Electronic properties of semiconductor quantum wires for shallow symmetric and asymmetric confinements

Quantum wires (QWs) and quantum point contacts (QPCs) have been realized in GaAs/AlGaAs heterostructures in which a two-dimensional electron gas resides at the interface between GaAs and AlGaAs layered semiconductors. The electron transport in these structures has previously been studied experimentally and theoretically, and a 0.7 conductance anomaly has been discovered. The present paper is motivated by experiments with a QW in shallow symmetric and asymmetric confinements that have shown additional conductance anomalies at zero magnetic field. The proposed device consists of a QPC that is formed by split gates and a top gate between two large electron reservoirs. This paper is focussed on the theoretical study of electron transport through a wide top-gated QPC in a low-density regime and is based on density functional theory. The electron–electron interaction and shallow confinement make the splitting of the conduction channel into two channels possible. Each of them becomes spin-polarized at certain split and top gates voltages and may contribute to conductance giving rise to additional conductance anomalies. For symmetrically loaded split gates two conduction channels contribute equally to conductance. For the case of asymmetrically applied voltage between split gates conductance anomalies may occur between values of 0.25(2e 2/h) and 0.7(2e 2/h) depending on the increased asymmetry in split gates voltages. This corresponds to different degrees of spin-polarization in the two conduction channels that contribute differently to conductance. In the case of a strong asymmetry in split gates voltages one channel of conduction is pinched off and just the one remaining channel contributes to conductance. We have found that on the perimeter of the anti-dot there are spin-polarized states. These states may also contribute to conductance if the radius of the anti-dot is small enough and tunneling between these states may occur. The spin-polarized states in the QPC with shallow confinement tuned by electric means may be used for the purposes of quantum technology.

Quantum wires (QWs) and quantum point contacts (QPCs) have been realized in GaAs/AlGaAs heterostructures in which a two-dimensional electron gas resides at the interface between GaAs and AlGaAs layered semiconductors. The electron transport in these structures has previously been studied experimentally and theoretically, and a 0.7 conductance anomaly has been discovered. The present paper is motivated by experiments with a QW in shallow symmetric and asymmetric confinements that have shown additional conductance anomalies at zero magnetic field. The proposed device consists of a QPC that is formed by split gates and a top gate between two large electron reservoirs. This paper is focussed on the theoretical study of electron transport through a wide top-gated QPC in a low-density regime and is based on density functional theory. The electron-electron interaction and shallow confinement make the splitting of the conduction channel into two channels possible. Each of them becomes spin-polarized at certain split and top gates voltages and may contribute to conductance giving rise to additional conductance anomalies. For symmetrically loaded split gates two conduction channels contribute equally to conductance. For the case of asymmetrically applied voltage between split gates conductance anomalies may occur between values of 0.25(2e 2 /h) and 0.7(2e 2 /h) depending on the increased asymmetry in split gates voltages. This corresponds to different degrees of spin-polarization in the two conduction channels that contribute differently to conductance. In the case of a strong asymmetry in split gates voltages one channel of conduction is pinched off and just the one remaining channel contributes to conductance. We have found that on the perimeter of the anti-dot there are spin-polarized states. These states may also contribute to conductance if the radius of the anti-dot is small enough and tunneling between these states may occur. The spin-polarized states in the QPC with shallow confinement tuned by electric means may be used for the purposes of quantum technology.
Keywords: quantum wire, electron transport, semiconductor heterostructure, two-dimensional electron gas, conductance anomalies (Some figures may appear in colour only in the online journal) * Author to whom any correspondence should be addressed.
