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Coupled donors in quantum ring in a threading magnetic field
Physica E 43 (2010) 559–566
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Physica E journal homepage: www.elsevier.com/locate/physe
Coupled donors in quantum ring in a threading magnetic field W. Gutie´rrez n, L.F. Garcı´a, I.D. Mikhailov Escuela de Fı´sica, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia
a r t i c l e in fo
abstract
Article history: Received 21 August 2010 Accepted 27 September 2010 Available online 1 October 2010
Electronic states of a singly ionized double donor system (D2+ ) confined in a nanostructure with a ring¨ like geometry in a threading magnetic field are calculated. Solutions of the Schrodinger equation are obtained by variational separation of variables in the adiabatic limit. Numerical results are shown for bonding and anti-bonding lowest-lying molecular states corresponding to different relative locations of the coupled impurities. We show that displacement of donors from the axis toward the ring and decrease in the angle between their position vectors may strongly affect the Aharonov–Bohm oscillations of the far-infrared spectra and produce a quenching of oscillations of the lower energy levels. & 2010 Elsevier B.V. All rights reserved.
1. Introduction There has been remarkable interest in semiconductor nanostructures during the past twenty years, motivated by possibilities of realization of new high-tech electronic device architectures. One of these possibilities is related to the building of an adequate microscopic two-level system, which can be used as the functional part in a wide range of device applications, including spintronics, optoelectronics, photovoltaics, and quantum information technologies. Coupled quantum wells or quantum dots have been considered previously to be basic candidates for this aim. Recently, it has been proposed to use instead the singly ionized double donor system (D2+ ) confined in a semiconductor quantum dot [1]. This system encodes logical information either on the spin or on the charge degrees of freedom of the single electron and allows us to manipulate conveniently its molecular properties, such as energy splitting between the bonding and anti-bonding lowest-lying molecular-like states or spatial distribution of carriers in the system [2]. This fact has raised interest in charge qubits, where the logical states are represented by the two lowest-lying electron orbital states positioned at the different donors. Theoretical analysis of the D2+ system in a spherical QD demonstrated that the confinement can greatly enhance the energy gap between the two lowest-lying electron states and other excited states, favouring the identification of the quasi-two-level system required for quantum computation purposes [3]. It can be expected that a quantum ring (QR) with single or two captured electrons, provided by a double donor system, will be a
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better candidate towards this aim than a spherical QD, because of the fact that the gaps between the lower energy levels in QR are larger and electron localization is stronger. Besides, the external magnetic field applied along the axis gives an additional possibility for controlling properties of molecular-like states in QRs, where due to the ring-like geometry, different quantum-size effects such as the Aharonov–Bohm (AB) oscillations [4] can be manifested in the presence of the external fields. Such peculiarity in QRs has stimulated in the last two decades much interest in the theoretical and experimental studies of their electronic and optical properties. Energy spectra of the one- and two-electron single and vertically coupled QRs have been studied by various authors [5]. It has been shown that the properties of the lowenergy electron states are very sensitive to any disorder that breaks the axial symmetry of the structure. Particularly, the nonuniformity leads to a transformation of the rotations around the axis of the lower energy electrons into their oscillations near the defects, to a flattening of the curves corresponding to energy dependencies on the magnetic field, and to quenching of the AB oscillations [6]. One could expect a similar transformation of the QR properties due to the presence of off-axis donors, which also break the axial symmetry. The effect of a positively charged impurity on energy spectra of one and two electrons in semiconductor nanorings under magnetic fields was studied in Ref. [7]. It has been shown that impurity-induced AB oscillations and FIR spectrum are strongly dependent on nanoring size and impurity position. To our knowledge, similar properties of singly ionized double donor system (D2+ ) confined in a semiconductor QR and its FIR spectrum have not been explored up to now. This system represents the simplest system confined in QR, which in contrast to the single donor exhibits energy splitting between the bonding and anti-bonding lowest-lying molecular-like states. Prompted by
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a possible important role of the double donors’ structure in nanotechnology, we present in this paper a theoretical study of the molecular properties of a D2+ system confined in a QR. In order to analyze the effect of the symmetry break, induced by the coupled donors, on AB oscillations of the QR FIR spectrum we calculate the energy levels and FIR spectrum of this structure under a magnetic field, for different impurity positions. On the other hand, study of the quantum behavior of the double donor system D2+ also allows us to understand better some properties of trions X + (positively charged excitons) confined in type II QRs in the limit case, as the effective mass of the hole is much larger than that of the electron. Recently, theoretical investigation of AB oscillations of the energy levels of neutral and charged excitons, in type I and type II QRs, has been a topic of great interest [8]. The organization of the remainder of this paper is as follows. In Section 2, the model and Hamiltonian are described and procedures to obtain the solutions are briefly outlined. The main results are shown and discussed in Section 3, followed by a summary in Section 4.
2. Theory We consider an infinite barrier model of a quantum dot with ring-like geometry, which represents an axially symmetrical very thin layer, in which the dependency of thickness h on distance r from the axis is given by the following simple relation: hðrÞ ¼ hw þ Dh WðRc þw=2rÞWðrRc þw=2Þ
ð1Þ
Here hw is the wetting layer (WL) thickness, Dh the increase of layer thickness inside the ring, w and Rc are the ring width and centreline radius, respectively, and W(x) is the Heaviside stepfunction, equal to zero for x o0 and to one for x 40. A singly ionized double donor system, schematically presented in Fig. 1, is ! ! characterized, besides, by donors’ position vectors n 1 and n 2 . In the framework of the effective-mass approximation, the Hamiltonian of this system in cylindrical coordinates is given by 2 2 ^ ¼ 1 @ r @ 1 @ @ þig @ H 2 2 @j r @r @r r @j @z2 X g2 r2 2 þ V ð3DÞ ðr,zÞ þ ! ! 4 i ¼ 1,2 9 r n 9
energies in units of the effective Rydberg, Ry ¼ e2 =2ea0 and strength of the external magnetic field applied along the z-axis is in units of dimensionless energy of the first Landau levelg ¼ e_B=2mc Ry, mn and e being the electron effective mass and the layer dielectric constant, respectively. In accordance with the experiment, the height-to-radius aspect ratio in self-assembled quantum dots is generally very small, i.e. (hw + Dh)/Rc 51. This fact allows us to use the advantage of adiabatic approximation (AA). Since electron motion in the z-direction is much faster than in-plane motion inside the layer, we follow the well-known adiabatic procedure of first finding the lowest energy Ez(r) and correspondent wave function fz(z,r) of the equation " # @2 2 þ V ð3DÞ ðr,zÞEz ðrÞ fz ðz, rÞ ¼ 0; @z pffiffiffiffiffiffiffiffiffi fz ðz, rÞ ¼ 2=psinðpz=hðrÞÞ; Ez ðrÞ ¼ p2 =h2 ðrÞ ð3Þ where r is treated as a parameter (cf. electronic motion for a fixed nuclear position in the molecular problem), and then represent the three-dimensional eigenfunctions of the Hamiltonian (2) C(3D)(r,z,f) as follows: pffiffiffiffi Cð3DÞ ðr,z, jÞ ¼ fz ðz, rÞcð2DÞ ðr, jÞ= r ð4Þ
c(2D)(r,f) being a variational two-dimensional wave function that describes only the electron in-plane motion (cf. nuclear ¨ motion in molecular problem). According to Schrodinger’s variational principle, a function C(2D)(r,f), for which the ð3DÞ ð3DÞ functional F½Cð2DÞ ¼ /c 9HE9c S has a stationary value is an eigenfunction of the Hamiltonian (2). Taking the functional derivative with respect to c(2D) gives a wave equation of the form 2 3 @2 1 @2 ð2DÞ ðrÞ 2 6 @r2 þ V 7 2 r @j 6 7 ð2DÞ 6 7c ðr, jÞ ¼ 0 ð5aÞ X 6 7 @ 4 þ ig Vi ðr, jÞ þ V0 E 5 @j i ¼ 1,2 V ð2DÞ ðrÞ ¼
ð2Þ
i
Here V(3D)(r,z) is the confinement potential, which is equal to zero and to infinity inside and outside the layer, respectively; all lengths are scaled in units of the effective Bohr radiusa0 ¼ _2 e=m e2 , all
Fig. 1. Scheme of a quantum ring with singly ionized coupled donor system.
