Effect of the tilted magnetic field on the magnetosubbands and conductance in the bilayer quantum wire

Physica B 499 (2016) 76–86

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Effect of the tilted magnetic field on the magnetosubbands and conductance in the bilayer quantum wire T. Chwiej AGH University of Science and Technology, al. A. Mickiewicza 30, 30–059 Cracow, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 5 January 2016 Received in revised form 17 June 2016 Accepted 18 June 2016 Available online 19 June 2016

We theoretically study the single electron magnetotransport in GaAs and InGaAs vertically stacked bilayer nanowires. In considered geometry, the tilted magnetic field is always perpendicular to the main (transport) axis of the quantum wire and, therefore its transverse and vertical components allow separately for changing the magnitude of intralayer and interlayer subbands mixing. We study the changes introduced to energy dispersion relation E(k) by tilted magnetic field of strength up to several tesla and analyze their origins for symmetric as well as asymmetric confining potentials in the growth direction. Calculated energy dispersion relations are thereafter used to show that the value of a conductance of the bilayer nanowire may abruptly rise as well as fall by few conductance quanta when the Fermi energy in nanosystem is changed. It is also shown that such conductance oscillations, in conjunction with spin Zeeman effect, may give a moderately spin polarized current in the bilayer nanowire. & 2016 Published by Elsevier B.V.

Keywords: Quantum wire Ballistic transport Spin polarized transport

1. Introduction The electron transport properties of two quantum wires set close to each other depends largely on the magnitude of their tunnel coupling [1–5]. For strong and moderate coupling strength, the electron wave functions originating from separate wires can hybridize. The magnitude of this hybridization depends naturally on that, if the energies of involved subbands are fitted and on whether their wave functions overlap or not. Both factors are particularly sensitive to the strength of the magnetic field, its orientation in relation to the direction of electron transport and depend on the spatial alignment of the wires as well [6,7]. The magnetic field directed along the transport axis (longitudinal direction) squeezes the electron wave functions leading to their stronger localization within a particular nanowire. Consequently, due to smaller overlap of the subbands' wave functions, their hybridization will be remarkably diminished [8,7]. However, in case of the magnetic field being perpendicular to the direction of electron motion, the Lorentz force pushes the electrons to the edges of the nanowires. The wave functions being pushed to the central barrier separating the wires can be in such case easily mixed. The main result of such interwire subbands mixing is the transformation of their crossings into anticrossings what opens the pseudogaps in the energy spectrum E(k) [9,3]. Within a single pseudogap, the conductance is at first lowered and then increased by single conductance quantum when the Fermi energy scans the anticrossing from the bottom. Generally, the mechanism lying behind the formation of pseudogaps in energy is the same for a nanosystem constituted by the two laterally or vertically coupled http://dx.doi.org/10.1016/j.physb.2016.06.017 0921-4526/& 2016 Published by Elsevier B.V.

transport layers. In practical applications, however, the latter gives more possibilities for changing the conductance of bilayer nanowire. First, it allows for fine tuning the Fermi energy in particular wire, independently of its value in the second one, by means of the top, the bottom and the side gates [10–13,5]. Second, each of the three components of magnetic field modifies in a different manner the conductance of a nanosystem [14,4]. For this reason, the nanosystem with two vertically aligned transport layers gives an opportunity to control even a single magnetosubband [14] what cannot be achieved if two wires are coupled laterally. In this paper, we theoretically study the effect of tilted magnetic field on a single electron transport in nanosystem constituted by two vertically coupled layers. The confining potential in considered nanosystem can be realized by application of the conventional heterojunctions like InAlAs/InGaAs or AlGaAs/GaAs . We assume the electron current flows in the wire (along x-axis) without scattering, that is, we consider the ballistic transport regime only. The layers are vertically aligned, one over another in the growth (z-axis) direction and the surrounding confining potential has the shape of the rectangular barrier. The direction of the magnetic field is perpendicular to the nanowire axis. Thus, we assume that only the vertical ( Bz ) and transverse ( By ) components of magnetic field have non-zero values and both can be changed independently with precision [15]. Such an approach gives us an ability to tune the interlayer and intralayer modes coupling by changing By and Bz , respectively. At first, we discuss the effect of the magnetical mixing of the vertical and transverse eigenmodes on the energy dispersion relation E(k). Then, the oscillating

T. Chwiej / Physica B 499 (2016) 76–86

well. Then, the positively ionized dopants effectively lower the confining potential near both edges of the well giving thus a system with two coupled layers within a single nanowire [7,20]. Depth of the upper well, or in other words the value of parameter α, can be tuned by adjusting the voltage applied: (i) to the central top gate [8] which may cover the whole structure [21], or, (ii) to the top split gate [7]. The example of a model confining potential is showed in Fig. 1(a). A bilayer system can be also formed by stacking two quantum wires one above the other during the epitaxial growth. Then the very narrow tunnel barrier separates them [5,14,12]. Since all effects we discuss here, depend mainly on the energy difference between two lowest eigenstates for vertical quantization, the actual shape of vertical confinement is of little importance. For that reason, our results presented below are representative for both types of confinements. The upper and lower layers are pierced by tilted homogeneous magnetic field which has only two non-vanishing components, → namely B = (0, By , Bz ). We assume that the values of By and Bz can be freely changed. Because the magnetic field lifts the spin degeneracy, considerations are limited mainly to subbands with spins set parallel to direction of magnetic field until otherwise is stated. The energies of subbands with antiparallel spins can be simply obtained by adding spin-Zeeman splitting energy ΔEZ = gμB B . In calculations we use a non-symmetric vector po→ tential A = [zBy − yBz , 0, 0], for which the single electron Hamiltonian reads

behavior of the magnetosubbands in vicinity of k = 0 and its influence on the nanowire's conductance is considered. We show that these subbands oscillations may substantially influence the conductance as it can be changed abruptly by a few conductance quanta when the Fermi energy level is successively shifted between two neighboring pseudogaps. This however depends on whether the bilayer nanowire works strictly in the ballistic regime or not. We also show how the effect of hybridization of magnetosubbands in the bilayer nanowire may potentially be utilized as a source of partly spin polarized current for moderate Fermi energies. The paper is organized as follows. In Section 2 we present the theoretical model used in calculations. Properties of magnetosubbands in presence of tilted magnetic field are studied in Section 3 while in Section 4 we discuss a potential application of pseudogaps appearing in energy spectrum for partial spin polarization of conductance. We finish with conclusions given in Section 5.

