Tuning of the spinorbit interaction in a quantum dot by an inplane magnetic field
Abstract
Using an exact diagonalization approach we show that one and twoelectron InAs quantum dots exhibit avoided crossing in the energy spectra that are induced by the spinorbit coupling in the presence of an inplane external magnetic field. The width of the avoided crossings depends strongly on the orientation of the magnetic field which reveals the intrinsic anisotropy of the spinorbit coupling interactions. We find that for specific orientations of the magnetic field avoided crossings vanish. Value of this orientation can be used to extract the ratio of the strength of Rashba and Dresselhaus interactions. The spinorbit anisotropy effects for various geometries and orientations of the confinement potential are discussed. Our analysis explains the physics behind the recent measurements performed on a gated selfassembled quantum dot [S. Takahashi et al. Phys. Rev. Lett. 104, 246801 (2010)].
pacs:
73.21.LaI Introduction
Over the past decade there has been a growing interest in the study of the spinorbit (SO) interaction in semiconductor lowdimensional systems motivated by the possibility of coherent spin manipulation.(1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13) The Hamiltonians describing the SO coupling resulting from the inversion asymmetry of the material (Dresselhaus(14) coupling) or the specific structure of the device (Rashba(15) interaction) are not invariant with respect to the rotation of the spin or the momentum operators separately, and consequently spinorbitcoupled systems posses intrinsic anisotropic properties. This anisotropy has been thoroughly studied for delocalized systems.(4); (5) In particular in transport experiments the dependence of the conductance of a narrow quantum wire on the direction of the external magnetic field can be used to determine the reciprocal strengths of the Rashba and Dresselhaus couplings.(6) The anisotropy of the spinorbit interaction is translated into anisotropic effective magnetic field(7) for a moving electron modifying the electron spin state. This effective magnetic field can be used to perform rotations of spin and thus to construct quantum gates (8) or a spinfield effect transistor.(9); (10) Moreover, the spinorbit coupling is responsible for anisotropic corrections (11) to the spin swap in a twoqubit quantum gate,(12) because it results into the precession of spinpackets tunneling between the two quantum dots.(13)
For electrons localized in a quantum dot the SO coupling results in avoided crossings (AC) in the energy spectra(16) and spin relaxation(17) mediated by phonons with a relaxation rate dependent on the orientation of the external magnetic field.(18) The energetic effects of the SO interaction are usually weak. Only recently SOinduced AC were experimentally measured on quantumdots that were situated in gated nanowires (19); (20) and in gated selfassembled quantum dots.(21) The latter experiment studied changes of the width of AC for different orientations of the magnetic field which extended the previous studies that were focused on a comparison of the spinsplittings for vertical and inplane alignment of magnetic field (22); (23) in circularly symmetric confinement potentials.
In the present work we explain the physics underlying the observations of Ref. (21). To the best of our knowledge the present paper explains for the first time the oscillatory dependence of the width of AC on the direction of the inplane magnetic field. The latter turns out to be the consequence of the influence of the individual SO couplings and the anisotropy of the confinement potential. This conclusion is supported by an exact threedimensional calculation of the energy spectra of one and twoelectron spinorbitcoupled quantum dots.
We show that for quantum dots with confinement potential elongated in direction for pure Rashba (pure Dresselhaus) coupling the AC disappears when the magnetic field is aligned along the short (long) axis of the dot. We show how this can be understood from the form of the SO Hamiltonians and the approximate parity of the oneelectron wave functions. The dependence of the AC width on the direction of the magnetic field turns to be a shaped function and when both couplings are present this function is shifted by an amount which depends on the relative strength of both interactions. This shift is affected by the orientation of the dot within the [001] plane due to the SO bulkinducedanisotropy (Dresselhaus term). For completeness we also study the influence of the dot shape. We show that for a squarebased quantum dot the anisotropic dependence of the AC width is only observed when both couplings are present.(23) Moreover we show that for increased height of the dot the orbital effect of the magnetic field modifies the energy spectrum but the shape of the dependence of the anticrossing width on the direction of the inplane magnetic field remains unaltered.
