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Density functional theorycalculations of electronic structurein silicon double quantum dots

P. Howard1, A. Andreev1,2,*, and D. A. Williams2

1 Advanced Technology Institute, University of Surrey, Guildford, GU2 7XH, United Kingdom2 Hitachi Cambridge Laboratory, Hitachi Europe Ltd., Cambridge, CB3 0HE, United Kingdom

Received 31 October 2007, revised 17 December 2007, accepted 28 December 2007Published online 13 June 2008

PACS 71.15.Mb, ,73.21.La,71.23.-b

∗ Corresponding author: e-mail a.andreev@surrey.ac.uk, Phone: +44-1483-686813, Fax: +44-1483-686781

The finite difference method and anisotropic effectivemass density functional theory (DFT) are used to cal-culate the electronic structure and wavefunctions of Sil-icon isolated double quantum dots. An optimised algo-rithm for determining the self-consistent DFT potentialwas developed.

The effects of applying electric fields to push the elec-trons from one dot to another dot in the double quantumdot are studied. We demonstrate that depending on thestructure parameters there are two regimes correspond-ing to the strong and weak coupling between the twodots.

1 Introduction Double Quantum Dot (DQD) struc-tures are a promising candidate for building qubits, the ba-sic unit of quantum computing. The use of DQDs as chargequbits was demonstrated recently [1,2]. In order to effi-ciently use a DQD for potential quantum computing ap-plications, a detailed knowledge of its electronic structureis required. The factors that are important for determina-tion of the electronic structure include the geometry of thedevice, many body effects and externally applied electricand magnetic fields. For quantum dots more than 10 nm indiameter, using sophisicated approaches based on ab ini-tio atomistic calculations is too slow and inefficient formodelling the device performance. Therefore, a simplerbut physically justified model that takes account of men-tioned important factors is required and this paper seeksto develop such an approach. The aim of this paper is todemonstrate an (anisotropic) effective mass density func-tional theory (DFT) model, which allows us to model dotssized between 5 and 100 nm.

This paper is focused on modelling a structure consist-ing of an Si/SiO2 isolated double quantum dot surroundedby electric gates and a Single Electron Transistor. We cal-culate the electronic structure of such a device, includingself-consistent consideration of many body effects through

DFT. The variation of the electron density inside the iso-lated DQD gives the qubit states.

The geometry of the structure determines the quan-tised states, and modelling these states provides an insightinto what is happening inside the quantum dots. This workseeks to model the system, and numerically determine theavailable electron wavefunctions in a given isolated DQDstructure.

1.1 DQDs as charge qubits The two basis statesrequired for use of a DQD as a qubit are related to the elec-trons existing in one dot or the other (more precisely, totwo states of the electron distribution in the DQD). Manip-ulation of the qubit may be done by varying the electricpotential of the gates to change the charge distribution inthe DQD system, i.e. to change the state of the qubit be-tween the two basis states. The state of the qubit can bedetermined by measuring the current through the SingleElectron Transistor (SET), which is very sensitive to thecharge distribution in the DQD. Long decoherence timeshave been observed in isolated DQDs [1] compared to thestructures with dots formed by the gates [2].

2 Model We use the Finite Difference Method tosolve the Schrodinger and Poisson equations for arbitrary

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

phys. stat. sol. (c) 5, No. 9, 3156–3158 (2008) / DOI 10.1002/pssc.200779304

Figure 1 Strong coupling regime. A: Total charge per quantumdot (unit e) vs gate-induced potential difference (meV) betweenthe dots in the strong coupling regime. Changes are fractions of 1eand smooth. B: Electron density (see Eqn. (1)) shows little changethroughout. C: Wavefunctions (1st level at 0 meV, 8th level at -30meV) show a high level of delocalisation.

potentials on a 3D grid. A novel method is used to solvethe Poisson equation in free space: numerical integrationis used to determine the boundary values, which are thenapplied as Dirichlet boundary conditions to the Poissonequation and solved as a linear algebra problem. The DQDpotential cross-section is approximated by a spiric section,with the effect of an applied electric field as a potential dif-ference between the dots. We use full anisotropic effectivemass Density Functional Theory, enabling the simulationof 5-100nm Quantum Dots. Surface charges around theDQD at the Si/SiO2 interface are taken into account as aregion of continuous charge near the interface.

2.1 Density functional theory The electron den-sity n(r) is given by:

n(r) =N∑i

|ψi(r)|2 (1)

where ψi is the wavefunction of the ith level, and the sumis taken over all occupied states.

Figure 2 Weak coupling regime. A: Total charge per quantumdot (unit e) vs gate-induced potential difference (meV) betweenthe dots in the weak coupling regime. A significant change ofthe total dot charge which is ∼1e in size is clearly visible. B:Electron density shows significant change around 0-15 meV. C:Wavefunctions (1st level at 0 meV, 8th level at -30 meV) show ahigh level of confinement.

