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Spintronics and exchange engineering in coupled quantum dots
Microelectronics Journal 34 (2003) 485–489 www.elsevier.com/locate/mejo
Spintronics and exchange engineering in coupled quantum dots Jean-Pierre Leburtona,b,*, Satyadev Nagarajaa,b, Philippe Matagneb, Richard M. Martinb,c a
b
Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Beckman Institute for Advanced Science and Technology, 405 N. Mathews Avenue, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA c Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Abstract By combining advanced device modeling with computational physics, microscopic changes in quantum many-body interactions of quantum dots are described as a function of macroscopic device parameters without a priori assumption on the confining potential. In planar coupled quantum dot structures, spin control can be achieved by modulating exchange interaction with an external gate as in conventional field effect transistors. These new features, coupled with recent advances in quantum dot physics constitute the basic ingredients for a solid state technology for quantum information processing. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Spintronics; Coupled quantum dots; Nanostructures
1. Introduction Quantum dots (QD) share with atoms the fundamental properties of charge and energy quantization; for this reason, they are often called ‘artificial atoms’ [1,2]. In cylindrical QDs, shell structures in the energy spectrum and Hund’s rule for spin alignment with partial shell filling of electrons have recently been observed [3]. One of the peculiarities of these nanostructures is the ability to control not only the shape of the QD but also the number of electrons through gate electrodes [4]. Since Hund’s rule is the manifestation of spin effects with shell filling in QDs, the electron spin can in principle, be also controlled by the electric field of a transistor gate [5]. The idea of controlling spin polarization independently of the number of electrons in QDs, has practical consequences since it provides the physical ingredients for processing quantum information and making quantum computation possible [6]. One of the greatest scientific and engineering challenges of this decade is the realization of a quantum computer. In this context, a solid-state system is highly desirable because of its compactness, scalability and compatibility with * Corresponding author. Address: Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. E-mail address: [email protected] (J.-P. Leburton).
existing semiconductor technology. In quantum computing, the basic information unit is a quantum bit or qubit, i.e. a physical object that can be represented as a superposition of two basis states in a 2D Hilbert space. In this context, the use of spin state (S) rather than charge-states as qubit in semiconductor materials is relatively appealing because of their relative insensitivity to electric noise in the device environment. There are presently several proposals for quantum computing with spins in semiconductors, but the Loss – DiVincenzo scheme [7] based on the manipulation of electron spins in coupled QDs offers several advantages over other schemes [8 –12]. Hence, the technology for planar and lateral QDs AlGAs/GaAs heterostructures is well established. Interfaces between III – V materials are very clean, and the electron effective mass is much smaller than in silicon, which eases the conditions for quantum confinement. This QD technology has already been successful in demonstrating a variety of spin effects such as the manifestation of Hund’s rule in artificial atoms [3] and a fully tunable Kondo effect [13]. Moreover, III –V materials enjoy long spin coherence times, as demonstrated by Awschalow and collaborators [14], which is of utmost importance for preserving quantum information over many qubit operations. However, the experimental realization of a solid-state quantum device remains a challenging issue for which the interplay between device geometry and
0026-2692/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0026-2692(03)00080-6
486
J.-P. Leburton et al. / Microelectronics Journal 34 (2003) 485–489
electrostatics, materials parameters and spin physics should be integrated into a realistic approach. Advances in computer simulation combine the sophistication of realistic spin-device modeling with the accuracy of computational physics of materials based on the density functional theory (DFT) [15 –20]. These powerful methods provide theoretical tools for analyzing fine details of manybody interactions in nanostructures in a three-dimensional (3D) environment made of heterostructures and doping, with realistic boundary conditions. Microscopic changes in the quantum states are described in terms of the variation of macroscopic parameters such as voltages, structure size and physical shape of the dots without a priori assumption on the confinement profile.
