Tunable wave-vector filtering in a two-dimensional electron gas modulated by magnetic barriers and δ-doping

We report a theoretical investigation on the control of wave-vector filtering (WVF) in a two-dimensional electron gas modulated by realistic magnetic barriers and δ-doping, which can be experimentally realised by depositing a ferromagnetic stripe on the surface of a GaAs/Al_xGa_1−xAs heterostructure and using atomic layer doping. Theoretical analysis demonstrates that a sizeable WVF effect still exists even if δ-doping is introduced into the device. Numerical calculation reveals that the WVF efficiency is related closely to the δ-doping. Thus, the WVF effect in a magnetic nanostructure can be conveniently manipulated by properly adjusting the weight and/or the position of the δ-doping, giving rise to a tunable momentum filter for nanoelectronics applications.

Keywords: Magnetic-barrier (MB) nanostructure; δ-doping; wave-vector filtering (WVF) effect; WVF efficiency; tunable momentum filter

It is well known that the interface of a modulation-doped semiconductor heterostructure contains a high-mobility two-dimensional electron gas (2DEG). Moreover, the motion of the 2DEG can be confined by an inhomogeneous magnetic field to the nanometer scale by means of modern nanofabrication techniques. For example, depositing a nanosized ferromagnetic (FM) stripe on the surface of a GaAs/AlxGa1−xAs heterostructure [[1]] can form the magnetic-barrier (MB) nanostructure [[2]]. Such a 2DEG nanostructure is a hybrid of the magnetic material and the semiconductor, where the former provides an inhomogeneous magnetic field influencing locally the motion of the electrons in the latter. Because of the small size, the low dimensionality and the particular magnetic confinement, the MB nanostructure possesses abundant quantum effects [[3]], for example, wave-vector filtering (WVF) [[4]], electron-spin polarisation (ESP) [[5]] and giant magnetoresistance (GMR) [[7]], which can be used to design new electronic devices [[10]].

With the development of materials growth techniques, such as molecular beam epitaxy (MBE) and metal-organic chemical vapour deposition (MOCVD), dimensional control has approached interatomic spacing, techniques sometimes referred to as atomic layer deposition [[11]]. Resorting to such technology, a tunable δ-potential can be intentionally doped into a structure for nanoelectronics applications. For example, resonant tunneling devices with δ-doping have been realised experimentally [[12]]. More recently, δ-doping was used by Lu et al. [[14]] to control an electron-spin filtering effect in MB nanostructures. It was found that the transmission, the conductance and the spin polarisation were dependent strongly on the δ-doping. As a consequence, spin filtering can be manipulated expediently by adjusting the weight and/or the position of the δ-doping to produce structurally tunable spin filters for spintronics applications. Later, several groups [[16]] supported the results of Lu et al. by investigating the manipulation of the δ-doping to effect spin filtering in other MB nanostructures.

Very recently, δ-doping technology has been applied to manipulate the GMR of MB nanostructures for magnetic information storage. Kong et al. [[23]] first studied the influence of δ-doping on the GMR effect in a δ-MB nanostructure. The magnetoresistance ratio (MR) was found to be closely related to the weight and/or the position of the δ-doping. A controllable GMR device based on such a δ-MB nanostructure was proposed for magnetoelectronics applications. Subsequently, the manipulation of δ-doping in other MB nanostructures has been explored and corresponding structurally controllable GMR devices proposed [[24]]. Motivated by these reports, in the present work, we investigate the control of δ-doping to influence wave-vector filtering (WVF) in a MB nanostructure and propose manipulable momentum filters for nanoelectronics applications.

The MB nanostructure under consideration is schematically shown in Figure 1a, which can be experimentally realised [[28]] by depositing a FM stripe with a horizontal magnetisation on the top of a GaAs/AlxGa1−xAs heterostructure. When the distance between the FM stripe and the 2DEG is very small, the magnetic field produced by the magnetised FM stripe acting perpendicularly to the 2DEG in (x, y) plane can be approximated as a δ-function MB [[30]], i.e.

(1)

Graph

Graph: Figure 1. (a) Schematic illustration of the MB nanostructure: one FM stripe is deposited on the top of the GaAs/AlxGa1−xAs heterostructure, and (b) the magnetic field profile, where a δ-doping Vδ(x − x0) is comprised by the atom layer doping technology.

where B is the magnetic strength of two MBs and L is the width of the FM stripe. Thus, the magnetic vector potential can be written as , in Landau gauge, viz [[31]]

(2)

Graph

The δ-doping, Vδ(x − x0), can be introduced into the above system with the help of the atomic layer doping technique [[11]], as shown in Figure 1b. The Hamiltonian describing such a modulated 2DEG system, within the single-particle and effective-mass approximation, reads [[32]]

(3)

Graph

where , m0 and are the effective mass, the free mass and the momentum of the electron, respectively.

