Nonlocal damping of helimagnets in one-dimensional interacting electron systems
We investigate the magnetization relaxation of a one-dimensional helimagnetic system coupled to interacting itinerant electrons. The relaxation is assumed to result from the emission of plasmons, the elementary excitations of the one-dimensional interacting electron system, caused by slow changes of the magnetization profile. This dissipation mechanism leads to a highly nonlocal form of magnetization damping that is strongly dependent on the electron-electron interaction. Forward scattering processes lead to a spatially constant damping kernel, while backscattering processes produce a spatially oscillating contribution. Due to the nonlocal damping, the thermal fluctuations become spatially correlated over the entire system. We estimate the characteristic magnetization relaxation times for magnetic quantum wires and nuclear helimagnets.
Recently, intense interest has developed in the helical magnetic ordering of one-dimensional (1D) systems of local moments coupled to itinerant electrons. Such systems exhibit a variety of intriguing many-body phenomena, such as spin-Peierls instabilitiesPincus:SSC1971 (); Braunecker:PRB2010 () and induced topological superconductivity,Choy:PRB2011 (); Kjaergaard:PRB20120 (); Klinovaja:PRL2013 () which result from the details of magnetism, electronic structure, and electron-electron interactions. These phenomena may be relevant for a wide variety of physical systems, ranging from magnetic atoms on superconducting Choy:PRB2011 (); Martin:PRB2012 (); Nadj-Perge:PRB2013 (); Vazifeh:PRL2013 (); Simon:PRL2013 (); Nadj-Perge:Science2014 () and normal metal Menzel:PRL2012 () substrates, to single walled carbon nanotubes Braunecker:PRL2009 () and semiconductor-based quantum wires. Braunecker:PRB2009 (); Klinovaja:PRL2013 (); Scheller:PRL2014 ()
While much of the work in this area so far has focused on static and thermodynamic properties of the 1D helimagnets, a richer understanding may be gained by developing and employing new dynamical probes for assessing the behaviors of these systems. For example, an interesting self-tuning effect was proposed for systems dominated by a Ruderman-Kittel-Kasuya-Yoshida (RKKY)-type interaction: Braunecker:PRL2009 (); Braunecker:PRB2009 (); Braunecker:PRB2012 (); Scheller:PRL2014 () the local moments are predicted to order into a spiral arrangement which, through coherent backscattering, gaps out one spin channel of the itinerant electron system for any value of the electron density. This remarkable phenomenon was even suggested as providing a route towards realizing topologically protected Majorana bound states in quantum wires. Vazifeh:PRL2013 (); Klinovaja:PRL2013 (); Simon:PRL2013 () However, because direct probes of magnetization are unavailable for many systems, it can be challenging to positively identify this intriguing magnetic state (necessarily via indirect means). Scheller:PRL2014 () With further theoretical understanding of dynamical responses, such as typical damping or relaxation times, additional tests (e.g., density quenches which change the preferred ordering wave vector) could be used to clarify the natures of the underlying states.
More generally, magnetization relaxation processes determine the magnetic response to external perturbations as well as to spontaneous thermal fluctuations. Furthermore, the nature of the magnetic response is crucially important for noise and magnetization dynamics in magnetoelectronic devices. Bland:book (); Reviews:STT () A better understanding of the spin dynamics in 1D helimagnets may pave the way for exploring phenomena such as current-driven magnetization dynamics, with potential practical applications beyond those envisaged so far. Thus, the investigation of microscopic damping mechanisms is essential for developing a thorough fundamental and practical understanding of these exciting new magnetic systems.
Given the motivations above, in this work we investigate the relaxation of 1D helimagnets via the emission of collective excitations into the interacting itinerant electron system. Note that the 1D nature of the itinerant system is important – our theory is meant to describe quasi-1D systems with a single transverse mode at the Fermi energy (e.g., semiconductor quantum wires Scheller:PRL2014 ()). The elementary excitations of these 1D electronic systems are plasmons, which describe density waves. Interestingly, previous theoretical works have predicted that electron-electron interactions in such 1D systems may play important roles both in establishing ordering Braunecker:PRL2009 (); Braunecker:PRB2009 () and in the relaxation dynamics of weakly-coupled (non-ordered) nuclear spins. Kiss:PRL2011 (); Zyuzin:arxiv () In this work, we use a bosonization approach to study the non-perturbative effects of electron-electron interactions on the damping of ordered spins. We find that interactions have a profound effect on damping, leading to an enhancement of the damping by several orders of magnitude. The damping has a highly non-local character. Consequently, the thermal fluctuations become spatially correlated over the entire sample. We estimate the characteristic magnetization relaxation times due to this mechanism for two classes of systems: (Ga,Mn)As quantum wires and nuclear helimagnets formed in GaAs quantum wires.
