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One of the central features of magnetic semiconductors (both the new III-Mn-V materials and the more established II-Mn-VI system is that a relatively small external field causes enormous Zeeman splitting (into spin-up and spin-down states) of the electronic energy levels, such as the band edges. This band bending arising from nanoscale magnetic fields allows the possibility of creating traps for single electrons in the semiconductor, a new concept in semiconductor physics.  Producing the required nonuniform magnetic field with periodic nano-scale spatial variation is a formidable problem beyond present nano-lithographic capabilities.  We overcome this challenge with hybrid structures coupling a magnetic semiconductor with a superconductor or an array of permalloy disks and using the corresponding vortex field to create the trap for the electrons in the semiconductor.

This material is based upon work supported by the National Science Foundation under Grant No. 0210519.

 

DMS-Superconductors hybrids

Nature 435 Number 7038, 71 (2005)

Phys. Rev. Lett 90, 246804 (2003)

Appl.  Phys.  Lett. 86, 113103, (2005)

Phys.  Rev. B 72, 08520, (2005)

 

 

DMS-Superconductor hybrids

 



 

 

Sketch of the hybrid structure, containing a SC layer in the vortex phase and a DMS quantum well. The field inhomogeneity of the vortex state is imprinted onto the DMS layer. Spin and charge textures are trapped in the high-field regions inside the quantum well. For an overview, see the animation following the link (quicktime anim).

 

The magnetic properties of superconductors (SCs) and DMSs are vastly different. When placed in a magnetic field, a type-II SC allows the field to penetrate in a non-uniform fashion, as Abrikosov vortices—the field lines are bundled up into flux quanta surrounded by superfluid eddy currents5. The length scale characterizing the field inhomogeneity is the decay length of the superfluid eddy currents surrounding the core of an isolated vortex. This is known as the penetration depth, and is a few tens of nanometres for several SCs (for example, 39 nm in Nb). In contrast, magnetic impurities (typically Mn) embedded in a paramagnetic DMS (such as Ga1-xMnxAs, x≈1.5–5%) align their spins along the external field B(r). The strong exchange J(r) between the spins of these impurities and the spins of free charge carriers present in the DMS induces a local spin polarization of the charge carriers. The end result is a giant Zeeman splitting between charge carrier spin states aligned parallel and antiparallel to the magnetic field.
A novel situation arises when these two materials are brought into close proximity (see Figure bellow). The inhomogeneous magnetic field of a SC vortex creates a Zeeman potential landscape in the DMS quantum well. If a large enough field variation occurs on small length scales, magnetic field induced localization of charge carriers occurs in the DMS. These localized states exhibit spin textures, which reflect the local magnetic field distribution. Thus, the vortex acts as a spin and charge tweezer: control of the vortices’ locations and dynamics translates into controlled manipulation of the spin and charge textures in the DMS. A further advantage is that these effects are present in the paramagnetic DMS, and therefore a low Curie temperature is not a hindering factor. The devices, however, must operate below the superconducting transition temperature.

From left to right: charge texture for spin-up states; charge texture for spin-down states (note the smaller scale); total spin-charge texture: arrows indicate the direction and relative magnitude of the local spin in the texture. The DMS-SC distance is z=10 nm and an effective mass m=0.5me and a moderate geff=500 for the DMS charge carriers.


The charge distribution bound by the vortex is made predominantly of spin-up contributions, their maxima differ by about a factor of 10. The spin-down components appear because of the finite radial component of the vortex field (Fig. 1d). The spin-up components are concentrated right under the vortex core where the perpendicular field is largest, whereas the spin-down components are pushed away from the vortex core, close to where the radial magnetic field is largest. The total charge and spin distributions plotted in the figure above show that the spin texture is strongly polarized along the axis of the vortex. These textures should be observable with spin-polarized scanning tunnelling spectroscopy.


