Electron exchange coupling for single-donor solid-state spin qubits

Electron exchange coupling for single-donor solid-state spin qubits

C. J. Wellard,¹L. C. L. Hollenberg,¹F. Parisoli,^1,2L. M. Kettle,³H.-S. Goan,⁴J. A. L. McIntosh,¹and D. N. Jamieson¹

1Centre for Quantum Computer Technology, School of Physics, University of Melbourne, Victoria 3010, Australia

2Department of Physics, University of Bologna, Bologna 40126, Italy

3University of Queensland, QLD 4072, Australia

4University of New South Wales, Sydney NSW 2052, Australia

共Received 18 March 2003; revised manuscript received 24 June 2003; published 24 November 2003兲 Intervalley interference between degenerate conduction band minima has been shown to lead to oscillations in the exchange energy between neighboring phosphorus donor electron states in silicon关B. Koiller, X. Hu, and S. Das Sarma, Phys. Rev. Lett. 88, 027903共2002兲; Phys. Rev. B 66, 115201 共2002兲兴. These same effects lead to an extreme sensitivity of the exchange energy on the relative orientation of the donor atoms, an issue of crucial importance in the construction of silicon-based spin quantum computers. In this article we calculate the donor electron exchange coupling as a function of donor position incorporating the full Bloch structure of the Kohn-Luttinger electron wave functions. It is found that due to the rapidly oscillating nature of the terms they produce, the periodic part of the Bloch functions can be safely ignored in the Heitler-London integrals as was done by Koiller, Hu, and Das Sarma, significantly reducing the complexity of calculations. We address issues of fabrication and calculate the expected exchange coupling between neighboring donors that have been implanted into the silicon substrate using an 15 keV ion beam in the so-called ‘‘top down’’ fabrication scheme for a Kane solid-state quantum computer. In addition, we calculate the exchange coupling as a function of the voltage bias on control gates used to manipulate the electron wave functions and implement quantum logic operations in the Kane proposal, and find that these gate biases can be used to both increase and decrease the magnitude of the exchange coupling between neighboring donor electrons. The zero-bias results reconfirm those previously obtained by Koiller, Hu, and Das Sarma.

DOI: 10.1103/PhysRevB.68.195209 PACS number共s兲: 71.55.Cn, 03.67.Lx, 85.30.De I. INTRODUCTION

Solid state systems have emerged as a promising candi- date for the construction of a large scale quantum computer 共QC兲. In particular, spin based architectures take advantage of the relatively long spin dephasing times of donor electrons or nuclei in silicon. Single qubit operations are performed by tuning the spin Zeeman energy to be resonant with an oscil- lating field which drives the transition while neighboring qu- bits are coupled via the electron exchange interaction, whether it be directly in the case of electron-spin proposals,¹ or indirectly in the case of nuclear spin quantum computers.

In this work we concentrate on the Kane² concept of single phosphorus donor nuclear spin qubits embedded in a silicon substrate, which is a leading candidate in the search for a scalable QC architecture. The Kane architecture共shown in Fig. 1兲 calls for the placement of phosphorus donors at substitutional关face-centered cubic 共fcc兲兴 sites in the host sili- con matrix, with interdonor spacings of order 200 Å. Quan- tum logic operations on the nuclear-spin qubits are imple- mented through coherent control of the donor electron wave functions which are coupled to the donor nuclei through the contact hyperfine interaction. This control of the electron wave functions is achieved through application of voltage biases to control gates placed on the substrate surface above (A gate兲, and between (J gate兲 the phosphorus donors, which create electrostatic potentials within the device thus altering the form of the electron wave functions.

In a recent paper, Koiller, Hu, and Das Sarma 共KHD兲³ presented theoretical evidence for oscillations in the electron-mediated exchange coupling as a function of inter-

donor separation, and a strong dependence of this coupling on the relative orientation of the two donors with respect the structure of the silicon substrate. They calculate the ex- change coupling using an approximate version of the Heitler- London formalism and by essentially ignoring the periodic part of the Bloch wave functions in the expression for the donor electron wave functions. In this article we eliminate both these approximations, by calculating the exchange cou- pling in the full Heitler-London formalism and by including the full Bloch structure of the donor electron wave function.

