- Email: [email protected]
Two vertically coupled quantum dots in a magnetic field
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Physica B 298 (2001) 282}286
Two vertically coupled quantum dots in a magnetic "eld B. Partoens*, F.M. Peeters Departement Natuurkunde, Universiteit Antwerpen (UIA), Universiteitsplein 1, B-2610 Antwerpen, Belgium
Abstract We studied the ground state of two vertically coupled quantum dots as a function of the interdot distance in the presence of a perpendicular magnetic "eld. The theoretical approach is based on the current spin density functional theory. Increasing the interdot distance leads to the population of antibonding levels, resulting in molecule-type phases. However, for increasing magnetic "eld strengths, we "nd that electrons hop back from antibonding to bonding levels, which is a consequence of electron interactions. The maximum density droplet is much less stable than in a single quantum dot. 2001 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Density functional theory; Maximum density droplet
1. Introduction Quantum dots are small man-made objects in which electrons are con"ned in all three spatial directions like in real atoms [1,2]. Therefore, they are also called artixcial atoms. Due to the increased role of electron interactions these arti"cial atoms exhibit new physics which has no analogue in real atoms. Experimentally, these quantum dots can be probed through tunneling experiments [3] which give us information on the position of the energy levels: the separation between the measured tunneling peaks re#ects that the shell structure and Hund's rules determine the electronic properties. Moreover, these tunneling peaks shift in a magnetic "eld and clearly show transitions in the many-electron ground states as a result of the
* Corresponding author. Fax: #32-3-820-22-45. E-mail addresses: [email protected] (B. Partoens), [email protected] (F.M. Peeters).
competition between the kinetic energy and the Coulomb energy. All these single quantum dot properties were successfully reproduced using the density functional theory formalism [4}12]. When two such arti"cial atoms are placed on top of each other, i.e. two quantum dots which couple vertically, we have an artixcial molecule. In such a system, we can also control the interdot distance and consequently the interdot Coulomb interaction and the tunneling between the dots. In this way one can play with the competition between intradot and interdot correlations [13}16]. Recently, this system was experimentally realized by Austing et al. [17]. Exact diagonalizations of coupled dots in a magnetic "eld, which are limited to a small number of electrons, were performed in Refs. [18}21]. In this paper we show the results obtained using the current spin density functional theory (CSDFT) for the ground state phases for two vertically coupled quantum dots in a magnetic "eld. In comparison to Ref. [18,19], we focus on much shorter interdot distances.
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 3 1 9 - 2
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2. Theoretical model We consider two circularly symmetric dots, placed at a distance d vertically from each other. It is often a good approximation to assume a parabolic con"nement <(r)"m r/2 of frequency , with m the e!ective electron mass. In the z-direction the quantum dots are created by two coupled quantum wells, as schematically shown in Fig. 1. A homogeneous magnetic "eld B"Be is applied X perpendicular to the xy-plane in which the electrons are con"ned. A typical experimental value for the con"nement energy is "3 meV [3]. Be cause the energy scale in such systems is substantially smaller than in real atoms, an external magnetic "eld will have a more profound in#uence on the ground state of these arti"cial structures. For example, the magnetic energy " eB/mc equals this con"nement energy already for B"2 T, and therefore in typical laboratory experiments the magnetic "eld can be large. We used CSDFT, which is an extension of spin density functional theory, and which is able to account for the e!ect of strong magnetic "elds. For the details of CSDFT we refer to the original work of Vignale and Rasolt [22]. Here we recall only the Kohn}Sham equation. In a symmetric gauge, the external vector potential acting on the electrons is A"Bre /2. The ground state energy and the denF sity (r) are expressed in terms of a set of Kohn}Sham orbitals (r)"exp(!il) (r)Z(z), (1) HJN HJN which are the eigenstates of the z-component of the angular momentum !l, and satisfy the Kohn} Sham equation in CSDFT
R 1 R l R l ! # ! # ! 2m Rr r Rr r Rz 2 mr e l A #<(z)#< (r) # ! & 2 mc r #< (r)#< ] (r)Z(z)" (r)Z(z), & N HJN HJN HJN (2)
with the z-component of the spin, "eB/mc the cyclotron frequency, and "( #/4. A is the exchange-correlation vector potential and
Fig. 1. The inset shows the con"nement potential in the zdirection. The symbols are the calculated splitting between the lowest bonding and antibonding level and the curve is an exponential "t to it. The corresponding wavefunctions are shown schematically in the inset.