The location of the vortices can be constrained spatially using nano-engineered grooves in the SC layer: the vortices are trapped in the regions where the SC film is thinnest. The position of the vortex in the groove can be controlled by sending a current through the SC layer. The current exerts a transversal force, which moves the vortex, together with the texture it created in the DMS. The spin textures attached to single vortices can therefore be used as operational units, or as building blocks in systems with multiple vortices. A simple device controlling the distance between two spin textures is sketched in ther figure below. The basic mechanism is also illustrated by a quicktime animation.

(a) A finite-sizegroove is produced in the SC film by etching and two vortices are trapped into it. Due to repulsion, in equilibrium the vortices will stay at maximum possible separation; (b) if acurrent flows through the SC layer, its transversal force on the vortices will move them (and their spin textures) closer together


In uniform SC films there is a simpler way to control the location of the vortices. The repulsion between vortices leads to appearance of regular vortex lattices. Since each vortex carries a magnetic flux the distance between vortices can be continuously tunedwith the external magnetic field. Even more interesting is the two-dimensional case. For an external magnetic field $ B > B_{c1}$, the SC exhibits a finite density of vortices arranged in an ordered triangular lattice. Since each unit cell encloses the magnetic flux $ B a^2 \sqrt{3}/2= \phi_0/2$, the lattice constant $ a(B) \sim
1/\sqrt{B}$ is again controlled by the external field. In the limit $ a(B) \gg \lambda$, magnetic fields of different vortices do not overlap within the DMS layer. The ``atom-like'' bound states of the isolated vortex widen into energy bands, typical of an ordered crystal. The width of the bands is given by the hopping amplitude t between neighboring bound states. A 2D electron gas in a periodic potential subject to an homogeneous magnetic field has a fractal-like energy spectrum as a function of the magnetic flux per unit cell. (Because the plot of this spectrum resembles to a butterfly, the spectrum itself is also frequently called Hoftadter butterfly.) Since for the superconductor the flux through each unit cell is $ \phi/\phi_0=q/p=1/2$, the energy bands of the DMS QW will be those of a triangular Hofstadter butterfly . As long as the hopping t is small compared to the spacing between consecutive bound states, this Hofstadter problem corresponds to the regime of a dominant periodic modulation, which can be treated within a simple tight-binding mode and each band is expected to split into p (here p=2) magnetic subbands. This band structure has unique signatures in the magnetotransport, as we discuss below. Its measurement would provide a clear signature of the Hofstadter butterfly, which is currently a matter of considerable experimental interest.

In current devices, the periodic modulation is always perturbationally small compared to the cyclotron energy; and (ii) the applied magnetic field plays a very unusual role here. In general when discussing the Hofstadter butterfly, one assumes a fixed lattice cell. Varying the magnetic field changes the flux through the unit cell, resulting in the self-similar eigenspectrum of the Hofstadter butterfly. By contrast, here a change in B implies a change in the lattice constant, while the flux through the unit cell is always the same due to the SC layer. As long as there is a vortex lattice, the setup corresponds to a $ \phi/\phi_0 =1/2$ Hofstadter butterfly irrespective of the value of the external magnetic field. Instead, B controls the amplitude of the periodic modulation, from being the large energy scale (small B) to being a small perturbation (large B).

In the above figure we show the evolution of the charge carrier energy spectra as B is varied. The large-B limit corresponds to a small periodic modulation, and indeed we see the emergence of equally spaced Landau levels with a weak dispersion due to the weak periodic modulation. As B is lowered, the band structure evolves into that corresponding to a strong modulation, and for the lowest B we see the flat bands corresponding to the isolated vortex bound levels. This significant change in the dispersion as B is varied is reflected in magnetotransport measurements, in particular in the Hall conductivity. In the figure below we show the values of the hall conductivity as a function of the charge carrier density n and the magnetic field B.