We show that while the first approximation breaks down for small donor separations, as discussed by KHD themselves,⁴ the second approximation, that of ignoring the periodic part of the Bloch functions, is in excellent agreement with the full calculations regardless of the donor orientation.

FIG. 1. The Kane architecture based on buried phosphorus dop- ants in a silicon substrate.

This paper is organized as follows. In Sec. II we review the calculation of the valence electron wave functions for phosphorus donors in silicon, in the Kohn-Luttinger effective mass formalism. In Sec. III we discuss the Heitler-London formalism used to calculate the exchange coupling between neighboring donor electrons, and we show that the terms that arise due to the periodic part of the Bloch functions oscillate sufficiently rapidly so as to average to zero over the range of the integrals. Section IV contains a discussion of some of the fabricational issues that arise from the strong position depen- dence of the exchange energy on the production of a phos- phorus donor based solid state quantum computer.

Section V is devoted to a calculation of the dependence of the exchange coupling between neighboring donor electrons on voltage biases applied to the control J gate, used to tune the interqubit coupling. By extending the conventional effec- tive mass formalism we construct a basis of nonisotropic hydrogenlike envelope functions in which we expand the donor-electron wave function in the presence of the electric potentials. The potential created inside the device due to the J-gate bias is calculated using a commercial software pack- age specifically designed for the modeling of semiconductor devices. Donor wave functions are obtained by direct diago- nalization, and the Heitler-London formalism is used to de- termine the exchange coupling for various gate biases and donor separations oriented in the silicon关100兴 direction.

II. THE DONOR ELECTRON WAVE FUNCTION Although the qubits of the Kane quantum computer are encoded by the nuclear spins, it is the donor electron which

mediates both single and coupled gate operations. Therefore, the crucial element in the quantum description of the device is the donor electron wave function. Initial attempts to de- scribe the operation of the device, particularly in response to external time dependent gate potentials which complicate the situation considerably, have focused on effective hydrogenic- type approximations for the donor electron wave function,^5,6 or the use of simplified potentials.⁷ These calculations pro- vide some useful estimates, however, more detailed calcula- tions are required, using realistic models of both the donor electron wave function and the potentials induced inside the device by the application of control gate biases.

In going beyond the effective hydrogenic treatment of the ground-state electron wave function for phosphorus donors in silicon it is recognized that the underlying crystal structure of the silicon must have an effect. The wave function is thus expanded in the basis of the Bloch functions for silicon,^8,9

⌿共r兲⫽

冕

The coefficients F(k) are obtained by substituting the above form into the Schro¨dinger equation, with the Hamiltonian H⫽H0⫺U(r), where H0 is the Hamiltonian for the pure silicon crystal and U(r) is the donor-potential for phos- phorus. The Bloch functions can be written in the form

␾k(r)⫽e^ik•ru_k(r), where u_k(r) is a function that shares the periodicity of the lattice, and can be expanded in a basis of plane waves with wave vectors equal to the reciprocal lattice vectors for silicon G, u_k(r)⫽兺GA_k(G)e^iG•r. The result is that the Schro¨dinger equation can be written as

EF共k兲⫽Ek

0F共k兲⫹_G,G

兺

⬘

冕

⫽Ek

0F共k兲⫹_G,G

兺

冕

where U˜ (k)⫽兰U(r)e^⫺ir•kdr is the Fourier transform of the impurity potential. The E_k⁰are the eigenenergies of the Bloch functions ␾k(r), for the pure silicon lattice. We make the approximation that due to the increased energies of the higher bands, only conduction band states contribute to the impurity wave function. Further, in silicon there are six de- generate conduction band minima, located along the 具¹⁰⁰典 directions in k space, 85% of the way between the center (⌫ point兲 and the zone boundary (X point兲. Because of the re- duced energies in these regions the envelope functions can be expressed as a sum of functions localized around each of the conduction band minima F(k)⫽兺_␮F_␮(k).

