Rikio 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: Rikio Konno (2023) Self-Consistent Renormalization Theory of Anisotropic Spin Fluctuations in Nearly Antiferromagnetic Metals. Nano Technol & Nano Sci J 5: 154. 

Received: July 5, 2023; Accepted: July 11, 2023; Published: July 15, 2023.

We investigated the temperature dependence of the inverse of the staggered magnetic susceptibility, the nuclear magnetic relaxation rate, and the T-linear coefficient of the specific heat in nearly antiferromagnetic metals by using the self-consistent renormalization theory of anisotropic spin fluctuations. At low temperatures, the inverse of the staggered magnetic susceptibility has T²-linear dependence. In elevated temperatures, the inverse of the staggered magnetic susceptibility has T-linear dependence. The nuclear magnetic relaxation rate has T-linear dependence at low temperatures. It has T^1/2-linear dependence in elevated temperatures. The T-linear coefficient of the specific heat has  where  is the inverse of the reduced staggered magnetic susceptibility at the zero temperature.

The magnetic properties of nearly antiferromagnetic metals have attracted the interest of many experimental and theoretical researchers [1-13]. Recently, the anisotropic spin fluctuations were investigated in quasi-one dimensional nearly antiferromagnetic metal beyond the random phase approximation [14]. However, in the three dimensional nearly antiferromagnetic metals, the influence on the anisotropic spin fluctuations has not been resolved. Therefore, the self-consistent renormalization theory of anisotropic spin fluctuations in the three dimensional nearly antiferromagnetic metals is constructed beyond the random phase approximation in this paper. The inverse of the staggered 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  = 1, k_B = 1, and gµ_B = 1 where g is the 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 [13] based on the single band Hubbard model, the non-interacting dynamical susceptibility as follows:

                           (1)

Q is the antiferromagnetic staggered wave vector. The square of the local spin amplitude  is

                                          (2)

                                                                             (3)

with                                                                                                                                                                                                                         

is the non-interacting staggered magnetic susceptibility. q_B is the zone boundary wavelength. From Eq. (2),is

                        (6)

where ψ(u_ν) is the digamma function,                                                                                                                                           

                                                                         (7)

                                                                (8)

By following Ref. [6]

                        (9)

                         (10)

The following inverse of the reduced staggered magnetic susceptibility is introduced., are parallel to the c-axis and perpendicular to c-axis, respectively.        

                  (11)

           (12)

where,                          

Figure 1 shows the temperature dependence of with (the red line),(the orange line),andFig.2 shows the temperature dependence ofwith  (the red line), (the orange line), , and. The inverse of the reduced staggered magnetic susceptibility has T-linear dependence from Figure 1 and Figure 2. At low temperatures t << 1, we use the following asymtotic expansion of the digamma function in the integrand of Eqs. (11) and (12). 

                                         (18)

At low temperatures the inverse of the staggered magnetic susceptibility has T²-linear dependence.

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

 (19)

Figure 1: The temperature dependence of the inverse of the reduced staggered magnetic susceptibility when (the red line),(the orange line),, respectively.

where 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 magnetic atoms. The nuclear magnetic relaxation rate in the nearly antiferromagnetic metal is

        (20)

where g is the g-factor of the conduction electron, and µ_B is the Bohr’s magneton. Fig. 3 shows the temperature dependence of the nuclear magnetic relaxation rate with  (the red line), (the orange line), and Fig. 4 shows the temperature dependence of the nuclear magnetic relaxation rate with  (the red line), (the orange line), and .From Eq. (20),  has T-linear dependence at low temperatures because the inverse of the staggered magnetic susceptibility has T²-dependence at low temperatures. It has T^1/2-linear dependence in elevated temperatures because the inverse of the staggered magnetic susceptibility has T-linear dependence in elevated temperatures. 

Figure 2: The temperature dependence of the inverse of the reduced staggered magnetic susceptibility when  (the red line)(the orange line), , , respectively.

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

                             (21)

with

                                             (22)

where Γ_qν is the damping constant of spin fluctuations. The specific heat of spin fluctuations is

                                                                                (23)

The T-linear coefficient of the specific heat γ_m is obtained

          (24)

From Eq.(24), γ_m increases when y_0ν increases

Figure 3: The temperature dependence of when (the red line),(the orange line),, ,respectively.

Figure 4. The temperature dependence ofwhen (the red line), (the orange line),,respectively.

We have made the self-consistent renormalization theory of anisotropic spin fluctuations in three dimensional nearly antiferromagnetic metals beyond the random phase approximation. We have investigated the temperature dependence of the inverse of the staggered magnetic susceptibility, nuclear magnetic relaxation rate, and the T-linear coefficient of the specific heat in nearly antiferromagnetic metals. We have found that the temperature dependence of the inverse of the staggered magnetic susceptibility has T²-linear behavior at low temperatures. With increasing temperatures, it has T-linear dependence. The nuclear magnetic relaxation rate has T-linear dependence at low temperatures. With increasing temperatures, it has T^1/2-linear dependence. The anisotropy appears in the inverse of the staggered 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 for fruitful 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, and S. Murayama for stimulating conversations.

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