2. Theoretical model The confining potential in a conventional semiconductor quantum wire can be formed electrostatically by gating two-dimensional electron gas (2DEG) [13], etching of nanogrooves [5] on the layered nanostructure that holds 2DEG few tens of nanometers beneath the surface or by the cleaved-edge overgrowth [16]. In all cases the 2DEG is formed within a square well created by double inverted heterojunction [5]. The first three methods give soft lateral confining potential which was widely used in theoretical works before [17,18] while the last one gives the rectangular confinement that we have adopted for this work. We consider a quantum wire in which the electrons move freely along x-axis but their motion in y–z plane is quantized since it is limited to a small area of rectangle quantum well, that is, y ∈ [ − a, a] and z ∈ [ − b , b]. Throughout this paper we use a = 50 nm and b = 15 nm . For simplicity we assume that the barrier surrounding the wire is infinite while the confining potential inside it depends only on the position in the growth direction (z-axis) i.e. → V ( r ) = V (z ). We model the confining potential with the following formula [19]:

V (z ) = Vmax[sin( (1 + z/b)π /2) + αsin( π (1 + z/b))].

H=−

→ Ψp, k( r ) = eikx

N

(3)

b

120

α=0 α=0.02

|ψ(z)|2

V(z) [ meV ]

M

∑ ∑ cm(p,)j·fm (z)·φj(y).

In Eq. (3) fm(z) and φj(y ) are the basis functions that describe the quantized motion of an electron in z and y directions, respectively. Since the confining potential in y direction [Eq. (1)] is rectangular with infinite barriers surrounding the quantum wire, we

α=0 α=0.1

160

(2)

m=1 j=1

(1)

a

= 2 2 iq= ∂ q2 2 ∇ + ⁎ Ax + Ax , 2m⁎ m ∂x 2m⁎

where i is an imaginary unit, q = − e is an electron charge and m⁎ is its effective mass (m⁎ = 0.067 for GaAs and m⁎ = 0.04 for InGaAs). The eigenstates of Hamiltonian given in Eq. (2) can be expressed as the linear combination of products of the plane waves for x direction and the eigenstates for the transverse (y-axis) and vertical (z-axis) directions. We define the electron wave function for the p − th subband and the wave vector k as follows:

Such confining potential has single maximum localized in its central region. It can be formed within a wide quantum well when one δ − doped layer is placed below and another one above the

200

77

80 40 0 -15

0 z [ nm ]

15

-15

0 z [ nm ]

15

Fig. 1. (a) The cross section of the confining potential in z direction for α = 0 and α = 0.1 (the left and right axes are the infinite barriers). (b) The probability density distributions of two lowest vertical eigenstates for symmetric ( α = 0 – black color) and asymmetric ( α = 0.02 – red color) confining potentials. The continuous curves stand for the ground state while the dashed lines for the first excited one. In (a) and (b) Vmax = 150 meV . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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T. Chwiej / Physica B 499 (2016) 76–86

immediately get the normalized basis functions for this direction

φj(y) = sin⎡⎣ jπ (1 + y /a)/2⎤⎦/ a ,

j = 1, 2, …, N .

(4)

The basis functions fm(z) are the eigenstates of Hamiltonian hz = − (= 2/2/m⁎)∂ 2/∂z 2 + V (z ) and have been found numerically by solving an eigenproblem of hz on a spatial mesh of nodes. The (p ) coefficients of linear combination cm , j appearing in Eq. (3) are the real numbers which are strictly 0 or 1 when magnetic mixing is (p ) absent (By = Bz = 0), otherwise they fulfill condition |cm , j| ≤ 1. The maximum in the confining potential divides the original well into two narrower tunnel-coupled wells, which for α = 0 have the same energy levels. Fig. 1(b) shows the probability densities for the ground [f1(z )] and the first excited [f2 (z )] eigenstates of Hamiltonian hz. For large amplitude of V(z) [see Eq. (1)], the densities are localized mainly in two narrower wells but to some extent they mutually overlap. As the energy difference between two (z ) lowest eigenenergies, that is ΔE21 = E2(z ) − E1(z ), is quite small, even a small distortion in the confining potential [ α = 0.02 in Figs. 1(b)] can effectively mix them. The result of mixing is their spatial separation [red curves in Fig. 1(b)]. The upper and lower narrow wells form then two coupled transport layers and the nanowire becomes in fact a bilayer system. To get deeper insight into the (z ) nature of this process, we calculate ΔE21 in dependence on Vmax. (z ) This dependency is showed in Fig. 2(a). In spite of α′s value, ΔE21 (z ) decreases when Vmax grows but the lowest ΔE21 we get always for α = 0. For α > 0, the ground state is localized in the upper well, what minimizes its energy, while the first excited one occupies the less energy favorable lower well [21]. Due to the large energy separation of the higher vertical eigenstates fm(z) with m = 3, 4, … from f1(z ) and f2 (z ) states, their contributions to the results can be neglected. Therefore in calculations we have used M¼ 2 in Eq. (3) while the number of transverse modes φj(y ) was limited to N ¼30. Having the wave functions defined in Eq. (3), we used them for transformation of Hamiltonian given by Eq. (2) to much simpler algebraic form. For this purpose, we first eliminate x variable from Hamiltonian H by averaging it over this variable Hy, z = 〈e−ikx|H|eikx〉. The reduced Hamiltonian then reads