2. Coupled quantum dot structure Coupled QDs with variable inter-dot barrier for controlling electron –electron interactions, can be fabricated with a planar geometry made of GaAs/AlGaAs hetero-structures that contain a two-dimensional electron gas (2DEG) [21]. The dots are defined by a system of gate pads and stubs that are negatively biased to deplete the 2DEG, leaving two pools of electrons that forms two QDs connected in series (Fig. 1) [22]. In these ‘artificial diatomic molecules’,
electron states can couple to form covalent states that are delocalized over the two dots, with electrons tunneling between them without being localized to either [23]. These bonding states have lower energy than the constituent dot states by an amount that is equivalent to the binding energy of the molecule. In our case, the dimensions are such that the electron – electron interaction energy is comparable to the single-particle energy level spacing. The number of electrons in the QD, N; is restricted to low values in a situation comparable to a light diatomic molecule such as H –H or B – B. The coupling between QDs can be adjusted by varying the voltage on the tuning gates Vt to change the height of the barrier between the two QDs. The number of electrons N in the double QD is varied as the 2DEG density, with the back gate voltage Vback for a fixed bias on the top gates. Hence, controllable exchange interaction that gives rise to spin polarization, can be engineered with this configuration by varying N and the system spin, independently [24]. Our structure consists of a 22.5 nm layer of undoped Al0.3Ga0.7As followed by a 125 nm layer of undoped GaAs, and finally a 18 nm GaAs cap layer (Fig. 1a). The latter is uniformly doped to 5 £ 1018 cm23 so that the conduction band is just above the Fermi level at the boundary between the GaAs cap layer and the undoped GaAs. The inverted heterostructure is grown on the GaAs substrate. The lateral
Fig. 1. Schematic representation of the planar coupled quantum dot device. (a) Layer structure with top and back gates; (b) top view of the metal gate arrangement with sizes and orientations; (c) schematic representation of the six lowest orbitals in the weak (left hand side) and strong (right hand side) coupling regimes. In both cases, the s-states are strongly localized in their respective dot. Left: px and pz-like orbitals are degenerate within each dot and decoupled from the corresponding state in the other dot. Right: Increasing coupling lifts the p-orbitals degeneracy with a reordering of the states. The red and blue orbitals indicate positive and negative parts of the wave functions, respectively.
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dimension of the gates and spacing are shown in Fig. 1b. Fig. 1c shows the schematic of the lowest four states with their wave functions in the double dot for two different tuning voltages [22]. For both values of Vt ; the ground state in the individual dots is s-like and forms a degenerate pair. Here, we borrow the terminology of atomic physics to label the quantum dot states. The first excited states, which are pxand pz-like, are degenerate for weak inter-dot coupling, whereas for strong coupling, the pz-like states mix to form symmetric (bonding) and anti-symmetric (anti-bonding) states that are lower in energy than the px-like states as seen in Fig. 1c. This re-ordering of the states has an important bearing on the spin-polarization of the double-dot system, as shall be fully explained later.
3. Computational model The implementation of a spin-dependent scheme for the electronic structure of artificial molecules involves the solution of the Kohn – Sham equation for each of the spins, i.e. up ð"Þ and down ð#Þ: Under the local spin density approximation within the DFT, the Hamiltonian H"ð#Þ for the spin " ð#Þ electrons reads [15,22] 2 1 ^ "ð#Þ ¼ 2 " 7 7 þ Ec ð~rÞ þ m"ð#Þ H ð1Þ xc ½n mp ð~rÞ 2 where mp ðrÞ is the position dependent effective mass of the electron in the different materials, Ec ðrÞ ¼ efðrÞ þ DEos ; is the effective conduction band edge, and fðrÞ is the electrostatic potential which contains the coulomb interaction between electrons, and DEos is the conduction band offset between GaAs and AlGaAs. The respective Hamiltonians are identical in all respects, except for the exchangecorrelation potential, which is given by,
m"ð#Þ xc
dðn1xc ½nÞ ¼ dn"ð#Þ
ð2Þ
where 1xc is the exchange-correlation energy as a function of the total electron density nðrÞ ¼ n" ðrÞ þ n# ðrÞ and the fractional spin polarization j ¼ ðn" 2 n# Þ=n; as parametrized by Ceperley and Alder [25]. While it is known that the DFT underestimates the exchange interaction between electrons, which leads to incorrect energy gaps in semiconductors, it provides a realistic description of spin – spin interactions in quantum nanostructures, as shown in the prediction of the addition energy of vertical QDs [26]. The 3D Poisson equation for the electrostatic potential fðrÞ reads 7½1ð~rÞ7fð~rÞ ¼ 2rð~rÞ
ð3Þ
Here 1ðrÞ is the permittivity of the material, and the charge density r is comprised of the electron and hole concentrations as well as the ionized donor and acceptor concentrations present in the respective regions of the device. The QD region itself is undoped or very slightly
487
