLL + SLR is no more a conserved quantity.

By measuring the correlations of the outcome spin current of our device, we can make a fundamental prediction. If the auto- and the cross-correlation differ in their absolute value, the ratchet polarizes the current through the scattering region. Another important sign is a positive cross-correlation, that reveals bosonic behaviour of the involved electrons. This effect will show up only in the case of spin-flip processes, too.

Results for the Spin Ratchet

PIC Cross- and auto-correlation for a system without SO interaction. Even for vanishing spin current the output shows noise contributions. Auto- and cross-correlation have the same absolute value. The red (solid) curve shows T++ for comparison.
PIC Correlations in a system with both Rashba coupling and a scattering barrier equal to fig. 4.7. As soon, as the second channel opens, spin polarization occurs (compare chapter 4). In this region the correlations take on different absolute values. The red (solid) curve shows T++ for comparison.
PIC Cross- and auto-correlation for a multimode wire. The correlation functions are plotted against the number of open modes in the device. Whilst the cross-correlation remains around 0, taking on even positive values, the auto-correlation rises with the number of channels.

We return to the system described in section 4.1.4. The width is assumed to be W = 12a, therefore we will observe two open channels for injection energies higher than 0.274. We introduce again an adiabatic scattering barrier, that closes the second transmission channel, as well as Rashba SO coupling kαa = 0.16.

For the first examination, the SO coupling will be neglected. Then the transmission functions take a form that was depicted in figure 4.4. We find in figure 5.1 our prediction, that auto- and cross-correlation of the spin shot noise are present even in the case of vanishing spin current, well confirmed. In this case, the absolute values of both correlations are identical, as long as no spin current is produced.

Figure 5.2 shows however, that as soon, as the second channel opens, the two correlation functions start to differ. We have seen in section 4.1.4, that in this regime the ratchet begins to produce spin polarized current. The auto- and cross-correlation (5.6) and (5.7) indicate therefore the energies, where one can expect spin current.

In the last plot 5.3 the dependence of the correlations from the number of open transverse channels is illustrated. Recalling the rise of TS in fig. 4.19 we observe a similar behaviour for the auto-correlation. The cross-correlation however remains fluctuating around 0. Indeed for certain values it achieves positive values. This behaviour was above mentioned to be only possible for bosons. However, positive cross-correlations for charge current have been observed in N-N-S beam splitters, where the bosonic contribution of the Cooper pairs in the superconducting arm takes over the leading role in a wide parameter range [90]. In the case of two normal leads coupled capacitively to a conductor, the dynamic screening can as well lead to positive cross-correlation [91].

For spin shot noise a positive cross-correlation as well as the differing of the absolute values of auto- and cross-correlation were found numerically in systems of two-level quantum dots and two coupled quantum dots, respectively [84].

Conclusions

Summary

In the last 70 years electronics evolved to one of the most sophisticated areas of human productivity. In even less time, it will reach its final frontier drawn by thermodynamics and quantum mechanics. In addition to other approaches like molecular electronics [92] the field of Spintronics aims to keep the development of more powerful devices unbroken.

In this context the spin of particles plays a crucial role. Creation of spin current as a starting point for further applications is a central topic. We presented in this work a ratchet mechanism that could serve as spin current generator inside an all-semiconductor structure. Ratchets have been proven to work effectively as rectifiers of current in nanoscale environments. The application of a spin-selective potential is therefore a straight-forward advancement and has shown its potential in previous results [18]. We have introduced the basic concepts of spin ratchets in chapter 1.

The numerical adaption of a promissing system has to take into account several model assumptions. We outlined transport in quantum wires, the general shape of the ratchet device, in chapter 2, turning the attention especially on the connection between the transmission through a scattering region and the current found by Landauer and Büttiker. Particles with spin in a 2DEG are subject to spin-orbit like effects. We illustrated the Rashba and the Dresselhaus SO coupling as well as their interplay leading to a new conserved quantity. Chapter 3 was dedicated to the physical approximations applied for the numerical implementation of the device. We concentrated on the Lattice Green’s Function and showed the derivation of the transmission probabilities via the Fisher-Lee relations.