Hyz = −

⎞ q=k =2 ⎛ 2 q2 2 2 A x + V (z ). ⁎ ⎜ ∇ y, z + k ⎟ − ⁎ Ax + 2m ⎝ m 2m⁎ ⎠

Next,

we

calculate

Hz = 〈φi(y )|Hyz|φj(y )〉

and

⎛ = 2γ 2 ⎞ 1 (2)⎟ j z H0 = ⎜⎜ ⁎ δi, j + Emδi, j + ⁎ Yi, j ⎟δl, m, 2m ⎝ 2m ⎠

H1 =

m⁎ωy2 2

(7)

Zl(,2m)δi, j,

(8)

H2 = − ωy=kδi, jZl(,1m),

(9)

H3 = − m⁎ωyωz Yi(,1j )Zl(,1m).

(10)

Symbols used in the above equations are defined as follows: γj = jπ /2/a, j = 1, 2, …, N , ωy = qBy /m⁎, ωz = qBz /m⁎, Zl(,1m) = 〈fl |z|fm 〉,

Zl(,2m) = 〈fl |z 2|fm 〉, Y i(,1j ) = 〈φi|y|φj〉, Y i(,2j ) = 〈φi|(m⁎ωzy + =k )2|φj〉 and δi, j is a Kronecker's delta. The indices (i,j) stand for the transverse modes [defined in Eq. (4)] while the pair of (l,m) mark the vertical ones. The effective Hamiltonian has now the matrix form which for only two-element basis {fm (z )} becomes 2  2 block matrix with real elements

⎡ H11 H12 ⎤ ⎥. Heff = ⎢ ⎣ H21 H22 ⎦

(11)

As all the terms appearing in Eqs. (7)–(10) are real, we imT . The only term in Eq. (6) that mediately get condition H21 = H12 depends on δl, m is H0. Therefore it contributes only to the diagonal submatrices H11 and H22 (their rank equals N, that is, the number of the basis states φj). Consequently, H0 does not mix vertical eigenstates. If the confining potential Vz is symmetric (α = 0) then the wave function f1(z ) has even parity (ground or bounding state) while f2 (z ) is odd (the first excited or antibonding state). Due to the even parity of the term Zl(,2m) , the matrix elements H1 give

(5)

contributions to the diagonal submatrices only if α = 0. Otherwise (α > 0) they contribute also to the off-diagonal submatrices H12

thereafter

and H21. However, the term Zl(,1m) has an odd parity for α = 0.

Heff = 〈fl (z )|Hz|fm (z )〉. Finally, the effective Hamiltonian gets the following form:

Heff = H0 + H1 + H2 + H3,

(6)

Therefore, it gives contribution to H12 due to H2 and H3 terms, or in other words, it always hybridizes the vertical states f1 and f2. Note that for α > 0, both terms Zl(,1m) and Zl(,2m) give contributions to the diagonal matrices H11 and H22.

with

8

α=0.02 α=0.01 α=0

4

0 50

100 Vmax [ meV ]

150

10 E1(0) - E1(km) [ meV ]

a

(z)

ΔE21 [ meV ]

12

b

ΔEz,12

8

2 meV 5 meV 10 meV

6 4 2 0

0

2

4 6 By [ T ]

8

10

(z ) Fig. 2. (a) The energy difference ΔE21 between the first and second vertical eigenstates in dependence on Vmax for several values of α. (b) The difference between maximum (k¼ 0) and minimum (km) of energy of the lowest subband in dependence on By. Results in (b) were obtained for Vmax = 150 meV and α = 0 .

T. Chwiej / Physica B 499 (2016) 76–86

3. Results

A3 = 3.1. Two-state model for Bz = 0 For Bz = 0 the basis states {φi(y )} are not mixed because the term

Y i(,1j ) disappears in Heff, while the term

Y i(,2j )

2 2

Y i(,2j ) reduces to

= = k δi, j due to ωz = 0 [see Eqs. (7)–(10)]. This gives us pos-

sibility to limit our considerations for a moment to the case with N ¼1. Then, the energy dispersion E(k) for the transverse modes φj with indices j > 2 are strictly the replicas of that with j¼ 1. It means that the diagonal elements of Heff are shifted towards the higher energies in the same manner. The matrix form of two-state Hamiltonian now reads

⎡ H0 + H1 H2 ⎤ ⎥ Heff = ⎢ H0 + H1⎦ ⎣ H2

(12)

⎤ ⎡ = 2(γ 2 + k 2) m⁎ωy2 (2) (2) 1 ⎥ ⎢ + E1(z ) + Z1,1 −ωy=kZ1,2 ⁎ ⎥ ⎢ 2m 2 ⎥ ⎢ 2 2 2 = ( γ + k ) 1 (2) (z ) ⎥ . =⎢ − ω = kZ + E y 2 ⎥ 1,2 ⎢ 2m⁎ ⎥ ⎢ ⁎ 2 m ωy (2) ⎥ ⎢ + Z 2,2 ⎥⎦ ⎢⎣ 2 The eigenvalues of Heff can be written as

E1,2 =

⎛ A ⎞2 2 |A1| = 2k 2 + A ± + 1 ⎜ 2⎟ k 3 2m⁎ 2 ⎝ A1 ⎠

(14)

where we have used the following abbreviations:

⎛ ⎞ m⁎ω 2 ⎛ ⎞ y ⎜ Z (2) − Z (2)⎟, A1 = ⎜⎜ E2(z ) − E1(z )⎟⎟ + 2,2 1,1⎟ ⎜ 2 ⎝ ⎝ ⎠ ⎠ A2 =

