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Whenever a radiating atom, an ion, or a molecule is placed in a spatial region under the influence of an external electric field, its spectral lines are observed to split [Sta14]. While the Zeeman effect distinguishes different spin eigenstates because of a magnetic field, the Stark effect splits orbital energy levels that couple in a different way to the electrostatic perturbation.

The solution of the problem of a hydrogen atom in vacuum within a uniform and static external electric fieldEhas long been provided by quantum mechanics [Fri90]: the Hamil- tonian is simply modified by an extra potential term−eE ·r, whereris the coordinate of the electron.

Textbook level time-independent perturbation theory correctly predicts, for small fields, quadratic shifts of the ground state energy (we assumeE kz):ˆ

∆E(2) = (eE)2X m6=1

|hψ1|z|ψmi|2

E1−Em

, (5.1)

where{|ψni}and{En}are respectively the orbital unperturbed eigenstates and eigenval-

ues, and the spin degree of freedom is neglected. Linear perturbative terms in |E| ≡ E are prevented by the parity symmetry of the Bohr eigenstates. The energy shifts thus pre- dicted, if opportunely extended to more complicated atoms and molecules, account for their dipole polarizabilities: the perturbed ground eigenstate

|ψ0 1i=|ψ1i −eE X m6=1 hψm|z|ψ1i E1−Em |ψmi (5.2)

CHAPTER5: ELECTRIC AND MAGNETIC FIELD MANIPULATIONS OF DONORS

is no longer a parity eigenstate: on the contrary, it shows an induced dipole moment aligned with theEdirection.

The exact curvature∆E(2)_/E2_{in Eq. 5.1 can only be calculated precisely, though, after an}

infinite sum over all excited orbital states is admixed into the perturbed1sstate in Eq. 5.2. This problem is a reflection of the fact that the Stark potential is not bounded from below (i.e.−eE ·r→ ∞for very large electronic coordinates): the previously bound hydrogen eigenfunctions become states with a finite lifetime that is in fact large enough, for small enough principal quantum number n, to make them still bound states for experimental purposes.

An alternative treatment is supplied by variational theory: theansatzfor the ground state under a uniform electric field [PHSP13] is inspired by the first order perturbative correc- tion to the wavefunction:

ψ(r) = [1 + (q1+q2r)z]e−r/aB, (5.3)

with aB the Bohr radius, r =

p

x2_+y2_+z2_{, while the variational coefficients q} 1 and

q2 represent the weight of higher orbital states coupled to the fundamental one, and are

determined via the principle of minimization of the binding energy of the state in Eq. 5.3. The Stark effect in vacuum is complicated, in the framework of shallow donor states in silicon, by two factors: i) as shown in chapter 3, the zero-field potential felt by the donor electron has a short-ranged impurity potential component on top of the (screened) Coulomb interaction. This modifies the ‘hydrogenic’ response to the field that would be typical of each separate valley as treated within a single-valley approach: there areintra- valley corrections; ii)the non-trivial structure of the silicon conduction band introduces the extra valley degree of freedom, hence it becomes important to account for the overall rearranging of theinter-valley interactionsunderE 6= 0. Our multi-valley EMT provides one of the most straightforward schemes that can capture the interplay between those two features, and also leads to physical insight.

For the purposes of donor electron quantum computing, ESR spin resonance frequencies are related to the energy distance between the logical|0iand|1i, as explained in chapter 2: coherent modifications of such frequencies allow, in principle, for single qubit oper- ations. The ESR spectrum of a group V donor electron in silicon is determined by the Hamiltonian

H =γeB0·S−γnB0·I+AS·I, (5.4)

whereAis the hyperfine interaction between the nuclear spin Iand the electron spinS; γe=

geµB

~

= 27.997 GHz/Tis the magnetic moment of the electron, leading to Zeeman splitting of the states|↑i,|↓iin the microwave frequency range, in magnetic fields of or- der 1 T;γn = 0.007 GHz/Tis the nuclear magnetic moment, yielding Zeeman splittings

5.2. STARK PHYSICS

in the radiofrequency range under the same conditions.

The hyperfine interaction between the electron spinSand the nuclear spinIis more gen- erally described though a coupling tensorA

HHF=I·A·S, (5.5)

but the most relevant part of HHF, which is usually exploited in quantum computing

schemes, is the Fermi contact scalar termAI·S[Kan98], whose values we list in Table 5.2 for all four group V donors. ESR and NMR donor spectra are determined primarily by the interplay between hyperfine and Zeeman splittings, which can result in non-trivial depen- dence of spin transition frequencies on the background magnetic fieldB0, with interest-

ing applications in single qubit control [WTG+_{13]. In particular, the ability to tune these}

resonant frequencies with external electrostatic gates has been exploited in numerous pro- posals [MTB+08, GLR08], with successful experimental realizations already achieved in some cases [MMM10, WUT+14, MWS+10]. In the presence of a modified electrostatic environment, the electronic density can be pulled off the impurity site, and thus the hyper- fine coupling can be altered. The pertinent regime for quantum computing schemes, and which has also been explored by the measurements reported in Ref. [PWU+14], is one of weak fields – of order a few tenths of a V/µm, which is well below ionization thresh- old, as will be demonstrated in Sec. 5.6. As stated previously, in this regime we expect a quadratic dependence onE: ∆A A0 ≡ |Ψ(E 6= 0,r= 0)| 2 |Ψ(E = 0,r= 0)|2 −1≡ηaE 2_{. (5.6)}

Modification of magnetic properties caused by an electric field is not only due to a chang- ing hyperfine coupling: there can also be a spin-orbit Stark effect. The spin-orbit interac- tion can weakly couple the donor ground state to the valley-orbit1s Edoublet; its strength depends on the angular momentum of the donor wavefunction, which is sensitive to (ap- plied gate voltages or strain) perturbations that break the tetrahedral symmetry inherent to the substitutional impurity within the silicon lattice [WF61]. Thegtensor, that defines the response of the donor electron spinSto some external magnetic fieldB,

Hmag=S·g·B, (5.7)

is affected as the weight of different conduction band minima (valleys) in the ground state is reorganized: even this variation is quadratic with the applied electric field. Before this work, the physical mechanisms underlying the Stark effect for donors in silicon were not yet fully understood: let us review the state of the existing knowledge before my new contributions.

CHAPTER5: ELECTRIC AND MAGNETIC FIELD MANIPULATIONS OF DONORS