The present work is organized as follows: we start with an outline of our theoretical approach in section II. In section III we present our numerical results starting from the singleelectron case which provides us with physical insight in the reasons for the SO coupling anisotropy. We continue by studying different orientations and geometries of the dot and we end the section with the twoelectron case that allows for a direct comparison with the recent experimental data of Ref. (21). We end with a concluding discussion in section IV and a summary in section V.
Ii Theory
ii.1 Model
Our aim is to calculate the energy spectra of the one and twoelectrons confined in a threedimensional quantum dot in the presence of SO coupling and a magnetic field oriented within the quantum dot plane. The effect of the spinorbit coupling on the energy are very small requiring a very high numerical precision when evaluating the energy spectrum. We assume that the quantum dot is cuboid in shape and that the confinement potential is separable, namely . Moreover, we assume that the onedimensional confinement potentials and can be described by an infinite quantum well model. This is a reasonable approximation for not to small quantum dots. Under these assumptions one can construct a sufficiently precise solver for the twoelectron problem. We consider quantum dot with varied inplane orientation with respect to the crystal host. The axis is taken along the crystal direction which is also the vertical axis of the dot. The orientation of the dot is described by a rotations of the and directions, which are the axes of the dot with respect to the and crystal directions. The outline of our quantum dot and the coordinate system used is depicted in Fig. 1.
ii.2 Method
We employ the effective mass approximation with the singleelectron Hamiltonian of the form
(1) 
where , 1 is the identity matrix, defines the confining potential and , are the spinorbit coupling Hamiltonians. The and directions are taken along the axes of the dot. But notice that the SO interaction Hamiltonians are defined in the coordinate system with axes parallel to the , and which we denote with and . Both coordinate systems are transformed into each other by an inplane rotation over an angle .
We introduce the Rashba coupling with Hamiltonian,
(2) 
where defines the coupling strength. For infinite quantum well confinement the term within the dot equals the external electric field. We neglect the influence of the inplane component of the electric field(24) and obtain the Rashba Hamiltonian in the form,
(3) 
Thus the electric field is in the direction which is incorporated by taking a nonzero slope of the bottom of .
Inversion asymmetry of the crystal lattice results in a Dresselhaus SO coupling that is described by the Hamiltonian
(4) 
where is the coupling constant.
The coordinate system used for the SO coupling can be transformed into the coordinate system used for the quantum dot through the transformation
(5) 
which applies both to the Pauli matrices and the coordinates of the momentum operator.
We include an inplane magnetic field of orientation which is described by the gauge . The magnetic field vector for is oriented along the direction (see Fig. 1).
The oneelectron Hamiltonian (1) can be rewritten in the form , where
(6) 
(7) 
(8) 
are spin independent parts separable in the and directions, and
(9) 
is the nonseparable part that contains the spin dependent terms,
The eigenenergies and the eigenvectors of the Hamiltonians are calculated separately on onedimensional meshes of points. In a next step we diagonalize in a basis of products of the eigenstates resulting in three–dimensional spinorbitals . We typically take , onedimensional eigenstates (we assumed ), which including the degeneracy of the spin gives a basis consisting of elements which results in an accuracy better than eV.
We solve the twoelectron problem as described by the Hamiltonian
(10) 
using the configurationinteraction approach. In our numerical calculation we take the dielectric constant for InAs . Hamiltonian (10) is diagonalized in a basis constructed of antisymmetrized singleelectron spinorbitals
(11) 
where are the spatial and spin coordinates of the corresponding electron. The electronelectron interaction matrix element requires the calculation of integrals of the form,
(12) 
A direct calculation of these dimensional integrals requires an enormous numerical cost. Therefore, we use the method(25) in which the innermost integral is attributed to an electric potential originating from an electric charge distribution . We calculate the electric potential by solving the Poisson equation with the boundary condition
(13) 
where lays within the boundary of the the computational box. The Poisson equation is solved on a grid that covers the dot area. The calculation accuracy is carefully monitored(26) and the configurationinteraction calculation convergence better than eV is reached for .
ii.3 Parameters
The bulk of our results presented in the following sections are obtained for parameters described below. In the experiment of Ref. (21) an anisotropic InAs selforganizedquantumdot (SOQD) grown on a GaAs substrate is studied with a confinement potential that is elongated due to the source and drain electrodes placed above the dot. The orientation of the dot with respect to the inplane crystal directions is not well resolved and in the present work this is taken as an additional parameter which is studied. We take nm as the long and nm as the short size of the dot.(21) We take nm as a reasonable estimate of the dot height (note that the SOQD has a nominal pyramidal shape(21) with height nm, but our model is limited to a potential with rectangular shape of vertical cross section). influences the effective strength of the Dresselhaus coupling constant and the orbital effects of the inplane magnetic field. Results for nm are also discussed below.