p-doped. At equilibrium, the electron concentrations for each spin in the QDs are computed from the wave functions obtained from the respective Kohn – Sham equations, i.e. 2 rðrÞ ¼ enðrÞ with n"ð#Þ ¼ Si jc"ð#Þ i ðrÞj : In the region outside the QDs a Thomas– Fermi distribution is used, so that the electron density outside the QD is a simple local function of the position of the conduction band edge with respect to the Fermi level, 1F : The various gate voltages—Vback ; Vt and those on the metallic pads and stubs-determine the boundary conditions on the potential fðrÞ in the Poisson equation. For the lateral surfaces in the x – y plane on Fig. 1, vanishing electric fields are assumed. Self-consistent solution of the Kohn – Sham and Poisson equations proceeds by solving the former for both spins, calculating the respective electron densities and exchange correlation potentials, solving the Poisson equation to determine the potential fðrÞ and repeating the sequence until the convergence criterion is satisfied [19]. Typically, this criterion is such that variations in the energy levels and electrostatic potential between successive solutions are below 1026 eV and 1026 V, respectively. The conduction band edge of an empty planar double dot shows two dots, identified as low potential energy regions separated by a barrier of approximately 3 meV at the hetero-interface for Vt ¼ 20:65 V (the lateral pad gates are at Vpad ¼ 20:4 V and Vback ¼ 0:975 VÞ: In the x – z plane, the variation of the bare potential is quasi-parabolic about the bottom of the two potential wells; therefore the first few states have features of the harmonic oscillator wave functions. The determination of Neq ; the number of electrons in the QDs at equilibrium for each value of the gate and tuning voltages, is achieved by using Slater’s transition rule [27] ð1 1 ð4Þ ET ðN þ 1Þ 2 ET ðNÞ ¼ 1LOA ðnÞdn < 1LOA 2 1F 2 0 where ET ðNÞ is the total energy of the dot for N electrons and 1LOA ð1=2Þ is the eigenvalue of the lowest-availableorbital when it is occupied by 0.5 electron. From the latter equation, it is seen that, if the right hand side is positive Neq ¼ N otherwise Neq ¼ N þ 1: Thus the N ! N þ 1 transition points are obtained by populating the system with N þ 0:5 electrons and varying Vback until 1LOA ð1=2Þ 2 1F becomes negative. It should be noted that the approximation made in the latter equation is valid only if 1LOA varies linearly with N: This approach has been very successful in the analysis of the electronic spectra and charging characteristics of vertically confined QDs [26].
4. Spintronics in coupled quantum dots In the present double QD structure, we focus on the spin states of the electron system and allow N to vary from zero to eight for two values of Vt : Vt ¼ 20:67 V defined as the weak coupling regime, and Vt ¼ 20:60 V defined as the strong coupling regime. Electron spin states that are relevant
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"ð#Þ in this analysis, are designated by s"ð#Þ 1 ; s2 (lower energy s"ð#Þ "ð#Þ "ð#Þ states in dots 1 and 2) and px1 ; px2 pz1 and p"(z2 # )-(higher energy p-states where the x- and z-indices indicate the orientation of the wave functions). For N ¼ 0; in the weak coupling limit, the computer model shows that s-states in dots 1 and 2, have negligible overlap because of the relatively high and wide barrier. Indeed, the bonding –antibonding energy separation resulting from the coupling between these states is orders of magnitude smaller than the coulomb charging energy so that s-electrons are practically localized in each dot. A similar situation arises for the px and pz-states, which although experiencing slight overlap because higher in energy, they see a lower and thinner barrier, are quasi-degenerate within each dot. In fact, in the weak coupling limit, pz1- and pz2-states that are oriented along the coupling direction between dots, experience a bigger overlap than the corresponding px-states, and consequently lie slightly lower in energy than the latter. Hence, as far as the lower s- and p-states are considered, the double dot system behaves as two quasi-independent dots (Fig. 1c, left). In addition, because of the large distance separating the two lower s-states for Vt ¼ 20:67 V; Coulomb interaction between electrons in dots 1 and 2 is negligibly small, and both dots can be charged simultaneously through double charging [22] to completely fill the s1 and s2 states. Therefore, for N ¼ 4; there is no net spin polarization in the double dot, since both contains equal number of spin " and spin # electrons. When the double dot is charged with a fifth electron, the latter occupies either the pz1 or pz2-state (e.g. " spin, i.e. p"z1or p"z2) which have the lowest available energy. The sixth electron takes advantage of the non-zero p-state overlap and occupies the other p"z-state with a parallel " spin. The seventh and eighth electrons find it energetically favorable to occupy successively p"x1 and p"x2, but not any of the spin # states, because of the attractive nature of the exchangecorrelation energy among the spin " electrons that results from the non-zero p-state overlap, and lowers the energy of the double QD. This particular high spin configuration among p-orbitals in the ‘artificial’ diatomic molecule deserves special attention because it appears to violate one of Zener principles on the onset of magnetism in transition elements; this principle forbids spin alignment for electrons on similar orbitals in adjacent atoms [28]. Therefore, it could be argued that the high spin configuration obtained in
