Riki o Konno^1*

¹Kindai University Technical College, 7-1 Kasugaoka, Nabari-shi, Mie 518-0459, Japan.

*Corresponding Author:Riki o Konno, Kindai University Technical College, 7-1 Kasugaoka, Nabari-shi, Mie 518-0459, Japan, Tel:81-595-41-0111; Fax:81-595-41-0111

Citation: Riki o Konno (2024) Self-Consistent Renormalization Theory of Anisotropic Spin Fluctuations in Nearly Ferromagnetic Metals. Nano Technol & Nano Sci J 6: 160.

Received: July 10, 2024; Accepted: July 17, 2024; Published: July 20, 2024.

Copyright: © 2024 Rikio Konno, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

We investigated the temperature dependence of the inverse of the magnetic susceptibility, the nuclear magnetic relaxation rate, and the T-linear coefficient of the specific heat in nearly ferromagnetic metals by using the self-consistent renormalization theory of anisotropic spin fluctuations. At low temperatures, the inverse of the magnetic susceptibility has T²-linear dependence. In elevated temperatures, the inverse of the magnetic susceptibility has T-linear dependence. The nuclear magnetic relaxation rate has T/Y_V - linear dependence where Y_V is the inverse of the reduced magnetic susceptibility. The T-linear coefficient of the specific heat has ln (1+1/Y_0V ) (V = || or ⊥) where Y_0V is the inverse of the reduced magnetic susceptibility at the zero temperature.

The magnetic properties of nearly ferromagnetic metals have intrigued the interest of many experimental and theoretical researchers [1-14]. Recently, the anisotropic spin fluctuations were investigated in nearly antiferromagnetic metals beyond the random phase approximation [15, 16]. However, in the nearly ferromagnetic metals, the influence of the anisotropic spin fluctuations has not been resolved. Therefore, the self-consistent renormalization theory of anisotropic spin fluctuations in the nearly ferromagnetic metals is constructed beyond the random phase approximation in this paper. The inverse of the magnetic susceptibility is investigated. The nuclear magnetic relaxation rate is studied. The T-linear coefficient of the specific heat is examined. Throughout this paper, we use units of energy, such that h=1, k_B = 1, and gμ_B where g-factor of the conduction electron, unless explicitly stated. We assume that the c-axis is the axis of easy magnetization.

This paper is organized as follows: the formulation will be provided in section 2. The numerical results will be supplied in section 3. The conclusions will be given in section 4.

Let's begin the non-interacting dynamical susceptibility. By using Moriya's expression [14] based on the single band Hubbard model, the non-interacting dynamical susceptibility x_0v as follows:

­x_0v (0)is the non-interacting magnetic susceptibility.  q_B is the magnitude of the zone boundary wave vector. From Eq. (2), is is

where Ψ(μ_v) is the digamma function,

where y_v (v = || or ⊥) is the inverse of the reduced magnetic susceptibility.  y ||, y ⊥ are parallel to the c-axis and perpendicular to c-axis, respectively. By following Ref. [6]

From Eqs. (9) and (10), the equations of the inverse of the reduced magnetic susceptibility are obtained.

Figure 1: The temperature dependence of the inverse of the reduced magnetic susceptibility y_v (v = || or ⊥) when y_0|| = 0.01 (the red line),  y_0⊥ = 0.1 (the orange line),  y_1|| = y_2|| =3, and y_1⊥ ,=  y_2⊥ = 1 respectively.

Fig.1 shows the temperature dependence of  y_v (v = ||, or ⊥) with y_0|| = 0.01 (the red line), y_0⊥ = 0.1 (the orange line), y_1|| = y_2|| =3, and y_1⊥ ,=  y_2⊥ = 1.  Fig.2 shows the temperature dependence of  y_v (v = ||, or ⊥) with y_0|| = 0.01 (the red line),  y_0⊥ = 0.1 (the orange line), y_1|| = y_2|| =6 and y_1⊥ ,=  y_2⊥ = 1.

