- Email: [email protected]
Deformed molecular states of quantum dots in strong magnetic fields
Physica B 298 (2001) 246}253
Deformed molecular states of quantum dots in strong magnetic "elds P.A. Maksym* Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK
Abstract The e!ect of a shearing electric "eld on molecular states of electrons in quantum dots in strong magnetic "elds is investigated theoretically. The electron density in circularly symmetric dots is circularly symmetric but the pair correlation function has peaks whose positions have the same symmetry as the classical energy minimum. For small numbers of electrons this is analogous to a rotating ring of electrons. The shearing electric "eld breaks the circular symmetry and stretches the ring, leading to states with linear symmetry. This transition is found to have a strong e!ect on the addition energy of a dot and can be either sudden or continuous, depending on the value of the magnetic "eld. It may be possible to use this e!ect as an experimental probe of the physical nature of dot states. 2001 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Nanostructure; Electron correlation
1. Introduction Molecular states in quantum dots have been studied theoretically for some time and ordered states, analogous to Wigner crystals, have been predicted to occur in large dots at zero magnetic "eld [1,2] and in smaller dots in a strong magnetic "eld [3}5]. Although the ground states and low lying excitations of molecular states in a strong "eld are relatively well understood (see Maksym et al. [6] for a review), experimental studies of molecular states remain a challenge. One particular di$culty is that because of the generalised Kohn theorem [7,8] far infra-red absorption spectra of circular dots are insensitive to electron correlation
* Tel.: #44-116-252-3579; fax: #44-116-252-2770. E-mail address: [email protected] (P.A. Maksym).
so it is di$cult to probe the physical nature of dot states spectroscopically. The aim of this work is to show that it might be possible to obtain more information by studying the response of the quantum states to a shearing electric "eld applied to a dot. It has recently become possible to fabricate electrically deformable quantum dots [9] so it should be possible to study the e!ects described here experimentally. Physically, electron correlation in few-electron dots (less than 6 electrons) favours ring-like arrangements of electrons. A system with non-zero angular momentum corresponds to a rotating ring and its quantum states are well approximated by the rotational}vibrational states of the corresponding molecule. A shearing electric "eld distorts the ring and when the "eld is strong enough linear correlation becomes favourable. As the "eld increases further the state separates into two pieces,
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 1 - 8
P.A. Maksym / Physica B 298 (2001) 246}253
corresponding to `fracturea of the electron molecule. Numerical calculations of the addition energy as a function of electric "eld show that it changes dramatically as the transition occurs. Further, the nature of the transition can be tuned by adjusting the magnetic "eld. If the magnetic "eld chosen is such that the initial ring-like state has the opposite parity to the fractured state the transition is sudden and is accompanied by a slope discontinuity in the electro-chemical potential di!erence. Otherwise the transition is continuous and the addition energy changes smoothly. These e!ects are clearly associated with molecular states and could perhaps be used as an experimental probe of them. The physical origin of these e!ects is the reduction of symmetry caused by the presence of the shearing "eld and the introduction of linear symmetry is particularly important. Similar e!ects have been found to occur when the dot is deformed in other ways, for example by compression along one axis, and a comparative study of various dot deformations will be presented in a separate publication. The aim here is to illustrate these e!ects by focussing on shear, a deformation for which the physics is particularly clear. This work opens with a review of molecular state properties (Section 2). This is followed by results on the fracture of 3, 4 and 5-electron systems, which have been obtained by exact diagonalization of the Hamiltonian in the zeroth Landau level (Section 3). In addition, results of classical Monte-Carlo simulations at zero magnetic "eld are given (Section 4). These results provide additional insight into the way the fracture occurs and suggest that the structure in the addition energy might be even richer in systems with more electrons. 2. Molecular states The pair correlation function provides direct evidence for the molecular nature of ground states of interacting electrons in circular dots in strong magnetic "elds. For dots with a parabolic con"ning potential this function is de"ned as the ground state expectation value (2) P(r, r )" N(N!1)
(r !r)(r !r ) , G H G$H
247
where N is the number of electrons, is a length parameter de"ned by " /(2mH) and is related to the cyclotron frequency, and the con "nement frequency, by " #/4. Phys ically, the correlation function is proportional to the probability of "nding an electron at r given that there is one at r . This function is illustrated in Fig. 1. The distance r is chosen to coincide with the maximum of the electron density and the resulting function of r is plotted for various electron numbers. The black spots indicate r . The length unit is 1.89 nm and the ground state angular momentum J and magnetic "eld at which each state occurs are given in the "gure. The con"nement energy, , is 4 meV. For each number of elec trons the correlation function has a highly symmetric form. Its shape is linear for 2 electrons and ring-like for up to 5 electrons: the spot and the peaks clearly form an N-fold polygon. For more than 5 electrons there is structure inside the ring and for each electron number the symmetry of the correlation function coincides with the symmetry of the classical energy minimum [10]. This demonstrates the molecular nature of the ground states in the strong "eld limit. Angular momentum is a good quantum number for circular dots and each form of molecular symmetry is associated with particular values of the ground state angular momentum, which are sometimes called magic numbers. For example, spin polarised ground states that correspond to the triangular symmetry of the 3 electron molecule must have angular momenta that are a multiple of 3, and in general, the angular momentum that corresponds to n-fold symmetry is J,0 mod n when n is odd and J,0 mod n#n/2 when n is even [4]. The reason for the connection between the angular momentum and the symmetry of the correlation is that electron molecular states must satisfy the Pauli exclusion principle and have correlations with molecular symmetry. Both conditions are satis"ed only when the angular momentum is magic. The connection between molecular symmetry and angular momentum has been investigated by various theoretical methods [4}6,11}13] and is known to occur in both single and vertically coupled dots [14,15] with circular symmetry. One particularly simple theoretical approach, although
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Fig. 1. Pair correlation functions for 2}7 spin polarised electrons.
