Żak, Robert Andrzej. Spin susceptibility of twodimensional electron systems. 2012, Doctoral Thesis, University of Basel, Faculty of Science.

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Abstract
A quantum computer, in contrast to traditional computers based on transistors, is a device that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform computation. One of possible realizations is a socalled spinqubit quantum computer which uses the intrinsic spin degree of freedom of an electron confined to a quantum dot as a qubit (a unit of quantum information that can be in a linear superposition of the basis states).
Electron spins in semiconductor quantum dots, e.g., in GaAs, are inevitably coupled via hyperfine interaction to the surrounding environment of nuclear spins. This coupling results in decoherence, which is the process leading to the loss of information stored in a qubit. Spontaneous polarization of nuclear spins should suppress decoherence in singleelectron spin qubits and ultimately facilitate quantum computing in these systems.
The main focus of this thesis is to study nonanalytic properties of electron spin susceptibility, which was shown to effectively describe the coupling strength between nuclear spins embedded in a two dimensional electron gas (2DEG), and give detailed insights into the issue of spontaneous polarization of nuclear spins.
In the first part we consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of secondorder corrections to the spin susceptibility in a 2DEG. We use the Fourier expansion of the scattering potential in the vicinity of the Fermi surface to find that each harmonic becomes renormalized independently. Since some of those harmonics are negative, the first derivative of the spin susceptibility is bound to be negative at small momenta, in contrast to the lowest order perturbation theory result, which predicts a positive slope. We present in detail an effective method to calculate diagrammatically corrections to the spin susceptibility to infinite order.
The second part deals with the effect of the Rashba spinorbit interaction (SOI) on the nonanalytic behavior of the spin susceptibility for a twodimensional electron liquid. A longrange interaction via virtual particlehole pairs between Fermiliquid quasiparticles leads to the nonanalytic behavior of the spin susceptibility as a function of the temperature, magnetic field, and wavenumber . Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of the outofplane spin susceptibility with temperature and the magnetic field saturates at the energy scale set by the SOI, that of the inplane spin susceptibility continues through this energy scale, until renormalization of the electronelectron interaction in the Cooper channel becomes important. We show that the Renormalization Group flow in the Cooper channel has a nontrivial fixed point, and study the consequences of this fixed point for the nonanalytic behavior of the spin susceptibility.
In the third part we analyze the ordered state of nuclear spins embedded in an interacting 2DEG with Rashba SOI. Stability of the ferromagnetic nuclearspin phase is governed by nonanalytic dependences of the electron spin susceptibility tensor on the momentum and on the SOI coupling constant. The uniform (zeromomentum) spin susceptibility is anisotropic (with the outofplane component being larger than the inplane one by a term proportional to the SOI coupling to second order in electronelectron interaction). For momenta larger than the SOI coupling, corrections to the leading, linearinSOIcoupling term scale linearly with momentum for the inplane component and are absent for the outof plane component of the spin susceptibility. This anisotropy has important consequences for the ferromagnetic nuclearspin phase: (i) the ordered state, if achieved, is of an Ising type and (ii) the spinwave dispersion is gapped at the vanishing momentum. To second order in electronelectron interaction, the dispersion is a decreasing function of the momentum, and the anisotropy is not sufficient to stabilize longrange order. However, we show that renormalization in the Cooper channel for momenta much larger than the SOI coupling is capable of reversing the sign of the momentumdependence of the inplane spin susceptibility and thus stabilizing the ordered state, if the system is sufficiently close to (but not necessarily in the immediate vicinity of) the KohnLuttinger instability.
Electron spins in semiconductor quantum dots, e.g., in GaAs, are inevitably coupled via hyperfine interaction to the surrounding environment of nuclear spins. This coupling results in decoherence, which is the process leading to the loss of information stored in a qubit. Spontaneous polarization of nuclear spins should suppress decoherence in singleelectron spin qubits and ultimately facilitate quantum computing in these systems.
The main focus of this thesis is to study nonanalytic properties of electron spin susceptibility, which was shown to effectively describe the coupling strength between nuclear spins embedded in a two dimensional electron gas (2DEG), and give detailed insights into the issue of spontaneous polarization of nuclear spins.
In the first part we consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of secondorder corrections to the spin susceptibility in a 2DEG. We use the Fourier expansion of the scattering potential in the vicinity of the Fermi surface to find that each harmonic becomes renormalized independently. Since some of those harmonics are negative, the first derivative of the spin susceptibility is bound to be negative at small momenta, in contrast to the lowest order perturbation theory result, which predicts a positive slope. We present in detail an effective method to calculate diagrammatically corrections to the spin susceptibility to infinite order.
The second part deals with the effect of the Rashba spinorbit interaction (SOI) on the nonanalytic behavior of the spin susceptibility for a twodimensional electron liquid. A longrange interaction via virtual particlehole pairs between Fermiliquid quasiparticles leads to the nonanalytic behavior of the spin susceptibility as a function of the temperature, magnetic field, and wavenumber . Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of the outofplane spin susceptibility with temperature and the magnetic field saturates at the energy scale set by the SOI, that of the inplane spin susceptibility continues through this energy scale, until renormalization of the electronelectron interaction in the Cooper channel becomes important. We show that the Renormalization Group flow in the Cooper channel has a nontrivial fixed point, and study the consequences of this fixed point for the nonanalytic behavior of the spin susceptibility.
In the third part we analyze the ordered state of nuclear spins embedded in an interacting 2DEG with Rashba SOI. Stability of the ferromagnetic nuclearspin phase is governed by nonanalytic dependences of the electron spin susceptibility tensor on the momentum and on the SOI coupling constant. The uniform (zeromomentum) spin susceptibility is anisotropic (with the outofplane component being larger than the inplane one by a term proportional to the SOI coupling to second order in electronelectron interaction). For momenta larger than the SOI coupling, corrections to the leading, linearinSOIcoupling term scale linearly with momentum for the inplane component and are absent for the outof plane component of the spin susceptibility. This anisotropy has important consequences for the ferromagnetic nuclearspin phase: (i) the ordered state, if achieved, is of an Ising type and (ii) the spinwave dispersion is gapped at the vanishing momentum. To second order in electronelectron interaction, the dispersion is a decreasing function of the momentum, and the anisotropy is not sufficient to stabilize longrange order. However, we show that renormalization in the Cooper channel for momenta much larger than the SOI coupling is capable of reversing the sign of the momentumdependence of the inplane spin susceptibility and thus stabilizing the ordered state, if the system is sufficiently close to (but not necessarily in the immediate vicinity of) the KohnLuttinger instability.
Advisors:  Loss, Daniel 

Committee Members:  Maslov, Dmitrii 
Faculties and Departments:  05 Faculty of Science > Departement Physik > Physik > Theoretische Physik Mesoscopics (Loss) 
UniBasel Contributors:  Loss, Daniel 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  10175 
Thesis status:  Complete 
Bibsysno:  Link to catalogue 
Number of Pages:  149 Bl. 
Language:  English 
Identification Number: 

Last Modified:  22 Jan 2018 15:51 
Deposited On:  26 Nov 2012 15:53 
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