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Introduction
High-quality modulation-doped semiconductor heterostructures are commonplace in nanoscience and nanotechnology. The idea to confine electrons in extremely thin layers of semiconductors and to study their quantization of energy and transport properties came from an early theoretical paper by Sakaki [1]. Later the first working one-dimensional (1D) lateral confinement scheme in GaAs/AlGaAs heterostructure was realized experimentally by Thornton et al [2]. Since then, the split-gate semiconductor devices, like quantum wires (QWs) and quantum point contacts (QPCs), have become genuine quantum laboratories to study a number of fundamental issues in low-dimensional physics [3]. Many physical phenomena have been discovered in the following years such as, for example, magnetic depopulation of 1D sub-band in twodimensional electron gas (2DEG) [4], quantum interference effects in electron waveguides [5], quantization of conductance [6,7], plasmons in a magnetic field in 1D GaAs/AlGaAs QW [8], effects of spin-charge separation and localization in one dimension [9], magneto-plasmons excitation in Rashba spintronics QWs [10] and many others (for further references, see [11]). The fields of solid state electron optics [12] and semiconductor spintronics [13] have been developed with possible applications in nanotechnology and quantum information processing.
The lateral confinement technique is free from imperfections and allows one to change the geometry of the system and the density of the underlying 2DEG continuously up to a very low electron density by changing the negative voltages applied to split gates. Systems fabricated in this way are very versatile: QWs can be connected to electron reservoirs that serve as leads and conductance measurements can be used as a tool for the identification of different electron configurations. Studies of electron transport in split-gate devices in the ballistic regime are of particular fundamental interest [3]. For example, conductance through a QPC is quantized in units of the fundamental conductance quanta 2e 2 /h under changing split gates voltage, which has been discovered experimentally by van Wees et al [6] and Wharam et al [7]. In theory, the existence of this phenomenon follows from the general Landauer-Büttiker formula [14] derived within the frame of a simple model of non-interacting electrons. Total conductance is N(2e 2 /h), where N is the number of open sub-bands in the QPC which is defined by the value of applied voltage on split gates. In the presence of an external magnetic field the conductance step becomes e 2 /h in magnitude pointing to the role of electron spin in this phenomenon.
Furthermore, many-body effects, like electron correlation, spontaneous spin polarization and localization of electrons in the form of Wigner crystal and spin lattices, leave welldefined marks on quantized conductance patterns known as conductance anomalies. The first experimental observation of a conductance anomaly near 0.7(2e 2 /h) was reported by Thomas and co-workers in the pioneering papers [15,16] for a QW at zero magnetic field. This so-called 0.7 conductance anomaly has been associated with spontaneous spin polarization of the electron gas and has been studied extensively over the years (for a survey of different measurements and their interpretations, see [17,18]). Berggren and Wang have for the first time in their theoretical papers [19,20] found the effect of spontaneous spin polarization in a QW by taking into account exchange interaction between electrons. Later in extensive theoretical studies of electron transport through QPCs [21][22][23][24] this model has been generalized by including the full exchange-correlation potential and realistic potentials from split gates. As a result, it has been shown that the 2DEG in a QPC at low densities becomes spin-polarized, which gives rise to the 0.7 conductance anomaly. This model has been called the static spin polarization model. However, there exists another model for the explanation of the 0.7 conductance anomaly, known as the dynamic spin polarization model, which is based on the formation of a bound state within a QPC due to which the conductance can be thermally activated in a Kondo-like fashion [25]. However, in [26] it has been found that conductance increases to 0.85(2e 2 /h) in applied dc bias which contradicts the Kondo-like picture, since Kondo correlations should be destroyed by a finite bias. Moreover, modeling with common realistic device parameters did not clearly recognize any bound states within a QPC [27,28].
As it has been pointed out in [28], these two models should not be mutually exclusive. The dynamic spin polarization model may be applicable at very low temperatures while a static spin polarization model prevails with increasing temperatures, source-drain bias and magnetic field. To resolve the debate on which model is more relevant for the 0.7 conductance anomaly it was evidently very important to develop a technique that allows direct measurement of a spin polarization. For this purpose, the technique of transverse electron focussing has been proposed in [29][30][31] to study the electron transmission through a QPC by investigation of the position, shape and height of focussing peaks. This technique has been further developed for n-type GaAs based devices consisting of two QPCs served as an injector and a detector of electrons [32,33]. Direct measurement of spin polarization has provided evidence for the spin polarization mechanism behind the 0.7 conductance anomaly.