Vi ðr, jÞ ¼
g2 r2 4 Z
hðrÞ 0
1 p2 þ 4r2 h2 ðrÞ
2fz2 ðz, rÞdz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ x2i 2rxi cosðjji Þ
ð5bÞ
ð5cÞ
In the wave equation (5a) two first terms describe the electron motion in the radial direction under effective two-dimensional confinement potential V(2D)(r), the following two terms correspond to electron rotation around the axis, and the terms Vi(r,f) give energies of interaction of electron with donors. According to relation (1) the height of the layer h(r) jumps from hw to hw + Dh at the frontiers of the ring with the WL, i.e. when the radius is equal to r ¼Rc w/2 or r ¼Rc +w/2. Therefore at these points the potential V(2D)(r), defined by relation (5b), has a jump of the value V0 ¼ p2 =h2w p2 =ðhw þ DhÞ2 , which usually is significantly larger than the averaged energy of the electron attraction to donors. For example, if we take typical values for the effective Bohr radius to be about 10 nm, thickness of the WL of about 2 nm, and DhE2 nm, then the barrier height between QR and WL in our model is about V0 E 200Ryn, while the averaged energy of attraction to donors is close to 1Ryn. This results in the fact that the electron is mainly located inside the ring (Rc w/2o r oRc + w/2) independent of the donors’ location. The general equation (5) can be applied for calculation of lowlying energies of any QD with axial symmetry and with different morphologies, such as disk-shaped, lens-shaped, etc. But in this
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work we want to emphasize on the analysis of properties typical for structures with ring-like geometry and for the particular case when the width of the ring is sufficiently small. Therefore , for the sake of mathematical convenience, we consider below only very narrow QRs, whose widths are much smaller than their radii (w5Rc). This we call the adiabatic limit for the following reason. As the scales of displacement in the radial and azimuthal directions are strongly different it is convenient to replace the variable r by a new dimensionless variable x that is changes inside the ring from 1 to+ 1, by using the substitution r ¼Rc +xw/2. Then Eq. (5) can be written as h ð2DÞ i ð2DÞ ^ H E c ðx, jÞ ¼ 0; H
ð2DÞ
4 @2 ð2DÞ 1 @2 @ ¼ 2 2 þ V~ ðxÞ þ ig 2 @j2 @j w @x ðRc þwx=2Þ X V~ i ðx, jÞ 1 o x o þ 1; 0 o j o 2p;
corresponding to the solutions of Eq. (10) for the lowest adiabatic potential curve E0(f). In our numerical work we solve Eq.(7) using the trigonometric sweep described in Ref. [10] and Eq. (10) by means of Fourier series expansion in the form
FðjÞ ¼
N X
cm eimj
ð11Þ
m ¼ N
The variational principle (9) then reduces to the linear system of algebraic equations N X
½Edmu,m Amu,m cm ¼ 0;
mu ¼ 0, 7 1, 7 2, . . ., 7 N;
m ¼ N
Amu,m ¼ mumbmum þ vmum gmdmu,m
ð12Þ
with
i ¼ 1,2
cð2DÞ ð1, jÞ ¼ cð2DÞ ð þ 1, jÞ ¼ 0;
cð2DÞ ðx,0Þ ¼ cð2DÞ ðx,2pÞ
ð6Þ
ð2DÞ ðxÞ ¼ V ð2DÞ ðRc þxw=2Þ and V~ i ðx, jÞ ¼ Here the notations V~ Vi ðRc þ xw=2, jÞ are used. Eq. (6) may be regarded as the ¨ Schrodinger equation describing two one-dimensional particles of masses w2/4 and R2c , respectively, interacting through the P ~ potential i ¼ 1,2 V i ðx, jÞ. Since one of these masses is much smaller than the other, we follow the well-known adiabatic procedure of first solving the equation " # X 4 @2 ð2DÞ V~ i ðx, jÞE0 ðjÞ f ðx, jÞ ¼ 0 2 2 þ V~ ðxÞ ð7Þ w @x i ¼ 1,2
where f is treated as a parameter. Once Eq. (7) is solved and the energy dependence on angle E0(j) is found then the solution of Eq. (6) may be presented in the form
cð2DÞ ðx, jÞ ¼ f ðx, jÞFðjÞ
ð8Þ
with variational function F(f) describing electron rotation around the axis. Energies of the low-lying states are found by minimizing the functional Z 1 Z p h ð2DÞ ^ F½F ¼ dx djf ðx, jÞFðjÞ H ðx, jÞEf ðx, jÞFðjÞ-min 0
561
p
ð9Þ Taking the functional derivative with respect to F gives a wave equation of the form d dFðjÞ dFðjÞ ig BðjÞ þ E0 ðjÞE FðjÞ ¼ 0; dj dj dj Z þ1 f 2 ðx, jÞ BðjÞ ¼ dx 2 1 ðRc þ wx=2Þ ð10Þ p o j o p; FðpÞ ¼ FðpÞ; FuðpÞ ¼ FuðpÞ
It is seen that the function E0(f), found as a solution of Eq. (7), appears in Eq. (10) as an effective adiabatic potential that governs electron rotation around the axis. The structure of energy spectrum given by the pair of Eqs. (7) and (10) is similar to that of vibro-rotational spectra of a molecule. The set of the vibrational levels corresponding to electron motion in the radial direction, given by different adiabatic potentials E0(j), is characterized according to Eq. (7) by gaps of order 1/w2. These levels are split into a series of rotational sublevels with gaps between them of the order 1=R2c according to Eq. (10). In what follows we analyze only the set of the lower rotational sublevels
Z p 1 E0 ðjÞeimj dj; 2p p Z p 1 BðjÞeimj dj; m ¼ 0, 7 1, 7 2, . . ., 7 N bm ¼ 2p p
vm ¼
ð13Þ
The condition that this set of equations has a nonzero solution leads to a secular equation of order N: detJEdmu,m Amu,m J ¼ 0;
mu,m ¼ 0, 71, 7 2, . . ., 7N
ð14Þ
