D 5 Spintronics and Emergent Qubits 1 Thomas Schapers¨ Peter Grunberg¨ Institut Forschungszentrum Julich¨ GmbH Contents 1 Introduction 2 2 Low dimensional semiconductor systems 2 3 Spin-orbit coupling in semiconductor structures 6 4 Electron interference effects 13 5 Rashba effect in quasi-one-dimensional structures 17 6 Spin-orbit quantum-bits 22 7 Andreev level qubits 29 1Lecture Notes of the 44th IFF Spring School “Quantum Information Processing” (Forschungszentrum Julich,¨ 2013). All rights reserved. D5.2 Thomas Schapers¨ 1 Introduction In this lecture the basic concepts of spintronics will be introduced. The main focus lies on effects related to spin-orbit coupling in semiconductor heterostructures, i.e. the Rashba and Dressel- haus effect. Especially the former allows to control the spin orientation by electrical means. The Rashba effect is essential for the realization of the spin field-effect transistor [1]. By reduc- ing the structure size to one dimension, the properties of spintronic devices can be improved. Here, we will particularly discuss cylindrical semiconductor nanowires, which are prepared by the so-called bottom up approach. By confining the electrons in a gate-defined quantum dot based on semiconductor nanowires, a spin qubits can be realized. It will be discussed, how spin-orbit coupling is employed to manipulate the spin orientation in these qubits. Semicon- ductor nanowires are also very interesting elements to realize Andreev level qubits. Here, the nanowire is used as a normal conducting link between two superconducting electrodes. Owing to phase-coherent Andreev reflection, a Josephson supercurrent can be observed. 2 Low dimensional semiconductor systems InAs-based 2-dimensional electron gases Semiconductor heterostructures based on InAs or its alloys are particularly suited for observing a strong spin-orbit coupling. A typical layer sequence is depicted in Fig. 1(a) [2]. Here, the 2-dimensional electron gas is located in a Ga0:47In0:53As layer at the boundary to the InP spacer layer. The Ga0:47In0:53As semiconductor has a smaller band gap compared to InP. At an indium content of 53% of the GaInAs layer the lattice constant fits to the lattice constant of InP. The electrons in the 2DEG are provided by the n-type doped InP layer which is separated from the 2DEG by a spacer layer (modulation doping). The whole structure is grown on an InP wafer. Heterostructures containing phosphorus are mostly grown by means of metal-organic vapor phase epitaxy (MOVPE). Here, the metal-organic precursors, e.g. trimethylindium (TMIn), trimethylgallium (TMGa), AsH3, or PH3, are added to the carrier gas, i.e. H2 or N2, which flows in a reactor over the surface of the wafer. In Fig. 2 the temperature dependence of the mobility of a 2-dimensional electron gas in an InGaAs/InP heterostructure is shown. Down to temperatures of around 40 K the mobility is limited by phonon scattering. At temperatures below 40 K the mobility saturates, which can be attributed to alloy scattering. Mobilities above 100000 cm2/Vs can be achieved [2]. In contrast to a 2-dimensional electron gas in a GaAs/AlGaAs heterostructures, the electron gas is located in a ternary material, where the In and Ga atoms are statistically distributed in the crystal. The corresponding disorder potential results in an additional scattering contribution, which limits the mobility at low temperatures. A typical conduction band profile of a GaInAs/InP heterostructure with an additional strained Ga0:23In0:77As layer is depicted in Fig. 3. The larger In content of 77% leads to a smaller band gap and a smaller effective electron mass compared to Ga0:47In0:53As. Furthermore, due to the smaller disorder potential the alloy scattering contribution is lowered. As can be seen in Fig. 3, the electron wave function is mainly located in the strained Ga0:23In0:77As layer. Owing to the different barrier materials and the doping at only one side of the Ga0:23In0:77As well, the rectangular quantum well is tilted. The tilted potential profile results in an electric field in the quantum well. The presence of an electric field in the 2-dimensional electron gas is an important prerequisite for the Rashba effect introduced below. Spintronics D5.3 Fig. 1: (a) Layer sequence of a GaInAs/InP heterostructure. The 2DEG is located in the GaInAs layer at the boundary to the InP spacer. (b) Conduction band profile and electron wave func- tions of the 2DEG at the boundary of the GaInAs/InP interface. Fig. 2: Temperature dependence of a 2-dimensional electron gas in a GaInAs/InP heterostruc- ture. At low temperatures the mobility is limited by alloy scattering. D5.4 Thomas Schapers¨ Fig. 3: Conduction band profile and squared amplitude jj2 of the electron wave function in a GaInAs/InP heterstructures with an inserted strained Ga0:23In0:77As layer. By employing etching methods the dimension of 2-dimensional