The spontaneous spin magnetization may lead to the design of spin-sensitive devices such as spin transistors, spin filters, spin polarizers and spin valves [34][35][36]. These devices allow the manipulation of electron spin by electric means with future envisaged applications in semiconductor spintronics and quantum information processing [37]. However, precise control of electronic states and spin-polarized electron transport is required in order to build integrated quantum circuits.
The split-gate devices remain physically very rich and constantly bring new experimental data such as unusual conductance behavior in the case of shallow potential in a low electron density regime pointing to the incipient formation of a Wigner crystal in the 2DEG [38,39]. Splitting of a focussing peak in two has been associated with the formation of two conduction channels, which has been observed in focussing experiments [40].
In a recent experimental study by Kumar et al [41,42] odd and even denominator fractions of conductance have been observed in a weakly confined 1D QW formed in GaAs/AlGaAs heterostructure, consisting of split gates and a top gate, for a low electron density and in the absence of a magnetic field. It has been suggested that a zig-zag array of electrons may be formed with ring paths that leads to a cyclic current and a resultant lowering of energy. The asymmetry of the split gates voltages results in further weakening the confinement potential (i.e. making the conduction channel wider) and enhances the appearance of new fractional quantum states. The states in question are in analogy with the fractional Hall effect, however they appear at zero magnetic field, which is a very profound discovery. It has been suggested that the interaction between two channels of conduction is responsible for the non-magnetic fractional quantization of conductance. In-plane magnetic field induces new even denominator fractions possibly indicative of electron pairing as a result of entanglement and formation of the single wave function between two charge centers. These new fractional quantum states are believed to be important both for the physics of low-dimensional electron systems and for quantum technologies.
In a recent paper [43] electron transport in quasi-1D QW in GaAs/AlGaAs heterostructure has been studied in the case of asymmetric confinement potential. The formation of two conduction channels has been confirmed by measurement of differential conductance. Anti-crossing of ground and first excited levels has been found by varying the asymmetry of the confinement potential. As it has been pointed out in [43], asymmetric confinement potentials have not been investigated to the same degree as symmetric confinement potentials. This investigation becomes very important in the case when electrons relax into two dimensions and their energies are determined both by the confinement and electron-electron interaction. These latest experimental results motivate our research in this direction.
In the present paper we study a wide top-gated QPC in the case of shallow symmetric and asymmetric confinements that give rise to electron densities close to experimentally observed ones. We consider effects of confinement and electron-electron interaction on conduction through the QPC within density functional theory. By varying the top gate voltage we have achieved the regime when confinement potential becomes flat (non-parabolic) within the QPC. When this region is filled with electrons two channels of conduction emerge due to electron-electron interactions. At the same time a ring structure appears in the middle of the QPC resembling an anti-dot embedded into the QPC. Asymmetry on the split gates helps to achieve this regime. Our findings fit well with the results of experimental studies [38,39,43] and previous theoretical predictions of a formation of a zig-zag of localized electrons [44][45][46].
The present paper is organized as follows: in section 2 we describe a model of a top-gated QPC, in section 3 we present the results of numerical calculations, in section 4 we discuss our findings and their importance for the explanation of experimental results, in section 5 we outline conclusions of our study.

Model and structure
Gated modulation-doped GaAs/AlGaAs heterostructure studied in this paper is represented in figure 1. By appropriate combination of semiconductor materials, patterned gates, doping and applied gate voltages the electrons can be trapped at the GaAs/AlGaAs interface forming the 2DEG lying below the surface. 2DEG is then properly shaped in the form of a QPC by means of lithography and application of a negative voltage to the metallic gates on top of the structure. As shown in figure 1(b) two extended semi-infinite metallic plates on top of the heterostructure, loaded by the negative voltage V g , are used to create a wide QW that serves as an electron reservoir. The two split gates, loaded by the voltages V sg1 and V sg2 , have a length of 200 nm, a width of 200 nm and they are 600 nm apart from each other. These split gates are used to create a quantum constriction in the transversal y-direction resembling a QPC. Voltages on these two split gates may be applied asymmetrically. 2DEG resides 300 nm underneath the split gates in the z-direction. An additional top gate with a length 600 nm and a width of 800 nm partially covers the spilt gates and is separated from them by a 200 nm thick dielectric layer. By applying a negative top gate voltage V tg one can vary electron density within the QPC down to 10 10 cm −2 allowing the regime of low electron density. In our calculations the donor layer has a thickness of 200 nm and the uniform donor density is chosen as 3.2 × 10 16 cm −3 .