one that only gives a su$cient condition, is to assume that the wave function is non-zero when the electron positions coincide with an equilibrium con"guration of the molecule and study the e!ect of symmetry operations on the resulting function. For example, in the spin polarised 3-electron case the equilibrium positions are the corners of an equilateral triangle. Rotation by 2/3 changes the phase of the wave function by exp(2iJ/3) in this case. But this transformation is equivalent to a cyclic permutation of all 3 electrons. This permutation has even parity so the two transformations are compatible only when exp(2iJ/3)"1, that is J,0 mod 3, one of the magic values mentioned in the previous paragraph. The symmetry argument only restricts the total angular momentum to an allowed set of values. The actual value that occurs at a particular magnetic "eld is the value that minimises the total energy. As the "eld increases it compresses the wave function. This increases the Coulomb energy and when the Coulomb energy increase is su$ciently large the system responds by increasing its angular momentum*this is energetically favourable because an increase of angular momentum is always accom-
Fig. 2. Ground state angular momentum of 3 electrons.
panied by an increase in the spatial extent of the quantum state [16]. Thus, the ground state angular momentum always increases with magnetic "eld but it is always one of the magic values. For example, in the case of 3 spin-polarised electrons the ground state angular momentum is always a multiple of 3. This is illustrated in Fig. 2 for 3 spin-polarised electrons in a GaAs dot with
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"4 meV. The change of angular momentum with magnetic "eld is very relevant to the response of the dot to a shearing electric "eld. The ground state parity depends on whether the angular momentum is odd or even. Therefore, it is possible to deform states of either odd or even parity and this can result in a radically di!erent response of the dot to shear (see next section). Further evidence of the molecular nature of the strong "eld states comes from the form of the low lying excitations. Molecules have excitations that are mostly of a rotational or vibrational nature, although there is some coupling between these two forms of motion. Similar excitations are found in electron molecular states of dots. In fact, there is very good agreement between excitation energies calculated by exact diagonalization of the Hamiltonian, with inclusion of higher Landau levels, and a theory based on a harmonic expansion of the Hamiltonian in the moving reference frame normally used to study real molecules (the Eckart frame) [4}6]. Thus, the picture of a rotating}vibrating electron molecule gives an intuitive way of understanding the physics in the regime in which it applies. A similar picture is helpful for understanding the e!ect of shear.
3. E4ect of shear on quantum states An ideal shearing electric "eld has the form F"Fi, y'0, F"!Fi, y(0, where i is a unit vector in the x direction. Physically, the important features of this "eld are that opposite sides of the dot are subjected to "elds in opposite directions and that the sign of the "eld changes over a relatively short distance in the centre of the dot. One way of realising this situation experimentally might be to use a mesa dot surrounded by a split gate. Dots surrounded by a gate split into four quadrants have been made [9] so that it would be possible to apply "elds in opposite directions to opposite sides of a dot. In this situation, there would be some fringing "elds at the edges of the dot but quantum states located inside the dot would be subjected to shear. To avoid the complications of the fringing "elds in a real dot the electric "eld considered in this work is derived from the model potential
249
!Fx tanh\(y/a), which gives nearly uniform "elds in opposite directions on opposite sides of the dot together with a small region near the x axis where the "eld, of magnitude F, changes sign. The width of this region, de"ned as the zone in which 95% of the changes take place, is about 5a. The e!ect of changing the width parameter a, was investigated with classical Monte-Carlo calculations of the minimum energy con"guration, similar to those described in Section 4. The shapes of the minimum energy con"gurations were found to be insensitive to a provided the transition region did not extend over an appreciable fraction of the dot and a"0.5 nm was used to obtain the results in Section 4. The e!ect of the shearing "eld is to stretch the dot in the x direction and this distorts the ring-like molecular states. When the "eld is strong enough, a linear con"guration becomes energetically favourable and the form of the quantum ground state becomes very di!erent from its form at zero electric "eld. Angular momentum is not a good quantum number when the shearing "eld is non-zero but the potential associated with the shearing "eld is invariant under inversion so parity is a good quantum number at all values of the shearing "eld. The symmetry changes induced by shear lead to changes in the form of the ground state but the changes are sudden only if they are accompanied by a change in the parity of the ground state. The important physics is not that the dot is sheared but that its symmetry is reduced in such a way that a linear electron con"guration is favourable when the deformation is high. Transitions, similar to those that occur as a result of shear also occur in other situations where a linear con"guration is favourable, for example when the dot is compressed in the y axis, details will be presented elsewhere. The ground state parity of molecular states in the linear con"guration can be found by the same methods that are used to study the circular con"guration. For example, the equilibrium con"guration of a 3-electron linear system has one electron at the origin and the other two electrons are diametrically opposite each other. The two diametrically opposite electrons interchange when the con"guration is inverted. This is equivalent to a single transposition so the linear 3-electron state