the calculation is the consequence of a DFT artifact. Recently, however, Wensauer et al. confirmed the DFT results based on a Heitler– London approach [29]. Similar conclusions have also been obtained by ‘exact’ diagonalization technique on vertically coupled quantum dots for N ¼ 6 electrons [30,31]. Let us point out that Zener’s principle is purely empirical, as it is based on the observation of the magnetic properties of natural elements that lacks the tunability of ‘artificial’ systems. Therefore, the total spin of the double dot can possibly steadily increase by 1=2" for each electron added after the fourth electron to 2" for N ¼ 8; and there is no contradiction with Zener’s principle applied to natural elements. After all, high spin configurations have been shown to compete for the ground state of light diatomic molecules such as B2 [32]. The variation of the total spin S in the double dot with N; is shown in Table 1. It is also seen that as N increases above eight electrons, the spin # states start to be occupied, thereby decreasing S by 1=2" for each additional electron forming anti-parallel pairs to complete the shell until N ¼ 12 when S is reduced to zero. The sequence of level filling with the occupation of degenerate states by electrons of parallel spin, is observed in atoms, and is governed by Hund’s rules; it is therefore impressive that similar rules successfully govern level filling in the double QD in the weak coupling regime. Let us point out that, even though S ¼ 2" is the most favored state of the double dot, energetically it is not significantly lower than other competing states for N ¼ 8: For instance, the excited states with S , 2" for N ¼ 8; are only about 0.1 meV higher in energy. Consequently, for this particular double dot structure, any attempt to observe experimentally the parallel alignment of the spins of unpaired electrons is restricted to low temperatures for which kB T p 0:1 meV; or any kinds of electrostatic fluctuations smaller than this value. However, it must be noted that the structure is not optimized, and that the evidence of spin polarization among p-states in the double dot system may be achieved in smaller dots with stronger exchange interaction. The key issue here, is the fact that the quantum mechanical coupling between the two dots in this bias regime, is not strong enough to lift the spatial ‘quasi-degeneracy’ among px- and pz-states which, for our particular configuration, were separated by no more than a few micro-electron volts. Stronger coupling between the quantum dots eliminates this effect. Accordingly, if Vt
Table 1 Total spin sequences as a function of the occupation number in the two coupling regimes of the double dot Vt (V)
Spin ð"Þ
20.67 20.60
N 1
2
3
4
5
6
7
8
9
10
11
12
1/2 1/2
? 0
1/2 1/2
0 0
1/2 1/2
1 0
3/2 1/2
2 0
3/2 1/2
1 0
1/2 1/2
0 0
The question mark at N ¼ 2 in the weak coupling regime indicates that the spins are uncorrelated.
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profiles, the distance between the GaAs/AlGaAs heterojunction and the control gates, and possibly choosing other III – V semiconductor systems to optimize the energy separation between singlet and triplet states.
Acknowledgements This work is supported by the DARPA-QuIST program DAAD10-01-1-0659, and the NSF grant DMR 99-76550
References
Fig. 2. Schematic representation of a singlet– triplet transition induced electro-statically in a double quantum dot with six electron.
increases to 2 0.60 V, also referred to the strong coupling regime, the p-state spatial degeneracy is completely lifted, while deeper s-states also couple, although to a slighter extent to lead to the spectrum of Fig. 1c, right. Therefore, the spin sequence as a function of N is alternatively S ¼ 1=2" for odd N when the last occupying electron is unpaired, and S ¼ 0 for even N; when it pairs up with an electron of the opposite spin (Table 1). The variation of inter-dot coupling by varying Vt provides a control of direct exchange interaction between p-like electrons in the two dots that may be more robust than for s-electrons. Hence, a lowering of the inter-dot barrier results in a reordering of the single-particle levels, thereby transforming the double-dot (for N ¼ 8) from a spin polarized S ¼ 2" to an unpolarized state S ¼ 0. An important result from Table 1 is that singlet – triplet transitions in double dots as shown in Fig. 2 could also be achieved for N ¼ 6 electrons within the Loss –Di Vincenzo scheme for quantum computing, where the control of qubit entanglement for a quantum control-not (XOR) gate operation could be realized with the S ¼ 1=2" spin states of two p-electrons instead of two single electrons ðN ¼ 2Þ in the original scenario [7]. The electrostatic nature of the confinement potential, specifically the coupling barrier, is central to the occurrence of the effects mentioned. Indeed, the barrier is not uniform, but wider (and higher) for the lower QD states than for the higher p-states. This situation is similar to the electronic properties of natural diatomic molecules where the strongly localized s-states correspond to atomic core states and the delocalized p-states to covalent bonding states. It is therefore possible to engineer exchange interaction in the ‘artificial molecule’ by suitably tailoring the coupling barrier between QDs. This is achievable by proper device design, i.e by adjusting gate size and shape, the doping
