Coherent Spin Ratchets: Transport and Noise Properties

Preface

Throughout the 20^th 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 [5, 6]. 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 [12, 13, 14], 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 [15, 16], 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

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 [6, 21]. 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 [23, 24, 25].

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 [15, 16]. 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 20^th 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 [15, 29].

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

mẍt + ηẋt = −V′x + 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,

ξ t ξ (t') = 2 η k B T δ (t - t') .

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

ẋ = lim t \to 0 x t - 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 [32, 33, 34]. 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

Ĥ t = p ̂ 2 2 m + V (x ̂) - x ̂ F t + Ĥ B (x ̂, 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 T→0 the ratchet still produces current. This result can not be understood in terms of classical dissipative systems.

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 [37, 38, 39]. 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.

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

I V 0 = 12 I + V 0 + I - V 0 .

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 T→0 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=limV→0IV∕V. For this case, the currents in the two rocking situations would cancel out, I+V=I−V, so that the average current can only be diﬀerent from zero, if higher order terms are respected,

I V = G 0 V + G 1 V 2 + G 2 V 3 + \dots

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 [17, 41]. 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 [42, 43], 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” [6, 5].

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 = ∑nσ∈L anσ χnσ ψnσ+ + ∑nσ∈L 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

ℋ Fock = ℋ (0) \oplus ℋ (1) \oplus \dots \oplus ℋ 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

âLn†E b̂Ln†E creation operators

These operators obey the anticommutation rule

â α m † E, â β n E' + = δ α β δ m n δ E - E'

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

b ̂ m ς = \sum n σ \in L r m ς, n σ â n σ + \sum n σ \in R t m ς, n σ' â n σ

where r and t′ refer to submatrices

S = rt′tr′ .

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

Tm←n,ς←σ = |tmς←nσ|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

â † â = b ̂ † b ̂ \Rightarrow â † 1 - S † S â = 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† =S−1 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

∑mς∈L⋃R|smς,nσ|2 =∑mς∈L|rmς,nσ|2 +∑mς∈R|tmς,nσ|2 = 1 .

Analogous a rule for the incoming amplitudes exists:

∑nσ∈L⋃R|smς,nσ|2 =∑nσ∈L|rmς,nσ|2 +∑nσ∈R|tmς,nσ′|2 = 1 .

We deﬁne the transmission function TE as

TE =∑mς,nσ|tmς∈R,nσ∈LE|2

and can relate this to functions RE and NLE =∑nσ∈L1, the number of modes in the lead L, via eqn. (2.8) and eqn. (2.9):

TE = NLE−RE

and

T E = N R E - R' E .

Comparing these with the corresponding relations for T′E, 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 = T′E .

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

K ̂ ≔ - i σ y Ĉ,

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

K ̂ p ̂ x ∕ z K ̂ - 1 = - p ̂ x ∕ z

and the spin of the state is ﬂipped,

K ̂ σ ̂ x ∕ z K ̂ - 1 = - σ ̂ x ∕ z .

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

K ̂ b n, σ = σ a n, - σ * .

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

σ a n, - σ * = \sum m \in L ⋃ R \sum ς = \pm 1 σ s - 1 n m, - σ ς * b m, ς * .

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

K ̂ b n, σ = \sum m \in L ⋃ R \sum ς = \pm 1 - σ ς s - 1 n m, - σ ς * K ̂ a m, - 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:

K ̂ b n, σ = \sum m \in L ⋃ R \sum ς = \pm 1 s n m, σ ς K ̂ a m, 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 =ℏ2mi ∂∂x⃗ −∂∂x⃖

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

ÎPx∈L,t=∫dzΨL†x,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,x∈L ,

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 = ∫dEe−iEt∕ℏχnz ∑nσ∈L âLnσEΨ̂Lnσ,E+xz +b̂LnσEΨ̂Lnσ,E−xz .

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

k n E \approx k n E',

that allows us to derive the expression

ÎPx ∈ L,t = eh∫dEdE′eiE−E′t∕ℏ ∑nσ∈L âLnσ†EâLnσE′ −b̂Lnσ†Eb̂LnσE′ .

