Collective hydrodynamic magnon transport study in magnetically ordered insulator based on Boltzmann approach

Spintronics¹ is a subject that uses the spin degrees of freedom of electrons, instead of electronic charges, to achieve information transport, processing and storage. Since the 1980s, the development of spintronics has completely changed the modern semiconductor and information industries. The first milestone discovery in spintronics was the giant magnetoresistance (GMR) effect² that was discovered between 1986 and 1989. The three-layer film structure of ferromagnetic metal/paramagnetic metal/ferromagnetic metal generates and controls self-rotational polarization current. Another milestone in spintronics was the 1999 discovery of the Spin Hall Effect (SHE),³ which causes the charge current to generate a transverse spin-polarized current under the spin–orbit coupling of heavy metals. The year 2005 saw the discovery of the reverse spin Hall effect (ISHE), in which a lateral charge current is induced by a spin-polarized current.

Today, spintronics has begun to develop in more diversified directions. With the further development of magnetic dynamics in nanostructures, the interaction between spintronics and magnetic moments has become an extremely important issue. This has opened up a new direction in spintronics: magnon spintronics.

Magnons, or spin waves, have been described as coherent fluctuations in a magnetically ordered structure. However, with the introduction of the quantum form of spin waves, they were shown also to exhibit “quasi-particle” properties similar to phonons and photons. It is precisely the unique and rich physical properties of magnons that make them exhibit great advantages in spintronics devices, and related fields have received extensive attention.

Magnon spintronics research is extremely rich. Within this discipline, the problem of magnon transport is one of the most important topics. Although magnons have both particle and wave properties, as a phenomenon of particle diffusion transport, they have attracted more and more attention in recent years, and this has become one of the most important issues in magnon spintronics.

In 2015, L. J. Cornelissen and B. J. van Wees et al.⁴ found that high-frequency magnons can be excited directly through electron current, and magnons can be transported over long distances in magnetic insulator yttrium iron garnet (YIG). A non-local structure was proposed, which can simultaneously stimulate the “electrically excited” magnon signal in proportion to the current I and the “thermally excited” magnon signal in proportion to I², as can be detected separately. The transport properties of both signals can be described by the one-dimensional diffusion equation

where n_m represents the non-equilibrium magnon density and λ represents its diffusion length. The transport properties can be considered as two stages, diffusion transport (d < λ) represented by 1/d attenuation and the relaxation transport (d > λ) represented by exponential decay.⁵ The limitation of the one-dimensional diffusion equation is that it is only effective when the YIG is thin enough and its length is much larger than λ.

In addition to the above limitations, the diffusion equation cannot explain the suppression of the magnon spin current by the external magnetic field⁶ or the abnormal magnon transport phenomenon when temperature is close to zero.⁷ According to the diffusion model, the boundary permeability of the magnon is almost zero when the temperature is close to zero. The electron spin is almost completely reflected at the boundary. There is no spin angular momentum transfer, and sufficient magnons cannot be excited. As shown in Figure 1, when the temperature is lower than 25 K, the diffusion length of the magnon does not decrease significantly, but there is an extreme point near 7 K.⁷ The transport of the magnon is affected by the thickness of the YIG film, and the sign inversion phenomenon of the “thermally excited” magnon signal measurement voltage is also difficult to explain.⁸ To solve the above problems, we put forward a more general magnon Boltzmann theory based on the existing theories and experimental phenomena, so as to make the necessary corrections and supplements to the magnon diffusion model.

The Hamiltonian of a magnetically ordered system is usually expressed by the Heisenberg model

where S_i is the magnetic moment at any atom i, J represents the exchange constant, g is the Langde factor, μ_B is the Bohr magneton, H_z is the intensity of the external magnetic field along the z-axis. According to J. H. Van Vleck’s Harmonic Oscillator Approximation,⁹ a magnetically ordered system can be considered as harmonic oscillators coupled with their neighbors, where each oscillator has its own coordinate and energy. Using this approximation, and ignoring the high order terms in Eq. (2), we get the magnon’s Hamiltonian

where S is the total magnetic moment of the atom, the vector a connects an atom with its neighbors, n_k is the quantum magnon numbers associated with the magnon with the k-wave vector of the crystal, H_loc, as the equivalent field, includes the external magnetic field H_z and related molecular fields, and E₀ is the ground energy constant. Under the long-wave approximation, the energy of the magnon is expressed as ɛ_k = J_sk² + △, the exchange stiffness is associated with the Curie temperature via J_s = 3k_BT_ca²/π²(S + 1), and △ is the spin-wave gap due to magnetic anisotropy.¹⁰ 

We shall utilize the Boltzmann approach to deal with these magnons’ non-equilibrium phenomena. Considering that the classical distribution function contains the momentum variable in the presence of spatially varying field or temperature gradients, but quantum mechanics says the position of a particle is completely uncertain if the particle processes a definite momentum, the use of classical distribution functions requires that spatial variations occur over length scales large compared with a typical de Broglie wavelength. In cases where this condition is not fulfilled, the density matrix method should be introduced to build a semiclassical Boltzmann equation.