For the purpose of the present study it is important to notice that the electric field in the growth direction defines the strength of the Rashba coupling. The electric field is influenced by the potential profile within the dot,(27) the Schottky barrier at the dot/electrode interface, surface charges and applied potentials.(24) The electrostatics of the actual device is complex and its complete description is out of the scope of the present work. Nevertheless, we are able to estimate the external electric field present in the system by considering the stability diagram and the width of the systems.(28) We estimated the maximal value of the external electric field to be of order kV/cm for which the electrons are still present in the dot.(29) From the gate voltage V of twoelectron spectroscopy we estimated kV/cm and this value is used in our numerical calculation. Finally, in this paper we indicate that the ratio of the Rashba coupling strength (that is proportional to ) to the strength of the Dresselhaus coupling can be extracted from the experimentally measured orientation of the magnetic field for which the SOinduced AC vanishes.
Iii Results
iii.1 Without SO coupling
We consider first the dot aligned such that the axis is oriented along (axis along ), namely . The energy spectrum obtained in the absence of the SO coupling (we take ) for a singleelectron anisotropic quantum is presented in Fig. 2 by the black solid curves. In the absence of the magnetic field the ground state is doubly degenerate with respect to spin and the spatial wave function is of even symmetry with respect to plane inversions: , , and . We denote the state of even symmetry with respect to all inversions by . The firstexcited state is a spindoublet with wavefunctions meeting the symmetry conditions: , , and . We will refer to this state as . The nonzero magnetic field lifts the spin degeneracy splitting of the states of the same parity by the Zeeman energy. The energy levels depicted by the black lines in Fig. 2 are obtained regardless of the value in spite of the lateral anisotropy of the dot. Due to the small value and the inplane alignment of , no orbital effects of the magnetic field are observed (the influence of the height of the dot is studied in subsection F).
Generally, in the presence of an inplane magnetic field the Hamiltonian (1), even without SO interaction, does not commute with the plane inversion operators and [defined as and ]. However, due to the insignificance of the orbital effect of the magnetic field for this flat quantum dot, the parity with respect to reflection through the and plains is approximately preserved (with and above ) even for nonzero . For the following discussion we denote the four lowestenergy states for small magnetic field aligned parallel to the direction as , , , with corresponding energies where the arrow denotes the spin state aligned parallel or antiparallel to the magnetic field vector.
iii.2 Single type of SO coupling
Inclusion of the SO interaction lifts the spin polarization of the states and changes the crossing observed between the energy levels of , around T into an anticrossing. The inset of Fig. 2 shows the anticrossing energy levels for ( parallel to the axis) by the red curves when only Dresselhaus coupling with is included. We denote the minimal energy difference between the anticrossing levels as . For applied parameters we obtain eV. Outside the anticrossing the SO interaction does not modify the energy spectrum in a noticeable way i.e. the black and red curves approximately coincide.
In the presence of the SO coupling the anticrossing energy levels depend on the orientation of the magnetic field. In Figs. 3(a) and 3(b) we plot the energy levels obtained for pure Dresselhaus and pure Rashba interaction, respectively, for three different values. In both cases clear dependence of the anticrossing width is observed with respect to the orientation. For pure Dresselhaus coupling the anticrossing is the widest when the magnetic field vector is perpendicular to the direction () [the red curve in Fig. 3(a)]. When the field is aligned along the direction () the mixing between levels vanishes [the black curve in Fig. 3(a)] and there is crossing of levels. With pure Rashba coupling the dependence is opposite – the anticrossing vanishes when is aligned along and is largest when is aligned along .
