La influencia de los Antecedentes Penales

separations as large as15 nm can be explored within reasonable timescales. Moreover, physical insight is inevitably lost: the authors find that a six-parameter pseudopotential is needed to fit the ground state energy, while the binding of the excited1slevels is signifi- cantly underestimated.

Not only will the multi-valley EMT presented in this chapter allow the entire 1s man- ifold of the binding energies of the isolated donor electron to be accurately described, but crucially it also provides a correct description of the hyperfine coupling to the donor nucleus, as measured in experiments. The most important consequence is our finding that the spread of the electronic wavefunction was significantly overestimated by previ- ous treatments, which only relied on fitting of orbital energies. Building on this result, we will show how the anisotropy of the donor wavefunction leads to a suppression in the oscillatory nature of the exchange coupling, especially for certain geometries. In addition, our theory is much less numerically intensive and easily adaptable to more complicated electromagnetic environments and different donor species.

4.4

Our method

Our method proceeds as follows: first of all we solve our multi-valley EMT equation, Eq. 3.41 variationally to find the donor stateΨ(r)as defined by Eq. 3.24. For each trial calculation, we first fixb and B in the pseudopotential 3.30, then we use the following ansatz for the envelopes [KL55b]:

F±z = s 1 πa2 DbD exp − s x2_+y2 a2 D + z 2 b2 D ! , F±x = s 1 πa2 DbD exp − s z2_+y2 a2 D + x 2 b2 D ! . (4.20)

We now minimize the expectation values of the energies of the three split 1s levels ac- cording to Eq. 3.41 by varying aD and bD separately for each of the singletA1, triplet

T2 and doubletE eigenstates. We then find the best values ofb andB by finding a good

match between our predictions and measured ground state donor energy (Table 3.1) and hyperfine coupling (Table 3.3) for Si:P. Each valley-orbit state is separately minimized, but the pair of model parameters is the same for all the states of one given donor.

Fitting the contact Fermi interaction is a novel requirement of our multi-valley EMT solu- tion of the donor Schr¨odinger equation, that had been previously overlooked by all EMT [LK55, NS71, SN76, ABG+_{81, KHDS02b, KR14] and ab initio [WH05] approaches.}

It was suggested first by Ref. [Hui13], but the envelopes used there were spherically isotropic, i.e. simpler than those in Eq. 4.20. We will see how the two effects, anisotropy

CHAPTER4: EXCHANGE COUPLING BETWEEN DONORS IN SILICON

inherent in the silicon conduction band and the central cell correction, the latter imple- mented by accounting for the hyperfine couplingA, conspire to modify significantly the state-of-the-art predictions of the exchange splitting J. In fact, it is clear that the sys- tem has a broken symmetry along the vector connecting the two donors, thus it can be anticipated that isotropic envelopes provide predictions of exchange coupling that can be even qualitatively different to those we present here. Moreover, for the sake of evaluation of inter-donor exchange, it is obviously important to have a correct picture of the ‘far’ wavefunctions, that is the spatial region of electronic density with radius larger than a few nm from the nuclear position. This is both ensured by the EMT treatment, originally designed exactly for such a goal, and by the successful description of the binding energy of the ground electronic state. But it is also essential to capture a correct average of the short-range behaviour, which will clearly affect the weight of the tail of the wavefunction that contributes to the overlap between two neighbouring electronic densities.

The integrals involved in the computation have been reduced to quasi-analytical shape, which ensures really fast computing times and numerical precision up to 0.5%. Every nu- merical integration in this stage has been performed with Wolfram Mathematica software. Withb = 19.96 nm−1andB = 246.1 nm−1the optimized Bohr radiiaD = 1.15 nm,bD = 0.61 nmlead toEA1 = −45.5 meV,ET2 =−36.0 meV, EE = −33.0 meV, which must

be compared with the experimental [RR81]EA1 = −45.59 meV, ET2 = −33.89 meV,

andEE =−32.58 meV: the agreement is excellent for the singlet, very good for the dou-

blet, more modest but not unacceptable for the triplet.