In this framework we introduced our model system in chapter 4. We considered a constricted region in a 2DEG creating a quasi one-dimensional wire (fig. 4.1). A symmetric scatterer and the presence of Rashba SO coupling take impact on the dynamics of electrons injected through one of the two leads. We applied a voltage offset between the leads and evaluated the average spin current with regard to two situations, where the offset was tilted.

The proper definition of spin current is a still open issue in literature [4950]. We followed the common definition of the spin current operator (2.23) and pointed out, that the leads to measure the net spin current of the system in the two opposite rocking situations have to be fixed, hence leading to two possible definitions of this quantity.

The ratchet functionality was examined for the simple case of two open transverse modes. The difference between the spin resolved transmissions T++ and T leading to the spin rectification was explained in the descriptive picture of Landau-Zener transmission probabilities. Both the adiabatic scatterer and the Rashba SO coupling were identified as essential ingredients, but due to the spatial symmetry of their assumed distribution the net charge current through the device vanishes. We showed therefore the creation of pure DC spin current in this simple case.

Realistic scenarios demand the examination of additional effects. Firstly, we expanded the ansatz, that was successful for two transmission channels, to a broader system. We compared numerical results for three modes to fittings based on the simple picture and found a qualitatively good agreement. The investigation of arbitrary numbers of open channels revealed a linear connection to the spin transmission, which we could reproduce by considering the “motion” of the interband crossing points on widening the wire.

Then, the Dresselhaus SO coupling was taken into account. We found a dependence of the functionality of the ratchet effect on the direction of the wire relatively to the underlying crystal lattice and confirmed the prediction of vanishing rectification in the case of Dresselhaus and Rashba effect being of equal strength.

Going beyond the average current we introduced the concept of spin shot noise in chapter 5. We examined the noise for vanishing spin transmission and found that the contributions of auto- and cross-correlation show the same absolute value. In the case of polarization in the device this holds no more, which can be used to experimentally detect regimes, where spin current can rise.

Perspectives

The system in question is based on a rather simple set-up. Therefore an experimental realization of the system and examination of effects should be possible with the state-of-the-art semiconductor technology. On the other hand, since the publication of Datta and Das’ spin field-effect transistor [8] 17 years went by without a published experimental success. However, an experimental try could give valuable information on the assumed principles and the validity of used approximations. The detection of the created spin rectification is an open issue, as it is in general for spin current in semiconductors [6], but considering shot noise properties this task could be managable.

These simplifications can also be seen as the starting point for further investigations. A central factor for the effectivity of the device was the adiabaticity and therefore the length of the scattering region. Ratchets however are considered generally as periodic structures rather than a saw with only one tooth. To stay within the regime of ballistic transport, characterized by the phase coherence length, barriers in an array have to be much shorter giving rise to resonance phenomena.

Ref. [78] shows results for a ratchet system with up to 100 scatterers revealing under certain circumstances a strong amplification of the rectification. The findings cannot be explained within the simple model applied for the adiabatic barrier. Furthermore the resonant states could possibly have a life time that reaches the magnitude of the switching frequency of the external driving. In this case the interplay between the applied AC voltage and the transmission probabilities needs further investigation.

The shot noise is also an open issue. The rise of the auto-correlation in broader wires and the regimes with positive cross-correlation can yield additional information about the working and the magnitude of spin rectification. Finally, the two rectification systems of Zeeman and Spin-Orbit ratchets were developed separately. An interplay between these two effects can give rise to interesting effects and may be the source of a new, combined ratchet mechanism.

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Erklärung

Hiermit erkläre ich, dass ich die Diplomarbeit selbstständig angefertigt und keine Hilfsmittel außer den in der Arbeit angegebenen benutzt habe.

Regensburg, Juli 2007

Manuel Strehl