The electrostatic confinement potential has been calculated as: where r = (x, y) is a position vector in two-dimensional space. Potential from donors eV d is −0.95 eV and potential from surface states eV s is 0.8 eV. The electron density of 2DEG is 3.8 × 10 10 cm −2 for unbiased top and split gates. Potential eV g (y) from the two extended metallic plates has been calculated using the analytical expression in [47]: where d = 980 nm is the width of the wide QW and z 0 = 300 nm. By varying the negative voltage V g on these two metallic plates one can reach a strong or a weak confinement regimes. The potentials on split gates eV sg1 and eV sg2 as well as on the top gate eV tg have been calculated by numerical integration over the surface of these gates using the expression in [48]: where V(r , 0) is the distribution of a potential on split (or top) gates, z is the distance from the gates to the 2DEG, i.e. 300 nm from the split gates and 500 nm from the top gate. Typical confinement potential for a weak confinement is shown in figure 2 which corresponds to a strong confinement. In the latter case, the confinement potential looks like a saddle potential that is parabolic in the transversal y-direction. The case of a strong confinement may be realized either by reducing the distance between the two extended metallic plates or the distance between two gates forming the split gate. This can also be achieved by loading these gates with stronger negative voltage. The confinement potential becomes more shallow if the 2DEG resides farther from the top of the structure as has been studied in [49]. The effective potential V σ eff includes the electrostatic confinement V conf from equation (1), the electron-electron interaction V ee , and exchange-correlation V σ xc potentials: where σ is electron spin. The electron-electron interaction potential V ee is calculated numerically including the contribution from mirror charges: where is the dielectric constant for GaAs, which is equal to 12.9, ρ(r) is total electron density which will be identified below in equation (9) and z 0 = 300 nm. To guarantee the convergence of the iterative procedure, the value of the electron-electron interaction potential (5) at each iteration is mixed with the one at the previous step. For the exchangecorrelation potential V σ xc we apply the parametrization expression from [50]. In order to break the spin symmetry we add a Zeeman term associated with a very tiny in-plane magnetic field (initially it is ∼ 10 −3 T but it is turned off after a few iterations). The direction of this additional magnetic field gives the axis of quantization and a preferable direction for the electron spin. Another way to get a spin polarization is to take into account the random noise (for example, from randomly distributed donors) as has been discussed in [51]. In that paper we also studied the effect of lateral spin-orbit interaction (LSOC) for asymmetrically biased QW on the formation of spin polarization. It has been shown that LSOC can also be used to trigger spin polarization in the presence of a non-zero source-drain bias. However, in the present study we do not consider the effect of LSOC.
As in our previous papers [24,28,49] we solve the twodimensional spin-relaxed Kohn-Sham equations with effective potential from equation (4): We solve the Kohn-Sham equations (6) self-consistently by discretization on a grid with the step of discretization of 10 nm using the finite difference method, as described in [28]. We use periodic boundary conditions: where T is the period in the longitudinal x-direction (1000 nm in our calculations), and k is the wave number: where m are integer numbers belonging to the first Brillouin zone (−M/2 < m M/2), M being the number of unit cells (10 in our calculations). At each iterative step we find the eigen-energies E σ k and the eigenstates φ σ k (r), for each spin   (9) and update the effective potential in equation (4). Its value is then used for the calculation of electron densities at each iterative step. In our calculations we choose the chemical potential μ = 0. The iterative procedure is finished when electron densities become identical within a given tolerance 10 −4 . The spin polarization of the system is calculated as a difference between densities for up-and down-spin electrons at the end of the iterative process: Probability current density for each spin direction is calculated using the quantum mechanical definition as described in [49], which allows us to calculate the total current and conductance. To perform the numerical simulations we have implemented an algorithm based on equation (6) and adapted it for parallel high performance cluster with distributed memory.