In addition, we can fit the value of the squared electron wavefunction at the donor nu- cleus|ψ(0)|2_{, which is proportional to the hyperfine coupling between the impurity nu-}

cleus and the donor electron, by expressing it as|ψ(0)|2 _{≈ 6η|F(0)|}2_{: here, following}

Ref. [KL55a],

η=|u0(k0,0)|2/h|u0(k0,r)|2iunit cell = 186±18.

We set this to match the |ΨA1(0)|

2 _{= 4.4×10}29_m−3 _{[FYPM54] extracted from exper-}

imental measurements of the hyperfine constant. EMT is in fact unreliable for the uk part of the total wavefunction Ψ(r), since near the nucleus the assumptions of smooth potential fail completely, as discussed in the previous chapter. However, we are only in- terested here in the specific value|ψ(0)2_{| at the nuclear site, which has been estimated}

through several different methods: in Ref. [KL55a] Kohn provides an approximate value ofη=|ψ(0)|2_/|ψ(r)|2

Av≈186, where|ψ(r)|2Avis the mean value of the squared modulus

of the electron wavefunction over one primitive cell of the Si lattice (usually normalized to 1). We adopt this value ofηhere. This result is based on a rough estimate of the value of the conduction band Bloch functions at the doping nuclear position, as dictated from an Hartree-Fock theory of undoped silicon, combined with a piecewise crude description of the donor wavefunction both close and far from the impurity cell.

4.4. OUR METHOD

That work however does not rely on any assumption about the nature of the binding poten- tial. Nonetheless, differentηvalues proposed by more recent Density Functional Theory [APC+11], that are further corroborated by old experiments [Wil64], are only shifted to η= 159.4±4.5. With reference to our calculations, such an update would lead to slightly different Bohr radiiaDandbD of the ground donor state (about 5%), but would not change

any of the conclusions presented later: thus we assumeη = 186.

A density plot of the resulting electronic cloud corresponding to the donor ground state is presented in Fig. 4.1, where the two cases are considered of a donor pair displaced along a [100] and a [110] crystallographic direction. The pictures thus obtained are directly compared to analog calculations based on the much simpler Kohn-Luttinger envelopes: the large discrepancies between the spatial extent of the two wavefunctions are evident.

Figure 4.1: Plots of the spatial electronic densities around two adjacent implanted donor nuclei, in a plane containing the separation vectord. The two panels above refer todalong [100], those below to [110]. Left panels are calculated with the wavefunctions used in the MV EMT theory, and show stronger concentration of the density around nuclei (hence larger hyperfine coupling) than the right panels which use KL wavefunctions [KL55b]. Red dots highlight the positions of the Si nuclei of the underlying lattice. The mismatch between their locations and the extrema of the density is a result of the nontrivial structure of the Si conduction band.

CHAPTER4: EXCHANGE COUPLING BETWEEN DONORS IN SILICON

Figure 4.2: The exchange splitting between electrons pertaining to adjacent Si:P donors is shown as a function of their separation d along the crystallographic [100] axis. The solid lines are guides for the eye that join the calculated points. Comparison between the data obtained with a simple donor wavefunction with KL envelopes [KHDS02b] (blue crosses) and our multi-valley EMT equation (red stars) is provided. The range of different drefers to the realistic uncertainty in the resolution of the placement of donors in the Si layer during the implantation process. Two kinds of discrepancy are evident: 1) the KLJ is always larger than the equivalent MV EMT point by more than one order of magnitude, and 2) oscillations are washed out completely in the MV EMT case.

4.5

Results

The single electron states just determined are ready to be applied to the Heitler-London calculations discussed in Sec. 4.2.2. The two-particle integrals were computed with a fast Monte-Carlo algorithm for adaptive multidimensional integration (cubature) [Joh10], and each data point took only a few minutes to compute. Figs. 4.2 and 4.3 show our evaluation ofJfor donor separationdin the [110] and [100] spectroscopic directions, compared with corresponding values we determined using KL envelopes [KL55b].