The direction of the magnetic field for which the mixing between the states disappears can be infered from the analytic form of the SO Hamiltonians utilizing the approximate symmetries of the wave functions of the confined electron. Let us first inspect the case of pure Dresselhaus coupling and remind that for the Hamiltonian (4) has the same form in the and coordinate system. Averaging the Hamiltonian (4) over the direction one obtains,
(14) 
The second term is the socalled cubic Dresselhaus term which is negligible as long as the height is much smaller than the lateral size of the dot [i.e. until the value of or becomes comparable with ]. For an infinite quantum well groundstate wave function in the direction the last term in (14) vanishes(33) and
(15) 
The simplified Dresselhaus Hamiltonian takes now the form
(16) 
Let us now consider the case of a magnetic field aligned paralel to the direction. In our basis we include only the lowenergy states , that exhibit an energy crossing without SO coupling. The matrix of the Hamiltonian limited to this basis is given by,
(17) 
The states , are separable into an orbital and a spin part. Due to the action of the Pauli matrices on the states with definite spin one gets,
(18) 
For the magnetic field vector aligned parallel to the direction the components of the momentum operator vector are . Due to parity one obtains,
(19) 
The nonvanishing offdiagonal matrix elements mix the states which results in an avoided crossing between the corresponding energy levels. By the red crosses in Fig. 3 we plot numerically calculated eigenvalues of the matrix (19). Note that the crosses and lines are in perfect agreement proving that for our dot with the assumed geometry the is in fact a good approximation to .
Let us now consider the case of a magnetic field aligned parallel to the () direction. In this case the lowenergy states which energy levels cross without SO coupling are , , where the arrow denotes the electron spin aligned parallel () and antiparallel () to the magnetic field vector . The matrix of the Hamiltonian in this two state basis is
(20) 
Due to spin one gets,
(21) 
For the magnetic field aligned along the direction the components of the momentum operator vector are, . All integrals in Eq. (21) vanish due to the parity of the states and we finally obtain,
(22) 
The matrix (22) consists only of diagonal elements that are equal to the energy of the basis states. Thus the , states are not mixed by the Dresselhaus coupling in this configuration and there is no anticrossing of energy levels. We plot the eigenvalues of the matrix (22) by the black crosses in Fig. 3(a).
A similar analysis can be made for the Rashba Hamiltonian (2). Due to the fact that the analytic form of both Hamiltonians and are similar, i.e. only the and are swapped (and the coupling constants are different), it is clear that the dependence of AC width on magnetic field direction is opposite – the mixing between the states vanishes when the magnetic filed is aligned along the direction.
iii.3 Anisotropy in the presence of both SO couplings
Let us now consider the effect of both Dresselhaus and Rashba coupling. Figure 4 presents the avoided crossing energy as a function of the angle between the axis and the magnetic field. For pure Dresselhaus (the blue dashed curve in Fig. 4) and pure Rashba (the green dotted curve in Fig. 4) coupling the extrema are shifted by in agreement with our previous analysis. The curves in Fig. 4 are accurately described by which is the same functional form as the one observed in the experimental work of Ref. (21) in Fig. 3(f) [where the behavior was described by ]. Moreover the maximal value of is of the same order as the magnitude observed experimentally. When both SO interactions are present the dependence of the anticrossing width is plotted in Fig. 4 by the red curve. The shape of the latter is the same as for pure Dresselhaus/Rashba coupling with pronounced minima where is zero. When the magnetic field is aligned along the or direction the equals the value for pure SO coupling. Note that the maxima are larger than the ones observed for pure couplings and its positions are now shifted and are no longer aligned along the axes of the dot. For and the shift of the dependence is . The latter value can be understood as follows. Let us denote the direction of the magnetic field for which the AC vanishes for pure Dresselhaus and pure Rashba couplings by the vectors and , respectively. Next, we estimate the strength of each interaction. Maximal induced anticrossing width is eV and eV for Dresselhaus and Rashba coupling, respectively. Thus the Dresselhaus interaction is times larger than the Rashba coupling what makes the vector times longer than . Let us denote the magnetic field for which the effect of both spinorbit couplings is zero by the vector . It is easy to show that this vector forms an angle with the axis. Thus when both couplings are present, the effect of the total spinorbit coupling disappears when the external magnetic field is directed along this vector. In fact that is exactly what we observe in our calculation (see position of the minimum of the dependence depicted with the red curve in Fig. 4). The formula reflect the fact that the dependency obtained for both SO couplings present can be considered as an absolute value of a sum of the dependencies obtained for pure SO couplings described by and for pure Rashba and Dresselhaus couplings respectively.