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must have odd parity. Similarly, 4 and 5-electron linear ground states have even parity. These parity assignments are independent of the "eld strengths. In contrast, the parity of ring-like states is dependent on both the magnetic "eld and the electron number. Thus, two types of evolution are possible when the shearing "eld is increased. In one case the initial ring-like state has the same parity as the linear state that occurs in the strong electric "eld regime. In the other case the initial and strong electric "eld states have opposite parity. Energy levels of the same parity cannot cross. So when the ring-like and linear states have the same parity the ground state can evolve continuously. But when the ring-like and linear states have opposite parity, level crossings occur when the "eld increases and there are sudden transitions in the form of the ground state. In principle, more complicated evolution could take place with several structural changes occuring as the limit of high deformation is approached. The parity rules apply in this case as well: if the ring-like and linear states have the same parity the number of level crossings is either zero or even but if these states are of opposite parity the number level crossings is odd. These transitions are found to a!ect the change in the electro-chemical potential, E !2E # ,> , E , where E is the ground state energy of the ,\ , N-electron system. The ground state energies have been computed by exact diagonalization in a Fock}Darwin basis [5,6,8]. The con"ning potential of the dot is taken to be parabolic with
"4 meV. The inverse hyperbolic tangent in the model potential is approximated by a step function so the potential matrix elements can be found analytically and this approximation is good when a is small. The classical Monte-Carlo simulations have shown that the form of the minimum energy con"gurations is insensitive to a so the approximation contains all the essential physics. Because of the lack of circular symmetry a fairly large basis is needed to ensure adequate convergence. All many electron basis states up to J"55 are taken into account. The estimated convergence of ground state energies within the zeroth Landau level for F"0.1 V m\ (the worst case) is better than 0.1% and the estimated convergence of electro-chemical potential di!erences is better than 1%. The inclu-
Fig. 3. Electro-chemical potential di!erence for sudden (B"17.5 T, solid line) and continuous (B"20.5 T, dashed line) transitions of the 4-electron system.
sion of one extra Landau level changes the electrochemical potential di!erence by about 0.6%. The many electron basis states are classi"ed according to parity to reduce the basis size by a factor of about 2 and this leads to useful savings in both CPU time and memory [17]. The maximum basis size is about 50,000 for the zeroth Landau level calculation and about 120,000 when one extra Landau level is included. All the calculations are done for spin polarised states. Results for the electro-chemical potential di!erence for N"4 as a function of applied electric "eld are shown in Fig. 3. When the magnetic "eld is 20.5 T the electro-chemical potential di!erence evolves smoothly but there are clear breaks at about 0.35 and 0.65 V m\ when the magnetic "eld is 17.5 T. The 20.5 T "eld case is one where the ring-like and linear states have the same parity for N"3, 4, 5 so no structure appears in the electrochemical potential di!erence. The structure in the 17.5 T case can be understood by considering the parities of the ring-like and linear states. All the ring-like states of the 4-electron system have even parity and this is the same as the parity of its linear states. But the parity of the ring-like states of the 3-electron system depends on magnetic "eld. At 17.5 T, the ground state angular momentum is J"12 (see Fig. 2) which corresponds to even parity but the linear state has odd parity. The structure in
P.A. Maksym / Physica B 298 (2001) 246}253
Fig. 4. Electro-chemical potential di!erence for 16 interacting classical particles.
the electro-chemical potential di!erence near F"0.65 V m\ corresponds to the crossing of the opposite parity levels of the 3-electron system. Similarly, the ground state of the 5-electron system at zero electric "eld has J"35, which is odd parity, and the structure in the electro-chemical potential di!erence near F"0.35 V m\ corresponds to an odd to even parity transition of the 5-electron system. The evolution of the electron density with electric "eld clearly demonstrates how the parity transitions are associated with the molecular nature of the electron states. Fig. 4 shows 3-electron densities at 17.5 T and zero electric "eld, a weak electric "eld, 0.03 V m\, "elds on either size of the transition, 0.06 and 0.07 V m\, and a strong electric "eld, 0.1 V m\. The circularly symmetric density at zero electric "eld corresponds to a ringlike molecular state. As the "eld increases the ring distorts and then fractures at the transition. Further increases of electric "eld then separate the three fragments leading to a state that is clearly linear when the electric "eld is 0.1 V m\.
