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arXiv:1005.1913 · Physical Review B

Spin susceptibility of interacting two-dimensional electrons in the presence of spin-orbit coupling

Robert Andrzej Żak, Dmitrii L. Maslov, Daniel Loss

Published Sep 9, 2010• 43 citations
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Abstract

A long-range interaction via virtual particle-hole pairs between Fermi-liquid quasiparticles leads to a nonanalytic behavior of the spin susceptibility $\ensuremath{\chi}$ as a function of the temperature $(T)$, magnetic field $(\mathbf{B})$, and wave number. In this paper, we study the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of $\ensuremath{\chi}$ for a two-dimensional electron liquid. Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of ${\ensuremath{\chi}}_{zz}$ with $T$ and $|\mathbf{B}|$ saturates at the energy scale set by the SOI, that of ${\ensuremath{\chi}}_{xx}$ $(={\ensuremath{\chi}}_{yy})$ continues through this energy scale until renormalization of the electron-electron 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 $\ensuremath{\chi}$. An immediate consequence of SOI-induced anisotropy in the nonanalytic behavior of $\ensuremath{\chi}$ is a possible instability of a second-order ferromagnetic quantum phase transition with respect to a first-order transition to an $XY$ ferromagnetic state.

DOI
10.1103/physrevb.82.115415

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A long-range interaction via virtual particle-hole pairs between Fermi-liquid quasiparticles leads to a nonanalytic behavior of the spin susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ as a function of the temperature (T)(T)(T), magnetic field (B)(\mathbf{B})(B), and wave number. In this paper, we study the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of \ensuremathχ\ensuremath{\chi}\ensuremathχ for a two-dimensional electron liquid. Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of \ensuremathχzz{\ensuremath{\chi}}_{zz}\ensuremathχzz​ with TTT and ∣B∣|\mathbf{B}|∣B∣ saturates at the energy scale set by the SOI, that of \ensuremathχxx{\ensuremath{\chi}}_{xx}\ensuremathχxx​ (=\ensuremathχyy)(={\ensuremath{\chi}}_{yy})(=\ensuremathχyy​) continues through this energy scale until renormalization of the electron-electron 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 \ensuremathχ\ensuremath{\chi}\ensuremathχ. An immediate consequence of SOI-induced anisotropy in the nonanalytic behavior of \ensuremathχ\ensuremath{\chi}\ensuremathχ is a possible instability of a second-order ferromagnetic quantum phase transition with respect to a first-order transition to an XYXYXY ferromagnetic state.
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 so-called spin-qubit 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). \n \nElectron 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 single-electron spin qubits and ultimately facilitate quantum computing in these systems. \n \nThe 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. \n \nIn the first part we consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of second-order 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. \n \nThe second part deals with the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of the spin susceptibility for a two-dimensional electron liquid. A long-range interaction via virtual particle-hole pairs between Fermi-liquid 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 out-of-plane spin susceptibility with temperature and the magnetic field saturates at the energy scale set by the SOI, that of the in-plane spin susceptibility continues through this energy scale, until renormalization of the electron-electron interaction in the Cooper channel becomes important. We show that the Renormalization Group flow in the Cooper channel has a non-trivial fixed point, and study the consequences of this fixed point for the nonanalytic behavior of the spin susceptibility. \n \nIn the third part we analyze the ordered state of nuclear spins embedded in an interacting 2DEG with Rashba SOI. Stability of the ferromagnetic nuclear-spin phase is governed by nonanalytic dependences of the electron spin susceptibility tensor on the momentum and on the SOI coupling constant. The uniform (zero-momentum) spin susceptibility is anisotropic (with the out-of-plane component being larger than the in-plane one by a term proportional to the SOI coupling to second order in electron-electron interaction). For momenta larger than the SOI coupling, corrections to the leading, linear-in-SOI-coupling term scale linearly with momentum for the in-plane component and are absent for the out-of plane component of the spin susceptibility. This anisotropy has important consequences for the ferromagnetic nuclear-spin phase: (i) the ordered state, if achieved, is of an Ising type and (ii) the spin-wave dispersion is gapped at the vanishing momentum. To second order in electron-electron interaction, the dispersion is a decreasing function of the momentum, and the anisotropy is not sufficient to stabilize long-range 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 momentum-dependence of the in-plane spin susceptibility and thus stabilizing the ordered state, if the system is sufficiently close to (but not necessarily in the immediate vicinity of) the Kohn-Luttinger instability.