This equation can be rewritten to

ÎPx ∈ L,t = eh∫dEdE′eiE−E′t∕ℏ ∑αβ∑mς,kς′âαmς†E𝒜αmς;βkς′L;E,E′âβkς′E′

by introducing a matrix 𝒜 with

𝒜αmς;βkς′L;E,E′ = δmkδςς′δαLδβL −∑nσ∈LsLnσ;αmς†EsLnσ;βkς′E′

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

â α m ς † E â β k ς' E' = δ α β δ m k δ ς ς' δ E - E' f α E,

with fαE the Fermi-Dirac distribution in the lead α

fαE =11 +expE−μα∕kBT .

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

Î C x \in L, t = e h \int d E N L E - R E f L E - T' E f R E .

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

Î C x \in L, t = e h \int d E T' E f L E - f R E,

Since TE = T′E, 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 [49, 50]

ĴS,x = ℏ 2σzĴx = ℏ2 4m⋆iσz ∂∂x⃗ − ∂ ∂x⃖ .

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

σ z = 10 0 - 1 .

With this deﬁnition the spin current takes the form

Î S x \in L, t = \int d z Ψ ̂ L † (x, z, t) Ĵ S, x Ψ ̂ L (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 4π∫dEdE′eiE−E′t∕ℏ ∑nσ∈L σ âLnσ†Eâ LnσE′ −b̂ Lnσ†Eb̂ LnσE′ .

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

𝒜αmς;βkς′(L,σ;E,E′) = σ δ mkδςς′δαLδβL −∑n∈LsLnσ;αmς†Es Lnσ;βkς′E′

we arrive at

ÎSx ∈ L,t = 1 4π∫dEdE′eiE−E′t∕ℏ ∑αβ∑mς,kς′∑σâαmς†E𝒜 αmς;βkς′(L,σ;E,E′)â βkς′E′ .

In a last step we take again the thermal average:

ÎSx∈L = 1 4π∫ dE∑α∑mς∑σσ 1 −∑n∈LsLnσ;αmς†s Lnσ;αmς fαE = 1 4π∫ dE∑σ σ∑mς∈L 1 −∑n∈LrLnσ;αmς†r Lnσ;αmς fLE+ −σ∑mς∈R ∑n∈Lt Lnσ;αmς′†t Lnσ;αmς′fRE

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

T ς σ E = \sum m, n | t m ς, n σ E | 2

and

R ς σ E = \sum m, n | r m ς, n σ E | 2 .

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

T E = \sum ς σ T ς σ E .

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

TSE =∑ςσςTςσE = T++E + T+−E−T−+E−T−−E .

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

R ς + E + R ς - E = 12 N L E - T ς +' E - T ς -' E,

and

R ς +' E + R ς -' E = 12 N L E - T ς + E - T ς - E .

So we arrive at

T S' E = - R S E .

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

ÎSx∈L = 1 8π∫dE fLE−fRETS′E .

We can also see a very important property of the spin current. Since in general TSE≠TS′E 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

\partial ρ \partial t + \nabla \to \cdot j \to = 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:

\partial ρ S \partial t + \nabla \to \cdot j \to S = τ S .

This torque density takes the form [50]

τ S = Re Ψ ̂ † 1 i ℏ σ ̂ z Ĥ Ψ ̂ .

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

ℐS = 12 ÎSx∈R,+V0 + ÎSx ∈ L,−V0 =18π∫dEΔfEV0 TSEV0−TS′E−V0 ,

with ΔfEV0 = fEεF+V0∕2−fEεF−V0∕2 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 TSE0−TS′E0, 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,

ISx∈L = 12 ÎSx∈L,+V0 + ÎSx ∈ L,−V0 =18π∫dEΔfEV0 TS′EV0−TS′E−V0 .

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 [51, 52]. 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 [51, 55, 56].

Origins of Spin-Orbit Coupling

An electron in the electrostatic ﬁeld of a proton moves with a velocity v = p∕me [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 c 2 v \times E (r),

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

H S O \sim \nabla Φ (r) σ \times p .