< the exchange-correlation scalar potential, < the intradot and < the interdot Hartree & & potential. The density in the z-direction is approximated by functions. For the material parameters we choose typical GaAs values for the e!ective mass m"0.067 m and the dielectric constant "12.4 This yields an e!ective Bohr radius of aH"9.79 nm. The strength of the external con"nement is set to " 5.78/N meV, where N is the total number of electrons in the two dot system. For di!erent N, this choice keeps the average electron density in each dot approximately constant, in this case roughly corresponding to a (two-dimensional) Wigner}Seitz radius r "1.75aH, as in a typical vertical quantum dot structure [17]. The con"nement in the z-direction consists of two coupled quantum wells as shown in Fig. 1. Due to the "nite barrier between both dots the lowest level in the z-direction is split into a symmetric bonding and an anti-symmetric antibonding level. For our numerical calculation we take the dimensions of the quantum wells corresponding to the experimental realization of Ref. [17]: ="120 As for the width of both dots, < "250 meV for the height of the barrier between the dots. The energy splitting between the lowest bonding and antibonding levels was then calculated numerically and could be "tted to an exponential decaying function
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
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B. Partoens, F.M. Peeters / Physica B 298 (2001) 282}286
"22.86 exp(!d(As )/13.455) meV (solid curve in Fig. 1).
3. Phase diagrams The obtained ground state phase diagrams for 3 and 6 electrons are shown in Fig. 2 in the parameter space of the level splitting (or equivalently, the interdot distance d) and the magnetic "eld B. The levels are labeled by the three quantum numbers (S , M , I ): spin S , angular momentum M , X X X X X and an isospin quantum number I , which is the X di!erence between the electrons in the bonding level and those in the antibonding level, divided by 2. The isospin was "rst introduced by Palacios and Hawrylak [21,23] who concentrated on the high magnetic "eld case. In the phase diagram for 3 electrons (see Fig. 2(a)) the occupation of the corresponding e!ective single particle states are shown schematically: the single particle energy levels are E " (2n#l#1) (with n the radial quantum number), which due to the splitting of the energy level in the z-direction in a bonding and an antibonding level are shifted by an energy . To understand these phase diagrams let us see what one expects from the single particle picture. In a magnetic "eld the single particle energy levels of a two-dimensional harmonic oscillator are altered into E "(2n#l#l) !l /2. Notice that the degeneracy in the angular momentum l and !l is lifted. To predict when the molecule-type phases appear we have to compare the single particle level splitting of a 2D harmonic oscillator in a magnetic "eld " ( #/4! /2 with the level splitting between bonding and antibonding levels . As decreases with increasing magnetic "eld, while does not change, we expect that the bonding levels will stay occupied for larger interdot distances. On the calculated phase diagrams one can indeed see that the bonding levels stay longer occupied when the magnetic "eld increases, even much longer than one would expect from the single particle condition ". This is again a consequence of the electron interactions. As a magnetic "eld
Fig. 2. Phase diagrams for (a) 3 and (b) 6 electrons in the two coupled dots. The phases are labeled with the quantum numbers (S , M , I ). The insets in (a) show the single particle con"gura X X tions where the occupation of bonding (left side of slash) and of antibonding levels (right side of slash) are shown.
pushes the electrons to the center of the dot, the Coulomb repulsion increases. Therefore, at a certain magnetic "eld value it is energetically favorable to increase the angular momentum, which costs con"nement energy but reduces the Coulomb repulsion. The phase diagrams also show that in coupled dots this increase in angular momentum is often realized by isospin #ips in which an electron hops back from an antibonding to a bonding level (corresponding with an increasing isospin).
