J. Hugo Dil, Physik-Institut, Universität Zürich, 8057 Zürich, and Swiss Light Source, Paul Scherrer Institute, 5232 Villigen PSI
Spintronics is the vision of using the spin of the electron rather than its charge to control information flows, which can be exploited in a future quantum computer. In the field of data storage, spintronics has already become reality; the giant magnetoresistance effect, for the discovery of which Fert and Grünberg received the 2008 Nobel prize, is used in modern memory devices. Currently many researchers are working on systems where small magnetic structures are used to design a complete spintronics ensemble. The magnetic moments of the electrons, i.e. the spin, lead to a net magnetization in these structures. As we know from our basic physics lectures the spin of an electron can in the simplest approach be manipulated by magnetic fields. Here we will discuss a possible alternative which is fully based on non-magnetic materials and utilizes the spin-orbit interaction in low-dimensional structures to control the electron spin. The advantage of this approach is that it can lead to reduced information processing time at even lower energy consumption and is as such highly interesting and promising for future applications.
For any useful spin computer one needs at least the following components:
Because spin-orbit interaction is a dynamic process it is diffcult to envisage spin-storage based on such an effect, as data storage requires a steady state solution and thus will most likely remain based on magnetic structures. The possible realization of the other three components without magnetic materials is discussed in the next sections.
The Rashba-Bychkov effect is a spin-splitting of bands in a 2D electron gas due to a net electric field and spin-orbit coupling. Although initially described for semiconductor heterostructures it was first observed on the surface of gold. The spin splitting does not result in a net magnetization of the system.
Spin manipulation; the Rashba effect
Spin manipulation without magnetic fields relies on spin-orbit coupling. A promising candidate to achieve a controlled manipulation is the Rashba effect, which relies on the breaking of space inversion symmetrie, as will be explained below.
In many cases the surface of a metal (or semiconductor) crystal is not just a truncation of the bulk. For instance at semiconductor surfaces the dangling bonds which are formed due to the sudden truncation of the crystal, induce a rearrangement of the atoms to lift these free bonds. Furthermore due to the confinement between a projected band gap and the image potential, surface states can form, which have totally different properties as the electronic states in the bulk material. In both cases the main cause of the differences is that at the surface the translational symmetry is broken along the surface normal, or, more general, that the space inversion symmetry is broken.
For an electron with a given momentum and spin space inversion symmetry means that it is equivalent whether the electron moves in one direction or the other; i.e. . In the absence of a magnetic field, time inversion symmetry holds; i.e. . In the bulk of a non-magnetic, centrosymmetric metal both time and space inversion symmetry are observed, resulting in the formation of spin-degenerate states For states located at a surface or interface (e.g surface states or quantum well states) the space inversion symmetry is actually broken, which means that the spin degeneracy does not necessarily hold for these states.
That the spin degeneracy is actually lifted for surface states can be understood by the following simple relativistic argument. The sudden termination of the crystal at the surface creates a potential gradient perpendicular to the surface, which can also be regarded as a local electric field. In the rest frame of a moving valence electron this electric field becomes a magnetic field through a Lorentz transformation. This magnetic field causes a Zeeman splitting of the electronic states and thus an energy difference between the states with different spin orientations. The magnitude of this magnetic field and thus also the energy splitting depends on the momentum of the electron and changes sign for opposite momenta. At zero momentum the splitting disappears and the bands are degenerate.
This momentum dependent energy splitting of surface or interface states is typically referred to as the Rashba (or Rashba-Bychkov) effect (see Ref.  and references therein). The detailed mathematical description of this effect goes beyond the scope of this article but the effect on the band structure of a two-dimensional nearly free electron gas is shown in Fig. 1(a) and can be summarized as follows. The original spin-degenerate parabola is split in two spin-polarized parabolae which are shifted by k0 from the zone centre. A cut along a constant energy plane shows two concentric circles with opposite spin rotation directions. In this simple model the spin will always be perfectly tangential to the constant energy contour, for more complex systems the spin may however deviate from this and can even show a sizeable out-of-plane polarization.
The use of this Rashba-type spin-splitting to manipulate the spin of an ensemble of electrons is based on basic quantum mechanical arguments and follows the lines of a proposal from 1990 . An electron with its spin along the positive z axis can be decomposed into electrons with spins along the y axis, reading .
Similarly, a spin along x can be written as
A phase difference between thus causes a rotation in the xz plane. This phase difference is exactly what is accumulated along an electrons path through a solid due to the different vectors at a given energy (typically Fermi energy) of the bands with opposite spin. As a result, the spin of an electron travelling through this Rashba medium will rotate as a function of distance travelled, where the amount of rotation depends on the momentum difference between the two parabolae; i.e. 2k0. The goal is now to manipulate this momentum splitting by an external parameter so that one can directly control the spin of an electron at the end of this medium. While this can be done by a gate voltage for some structures, we will here discuss the influence of structural changes. In our group we have shown by spin and angle-resolved photoemission that the Rashba splitting can be tuned by mixing different amounts of Pb and Bi on a Ag(111) surface . Furthermore we have found that the spin-splitting of quantum well states in thin Pb films on Si(111) shows a strong dependence on the interface barrier  and may thus be tuned through the charge density in the substrate.