4. E4ect of shear on classical states Quantum mechanical electro-chemical potential di!erences can only be calculated reliably for electron numbers less than about 6 because the lack of
251
symmetry makes exact diagonalization prohibitively expensive. Intuitively, however, it seems likely that structural transitions of a more complicated and richer nature would occur in larger systems and this possibility is investigated by Monte-Carlo simulation of classical systems of interacting charged particles in a parabolic potential and a shearing electric "eld. The mean potential energy of the N-particle system is found by Monte-Carlo with simulated annealing and in addition further studies have been done where the electric "eld applied to the simulated system has been swept up and down. No magnetic "eld is included but this is not an issue as the main aim of the calculation is to study how the form of the classical energy minimum evolves with electric "eld. Minima with similar forms are expected to occur in molecular states in the strong magnetic "eld regime. The parameters of the classical system correspond to those of the parabolic quantum dot: "4 meV, the electric "eld range is 0}0.1 V m\, the particle charge is !e and the temperature is k ¹"10\ meV. Fig. 5 shows results for the mean potential energy di!erence, < !2< #< , of a 16-par,> , ,\ ticle system. The statistical error is smaller than the symbol size but there are a few points that do not fall onto the line (for example, just below 0.1 V m\). This re#ects a tendency for the system to become trapped in metastable states which is present to a small extent although the data was averaged over a very large number of Monte-Carlo steps (10). Breaks in the energy di!erence are clearly visible and the structure is generally richer than for the small quantum system, for example there is a pronounced change in the sign of the slope of the energy di!erence close to F"0.02 V m\. Results of additional runs in which the electric "eld was swept during the simulation show strong hysteresis e!ects in which the energy di!erence on the down sweep follows a different curve from the di!erence on the up sweep. The hysteresis disappears when the temperature is raised to k ¹"0.2 meV. This is on the high temperature side of an order}disorder transition that is known to occur in the classical system [18]. Thus, the presence or absence of structure in the energy di!erence appears to be sensitive to the phase of the
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P.A. Maksym / Physica B 298 (2001) 246}253
Fig. 5. Evolution of electron density for 3 interacting electrons. The contour plots cover an area of 10;10.
Fig. 6. Minimum energy con"guration of 16 con"ned classical particles. Each plot covers an area of 80;80 nm.
few particle system and this leads to the tentative conclusion that experimental studies of the addition energy of deformed dots might give information about the phase diagram of con"ned few-electron systems. The evolution of the 16-particle con"gurations with electric "eld is particularly rich (Fig. 6) and the structural changes are connected with the structure of the energy di!erence as a function of electric "eld. At zero "eld, the con"guration consists of two concentric rings with a particle at the centre. This form persists up to about 0.02 V m\ when the central particle disappears and the central ring becomes elongated. The next structural change occurs between 0.062 and 0.064 V m\ when the elongated central ring is replaced by a line of 6 particles. Further increase of "eld then results in separation into two pieces, each containing 8 particles. Analogous structural changes occur in the 15 and 17particle systems and it is not possible to ascribe the structure in the energy di!erence around
0.06 V m\ unambiguously to a system with a one particular number of particles. However, the small jump in the energy di!erence around 0.03 V m\ is probably connected with the disappearance of the central ring of the 15-electron system.
5. Conclusion The e!ect of dot shape deformation on molecular states of interacting electrons in quantum dots in strong magnetic "elds has been investigated. A shearing electric "eld in systems of 3, 4, and 5 electrons induces a structural transition in which the quantum ground state changes from one based on a ring-like con"guration to one based on a linear con"guration. This change is accompanied by dramatic and in principle measurable changes in the addition energy. A particularly interesting feature of the transition is that its nature can be changed from sudden to continuous by changing
P.A. Maksym / Physica B 298 (2001) 246}253
the magnetic "eld. Similar changes are predicted to occur in larger systems of interacting classical charged particles and this suggests that richer structure might be observable in the addition energy of quantum dots with larger numbers of electrons when their shape is deformed. The structural transitions are a consequence of the reduction in the symmetry of the dot and similar transitions have been found to occur when a dot is compressed. It may be possible to use these e!ects to probe the physical nature of quantum states and phase transitions in correlated few-electron systems con"ned in quantum dots. The strong magnetic "eld regime could be particularly advantageous for experimental studies because molecular states should be found in relatively small dots and hence, there should be less sensitivity to impurity e!ects.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Acknowledgements I would like to thank Prof. S. Tarucha for asking a question which led to the study of the e!ect of shear and I am grateful to Prof. H. Aoki for hospitality at the University of Tokyo where this work was started. This work was supported by the European Union and Leicester University.
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[14] [15] [16] [17] [18]
G.W. Bryant, Phys. Rev. Lett. 59 (1987) 1140. W. HaK usler, B. Kramer, J. Ma\ sek, Z. Phys. B 85 (1991) 435. R.B. Laughlin, Phys. Rev. B 27 (1983) 3383. P.A. Maksym, Physica B 184 (1993) 385. P.A. Maksym, Phys. Rev. B 53 (1996) 10871. P.A. Maksym, H. Imamura, G.P. Mallon, H. Aoki, J. Phys. Condens. Matter 12 (2000) R299. L. Brey, N.F. Johnson, B.I. Halperin, Phys. Rev. B 40 (1989) 10647. P.A. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. D.G. Austing, T. Honda, S. Tarucha, Semicond. Sci. Technol. 12 (1997) 631. F. Bolton, U. RoK ssler, Superlatt. Microstruct. 13 (1993) 139. W.Y. Ruan, Y.Y. Liu, C.G. Bao, Z.O. Zhang, Phys. Rev. B 51 (1995) 7942. T. Seki, Y. Kuramoto, T. Nishino, J. Phys. Soc. Japan 65 (1996) 3945. C.G. Bao, W.Y. Ruan, Y.Y. Liu, Phys. Rev. B 53 (1996) 10820. H. Imamura, P.A. Maksym, H. Aoki, Phys. Rev. B 53 (1996) 12613. H. Imamura, P.A. Maksym, H. Aoki, Phys. Rev. B 59 (1999) 5817. P.A. Maksym, L.D. Hallam, J. Weis, Physica B 212 (1995) 213. P.A. Maksym, Physica B 249 (1998) 233. V.M. Bedanov, F.M. Peeters, Phys. Rev. B 49 (1994) 2667.
