Temperature and Field Dependence of Ferromagnetic Magnon in Monolayer Honeycomb Spin Lattice

We consider a monolayer honeycomb spin-lattice consisting of 2N spins residing in two triangular $a$ and $b$ spin sub-lattices shown in Fig. 1LABEL:sub@fig:1a. Position vectors of these spins in the sub-lattices are denoted by two-dimensional lattice vectors $\bf j$ and $\bf g$. Three nearest-neighbors (NN) of an $a-$spin at site j are $b-$spins defined by three vectors $\bm{\Delta}$ which are ${\bm{\Delta}_{1}}={a_{0}\over 2}(3,\sqrt{3})$, ${\bm{\Delta}_{2}}={a_{0}\over 2}(3,-\sqrt{3})$, ${\bm{\Delta}_{3}}={a_{0}\over 2}(-1,0)$ with the length of the hexagonal edge $a_{0}$. Similarly, three NN of a $b-$spin at site g are $a-$spins positioned by vectors $-\bm{\Delta}$. The six next-nearest-neighbors (NNN) of the $a$($b$)-sub-lattice are the same spin types defined by vectors ${\bm{\rho}}_{1}={a_{0}\over 2}(3,\sqrt{3})=-{\bm{\rho}}_{4}$, ${\bm{\rho}}_{2}={a_{0}\over 2}(3,-\sqrt{3})=-{\bm{\rho}}_{5}$, ${\bm{\rho}}_{3}={a_{0}\over 2}(0,-\sqrt{3})=-{\bm{\rho}}_{6}$.

The Hamiltonian of the XZ$-$Heisenberg model for the honeycomb monolayer spin-lattice in the external fields is written by

Here $h_{0}$, $\Omega_{0}$ are the external longitudinal and transverse field intensity given in the energy unit. The components of sub-lattice spin operators in the crystallographic $xyz$ frame, where the $z–$axis is perpendicular to the honeycomb spin-lattice plane, are denoted by $s^{z}_{a\bf j}$, $s^{x}_{a\bf j}$, $s^{z}_{b\bf g}$, $s^{x}_{b\bf g}$. $J$ and $L$ are NN exchange parameters between spin-pair components along the out-of-plane $z$ and in-plane $x$ directions, respectively. Since long-range order generally does not exist in a 2D isotropic exchange Heisenberg model [14], the exchange parameters are chosen as $J\neq L$. $J’$ is the NNN exchange between spins of the same types.

Using the mean-field approximation (MFA), we rewrite the Hamiltonian (1) in terms of spin fluctuation operators $\delta s^{z}_{a(b)\bf j}=s^{z}_{a(b)\bf j}-m_{az(bz)}$,

where

and

Here $m_{\nu x}=<s^{x}_{\nu}>$, $m_{\nu z}=<s^{z}_{\nu}>$ with $\nu=a,b$ are thermodynamic average of the spin-moment components per site and $m_{\nu}=\sqrt{m^{2}_{\nu x}+m^{2}_{\nu z}}$, and $<...>={\rm Tr}(e^{-\beta H}...)/{\rm Tr}(e^{-\beta H})$. The formula (5) produces the longitudinal and transverse components, $h_{\nu}$ and $\Omega_{\nu}$, of the net field acting on the $\nu$-sub-lattice. The Hamiltonian $H_{0}$ is diagonalized using the transformation,

During the transformation, spin fluctuation operators $\delta s^{\alpha}_{\nu\bf j}$ ($\rm\alpha=x,z$) are transformed to new operators $\delta S^{\alpha}_{\nu\bf j}=S^{\alpha}_{\nu\bf j}-<S^{\alpha}_{\nu}>$, ($\alpha=X,Z$). Performing Fourier transformation $\delta S^{\alpha}_{\nu\bf j}=N^{-1/2}\sum_{\bf k}\delta S^{\alpha}_{\nu}({\bf k})e^{i{\bf kj}}$ with two-dimensional wave vector $\bf k$ defined in the first Brillouin zone (shown in Fig. 1(b)), we rewrite the Hamiltonian (3) and (4) as

and

where $\lambda$ and $\lambda^{\prime}$ run from 1 to 4 meaning $1\equiv aX$, $2\equiv aZ$, $3\equiv bX$, $4\equiv bZ$. The anisotropic exchange interaction is a Hermitian matrix defined by