By diagonalizing the secular equation (14), we can obtain the kth rotational energy sublevel Ek and the corresponding set of ðkÞ ðkÞ coefficients cm , m ¼ 0, 71, 7 2, . . ., 7N. If the coefficient cm is ðkÞ given by the Kronecker symbolcm ¼ dm,k , the corresponding sublevel is a pure rotational state with angular momentum m, otherwise it is superposition of various rotational states with different angular momenta. The total 3D wave functions then are given by the following expression:
CðkÞ ðr,z, jÞ ¼
N pffiffiffiffiffiffiffiffiffi pffiffiffiffi X ðkÞ imj 2=psinðpz=hðrÞÞf ð2ðrRc Þ=w, jÞ= r cm e m ¼ N
ð15Þ To analyze the properties of the solutions, two useful quantities can be extracted from C(k)(r,z,f), namely the electron density, ! ! 2 Nk ð r Þ ¼ 9CðkÞ ð r Þ9 and the absorption coefficient a(o), which in the electric dipole approximation at low temperature is given by [9] X ! 2 aðEÞ ¼ cE 9/CðkÞ 9! e U d 9Cð0Þ S9 dðEDEk Þ; DEk ¼ EðkÞ Eð0Þ ð16Þ k
! where d is the electric dipole operator, c a constant factor, and DEk the energy difference between the initial and the final states. For circularly polarized light, the complex polarization vector of pffiffiffi pffiffiffi the constant external electronic field -e 7 ¼ ð1= 2, 7 i= 2Þ corresponds to right (+) and left ( ) circularly polarized light, respectively.
3. Results and discussion We calculate low-lying energies of the singly ionized double donor system confined in a semiconductor QR for different donors’ positions and magnetic fields g. The studied rings have a centreline radiusRc ¼ 10a0 , width w ¼ 1a0 , and wetting layer thickness and dot height over the wetting layer hw ¼ 0:2a0 and Dh ¼ 0:2a0 , respectively. In order to help study the effect of donors’ positions on the structure of energy spectrum, we use
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below as parameters two position indices bi ¼ xi/Rc, i¼1,2, xi being the distances from donors’ locations up to the axis, and the angle Df between vectors of donors’ positions. In our calculations, in the beginning we solved Eq. (7) for 100 different angles fi, i¼1, 2, 3, y, 100 and then defined the unknown adiabatic potential curve E0(f) as an interpolating cubic spline fitted to 100 calculated lowest energies E0(fi). Some examples of the calculated adiabatic potential curves for donors located at diametrically opposite positions (Df ¼ p) and for different values of position indices are displayed in Fig. 2(a) for the case when position indices of donors are identical and in Fig. 2(b) when they are different. It is seen that with the donors’ displacement from the axis towards the ring, the adiabatic potential curves acquire two pronounced minima, which can be regarded as the potential profile of a double quantum well, symmetrical in the first case and asymmetrical in the second one. The potential minima depths become very small as the position index tends to zero and they grow drastically when the value of position index approximates one. Thus, it is possible to change the depth of the wells and the symmetry of the effective potential curves by varying the position indices of donors.
However, the height and the width of the barrier between wells of the adiabatic potential can be changed only by varying the angle Df. It is clear that barrier width always coincides with angle Df between donors’ position vectors. Therefore, by decreasing this angle one can reduce the effective width of the barrier between wells. In Fig. 3 we show the evolution of the shape of the effective potential curves E0(f) with the change in angle Df. It is seen that the width and the height of the barrier between wells are decreased when the angle between the position vectors tends to zero, both for symmetrical (b1 ¼ b2 ¼0.8) and for asymmetrical (b1 ¼0.7, b2 ¼0.8) arrangements of donors. The shape of the adiabatic potential affects essentially the character of electron motion along the ring and one can observe different types of electron motion along the ring. If the potential minima are sufficiently deep and the barrier between them is sufficiently high, the electron is localized inside one of the minima, and no persistent current occurs. When the barrier height or width is reduced, electron tunneling between the potential minima of the adiabatic potential becomes more probable and, finally, if the potential minima are shallow enough, the electron can rotate around the axis, making possible the existence of the persistent currents.
Fig. 2. Curves of adiabatic potentials E0(f) corresponding to electron rotation around the axis for different position indices b of donors with diametrically opposite positions, when their arrangement is (a) symmetric (b1 ¼ b2 ¼ b) and (b) asymmetric (b1 ¼ b, b2 ¼0.9b).
Fig. 3. Curves of adiabatic potentials E0(f) corresponding to electron rotation around the axis for different angles Df between vectors of donors’ positions, when their arrangement is (a) symmetric (b1 ¼ b2 ¼0.8) and (b) asymmetric (b1 ¼ 0.7, b2 ¼ 0.8).