This form was derived by quadratic v∕c 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

Ĥ α = α ℏ σ \times p z .

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 = ℏkα m⋆ (σxpz−σzpx) ,

with α ℏ = ℏkα 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 = Ĥ 0 + Ĥ p Ĥ 0 = p ̂ x 2 2 m ⋆ - ℏ k α m ⋆ σ ̂ z p ̂ x + p ̂ z 2 2 m ⋆ + V c z Ĥ p = ℏ k α m ⋆ σ ̂ x p ̂ z

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⋆(p̂x−ℏkασ̂z)2 −ℏ2kα2 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

p ̂ z 2 2 m ⋆ χ n z + V c z χ n z = ɛ n χ n z

with transverse wave functions χnz = 2∕πsin(nπz∕W), 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:

Ψ n, k, σ x z = 1 2 π e i k x χ n z 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 = ℏ 2m⋆(k−σkα)2 −ℏ2kα2 2m⋆ + ɛn .

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, σ Ĥ p n', k', σ' = ℏ k α m ⋆ δ k - k' δ σ, - σ' n p ̂ z 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 n≠n′ 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 [42, 60]. 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

E n, - (k n = m) =! E m, + (k n = m) \Leftrightarrow k n = m = 2 m ⋆ ℏ 2 ɛ m - ɛ n 4 k α .

The weak coupling regime is identiﬁed with the condition kn=m ≫ kα. We rewrite the level energies as ɛn = ℏ2kn2 2m⋆ and introduce the spin-precession length Lp = π∕kα [42]. In a one-dimensional quantum waveguide with hard wall conﬁnement kn ∝ nπ∕W holds. Then the condition takes the form

kα2 = πLp 2 ≪ 14m−n2 πW2 .

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

W ∕ L p ≪ 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 [12, 54] 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 ℏ 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, e→x ∥ 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:

k y \sim π d y 2 .

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

Ĥ k 3 \approx γ ℏ K ̂ y 2 (σ ̂ x p ̂ x - σ ̂ z p ̂ z) - γ ℏ (σ ̂ z p ̂ z K ̂ x 2 - σ ̂ x p ̂ x K ̂ z 2) .

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 kx∕z can be neglected. A good approximation for the Dresselhaus Hamiltonian reads then

ĤD = ℏkβ m⋆ (σ̂xp̂x −σ̂zp̂z) .

In this equation we introduced a factor kβ = m⋆ ℏ2 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 = Ĥ wire + Ĥ α + Ĥ β \Rightarrow P ̂ Ĥ p ̂ - 1 = 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 = Ĥ wire + Ĥ α + Ĥ β =! Ĥ 0 + Ĥ p,

Ĥ 0 = p ̂ x 2 2 m ⋆ - ℏ k α m ⋆ σ ̂ z p ̂ x + p ̂ z 2 2 m ⋆ - ℏ k β m ⋆ σ ̂ z p ̂ z + V c z, Ĥ p = ℏ k α m ⋆ σ ̂ x p ̂ z + ℏ k β m ⋆ σ ̂ x p ̂ x .

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

Ψ n, k, σ (0) x z = 1 2 π e i k x ψ n, σ z σ,

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

ψ n, σ z = e i σ k β z χ n z .

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

E n, σ (0) k = ℏ 2 2 m ⋆ (k - σ k α) 2 + ɛ n - ℏ 2 2 m ⋆ (k α 2 + k β 2) .

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, σ Ĥ p n', k', σ' = ℏ k α m ⋆ δ (k - k') δ σ, - σ' n p ̂ z n' + ℏ k β m ⋆ ℏ k δ (k - k') δ n, n' δ σ, - 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 = p ̂ x 2 2 m ⋆ + p ̂ z 2 2 m ⋆ + V c z + ℏ k S O m ⋆ p ̂ x (σ ̂ x - σ ̂ z) + p ̂ z (σ ̂ x - σ ̂ z) = p ̂ x 2 2 m ⋆ + p ̂ z 2 2 m ⋆ + V c z + ℏ k S O m ⋆ p ̂ x 2 Σ ̂ + p ̂ z 2 Σ ̂ .