In the effective-mass treatment^{8 –10}the Bloch energies are expanded to second order around the conduction band minima E_k⁰⬇(ប²/2)(k_⬜²/m_⬜⫹k²_储/m_储), where k_Žis the com- ponent of k⫺k_␮perpendicular to k_␮ and k_储 is the parallel

component. The m_⬜,m_储 are effective masses and the in- equality of these two values reflects the anisotropy of the conduction band minima. An additional approximation is made whereby only the terms with G⫽G⬘ in the potential term of the Schro¨dinger equation are included. The assump- tion is that U(k⫺k⬘^⫹G⫺G⬘^)ⰆU(k⫺k⬘^{) for G⫽G}⬘^{. This} approximation is well satisfied for a Coulombic potential U(k)⫽1/(␬␲²兩k兩²), with␬⫽11.9 the dielectric constant for silicon, and with the reciprocal lattice vectors of magnitude 兩G兩⫽n2␲/d, where n is an integer and d⫽5.43 Å is the lattice spacing for silicon. Another way of viewing the ap- proximation is that the terms e^{i(G⫺G⬘)•r}in the first line of Eq.

共2兲 oscillate on a scale sufficiently short compared to varia- tions in U(r), that they average the integrand to zero. We will show in the next section that the same approximation allows us to ignore the periodic part of the Bloch functions in

C. J. WELLARD et al. PHYSICAL REVIEW B 68, 195209 共2003兲

the donor electron wave functions when performing Heitler- London integrations.

One further approximation is necessary, A_k⬘,G⬇Ak,G, which coupled with the relation 兺G兩Ak,G兩²⫽1 gives the ef- fective mass Schro¨dinger equation

ប²

2

冉

m_储

冊 ^兺

^冕

^兺

⫽E

兺

In the standard effective mass treatment the so-called valley- orbit terms, which couple the envelope functions at different conduction band minima are ignored, and we are left with six independent equations, one for each minimum. With a Cou- lombic impurity potential the solutions are nonisotropic hy- drogenic wave functions of the form

F_⫾z共r兲⫽exp关⫺

冑

冑

, 共4兲

where F_␮(r)⫽兰F_␮(k⫺k_␮)e^ik•rdk, and we have used the example of the envelope function localized around the z minima of the conduction band. The Kohn-Luttinger^8,9form for the electron wave function of a donor situated at any position R is then given by

⌿共rÀR兲⫽

兺

where the periodic part of the Bloch function is independent of the position of the substitutional impurity. The values a_⬜

⫽25.09 Å, a_储⫽14.43 Å are the effective Bohr radii, and are determined variationally.^3,8

It is well known that the valley orbit coupling does con- tribute to the energy of the state,¹¹ particularly if the donor potential is not Coulombic as is the case in the immediate vicinity of the donor nucleus. The consequence of this is that the donor electron binding energy given by this wave func- tion is E⫽28.95 meV, significantly lower than the experi- mental value of E⫽45.5 meV.^12,13 This discrepancy is thought to arise from the deviation of the donor potential from a purely Coulombic potential as well as the effect of a nonstatic dielectric constant in silicon.¹⁴It is expected, how- ever, that at distances of more than approximately an effec- tive Bohr radius from the donor nucleus, the donor potential should be approximately Coulombic and so Eq.共5兲 will pro- vide a good description of the true donor electron in this region.⁸Thus the Kohn-Luttinger form of the donor-electron wave function should be adequate for the purposes of calcu- lating exchange energies for donor separations in the range considered in this article.

