Manipulation of a Nuclear Spin by a Magnetic Domain Wall in a Quantum Hall Ferromagnet

The manipulation of a nuclear spin by an electron spin requires the energy to flip the electron spin to be vanishingly small. This can be realized in a many electron system with degenerate ground states of opposite spin polarization in different Landau levels. We present here a microscopic theory of a domain wall between spin unpolarized and spin polarized quantum Hall ferromagnet states at filling factor two with the Zeeman energy comparable to the cyclotron energy. We determine the energies and many-body wave functions of the electronic quantum Hall droplet with up to N = 80 electrons as a function of the total spin, angular momentum, cyclotron and Zeeman energies from the spin singlet ν = 2 phase, through an intermediate polarization state exhibiting a domain wall to the fully spin-polarized phase involving the lowest and the second Landau levels. We demonstrate that the energy needed to flip one electron spin in a domain wall becomes comparable to the energy needed to flip the nuclear spin. The orthogonality of orbital electronic states is overcome by the many-electron character of the domain - the movement of the domain wall relative to the position of the nuclear spin enables the manipulation of the nuclear spin by electrical means.

interaction of electronic and nuclear spins of the domain wall in a QHF which we treat exactly 25,[31][32][33] , beyond the variational mean field description of the domain wall [26][27][28][29][30] . We hence model the ν = 2 state by N e electrons confined to a finite size quantum Hall droplet (QHD) in a perpendicular magnetic field B [31][32][33] . The electrons interact via the contact hyperfine interaction with a nuclear (impurity) spin Mat a position R. The single electron states are σ n m , , with energies ε(nmσ), where n is the Landau level (LL) index, m the intra-LL quantum number, σ = ± 1 the electron spin, and the electron Zeeman energy is comparable to the cyclotron energy, c (see Supplementary Material for details). We note that the orbitals σ n m , , form rings, whose radii increase as m 2 within each LL. With σ + c i ( σ c i ) the electron creation (annihilation) operators on the orbital i ≡ (n, m) and  → =ˆˆM M M M ( , , ) x y z the spin operator of the nuclear spin, the Hamiltonian of electrons and a localized nuclear spin M can now be written as 31 The first term is the electron energy, the second term describes the electron-electron Coulomb interactions and the third term is the Zeeman energy E z IMP of nuclear spin. The last three terms describe the hyperfine interaction of the electron and the nuclear spin, with the matrix elements i j 0 31 . Finally, the term ε σ ∆ i ( ) accounts both for the interactions with the positive background and removal of the finite-size effects. This correction is chosen by ensuring that the Coulomb exchange energy is uniform across the QHD and by balancing the total negative charge of the system by an equivalent number of positive charges (see Supplementary Material for details). For clarity, we restrict here the single-particle spectrum to two lowest Landau levels, shown in Fig. 1(b). The lowest Landau level (LLL) orbitals ε(n = 0, m) are drawn in green and blue, while the second Landau level (2LL) orbitals ε(n = 1, m) are drawn in red and black. With the quasi-degeneracy of the LLL spin up The red curve represents the charge density of a single spin-down electron orbital, while the nuclear spin is marked in black. Right: The simultaneous flipping of nuclear spin and electron spin involving the electron orbital transition, from blue to red, spin, to match the nuclear and electron spin Zeeman energies. (b) The red and blue single-particle electronic states realized in a two-dimensional quantum dot with weak parabolic confinement in a large perpendicular magnetic field, with the cyclotron energy Ω c comparable to the Zeeman splitting E z due to the large electronic Lande factor.
Scientific RepoRts | 7:43553 | DOI: 10.1038/srep43553 orbitals ε(n = 0, m, ↑ ) (blue in Fig. 1(b)) and the 2LL spin down orbitals ε(n = 1, m, ↓ ) (red), the energy to flip the spin and change LL orbitals of one electron is comparable with the energy to flip the nuclear spin. However, we have not one but N e electrons, with the spin-down LLL completely filled, and the quasi-degenerate orbitals of spin-up LLL and spin-down 2LL populated partially.

Construction of spin domain states
We start by constructing two states, shown in Fig. 2

Energy spectra of spin domain states
In the following, we present results of model calculations for the QHD with N e = 80, confining energy Ω = .   , shown in Fig. 3(b) in different levels of approximation. The effective Knight field is large in a spin polarized domain in the center of the QHD, and decreases to zero towards the spin unpolarized domain. Interestingly, we find that the domain wall in the Knight shift is much broader than what might be expected from the electron spin alone, shown in Fig. 3(a).

Spin domain states interacting with a nuclear spin
Let us now discuss the electronic spin flip. We are interested in the energy to flip one electron spin in the domain wall state S 2 z , i.e., the difference of energies corresponding to S 2 z and S z + 2 configurations. This energy difference is smallest close to the critical value of = − ⁎ S 2 16 z in this illustration, close to the top of the energy barrier. Hence the degeneracy of the domain wall states at the top of the energy barrier, not the degeneracy of the two electronic domains, gives the electron spin flip energy commensurate with the energy needed to flip the nuclear spin, thus enabling the flip-flop process between the electron and the nuclear spins. In Fig. 4 we switch from the initial state = − ⁎ S 2 1 6 z , depicted schematically in the right-hand diagram of Fig. 4(b), to the final state = − S 2 1 4 z f , corresponding to one electron spin flip Fig. 4(a). The final state, depicted schematically in the left-hand diagram of Fig. 4(b), is also a domain-wall state, but with the domain wall shifted by one orbital towards the center of the QHD. In this transition, the energy of the electronic system decreases, as shown in Fig. 4(a). As a result, the energy of nuclear spin, residing at position R, is increasing with its spin rotating up, as depicted schematically in Fig. 4(b). The probability of this flip-flop process is given by the matrix element of the electron-nuclear interaction part of the Hamiltonian (1)  . We note that this amplitude is exactly zero at the orbital corresponding to the center of the domain wall (m R = m * = 7). This results from the orthogonality of the single-particle orbitals corresponding to the initial occupied and final empty electronic state. As the nuclear spin moves away from the domain wall the tail of the wavefunction leads to finite transition probability. Figure 5 shows the amplitude of the electron-nuclear spin flip-flop as a function of position R of the nuclear spin. The black line gives the amplitude calculated for the HF single spin-domain configurations only. As we add the correlations (blue line), we see that the amplitude is also zero when the nuclear spin is placed at the center of the domain wall, but the amplitude is significantly enhanced for all other positions of the nuclear spin due to electronic correlations, i.e., transitions within the width of the domain wall are contributing.

Summary
We presented here a microscopic theory of hyperfine coupling of a nuclear spin with the spins of electrons in a domain wall of a quantum Hall ferromagnet. We showed that the energy of the electronic spin transition in the domain wall can be brought down to the energy needed to flip the nuclear spin while the amplitude, related to the movement of the domain wall, is enhanced by electronic correlations. This understanding opens the way towards predictive theories of nuclear spin manipulation with electron spin, accounting for material parameters, improved treatment of electron-electron interactions, spin-orbit coupling and strong coupling between many nuclear and electron spins 30,31 .