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Spintronics and exchange engineering in coupled quantum dots Jean-Pierre Leburtona,b,*, Satyadev Nagarajaa,b, Philippe Matagneb, Richard M. Martinb,c a
b
Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Beckman Institute for Advanced Science and Technology, 405 N. Mathews Avenue, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA c Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Abstract By combining advanced device modeling with computational physics, microscopic changes in quantum many-body interactions of quantum dots are described as a function of macroscopic device parameters without a priori assumption on the confining potential. In planar coupled quantum dot structures, spin control can be achieved by modulating exchange interaction with an external gate as in conventional field effect transistors. These new features, coupled with recent advances in quantum dot physics constitute the basic ingredients for a solid state technology for quantum information processing. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Spintronics; Coupled quantum dots; Nanostructures
1. Introduction Quantum dots (QD) share with atoms the fundamental properties of charge and energy quantization; for this reason, they are often called ‘artificial atoms’ [1,2]. In cylindrical QDs, shell structures in the energy spectrum and Hund’s rule for spin alignment with partial shell filling of electrons have recently been observed [3]. One of the peculiarities of these nanostructures is the ability to control not only the shape of the QD but also the number of electrons through gate electrodes [4]. Since Hund’s rule is the manifestation of spin effects with shell filling in QDs, the electron spin can in principle, be also controlled by the electric field of a transistor gate [5]. The idea of controlling spin polarization independently of the number of electrons in QDs, has practical consequences since it provides the physical ingredients for processing quantum information and making quantum computation possible [6]. One of the greatest scientific and engineering challenges of this decade is the realization of a quantum computer. In this context, a solid-state system is highly desirable because of its compactness, scalability and compatibility with * Corresponding author. Address: Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. E-mail address: [email protected] (J.-P. Leburton).
existing semiconductor technology. In quantum computing, the basic information unit is a quantum bit or qubit, i.e. a physical object that can be represented as a superposition of two basis states in a 2D Hilbert space. In this context, the use of spin state (S) rather than charge-states as qubit in semiconductor materials is relatively appealing because of their relative insensitivity to electric noise in the device environment. There are presently several proposals for quantum computing with spins in semiconductors, but the Loss – DiVincenzo scheme [7] based on the manipulation of electron spins in coupled QDs offers several advantages over other schemes [8 –12]. Hence, the technology for planar and lateral QDs AlGAs/GaAs heterostructures is well established. Interfaces between III – V materials are very clean, and the electron effective mass is much smaller than in silicon, which eases the conditions for quantum confinement. This QD technology has already been successful in demonstrating a variety of spin effects such as the manifestation of Hund’s rule in artificial atoms [3] and a fully tunable Kondo effect [13]. Moreover, III –V materials enjoy long spin coherence times, as demonstrated by Awschalow and collaborators [14], which is of utmost importance for preserving quantum information over many qubit operations. However, the experimental realization of a solid-state quantum device remains a challenging issue for which the interplay between device geometry and
0026-2692/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0026-2692(03)00080-6
486
J.-P. Leburton et al. / Microelectronics Journal 34 (2003) 485–489
electrostatics, materials parameters and spin physics should be integrated into a realistic approach. Advances in computer simulation combine the sophistication of realistic spin-device modeling with the accuracy of computational physics of materials based on the density functional theory (DFT) [15 –20]. These powerful methods provide theoretical tools for analyzing fine details of manybody interactions in nanostructures in a three-dimensional (3D) environment made of heterostructures and doping, with realistic boundary conditions. Microscopic changes in the quantum states are described in terms of the variation of macroscopic parameters such as voltages, structure size and physical shape of the dots without a priori assumption on the confinement profile.