A plot of the donor electron wave function along three directions of high symmetry, calculated using effective Bohr radii as determined by Koiller et al.,³ is shown in Fig. 2 for a donor placed at a substitutional lattice site. The coefficients A_k

␮,G, were calculated using the simple local empirical pseudopotential method as outlined in Ref. 15, and a basis of 125 states. This method accurately reproduces the electronic FIG. 2. The solid line shows Kohn-Luttinger wave function for a phosphorus donor electron in silicon, plotted along directions of high symmetry within the crystal. The dotted line shows an isotropic 1s hydrogenic wave function, with an Bohr radius of 20.13 Å.

band structure for silicon, particularly in the region of inter- est for this calculation, the lowest conduction bands, as ob- tained using more complicated nonlocal pseudopotential techniques. The donor electron wave functions obtained in this manner clearly display oscillations produced by the in- terference between the Bloch functions located at the differ- ent conduction band minima. The wave functions are real, and in the关111兴 direction slightly asymmetric, the asymme- try being a consequence of the presence of the second sub- lattice. In the 关111兴 orientation the silicon atoms are not evenly spaced, and so the neighboring silicon atom on one side of the phosphorus donor is much closer than that on the other side. Superimposed over the donor wave function is an isotropic 1s hydrogenic envelope with a Bohr radius of 20.13 Å, reflecting the effect of the superposition of the nonisotropic envelope functions F_␮.

III. THE HEITLER LONDON FORMALISM The two electron Hamiltonian for a system of two donors separated by a vector R, in effective Rydberg units, is

H⫽⫺ ប 2m²“1

2⫺ ប 2m²“2

2⫺ e²

4␲⑀^r1⫺ e² 4␲⑀兩R⫺r1兩 ⫺

e² 4␲⑀^r2

⫺ e²

4␲⑀兩R⫺r2兩 ⫹ e² 4␲⑀兩r1⫺r2兩 ⫹

e²

4␲⑀^{R⫹VSi共r1,r2兲.}

共6兲 In the standard Heitler-London approximation¹⁶one assumes that the lowest energy two electron wave function of the two-donor system is simply a correctly symmetrized super- position of single electron wave functions centered around each of the two donors

⌿_⫾共r1,r₂兲⫽ 1

冑

⫾⌿共r1⫹R/2兲⌿共r2⫺R/2兲兴, 共7兲 where the two donors are located at positions ⫾R/2. This approximation is asymptotically exact and should hold for separations greater than the effective Bohr radii 兩R兩 Ⰷa_⬜,a_储. The antisymmetry of the fermion wave function then tells us that the exchange splitting, the difference in energies between the spin singlet and triplet spin states, is simply equal to the difference in energy between the states

⌿_⫾, that is E_triplet⫺Esinglet⫽E_⫺⫺E_⫹.

In this article we present our results in terms of the ex- change, or J coupling, in the exchange term of the effective spin Hamiltonian J␴ជ1

e•␴ជ2

e, which is common to the quantum computing literature. We make this decision despite the ob- servation that it is the exchange splitting, the energy differ- ence between the single and triplet states, that is most com- monly presented in the solid-state literature. These quantities are related by the expression J⫽(Etriplet⫺Esinglet)/4.

In the Heitler-London formalism the exchange coupling can be expressed in terms of matrix elements of the Hamil- tonian one can rewrite this as

J共R兲⫽1

2关S共R兲²I₁共R兲⫺I2共R兲兴/关1⫺S共R兲⁴兴, 共8兲 where the overlap integrals are given explicitly by

I₁共R兲⫽

冕

⫻H⌿共r1⫹R/2兲⌿共r2⫺R/2兲,

I₂共R兲⫽

冕

⫻H⌿共r2⫹R/2兲⌿共r1⫺R/2兲,

S共R兲⫽

冕

Computation of these expressions using the Kohn-Luttinger wave function without approximation is rather tedious, given the large number reciprocal lattice vectors it is necessary to sum over to evaluate the periodic part of the Bloch functions for each of the six Bloch states in the donor electron wave function. We have used an adaptive Monte Carlo quadrature program to evaluate all overlap integrals, taking due care to obtain reasonable precision.