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4. Stability of the maximum density droplet The study of the maximum density droplet (MDD) phase in a single quantum dot attracted much interest [24}28]. This state is the most dense spin-polarized con"guration and it is the ground state in a rather large magnetic "eld region. In the MDD the electrons occupy adjacent orbitals with consecutive angular momentum. The stability of the MDD is determined by the competition between the kinetic energy and the Coulomb repulsion between the electrons. As all electrons are spin-polarized, the exchange also acts as a binding force. The kinetic energy would favor the MDD up to in"nite "elds. However, increasing the magnetic "eld shrinks the electron orbits and thus increases the Coulomb repulsion. Therefore, the Coulomb interaction favors a di!use occupation. At some threshold magnetic "eld, the direct Coulomb interaction has become so large that the MDD breaks apart into a lower density droplet, so increasing the total angular momentum. Similarly, we expect a spin-polarized state for su$ciently large magnetic "elds for two coupled quantum dots, but now with the additional complication that the electrons populate bonding and antibonding levels. The electrons in the antibonding states have smaller angular momenta than the bonding states as they are less populated. Thus the angular momentum can be increased for increasing magnetic "eld through a one-by-one depopulation of antibonding states, until the MDD is formed where only bonding levels are populated. However, we "nd that the MDD of bonding levels is only the ground state for the 3 electron case in a small magnetic "eld region. Note indeed that the MDD phase of bonding levels with angular momentum M "N(N!1)/2 is absent for 6 elecX trons. Although the MDD is stable in the 3 electron coupled dot system, the magnetic "eld region B"5.7 to 6.4 T is much smaller than in the single quantum dot case, where the MDD is the ground state from B"3.7 to 8 T. The energy di!erence between the ground state and the MDD can be very small. This is illustrated in Fig. 3 for N"6. The energy per electron is shown for the three phases which become the ground state around 5.5 T, together with the energy
Fig. 3. The ground state energy per particle for the 6 electron arti"cial molecule around 5 T, together with the energy per particle for the MDD phase. The energy per particle was subtracted.
per particle for the MDD phase. These curves are obtained by "xing the total angular momentum in our CSDFT program. Notice that they practically coincide around 5.5 T. Therefore, it is also possible that in an exact diagonalization the energies will shift in such a way that the MDD will become the ground state. However, what is most important is that the MDD, which is the ground state in a single dot in a large magnetic "eld window, is certainly not as stable in our two vertically coupled dots. The reason that the MDD in coupled dots is much less stable than in a single quantum dot is because of the much smaller exchange energy contribution, due to the potential barrier, which acts as a binding force for the MDD. Acknowledgements Part of this work is supported by the Flemish Science Foundation (FWO-Vlaanderen), the `Onderzoeksraad van de Universiteit Antwerpena and IUAP-IV. B. Partoens is a research assistant and F.M.P. A research director with the FWO-VI. Discussions with A. Gonzalez and D.G. Austing are gratefully acknowledged. References [1] L. Jacak, P. Hawrylak, A. WoH js, Quantum Dots, Springer, Berlin Heidelberg, 1998.
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[2] T. Chakraborty, Quantum Dots, Elsevier, Amsterdam, 1999. [3] S. Tarucha, D.G. Austing, T. Honda, R.J.V. der Hage, L.P. Kouwenhoven, Phys. Rev. Lett. 77 (1996) 3613. [4] M. Koskinen, M. Manninen, S.M. Reimann, Phys. Rev. Lett. 79 (1997) 1389. [5] K. Hirose, N.S. Wingreen, Phys. Rev. B 59 (1999) 4604. [6] S. Nagaraja, P. Matagne, V.-Y. Thean, J.-P. Leburton, Phys. Rev. B 60 (1999) 8759. [7] I.-H. Lee, V. Rao, R.M. Martin, J.-P. Leburton, Phys. Rev. B 57 (1997) 9035. [8] M. Stopa, Phys. Rev. B 54 (1996) 13767. [9] M. Ferconi, G. Vignale, Phys. Rev. B 50 (1994) 14722. [10] O. Ste!ens, M. Suhrke, Phys. Rev. Lett. 82 (1999) 3891. [11] M. Pi et al., Phys. Rev. B 57 (1998) 14783. [12] S.M. Reimann et al., Phys. Rev. Lett. 83 (1999) 3270. [13] B. Partoens, V.A. Schweigert, F.M. Peeters, 79 (1997) 3990. [14] B. Partoens, A. Matulis, F.M. Peeters, Phys. Rev. B 57 (1998) 13039.