iii.4 Dependence on the quantum dot orientation
Different inplane orientations of the anisotropic potential of the dot with respect to the crystal host are now considered where the long axis of the dot forms an angle with . In Figs. 5(a,b,c) we present the size of the avoidedcrossing as a function of the direction of the rotated magnetic field (note that the angle is defined as an angle between the magnetic field vector and the long axis of the dot) for six different orientations of the dot. The dotted curves in Fig. 5(a) presents the result obtained for pure Rashba coupling. We observe that the dependencies are exactly the same as in Fig. 4 regardless of the dot alignment. The minimum of the does not change its position and the energy levels are not affected by the orientation of the dot. We show in Fig. 5(d) the angle for which the as a function of the angle by the green diamonds.
For pure Dresselhaus coupling the dependencies [depicted by dashed curves in Fig. 5(b)] are shifted as the dot is rotated. For the case studied in previous subsections (where ) the AC vanished when the magnetic field was aligned along the long axis of the dot (). When the dot is oriented by (long axis oriented along the direction), the anticrossing vanishes when the magnetic field is aligned along the short axis of the dot [see green dashed curve in Fig. 5(b)] – – the same as for pure Rashba case. We plot in Fig. 5(d) the angle for pure Dresselhaus coupling by the black squares for different orientations of the dot. We find that the angle exhibits a dependence [black solid lines in Fig. 5(d)]. Moreover we observe that for both cases when only a single type of SO coupling is present the maximal value of the AC width remains unchanged.
In Fig. 5(c) we show the results when both SO couplings are present by the solid curves. The maximal values of and the angle for which the minima are observed change when the dot orientation is varied. Both facts can be understood similarly as discussed in subsection C. We can justify the values considering the orientation of the vector. But now the orientation of the vector assigned with Dresselhaus coupling is changed as the dot is rotated, ie. the vector forms an angle with the long axis of the dot. The rotation of the dot does not change the maximal value of when only a single type of SO coupling is present and the previously derived value for the relative strength of both couplings remains unchanged (and thus also the ratio of the length of the and vectors). We take as the length of and as the length of . In Fig. 6 we schematically present the considered vectors and the angles they form with the axes of the dot. The angle between the vector (red arrow in Fig. 6) and the direction can be easily calculated
(23) 
With the red dots in Fig. 5(d) we plot the angle obtained from our numerical calculation in the presence of both couplings for different orientations of the dot which agree very well with the values (red curves) obtained from Eq. (23). Along with the changes of the orientation the length of the vector is changed which results in different values of the maximal AC width observed in Fig. 5(c).
A systematic study of the value of the angle dependence on the SO coupling strengths and the dot alignment is given in subsection G where the twoelectron case is studied.
iii.5 Quantum dot with square base
The above discussion was for a lateral anisotropic quantum dot. Now we study the case of a dot with symmetrical base (we assume nm) and and investigate if this has an influence on the anisotropy induced by the SO coupling. In the absence of the SO interaction and a magnetic field the firstexcited state is spindoubly degenerate due to parity. The magnetic field lifts the spin degeneracy but the degeneracy due to parity is not removed. The inclusion of a single type of SO interaction induces a repulsion between the energy levels of the groundstate and one of the states from the parity doublet [see the red dashed curves in Fig. 7(a) for the case of pure Dresselhaus coupling and Fig. 7(b) for pure Rashba coupling]. The same configuration of energy levels is obtained regardless of the angle . In both Figs. 7(a) and 7(b) the black (), blue (), yellow dotted () and red dashed curves () coincide. The dependence of the energy levels on the magnetic field orientation starts to appear already when the dot is elongated by a factor of .