Deformed molecular states of quantum dots in strong magnetic "elds P.A. Maksym* Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK
Abstract The e!ect of a shearing electric "eld on molecular states of electrons in quantum dots in strong magnetic "elds is investigated theoretically. The electron density in circularly symmetric dots is circularly symmetric but the pair correlation function has peaks whose positions have the same symmetry as the classical energy minimum. For small numbers of electrons this is analogous to a rotating ring of electrons. The shearing electric "eld breaks the circular symmetry and stretches the ring, leading to states with linear symmetry. This transition is found to have a strong e!ect on the addition energy of a dot and can be either sudden or continuous, depending on the value of the magnetic "eld. It may be possible to use this e!ect as an experimental probe of the physical nature of dot states. 2001 Elsevier Science B.V. All rights reserved. Keywords: Quantum dots; Nanostructure; Electron correlation
1. Introduction Molecular states in quantum dots have been studied theoretically for some time and ordered states, analogous to Wigner crystals, have been predicted to occur in large dots at zero magnetic "eld [1,2] and in smaller dots in a strong magnetic "eld [3}5]. Although the ground states and low lying excitations of molecular states in a strong "eld are relatively well understood (see Maksym et al. [6] for a review), experimental studies of molecular states remain a challenge. One particular di$culty is that because of the generalised Kohn theorem [7,8] far infra-red absorption spectra of circular dots are insensitive to electron correlation
* Tel.: #44-116-252-3579; fax: #44-116-252-2770. E-mail address: [email protected] (P.A. Maksym).
so it is di$cult to probe the physical nature of dot states spectroscopically. The aim of this work is to show that it might be possible to obtain more information by studying the response of the quantum states to a shearing electric "eld applied to a dot. It has recently become possible to fabricate electrically deformable quantum dots [9] so it should be possible to study the e!ects described here experimentally. Physically, electron correlation in few-electron dots (less than 6 electrons) favours ring-like arrangements of electrons. A system with non-zero angular momentum corresponds to a rotating ring and its quantum states are well approximated by the rotational}vibrational states of the corresponding molecule. A shearing electric "eld distorts the ring and when the "eld is strong enough linear correlation becomes favourable. As the "eld increases further the state separates into two pieces,
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 1 - 8
P.A. Maksym / Physica B 298 (2001) 246}253
corresponding to `fracturea of the electron molecule. Numerical calculations of the addition energy as a function of electric "eld show that it changes dramatically as the transition occurs. Further, the nature of the transition can be tuned by adjusting the magnetic "eld. If the magnetic "eld chosen is such that the initial ring-like state has the opposite parity to the fractured state the transition is sudden and is accompanied by a slope discontinuity in the electro-chemical potential di!erence. Otherwise the transition is continuous and the addition energy changes smoothly. These e!ects are clearly associated with molecular states and could perhaps be used as an experimental probe of them. The physical origin of these e!ects is the reduction of symmetry caused by the presence of the shearing "eld and the introduction of linear symmetry is particularly important. Similar e!ects have been found to occur when the dot is deformed in other ways, for example by compression along one axis, and a comparative study of various dot deformations will be presented in a separate publication. The aim here is to illustrate these e!ects by focussing on shear, a deformation for which the physics is particularly clear. This work opens with a review of molecular state properties (Section 2). This is followed by results on the fracture of 3, 4 and 5-electron systems, which have been obtained by exact diagonalization of the Hamiltonian in the zeroth Landau level (Section 3). In addition, results of classical Monte-Carlo simulations at zero magnetic "eld are given (Section 4). These results provide additional insight into the way the fracture occurs and suggest that the structure in the addition energy might be even richer in systems with more electrons. 2. Molecular states The pair correlation function provides direct evidence for the molecular nature of ground states of interacting electrons in circular dots in strong magnetic "elds. For dots with a parabolic con"ning potential this function is de"ned as the ground state expectation value (2) P(r, r )" N(N!1)
(r !r)(r !r ) , G H G$H
247
where N is the number of electrons, is a length parameter de"ned by " /(2mH) and is related to the cyclotron frequency, and the con "nement frequency, by " #/4. Phys ically, the correlation function is proportional to the probability of "nding an electron at r given that there is one at r . This function is illustrated in Fig. 1. The distance r is chosen to coincide with the maximum of the electron density and the resulting function of r is plotted for various electron numbers. The black spots indicate r . The length unit is 1.89 nm and the ground state angular momentum J and magnetic "eld at which each state occurs are given in the "gure. The con"nement energy, , is 4 meV. For each number of elec trons the correlation function has a highly symmetric form. Its shape is linear for 2 electrons and ring-like for up to 5 electrons: the spot and the peaks clearly form an N-fold polygon. For more than 5 electrons there is structure inside the ring and for each electron number the symmetry of the correlation function coincides with the symmetry of the classical energy minimum [10]. This demonstrates the molecular nature of the ground states in the strong "eld limit. Angular momentum is a good quantum number for circular dots and each form of molecular symmetry is associated with particular values of the ground state angular momentum, which are sometimes called magic numbers. For example, spin polarised ground states that correspond to the triangular symmetry of the 3 electron molecule must have angular momenta that are a multiple of 3, and in general, the angular momentum that corresponds to n-fold symmetry is J,0 mod n when n is odd and J,0 mod n#n/2 when n is even [4]. The reason for the connection between the angular momentum and the symmetry of the correlation is that electron molecular states must satisfy the Pauli exclusion principle and have correlations with molecular symmetry. Both conditions are satis"ed only when the angular momentum is magic. The connection between molecular symmetry and angular momentum has been investigated by various theoretical methods [4}6,11}13] and is known to occur in both single and vertically coupled dots [14,15] with circular symmetry. One particularly simple theoretical approach, although
248
P.A. Maksym / Physica B 298 (2001) 246}253
Fig. 1. Pair correlation functions for 2}7 spin polarised electrons.