[15] B. Partoens, A. Matulis, F.M. Peeters, Phys. Rev. B 59 (1999) 1617. [16] B. Partoens, F.M. Peeters, Phys. Rev. Lett. 84 (2000) 4433. [17] D.G. Austing, T. Honda, K. Muraki, Y. Tokura, S. Tarucha, Physica B 249}251 (1998) 206. [18] H. Imamura, P.A. Maksym, H. Aoki, Phys. Rev. B 53 (1996) 12613. [19] H. Imamura, P.A. Maksym, H. Aoki, Phys. Rev. B 59 (1999) 5817. [20] M. Rontani et al., Solid Stat. Comm. 112 (1999) 151. [21] Y. Tokura, D.G. Austing, S. Tarucha, J. Phys.: Condens. Matter 11 (1999) 6023. [22] G. Vignale, M. Rasolt, Phys. Rev. Lett. 59 (1987) 2360. [23] J.J. Palacios, P. Hawrylak, Phys. Rev. B 51 (1995) 1769. [24] T.H. Oosterkamp et al., Phys. Rev. Lett. 82 (1999) 2931. [25] S.-R.E. Yang, A.H. MacDonald, M.D. Johnson, Phys. Rev. Lett. 71 (1993) 3194. [26] M. Ferconi, G. Vignale, Phys. Rev. B 56 (1997) 12108. [27] C. de C. Chamon, X.G. Wen, Phys. Rev. B 49 (1997) 8227. [28] S.M. Reimann et al., Phys. Rev. Lett. 83 (1999) 3270.
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Physica B 298 (2001) 282}286
Two vertically coupled quantum dots in a magnetic "eld B. Partoens*, F.M. Peeters Departement Natuurkunde, Universiteit Antwerpen (UIA), Universiteitsplein 1, B-2610 Antwerpen, Belgium
Abstract We studied the ground state of two vertically coupled quantum dots as a function of the interdot distance in the presence of a perpendicular magnetic "eld. The theoretical approach is based on the current spin density functional theory. Increasing the interdot distance leads to the population of antibonding levels, resulting in molecule-type phases. However, for increasing magnetic "eld strengths, we "nd that electrons hop back from antibonding to bonding levels, which is a consequence of electron interactions. The maximum density droplet is much less stable than in a single quantum dot. 2001 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Density functional theory; Maximum density droplet
1. Introduction Quantum dots are small man-made objects in which electrons are con"ned in all three spatial directions like in real atoms [1,2]. Therefore, they are also called artixcial atoms. Due to the increased role of electron interactions these arti"cial atoms exhibit new physics which has no analogue in real atoms. Experimentally, these quantum dots can be probed through tunneling experiments [3] which give us information on the position of the energy levels: the separation between the measured tunneling peaks re#ects that the shell structure and Hund's rules determine the electronic properties. Moreover, these tunneling peaks shift in a magnetic "eld and clearly show transitions in the many-electron ground states as a result of the
* Corresponding author. Fax: #32-3-820-22-45. E-mail addresses: [email protected] (B. Partoens), [email protected] (F.M. Peeters).
competition between the kinetic energy and the Coulomb energy. All these single quantum dot properties were successfully reproduced using the density functional theory formalism [4}12]. When two such arti"cial atoms are placed on top of each other, i.e. two quantum dots which couple vertically, we have an artixcial molecule. In such a system, we can also control the interdot distance and consequently the interdot Coulomb interaction and the tunneling between the dots. In this way one can play with the competition between intradot and interdot correlations [13}16]. Recently, this system was experimentally realized by Austing et al. [17]. Exact diagonalizations of coupled dots in a magnetic "eld, which are limited to a small number of electrons, were performed in Refs. [18}21]. In this paper we show the results obtained using the current spin density functional theory (CSDFT) for the ground state phases for two vertically coupled quantum dots in a magnetic "eld. In comparison to Ref. [18,19], we focus on much shorter interdot distances.