The new spin operator

Σ ̂ : = 12 (σ ̂ x - σ ̂ z)

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

Ĥ p ̂ x = Ĥ Σ ̂ = 0

we ﬁnd, that there exists a set of exact eigenstates

Ψ n, k, Σ (α = β) x z = 12 e i k x ψ n, Σ z \pm,

ψ n, Σ z = e - i 2 Σ k S O z χ n z,

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 ⋆ = ℏ 2 d 2 ɛ d k 2 - 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

E - Ĥ Ψ = 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

E−ĤGE = 1 .

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

G E = 1 E - 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 \pm = lim η \to 0 + G (E \pm i η) .

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 = ∑nΦnΦnE−En ± 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

i G + t = i 2 π \int d E G + E e - i E t ∕ ℏ = θ t e - i Ĥ t ∕ ℏ e - η t ∕ ℏ,

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

Φ t 2 = U t 2, t 1 Φ t 1 .

For the spatial representation of the state this yields

Φ r 2, t 2 = \int d 3 r 1 r 2 U t 2, t 1 r 1 r 1 Φ t 1,

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

K r 2, t 2; r 1, t 1 = r 2 U t 2, t 1 r 1 θ t 2 - t 1 .

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 = \sum r r ɛ r r + \sum r, r' r V 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) = \sum r δ (x - r) Ψ (r) .

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

â r, â r' † + = δ r, r' â r, â r' + = 0 = â r †, â r' † +

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

Ĥ =∑rɛrâr†âr +∑r,r′ V r,r′âr†âr′ + h.c. .

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

x \to l a z \to t a,

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 = p ̂ x 2 + p ̂ z 2 2 m ⋆ + U x z + ℏ k α m ⋆ (σ ̂ x p ̂ z - σ ̂ z p ̂ x) + ℏ k β m ⋆ (σ ̂ x p ̂ x - σ ̂ x p ̂ x) .

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,

x 0, z 0; σ p ̂ x Ψ = ℏ i \partial \partial x x 0 Ψ σ (x, z 0) = ℏ i Ψ l + 1, t σ - Ψ l - 1, t σ 2 a + 𝒪 (a 2) x 0, z 0; σ p ̂ z Ψ = ℏ i \partial \partial z z 0 Ψ σ (x 0, z) = ℏ i Ψ l, t + 1 σ - Ψ l, t - 1 σ 2 a + 𝒪 (a 2) x 0, z 0; σ p ̂ x 2 Ψ = - ℏ 2 \partial 2 \partial x 2 x 0 Ψ σ (x, z 0) = - ℏ 2 Ψ l + 1, t σ - 2 Ψ l, t σ + Ψ l - 1, t σ 2 a + 𝒪 (a 4) x 0, z 0; σ p ̂ z 2 Ψ = - ℏ 2 \partial 2 \partial z 2 z 0 Ψ σ (x 0, z) = - ℏ 2 Ψ l, t + 1 σ - 2 Ψ l, t σ + Ψ l, t - 1 σ 2 a + 𝒪 (a 4) .

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

x 0, z 0 Ĥ Ψ = - ℏ 2 2 m ⋆ Ψ l + 1, t + - 2 Ψ l, t + + Ψ l - 1, t + Ψ l + 1, t - - 2 Ψ l, t - + Ψ l - 1, t - - ℏ 2 2 m ⋆ Ψ l, t + 1 + - 2 Ψ l, t + + Ψ l, t - 1 + Ψ l, t + 1 - - 2 Ψ l, t - + Ψ l, t - 1 - + U (l a, t a) Ψ l, t + Ψ l, t - + ℏ k α m ⋆ ℏ 2 i a σ ̂ x Ψ l, t + 1 + - Ψ l, t - 1 + Ψ l, t + 1 - - Ψ l, t - 1 - - ℏ k α m ⋆ ℏ 2 i a σ ̂ z Ψ l + 1, t + - Ψ l - 1, t + Ψ l + 1, t - - Ψ l - 1, t - + ℏ k β m ⋆ ℏ 2 i a σ ̂ x Ψ l + 1, t + - Ψ l - 1, t + Ψ l + 1, t - - Ψ l - 1, t - - ℏ k β m ⋆ ℏ 2 i a σ ̂ z Ψ l, t + 1 + - Ψ l, t - 1 + Ψ l, t + 1 - - Ψ l, t - 1 - .