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In what follows, we analyze the energy spectrum only for the case when the donors’ position indices coincide (b1 ¼ b2 ¼ b) and the effective potential has the form of a symmetrical double well. The non-uniform effective potential E0(f) in Eq. (10), provided by the impurities, breaks the rotational invariance, mixing the pure rotational states with different angular momenta for some lowlying levels and transforming them in vibration close to the donors’ positions. The nearer the donors’ positions to the ring centreline, the higher the barrier between the wells, the stronger the mixing of the rotational states and the greater the number of localized low-lying levels. The energies of some low-lying levels, corresponding to the bonding states (solid lines) and the anti-bonding states (dashed lines), as functions of the parameter b, for donors with diametrically opposite positions, are plotted in Fig. 4(a) and for donors with the angle Df ¼ p between their position vectors are presented in Fig. 4(b). It is seen that in both cases, energies of the ground and only a few low-lying excited states depend appreciably on the impurities’ positions. As donors are displaced from the centre toward the ring, the averaged separation between
electron and ions diminishes, the energies of these states decrease due to increased electron–ion attraction and they attain minimum values when the donors come close to the centreline, i.e. the position index value b approaches one. The growing electron–ion attraction when donors are displaced toward the centreline can localize only a few electronic states in which the electrons’ kinetic energies are not very large. The lower the electron kinetic energy, the stronger the localization and the larger the lowering of energy level. Therefore, the largest gap can be observed in Fig. 4 between the ground and the first excited state, as the donors are placed on the centreline. Also, it can be seen that the change in impurities’ positions has a weak effect on higher levels, whose kinetic energies are larger than the energy of attraction between electrons and ions. Energies of these states, corresponding to an above-barrier rotation, are positive and the respective wave functions are similar to those for unmixed pure rotational states. Due to the twofold rotational symmetry of the structure, all energy levels are split into pairs of sublevels, presented in Fig. 4 by solid lines for bonding sublevels and by dashed lines for antibonding sublevels. As seen from Fig. 4(a), the splitting of lower energy levels in donors with diametrically opposite positions is almost indistinguishable, while when the angle between the donors’ position vectors Df decreases up to p/5 the splitting
Fig. 4. Energies of some low-lying states of a singly ionized double donor system confined in a nanoring as functions of positions index b for two different values of angle Df.
Fig. 5. Energies of some low-lying states of a singly ionized double donor system confined in a nanoring as functions of angle Df for two different values of the position indices.
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becomes noticeable and it depends strongly on the donor’s position index, as seen from Fig. 4(b). This is due to the fact that the barrier width between wells reduces with decrease in Df the
splitting related to electron tunneling through the barrier becomes strongly dependent on the barrier height, which is sensible to the position index b as seen from Figs. 2 and 3.
Fig. 6. Energies of some lower states of a singly ionized double donor system confined in a nanoring with different position indices b and angles Df between the donors’ position vectors, as functions of the magnetic field strength.
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In Fig. 5 we display the dependencies of the lower energies on the angle between the donors’ position vectors for two different values of the position index. It is seen that the gaps between bonding and anti-bonding lower levels are very small for donors with diametrically opposite positions when the width of the barrier between two minima in the adiabatic potential is large and electron tunneling is almost improbable. As the angle between the donors’ position vectors is reduced, the barrier width diminishes, increasing the tunneling probability and the splitting of the energy levels. The larger the position index, the higher the barrier height and the smaller should be the angle for providing the same levels’ splitting, as seen from Fig. 5(a) and (b). The calculated energy levels for different position indices b and angles Df between donors’ position vectors as a function of magnetic field strength are given in Fig. 6. It is well known that in perfect narrow rings the energy levels are intersecting parabolas. Similar behavior of energy levels should be observed in the presence of donors located very close to the axis, since the adiabatic potential in this case remains almost homogeneous. As donors are displaced from the axis toward the ring and the correspondent position index increases, the adiabatic potential acquires the shape of a double quantum well. It is seen from Fig. 6 that in the presence of off-axis donors, gaps are opened at the points of intersection of the lower parabolas. For a fixed position index b and angle Df, the gap decreases for larger values of intersection point energy. At a fixed point of intersection, the gap increases on decreasing the angle Df and increasing the position index b. It is also seen that decreases in angle Df and increase in position index b make the potential minima deeper, producing negative shifts of the energy levels, which are larger for lower states. The closer the donors’ location to the ring’s centreline or the smaller the angle Df, the deeper the potential minima and the
565
greater the number of bound states, whose energies are almost independent of the magnetic field due to their localization inside the wells. It is also seen from Fig. 6 that the transition from unbound to bound states is smooth; the shallow bound states are still weakly sensitive to the magnetic field while the deep bound states are almost insensitive to the magnetic field. When the position index b tends to zero the rotational symmetry of the adiabatic potential is restored, and the corresponding energy sublevels are pure rotational states with different angular momenta m¼0, 71, 72, y It can be seen from Fig. 6(d) that in this case the ground-state energy of the singly ionized double donor system in a nanoring oscillates smoothly with the magnetic field, while the angular momentum m suffers step-wise evolution, jumping initially from 0 to 1, afterward to 2, etc. According to the selection rule, the dipole transition is possible only from the ground state with angular momentum m to the excited states with m71. This is the reason why the FIR spectra in Fig. 7(a) for the case b ¼0 show the discontinuous drops with increase in magnetic fields due to the selection rule and the AB oscillations of the energy levels. The peak position jumps down as the magnetic field strength acquires values g ¼ 1=R2c , 3=R2c , 5=R2c , . . .. As donors are displaced from the axis toward the ring and the position index b increases, the rotational symmetry breaks and the energy sublevels transform to a mixture of the pure rotational states with different m. The selection rule for dipole transitions is then changed and more peaks appear in the FIR spectra. In Fig. 7(b)–(d) it is seen that increases in b lead to the emergence of new transitions that can exist between the ground state and the first excited states that may or may not oscillate with the magnetic field. Also it is seen that for impurities in a strongly interacting regime, b Z0.5; oscillations in the lowest optically active state are suppressed due to the strong Coulomb
Fig. 7. FIR spectra of left circularly polarized light as a function of energy of a singly ionized double donor system confined in a nanoring with centreline radius Rc ¼ 10an0 of width w¼ 1an0, with diametrically opposite donor locations with positions Df ¼ p and different position indices b.
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interaction between the electron and impurities, whereas AB oscillations are manifested in the transitions to higher excited states.
Novel Materials ECNM, under Contract no. 043-2005, and Cod. no. 1102-05-16923 subscribed with COLCIENCIAS. References
4. Summary and conclusions We theoretically investigate the low-lying energy levels and FIR spectra of a singly ionized double donor system (D2+ ) confined in a nanoring in 3D under the influence of a perpendicular magnetic field. One of the most interesting results obtained is that the strong confinement allows us to obtain artificial molecules more stable than the natural molecule, with the additional feature that the energy gap between the two lowest-lying electron states and other excited states favors the building of an adequate microscopic twolevel system required for quantum computation purposes. Also, we show that the presence of impurities leads to AB oscillations of absorption as a function of the magnetic field, enriching the FIR spectra and hence optical properties of the system.