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 3 1 9 - 2
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2. Theoretical model We consider two circularly symmetric dots, placed at a distance d vertically from each other. It is often a good approximation to assume a parabolic con"nement <(r)"m r/2 of frequency , with m the e!ective electron mass. In the z-direction the quantum dots are created by two coupled quantum wells, as schematically shown in Fig. 1. A homogeneous magnetic "eld B"Be is applied X perpendicular to the xy-plane in which the electrons are con"ned. A typical experimental value for the con"nement energy is "3 meV [3]. Be cause the energy scale in such systems is substantially smaller than in real atoms, an external magnetic "eld will have a more profound in#uence on the ground state of these arti"cial structures. For example, the magnetic energy " eB/mc equals this con"nement energy already for B"2 T, and therefore in typical laboratory experiments the magnetic "eld can be large. We used CSDFT, which is an extension of spin density functional theory, and which is able to account for the e!ect of strong magnetic "elds. For the details of CSDFT we refer to the original work of Vignale and Rasolt [22]. Here we recall only the Kohn}Sham equation. In a symmetric gauge, the external vector potential acting on the electrons is A"Bre /2. The ground state energy and the denF sity (r) are expressed in terms of a set of Kohn}Sham orbitals (r)"exp(!il) (r)Z(z), (1) HJN HJN which are the eigenstates of the z-component of the angular momentum !l, and satisfy the Kohn} Sham equation in CSDFT
R 1 R l R l ! # ! # ! 2m Rr r Rr r Rz 2 mr e l A #<(z)#< (r) # ! & 2 mc r #< (r)#< ] (r)Z(z)" (r)Z(z), & N HJN HJN HJN (2)
with the z-component of the spin, "eB/mc the cyclotron frequency, and "( #/4. A is the exchange-correlation vector potential and
Fig. 1. The inset shows the con"nement potential in the zdirection. The symbols are the calculated splitting between the lowest bonding and antibonding level and the curve is an exponential "t to it. The corresponding wavefunctions are shown schematically in the inset.
< the exchange-correlation scalar potential, < the intradot and < the interdot Hartree & & potential. The density in the z-direction is approximated by functions. For the material parameters we choose typical GaAs values for the e!ective mass m"0.067 m and the dielectric constant "12.4 This yields an e!ective Bohr radius of aH"9.79 nm. The strength of the external con"nement is set to " 5.78/N meV, where N is the total number of electrons in the two dot system. For di!erent N, this choice keeps the average electron density in each dot approximately constant, in this case roughly corresponding to a (two-dimensional) Wigner}Seitz radius r "1.75aH, as in a typical vertical quantum dot structure [17]. The con"nement in the z-direction consists of two coupled quantum wells as shown in Fig. 1. Due to the "nite barrier between both dots the lowest level in the z-direction is split into a symmetric bonding and an anti-symmetric antibonding level. For our numerical calculation we take the dimensions of the quantum wells corresponding to the experimental realization of Ref. [17]: ="120 As for the width of both dots, < "250 meV for the height of the barrier between the dots. The energy splitting between the lowest bonding and antibonding levels was then calculated numerically and could be "tted to an exponential decaying function
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
284
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
B. Partoens, F.M. Peeters / Physica B 298 (2001) 282}286
"22.86 exp(!d(As )/13.455) meV (solid curve in Fig. 1).
3. Phase diagrams The obtained ground state phase diagrams for 3 and 6 electrons are shown in Fig. 2 in the parameter space of the level splitting (or equivalently, the interdot distance d) and the magnetic "eld B. The levels are labeled by the three quantum numbers (S , M , I ): spin S , angular momentum M , X X X X X and an isospin quantum number I , which is the X di!erence between the electrons in the bonding level and those in the antibonding level, divided by 2. The isospin was "rst introduced by Palacios and Hawrylak [21,23] who concentrated on the high magnetic "eld case. In the phase diagram for 3 electrons (see Fig. 2(a)) the occupation of the corresponding e!ective single particle states are shown schematically: the single particle energy levels are E " (2n#l#1) (with n the radial quantum number), which due to the splitting of the energy level in the z-direction in a bonding and an antibonding level are shifted by an energy . To understand these phase diagrams let us see what one expects from the single particle picture. In a magnetic "eld the single particle energy levels of a two-dimensional harmonic oscillator are altered into E "(2n#l#l) !l /2. Notice that the degeneracy in the angular momentum l and !l is lifted. To predict when the molecule-type phases appear we have to compare the single particle level splitting of a 2D harmonic oscillator in a magnetic "eld " ( #/4! /2 with the level splitting between bonding and antibonding levels . As decreases with increasing magnetic "eld, while does not change, we expect that the bonding levels will stay occupied for larger interdot distances. On the calculated phase diagrams one can indeed see that the bonding levels stay longer occupied when the magnetic "eld increases, even much longer than one would expect from the single particle condition ". This is again a consequence of the electron interactions. As a magnetic "eld
Fig. 2. Phase diagrams for (a) 3 and (b) 6 electrons in the two coupled dots. The phases are labeled with the quantum numbers (S , M , I ). The insets in (a) show the single particle con"gura X X tions where the occupation of bonding (left side of slash) and of antibonding levels (right side of slash) are shown.