However, when both Rashba and Dresselhaus interactions are present the AC width varies with the rotation of the magnetic field – Figs. 7 (c,d). We observe that the anisotropy is most pronounced when is increased by a factor of two – the case when both couplings have comparable strengths.(23); (34) In such a case when the magnetic field is directed along the diagonal, i.e. [the blue curves in Fig. 7(d)] the anticrossing between the energy levels of the groundstate and both states from the paritydoubled becomes very small.
iii.6 Larger dot height
Let us now return to the case of the quantum dot with rectangular base with nm and nm. For the previous dot with nm no orbital effects from the magnetic field on the energy spectrum was observed [see the black curves in Fig. 2]. However this is no longer true for larger values. This can be seen from Figs. 8(a) and 8(b) where we plot the energy levels of a quantum dot with height nm and nm, respectively, in the presence of SO coupling (with both SO interactions present). The energy levels depend on the magnetic field orientation even outside the anticrossing region. This is due to the elongation of the confinement potential in the direction. The SOinduced anticrossing is shifted to lower magnetic fields as the value of the angle becomes closer to [this is analogous to the experimental observation – compare with Fig. S7(a) from Ref. (28)]. We calculated the anticrossing widths for different values of and plot them as red dots in the insets of Fig. 8(a,b). Then we fitted the points with the function where eV, for nm and eV, for nm. Notice the agreement between the fitted curve and the data points. From this fact we conclude that in spite of the presence of orbital effects, previously found dependence of the anticrossing width on the angle still holds, but with modified and values. The latter fact can be attributed to the reduction of the Dresselhaus coupling strength. This can be accounted for by considering the Dresselhaus coupling Hamiltonian (16) in which the coupling strength decreases as . In the calculation performed for pure Dresselhaus interaction we obtain the maximal values eV, eV and eV for nm, nm and nm, respectively. For increased dot height the obtained values decrease approximately as with the largest discrepancy for large value (i.e. when the approximation of the coupling strength by Eq. (15) becomes inaccurate). The decrease of the Dresselhaus coupling strength for increased height of the dot results in a shift of the dependency on towards the one obtained for a flat quantumdot with only Rashba interaction present (compare the black curve in the inset of Fig. 8(b) with the green dotted curve in Fig. 4) – becomes close to . Also the maximal value becomes closer to the one obtained for pure Rashba coupling – tends to eV with increasing . The shift in the value [see insets of Figs. 8 (a,b)] can be understood from the relative strengths of the Rashba and Dresselhaus coupling as discussed in section III. C.
iii.7 Two electron results
In a recent experiment [(21)] the groundstate and excited states were measured provided that the latter entered into a finite but narrow transport window determined by the voltages applied to the source and drain electrodes. The avoided crossings that appear for a singleelectron in the excited part of the spectrum, which we described above, were outside the transport window.
In the twoelectron regime and in the absence of both the magnetic field and the SO interaction the groundstate is a spinsinglet and the first excited state is a spintriplet. Under the presence of an external magnetic field the groundstate singlet energy crosses the triplet energy. When we turn on the SO coupling it induces an avoided crossing between the states of opposite spin which was well resolved in the experiment [(21)].
Similarly to the oneelectron case the SO coupling is responsible for changes in the size of the anticrossing energy when the orientation of the magnetic field is varied. Figure 9 presents the lowenergy spectrum of the twoelectron quantum dot in the presence of both Rashba and Dresselhaus coupling for a dot aligned with its long axis along the direction (). In the inset we plot the energy levels in the vicinity of the avoided crossing. The anticrossing vanishes for exactly the same angle as for the oneelectron case discussed above (see the green curves in the inset of Fig. 9).
In Fig. 10(a) we plot the angular dependence of the anticrossing width for pure Dresselhaus, pure Rashba and when both couplings are present by the blue dashed, green dotted and solid red curves, respectively. Notice that all three dependencies have the same shape as for the case of the oneelectron considered in subsection B (compare with Fig. 4), only the maximal values are about times smaller.