2. Coupled quantum dot structure Coupled QDs with variable inter-dot barrier for controlling electron –electron interactions, can be fabricated with a planar geometry made of GaAs/AlGaAs hetero-structures that contain a two-dimensional electron gas (2DEG) [21]. The dots are defined by a system of gate pads and stubs that are negatively biased to deplete the 2DEG, leaving two pools of electrons that forms two QDs connected in series (Fig. 1) [22]. In these ‘artificial diatomic molecules’,
electron states can couple to form covalent states that are delocalized over the two dots, with electrons tunneling between them without being localized to either [23]. These bonding states have lower energy than the constituent dot states by an amount that is equivalent to the binding energy of the molecule. In our case, the dimensions are such that the electron – electron interaction energy is comparable to the single-particle energy level spacing. The number of electrons in the QD, N; is restricted to low values in a situation comparable to a light diatomic molecule such as H –H or B – B. The coupling between QDs can be adjusted by varying the voltage on the tuning gates Vt to change the height of the barrier between the two QDs. The number of electrons N in the double QD is varied as the 2DEG density, with the back gate voltage Vback for a fixed bias on the top gates. Hence, controllable exchange interaction that gives rise to spin polarization, can be engineered with this configuration by varying N and the system spin, independently [24]. Our structure consists of a 22.5 nm layer of undoped Al0.3Ga0.7As followed by a 125 nm layer of undoped GaAs, and finally a 18 nm GaAs cap layer (Fig. 1a). The latter is uniformly doped to 5 £ 1018 cm23 so that the conduction band is just above the Fermi level at the boundary between the GaAs cap layer and the undoped GaAs. The inverted heterostructure is grown on the GaAs substrate. The lateral
Fig. 1. Schematic representation of the planar coupled quantum dot device. (a) Layer structure with top and back gates; (b) top view of the metal gate arrangement with sizes and orientations; (c) schematic representation of the six lowest orbitals in the weak (left hand side) and strong (right hand side) coupling regimes. In both cases, the s-states are strongly localized in their respective dot. Left: px and pz-like orbitals are degenerate within each dot and decoupled from the corresponding state in the other dot. Right: Increasing coupling lifts the p-orbitals degeneracy with a reordering of the states. The red and blue orbitals indicate positive and negative parts of the wave functions, respectively.
J.-P. Leburton et al. / Microelectronics Journal 34 (2003) 485–489
dimension of the gates and spacing are shown in Fig. 1b. Fig. 1c shows the schematic of the lowest four states with their wave functions in the double dot for two different tuning voltages [22]. For both values of Vt ; the ground state in the individual dots is s-like and forms a degenerate pair. Here, we borrow the terminology of atomic physics to label the quantum dot states. The first excited states, which are pxand pz-like, are degenerate for weak inter-dot coupling, whereas for strong coupling, the pz-like states mix to form symmetric (bonding) and anti-symmetric (anti-bonding) states that are lower in energy than the px-like states as seen in Fig. 1c. This re-ordering of the states has an important bearing on the spin-polarization of the double-dot system, as shall be fully explained later.
3. Computational model The implementation of a spin-dependent scheme for the electronic structure of artificial molecules involves the solution of the Kohn – Sham equation for each of the spins, i.e. up ð"Þ and down ð#Þ: Under the local spin density approximation within the DFT, the Hamiltonian H"ð#Þ for the spin " ð#Þ electrons reads [15,22] 2 1 ^ "ð#Þ ¼ 2 " 7 7 þ Ec ð~rÞ þ m"ð#Þ H ð1Þ xc ½n mp ð~rÞ 2 where mp ðrÞ is the position dependent effective mass of the electron in the different materials, Ec ðrÞ ¼ efðrÞ þ DEos ; is the effective conduction band edge, and fðrÞ is the electrostatic potential which contains the coulomb interaction between electrons, and DEos is the conduction band offset between GaAs and AlGaAs. The respective Hamiltonians are identical in all respects, except for the exchangecorrelation potential, which is given by,
m"ð#Þ xc
dðn1xc ½nÞ ¼ dn"ð#Þ
ð2Þ
where 1xc is the exchange-correlation energy as a function of the total electron density nðrÞ ¼ n" ðrÞ þ n# ðrÞ and the fractional spin polarization j ¼ ðn" 2 n# Þ=n; as parametrized by Ceperley and Alder [25]. While it is known that the DFT underestimates the exchange interaction between electrons, which leads to incorrect energy gaps in semiconductors, it provides a realistic description of spin – spin interactions in quantum nanostructures, as shown in the prediction of the addition energy of vertical QDs [26]. The 3D Poisson equation for the electrostatic potential fðrÞ reads 7½1ð~rÞ7fð~rÞ ¼ 2rð~rÞ
ð3Þ
Here 1ðrÞ is the permittivity of the material, and the charge density r is comprised of the electron and hole concentrations as well as the ionized donor and acceptor concentrations present in the respective regions of the device. The QD region itself is undoped or very slightly
487