Symmetric split gates potentials
The effective potential for the case when V tg = −0. 48 figure 3(a). One can observe a double-well potential formed in the transverse y-direction of the conduction channel. The density of electrons is plotted in figure 3(b) for the same set of gate voltages. The conduction channel effectively splits in two and an anti-dot like structure arises in the middle of the channel.
However, in the opposite case for a strong confinement we have observed just one channel of conduction until it becomes pinched-off. In figure 4(a) we show the effective potential and in figure 4(b) the electron density for the case In the transversal y-direction, there is no barrier on the effective potential in the middle of the constriction. There is only a single conduction channel until the QPC becomes pinched off.
Let us now take a closer look at the case of a weak confinement. We plot in figure 5 the spin polarization calculated from equation (10), i.e. the excess of up-spin electrons over down-spin electrons, for the cases of a symmetric split gates voltage. Three cases are shown for figure 5(c). Top gate voltage for these three cases is set to V tg = −0.48 V. One can see that for the case in figure 5(a) there are four rows of spin polarization in x-direction (marked as 1, 2, 3 and 4), two on either side of the constriction, that correspond to a second sub-band in transversal y-direction contributing to conduction. For the case shown in figure 5(b) there is one row of spin polarization on each side of the constriction. There are pronounced spin polarization peaks in the middle of each conduction channel (marked as 1 and 2) and two additional spin polarization peaks on the two edges of the channels in the longitudinal xdirection (marked as 3 to 6 and 7 to 10 for each of these two channels). The system looks like an anti-dot with spin polarization on its perimeter. On further increasing the negative split gates voltage as in figure 5(c) only two spin-polarized states can be found on the two edges of the channels in the longitudinal x-direction (marked as 1 and 2 and 3 and 4 for each of the two channels). This spin polarization behavior should have an influence on conductance giving rise to additional anomalies. The detailed evolution of spin polarization has been studied in [28] for a single channel of conduction. It has been found that the spin-polarized states near the edges of the channel in the longitudinal x-direction give rise to a conductance anomaly near 0.25(2e 2 /h). In our case of a weak confinement the two rows of spin polarization are formed across the constriction (as seen in figure 5(b)) and each of them contributes 0.25(2e 2 /h) to conductance, so that the anomaly of this kind may be seen near 0.5(2e 2 /h). The degree of spin polarization p is about 10 −5 nm −2 within the conduction channels which is a typical value for QPCs obtained earlier in papers [24,28,49].

Asymmetric split gates potentials
Next, we study the case of asymmetrically applied split gates voltages that is shown in figure 6. We vary the negative voltages V sg1 and V sg2 on two splits gates by the same amount such that the difference ΔV sg = V sg1 − V sg2 remains constant. In figure 6 asymmetry between split gates voltages is ΔV sg = 0.1 V. In figure 6(a) we show the spin polarization for the split gates voltages set as: V sg1 = −0.23 V and V sg2 = −0.33 V. In this case there are three conduction channels (marked as 1, 2 and 3). In figure 6(b) we show the spin polarization for the split gates voltages: V sg1 = −0.28 V and V sg2 = −0.38 V when one channel is pinched off and conduction is governed by one remaining channel (marked as 1) producing a small conductance anomaly seen near 0.6(2e 2 /h) (seen in figure 7 for the curve corresponding to ΔV sg = 0.1 V). In this case one spin polarization peak appears in the middle of this channel. If we further increase negative voltage on the split gates as in figure 6(c) for V sg1 = −0.33 V and V sg2 = −0.43 V, we see that the spin polarization peak splits into two peaks (marked as 1 and 2) on the longitudinal edges of the channel that give rise to a small conductance anomaly near 0.25(2e 2 /h) that can be seen in figure 7 for the curve corresponding to ΔV sg = 0.1 V. The conductance anomaly occurs in the interval of split gates voltages V sg1 of about 5 × 10 −3 V. Therefore, the conductance anomaly due to spin polarization peaks on the longitudinal edges moves from 0.5(2e 2 /h) in the symmetric split gates case to 0.25(2e 2 /h) when asymmetric split gates voltages are applied. This happens because of different contributions to conduction from the two conduction channels. For larger split gates voltages difference as, for example, for ΔV sg = 0.2 V the 0.7(2e 2 /h) conductance anomaly re-appears (as in figure 7 for the curve corresponding to ΔV sg = 0.2 V), which points to a spin polarization peak in the middle of a single conduction channel     since the second conduction channel becomes pinched off very quickly. Conductance anomaly at 0.25(2e 2 /h) can be seen in figure 7 for the curve, corresponding to ΔV sg = 0.2 V, due to two spin-polarized states on the two edges in longitudinal x-direction.