electron gas can be reduced to a 1-dimensional wire structure. The etching process can either be performed by wet chemical etching or by dry etching methods. As described in the next section, a more elegant way is to prepare these wire structures directly by means of epitaxial growth. Semiconductor nanowires Nano-scaled wire structures can also be fabricated by self-organized growth without using elab- orate lithographical means. One possible approach is to use the vapor-liquid-solid (VLS) growth mode [3]. Here, a nanometer-size gold particle is used as a seed particle for the nanowire growth. In many cases InAs is used as a semiconductor material. The underlying reason is that at the surface an electron accumulation layer is formed due to the Fermi level pinning within the conduction band [cf. Fig. 4(a)]. This property is only found in a few semiconductors, e.g. InAs, InN, or InSb. It ensures that even for low nanowire diameters the nanowires contain conductive electrons. For most other semiconductors a depletion layer is formed at the surface, e.g. for GaAs or Si, owing to the Fermi level pinning within the band gap [cf. Fig. 4(b)]. As an alternative to the VLS method, nanowires can also be fabricated by selective-area metal- organic vapor phase epitaxy (SA-MOVPE). Here, a pre-patterned substrate is used, where the nanowires are grown only at those positions where holes are present in a dielectric mask layer. By this method the location of the nanowires can be determined, which is advantageous for subsequent device processing [cf. Fig. 5(b)]. For transport measurements the as-grown nanowires are usually separated from the substrate and transferred to another substrate with alignment markers. By employing electron beam lithography these nanowire are contacted. A scanning electron micrograph of a typical con- tacted semiconductor nanowire is shown in Fig. 5(a). Spintronics D5.5 Fig. 4: (a) Semiconductor nanowire with a surface accumulation layer. (b) Nanowire with a depletion layer at the surface. Fig. 5: (a) Scanning electron beam micrograph of a field of InAs nanowires grown by selective- area MOVPE. (b) InAs nanowire contacted with ohmic contacts on both terminals. In addition, gate fingers are placed in the center, in order to control the electron concentration. D5.6 Thomas Schapers¨ Fig. 6: Schematics of a spin transistor: (a) No gate voltage is applied. The spin precession due to the presence of the Rashba effect leads to a spin orientation which is opposite to the magnetization of the drain electrode. No current can flow in this case, corresponding to a red traffic light. (b) A gate voltage is applied, which enhances the Rashba effect. Owing to stronger spin precession, the spin orientation of the electrons arriving at the drain electrode fits to its magnetization. A current can flow, corresponding to a green traffic light. 3 Spin-orbit coupling in semiconductor structures Spintronics: Basic concept A well-known example of a spintronic device is the spin field-effect transistor. In Fig. 6 a schematics of the spin field-effect transistor is shown [1, 4]. The spin transistor consist of three components: a spin injector as the source electrode, a semiconducting area where the spin orientation is controlled, and a spin detector as the drain electrode. Here, we will focus on the properties of the semiconductor channel located in between the magnetic contacts. In Fig. 6(a) the spin precession in a spin transistor is shown, when no voltage is applied to the gate electrode on top of the semiconductor channel. In the channel the spin is precessing owing to the presence of the Rashba spin-orbit coupling [5]. For the sake of simplicity we assumed that the electron arrives at the drain contact with a spin orientation opposite to the spin orientation in the detector electrode. In this case the electrons cannot enter the detector, since there are no states available the incoming electrons can occupy. As a consequence, the current through the spin transistor is blocked. As will be explained below in detail, by applying a gate voltage the strength of the Rashba spin-orbit coupling can be enhanced. This leads to a stronger spin precession so that the spin orientation of the electrons arriving at the drain electrode fits to the magnetization of the detector electrodes. Now a current can flow in the structure, since the electrons can enter the drain electrode. Below we will explain the physical background of spin-orbit coupling in a semiconductor het- erostructure in detail. Furthermore, a method will be introduced to determine the strength of spin-orbit coupling by transport experiments. Finally, spin-orbit coupling in wire structures will be discussed. Spintronics D5.7 Fig. 7: (a) Schematic illustration of the Rashba spin-orbit coupling in a 2-dimensional electron gas (b) Energy-momentum dispersion with an energy splitting due to the Rashba effect.