⁎

2 ⎛ m⁎ ⎞2 2 ⎛ A1 ⎞ ⎜ 2 ⎟ A2 − ⎜ ⎟ . ⎝ 2= ⎠ ⎝ A2 ⎠

km = ±

a

(18)

Value of km depends on A1 and A2. Therefore, for a fixed geometry and values of material parameters of nanowire, km depends directly on By because expression in square root in Eq. (18) has to be non-negative. Minimal value of By that gives k m > 0 can be estimated from the following formula:

(

)

E2(z ) − E1(z ) 2m⁎ . 2 q 4 Z (1) 2 − ⎛⎜ Z (2) − Z (2)⎞⎟ 2,2 1,2 1,1 ⎝ ⎠

( )

(19)

(z ) For α = 0, when the energy difference ΔE21 is determined by the (z ) value of Vmax, the increase of Vmax results in decreasing of ΔE21 . Consequently, smaller Bymin is then needed for lateral minima to appear. The example of energy difference between the maximum and minimum of energy depending on By for the first subband is (z ) shown in Fig. 2(b). Let us notice that, the lower ΔE21 is, the lower By is needed for this difference to have the non-zero value. In the vicinity of k ¼0 the squared term in Eq. (14) can be approximated by its power series of k2 terms. Then, by neglecting the terms kn for n > 2, we get the simplified expression for the energy dispersion

k→0

= A3 ±

(15) (16)

(17)

The energy dispersion relation E(k) for these two subbands are displayed in Fig. 3(a) (black color). In first subband there are three extrema, two minima separated by a maximum localized at k ¼0. Localizations of these two energy minimums can be found by imposing condition on a dispersion relation ∂E /∂k|k = km = 0 what gives

E1,2

(1) 2ωy=Z12 ,

20

⎛ ⎞ ⎞ m⁎ωy2 ⎛ (2) ⎜ Z + Z (2)⎟. + ⎜⎜ E1(z ) + E2(z )⎟⎟/2 + 2,2⎟ 1,1 ⎜ 2m 4 ⎝ ⎝ ⎠ ⎠

= 2γi2

Bymin >

(13)

79

⎛ =2 |A1| 1 A22 ⎞ ⎟⎟. + k 2⎜⎜ ⁎ ± 2 4 |A1| ⎠ ⎝ 2m

(20)

where the sign +( − ) corresponds to the upper (the lower) subband. It turns out that both subbands have parabolic shapes near k¼ 0. However, due to the magnetic hybridization, the parabola

80

2-nd

b

4 0

-km -0.2

0 k [ nm-1 ]

40

km 0.2

20

0

1-st

8

60 2 -n d

1-st

12

E [ meV ]

E [ meV ]

16

0.2 0.4 k [ nm-1 ]

Fig. 3. Energy dispersion relation in function of the canonical wave vector k (black line) and the kinetic wave vector kkin (red line) for two lowest subbands. Figure (b) is a continuation of (a) but for the higher energy and shows the degeneracy of both subbands as function of kkin. Horizontal arrows on (a) show directions of the wave vector (z ) transformation k → kkin , while the vertical ones mark the energy minima in the first subband. Parameters used in calculations: By = 10 T , Bz = 0 , α = 0 , ΔE21 = 5 meV . All energies are given with respect to a bottom of the lowest subband. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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T. Chwiej / Physica B 499 (2016) 76–86

corresponding to the local energy maximum in the lowest subband is inverted. For k ¼0, the energy difference between both subbands equals |A1| and thus depends on the sum of two terms: (2) (2) − Z1,1 ( E2(z) − E1(z)) and ( Z2,2 ). It means that, if the basis states

f1(z )

and f2 (z ) become degenerated for increasing Vmax [Fig. 2(a)] with similar densities [Fig. 1(b)], then the energy difference between two lowest subbands gets much smaller. The presence of two lateral energy minima in the magnetosubbands considerably modifies the electron transport in the nanowire. For the canonical wave vector's interval k ∈ (0, k m), the kinetic wave vector kkin = 〈k + qAx /= 〉 corresponding to the electron's group velocity becomes negative and vice versa for k ∈ ( − k m, 0). This negative energy dispersion relation is exclusively caused by the effect of the magnetic hybridization of the subbands in the bilayer nanosystem. We will analyze this problem in detail. In Fig. 3 we have plotted the energies of two lowest subbands in dependence of kkin. After transformation k → kkin , the dependence of the electron's energy on kkin becomes ambiguous for the first subband if kkin is small. It consists of two curves: the closed loop surrounded by two parabolic branches. The horizontal arrows in Fig. 3 indicate the directions of the wave vector's transformations. First, let us notice that for the closed loop, k ∈ [ − k m, 0] transforms to kkin > 0 and due to the symmetry of E (k), k ∈ [0, k m] transforms to kkin < 0. Second remark concerns the scalability of kkin. In Fig. 3(a) we see that the value of the kinetic wave vector is compressed for the lower subband and expanded for the second one with respect to the canonical wave vector's value. And last, the two lowest subbands become degenerated for much larger Fermi energies what is showed in Fig. 3(b). It means that the electrons in both subbands move with the same group velocity vgr = =kkin/m⁎. This fact can be easily explained if the energy dispersion relation given by Eq. (20) will be expressed as function of kkin variable instead of k. For this purpose, we first calculate an expectation value of the kinetic wave vector for the p − th subband M (p ) 〈kkin 〉=k−

M

cl(,1p)cm(p,1)

∑∑ l= 1 m= 1

m⁎ωy

=

Zl(,1m).