pushes the electrons to the center of the dot, the Coulomb repulsion increases. Therefore, at a certain magnetic "eld value it is energetically favorable to increase the angular momentum, which costs con"nement energy but reduces the Coulomb repulsion. The phase diagrams also show that in coupled dots this increase in angular momentum is often realized by isospin #ips in which an electron hops back from an antibonding to a bonding level (corresponding with an increasing isospin).
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4. Stability of the maximum density droplet The study of the maximum density droplet (MDD) phase in a single quantum dot attracted much interest [24}28]. This state is the most dense spin-polarized con"guration and it is the ground state in a rather large magnetic "eld region. In the MDD the electrons occupy adjacent orbitals with consecutive angular momentum. The stability of the MDD is determined by the competition between the kinetic energy and the Coulomb repulsion between the electrons. As all electrons are spin-polarized, the exchange also acts as a binding force. The kinetic energy would favor the MDD up to in"nite "elds. However, increasing the magnetic "eld shrinks the electron orbits and thus increases the Coulomb repulsion. Therefore, the Coulomb interaction favors a di!use occupation. At some threshold magnetic "eld, the direct Coulomb interaction has become so large that the MDD breaks apart into a lower density droplet, so increasing the total angular momentum. Similarly, we expect a spin-polarized state for su$ciently large magnetic "elds for two coupled quantum dots, but now with the additional complication that the electrons populate bonding and antibonding levels. The electrons in the antibonding states have smaller angular momenta than the bonding states as they are less populated. Thus the angular momentum can be increased for increasing magnetic "eld through a one-by-one depopulation of antibonding states, until the MDD is formed where only bonding levels are populated. However, we "nd that the MDD of bonding levels is only the ground state for the 3 electron case in a small magnetic "eld region. Note indeed that the MDD phase of bonding levels with angular momentum M "N(N!1)/2 is absent for 6 elecX trons. Although the MDD is stable in the 3 electron coupled dot system, the magnetic "eld region B"5.7 to 6.4 T is much smaller than in the single quantum dot case, where the MDD is the ground state from B"3.7 to 8 T. The energy di!erence between the ground state and the MDD can be very small. This is illustrated in Fig. 3 for N"6. The energy per electron is shown for the three phases which become the ground state around 5.5 T, together with the energy
Fig. 3. The ground state energy per particle for the 6 electron arti"cial molecule around 5 T, together with the energy per particle for the MDD phase. The energy per particle was subtracted.
per particle for the MDD phase. These curves are obtained by "xing the total angular momentum in our CSDFT program. Notice that they practically coincide around 5.5 T. Therefore, it is also possible that in an exact diagonalization the energies will shift in such a way that the MDD will become the ground state. However, what is most important is that the MDD, which is the ground state in a single dot in a large magnetic "eld window, is certainly not as stable in our two vertically coupled dots. The reason that the MDD in coupled dots is much less stable than in a single quantum dot is because of the much smaller exchange energy contribution, due to the potential barrier, which acts as a binding force for the MDD. Acknowledgements Part of this work is supported by the Flemish Science Foundation (FWO-Vlaanderen), the `Onderzoeksraad van de Universiteit Antwerpena and IUAP-IV. B. Partoens is a research assistant and F.M.P. A research director with the FWO-VI. Discussions with A. Gonzalez and D.G. Austing are gratefully acknowledged. References [1] L. Jacak, P. Hawrylak, A. WoH js, Quantum Dots, Springer, Berlin Heidelberg, 1998.
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