 

For details, see Mona Berciu, Tatiana G. Rappoport and Boldizsar Jankó, Nature 435 Number 7038, 71 (2005)

 

Quicktime movies : hybrid system and vortex dynamics

Image gallery:

DMS /SC hybrid 1

DMS /SC hybrid 2

Animations: Copyright - Ovidiu Toader

 

 

 

 

 

DMS - micromagnets hybrids

 

We have succeeded to obtain, for the first time, the energy spectrum, the structure of the eigenstates describing the localized states of the MS material under the magnetic singularity of the permalloy. We studied in detail the properties of a magnetic semiconductor-permalloy disk hybrid system. We found that the giant Zeeman response of the magnetic semiconductor in conjunction with the highly nonuniform magnetic field created by the vortex state of a permalloy disk can lead to Zeeman localized states at the interface of the two materials. These trapped states are chiral, with chirality controlled by the orientation of the core magnetization of the permalloy disk. We calculate the energy spectrum and the eigenstates of these Zeeman localized states, and discuss their experimental signatures in spectroscopic probes. For details, see M. Berciu and B. Jankó, Phys. Rev. Lett 90, 246804 (2003) .


We have recently extended this work to consider a hybrid structure of a CdMnTe/CdMgTe quantum well (QW) at tens of nanometers below a rectangular or cylindrical ferromagnetic island. We find that due to the giant Zeeman interaction, the nonhomogeneous magnetic field produced by the rectangular ferromagnetic island acts as an effective potential that can efficiently “trap” spin-polarized quasi-particles in the QW. We present quantitative predictions for the optical response of the DMS where the localization of the quasi-particles is evident. We also discuss how these predictions are sensitive to the variation of certain parameters, such as the distance between the QW and the ferromagnetic island, the thickness of the micromagnet and the electronic g factors. For the calculations involving both cylindrical and rectangular micromagnets, we study the properties of the electron in the valence band and in the conduction band of a zinc-blende semiconductor by considering a Luttinger Hamiltonian with the appropriate set of Luttinger parameters and electron mass both the QW and the barriers.

Cylindrical micromagnet: a zero-dimensional trap

 


We investigate a Fe micro-disk illustrated in the above figure. In this state, due to the competition between the exchange interaction and the shape asymmetry, the magnetization lies in the plane of the micro-disk except near the center, where the local magnetization is forced to point out of the plane to form a magnetic vortex. The diameter of the core (Rc) to which the non-zero perpendicular magnetization is confined extends over only about 60 nm. In the figure below we show the absorption coefficient for three distances d between the micro-disk and the QW: d=5 nm, 10 nm and 15 nm. The energy of the photon is measured relative to the energy of the main absorption peak in the QW in the absence of a micromagnet. Vertical bars represent the optical oscillator strength of the transitions, and the numbers (nm) show that the corresponding line is a transition between the n th electron state and the m th hole state.

 

Rectangular micromagnet: a one-dimensional trap

 

Z component of the magnetic field for the rectangular micromagnet

We consider a rectangular, flat Fe micromagnet in the single domain state , with magnetization pointing in the x-direction. The single-domain state of the micromagnet with the mentioned size was investigated by micro-magnetic simulation using the OOMMF package. In the next figurewe present the absorption spectrum at three distances d between the QW and the micromagnet: d=10, 20, and 30 nm. In all three cases the micromagnet thickness was kept at a constant value of Dz=150 nm. As before, zero energy is chosen at the main absorption peak. At d=10 nm, the binding energy is 66 meV, while at d=60 nm it is smaller by a factor of 2. This follows from the fact that the further the micromagnet is from the QW, the smaller is the magnetic field at the QW position. Non-diagonal transitions are relatively strong both in the case of the micro-disk and of the rectangular micromagnet. The separation between the peaks decreases as d increases, because the gradient of the magnetic field (and equivalently, the gradient of the potential) also decreases with increasing d.

For details, see : P. Redlinski, T. G. Rappoport, A. Libal, J. K. Furdyna, B. Jankó and T. Wojtowicz; Appl.  Phys.  Lett. 86, 113103, (2005) and condmat/0505018

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