This expression can be simpliﬁed by collecting terms. To receive the Hamiltonian dimensionless, we additionally divide by ℏ2 2m⋆a2. With ɛ0 = 4 the on-site energy and Ũ = U∕ ℏ2 2m⋆a2 the dimensionless potential we arrive at

H ̃ a = Ĥ ∕ ℏ 2 2 m ⋆ a 2 = \sum l, t ɛ 0 + Ũ l t a l t l t + \sum l, t - 1 + i k α a - i k β a - i k β a - 1 - i k α a l, t l + 1, t + h . c . + \sum l, t - 1 + i k β a - i k α a - i k α a - 1 - i k β a 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:

lengths with a,
energies with a2,
SO coupling constants with 1∕a.

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)𝟙−ĤL τLS τLS† (E + iη)𝟙−ĤS −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 [64, 19]. 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 + i η) 𝟙 - Ĥ L G L S + τ L S G S = 0

and ﬁnd

G L S = - g L τ L S G S

with gL = (E + iη)𝟙−ĤL −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), E𝟙−ĤS GS + τLS†GLS = 𝟙 , and ﬁnd ﬁnally

GS = E𝟙−ĤS−τLS†gLτLS −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 := τLS†gLτLS, 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 + i η) G S - H S G S = 𝟙

into an expression in the form of the Dyson equation

G S = G L + G L (τ L S † g L τ L S) G S,

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)

H ̃ a = \sum l = 1 \infty \sum t = 1 M ɛ 0 l t l t - \sum l = 1 \infty \sum t = 1 M l, t l + 1, t + h . c . - \sum l = 1 \infty \sum t = 1 M 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

Σ S E = \sum n = 1 M χ n t χ n † t' F E

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

Ψ n, k x z = 2 L 0 sin (k x) 2 W sin (π n W z) = Φ k x χ n z .

The eigen-energies are simply

Enk = ɛn + ℏ2 2m⋆k2 = ℏ2 2m⋆ n π W2 + ℏ2 2m⋆k2 ,

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

Ψ n, k (l, t) = 2 a L 0 sin (k l a) 2 M + 1 sin (π n M + 1 t) = Φ k l χ n t .

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

Enk = ɛ0−2cos(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 ẋ=iℏĤx. If we use now the dispersion relation (3.10), that we found above, we arrive at the expression

vnk = nkẋnk = 1ℏ ∂∂kEnk = 2aℏsin(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 = r t' t r'

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,

Ψ n, k, σ (l, t) = a 2 π e i k l a 2 M + 1 sin (n π M + 1 l) 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 [67, 68]

R m ς, n σ = - δ m n δ ς σ + i ℏ a v m ς E v n σ E \sum j \in A \sum i \in A Ψ m ς, E † (0, j) G j ς, i σ E Ψ n σ, E (0, i)

and for the transmission

t m ς, n σ = i ℏ a v m ς E v n σ E \sum j \in B \sum i \in A Ψ m ς, E † (x B, j) G j ς, i σ E Ψ n σ, 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 GC−1 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

G C - 1 = ℏ 4 π e 2 1 N (ɛ F),

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

G - 1 = ℏ 4 π e 2 1 T (ɛ F)

with T(ɛF) the total transmission probability (2.10). With this knowledge we can determine the resistance GS−1 of the central region in the device:

G S - 1 = G - 1 - G C - 1 = ℏ 4 π e 2 1 N (ɛ F) R (ɛ F) T (ɛ F) .

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:

I S R = 12 (I S (R, + V 0) + I S (R, - V 0)) .

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

ISR=18π∫dEΔf(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= fR−V0. 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 h∫dETE(fRE−fLE) .