In a non-interacting bosonic system, the destribution of the particles can be expressed as a Bose-Einstein distribution $Nm0=[exp(εk/kT)−1]−1$⁠, where ɛ_k is the single particle energy. When not considering equilibrium, we use the density matrix to express the occupied number

which we can Fourier transform to obtain the distribution function

and equivalent single particle energy

while ɛ(r, r′) is the energy density matrix, which is given by the Hamiltonian Eq. (2).

The equation of motion of the density matrix can be expanded as follows. We use $ρ$ and ⟨H⟩ to represent the density matrix element ρ(r, r′) and the energy matrix element ɛ(r, r′)

which gives us the expression of semiclassic Boltzmann equation

where $vk=1ℏ∂εk∂k$ is the group velocity of the magnon, and U is the potential energy of the magnon, which is only related to the real space coordinates. For the magnon system, the commutative form in the above equation vanishes. An additional term, not included in H of the above equation, will be treated as a collision term and be supplemented on the right-hand side.

Generally speaking, the mechanism of the multi-particle collision process of the magnon can be divided into two types. The first type of scattering redistributes the velocity or momentum of the magnon. This type of scattering process is mainly intrinsic magnon particle number conservation scattering. The second type of scattering is the coupling between the magnon and the environment, in which the magnon relaxes to an equilibrium state by interacting with the crystal lattice.

Among such scattering processes, three-magnon scattering caused by dipole-dipole interaction is a novel phenomenon. As we all know, as a kind of boson, the momentum of the magnon is strictly limited to the first Brillouin zone. If the momentum of the magnon exceeds the first Brillouin zone, a unit reciprocal lattice vector needs to be subtracted from the wave vector so that the scattered wave vector is still in the first Brillouin zone. This process makes the “momentum” of the quasiparticle not conserved, which is called the Umklapp process.¹¹ This will change the total momentum of the magnon subsystem and relax the magnon to the equilibrium state represented by the Bose-Einstein distribution $Nm0=[exp(εkkBT)−1]−1$⁠, thereby changing the overall flow direction of the magnon flow. In another process, the momentum of the magnon is always in the first Brillouin zone, and the total momentum of the magnon subsystem is conserved. This process will relax the magnon system to the displaced Planck distribution $N(λ)=[exp(εk−k⋅λkBT)−1]−1$⁠, which is called the Normal process.¹² At this time, there is an overall drift motion in the magnon subsystem, making the magnon similar to the overall velocity of the “fluid”, and the magnon appears as a “Poiseuille flow”.¹³ The actual magnon system includes two kinds of scattering processes. At this time, the magnon system is no longer in the pure “gas” state, but exhibits a certain “viscous fluid” property, and the transport properties of the magnon are therefore greatly affected. The ratio of these two processes is affected by temperature, because the momentum of the magnon excited at low temperature is not large enough to excite Umklapp scattering. At extremely low temperatures, Umklapp scattering can even be ignored.

Considering all the factors mentioned above, we construct a one-dimensional Boltzmann equation of temperature-dependent magnon transport in the steady state

in which ɛ_k = J_sk² is the energy of magnon, v_k = 2J_sk/ℏ denotes magnon group velocity, τ_N and τ_U represent relaxation time of Normal process and Umklapp process. The left-hand side of the equation describes the diffusion behaviors of the magnon affected by the temperature gradient, and the right-hand side is the relaxation time approximation of the scattering term. It should be pointed out that the first term on the right-hand side of the equation describes not only Umklapp scattering, but also other scatterings that also change momentum, such as boundary, impurity scattering, etc. The difference is reflected in the relaxation time, which will be discussed later. The solution in a semi-infinite interval of the one-dimensional equation is

in which $τc−1=τN−1+τU−1$⁠, δg denotes spin injection, which depends on boundary condition and will not be discussed in this article, and T_m and λ are the temperature of the magnon and the overall velocity of the magnon fluid respectively. We assume that the temperature gradient in the magnon body transport and the overall flow velocity of the magnon fluid vary little with position. Although there is a certain difference from the actual situation, this formulation can reflect the physical nature of one-dimensional magnon transport.