As was presented in subsection C for the anisotropic quantum dot the angle depends on the relative strength of both SO interactions and the inplane orientation of the dot (explained in subsection D). On the other hand the value can be measured experimentally(21) and the orientation of the quantum dot with respect to the crystal directions can be obtained by inspecting the facets of the dot. This opens the possibility to employ such a measurement to evaluate the relative strength of the Rashba and Dresselhaus couplings for a dot with given orientation with respect to the crystal host. Let us define the strength ratio of the SO interactions as the ratio of the effective coupling constants and . The Rashba coupling strength denoted with is calculated as and the Dresselhaus coupling is obtained from Eq. (15).
We previously derived the angle for given relative strength of the SO couplings for a given orientation of the dot [see Eq. (23)]. Let us substitute the value by in Eq. (23) from which we obtain
(24) 
This function is shown in Fig. 10(b) by the solid lines for different orientations of the quantum dot. With the black symbols we mark the angle obtained from our numerical calculations for dots with different geometries (see figure caption) with for different SO coupling strengths. For such case with (the dot oriented with its long axis along ) and pure Dresselhaus coupling () we obtain . When the Rashba coupling strength is increased the points move toward the angle obtained for pure Rashba SO coupling. The green, red and blue symbols in Fig. 10(b) are the values obtained from our twoelectron numerical calculation for different orientation of the quantum dot.
In the above discussion we assume that and describe the strength of the spinorbit interactions. For the Rashba coupling given by the Hamiltonian (3) [i.e. when an electric field is only present in the growth direction] the above expression is valid regardless of the dot geometry. However, due to the fact that originates from the Hamiltonian (14) it describes the strength of the Dresselhaus coupling correctly only when the cubic term is negligible, which is the case when and when the term with is close to zero i.e. for a dot with limited height.(33) All the symbols in Fig. 10(b) approximately coincide with the dependency given by Eq. (24). Discrepancy is seen in the limit of a narrow dot with nm (the triangles) and for increased height of the dot for nm (diamond symbols). We conclude that for anisotropic quantum dots with limited height the ratio is a good measure of the relative strength of the Rashba and the Dresselhaus spinorbit couplings which can be estimated from analytic expression (24).
The experiment of Ref. (21) found and we can use Eq. (24) to calculate the relative strength of the SO interactions. However, as the orientation of the anisotropic potential of the dot with respect to the crystal directions was not resolved in the experiment we need to assume a value for . We take and by matching the absolute value of the SO coupling constants (through the maximal value of ) we obtained by fitting the experimentally measured values for the AC width with our simulation results. In the inset to Fig. 10(a) we plot our results (purple curve) for the SO coupling constants , together with the data points from Ref. (21). However as the relation between the crystal directions and the long axis of the dot is not known the fit only proves the validity of the discussed process behind the anisotropy and not the exact value of the ratio . Moreover as the electrostatics of the actual device is complex the presented result is not the exact simulation of the experiment. Therefore, we present in Table I the strength ratios for different orientations of the dot. Note that Eq. (24) does not allow to calculate the relative strength of the couplings for a dot aligned with long axis exactly along or . In a such configuration for pure Dresselhaus as well as for pure Rashba coupling the AC vanishes for [compare dotted curves in Fig. 5(a) with greendotted curve in Fig. 5(b)] and by that for both couplings present simultaneously the minimum of the dependence on is not shifted irrespective of the coupling strength ratio.





0  1.66 
1.27  
–  
0.94  
1.66 
Iv Discussion
In the present paper we discussed the avoidedcrossings of energy levels as induced by the presence of different SO couplings. Only for the case of a squarebased quantumdot [see Figs. 7(a,b)] the dependence of AC width as function of the magnetic field direction was observed solely for both couplings present with comparable strength. This result is related to those of Ref. (23) where the spinsplitting of singleelectron energy levels in strictly twodimensional circular quantum dots in the presence of a small inplane magnetic field (before the crossings/avoidedcrossings appear) was calculated. When Dresselhaus and Rashba coupling strengths are equal a well known high symmetric case is found which is beneficial for many spintronics applications.(5); (10) For that special case the energy spectrum is not affected by SO interaction effects and the spin in the [110] direction is strictly defined. The Zeeman interaction lifts this symmetry and results in a spectrum that depends on the orientation of the magnetic field as discussed in Ref. (23). Since for equal coupling strengths the spins in the direction are well defined, the Zeeman interaction for oriented along does not produce any AC between energy levels of spinorthogonal states [see the blue curve in Fig. 7(d)].