one that only gives a su$cient condition, is to assume that the wave function is non-zero when the electron positions coincide with an equilibrium con"guration of the molecule and study the e!ect of symmetry operations on the resulting function. For example, in the spin polarised 3-electron case the equilibrium positions are the corners of an equilateral triangle. Rotation by 2/3 changes the phase of the wave function by exp(2iJ/3) in this case. But this transformation is equivalent to a cyclic permutation of all 3 electrons. This permutation has even parity so the two transformations are compatible only when exp(2iJ/3)"1, that is J,0 mod 3, one of the magic values mentioned in the previous paragraph. The symmetry argument only restricts the total angular momentum to an allowed set of values. The actual value that occurs at a particular magnetic "eld is the value that minimises the total energy. As the "eld increases it compresses the wave function. This increases the Coulomb energy and when the Coulomb energy increase is su$ciently large the system responds by increasing its angular momentum*this is energetically favourable because an increase of angular momentum is always accom-
Fig. 2. Ground state angular momentum of 3 electrons.
panied by an increase in the spatial extent of the quantum state [16]. Thus, the ground state angular momentum always increases with magnetic "eld but it is always one of the magic values. For example, in the case of 3 spin-polarised electrons the ground state angular momentum is always a multiple of 3. This is illustrated in Fig. 2 for 3 spin-polarised electrons in a GaAs dot with
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"4 meV. The change of angular momentum with magnetic "eld is very relevant to the response of the dot to a shearing electric "eld. The ground state parity depends on whether the angular momentum is odd or even. Therefore, it is possible to deform states of either odd or even parity and this can result in a radically di!erent response of the dot to shear (see next section). Further evidence of the molecular nature of the strong "eld states comes from the form of the low lying excitations. Molecules have excitations that are mostly of a rotational or vibrational nature, although there is some coupling between these two forms of motion. Similar excitations are found in electron molecular states of dots. In fact, there is very good agreement between excitation energies calculated by exact diagonalization of the Hamiltonian, with inclusion of higher Landau levels, and a theory based on a harmonic expansion of the Hamiltonian in the moving reference frame normally used to study real molecules (the Eckart frame) [4}6]. Thus, the picture of a rotating}vibrating electron molecule gives an intuitive way of understanding the physics in the regime in which it applies. A similar picture is helpful for understanding the e!ect of shear.
3. E4ect of shear on quantum states An ideal shearing electric "eld has the form F"Fi, y'0, F"!Fi, y(0, where i is a unit vector in the x direction. Physically, the important features of this "eld are that opposite sides of the dot are subjected to "elds in opposite directions and that the sign of the "eld changes over a relatively short distance in the centre of the dot. One way of realising this situation experimentally might be to use a mesa dot surrounded by a split gate. Dots surrounded by a gate split into four quadrants have been made [9] so that it would be possible to apply "elds in opposite directions to opposite sides of a dot. In this situation, there would be some fringing "elds at the edges of the dot but quantum states located inside the dot would be subjected to shear. To avoid the complications of the fringing "elds in a real dot the electric "eld considered in this work is derived from the model potential
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!Fx tanh\(y/a), which gives nearly uniform "elds in opposite directions on opposite sides of the dot together with a small region near the x axis where the "eld, of magnitude F, changes sign. The width of this region, de"ned as the zone in which 95% of the changes take place, is about 5a. The e!ect of changing the width parameter a, was investigated with classical Monte-Carlo calculations of the minimum energy con"guration, similar to those described in Section 4. The shapes of the minimum energy con"gurations were found to be insensitive to a provided the transition region did not extend over an appreciable fraction of the dot and a"0.5 nm was used to obtain the results in Section 4. The e!ect of the shearing "eld is to stretch the dot in the x direction and this distorts the ring-like molecular states. When the "eld is strong enough, a linear con"guration becomes energetically favourable and the form of the quantum ground state becomes very di!erent from its form at zero electric "eld. Angular momentum is not a good quantum number when the shearing "eld is non-zero but the potential associated with the shearing "eld is invariant under inversion so parity is a good quantum number at all values of the shearing "eld. The symmetry changes induced by shear lead to changes in the form of the ground state but the changes are sudden only if they are accompanied by a change in the parity of the ground state. The important physics is not that the dot is sheared but that its symmetry is reduced in such a way that a linear electron con"guration is favourable when the deformation is high. Transitions, similar to those that occur as a result of shear also occur in other situations where a linear con"guration is favourable, for example when the dot is compressed in the y axis, details will be presented elsewhere. The ground state parity of molecular states in the linear con"guration can be found by the same methods that are used to study the circular con"guration. For example, the equilibrium con"guration of a 3-electron linear system has one electron at the origin and the other two electrons are diametrically opposite each other. The two diametrically opposite electrons interchange when the con"guration is inverted. This is equivalent to a single transposition so the linear 3-electron state