Acknowledgements This work was financed by the Universidad Industrial de Santander (UIS) through the Vicerrectoria General de Investigaciones (DIF Ciencias, Cod. no. 5124), the Excellence Centre of
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Physica E journal homepage: www.elsevier.com/locate/physe
Coupled donors in quantum ring in a threading magnetic field W. Gutie´rrez n, L.F. Garcı´a, I.D. Mikhailov Escuela de Fı´sica, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia
a r t i c l e in fo
abstract
Article history: Received 21 August 2010 Accepted 27 September 2010 Available online 1 October 2010
Electronic states of a singly ionized double donor system (D2+ ) confined in a nanostructure with a ring¨ like geometry in a threading magnetic field are calculated. Solutions of the Schrodinger equation are obtained by variational separation of variables in the adiabatic limit. Numerical results are shown for bonding and anti-bonding lowest-lying molecular states corresponding to different relative locations of the coupled impurities. We show that displacement of donors from the axis toward the ring and decrease in the angle between their position vectors may strongly affect the Aharonov–Bohm oscillations of the far-infrared spectra and produce a quenching of oscillations of the lower energy levels. & 2010 Elsevier B.V. All rights reserved.
1. Introduction There has been remarkable interest in semiconductor nanostructures during the past twenty years, motivated by possibilities of realization of new high-tech electronic device architectures. One of these possibilities is related to the building of an adequate microscopic two-level system, which can be used as the functional part in a wide range of device applications, including spintronics, optoelectronics, photovoltaics, and quantum information technologies. Coupled quantum wells or quantum dots have been considered previously to be basic candidates for this aim. Recently, it has been proposed to use instead the singly ionized double donor system (D2+ ) confined in a semiconductor quantum dot [1]. This system encodes logical information either on the spin or on the charge degrees of freedom of the single electron and allows us to manipulate conveniently its molecular properties, such as energy splitting between the bonding and anti-bonding lowest-lying molecular-like states or spatial distribution of carriers in the system [2]. This fact has raised interest in charge qubits, where the logical states are represented by the two lowest-lying electron orbital states positioned at the different donors. Theoretical analysis of the D2+ system in a spherical QD demonstrated that the confinement can greatly enhance the energy gap between the two lowest-lying electron states and other excited states, favouring the identification of the quasi-two-level system required for quantum computation purposes [3]. It can be expected that a quantum ring (QR) with single or two captured electrons, provided by a double donor system, will be a
n
Corresponding author. Tel.: + 57 7 6323095; fax: + 57 7 6332477. E-mail address: [email protected] (W. Gutie´rrez).
1386-9477/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2010.09.015
better candidate towards this aim than a spherical QD, because of the fact that the gaps between the lower energy levels in QR are larger and electron localization is stronger. Besides, the external magnetic field applied along the axis gives an additional possibility for controlling properties of molecular-like states in QRs, where due to the ring-like geometry, different quantum-size effects such as the Aharonov–Bohm (AB) oscillations [4] can be manifested in the presence of the external fields. Such peculiarity in QRs has stimulated in the last two decades much interest in the theoretical and experimental studies of their electronic and optical properties. Energy spectra of the one- and two-electron single and vertically coupled QRs have been studied by various authors [5]. It has been shown that the properties of the lowenergy electron states are very sensitive to any disorder that breaks the axial symmetry of the structure. Particularly, the nonuniformity leads to a transformation of the rotations around the axis of the lower energy electrons into their oscillations near the defects, to a flattening of the curves corresponding to energy dependencies on the magnetic field, and to quenching of the AB oscillations [6]. One could expect a similar transformation of the QR properties due to the presence of off-axis donors, which also break the axial symmetry. The effect of a positively charged impurity on energy spectra of one and two electrons in semiconductor nanorings under magnetic fields was studied in Ref. [7]. It has been shown that impurity-induced AB oscillations and FIR spectrum are strongly dependent on nanoring size and impurity position. To our knowledge, similar properties of singly ionized double donor system (D2+ ) confined in a semiconductor QR and its FIR spectrum have not been explored up to now. This system represents the simplest system confined in QR, which in contrast to the single donor exhibits energy splitting between the bonding and anti-bonding lowest-lying molecular-like states. Prompted by
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a possible important role of the double donors’ structure in nanotechnology, we present in this paper a theoretical study of the molecular properties of a D2+ system confined in a QR. In order to analyze the effect of the symmetry break, induced by the coupled donors, on AB oscillations of the QR FIR spectrum we calculate the energy levels and FIR spectrum of this structure under a magnetic field, for different impurity positions. On the other hand, study of the quantum behavior of the double donor system D2+ also allows us to understand better some properties of trions X + (positively charged excitons) confined in type II QRs in the limit case, as the effective mass of the hole is much larger than that of the electron. Recently, theoretical investigation of AB oscillations of the energy levels of neutral and charged excitons, in type I and type II QRs, has been a topic of great interest [8]. The organization of the remainder of this paper is as follows. In Section 2, the model and Hamiltonian are described and procedures to obtain the solutions are briefly outlined. The main results are shown and discussed in Section 3, followed by a summary in Section 4.