(21)

(p ) where coefficients cl(,1p) and cm ,1 are the components of the twostate effective Hamiltonian (p ¼1,2) given by Eq. (13). By using the components of its eigenvectors, after some algebra we obtain the formula that describes the subband's kinetic wave vector

⎛ ⎞ A22 m⁎ ⎟. 〈kkin〉 = k⎜ 1 ∓ ⎜ 2= 2 A12 + A22 k 2 ⎟⎠ ⎝

(22)

where ( − ) stands for the first (lower) subband, while ( + ) for the second (upper) one. Next, by assuming a very large value of k we may replace the term with square root in Eq. (22) with its power series expansion. After leaving only the first term in this expansion m⁎ we get k ≈ kkin ± 2 |A2 |. Substitution of this approximate expres2=

sion for k into Eq. (14) finally gives

⎛ ⎞ =2 (p ) (p ) ⎟= E⎜ kkin ⁎ kkin ⎠ 2m ⎝

2

( )

+ A3 −

m⁎ 2 A2 . 8= 2

(23)

Now, it is easy to notice that, if the kinetic wave vectors for the first and second subbands have the same values, then these subbands are degenerated [see Fig. 3(b)]. 3.2. Interlayer subbands mixing in case of a non-symmetric confining potential In this section we consider the effect of the interlayer subbands mixing (Bz = 0) on the energy dispersion relation E(k) for α > 0 and

(z ) Fig. 4. Energy dispersion relation for α = 0 ( ΔE21 = 5 meV , left column) and (z ) = 5.88 meV , right column). All energies are given with respect to the α = 0.02 (ΔE21 bottom of the lowest subband.

on the conductance of the bilayer nanowire. In Fig. 4 we have plotted the low energy spectra calculated for our bilayer model for α = 0 and α = 0.02. If there is no the in-plane magnetic field (first row in Fig. 4), the vertical eigenmodes are not mixed and the symmetry of energy spectrum is kept i.e. E(k) = E( − k) irrespective of α′s value. For α = 0.02 the energy branches of subsequent subbands are only shifted upwards on energy scale in comparison to the case with α = 0. When value of By equals 5 T (second row in Fig. 4), then for α = 0 there appears the negative energy dispersion relation (k m > 0) in the three lowest energy subbands. These subbands correspond to the ground, the first and the second excited states in y direction, respectively. However, the difference between the maximum and minimum of energy for each of these subbands is very small as it equals only 0.25 meV . If the confining potential V(z) becomes slightly asymmetric [ α = 0.02 in Fig. 4(d)], it destroys the symmetry of energy subbands. In such case, the upper quantum well is wider than the lower one, which results in the larger energy separation of the basis states f1(z ) and f2 (z ). For (z ) α = 0.02 their energy difference grows up to ΔE21 = 5.88 meV . Although, this growth seems to be not large, it is sufficient to suppress the negative energy dispersion relation for the moderate value of an in-plane magnetic field (By = 5 T). On the other hand,

T. Chwiej / Physica B 499 (2016) 76–86

8

Bz

α

a

α

b

6 4 2

By

0

E [ meV ]

12

-0.1

0

-0.1

0.1

c

0

0.1

d

8

4

By

0

-0.2

20

-0.1

0

-0.2

0.1

e

-0.1

0

0.1

0.2

f

16 12 8 4 0

By

-0.2

0

-0.2

0.2

0

0.2 -1

-1

(z ) Fig. 5. Energy dispersion relation E(k) for ΔE21 = 2 meV (left column) and (z ) ΔE21 = 5.09 meV (right column) and Bz = 0 . The values of α are given at the top of each column. All energies are given with respect to the bottom of the lowest subband.

the remnant of the magnetic hybridization is still visible in Fig. 4 (d) as the minima in the three lowest subbands are shifted to towards the negative k. In this case, the electrons with k > 0 are

a

localized in the lower layer due to an action of the magnetic force. Moreover, they have higher energies than those with k < 0 [see localization of electron densities in Fig. 1(b)] which move in a wider upper well. For this reason, the right parts of the energy spectra in Figs. 4(d) and (f) are shifted upwards with respect to their left counterparts. The stronger magnetic field (By = 10 T) obviously enhances the effect of the magnetic hybridization. In Fig. 4(e) we notice that the difference between maximum and minimum of energy equals 5 meV for the three lowest subbands. It is thus much larger when compared to a fraction of meV we have got for By = 5 T . Additionally, the mixing of vertical eigenmodes is now so strong that the negative dispersion relation in energy spectrum is reconstructed also for α = 0.02 [cf. Figs. 4(d) and (f)]. (z ) , we As the extent of the vertical modes mixing depends on ΔE21 have repeated calculations for its smaller value. The left column in (z ) Fig. 5 displays the energy spectra for ΔE21 = 2 meV and α = 0. Again, in absence of the in-plane magnetic field component (By = 0) the subbands have parabolic shapes. Now however, in contrary to the previous case, a moderate in-plane magnetic field (By = 5 T) effectively mixes the vertical eigenmodes. There are formed two equally deep minima for α = 0 and one deep (the left one) and one shallow (the right one) for α = 0.02 [see Figs. 5(c–f)]. Some of these subbands, which lie higher on energy scale, cross with subbands of parabolic shape. These parabolic branches are formed when the electron is excited in the vertical direction. Value of m is then reduced to m = M = 2 in Eq. (3). As the parabolicshaped subbands have different parity than the oscillating ones, both types of subbands may cross each other what is shown in Fig. 5. Interestingly, these crossings survive also for α = 0.02 [cf. Figs. 5(d) and 5(f)]. However, let us notice here, that such crossings are absent in the low energy spectra presented in Fig. 4(c) due to (z ) the larger value of ΔE21 as well as due to the limited range of the energy scale in Figs. 4(e) and 4(f). For the strongest magnetic field considered here (By = 10 T) the energy minima for α = 0 become more than five times deeper than for moderate field (By = 5 T) and (z ) almost two times deeper than those for ΔE21 = 5 meV [cf. Figs. 5 (e) and 4(e)]. In Fig. 6 we compare the spin-up contributions to the conductance of the single- and the bilayer nanowires. The conductance has been calculated for temperature T¼ 0 by counting the crossings of the Fermi energy level with subbands for kkin > 0. The results obtained for a single-layer wire (M ¼1) show standard, well-known a step-like raising function (gray color). The height of these conductance steps equals the quantum of conductance G0 = e2/h (the spin degeneracy is lifted). On the other hand, the conductance of the bilayer wire [M ¼2, red color stands for α = 0 while the black for α = 0.02] also exhibits a step-like character.