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must have odd parity. Similarly, 4 and 5-electron linear ground states have even parity. These parity assignments are independent of the "eld strengths. In contrast, the parity of ring-like states is dependent on both the magnetic "eld and the electron number. Thus, two types of evolution are possible when the shearing "eld is increased. In one case the initial ring-like state has the same parity as the linear state that occurs in the strong electric "eld regime. In the other case the initial and strong electric "eld states have opposite parity. Energy levels of the same parity cannot cross. So when the ring-like and linear states have the same parity the ground state can evolve continuously. But when the ring-like and linear states have opposite parity, level crossings occur when the "eld increases and there are sudden transitions in the form of the ground state. In principle, more complicated evolution could take place with several structural changes occuring as the limit of high deformation is approached. The parity rules apply in this case as well: if the ring-like and linear states have the same parity the number of level crossings is either zero or even but if these states are of opposite parity the number level crossings is odd. These transitions are found to a!ect the change in the electro-chemical potential, E !2E # ,> , E , where E is the ground state energy of the ,\ , N-electron system. The ground state energies have been computed by exact diagonalization in a Fock}Darwin basis [5,6,8]. The con"ning potential of the dot is taken to be parabolic with
"4 meV. The inverse hyperbolic tangent in the model potential is approximated by a step function so the potential matrix elements can be found analytically and this approximation is good when a is small. The classical Monte-Carlo simulations have shown that the form of the minimum energy con"gurations is insensitive to a so the approximation contains all the essential physics. Because of the lack of circular symmetry a fairly large basis is needed to ensure adequate convergence. All many electron basis states up to J"55 are taken into account. The estimated convergence of ground state energies within the zeroth Landau level for F"0.1 V m\ (the worst case) is better than 0.1% and the estimated convergence of electro-chemical potential di!erences is better than 1%. The inclu-
Fig. 3. Electro-chemical potential di!erence for sudden (B"17.5 T, solid line) and continuous (B"20.5 T, dashed line) transitions of the 4-electron system.
sion of one extra Landau level changes the electrochemical potential di!erence by about 0.6%. The many electron basis states are classi"ed according to parity to reduce the basis size by a factor of about 2 and this leads to useful savings in both CPU time and memory [17]. The maximum basis size is about 50,000 for the zeroth Landau level calculation and about 120,000 when one extra Landau level is included. All the calculations are done for spin polarised states. Results for the electro-chemical potential di!erence for N"4 as a function of applied electric "eld are shown in Fig. 3. When the magnetic "eld is 20.5 T the electro-chemical potential di!erence evolves smoothly but there are clear breaks at about 0.35 and 0.65 V m\ when the magnetic "eld is 17.5 T. The 20.5 T "eld case is one where the ring-like and linear states have the same parity for N"3, 4, 5 so no structure appears in the electrochemical potential di!erence. The structure in the 17.5 T case can be understood by considering the parities of the ring-like and linear states. All the ring-like states of the 4-electron system have even parity and this is the same as the parity of its linear states. But the parity of the ring-like states of the 3-electron system depends on magnetic "eld. At 17.5 T, the ground state angular momentum is J"12 (see Fig. 2) which corresponds to even parity but the linear state has odd parity. The structure in
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Fig. 4. Electro-chemical potential di!erence for 16 interacting classical particles.
the electro-chemical potential di!erence near F"0.65 V m\ corresponds to the crossing of the opposite parity levels of the 3-electron system. Similarly, the ground state of the 5-electron system at zero electric "eld has J"35, which is odd parity, and the structure in the electro-chemical potential di!erence near F"0.35 V m\ corresponds to an odd to even parity transition of the 5-electron system. The evolution of the electron density with electric "eld clearly demonstrates how the parity transitions are associated with the molecular nature of the electron states. Fig. 4 shows 3-electron densities at 17.5 T and zero electric "eld, a weak electric "eld, 0.03 V m\, "elds on either size of the transition, 0.06 and 0.07 V m\, and a strong electric "eld, 0.1 V m\. The circularly symmetric density at zero electric "eld corresponds to a ringlike molecular state. As the "eld increases the ring distorts and then fractures at the transition. Further increases of electric "eld then separate the three fragments leading to a state that is clearly linear when the electric "eld is 0.1 V m\.