2. Theory We consider an infinite barrier model of a quantum dot with ring-like geometry, which represents an axially symmetrical very thin layer, in which the dependency of thickness h on distance r from the axis is given by the following simple relation: hðrÞ ¼ hw þ Dh WðRc þw=2rÞWðrRc þw=2Þ
ð1Þ
Here hw is the wetting layer (WL) thickness, Dh the increase of layer thickness inside the ring, w and Rc are the ring width and centreline radius, respectively, and W(x) is the Heaviside stepfunction, equal to zero for x o0 and to one for x 40. A singly ionized double donor system, schematically presented in Fig. 1, is ! ! characterized, besides, by donors’ position vectors n 1 and n 2 . In the framework of the effective-mass approximation, the Hamiltonian of this system in cylindrical coordinates is given by 2 2 ^ ¼ 1 @ r @ 1 @ @ þig @ H 2 2 @j r @r @r r @j @z2 X g2 r2 2 þ V ð3DÞ ðr,zÞ þ ! ! 4 i ¼ 1,2 9 r n 9
energies in units of the effective Rydberg, Ry ¼ e2 =2ea0 and strength of the external magnetic field applied along the z-axis is in units of dimensionless energy of the first Landau levelg ¼ e_B=2mc Ry, mn and e being the electron effective mass and the layer dielectric constant, respectively. In accordance with the experiment, the height-to-radius aspect ratio in self-assembled quantum dots is generally very small, i.e. (hw + Dh)/Rc 51. This fact allows us to use the advantage of adiabatic approximation (AA). Since electron motion in the z-direction is much faster than in-plane motion inside the layer, we follow the well-known adiabatic procedure of first finding the lowest energy Ez(r) and correspondent wave function fz(z,r) of the equation " # @2 2 þ V ð3DÞ ðr,zÞEz ðrÞ fz ðz, rÞ ¼ 0; @z pffiffiffiffiffiffiffiffiffi fz ðz, rÞ ¼ 2=psinðpz=hðrÞÞ; Ez ðrÞ ¼ p2 =h2 ðrÞ ð3Þ where r is treated as a parameter (cf. electronic motion for a fixed nuclear position in the molecular problem), and then represent the three-dimensional eigenfunctions of the Hamiltonian (2) C(3D)(r,z,f) as follows: pffiffiffiffi Cð3DÞ ðr,z, jÞ ¼ fz ðz, rÞcð2DÞ ðr, jÞ= r ð4Þ
c(2D)(r,f) being a variational two-dimensional wave function that describes only the electron in-plane motion (cf. nuclear ¨ motion in molecular problem). According to Schrodinger’s variational principle, a function C(2D)(r,f), for which the ð3DÞ ð3DÞ functional F½Cð2DÞ ¼ /c 9HE9c S has a stationary value is an eigenfunction of the Hamiltonian (2). Taking the functional derivative with respect to c(2D) gives a wave equation of the form 2 3 @2 1 @2 ð2DÞ ðrÞ 2 6 @r2 þ V 7 2 r @j 6 7 ð2DÞ 6 7c ðr, jÞ ¼ 0 ð5aÞ X 6 7 @ 4 þ ig Vi ðr, jÞ þ V0 E 5 @j i ¼ 1,2 V ð2DÞ ðrÞ ¼
ð2Þ
i
Here V(3D)(r,z) is the confinement potential, which is equal to zero and to infinity inside and outside the layer, respectively; all lengths are scaled in units of the effective Bohr radiusa0 ¼ _2 e=m e2 , all
Fig. 1. Scheme of a quantum ring with singly ionized coupled donor system.
Vi ðr, jÞ ¼
g2 r2 4 Z
hðrÞ 0
1 p2 þ 4r2 h2 ðrÞ
2fz2 ðz, rÞdz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ x2i 2rxi cosðjji Þ
ð5bÞ
ð5cÞ
In the wave equation (5a) two first terms describe the electron motion in the radial direction under effective two-dimensional confinement potential V(2D)(r), the following two terms correspond to electron rotation around the axis, and the terms Vi(r,f) give energies of interaction of electron with donors. According to relation (1) the height of the layer h(r) jumps from hw to hw + Dh at the frontiers of the ring with the WL, i.e. when the radius is equal to r ¼Rc w/2 or r ¼Rc +w/2. Therefore at these points the potential V(2D)(r), defined by relation (5b), has a jump of the value V0 ¼ p2 =h2w p2 =ðhw þ DhÞ2 , which usually is significantly larger than the averaged energy of the electron attraction to donors. For example, if we take typical values for the effective Bohr radius to be about 10 nm, thickness of the WL of about 2 nm, and DhE2 nm, then the barrier height between QR and WL in our model is about V0 E 200Ryn, while the averaged energy of attraction to donors is close to 1Ryn. This results in the fact that the electron is mainly located inside the ring (Rc w/2o r oRc + w/2) independent of the donors’ location. The general equation (5) can be applied for calculation of lowlying energies of any QD with axial symmetry and with different morphologies, such as disk-shaped, lens-shaped, etc. But in this
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work we want to emphasize on the analysis of properties typical for structures with ring-like geometry and for the particular case when the width of the ring is sufficiently small. Therefore , for the sake of mathematical convenience, we consider below only very narrow QRs, whose widths are much smaller than their radii (w5Rc). This we call the adiabatic limit for the following reason. As the scales of displacement in the radial and azimuthal directions are strongly different it is convenient to replace the variable r by a new dimensionless variable x that is changes inside the ring from 1 to+ 1, by using the substitution r ¼Rc +xw/2. Then Eq. (5) can be written as h ð2DÞ i ð2DÞ ^ H E c ðx, jÞ ¼ 0; H
ð2DÞ
4 @2 ð2DÞ 1 @2 @ ¼ 2 2 þ V~ ðxÞ þ ig 2 @j2 @j w @x ðRc þwx=2Þ X V~ i ðx, jÞ 1 o x o þ 1; 0 o j o 2p;
corresponding to the solutions of Eq. (10) for the lowest adiabatic potential curve E0(f). In our numerical work we solve Eq.(7) using the trigonometric sweep described in Ref. [10] and Eq. (10) by means of Fourier series expansion in the form
FðjÞ ¼
N X
cm eimj
ð11Þ
m ¼ N
The variational principle (9) then reduces to the linear system of algebraic equations N X
½Edmu,m Amu,m cm ¼ 0;
mu ¼ 0, 7 1, 7 2, . . ., 7 N;
m ¼ N
Amu,m ¼ mumbmum þ vmum gmdmu,m
ð12Þ
with
i ¼ 1,2
cð2DÞ ð1, jÞ ¼ cð2DÞ ð þ 1, jÞ ¼ 0;
cð2DÞ ðx,0Þ ¼ cð2DÞ ðx,2pÞ
ð6Þ
ð2DÞ ðxÞ ¼ V ð2DÞ ðRc þxw=2Þ and V~ i ðx, jÞ ¼ Here the notations V~ Vi ðRc þ xw=2, jÞ are used. Eq. (6) may be regarded as the ¨ Schrodinger equation describing two one-dimensional particles of masses w2/4 and R2c , respectively, interacting through the P ~ potential i ¼ 1,2 V i ðx, jÞ. Since one of these masses is much smaller than the other, we follow the well-known adiabatic procedure of first solving the equation " # X 4 @2 ð2DÞ V~ i ðx, jÞE0 ðjÞ f ðx, jÞ ¼ 0 2 2 þ V~ ðxÞ ð7Þ w @x i ¼ 1,2
where f is treated as a parameter. Once Eq. (7) is solved and the energy dependence on angle E0(j) is found then the solution of Eq. (6) may be presented in the form