The biggest difference between the two theories lies in the magnitude of the exchange splittings: the extra localization in real space provided by the strong short range potential of the impurity for MV EMT leads to a shrinking of the effective Bohr radii of the ground state envelopes (aD = 1.15 nm, bD = 0.61 nm), when compared to KL (aD = 2.509 nm,

bD = 1.443 nm [KHDS02a]). This is illustrated by the electron density plots shown in

Fig. 4.1. Another striking discrepancy arises between the amplitude of the oscillations of J(d)as predicted by the two theories: let us delay a justification of this effect until after the presentation of a comparison of the same quantity with the more recent BMB results of Ref. [WH05], as displayed in Fig. 4.4.

4.5. RESULTS

d(nm)

Figure 4.3: The exchange splitting between electrons pertaining to adjacent Si:P donors is shown as a function of their separationdalong the crystallographic [110] axis. The solid lines are guides for the eye that join the calculated points. As compared to Fig. 4.2, the ratio of the KL over MV EMT data is even larger, and the oscillations are now still visible in the MV EMT calculations, even though they are still significantly reduced.

For the [110] direction, the same qualitative behaviour predicted in [WH05] is apparent, but we find that the overall strength of the interaction was still overestimated by more than one order of magnitude. More importantly, we find again shallower oscillations: to explain this, consider again the form of the indirect exchangeCI(introduced in Eq. 4.19),

that has the same valley structure as the complete exchange splitting, but has the ad- vantage of clear analytical structure. The longitudinal C_Iµν, where either kµ or kν has

some component alongd, give oscillating contributions toJ(d) and are responsible for the large oscillations apparent in the KL (and BMB) cases. The transverse C_Iµν where

kµ·d=kν ·d= 0decrease monotonically withd.

Owing to the large difference between longitudinal and transverse effective masses in Si used in MV EMT, our envelopes are very anisotropic: we getaD/bD ≈ 1.90, compared

to KL’s 1.74. To explain why anisotropy gives a great suppression of the oscillating terms in MV EMT, let us simplify further and consider the overlap integralS(d), since it has been shown that the valley structure ofS2 _{follows closely that ofC}

I and thus J. More-

over, both the envelope overlap parts of the r1 andr2 integrands in Eq. 4.19 are peaked

in the region between the two donors - i.e. for the same values of r1 and r2. The de-

nominator of the integrand has its largest value whenr1 −r2is small; it can therefore be

shown that theddependence of the indirect exchange integralCI, is dominated by that of

S2_{(d)[KHDS02b]. This is true so long as {a}

CHAPTER4: EXCHANGE COUPLING BETWEEN DONORS IN SILICON

Figure 4.4: The exchange coupling between electrons pertaining to adjacent Si:P donors is shown as a function of their separationdalong the crystallographic [110] axis. The solid lines are guides for the eye that join the calculated points. Comparison of the data obtained in [WH05] using the BMB exact solution of the donor Hamiltonian and our multi-valley EMT equation is provided. The range of differentdrefers to the realistic uncertainty in the resolution of the placement of donors in the Si layer during the implantation process.

for all results presented here.

Thus, if we take e.g.dparallel to [110],

∂ ∂dlog(S l µµ) / ∂ ∂dlog(S t νν) ∝ p a2 D+b2D bD √ 2 >1, (4.21)

where the superscriptsl andtlabel respectivelylongitudinalandtransverseterms, as de- fined before. Hence the oscillating longitudinal terms decay more quickly withdthan the transverse ones; asdincreases, oscillations are smoothed out. Anisotropy plays a key role in this effect, and this is far more evident within MV EMT, where our fitting of hyperfine coupling results in a spread of the donor wavefunctions that is much smaller than those in KL or BMB. Withddirected along the [110] direction, even though 32 of the 36 terms in Eq. 4.19 are transverse, these are heavily suppressed and the oscillations then appear shallower in MV EMT than in the other theories.

Even more striking is the form ofJ(d)when the separation lies precisely along the [100] direction, as shown in Fig. 4.2. Thanks to the higher symmetry in this case, only four of the 36C_Iµν(d)are associated with oscillations, and these are suppressed to such an extent that the exchange is now monotonically decreasing.