p-doped. At equilibrium, the electron concentrations for each spin in the QDs are computed from the wave functions obtained from the respective Kohn – Sham equations, i.e. 2 rðrÞ ¼ enðrÞ with n"ð#Þ ¼ Si jc"ð#Þ i ðrÞj : In the region outside the QDs a Thomas– Fermi distribution is used, so that the electron density outside the QD is a simple local function of the position of the conduction band edge with respect to the Fermi level, 1F : The various gate voltages—Vback ; Vt and those on the metallic pads and stubs-determine the boundary conditions on the potential fðrÞ in the Poisson equation. For the lateral surfaces in the x – y plane on Fig. 1, vanishing electric fields are assumed. Self-consistent solution of the Kohn – Sham and Poisson equations proceeds by solving the former for both spins, calculating the respective electron densities and exchange correlation potentials, solving the Poisson equation to determine the potential fðrÞ and repeating the sequence until the convergence criterion is satisfied [19]. Typically, this criterion is such that variations in the energy levels and electrostatic potential between successive solutions are below 1026 eV and 1026 V, respectively. The conduction band edge of an empty planar double dot shows two dots, identified as low potential energy regions separated by a barrier of approximately 3 meV at the hetero-interface for Vt ¼ 20:65 V (the lateral pad gates are at Vpad ¼ 20:4 V and Vback ¼ 0:975 VÞ: In the x – z plane, the variation of the bare potential is quasi-parabolic about the bottom of the two potential wells; therefore the first few states have features of the harmonic oscillator wave functions. The determination of Neq ; the number of electrons in the QDs at equilibrium for each value of the gate and tuning voltages, is achieved by using Slater’s transition rule [27] ð1 1 ð4Þ ET ðN þ 1Þ 2 ET ðNÞ ¼ 1LOA ðnÞdn < 1LOA 2 1F 2 0 where ET ðNÞ is the total energy of the dot for N electrons and 1LOA ð1=2Þ is the eigenvalue of the lowest-availableorbital when it is occupied by 0.5 electron. From the latter equation, it is seen that, if the right hand side is positive Neq ¼ N otherwise Neq ¼ N þ 1: Thus the N ! N þ 1 transition points are obtained by populating the system with N þ 0:5 electrons and varying Vback until 1LOA ð1=2Þ 2 1F becomes negative. It should be noted that the approximation made in the latter equation is valid only if 1LOA varies linearly with N: This approach has been very successful in the analysis of the electronic spectra and charging characteristics of vertically confined QDs [26].
4. Spintronics in coupled quantum dots In the present double QD structure, we focus on the spin states of the electron system and allow N to vary from zero to eight for two values of Vt : Vt ¼ 20:67 V defined as the weak coupling regime, and Vt ¼ 20:60 V defined as the strong coupling regime. Electron spin states that are relevant
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"ð#Þ in this analysis, are designated by s"ð#Þ 1 ; s2 (lower energy s"ð#Þ "ð#Þ "ð#Þ states in dots 1 and 2) and px1 ; px2 pz1 and p"(z2 # )-(higher energy p-states where the x- and z-indices indicate the orientation of the wave functions). For N ¼ 0; in the weak coupling limit, the computer model shows that s-states in dots 1 and 2, have negligible overlap because of the relatively high and wide barrier. Indeed, the bonding –antibonding energy separation resulting from the coupling between these states is orders of magnitude smaller than the coulomb charging energy so that s-electrons are practically localized in each dot. A similar situation arises for the px and pz-states, which although experiencing slight overlap because higher in energy, they see a lower and thinner barrier, are quasi-degenerate within each dot. In fact, in the weak coupling limit, pz1- and pz2-states that are oriented along the coupling direction between dots, experience a bigger overlap than the corresponding px-states, and consequently lie slightly lower in energy than the latter. Hence, as far as the lower s- and p-states are considered, the double dot system behaves as two quasi-independent dots (Fig. 1c, left). In addition, because of the large distance separating the two lower s-states for Vt ¼ 20:67 V; Coulomb interaction between electrons in dots 1 and 2 is negligibly small, and both dots can be charged simultaneously through double charging [22] to completely fill the s1 and s2 states. Therefore, for N ¼ 4; there is no net spin polarization in the double dot, since both contains equal number of spin " and spin # electrons. When the double dot is charged with a fifth electron, the latter occupies either the pz1 or pz2-state (e.g. " spin, i.e. p"z1or p"z2) which have the lowest available energy. The sixth electron takes advantage of the non-zero p-state overlap and occupies the other p"z-state with a parallel " spin. The seventh and eighth electrons find it energetically favorable to occupy successively p"x1 and p"x2, but not any of the spin # states, because of the attractive nature of the exchangecorrelation energy among the spin " electrons that results from the non-zero p-state overlap, and lowers the energy of the double QD. This particular high spin configuration among p-orbitals in the ‘artificial’ diatomic molecule deserves special attention because it appears to violate one of Zener principles on the onset of magnetism in transition elements; this principle forbids spin alignment for electrons on similar orbitals in adjacent atoms [28]. Therefore, it could be argued that the high spin configuration obtained in