12

b

By

By

8

2

/h]

8

2

/h]

12

81

4

4

0

0

5

10

EF

15

20

25

0

0

5

10

15

20

25

EF

Fig. 6. The spin-up contribution to the conductance of the nanowire for: (a) By = 5 T , b) By = 10 T and Bz = 0 . The results were obtained for a single layer wire (M ¼ 1, gray (z ) (z ) region) and for a bilayer wire (M¼ 2, black and red colors). Black color marks the conductance for α = 0 (ΔE21 = 5 meV) , while the red color for α = 0.02 (ΔE21 = 5.88 meV). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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Now however, two additional features are present: (i) the heights of the steps may be equal to G0 or 2G0 and, (ii) the value of conductance may fall by G0. The second case takes place when the Fermi energy exceeds the central energy maximum of the particular subband. Due to the stronger coupling of the vertical modes for By = 10 T , the rising and the falling steps become better separated than for By = 5 T . This is easily noticeable if we compare the changes in conductance for EF < 10 meV in Figs. 6(a) and 6(b). When the confining potential looses its spatial symmetry in z − axis direction (α = 0.02), the conductance of bilayer wire shows has the tendency to be lower than for α = 0, which establishes its upper limit. From the bottom it is limited by the conductance of a single-layer wire. However, the conductance of bilayer wire achieves its lower limit only occasionally, generally when the Fermi energy is small. 3.3. Simultaneous mixing of the vertical and transverse modes in tilted magnetic field If we account the vertical component of magnetic field (Bz > 0) in our considerations, then the term Y i(,2j ) appearing in the diagonal part of effective Hamiltonian [Eq. (6)] shall mix the transverse eigenstates φi(y ). Moreover, the transverse and vertical states can be simultaneously mixed by the matrix elements H3 defined in Eq. (10), provided that ωyωz ≠ 0. The latter term contributes to the offdiagonal elements in Eq. (11) for α = 0 due to non-zero value of

(1) or simultaneously to the diagonal and the off-diagonal subZ1,2 matrices in Eq. (11) for α > 0. (z ) Magnetosubbands for ΔE21 = 2 meV and Bz = 1 T are plotted in Figs. 7(a–c) for α = 0 and in Fig. 7(d) for α = 0.02. For By = 0, subbands have the parabolic-like shapes and do not cross each other. Since ωy = 0, all the off-diagonal elements [see Eqs. (8)–(10)] in effective Hamiltonian [Eq. (11)] vanish and only the diagonal elements given by Eq. (7) survive. Diagonal elements cannot however hybridize the vertical modes. For this reason, subbands corresponding to an excitation of electron in vertical (z − axis) direction [dashed lines in Fig. 7(a)] are simply the replicas of those in which the electron stays in the ground state. They are only (z ) . Due to the different shifted upwards on energy scale by ΔE21 parities of the vertical basis states with respect to reflection z → − z for By = 0, both types of subbands may cross in the higher part of energy spectrum [not shown in Fig. 7(a)]. That however does not hold for the tilted magnetic field. If both By and Bz have non-zero values, the vertical and transverse modes are simultaneously mixed. This double-mixing results from the activation of the off-diagonal elements in effective Hamiltonian [Eq. (10)] through the products of the terms Y i(,1j ) and Zl(,1m) . In Fig. 7(b) we see that for By = 5 T there are two deep lateral minima in the lowest subband similarly as in the case with By > 0 and Bz = 0 [cf. Figs. 4 (c) and 5(c)]. However, for Bz = 1 T , due to mixing of the transverse modes, the subbands lying higher on energy scale are not the replicas of the lowest one any longer. In vicinity of k ¼0, the energy

(z ) Fig. 7. Energy dispersion relation E(k) for ΔE21 = 2 meV , Bz = 1 T and different values of By. In (a), (b) and (c) we show the results for α = 0 , while in (d) for α = 0.02. In (a) the solid lines mark those subbands in which an electron occupies the ground state in the growth direction [f1(z )], while the dashed ones correspond to the first excited state [f2 (z )]. All energies are given with respect to a bottom of the lowest subband.