Because of the translational invariance along the y axis for the MB nanostructure, the solution of the stationary Schrödinger equation, HΨ(x, y) = EΨ(x, y), can be given by Ψ(x, y) = ψ(x) exp (ikyy), where ky is the wave-vector component in the y direction. Thus, the wave function ψ(x) complies with the following one-dimensional (1D) Schrödinger equation

(4)

Graph

with the effective potential [[33]] for the electron in the MB nanostructure given by

(5)

Graph

Clearly, this effective potential depends not only on the longitudinal wave vector ky and the magnetic configuration Bz(x), but also on the δ-doping, Vδ(x − x0). Actually it is the δ-doping dependence of the Ueff(x, ky, V, x0) that results in the possibility to manipulate the WVF effect of the MB nanostructure [[14]]. Equation (4) can be solved exactly using the transfer-matrix method (TMM) [[34]]. Without loss of generality, the wave functions in the incident and outgoing regions of the MB nanostructure, can be written as ψleft(x) = exp (iklx) + γ exp (−iklx), x < −L/2 and ψright(x) = τ exp (ikrx), x > L/2, respectively, where and γ/τ is the reflection/transmission amplitude. In the MB-nanostructure region, the wave function can be expressed by a linear combination of the plane waves, ψj(x) = Cj exp (ikjx) + Dj exp (−ikjx). By means of the TMM, the transmission coefficient for the electron with the incident energy E across the MB nanostructure can be analytically obtained from

(6)

Graph

Once the transmission probability is obtained, the degree of the WVF effect can be characterised by the so-called wave-vector filtering efficiency, which can be defined by differentiating the transmission coefficient over the wave vector for a fixed incident energy as

(7)

Graph

For convenience, we express all relevant quantities in the dimensionless form by means of two characteristic parameters: the cyclotron frequency ωc = eB0/ and the magnetic length , e.g. x → ℓBx and E → (ℏωc)E = E0E. In our numerical calculation, we take the GaAs system as the material for the 2DEG, i.e. , and ne(GaAs) ≈ 1011 cm−2, which leads to the basic units ℓB = 57.5nm and E0 = 0.34meV for an estimated magnetic field B0 = 0.2T, and some structural parameters are chosen as B = 3.0 and L = 3.0 for simplicity. It should be noticed that, the second item in the effective potential Ueff (see Equation (5)) is the Zeeman splitting and coupling effect, which is associated with the quantity B/4m0. For GaAs and the B = 3.0, this equals 0.02214, which is much smaller compared to the other items in Ueff. Thus, Zeeman splitting and a coupling effect play minor roles in determining electronic transport property for GaAs, and will be ignored in the present work.

In a previous study [[4]], the MB nanostructure shown in Figure 1a was found to possess a sizable WVF effect. Now, we consider whether this system still has such a quantum if δ-doping is included. To answer this issue, Figure 2a shows the transmission coefficient as a function of the incident energy for an electron with wave vector ky = −1.0 (solid curve), 0.0 (dashed curve) and + 1.0 (dotted curve), where the δ-doping is set to be V = 2.0 and x0 = 0.5. From this figure, an obvious anisotropy with the wave vector can be seen clearly, owing to an essentially 2D process [[2]] for electrons tunneling through a MB nanostructure. In other words, there exists a great discrepancy of the transmission between different wave vectors. Thereby, an appreciable WVF effect occurs in the MB nanostructure even if the δ-doping is comprised. In order to observe the WVF effect more evidently, the wave-vector filtering efficiency as the function of the wave vector is presented in Figure 2b for incident energies E = 3.0 (solid line), 6.0 (dashed line) and 9.0 (dotted line), where the δ-doping is the same as in Figure 2a. Indeed, an evident WVF effect can be observed clearly, which can be understood from the fact that an essentially 2D process for the electron through a MB nanostructure is independent of the δ-doping. In addition, the wave-vector filtering efficiency η shows up a great dependence on not only the wave vector ky but also the incident energy E, especially for the small incident angle and the high incident energy. For a given incident energy, η changes drastically with ky. The ky value is enhanced and the η − ky curve shifts to the right when the E becomes large.

Graph: Figure 2. (a) Transmission coefficient vs. the incident energy for the wave vectors ky = −1.0 (solid curve), 0.0 (dashed curve) and + 1.0 (dotted curve), and (b) the wave-vector filtering efficiency as the function of the wave vector for the incident energies E = 3.0 (solid curve), 6.0 (dashed curve) and 9.0 (dotted curve), where the δ-doping is set to be V = 2.0 and x0 = +0.5.

After seeing an appreciable WVF effect when δ-doping is introduced into the MB nanostructure, we consider what impact such a doping has on the degree of the WVF effect (i.e. the wave-vector filtering efficiency). Undoubtedly, the δ-doping will yield a great influence on the WVF effect, because the effective potential Ueff of the MB nanostructure is associated closely with the δ-doping (cf. Equation (5)). Next, we investigate in detail how the weight V and the position x0 of the δ-doping affect the degree of the WVF effect η of the MB nanostructure as shown in Figure 1.