cð2DÞ ðx, jÞ ¼ f ðx, jÞFðjÞ
ð8Þ
with variational function F(f) describing electron rotation around the axis. Energies of the low-lying states are found by minimizing the functional Z 1 Z p h ð2DÞ ^ F½F ¼ dx djf ðx, jÞFðjÞ H ðx, jÞEf ðx, jÞFðjÞ-min 0
561
p
ð9Þ Taking the functional derivative with respect to F gives a wave equation of the form d dFðjÞ dFðjÞ ig BðjÞ þ E0 ðjÞE FðjÞ ¼ 0; dj dj dj Z þ1 f 2 ðx, jÞ BðjÞ ¼ dx 2 1 ðRc þ wx=2Þ ð10Þ p o j o p; FðpÞ ¼ FðpÞ; FuðpÞ ¼ FuðpÞ
It is seen that the function E0(f), found as a solution of Eq. (7), appears in Eq. (10) as an effective adiabatic potential that governs electron rotation around the axis. The structure of energy spectrum given by the pair of Eqs. (7) and (10) is similar to that of vibro-rotational spectra of a molecule. The set of the vibrational levels corresponding to electron motion in the radial direction, given by different adiabatic potentials E0(j), is characterized according to Eq. (7) by gaps of order 1/w2. These levels are split into a series of rotational sublevels with gaps between them of the order 1=R2c according to Eq. (10). In what follows we analyze only the set of the lower rotational sublevels
Z p 1 E0 ðjÞeimj dj; 2p p Z p 1 BðjÞeimj dj; m ¼ 0, 7 1, 7 2, . . ., 7 N bm ¼ 2p p
vm ¼
ð13Þ
The condition that this set of equations has a nonzero solution leads to a secular equation of order N: detJEdmu,m Amu,m J ¼ 0;
mu,m ¼ 0, 71, 7 2, . . ., 7N
ð14Þ
By diagonalizing the secular equation (14), we can obtain the kth rotational energy sublevel Ek and the corresponding set of ðkÞ ðkÞ coefficients cm , m ¼ 0, 71, 7 2, . . ., 7N. If the coefficient cm is ðkÞ given by the Kronecker symbolcm ¼ dm,k , the corresponding sublevel is a pure rotational state with angular momentum m, otherwise it is superposition of various rotational states with different angular momenta. The total 3D wave functions then are given by the following expression:
CðkÞ ðr,z, jÞ ¼
N pffiffiffiffiffiffiffiffiffi pffiffiffiffi X ðkÞ imj 2=psinðpz=hðrÞÞf ð2ðrRc Þ=w, jÞ= r cm e m ¼ N
ð15Þ To analyze the properties of the solutions, two useful quantities can be extracted from C(k)(r,z,f), namely the electron density, ! ! 2 Nk ð r Þ ¼ 9CðkÞ ð r Þ9 and the absorption coefficient a(o), which in the electric dipole approximation at low temperature is given by [9] X ! 2 aðEÞ ¼ cE 9/CðkÞ 9! e U d 9Cð0Þ S9 dðEDEk Þ; DEk ¼ EðkÞ Eð0Þ ð16Þ k
! where d is the electric dipole operator, c a constant factor, and DEk the energy difference between the initial and the final states. For circularly polarized light, the complex polarization vector of pffiffiffi pffiffiffi the constant external electronic field -e 7 ¼ ð1= 2, 7 i= 2Þ corresponds to right (+) and left ( ) circularly polarized light, respectively.
3. Results and discussion We calculate low-lying energies of the singly ionized double donor system confined in a semiconductor QR for different donors’ positions and magnetic fields g. The studied rings have a centreline radiusRc ¼ 10a0 , width w ¼ 1a0 , and wetting layer thickness and dot height over the wetting layer hw ¼ 0:2a0 and Dh ¼ 0:2a0 , respectively. In order to help study the effect of donors’ positions on the structure of energy spectrum, we use
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below as parameters two position indices bi ¼ xi/Rc, i¼1,2, xi being the distances from donors’ locations up to the axis, and the angle Df between vectors of donors’ positions. In our calculations, in the beginning we solved Eq. (7) for 100 different angles fi, i¼1, 2, 3, y, 100 and then defined the unknown adiabatic potential curve E0(f) as an interpolating cubic spline fitted to 100 calculated lowest energies E0(fi). Some examples of the calculated adiabatic potential curves for donors located at diametrically opposite positions (Df ¼ p) and for different values of position indices are displayed in Fig. 2(a) for the case when position indices of donors are identical and in Fig. 2(b) when they are different. It is seen that with the donors’ displacement from the axis towards the ring, the adiabatic potential curves acquire two pronounced minima, which can be regarded as the potential profile of a double quantum well, symmetrical in the first case and asymmetrical in the second one. The potential minima depths become very small as the position index tends to zero and they grow drastically when the value of position index approximates one. Thus, it is possible to change the depth of the wells and the symmetry of the effective potential curves by varying the position indices of donors.
However, the height and the width of the barrier between wells of the adiabatic potential can be changed only by varying the angle Df. It is clear that barrier width always coincides with angle Df between donors’ position vectors. Therefore, by decreasing this angle one can reduce the effective width of the barrier between wells. In Fig. 3 we show the evolution of the shape of the effective potential curves E0(f) with the change in angle Df. It is seen that the width and the height of the barrier between wells are decreased when the angle between the position vectors tends to zero, both for symmetrical (b1 ¼ b2 ¼0.8) and for asymmetrical (b1 ¼0.7, b2 ¼0.8) arrangements of donors. The shape of the adiabatic potential affects essentially the character of electron motion along the ring and one can observe different types of electron motion along the ring. If the potential minima are sufficiently deep and the barrier between them is sufficiently high, the electron is localized inside one of the minima, and no persistent current occurs. When the barrier height or width is reduced, electron tunneling between the potential minima of the adiabatic potential becomes more probable and, finally, if the potential minima are shallow enough, the electron can rotate around the axis, making possible the existence of the persistent currents.
Fig. 2. Curves of adiabatic potentials E0(f) corresponding to electron rotation around the axis for different position indices b of donors with diametrically opposite positions, when their arrangement is (a) symmetric (b1 ¼ b2 ¼ b) and (b) asymmetric (b1 ¼ b, b2 ¼0.9b).
Fig. 3. Curves of adiabatic potentials E0(f) corresponding to electron rotation around the axis for different angles Df between vectors of donors’ positions, when their arrangement is (a) symmetric (b1 ¼ b2 ¼0.8) and (b) asymmetric (b1 ¼ 0.7, b2 ¼ 0.8).