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crossings are replaced by the avoided crossings at positions where the subbands strongly mix with their neighbors. In consequence there appear additional pseudogaps in the energy spectrum. The widths of anticrossings are dependent on By. For its increasing value, spatial localizations of subbands in the lower and upper layers are enhanced. That in turn diminishes overlap of transverse modes belonging to different layers and may eventually decrease the widths of the anticrossings [cf. Figs. 7(b) and 7(c)]. Generally, the pattern of these avoiding crossings does not change much when th upper and lower quantum wells are slightly different [α = 0.02] besides an accuracy of additional positive slope in energy [cf. Figs. 7(c) and 7(d)]. Although, we might expect that the vertical modes shall be effectively mixed for By = 5 T , the pronounced two deep lateral minima are present only in two lowest subbands. For the remaining subbands these minima are replaced by bending points due to comparable mixing of the vertical and the transverse modes. If, however, the transverse component of the magnetic field becomes two times stronger (By = 10 T) then the coupling of the vertical modes dominates. In such case, the deep lateral minima are again present in all low energy subbands [cf. Figs. 7(c) and 7(d)]. It is worth to mention here that their localizations are then the same as in the lowest subband [Fig. 7(c)]. The appearance of additional pseudogaps in the energy spectrum obviously modifies the conductance of bilayer wire. An example dependence of wire's conductance on Fermi energy for tilted magnetic field ( Bz = 1 T and By = 10 T ) is shown in Fig. 8. For symmetric confinement (α = 0), the conductance raises by 2G0 when the energy exceeds the bottoms of the first three subbands. It is visible in Fig. 8 for E = 0 meV , E = 2.18 meV and E = 5.13 meV . When the Fermi energy exceeds one of these thresholds then subsequent subband is activated. It gives doubled contribution to conductance because it has two energy minima instead of one. On the other hand, conductance falls only by G0 when the Fermi energy grows above the local maximum of energy of particular subband. If small oscillations near k ¼0 are activated in subbands by tilted magnetic field then the conductance may abruptly change its value by more than 2G0. In such case, the heights of G steps depend strictly on the number of the local minimums belonging to the particular subband. The amplification factor equals then the subband's index plus one. For example, in Fig. 8 we see that the conductance grows by 4G0 when the energy increases from 9.1 meV to 9.4 meV and even by 6G0 when it is changed between 14.2 meV and 14.8 meV .

12

M=2 α α

10

2

8 6 4 2 0

M=1 0

4

8

12

16

20

Fig. 8. The spin-up contribution to the conductance of the single layer (M ¼ 1) and of the bilayer (M ¼2) wires in presence of the tilted magnetic field. Following parameters were used in calculations: By = 10 T and Bz = 1 T .

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This value may also be significantly decreased if the electron's energy is shifted upwards through the set of the maxima localized in proximity of k ¼0. Simply, it then enters a pseudogap region. We notice such a large drop, being equal to 5G0, when Fermi energy is increased from 15.95 meV to 16.7 meV . Generally, the conductance's steps for α = 0.02 are lower than for α = 0 mainly due to a considerable shift of energy levels in the narrower layer. This effectively suppresses the mixing of vertical modes because then (z ) is significantly growing. The conductance of a the value of ΔE21 single layer wire constitutes a lower bound for conductance of bilayer wire. Both are equal only if the Fermi energy localizes within a pseudogap. Such case is visible in Fig. 8 for both values of α and energies 13.6 meV and 16.7 meV .

4. Spin polarization of conductance in the bilayer nanowire In the previous sections we have discussed the mechanism of formation of pseudogaps in the electron's energy spectrum as well as the effect of pseudogaps on the nanowire's conductance. Those considerations did not account for the spin Zeeman effect and its contribution to an electron's energy. An interaction of the electron's spin with a strong magnetic field, what is the case considered here, splits the spin-down and spin-up subbands by ΔEZ = gμB B . For this reason, the conductance of the bilayer wire may, in fact, be partly spin polarized [22], i.e. η = (G↑ − G↓) /(G↑ + G↓) > 0. Let us note that the spin polarization shall be dependent not only on the number of the active spin-up and spin-down subbands as it is in the case of a single-layer wire but also on that, whether the Fermi energy is pinned within the pseudogap or not. In the latter case one may expect a larger value of η. In Fig. 9 we have plotted the conductance, its derivative with respect to energy (dG/dE ) and the value of the spin polarization of conductance (η) for the bilayer wire in function of By and energy for Bz = 1 T and T = 0 K . These outcomes were obtained for the wire made of GaAs [the first and second columns for (z ) ΔE21 = 2.0, 5.0 meV and g = − 0.44 ] and of InGaAs [23] [the third (z ) column for ΔE21 = 5 meV and g = − 4.0]. In Figs. 9(a), 9(d) and 9(g), which show the conductance, we notice two characteristic regions lying above and under the antidiagonal. In first region (above the antidiagonal) the conductance is changed very frequently. Its value is increased as well as decreased when the electron's energy grows. The second characteristic region appears rather for strong magnetic field (under the antidiagonal). It has more regular pattern resembling very much that of a single layer nanowire, as the value of conductance increases by G0 when a subsequent subband becomes active. Besides the conductance also the transconductance is very often measured in experiments as it is tightly connected with the dynamical properties of the nanosystem. The transconductance is sensitive to the variations of voltages applied to metallic gates [24] used e.g. to tune the Fermi energy in the wire. Figs. 9 (b), 9(e), and 9(h) show the dependence of dG/dE on the By and the Fermi energy. As the GaAs nanowire has relatively small g factor value, the energy splittings of subbands are small [Figs. 9 (b), and 9(e)]. Subsequent spin-up and spin-down subbands are gathered in pairs even for strong By. Consequently, the conductance can be, to a large extent, spin polarized but only for the very narrow energy stripes [see Figs. 9(c) and 9(f)]. For example, (z ) for ΔE21 = 2 meV [Fig. 9(c)] polarization may reach 60% and 50% for the following pairs of parameters: By = 5.94 T , E = 3.54 meV and By = 6.69 T , E = 6.63 meV . Regions with similar values of po(z ) larization can also be found in Fig. 9(f) for ΔE21 = 5 meV . For InGaAs nanowire, the energy splitting of subbands due to the Zeeman effect becomes even comparable with the energy