Ii Theory and model
Our approach is based on the theoretical framework developed for magnetization damping in metallic ferromagnets. Brataas:PRL2008 (); Brataas:PRB2011 () A key ingredient of the model is that the dynamics of the low-lying collective spin excitations are parametrized by a classical magnetization order-parameter field whose magnitude is assumed to be constant in time and homogeneous in space, while its local orientation is allowed to fluctuate. In this case, the evolution of the spin system can be described by the Landau-Lifshitz-Gilbert (LLG) phenomenology: Bland:book (); Reviews:STT (); comment:CoarseGrain ()
Here, the unit vector parametrizes the local spin-order and is oriented parallel to the magnetization vector , is the gyromagnetic ratio in terms of the g-factor of local spins and the Bohr magneton , and is the effective field found by varying the magnetic free energy functional with respect to the magnetization. The quantity in the first line is a stochastic magnetic field induced by the thermal fluctuations (to be discussed further below). We assume that the free energy functional stabilizes an equilibrium helimagnetic texture of the form Braunecker:PRL2009 (); Braunecker:PRB2009 (); Vazifeh:PRL2013 (); Klinovaja:PRL2013 (); Menzel:PRL2012 ()
where depends on the ordering mechanism. Throughout this work, we use coordinate axes with the -axis oriented along the 1D conductor (see Fig. 1).
Magnetization relaxation is described by the second-rank Gilbert damping tensor in Eq. (1). We consider magnetization relaxation via excitations of the itinerant electron system. In this case, the Gilbert damping tensor is given by Brataas:PRL2008 (); Brataas:PRB2011 () (see Appendix A for a derivation)
where is the Fourier transform of the spin susceptibility of the itinerant electrons, . Here, is the spin-density operator for itinerant electrons, taken in the interaction picture with respect to Hamiltonian (II) below, with the static magnetization (2). Above, is the vector of Pauli matrices and is the spinor-valued fermionic field operator.
We model the itinerant electrons via the Hamiltonian:
where is the momentum operator, is the electron-electron interaction potential, and is the magnetic coupling. Summation over repeated indices is implied.
In the calculation below, we aim to evaluate the Gilbert damping tensor in Eq. (3), using the spin susceptibility for the electronic system described by Eq. (II), with a fixed chemical potential. Linearizing around the Fermi points, we will develop a Luttinger liquid type description of the nearly helical system, allowing interactions to be taken into account non-perturbatively.
We now explicitly calculate the Gilbert damping tensor in Eq. (3). To facilitate calculation of the spin susceptibility, we transform to a non-uniformly rotated frame via the unitary transformation , with . This transformation “untwists” the helix, rendering the free electron part of the transformed Hamiltonian translationally invariant,
while the interaction term is unaffected. In this representation, the spin susceptibility and the Gilbert damping tensor transform to and , where is the SO(3) matrix associated with . The energy dispersion of is shown in Fig. 2; its eigenfunctions are , where is the band index, is the eigenspinor, and , for a system of length .
In this work, we set the chemical potential in the gap that separates the bands near , such that the single-particle dispersion in Eq. (5) features only a single branch of right and left moving modes at the Fermi energy. We neglect interband couplings and write an effective description within the lowest band, linearized about the Fermi wavevectors . We fix the spinor parts of the wave functions to their values at the Fermi energy (Fig. 2). Comment:OneBand ()
To compute the spin-spin susceptibility in the presence of electron-electron interactions, we employ a bosonic description. As a first step, we express the fermionic field operator (projected into the lowest band) as a superposition of fields representing right (+) and left (-) movers: . The fields () take the form , where and the spatial part (in terms of the destruction operators ) is . Substituting the fermionic field operator into the Hamiltonian , performing a Fourier transformation to k-space, and evaluating at momentum zero and for forward- and back- scattering processes, respectively, we obtain
where and are the bosonic fields, is the density wave velocity, and is the Luttinger parameter.
The bosonic representations of the fermionic fields are
where is an infinitesimal short distance cutoff comment1 () and are the Klein factors. The repulsive electron-electron interaction implies that , where for non-interacting electrons.