In figure 7 we plot conductance for cases when split gates voltages difference are ΔV sg = 0 V, ΔV sg = 0.05 V, ΔV sg = 0.1 V and ΔV sg = 0.2 V. The curves are shifted by 0.03 V for better resolution. For split gates voltages difference of ΔV sg = 0 V and ΔV sg = 0.05 V there are two channels of conduction producing the conductance anomaly between 0.5(2e 2 /h) and 0.65(2e 2 /h). When ΔV sg = 0.1 V, with increasing V sg2 eventually one channel is pinched off and a conductance anomaly occurs near 0.25(2e 2 /h). In the case when ΔV sg = 0.2 V there is only a single conduction channel from the very beginning and the conductance anomaly occurs near 0.7(2e 2 /h), which is typical for a conduction through a single channel. Furthermore, one can expect extra conductance anomalies because of tunneling between spin-polarized states on the perimeter of the anti-dot. The appearance of such anomalies depends on the size of the anti-dot, which in turn is determined by the top gate voltage.

Discussion
Electron-electron interaction plays an important role in the formation of two conduction channels. The described effect occurs at low electron density that can be achieved by changing the negative voltage on the top gate. Our findings of the formation of two conduction rows agree with the results of experimental papers [38,39] and with the results of the theoretical study [52] for symmetric split gates potentials. Owen and Barnes in [52] have shown that the exchange-correlation potential induces density modulations in a transverse direction of the QW. Eventually, under further weakening of the confinement the electron gas splits into two rows. It happens at low electron density when screening between electrons is weak. The authors of paper [52] produced a phase diagram that shows a transition from a single QW to two QWs in a structure with different top gate and split gates voltages. We emphasize that it is a very delicate interplay between the top and split gates confinement potentials and electron-electron interaction that gives rise to this phenomenon. We focus in this paper on the asymmetrically biased QPC that has previously not been studied theoretically. Study of spin polarization in asymmetric QPCs has gained a special interest because of the prediction of Wigner crystal formation in such structure as, for example, in recent experiments with fractional quantum states [41,42]. Moreover, an electron focussing lens has recently been employed to study an asymmetrically biased QPC [53]. A correlation has been found between conductance anomalies in asymmetric QPC and focussing peaks in a detector QPC.
It has been shown in paper [54] that by tuning the relative strength of electron interactions and reducing the electron density one can approach the Wigner crystal regime, where the Coulomb repulsion energy is much stronger than the kinetic energy. Matveev et al [44,55] have suggested that a Wigner crystal can evolve into a zig-zag chain as a function of electron density in a QW. Exchange interaction between neighboring spins can cause the formation of ferromagnetic and anti-ferromagnetic states, the dimer order etc. Pepper et al [39] have for the first time studied the possibility of the formation of a Wigner crystal in QPCs in the regime of weak confinement. In the strong confinement regime, the conductance turns out to be the usual step of integer 2e 2 /h. Near the transition to the intermediate regime the plateau at 0.5(2e 2 /h) appears and gets stronger as the confinement is weakened further. At the same time, the plateau 2e 2 /h starts to be suppressed while the higher plateaus are still present. In their study the suppression of the plateau at 2e 2 /h was associated with the formation of a double-row configuration where there are effectively two channels contributing to the conductance. Theoretical investigation in the frame of quantum Monte Carlo simulation of a quantum ring shows the existence of the anti-ferromagnetic spin states in the low electron density regime [56]. Furthermore, the simulation of a 1D QW in the frame of the same model has shown a possibility of a zig-zag chain formation [46]. In our previous paper [27] we studied a QW with periodical boundary conditions and recovered the regime of formation of the spin-polarized electron rows.