4. E4ect of shear on classical states Quantum mechanical electro-chemical potential di!erences can only be calculated reliably for electron numbers less than about 6 because the lack of
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symmetry makes exact diagonalization prohibitively expensive. Intuitively, however, it seems likely that structural transitions of a more complicated and richer nature would occur in larger systems and this possibility is investigated by Monte-Carlo simulation of classical systems of interacting charged particles in a parabolic potential and a shearing electric "eld. The mean potential energy of the N-particle system is found by Monte-Carlo with simulated annealing and in addition further studies have been done where the electric "eld applied to the simulated system has been swept up and down. No magnetic "eld is included but this is not an issue as the main aim of the calculation is to study how the form of the classical energy minimum evolves with electric "eld. Minima with similar forms are expected to occur in molecular states in the strong magnetic "eld regime. The parameters of the classical system correspond to those of the parabolic quantum dot: "4 meV, the electric "eld range is 0}0.1 V m\, the particle charge is !e and the temperature is k ¹"10\ meV. Fig. 5 shows results for the mean potential energy di!erence, < !2< #< , of a 16-par,> , ,\ ticle system. The statistical error is smaller than the symbol size but there are a few points that do not fall onto the line (for example, just below 0.1 V m\). This re#ects a tendency for the system to become trapped in metastable states which is present to a small extent although the data was averaged over a very large number of Monte-Carlo steps (10). Breaks in the energy di!erence are clearly visible and the structure is generally richer than for the small quantum system, for example there is a pronounced change in the sign of the slope of the energy di!erence close to F"0.02 V m\. Results of additional runs in which the electric "eld was swept during the simulation show strong hysteresis e!ects in which the energy di!erence on the down sweep follows a different curve from the di!erence on the up sweep. The hysteresis disappears when the temperature is raised to k ¹"0.2 meV. This is on the high temperature side of an order}disorder transition that is known to occur in the classical system [18]. Thus, the presence or absence of structure in the energy di!erence appears to be sensitive to the phase of the
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Fig. 5. Evolution of electron density for 3 interacting electrons. The contour plots cover an area of 10;10.
Fig. 6. Minimum energy con"guration of 16 con"ned classical particles. Each plot covers an area of 80;80 nm.
few particle system and this leads to the tentative conclusion that experimental studies of the addition energy of deformed dots might give information about the phase diagram of con"ned few-electron systems. The evolution of the 16-particle con"gurations with electric "eld is particularly rich (Fig. 6) and the structural changes are connected with the structure of the energy di!erence as a function of electric "eld. At zero "eld, the con"guration consists of two concentric rings with a particle at the centre. This form persists up to about 0.02 V m\ when the central particle disappears and the central ring becomes elongated. The next structural change occurs between 0.062 and 0.064 V m\ when the elongated central ring is replaced by a line of 6 particles. Further increase of "eld then results in separation into two pieces, each containing 8 particles. Analogous structural changes occur in the 15 and 17particle systems and it is not possible to ascribe the structure in the energy di!erence around
0.06 V m\ unambiguously to a system with a one particular number of particles. However, the small jump in the energy di!erence around 0.03 V m\ is probably connected with the disappearance of the central ring of the 15-electron system.
5. Conclusion The e!ect of dot shape deformation on molecular states of interacting electrons in quantum dots in strong magnetic "elds has been investigated. A shearing electric "eld in systems of 3, 4, and 5 electrons induces a structural transition in which the quantum ground state changes from one based on a ring-like con"guration to one based on a linear con"guration. This change is accompanied by dramatic and in principle measurable changes in the addition energy. A particularly interesting feature of the transition is that its nature can be changed from sudden to continuous by changing
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the magnetic "eld. Similar changes are predicted to occur in larger systems of interacting classical charged particles and this suggests that richer structure might be observable in the addition energy of quantum dots with larger numbers of electrons when their shape is deformed. The structural transitions are a consequence of the reduction in the symmetry of the dot and similar transitions have been found to occur when a dot is compressed. It may be possible to use these e!ects to probe the physical nature of quantum states and phase transitions in correlated few-electron systems con"ned in quantum dots. The strong magnetic "eld regime could be particularly advantageous for experimental studies because molecular states should be found in relatively small dots and hence, there should be less sensitivity to impurity e!ects.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Acknowledgements I would like to thank Prof. S. Tarucha for asking a question which led to the study of the e!ect of shear and I am grateful to Prof. H. Aoki for hospitality at the University of Tokyo where this work was started. This work was supported by the European Union and Leicester University.
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[14] [15] [16] [17] [18]
G.W. Bryant, Phys. Rev. Lett. 59 (1987) 1140. W. HaK usler, B. Kramer, J. Ma\ sek, Z. Phys. B 85 (1991) 435. R.B. Laughlin, Phys. Rev. B 27 (1983) 3383. P.A. Maksym, Physica B 184 (1993) 385. P.A. Maksym, Phys. Rev. B 53 (1996) 10871. P.A. Maksym, H. Imamura, G.P. Mallon, H. Aoki, J. Phys. Condens. Matter 12 (2000) R299. L. Brey, N.F. Johnson, B.I. Halperin, Phys. Rev. B 40 (1989) 10647. P.A. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. D.G. Austing, T. Honda, S. Tarucha, Semicond. Sci. Technol. 12 (1997) 631. F. Bolton, U. RoK ssler, Superlatt. Microstruct. 13 (1993) 139. W.Y. Ruan, Y.Y. Liu, C.G. Bao, Z.O. Zhang, Phys. Rev. B 51 (1995) 7942. T. Seki, Y. Kuramoto, T. Nishino, J. Phys. Soc. Japan 65 (1996) 3945. C.G. Bao, W.Y. Ruan, Y.Y. Liu, Phys. Rev. B 53 (1996) 10820. H. Imamura, P.A. Maksym, H. Aoki, Phys. Rev. B 53 (1996) 12613. H. Imamura, P.A. Maksym, H. Aoki, Phys. Rev. B 59 (1999) 5817. P.A. Maksym, L.D. Hallam, J. Weis, Physica B 212 (1995) 213. P.A. Maksym, Physica B 249 (1998) 233. V.M. Bedanov, F.M. Peeters, Phys. Rev. B 49 (1994) 2667.