We calculate the spin susceptibility following the standard approach for Luttinger liquids, see Ref. Giamarchi:book, for details. The resulting (imaginary) time-ordered spin-spin correlation function commentM1 (), , is diagonal and can be written as where
Here, , , , , and . The spin susceptibility is , where is the time-ordered correlation function in real time, which is obtained via the Wick rotation . For , diverges in the low-frequency limit. We regularize the divergence by evaluating the expression at the low-frequency cut-off set by the finite length of the system. comment2b ()
The analysis above gives the Gilbert damping tensor
Here, is the gamma function, , and , where is the modified Bessel function of the first kind, is the modified Struve function, and is the modified Bessel function of the second kind. To obtain the lab-frame damping tensor, the transformation must be applied. We continue the analysis in the rotated frame, where the expressions are much simpler.
The damping consists of two parts with very distinct position dependencies: a homogeneous term and a rapidly oscillating term . The corresponding highly nonlocal magnetization relaxation caused by the plasmon excitations differs markedly from the damping of conventional metallic ferromagnets, which is believed to be local. CommentM2 ()
Constraints of the model reduce the number of independent tensor elements. First, implies that the damping tensor is described by four independent coefficients: , , , and . Second, the constraint imposed by normalization implies that in the rotated reference frame (for small ). Thus, the damping is governed by only two coefficients: and .
What are the characteristics of these two independent damping coefficients? The tensor element originates from forward scattering processes and governs the damping of the long wave-length spin-wave modes. Similarly, is associated with electronic backscattering, and controls the relaxation of short wave-length modes.
Interestingly, is proportional to the momentum-momentum correlator , and is thus inversely proportional to the Luttinger parameter . Consequently, electron-electron interaction enhances as . A much more complex dependency of the electron-electron interactions is seen in the magnetization damping caused by backscattering processes. Remarkably, electron-electron interactions may increase by nearly four orders of magnitude compared to its value in the non-interacting limit (Fig. 3a). The tensor element reaches a maximum at before it drops quickly to zero in the strongly interacting regime . In this limit, the potential energy of the bosonic Hamiltonian (7) completely governs the electron dynamics and density variations of the Luttinger liquid become insusceptible to time variations in the magnetization. The dramatic enhancement of is related to the fact that electron-electron interactions make the damping extremely sensitive to , which measures the ratio of the short distance cut-off to the long distance cut-off (Fig. 3b). In the absence of interactions, approaches a constant value in the limit . However, with interactions, is singular in this limit and the singularity becomes stronger with increasing strength of the interactions.
To investigate the experimental consequences of Eqs. (11)-(13), we discuss thermal fluctuations and estimate the characteristic relaxation time for two classes of systems which are proposed to hold 1D helimagnetic states. Additionally, we propose a method for probing the relaxation time in nuclear wires via transport measurements. The magnetic order is assumed to be to stabilized by the RKKY interaction, implying . Comment_kf ()
The form of the damping tensor has remarkable implications for the statistical properties of thermal fluctuations. While the average of the stochastic magnetic field in Eq. (1) is zero, , its correlations (in accordance with the fluctuation-dissipation theorem) are given in the classical (Maxwell-Boltzmann) limit as Brown:PR1963 (); Brataas:PRB2011 ()
where is the temperature and the average is taken over an ensemble in thermal equilibrium. According to Eq. (14), the thermal fluctuations are correlated over the entire sample. The fluctuations divide into two distinct classes; one class characterized by a spatially constant correlation and a second class characterized by an oscillating correlation. Strongly correlated thermal fluctuations of this form have not been reported or investigated before in any magnetic system, and a thorough investigation of how the associated stochastic field in Eq. (1) influences the magnetization dynamics should be an interesting task for future studies.
For a magnetization precessing at frequency , Eq. (1) yields two characteristic relaxation times and , associated with the homogeneous and oscillatory dissipation terms: . We now estimate for two classes of systems, which are believed to host 1D helimagnetism: Klinovaja:PRL2013 () GaMnAs quantum wire and GaAs wire in which the nuclear spins are hyperfine coupled to the itinerant system (see Appendix B for material parameters). The (Ga,Mn)As and GaAs wire cross sections are assumed to contain unit cells. The characteristic magnon frequency of the first excited mode is , Braunecker:PRB2009 () where is the critical temperature and is the total spin of the cross section. We assume that each unit cell is fully spin polarized. For the magnetic (Ga,Mn)As wire, we find s and s, while for the nuclear wire the relaxation times are s and s.