In the present study spin-polarized electron states appear within each of conduction channels by a proper choice of voltages on the top and split gates. When split gates voltages are applied symmetrically, the effect of two conduction channels can be seen on the conductance dependencies where the anomaly at 0.5(2e 2 /h) is associated with the spin-polarized states on the two sides of the constriction in the longitudinal x-direction for two conduction channels (marked in figure 5(c) as 1-4). The 0.5(2e 2 /h) anomalies have been observed in the early experimental studies [38,39]. When split gates voltages are applied asymmetrically, this anomaly moves down to 0.25(2e 2 /h), which is associated with two spin-polarized states on the two edges in a single conduction channel (marked in figure 6(c) as 1 and 2). This case is realized when the voltage difference on split gates V sg1 and V sg2 is rather large and from the very beginning there is only a single conduction channel. The conductance anomaly at 0.7(2e 2 /h) can be seen in figure 7 for the case of ΔV sg = 0.2 V. It is caused by a formation of the spin polarization in the middle of this channel, while the second conduction channel is already pinched off. The reappearing of the 0.7(2e 2 /h) anomaly for the case of a strong asymmetric confinement has been shown in the recent experimental paper [43]. This fits well with our conclusion that only one conduction channel remains in this case contributing to the conductance. Additional conductance anomalies due to the spin-polarized electron states on the perimeter of the antidot were not clearly recognized for the present calculation but may be seen in experiments.

Conclusions
We have studied the top-gated semiconductor heterostructure in the case of shallow symmetric and asymmetric confinements. The effect of conduction channel splitting into two channels discussed in this paper is caused both by confinement and electron-electron interaction in the case of low electron density. Spontaneous spin polarization of electrons occurs at certain split gates voltages and gives rise to conductance anomalies near 0.5(2e 2 /h) associated with the contribution of spin-polarized states on the edges of two conduction channels in the longitudinal x-direction. With application of asymmetric split gates voltages this anomaly moves until it reaches 0.25(2e 2 /h), which corresponds to a single conduction channel with two spin-polarized states on its longitudinal edges. If asymmetry of split gates voltages increases, we have observed that at some rather large difference between split gates voltages conductance for a single channel is restored with anomaly realized at 0.7(2e 2 /h). One can observe the trace of conductance anomalies that are realized between 0.5(2e 2 /h) and 0.25(2e 2 /h) for different asymmetric split gates voltages. One can notice that 0.25(2e 2 /h) conductance anomaly corresponds to 1/2 fractional state. Different spin-polarized states found in the present study may be interesting for analysis of the fractional quantum states discovered in experiments [41,42]. As can be seen in figure 6(c), in the low electron density regime there are a few spin-polarized states on the anti-dot's perimeter. It may be predicted that the rich behavior of conductance anomalies observed in experiments may be a result of tunneling between these spin-polarized states. This requires a special study for different confinements and electron densities. This anti-dot structure itself may be interesting for the study of interference effects in 2DEG realized without application of a magnetic field and for the implementation of electron focussing devices, such as electron focussing lens.
One can extend the present model by developing the strictly correlated-electrons density functional theory to construct exchange-correlation potential in Kohn-Sham equations in order to take into account strong electron correlations. Such an approach has been implemented for 1D QW where the localized electron states have been recovered [57]. However, this approach has to be developed further in the case when the 1D wire relaxes into two dimensions.