On the other hand, in the presence of a vertically oriented magnetic field, the size of the Zeeman interaction induced lifting of the symmetry depends on the inplane orientation (34) and also the width(35) of the dot what results in changes in both the AC width and the effective factor which are solely observed when both SO interactions are present with comparable strength. However, changing the dot orientation is hardly achievable experimentally and therefore in the present work we considered an anisotropy that can be probed by changing the orientation of the magnetic field.
In the present work we investigated the anisotropic dependence of the avoidedcrossing width that occurs even for a single type of SO coupling [see Figs. 3(a) and 3(b)]. This effect is strictly connected both with the elongation of the confinement potential and the inplane alignment of the magnetic field [see the discussion in subsection B]. The exact shape of the confinement potential is not important for the studied phenomena which is a generic propriety of a spinorbitcoupled quantum dot. In our analysis we indicated the trends that determined the dependence of on , in particular the dependence on the dot geometry [for the dot with increased height and for different lateral sizes of the dot – see the black symbols in Fig. 10 that in spite of the different geometries of the dot still undergo the same analytical dependence Eq. (24)] or the orientation of the quantum dot with respect to the crystallographic directions (which influences the position of the minima of purely due to Dresselhaus coupling – see discussion in section III. D).
The present study shows that for an elongated quantum dot with pure Rashba coupling the anticrossing vanishes always when the magnetic field is aligned along the short axis of the dot [see the minima of the dotted curves in Figs. 4, 5(a) and 10(a)]. Only the presence of Dresselhaus coupling can result in a value that is different from . The magnetic field direction () for which the anticrossing vanished in the experiment of Ref. (21) suggests both SO couplings are present, contrary to the argumentation provided in Ref. (21). The authors suggested that the Dresselhaus coupling would not induce mixing between the two lowestenergy states due to their well defined and different values of the total angular momentum in a high magnetic field. However we found, that due to the inplane alignment of the magnetic field(36) the Dresselhaus coupling in fact induces avoidedcrossings in the energy spectrum of a flat quantum dot [see Fig. 2] and leads also to a shift in the dependence of the AC width on the magnetic field direction [see Fig. 3, Fig. 5(a,b) and insets to Fig. 8].
V Summary and conclusions
We presented a study of the energy spectrum of one and twoelectron spinorbitcoupled threedimensional quantum dots in the presence of an external inplane magnetic field. We found that the size of the avoidedcrossings in one and twoelectron energyspectrum oscillates as a function of the orientation of the magnetic field. The oscillatory behavior could accurately be described by which agrees with recent excitedstate spectroscopy measurements performed on InAs gated selforganizedquantum dot.(21)
For a quantum dot which is elongated in the direction and when only a single type of SO coupling is present the avoided crossing vanishes for (), i.e. when the magnetic field is aligned parallel to the long (short) axis of the dot for Dresselhaus (Rashba) coupling. We explain this behavior as a consequence of parity and spindependent mixing of the states caused by the SO interaction. When both couplings are present the value varies between and and the ratio of the relative strength of the interactions follows a dependence. The change of the inplane dot orientation results in a change of which is observed only when Dresselhaus coupling is present. We show that the experimentally measured value(21) along with the knowledge of the orientation of the dot can be used to determine the ratio of the strengths of the individual SO interactions in case of anisotropic quantum dots.
Acknowledgements
The authors thank S. Takahashi for helpful discussions. This work was supported by the ”Krakow Interdisciplinary PhDProject in Nanoscience and Advanced Nanostructures” operated within the Foundation for Polish Science MPD Programme cofinanced by the EU European Regional Development Fund, the Project No. N N202103938 supported by Ministry of Science an Higher Education (MNiSW) for 2010–2013 and the Belgian Science Policy (IAP). W.P. has been partly supported by the EU Human Capital Operation Program, Polish Project No. POKL.04.010100434/0800. Calculations were performed in ACK–CYFRONET–AGH on the RackServer Zeus.
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