In Fig. 3 we plot the exchange energy as a function of donor separation in each of the major lattice directions. For comparison the result using an isotropic hydrogenic wave function is also given. We compare our results with those obtained using the method of KHD in their original paper,³ who used two major approximations in order to make the calculation more tractable, some of which are discussed in detail in references.^4,17 First, the Heitler-London expression for the exchange energy is approximated by the Coulomb exchange integral alone, an approximation that is asymptoti- cally quite good, however it is this approximation that is responsible for the difference between our result and that obtained by KHD in Ref. 3. In Ref. 4, the authors have used the complete form for the Heitler-London expression, and obtained results that match those presented here.

The other approximation made is to effectively ignore the contribution of the periodic part of the Bloch functions u_␮(r)⫽1; this is an excellent approximation as can be seen from the figures in which it is almost impossible to distin- guish between the results obtained in this approximation and those for which the detailed Bloch structure was included.

The ability to ignore the periodic part of the Bloch structure in the computation of the Heitler-London integrals signifi- cantly reduces the complexity of the calculation by eliminat- ing the sums over reciprocal lattice vectors. It is worth ex- amining this approximation in more detail as we find it is effectively the same approximation as is made in effective mass formalism to obtain the Kohn-Luttinger form for the donor electron wave function.

The Heitler-London integrals are all of the form

C. J. WELLARD et al. PHYSICAL REVIEW B 68, 195209 共2003兲

I⫽

冕

⫻⌿共r1⫹R/2兲dr1dr₂

⫽

冕

^兺

兺

F_i共r1⫺R/2兲Fj共r2⫹R/2兲

⫻Fl共r2⫺R/2兲Fm共r1⫹R/2兲V共r1,r₂兲

⫻e^{i(ki⫹km⫺kj⫺kl)•R/2}e^{i(km⫺ki)•r1}e^{i(kl⫺kj)•r2}

⫻Ak*_i,G_␣A_k

j,G_␤

* A_k

l,G_␥A_k

m,G_␦e^{i(G␣⫺G␦)•r1}

⫻e^{i(G␤⫺G␥)•r2}dr₁dr₂. 共10兲 The various potentials represented in the above expression as V(r₁,r₂) are all impurity potentials of a similar nature to those encountered in the effective mass formalism, and so we expect that they are sufficiently slowly varying that we can safely ignore the rapidly oscillating terms in the above inte- grand, in exactly the same manner. This immediately gives

I⫽

冕 ^兺

F_i共r1⫺R/2兲Fj共r2⫹R/2兲Fi共r2⫺R/2兲

⫻Fj共r1⫹R/2兲V共r1,r₂兲e^{i(ki⫹km⫺kj⫺kl)•R/2}dr₁dr₂, 共11兲

which is equivalent to setting u_␮(r)⫽1 in Eq. 共10兲. The

‘‘rapidly oscillating’’ terms include the exponentials contain- ing the values of k at the conduction band minima, which are separated by a minimum magnitude of ␦^k⫽2

冑

⫻0.85␲/d. This separation is sufficiently large to allow us to ignore terms except those for which i⫽m,l⫽ j. Since 兺GA_k,G* A_k,G⫽1, this allows us to ignore the periodic part of the Bloch functions in the Heitler-London integrals.

IV. FABRICATION

The observed extreme sensitivity of the exchange cou- pling on the relative orientation of the two phosphorus do- nors sets stringent requirements on the placement of donors for any quantum computer architecture that is reliant on the exchange interaction to couple qubits. In this section we dis- cuss these implications for the construction of a Kane nuclear spin quantum computer.