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In what follows, we analyze the energy spectrum only for the case when the donors’ position indices coincide (b1 ¼ b2 ¼ b) and the effective potential has the form of a symmetrical double well. The non-uniform effective potential E0(f) in Eq. (10), provided by the impurities, breaks the rotational invariance, mixing the pure rotational states with different angular momenta for some lowlying levels and transforming them in vibration close to the donors’ positions. The nearer the donors’ positions to the ring centreline, the higher the barrier between the wells, the stronger the mixing of the rotational states and the greater the number of localized low-lying levels. The energies of some low-lying levels, corresponding to the bonding states (solid lines) and the anti-bonding states (dashed lines), as functions of the parameter b, for donors with diametrically opposite positions, are plotted in Fig. 4(a) and for donors with the angle Df ¼ p between their position vectors are presented in Fig. 4(b). It is seen that in both cases, energies of the ground and only a few low-lying excited states depend appreciably on the impurities’ positions. As donors are displaced from the centre toward the ring, the averaged separation between
electron and ions diminishes, the energies of these states decrease due to increased electron–ion attraction and they attain minimum values when the donors come close to the centreline, i.e. the position index value b approaches one. The growing electron–ion attraction when donors are displaced toward the centreline can localize only a few electronic states in which the electrons’ kinetic energies are not very large. The lower the electron kinetic energy, the stronger the localization and the larger the lowering of energy level. Therefore, the largest gap can be observed in Fig. 4 between the ground and the first excited state, as the donors are placed on the centreline. Also, it can be seen that the change in impurities’ positions has a weak effect on higher levels, whose kinetic energies are larger than the energy of attraction between electrons and ions. Energies of these states, corresponding to an above-barrier rotation, are positive and the respective wave functions are similar to those for unmixed pure rotational states. Due to the twofold rotational symmetry of the structure, all energy levels are split into pairs of sublevels, presented in Fig. 4 by solid lines for bonding sublevels and by dashed lines for antibonding sublevels. As seen from Fig. 4(a), the splitting of lower energy levels in donors with diametrically opposite positions is almost indistinguishable, while when the angle between the donors’ position vectors Df decreases up to p/5 the splitting
Fig. 4. Energies of some low-lying states of a singly ionized double donor system confined in a nanoring as functions of positions index b for two different values of angle Df.
Fig. 5. Energies of some low-lying states of a singly ionized double donor system confined in a nanoring as functions of angle Df for two different values of the position indices.
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becomes noticeable and it depends strongly on the donor’s position index, as seen from Fig. 4(b). This is due to the fact that the barrier width between wells reduces with decrease in Df the
splitting related to electron tunneling through the barrier becomes strongly dependent on the barrier height, which is sensible to the position index b as seen from Figs. 2 and 3.
Fig. 6. Energies of some lower states of a singly ionized double donor system confined in a nanoring with different position indices b and angles Df between the donors’ position vectors, as functions of the magnetic field strength.
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In Fig. 5 we display the dependencies of the lower energies on the angle between the donors’ position vectors for two different values of the position index. It is seen that the gaps between bonding and anti-bonding lower levels are very small for donors with diametrically opposite positions when the width of the barrier between two minima in the adiabatic potential is large and electron tunneling is almost improbable. As the angle between the donors’ position vectors is reduced, the barrier width diminishes, increasing the tunneling probability and the splitting of the energy levels. The larger the position index, the higher the barrier height and the smaller should be the angle for providing the same levels’ splitting, as seen from Fig. 5(a) and (b). The calculated energy levels for different position indices b and angles Df between donors’ position vectors as a function of magnetic field strength are given in Fig. 6. It is well known that in perfect narrow rings the energy levels are intersecting parabolas. Similar behavior of energy levels should be observed in the presence of donors located very close to the axis, since the adiabatic potential in this case remains almost homogeneous. As donors are displaced from the axis toward the ring and the correspondent position index increases, the adiabatic potential acquires the shape of a double quantum well. It is seen from Fig. 6 that in the presence of off-axis donors, gaps are opened at the points of intersection of the lower parabolas. For a fixed position index b and angle Df, the gap decreases for larger values of intersection point energy. At a fixed point of intersection, the gap increases on decreasing the angle Df and increasing the position index b. It is also seen that decreases in angle Df and increase in position index b make the potential minima deeper, producing negative shifts of the energy levels, which are larger for lower states. The closer the donors’ location to the ring’s centreline or the smaller the angle Df, the deeper the potential minima and the
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greater the number of bound states, whose energies are almost independent of the magnetic field due to their localization inside the wells. It is also seen from Fig. 6 that the transition from unbound to bound states is smooth; the shallow bound states are still weakly sensitive to the magnetic field while the deep bound states are almost insensitive to the magnetic field. When the position index b tends to zero the rotational symmetry of the adiabatic potential is restored, and the corresponding energy sublevels are pure rotational states with different angular momenta m¼0, 71, 72, y It can be seen from Fig. 6(d) that in this case the ground-state energy of the singly ionized double donor system in a nanoring oscillates smoothly with the magnetic field, while the angular momentum m suffers step-wise evolution, jumping initially from 0 to 1, afterward to 2, etc. According to the selection rule, the dipole transition is possible only from the ground state with angular momentum m to the excited states with m71. This is the reason why the FIR spectra in Fig. 7(a) for the case b ¼0 show the discontinuous drops with increase in magnetic fields due to the selection rule and the AB oscillations of the energy levels. The peak position jumps down as the magnetic field strength acquires values g ¼ 1=R2c , 3=R2c , 5=R2c , . . .. As donors are displaced from the axis toward the ring and the position index b increases, the rotational symmetry breaks and the energy sublevels transform to a mixture of the pure rotational states with different m. The selection rule for dipole transitions is then changed and more peaks appear in the FIR spectra. In Fig. 7(b)–(d) it is seen that increases in b lead to the emergence of new transitions that can exist between the ground state and the first excited states that may or may not oscillate with the magnetic field. Also it is seen that for impurities in a strongly interacting regime, b Z0.5; oscillations in the lowest optically active state are suppressed due to the strong Coulomb
Fig. 7. FIR spectra of left circularly polarized light as a function of energy of a singly ionized double donor system confined in a nanoring with centreline radius Rc ¼ 10an0 of width w¼ 1an0, with diametrically opposite donor locations with positions Df ¼ p and different position indices b.
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interaction between the electron and impurities, whereas AB oscillations are manifested in the transitions to higher excited states.
Novel Materials ECNM, under Contract no. 043-2005, and Cod. no. 1102-05-16923 subscribed with COLCIENCIAS. References
4. Summary and conclusions We theoretically investigate the low-lying energy levels and FIR spectra of a singly ionized double donor system (D2+ ) confined in a nanoring in 3D under the influence of a perpendicular magnetic field. One of the most interesting results obtained is that the strong confinement allows us to obtain artificial molecules more stable than the natural molecule, with the additional feature that the energy gap between the two lowest-lying electron states and other excited states favors the building of an adequate microscopic twolevel system required for quantum computation purposes. Also, we show that the presence of impurities leads to AB oscillations of absorption as a function of the magnetic field, enriching the FIR spectra and hence optical properties of the system.
Acknowledgements This work was financed by the Universidad Industrial de Santander (UIS) through the Vicerrectoria General de Investigaciones (DIF Ciencias, Cod. no. 5124), the Excellence Centre of
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