In Eq. (9), the magnon distribution function n_m is expressed as a function of ∇T_m and λ. These three factors are coupled with each other, and in order to separately resolve them, we need other relationships. As mentioned above, momentum is conserved in the N process

Through this condition, we multiply the Boltzmann equation by the wave vector k and integrate it in momentum space, and combine it with Eq. (6) to obtain the relationship between ∇T_m and λ

The relationship between n_m and T_m is connected through the local magnetic moment m_z¹⁴ 

where $V≈23ξ(5/2)4πJskBT03/2$ is the temperature dependence magnetic coherence volume, which leads to

Connecting all the relationships described above, we can get the magnon viscous fluid transport equation

where $δnm(x)=12π∫(nm−Nm0)dk$ is the density of non-equilibrium magnons in real space, and these coefficients are

We use the one-dimensional and semi-infinite interval of the viscous magnon fluid Boltzmann equation to elucidate the inexplicable extreme point of thermally excited magnon diffusion length when the temperature is less than 25 K, which was exhibited in Figure 1. The temperature of the extreme point is about 7 K. At this temperature, the Normal scattering in the magnon system is relatively strong, which means that the inherent viscous resistance of the magnon fluid is relatively weak, and the diffusion length of the magnon is relatively large. When the temperature rises, the momentum in the system is not conserved during scattering, and the impurity scattering, the inelastic boundary scattering, and the magnon-phonon scattering especially contribute to the loss of momentum. Under high temperature, the fluid properties of the magnon are eliminated, and then it appears in the form of gas. This is what we call “magnon gas”. These scattering processes can be combined by the Matthiessen method, and the total relaxation time τ_R can be obtained from the relaxation time of Umklapp scattering τ_U, boundary scattering τ_B and impurity scattering τ_I

According to the literature, these relaxtion times are^15–17 

in which $Td=gμB2/kBa3,Te=2SJ/kB$⁠. For YIG, S = 14.2, T_d = 0.0014 K, T_e = 37.37 K.

We calculated the ratio of the relaxation time of momentum non-conservative scattering, which depends on Eq. (17), and momentum-conserving scattering in a l₀ = 200 nm thick YIG film at low temperature. As we can see in Figure 2, there is a maximum value at 12.4 K. When the temperature is greater than 25 K, τ_R/τ_N → 0, that is, the momentum non-conservative scattering in the magnon system far exceeds the momentum conservative scattering, and the fluid nature of the magnon disappears. When the temperature is less than 25 K, there is a considerable proportion of Normal scattering in the magnon system. At this time, the magnon system can be regarded as a fluid system instead of a gas system. In our calculations, boundary scattering occupies the most important factor in non-conservative momentum scattering at extremely low temperatures. The group velocity of thermally excited magnons is approximately v_g = 3500 m/s, and the relaxation time is on the magnitude of 10⁻⁹ s. The thickness of YIG 200nm is much smaller than the mean free path of the magnon. Therefore, for non-one-dimensional systems, the YIG thickness has a significant effect on the fluid properties of the magnon.

After investigating the temperature dependence between the relaxation time of momentum non-conservative scattering τ_R and momentum conservative scattering τ_N, we substitute them into Eq. (14), and replace τ_U with τ_R to research temperature-dependent magnon transport properties. In Figure 3, we obtain the relationship between the diffusion length of the magnon and the temperature through exponential function fitting. First of all, there is an extreme point of the magnon at 12.40K. This phenomenon has been illustrated in Figure 2. It is mainly due to the existence of a maximum value of R/N at 12.4K, which greatly strengthens diffusion behavior of magnons. The maximum value of our data is at 12.4K, which is relatively close to the 7K in the experiment. Comparing the experimental data with the theoretical results in Figure 3, when the temperature is greater than the extreme point but less than 25K, the theoretical results are exactly the same as the experimental results. In our theory, from 12.4K to 25K, there is a minimum point in the diffusion length. This minimum point is the result of the competition among the relaxation times τ_N, τ_U and τ_c, instead of only relying on the ratio of τ_R/τ_N. It can be seen that our theory is completely reasonable in explaining the transport of magnons at extremely low temperatures.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.