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

I C R = 12 (I C (R, + V 0) + I C (R, - V 0))

and insert eqn. (4.2):

IC = e 2h∫dEΔfEV0 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.,

Ĥ Â = 0 .

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

Â Ĥ Ψ = Â E Ψ \Rightarrow Ĥ (Â Ψ) = E (Â Ψ),

so that ÂΨ is also an eigenfunction of the Hamiltonian Ĥ. We keep in mind, that application of Â 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:

P ̂ U s x p ̂ - 1 = U S (- x) = U s x P ̂ V c z p ̂ - 1 = V c (- z) = V c z P ̂ p ̂ x p ̂ - 1 = - p ̂ x P ̂ p ̂ z p ̂ - 1 = - p ̂ z P ̂ σ ̂ x p ̂ - 1 = - σ ̂ x P ̂ σ ̂ z p ̂ - 1 = - σ ̂ 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 )p̂−1 = 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

χ n (- z) = p n χ n z with p n = \pm 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

b ̂ m ς \in L = \sum n σ \in L r m ς, n σ â n σ + \sum n σ \in R t m ς, n σ' â n σ .

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:

Ψ L (x, z, t) = \int d E e - i E t ∕ ℏ \sum n σ \in L â n σ E ψ n σ, E + x z + b ̂ n σ E ψ n σ, E - x z

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

P ̂ b ̂ n \in L, σ = σ p n b n \in R, - σ = \sum m ς \in L σ p n s n, - σ; m ς (V) â m ς + \sum m ς \in R σ p n s n, - σ, m ς (V) â m ς n \in R

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:

P ̂ â n \in L, σ = σ p n â n \in R, - σ .

Inserting in the expression for b̂ leaves us with

P̂b̂n∈L,σ = ∑mς∈Lσςpnpmsn,−σ;mς(V )(P̂âmς)+ +∑mς∈Rσςpnpmsn,−σ,mς(V )(P̂âmς)|n∈R

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

P̂b̂n∈L,σ = ∑mς∈Lsn,−σ;mς(−V )(P̂âmς)+ +∑mς∈Rsn,−σ,mς(−V )(P̂âmς)|n∈R.

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

snm∈L;σς(−V ) = σςpnpmsnm∈R;−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

snm∈L;σς(−V )2 = s nm∈R;−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 2h∫dE fRE−fLE 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

snm∈L;σς 2 = s nm∈R;−σ,−ς 2 .

Taking a look at the spin mixing terms we ﬁnd

snm∈L;+−2 = s nm∈R;−+ 2 s nm∈L;−+ 2 = s nm∈R;+−2

Additionally the time-reversal symmetry (2.13) gives

snm∈L;+−2 = s mn∈L;+−2 s nm∈L;−+ 2 = s mn∈L;−+ 2

Therefore we can show

T + - E = parity T - +' 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 [69, 70, 71], e.g., in ferromagnetic structures. T++E and T−−E 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 T−−E.

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 T−−E 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 = ℏkαm⋆ σxp̂z−σzp̂x ,

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

Ĥ′t = −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

V t = + V 0 for 0 \leq t < T ∕ 2 - V 0 for T ∕ 2 \leq 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 = ℏ∕m⋆2πn ≈ 3 ⋅ 107cm/s one expects usual transmission times of about L∕vF ≈ 10−12s. Here a carrier density n = 5 ⋅ 10−11cm−2 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 R∕T. 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 [73, 74, 75] 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 + Ĥ′t .

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 −ℏkαm⋆ σ̂zp̂x and a perturbation part Ĥp = ℏkαm⋆(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, dUsx∕dx ≪ 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+
1+	1	0	0
1−	0	1−P122	1−P12P12
2+	0	P121−P12	P12P12
2−	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−− = 1−P122

T−+ = 1−P12P12

T+− = P121−P12

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 =∑σσ′σ′Tσσ′ = 1 + P122 + 1−P122 = 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 ),

T S V - T S' - V = 4.9 2 T S V,

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

ℐ S = 1 8 π \int d E Δ f E, V 0 T S E; + V = 1 4 π \int d E Δ f E, V 0 P

Coherent Spin Ratchets

Transport and Noise Properties