Fabrication of the Kane solid-state quantum computer is being pursued along two parallel paths.¹⁸ In the so-called

‘‘bottom up’’ approach, individual phosphorus donors are ef- fectively placed with atomic precision on a phosphorus sur- face by application of phosphine gas to a hydrogen termi- nated silicon surface in which individual hydrogen atoms have been removed at the proposed donor site using a scanning-tunneling microscope tip. The hydrogen monolayer FIG. 3. Calculated exchange couplings as a function of donor separations along three high symmetry directions. In each case we plot with a solid line the results obtained when the periodic part of the Bloch function u(r) is included, a dashed line indicates the results obtained with u(r)⫽1. In each plot these lines cannot be separately resolved. The dotted line gives the results obtained when using the approxima- tions of KHD and the dotted-dashed line gives the exchange coupling calculated using a 1s hydrogenic orbital with an effective Bohr radius of 20.13 Å. The asterisks denote the positions of lattice sites.

is then removed and the surface overgrown with phosphorus.¹⁹Although this approach has not been fully de- veloped, it is likely that it will be able to produce an array of donors located at precise fcc substitutional sites, to within a few lattice spaces. Small displacements are inevitable, how- ever, and can have dramatic effects on the exchange coupling between the donors. This is illustrated in Fig. 4 where we show a plot of the exchange coupling as a function of the magnitude of the displacement兩␦兩 of the second phosphorus dopant from its desired position at a fcc substitutional site a distance of 200.91 Å, in the 关100兴 direction, from the first dopant. These dopants have been displaced by no more than four fundamental lattice spacings on either of the two fcc lattices that make up the silicon matrix, however, the ex- change coupling varies by more than an order of magnitude emphasizing the need for precise placement.

The second approach, known as the top down approach calls for the implantation if the phosphorus donors into the silicon using an ion beam of phosphorus ions incident on the substrate.²⁰ The precision of such a technique in the place- ment of dopants is fundamentally limited by the phenom- enon known as ‘‘straggling,’’ whereby the implanted ions scatter from the nuclei of the host silicon atoms. Simulations of this process for a beam of 15 keV phosphorus ions im- planted into silicon, an energy appropriate for an implanta- tion depth of approximately 200 Å into the silicon substrate, give a roughly Gaussian distribution for the final position of the dopant with a variance of about 90 Å in the transverse direction, and 100 Å in the longitudinal direction. Using these data, for two dopants implanted 200 Å apart in the 关100兴 direction, we have calculated the distribution of ex- change couplings between the donor electrons, assuming that after thermal annealing the phosphorus donors take up a po- sition on a fcc substitutional site. Figure 5 shows a plot the integrated probability, 兰J⬁P(J⬘^)dJ⬘, that is the probability that the exchange coupling is greater than a value J, for two dopants implanted in the manner just described.

In the case of electron spin quantum computers where it is the exchange energy that directly couples neighboring qubits,

a coupling of 50 ␮eV corresponds to a gate time of the order of 10 GHz. The top down approach can in this case produce qubits coupled by extremely fast gates with a very high prob- ability. In nuclear spin quantum computers however, the ex- change interaction only mediates the interqubit coupling and the resulting gates are slower. The original Kane proposal calls for an adiabatic implementation of a controlled-NOT gate that requires a exchange coupling of approximately 50␮^eV.2,21 Additional proposals exist for nonadiabatic implementations^22,23for which controlled-NOT gates can be accurately implemented with a lower exchange coupling.

Figure 5 shows the probability that the bare exchange cou- pling for implanted donors being greater than 50␮^{eV, is} about 6%. What is more important than the bare coupling however is the values of the exchange coupling that can be achieved with the application of J-gate biases.

We note that KHD have performed calculations that sug- gest a method for overcoming the strong dependence of the exchange coupling on the donor orientation.⁴They calculate the exchange coupling, in the same approximations as dis- cussed earlier, for phosphorus donor electrons in uniaxially strained silicon. The strain is a product of the silicon host being fabricated on a layer of Si_1⫺␦Ge_␦, and is found to suppress the oscillations in the coupling for donors that lie within a plane perpendicular to the direction of the uniaxial strain. The coupling remains highly sensitive to displace- ments away from this plane.