Dynamical probes offer new routes for characterizing the nature of 1D helimagnetic systems, providing complementary information to that obtained in static (dc) measurements. For example, consider the recent experiment of Ref. Scheller:PRL2014, , in which the conductance of a GaAs-based quantum wire was observed to drop from to when the temperature was reduced to below 0.1 K. The appearance of an plateau hints at the lifting of spin-degeneracy, and was interpreted as evidence of the formation of a nuclear spin helix. Importantly, within the model used to interpret the experiment, the spatial period of the helix tunes itself to be equal to half of the Fermi wavelength, i.e., it is directly linked to the density of the electronic system. Furthermore, the appearance of an plateau is directly linked to this commensuration between the ordering and Fermi wavevectors. A rapid change of backgate voltage which alters the electron density should thus destroy the commensurability; just after such a quench, one would then expect to observe a conductance of , which would gradually return to a value of as the nuclear system finds its new equilibrium order, on a timescale set by the magnetic relaxation rate. Taking similar assumptions and parameter values to those of the model used to support the conclusions of Ref. Scheller:PRL2014, , we predict very long relaxation times for the nuclear magnetic order, on the order of milliseconds or more. Nuclear spin diffusion, another possible important relaxation mechanism, is expected to become effective on longer timescales (see Appendix C). Dynamics on such timescales should in principle be observable in experiments, and therefore may provide crucial independent means for verifying the interpretation of the experiment.
In conclusion, we have developed a theoretical formalism for describing magnetization dissipation of 1D helimagnets via the emission of plasmon excitations. The damping is found to be highly nonlocal and strongly dependent on the electron-electron interaction, differing markedly from the damping of conventional metallic ferromagnets.
Research supported the Danish National Research Foundation. MR acknowledges support from the Villum Foundation and the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement PIIF-GA-2013-627838.
Appendix A Derivation of damping tensor
We calculate the magnetization-damping rate by relating it to the energy absorption rate of the itinerant electron system subjected to small fluctuations about a static helimagnetic profile . In describing the interaction between the magnetic order parameter field and the itinerant electron system, we assume that the magnetization evolves very little over the characteristic timescales of electron dynamics. In this case, the response of the electron system can be calculated using linear response (Kubo) theory.
The total energy dissipation of the magnetic system is
The time derivative is determined by the LLG equation, which yields the energy dissipation
Due to energy conservation, the energy lost by the magnetic system must be gained by the itinerant electron system to which it is coupled. This implies that , where is the Hamiltonian (II) of the itinerant system coupled to the magnetization .
We now use linear response theory to obtain the rate of change of the electronic energy due to a slow evolution of the magnetization : Rammer:book ()
where is the Heaviside step function, and is the perturbing Hamiltonian produced by the small variation of the helimagnetic order. Here, is the spin-density operator , taken in the interaction picture with respect to the unperturbed Hamiltonian.
Fourier transforming Eq. (17) with respect to time and using yields
where is the Fourier transform of the spin susceptibility . To leading order in the precession frequency, the behavior of the energy change is captured by replacing by its value in the zero frequency limit. Transforming back to the time domain and using gives the energy absorption rate
Appendix B Material Parameters
Table 1 shows typical material parameters of (Ga,Mn)As quantum wires and nuclear helimagnets formed in GaAs quantum wires. These parameter values are used in the main text to estimate the characteristic relaxation times. Here, is the length of the quantum wire, is the chemical potential, is the short distance cutoff, comment1 () is the Luttinger parameter, is the density wave velocity, is the spin density of the 1D magnetic system, describes the coupling between the itinerant electrons and the ordered spin system, is the total spin contained in the cross section of the wire, and is the critical temperature of the spin system. Braunecker:PRB2009 ()
The short distance cutoff (given by the chemical potential) together with the length of the wire and the density wave velocity determine the value of the dimensionless parameter, . Because the damping becomes highly sensitive to the values of and in the presence of interactions (Fig. 3), we believe uncertainties in these two parameters govern the sensitivity in the relaxation times estimated from the values in Table 1. While only depends on , the backscattering term grows as becomes small (see Fig. 3), rising sharply for smaller (i.e., stronger interactions). For fixed , the relaxation is strongly enhanced by interactions up to a max value and then falls off sharply.
Appendix C Relaxation via Nuclear Spin Diffusion
Let denote the characteristic magnetization relaxation time induced by nuclear spin diffusion. The spin diffusion constant of GaAs has been estimated to be on the order of m/s. Paget:PRB1982 () The relaxation time is the time required to diffuse from an internal point of the quantum wire to the surrounding nuclei, i.e., a diffusion length of about the wire diameter (where is the lattice constant of GaAs). This leads to following estimate of the relaxation time
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