It has been shown that to take into account the many-body effects in the case of N electrons in the DQD it ispossible to use the Kohn-Sham equations [3]. In this casethe many-body equations can be reduced to an equationsimilar to the single electron Schroedinger equation withan effective Hamiltonian, which depends on the electrondensity (as a density functional). For electrons in Si con-duction band minima we can use the anisotropic effectivemass approximation. Consequently, the anisotropic versionof the Kohn-Sham equations in the effective mass approx-imation becomes:

−h2

2m0

[1

mx

∂2

∂x2+

1my

∂2

∂y2+

1mz

∂2

∂z2

]ψi(r)

+ Veff (r)ψi(r) = εiψn(r) (2)

where m0 is the electron rest mass, εi is the energy of theith level and m∗

x, m∗y , m∗

z are the effective masses in eachdirection. The equation is solved for each valley of theanisotropic effective mass, with one direction taking the

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Article

phys. stat. sol. (c) 5, No. 9 (2008) 3157

longitudinal value - e.g. mx = ml, my = mt, mz = mt).The values used are: (transverse) mt = 0.1905, (longitudi-nal) ml = 0.9163.

The effective potential Veff is given by

Veff (r) = Vext(r)

+1ε

∫ρe[n(r′)] + ρsc(r′) + ρimp(r′)

|r − r′| dr′

+∂Exc[n]

∂n(3)

where n is the electron density, Vext is the external po-tential of the system (for example, the electric potentialinduced by the nearby gates), ε is the dielectric constantof Si; ρe[n] is the charge density from the electron den-sity (ρe(r) = e.n(r)), ρsc is the surface charge densityand ρimp is the impurity charge density. The second termforms the direct Coulomb interaction contribution and thefinal term is that of the Exchange-Correlation contribution;Exc[n] is the Exchange-Correlation energy functional ofthe density.

We solve Equations (2)-(3) self-consistently to deter-mine the wavefunctions and electronic structure of the sys-tem. We use the Local Density Approximation for the Ex-change Correlation interactions, as given in [4]. An opti-mised algorithm is used to accelerate convergence towardsself-consistency, based on a system of differential weights.

2.2 Limitations of the method We briefly discussthe limitations of the Local Density Approximation andDensity Functional Theory with regard to the DQD struc-ture considered in this work. In the weak coupling scheme,the assumption that the behaviour will be similar to thatcalculated for a 2DEG [4] breaks down. Specifically, it hasbeen shown [8] that the LSDA (and by extension, the LDA)overestimate the energies due to a self-interaction error,which can be overcome in the Heitler-London [5] method.This method is, however, only able to cope with few elec-trons, whereas DFT is more appropriate for larger numbersof electrons [8]. We are also considering a system with nomagnetic field, and thus no spin effects.

DFT schemes are also known to fail to account forthe London dispersion [6] forces in the correlation inter-action. Correlation errors prevent reliable estimation of theinteraction energy, showing the inability of DFT to recovershort range second order contributions. However, unlikethe Heitler-London scheme, DFT makes no assumptionabout the wavefunctions involved and even in the weakcoupling regime considered here, the wavefunctions calcu-lated for the higher energy electrons are delocalised acrossthe two dots to some extent, as there is no potential bar-rier between the dots. The weak coupling here is thus notas weak as the ’weak coupling’ referred to in the litera-ture (where Heitler-London methods are employed), andthe distance between the dots is sufficiently small that theself-interaction errors [8] are unlikely to be an issue.

3 Results We look at the effects of DQD geometryon charge density re-distribution induced by the electricfield from the nearby gate on 60 nm diameter dots. We de-fine the charge per dot as:

Qpd =∫vd

[ρ(r) + ρimp(r) + ρsc(r)] dr (4)

where vd is the volume region of one dot, delineated by theslice of the x = 0 plane; ρimp is the impurity charge den-sity and ρsc is the surface charge density. Thus this gives avalue of the total charge in each dot.

Increasing the surface charge and/or narrowing theneck between the dots reduces the coupling between thedots, pushing the device towards the weak coupling regime(Fig. 2). In this regime the electrons move suddenly be-tween the dots after some threshold potential difference isreached — graph A in Fig. 2 changes significantly onlyonce, and the change is ∼1e of charge. By contrast in thestrong coupling regime, the wavefunctions are delocalizedacross the whole DQD (Fig. 1C), and only slight shiftsin the electron density are seen - the charge per dot neverchanges significantly (less than 0.1e) between values of thepotential difference, and the total charge variation is small.

4 Conclusions In this paper we have developed aphysical and numerical model to calculate the electronicstructure of DQD devices for quantum computing appli-cations. We have demonstrated that in the weak couplingregime electrons move wholly from one dot to the otherwhen changing the potential difference between the dotsby changing the gate voltage. In the weak coupling regimethe wavefunctions for a given state are highly confined toa single dot. In the strong coupling regime, however, theDQD acts more like one large dot, and the electrons flowsmoothly from one dot to the other under the variation ofthe potential difference between the two dots.

Acknowledgements We thank Hitachi Cambridge Labo-ratory and the EPSRC for financial support.

This work was partly supported by Special CoordinationFunds for Promoting Science and Technology.

References

[1] J. Gorman, D. G. Hasko and D. A. Williams, Phys. Rev. Lett.95, 090502 (2005).

[2] T. Hayashi, T. Fujisawa, H.-D. Cheong, Y.-H. Jeong, and Y.Hiragama, Phys. Rev. Lett. 91, 226804 (2003).

[3] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).[4] J. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).[5] C. Herring, Rev. Mod. Phys. 34, 631 (1962).[6] S. Kristyan and P. Pulay, Chem. Phys. Lett. 229, 175 (1994).[7] T. R. Walsh, Phys. Chem. Chem. Phys. 7, 443 (2005).[8] A. Wensauer, O. Steffens, M. Suhrke, and U. Rossler, Phys.

Rev. B 62, 2605 (2000).

3158 P. Howard et al.: DFT calculations of Si DQDs

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