First of all, we fix the position (such as x0 = +0.5) and take the wave vector ky = 0.0 (viz, the normal incidence) as an example to study the effect of the weight V of δ-doping on the WVF effect. Figure 3 exhibits the wave-vector filtering efficiency η as a function of the incident energy E for weights of δ-doping V = 1.0 (solid curve), 3.0 (dashed curve) and 5.0 (dotted curve), where the position of the δ-doping remains unchanged at the coordinate x0 = +0.5. The significant discrepancy of the wave-vector filtering efficiency can be seen for the different weights of δ-doping. With increasing the weight, the efficiency becomes small and the η–E curve shifts upwards. In other words, the WVF effect in the MB nanostructure can be controlled by properly tuning the weight of the δ-doping. The modulation of the wave-vector filtering efficiency of the MB nanostructure by the weight of the δ-doping can be observed more apparently from the inset of Figure 3, where the efficiency η is directly plotted as the function of the weight V for the wave vector ky = 0.0 and the particular incident energy E = 6.0. The wave-vector filtering efficiency varies drastically with the weight of the δ-doping, especially within the range of 0.0 < V < 5.0. According to Equation (5), the modulation of the weight V of the δ-doping to the WVF effect originates obviously from the dependence of the effective potential Ueff on the weight V.

Graph: Figure 3. Wave-vector filtering efficiency η changes with the incident energy E for the weights of the δ-doping V = 1.0 (solid line), 3.0 (dashed line) and 5.0 (dotted line), while in the inset the η is directly plotted as the function of the weight of the δ-doping for the E = 6.0, where the position of the δ-doping is fixed at x0 = +0.5 and the wave vector is taken to be ky = 0.0.

Apparently, the Ueff is dependent on not only the weight V but also the position x0 of the δ-doping. Therefore, the position x0 of the δ-doping also will impact on the WVF effect of the MB nanostructure as shown in Figure 1. Finally, in Figure 4, we give the variation of the wave-vector filtering efficiency η with the incident energy E for the positions of the δ-doping x0 = 0.0 (solid line), 0.5 (dashed line) and 1.0 (dotted line), where the wave vector ky = 0.0 and the incident energy E = 6.0 are chosen; while the weight of the δ-doping is fixed to be V = 2.0. When the position of the δ-doping changes, from this figure, we can see the corresponding variation of the wave-vector filtering efficiency. In particular, this change is more obvious if the δ-doping deviates far from the centre of the MB nanostructure (cf. the dashed and dotted lines). Such a feature implies that we can also control expediently the WVF effect of the MB nanostructure as shown in Figure 1 by adjusting the position of the δ-doping. In order to observe more clearly this manipulation of the position, in the inset we directly present the wave-vector filtering efficiency η as the function of the position x0 of the δ-doping, where the wave vector ky, the incident energy E and the weight V of the δ-doping are the same as in Figure 4. Indeed, the position of the δ-doping causes a strong modulation on the WVF effect in the MB nanostructure. Furthermore, such a modulation to the wave-vector filtering efficiency shows up a symmetric behaviour with respect to the position of the δ-doping, namely, η(−x0) = η(x0) for a given incident energy E and a particular wave vector ky. This symmetric behaviour can be understood in the light of the intrinsic symmetry [[35]] (the antisymmetric magnetic profile Bz(−x) = −Bz(x) and the symmetric magnetic vector potential Ay(−x) = Ay(x)) and the invariant transmission for a particle across a potential barrier in the opposite directions [[36]].

Graph: Figure 4. For the wave vector ky = 0.0, the wave-vector filtering efficiency η varies with the incident energy E for the position of the δ-doping x0 = 0.0 (solid curve), +0.5 (dashed curve) and +1.0 (dotted curve), where the weight of the δ-doping is set to be V = 2.0, while the inset gives the η as the function of the x0 for E = 6.0 and the ky = 0.0.

We have theoretically explored the control of δ-doping on the WVF effect in a MB nanostructure, which can be experimentally fabricated by depositing a FM stripe with the horizontal magnetisation on the top of a GaAs/AlxGa1−xAs heterostructure and using atomic layer doping. It is confirmed that a sizeable WVF effect still appears when δ-doping is included into the MB nanostructure. It is also demonstrated that the wave-vector filtering efficiency is associated with the δ-doping. Thus, the WVF effect can be manipulated not only by the weight but also by the position of the δ-doping. These interesting properties will be useful for understanding the modulation mechanism to the WVF effect in MB nanostructures and for designing structurally controllable momentum filters in nanoelectronics applications.

No potential conflict of interest was reported by the authors.

This work supported by the Scientific Research Fund of Hunan Provincial Education Department [grant number 16B243], [grant number 16A081], [grant number 17B256], the Opening Project of Key Laboratory of Comprehensive Utilization of Advantage Plants Resources in Hunan South, the Hunan University of Science and Engineering [grant number XNZW15C03], [grant number XNZW16C05], and the Construct Program of the Key Discipline (Circuits and Systems) in the Hunan University of Science and Engineering.