V. EXTERNAL CONTROL OF THE EXCHANGE COUPLING

Inherent in the Kane proposal for a solid state quantum computer is the ability to control the exchange coupling of neighboring donor electrons through the application of volt- age biases to a control J-gate placed on the substrate surface between the position of the phosphorus donors, as illustrated in Fig. 1. In this section we calculate the effect on the ex- change coupling on the J-gate bias. Some results have been reported on similar calculations. Fang et al.⁷ used an unre- stricted Hartree-Fock method, which avoids some of the ap- FIG. 4. Calculated exchange couplings for donors at fcc lattice

sites that are displaced by a vector␦from their ideal separation of 200.91 Å in the 关100兴 direction. The couplings are plotted as a fraction of the expected exchange coupling J(200.91 Å)

⫽0.18␮eV.

FIG. 5. A plot of the logarithm of the integrated probability for the exchange coupling between donor electrons for donors im- planted 200 Å apart in the 关100兴 direction, using a 15 keV phos- phorus ion beam. The donors are assumed to take a substitutional fcc lattice site.

C. J. WELLARD et al. PHYSICAL REVIEW B 68, 195209 共2003兲

proximations inherent in the Heitler-London approach, to calculate the exchange coupling between phosphorus donor electrons as a function of a simplified electric potential. They use a trial wave function of the form ⌿(r)⫽F(r)␾k_␮(r), including only one of the six degenerate conduction band minima, with a spherical envelope function, that is one of the same form as Eq.共4兲, but with a_⬜⫽a储⫽20 Å. As a result of their single minimum approximation they cannot possibly reproduce the oscillatory nature of the coupling which results from interference between the terms localized at the six minima. They model the potential produced inside the device by the control gate as one-dimensional potential of the form V_J(x)⫽␮关(x⫺R/2)(x⫹R/2)/(R/2)²兴⫻30 meV, where x is the distance is along the direction between the two donors which are situated at x⫽⫾R/2, as defined in Fig. 1.

In our calculation we compute the potential produced in- side the device due to the application of a voltage bias to a metallic J gate by numerical solution of the Poisson equation using a commercial packageTCAD,²⁴designed for the semi- conductor industry. In Fig. 6 we plot the potential parallel to the interdonor axis, for several values of the z coordinate, which denotes the distance below the oxide barrier, located at z⫽⫺200 Å. For more information on the details of the potential calculation see Ref. 25. We compare the potential obtained for a J-gate voltage of 1 V, with the one- dimensional potential of Fang et al. and see that the two potentials agree well for z⫽0, the plane of the donors, pro- vided we set␮⫽0.15. However, we see that for this value of

␮ the two potentials are quite different in the planes 50 Å above and below the donors. We must therefore conclude that the one-dimensional approximation is not a good one.

To calculate the effect of the applied potential on the elec- tron we expand the wave function as follows:

⌿共r⫺R/2,VJ兲

⫽_n,l,m

兺

兺

where the F_␮^n,l,m(r⫺R/2) are nonisotropic hydrogenic enve- lope functions defined by Faulkner.¹²The␮ is a label for the six degenerate conduction band minima and determines the direction of the anisotropy of the envelope function, for ex- ample, F_⫾z^n,l,m(r)⫽F^n,l,m(x,y ,␥z), is the hydrogenic function with a Bohr radius a⫽a_⬜and␥⫽a_储/a_⬜. The orthonormal- ity of this basis is enforced by the e^{i(k␮⫺k␯)•(r⫺R/2)} terms which appear in the matrix elements and due to their rapidly oscillating nature average to zero unless k_␮⫽k_␯. The coef- ficients c_n,l,m are determined by direct numerical diagonali- sation of the Hamiltonian in the presence of the electrostatic potential. We use a basis of 140 states, which includes all states with principle quantum number up to and including n⫽7, and we rescale the energies such that the unperturbed ground-state energy gives the experimentally observed value of 45.5 meV for phosphorus donors in silicon.