Spin injection and transport
A spin injector or spin filter can be defined as a medium which creates a current with a certain degree of spin-polarization from an initially unpolarized current. In a magnetic material this is achieved through the difference in density-of-states (DOS) of the majority and minority spin states at the Fermi level. As a result of the Rashba effect the density-of-states of the nearly two-dimensional electron gas is no longer constant, but shows a strong energy dependence as depicted in Fig. 1(b). Between the crossing point of the two parabolae and the band apex the DOS shows a Van Hove type singularity, which is predicted to have substantial influence on the electronic properties through the enhanced electron-phonon coupling. An other interesting situation arises exactly at the crossing point , here the DOS of one spin orientation vanishes whereas the other spin direction has a large DOS. At the interface between a Rashba medium and a normal metal the spin polarization of a current accross this interface can be as large as 80% . Depending on the polarity of a voltage accross the interface, a Rashba-spin-degenerate metal interface can either be used as a spin injector or acceptor.
Van Hove singularity
The density-of-states of a 3D free electron gas shows a square root dependence on the energy. For a 2D system the DOS is constant or steplike. For a 1D system the DOS scales with 1/√(E0 - E), this type of behaviour is refered to as a Van Hove type of singularity after the Belgian physicist Léon Van Hove (1924-1990).
This mechanism functions even better if the DOS of one spin direction would not only go to zero, but if the band would not be present around the Fermi level at all. This is exactly what happens for the novel class of materials called topological insulators . In a simple picture a topological insulator is a band insulator with a metallic surface which supports an odd number of spin polarized states for any momentum direction. Although some of these spin channels will ”cancel“ with channels with opposite spin, there will always be at least one spin polarized channel left over. Because time-inversion symmetry still hold this state forms a pair (Kramers pair) with a state with opposite momentum and spin. If an electron encounters an impurity it is typically forced to partly reverse its direction of motion, which causes an decrease in conductance. For an electron travelling in one of the channels of a topological insulator a reversal of direction means that it would also have to flip its spin, which is extremely unlikely in the absence of magnetic impurities. As a result, the conductance is hardly influenced by non-magnetic impurities and one could expect a very efficient spin transport, without the loss of information.
In our group we have studied several topological insulators or their parent compounds using spin and angle resolved photoemission. The advantage of this technique is that one can directly identify the number of Fermi level crossings including their spin polarization. First conclusive results were obtained for pure Sb and a BiSb alloy , where the correct topology could be identified but which has only a bulk band gap of 50 meV and as much as five Fermi crossings. The next generation of topological insulators is formed by BiSe and related compounds with only one spin polarized state at the Fermi level and a bulk band gap of 350 meV . The advantage of the latter compounds is that through bulk and surface doping the bands can be aligned such as to achieve the required conditions.
Topological insulators are band insulators with one or more metallic surface states which are spin-polarized due to the Rashba effect. These states are topologically protected, meaning that they can not be destroyed by continuous deformations. The spin orbit coupling thus takes the role of the magnetic field in the integer quantum Hall effect. See also S.-C. Zhang Physics 1, 6 (2008) for an informal description.
In these materials the electrons can move in the full two-dimensional plane of the surface yielding a finite scattering probability in directions where the spin is not orthogonal, which might eventually lead to a loss of information and an increase of resistance. We have taken a first step to also overcome this limitation by the identification of a one-dimesional topological metal; i.e. a topological surface with metallic bulk properties. It was found that the stepped surface of Bi(114) contains only two clearly one-dimensional spin polarized states, and each state contains electrons with exactly opposite momentum . If a similar situation can be created at the surface of a bulk insulating system, such as for example BiSb, a one-dimensional topological insulator will be obtained with loss free unidirectional spin transport. Furthermore such a system will show a one-dimensional Quantum Spin Hall Effect and as such could open a whole new realm of intriguing physics.
Figure 2 schematically summarizes the subjects addressed in this article. An unpolarized flow of electrons can be aligned using a Rashba system with the crossing point at the Fermi level or with a topological insulator. Coherent, loss free spin transport can be achieved through a topological insulator. Afterwards the spin can be manipulated according to need in a Rashba medium by tuning an external parameter.
In this article the direct implications of topological insulators on fault free quantum computing and their link to particle physics in condensed matter have not been discussed for the sake of simplicity. Furthermore, the systems described here most likely can’t directly be implemented in practical applications and some should primarily be regarded as model systems. However, the knowledge which is gained from our research and similar studies by other groups, does show that solutions based on spin-orbit interaction could provide an alternative to magnetic materials in many spintronic components.
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[Released: August 2010]