Figure 2: The temperature dependence of the inverse of the reduced magnetic susceptibility y_v (v = ||, or ⊥) when y_0|| = 0.01 (the red line), y_0⊥ = 0.1 (the orange line), y_1|| = y_2|| =6 and y_1⊥ ,=  y_2⊥ = 1 respectively.

The inverse of the reduced magnetic susceptibility has T-linear dependence from Fig.1 and Fig.2. At low temperatures t<< 1, we use the following asymtotic expansion of the digamma function in the integrand of Eqs. (11) and (12).

At low temperatures the inverse of the magnetic susceptibility has T²-linear dependence. In elevated temperatures, it has T-linear dependence.  

The nuclear magnetic relaxation rate is studied by using the dynamical susceptibility in the nearly ferromagnetic metals. It is obtained:

where T_1v (v = ||, or ⊥) is a nuclear magnetic relaxation time,  A_hf is the hyperfine coupling constant. γ_n is the nuclear gyromagnetic ratio, and N₀ is the number of the magnetic atom. The nuclear magnetic relaxation rate in the nearly ferromagnetic metal is where g is the g-factor of the conduction electron, and μ_B is the Bohr's magneton.

Figure 3: The temperature dependence of y_0|| = 0.01 (the red line),  y_0⊥ = 0.1 (the orange line),  y_1|| = y_2|| =3, and y_1⊥ ,=  y_2⊥ = 1 respectively.

Fig. 3 shows the temperature dependence of the nuclear magnetic relaxation rate with y_0|| = 0.01 (the red line),  y_0⊥ = 0.1 (the orange line),  y_1|| = y_2|| =3, and y_1⊥ ,=  y_2⊥ = 1 respectively. Fig. 4 shows the temperature dependence of the nuclear magnetic relaxation rate with y_0⊥ = 0.1 (the orange line),  y_1|| = y_2|| =6, and y_1⊥ ,=  y_2⊥ = 1.  From Eq. (20), the nuclear magnetic relaxation rate has t/y_v -linear dependence because of y_v << 1 in nearly ferromagnetic metals. In contrast to nearly ferromagnetic metals, the nuclear magnetic relaxation rate has  - linear dependence in nearly antiferromagnetic metals where y_sv is the inverse of the staggered magnetic susceptibility [16].

The T-linear Coefficient of the Specific Heat

The free energy of spin fluctuations is obtained as follows [5]:

Figure 4: The temperature dependence of  when y_0|| = 0.01 (the red line),  y_0⊥ = 0.1 (the orange line),  y_1|| = y_2|| =6, and y_1⊥ ,=  y_2⊥ = 1 respectively. 

where Γ_qv is the damping constant of spin fluctuations (v= || or ⊥) . The specific heat of spin fluctuations is

From Eq. (24),  γ_m increases when y_0vdecreases. In contrast to nearly ferromagnetic metals, γ_m increases in nearly antiferromagnetic metals when ys0 increases where ys0 is the inverse of the reduced staggered magnetic susceptibility at the zero temperature.

We have made the self-consistent renormalization theory of anisotropic spin fluctuations in three dimensional nearly ferromagnetic metals beyond the random phase approximation. We have investigated the temperature dependence of the inverse of the magnetic susceptibility, nuclear magnetic relaxation rate, and the T-linear coefficient of the specific heat in nearly ferromagnetic metals. We have found that the temperature dependence of the inverse of the magnetic susceptibility has T²-linear behavior at low temperatures. With increasing temperatures, it has T-linear dependence. The nuclear magnetic relaxation rate has t/y_v-linear dependence. The anisotropy appears in the inverse of the magnetic susceptibility and the nuclear magnetic relaxation rate by anisotropic spin fluctuations.

This work is supported by the Kindai University Technical College grants. The author would like to thank Y. Takahashi, D. Legut, and S. Khmelevskyi for fruitfull discussions. He would like to also thank Y. Tokunaga, Y. Haga, H. von Lohneysen, M. Brando, F. Steglich, C. Geibel, J. Flouquet, A. de Visser, S. Murayama, E. Bauer, M. Grosche, K. Gofryk, P. Canfield, F. Honda,K. Ishida and V. Sechovsky for stimulating conversations.

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