The donor electron wave functions obtained in this way are then used in the Heitler-London formalism to calculate the electron exchange energy, as a function of both the donor separation, and applied J-gate bias. The results are plotted in Fig. 7, for donor separations from 80–120 Å in the 关100兴 direction. We see that application of a positive bias will tend to increase the exchange coupling, while a negative bias de- creases the strength of the coupling. It is also worth noting that the relative increase in the exchange coupling produced by a given bias increases with the donor separation. Also the change in coupling is strongest at the peaks, the application of a positive bias enhances the oscillations in the exchange coupling whereas a negative bias of magnitude 1 V inverts the oscillations such that points that were originally at peaks become troughs.

The range of separations shown in these plots is far less than the approximately 200 Å donor separation called for in the original Kane proposal—we have plotted values over range for ease of comparison with our previous results, and more particularly those of KHD. In Fig. 8 we plot the ex- change coupling as a function of the J-gate bias for donors separated by 200.91 Å in the关100兴 direction. At this separa- tion a 1 V J-gate bias can increase the exchange coupling by over two orders of magnitude up to a value of approximately FIG. 6. Potential produced inside the device by a voltage bias of

1 V applied to a metallic J gate. Here x is the interdonor axis, the donors are located at x⫽⫾100 Å and at z⫽0. The potential is independent of the y coordinate, mimicking a very long electrode.

The dashed line is the one-dimensional potential used by Fang et al.

with␮⫽0.15 and shifted to match our potential at x,z⫽0.

FIG. 7. Exchange coupling as a function of donor separation for donors oriented along the关100兴 direction and various J-gate biases.

30␮eV. The exchange coupling can also be reduced via application of a negative bias.

Although it is difficult to compare our results with those of Fang et al. due to the different potentials, the shift of an order of magnitude obtained for separation of 200 Å and a bias of 1 V is greater than the shift they predict. For a value of␮⫽0.15, Fang et al. predict a shift of less than an order of magnitude. This discrepancy can in part be attributed to the 1D approximation for the gate potential which does not ac- count for the fact that the magnitude of the potential in- creases as the J gate is approached from below 共in the z direction兲. We note that it may not be possible to apply volt- ages much greater than one volt to these nanoscale devices without exceeding the breakdown fields of the various mate- rials involved.²⁵

VI. DISCUSSION AND CONCLUSIONS

In agreement with the approximations implicit in the cal- culations of donor electron exchange energy of Koiller, Hu, and Das Sarma,³we show explicitly that the periodic part of the Bloch functions in the donor electron wave functions can be ignored in the Heitler-London integrals, greatly reducing the complexity of the calculation. This has allowed the cal- culation of the intrinsic exchange coupling for a large num- ber of donor pairs distributed according to the probability distribution for donors implanted 200 Å apart by 15 keV ion beams. In this manner we have determined the probability distribution of the exchange coupling for the top-down ap- proach for the fabrication of a Kane solid-state quantum computer.

In addition we have investigated the application of control gate biases to increase the exchange coupling between donor pairs using both realistic potentials and realistic donor elec- tron wave functions. We find that as expected the exchange coupling can be increased by the application of a positive bias, and decreased with a negative bias. Over the range of donor separations investigated it was found that the relative increase of the exchange coupling produced by a given bias increases with the separation, and for donors separated by 200 Å in the 关100兴 direction a 1 V bias can increase the coupling by over two orders of magnitude.

ACKNOWLEDGMENTS

This work was supported by the Australian Research Council and the Victorian Partnership for Advanced Comput- ing Expertise Grants scheme. The authors would like to thank W. Haig 共High Performance Computing System Sup- port Group, Department Of Defense兲 for computational sup- port. H.S.G. would like to acknowledge the support of Hewlett-Packard.

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