We derive analytical expressions for the quasiparticle lifetime $\ensuremath{\tau}$, the effective mass m∗{m}^{*}m∗, and the Green's function renormalization factor ZZZ for a two-dimensional Fermi liquid with electron-electron interaction in the presence of the Rashba spin-orbit interaction. We find that the modifications are independent of the Rashba band index $\ensuremath{\rho}$, and occur in second order of the spin-orbit coupling $\ensuremath{\alpha}$. In the derivation of these results, we also discuss the screening of the Coulomb interaction, as well as the susceptibility and the self-energy in small $\ensuremath{\alpha}$.
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We consider the nonanalytic terms in the spin susceptibility arising as a result of rescattering of pairs of quasiparticles. We emphasize the importance of rescattering in the Cooper channel for the analysis of the temperature dependences in the two-dimensional electron systems in the ballistic regime. In the calculation of the linear in TTT term we use angular harmonics in the Cooper channel, because for each harmonic the interaction amplitude is renormalized independently. We observe, that as a consequence of strong renormalizations in the Cooper ladder, the temperature derivative of the spin susceptibility may change its sign at low temperatures.
We studied the influence of spin-orbit interaction (SOI) in interacting low dimensional quantum systems at zero temperature within the framework of the functional renormalization group (fRG). Among the several types of spin-orbit interaction the so-called Rashba spin-orbit interaction is especially intriguing for future spintronic applications as it may be tuned via external electric fields. We investigated its effect on the low energy physics of an interacting quantum wire in an applied Zeeman field which is modeled as a generalization of the extended Hubbard model. To this end we performed a renormalization group study of the two particle interaction, including the SOI and the Zeeman field exactly on the single particle level. Considering the resulting two band model, we formulated the RG equations for the two particle vertex keeping the full band structure as well as the non trivial momentum dependence of the low energy two particle scattering processes. In order to solve these equations numerically we defined criteria that allowed us to classify whether a given set of initial conditions flows towards the strongly coupled regime. We found regions in the models parameter space where a weak coupling method as the fRG is applicable and it is possible to calculate additional quantities of interest. Furthermore we analyzed the effect of the Rashba SOI on the properties of an interacting multi level quantum dot coupled to two semi innite leads. Of special interest was the interplay with a Zeeman field and its orientation with respect to the SOI term. We found a renormalization of the spin-orbit energy which is an experimental quantity used to asses SOI effects in transport measurements, as well as renormalized effective g factors used to describe the Zeeman field dependence. In particular in asymmetrically coupled systems the large parameter space allows for rich physics which we studied by means of the linear conductance obtained via the generalized Landauer-Büttiker formalism from the renormalized self energy. The multi level quantum dot exhibits Kondo physics at finite Zeeman fields which can be suppressed if the field and the SOI term are not aligned in parallel. Due to broken particle hole symmetry we observed a gate voltage dependence for finite field conductance plateaus. The computed dependencies have been observed in experiments and thus highlight the flexibility of the considered model. Finally we computed the linear conductance for a multi level quantum dot model that has been proposed to describe SOI and impurity scattering effects in carbon nanotubes (CNT). The SOI was seen to affect the already complicated level structure of the CNT and thus profoundly changes the linear conductance, especially the location of conductance resonances with respect to an applied magnetic field and gate voltage. Again asymmetries influence the conductance and the effects are explained by means of the effective tunnel couplings to the eigenstates of the CNT. We finished with an improved analysis of a reduced version of this model that includes two particle vertex renormalization. The computed linear conductance is in excellent agreement with experimental data.
The wave vector and temperature-dependent static spin susceptibility, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$(Q,T), of clean interacting Fermi systems is considered in dimensions 1\ensuremath{\leqslant}d\ensuremath{\leqslant}3. We show that at zero temperature ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$ is a nonanalytic function of |Q|, with the leading nonanalyticity being |Q${\mathrm{|}}^{\mathrm{d}\mathrm{\ensuremath{-}}1}$ for 13, and Q2{\mathbf{Q}}^{2}Q2ln |Q| for d=3. For the homogeneous spin susceptibility we find a nonanalytic temperature dependence ${\mathrm{T}}^{\mathrm{d}\mathrm{\ensuremath{-}}1}$ for 13. We give qualitative mode-mode coupling arguments to that effect, and corroborate these arguments by a perturbative calculation to second order in the electron-electron interaction amplitude. The implications of this, in particular for itinerant ferromagnetism, are discussed. We also point out the relation between our findings and established perturbative results for one-dimensional systems, as well as for the temperature dependence of ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$(Q=0) in d=3.