the calculation is the consequence of a DFT artifact. Recently, however, Wensauer et al. confirmed the DFT results based on a Heitler– London approach [29]. Similar conclusions have also been obtained by ‘exact’ diagonalization technique on vertically coupled quantum dots for N ¼ 6 electrons [30,31]. Let us point out that Zener’s principle is purely empirical, as it is based on the observation of the magnetic properties of natural elements that lacks the tunability of ‘artificial’ systems. Therefore, the total spin of the double dot can possibly steadily increase by 1=2" for each electron added after the fourth electron to 2" for N ¼ 8; and there is no contradiction with Zener’s principle applied to natural elements. After all, high spin configurations have been shown to compete for the ground state of light diatomic molecules such as B2 [32]. The variation of the total spin S in the double dot with N; is shown in Table 1. It is also seen that as N increases above eight electrons, the spin # states start to be occupied, thereby decreasing S by 1=2" for each additional electron forming anti-parallel pairs to complete the shell until N ¼ 12 when S is reduced to zero. The sequence of level filling with the occupation of degenerate states by electrons of parallel spin, is observed in atoms, and is governed by Hund’s rules; it is therefore impressive that similar rules successfully govern level filling in the double QD in the weak coupling regime. Let us point out that, even though S ¼ 2" is the most favored state of the double dot, energetically it is not significantly lower than other competing states for N ¼ 8: For instance, the excited states with S , 2" for N ¼ 8; are only about 0.1 meV higher in energy. Consequently, for this particular double dot structure, any attempt to observe experimentally the parallel alignment of the spins of unpaired electrons is restricted to low temperatures for which kB T p 0:1 meV; or any kinds of electrostatic fluctuations smaller than this value. However, it must be noted that the structure is not optimized, and that the evidence of spin polarization among p-states in the double dot system may be achieved in smaller dots with stronger exchange interaction. The key issue here, is the fact that the quantum mechanical coupling between the two dots in this bias regime, is not strong enough to lift the spatial ‘quasi-degeneracy’ among px- and pz-states which, for our particular configuration, were separated by no more than a few micro-electron volts. Stronger coupling between the quantum dots eliminates this effect. Accordingly, if Vt
Table 1 Total spin sequences as a function of the occupation number in the two coupling regimes of the double dot Vt (V)
Spin ð"Þ
20.67 20.60
N 1
2
3
4
5
6
7
8
9
10
11
12
1/2 1/2
? 0
1/2 1/2
0 0
1/2 1/2
1 0
3/2 1/2
2 0
3/2 1/2
1 0
1/2 1/2
0 0
The question mark at N ¼ 2 in the weak coupling regime indicates that the spins are uncorrelated.
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profiles, the distance between the GaAs/AlGaAs heterojunction and the control gates, and possibly choosing other III – V semiconductor systems to optimize the energy separation between singlet and triplet states.
Acknowledgements This work is supported by the DARPA-QuIST program DAAD10-01-1-0659, and the NSF grant DMR 99-76550
References
Fig. 2. Schematic representation of a singlet– triplet transition induced electro-statically in a double quantum dot with six electron.
increases to 2 0.60 V, also referred to the strong coupling regime, the p-state spatial degeneracy is completely lifted, while deeper s-states also couple, although to a slighter extent to lead to the spectrum of Fig. 1c, right. Therefore, the spin sequence as a function of N is alternatively S ¼ 1=2" for odd N when the last occupying electron is unpaired, and S ¼ 0 for even N; when it pairs up with an electron of the opposite spin (Table 1). The variation of inter-dot coupling by varying Vt provides a control of direct exchange interaction between p-like electrons in the two dots that may be more robust than for s-electrons. Hence, a lowering of the inter-dot barrier results in a reordering of the single-particle levels, thereby transforming the double-dot (for N ¼ 8) from a spin polarized S ¼ 2" to an unpolarized state S ¼ 0. An important result from Table 1 is that singlet – triplet transitions in double dots as shown in Fig. 2 could also be achieved for N ¼ 6 electrons within the Loss –Di Vincenzo scheme for quantum computing, where the control of qubit entanglement for a quantum control-not (XOR) gate operation could be realized with the S ¼ 1=2" spin states of two p-electrons instead of two single electrons ðN ¼ 2Þ in the original scenario [7]. The electrostatic nature of the confinement potential, specifically the coupling barrier, is central to the occurrence of the effects mentioned. Indeed, the barrier is not uniform, but wider (and higher) for the lower QD states than for the higher p-states. This situation is similar to the electronic properties of natural diatomic molecules where the strongly localized s-states correspond to atomic core states and the delocalized p-states to covalent bonding states. It is therefore possible to engineer exchange interaction in the ‘artificial molecule’ by suitably tailoring the coupling barrier between QDs. This is achievable by proper device design, i.e by adjusting gate size and shape, the doping
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