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Fig. 9. Conductance (first row), its derivative dG/dE (second row) and the spin polarization of conductance (third row) of a bilayer quantum wire. First and second columns (z ) present the results for GaAs while the third column for InGaAs. The energy difference ΔE21 is displayed at the top of each column. Value of the vertical component of magnetic field equals Bz = 1 T . All energies are shifted down so as to the lowest subband has zero energy for By = 0 . Therefore, in order to get the Fermi energy, the energy of the lowest subband must be subtracted before. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

difference between the bottoms of two neighboring subbands [see the right part of Fig. 9(h)]. In such case, the subbands which are characterized by the negative energy dispersion relation (blue curves) are significantly shifted on energy scale even for moderate magnetic field [e.g. By ≈ 4 − 8 T in Fig. 9(h)]. For this reason, the conductance can now be partially spin polarized in the much wider energy regions than it was for GaAs wire [see two distinct reddish stripes appearing near the central part of Fig. 9(i), which mark 60% and 50% conductance polarization, respectively]. This example show the advantage of the bilayer nanowires over, e.g., the Y-shaped nanostructures [25,26] in providing the partially spin polarized current for the moderate Fermi energies. Drawback of this solution is however that, it still requires strong magnetic fields to operate. We have repeated the calculations of spin polarization of conductance for InGaAs wire for temperature T = 4.2 K . The results are displayed in Fig. 10. As expected, the temperature

smearing of subbands considerably modifies the η's dependence on B and E values. Briefly, it lowers the spin polarization to about 30% and simultaneously shrinks the regions of large spin polarization of the conductance. In fact, these two unfavorable effects can be easily limited if the bilayer nanowire is made of the semiconductor material that has much larger gyromagnetic factor such as InSb [27] or InAs [28]. This however implies an existence of large spin–orbit interaction (SOI) in the nanowire, which would contribute to dispersion relation E(k). As it was predicted by Pershin et al. [29] and finally proved by Quay et al. [30], such mod→ ifications depend on the mutual orientation of the external (B ) and → the internal (BSO ) magnetic fields. Assuming only presence of Rashba SOI in our nanosystem, caused by the variations of the confining potential in the vertical direction [V = V (z )], from the → spin–orbital part of Hamiltonian HSO = α→ σ ·( k × ∇V ) we

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Fig. 10. The spin polarization of conductance for InGaAs bilayer quantum wire at T = 4.2 K . Other parameters are the same as for Fig. 9(i): g = − 4.0 and (z ) ΔE21 = 5 meV .

→ l BSO . Therefore, the comimmediately get the direction of BSO = y → bined effect of the transverse magnetic field (By) and BSO would rely only on enhancing of the spin Zeeman effect [29]. In such case, any fluctuations of Rashba SOI coupling, introduced e.g. due to fluctuations of dopants in δ layers [31–33], shall locally shift the subbands on energy scale, consequently smoothing the conductance steps in G(E) function. For tilted magnetic field, its ver→ tical component (Bz) is perpendicular to BSO , what implies mixing of the spin-up and spin-down subbands and leads eventually to formation of helical modes within the spin-pseudogaps. That, however, requires strong magnetic fields ( B = 9 T in Ref. [30] in comparison to Bz = 1 T in our paper) and one may expect its influence on the described here magnetic hybridization of the vertical modes in the electron's wave function shall not be completely destructive for weak fields.

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magnetic field, while the perpendicular component is responsible for the intralayer modes mixing. Irrespective of the actual coupling direction, if these small-amplitude oscillations appear in E(k), the conductance of a bilayer wire can abruptly change its value when the Fermi energy is changed. It can be increased as well as decreased by a few conductance quanta, provided that, the confining potential has low number of defects. These abrupt changes in conductance in conjunction with the Zeeman splitting of energy subbands give contribution to spin polarization of conductance. This polarization can achieve up to 60% at zero temperature and a half of this value for T = 4.2 T , even though several subbands are opened. As the magnitude of spin polarization can be tuned by changing the strength of the magnetic field and/or the Fermi energy, our results can easily be verified experimentally. Although there are no direct experimental results confirming our predictions yet, a number of experiments were performed for similar bilayer nanosystems. Thomas et al. [8] have measured the conductance of two vertically coupled wires in dependence of the top gate voltages for both the parallel and perpendicular magnetic fields. Also in work of Fischer et al. [7], the conductance of similar bilayer nanowire with asymmetric vertical confining potential was experimentally investigated. In both experiments however, the lateral confinement was smooth whereas our predictions concern the nanowires with rectangular-like lateral confinement. Since the shape of the lateral confining potential to a large extent influences the energy dispersion relation E(k) in bilayer nanowire, mainly due to the electron–electron interaction [34], the direct comparison of our results and the experimental data cannot be done.

Acknowledgments The work was financed by Polish Ministry of Science and Higher Education (MNiSW).

References 5. Conclusions We have theoretically investigated an effect of tilted magnetic field on the interlayer and intralayer subbands mixing in a system consisting of two vertically aligned nanowires with a rectangularlike external confining potential. It has been shown, that the transverse component of the magnetic field, which is perpendicular to the wire's axis but parallel to the layers, can effectively mix two lowest vertical eigenmodes what transforms the low energy parabolas into slowly oscillating curves with two deep lateral energy minima. If besides the transverse component of the magnetic field, the vertical one is also taken into account, then both the vertical and transverse modes are mixed simultaneously. In such case, the degeneracy of energy subbands in vicinity of k¼ 0 is lifted, the crossings between subbands are replaced by the avoided crossings, and consequently, the small pseudogaps appear in the energy spectrum. A qualitatively similar behavior of magnetosubbands were predicted for two laterally aligned wires by Shi and Gu [9]. They showed that only the perpendicular component of magnetic field is needed for effective mixing of magnetosubbands. That resulted from the fact that the directions of the intralayer and interlayer subbands' couplings were the same. Consequently, such unrestricted hybridization of magnetosubbands gives then simultaneously both types of oscillations in E(k), that is, two deep lateral minima as well as the small-amplitude oscillations near k ¼0. In contrary to the flat geometry of nanosystem considered by Shi and Gu, the interlayer subbands hybridization in the vertically aligned bilayer wire depends only on the transverse component of

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