where

and

The temperature and field dependence of the magnon spectra is obtained from the poles of the imaginary time Green’s function (GF) as carried out in Ref. [9]. The Green’s function basing on the spin fluctuation operators $\delta s^{\alpha}_{\nu j}$ ($\alpha=x,z$; $\nu=a,b$) given in the crystallographic spin-lattice coordinates and in the Heisenberg picture is written as

where

Taking Fourier transformation of the GF, we have

where $\omega=2\pi n/\beta$; $n=0,\pm 1,\pm 2,...$. Denoting $(\bf k,\omega)$ as the three-component wave vector $\bf q$ and putting $\bf R=j-j^{\prime}$, the Fourier image of the GF writes

where

The unitary transformation (6) produces the connection between the Fourier images of the GF in the crystallographic and local XZ coordinates,

The Fourier transform of Green’s function in the local XZ coordinate presented in Eqs. (19)-(22) are obtained analogously to Eqs. (15)-(16) using indexes $\lambda$, $\lambda^{\prime}$,

The GF in Eq. (24) is defined by

where $<...>_{0}={\rm Tr}(e^{-\beta H_{0}}...)/{\rm Tr}(e^{-\beta H_{0}})$ implies the thermodynamic average with the mean-field Hamiltonian in Eq. (7) and $\sigma(\beta)=\hat{T}{\rm exp}[-\int_{0}^{\beta}H_{1}(\tau)d\tau]$ is the scattering matrix. The Green’s function matrix $\hat{\Gamma}({\bf q})$ is presented by,

In Ref. [9], we developed a procedure to calculate $\hat{\Gamma}({\bf q})$ in the Gaussian approximation using the functional integral representation for the scattering matrix, which is

Here $I^{-1}_{\lambda\lambda^{\prime}}({\bf k})$ denotes the inverse matrix elements of $\hat{I}({\bf k})$ indicated in Eq. (9). The functional integration over the complex field variable $\psi_{\lambda}({\bf q})=\psi^{c}_{\lambda}({\bf q})+i\psi^{s}_{\lambda}({\bf q})$ takes the following form,

Here $\psi^{c}_{\lambda}({\bf q})$ and $\psi^{s}_{\lambda}({\bf q})$ are the real and imaginary parts of $\psi_{\lambda}({\bf q})$, respectively and $\psi^{c}_{\lambda}({-\bf q})=\psi^{c}_{\lambda}({\bf q})$, $\psi^{s}_{\lambda}({-\bf q})=-\psi^{s}_{\lambda}({\bf q})$.

In the Gaussian approximation, Green’s function $\hat{\Gamma}({\bf q})$ satisfies the matrix equation:

where

and

Here, the Brillouin function $\rm b_{s}(y_{\nu})$ and its derivative $b^{\prime}_{s}(y_{\nu})$ are given in Ref. [15].

Using Eqs.(9)-(12), (29)-(32), we evaluate the GFs $\Gamma^{\lambda\lambda^{\prime}}({\bf q})$ and insert them into Eqs. (19)-(22) to obtain the explicit expressions for the GFs in the original spin-lattice coordinate. Performing an analytic continuation, the transverse GF is written by

with the k-dependent poles $\epsilon^{\pm}_{\bf k}$ of the GF known as elementary excitation energies or magnons,

We note that the formula (34) better describes the temperature and field dependence of the magnon band structure compared with that obtained by Ref. [2]. To analyze the magnon spectrum in Eq. (34), we use the MFA result for magnetic moment per site derived from the unitary transformation in Eq. (6), which are

We can also obtain various elementary excitations (ferromagnetic, antiferromagnetic, ferromagnetic magnon) in the honeycomb spin-lattice using Eq. (34). In the following parts, we concentrate only on ferromagnetic magnon which has been experimentally observed in the typical monolayer CrI₃ and the Van der Waals structures [5, 7].

In this part, we examine the magnon spectra in the presence of the spin orientation (SR) effect by virtue of the TF. To emphasize the special role of the transverse field, the longitudinal field is supposedly turned off, $h_{0}=0$. The ferromagnetic (FM) magnon case corresponds to a homogeneous molecular field where $\gamma_{a}=\gamma_{b}\equiv\gamma$, $h=3(J+2J^{\prime})m_{z}$, $\Omega=\Omega_{0}+3(L+2J^{\prime})m_{x}$ and $C=(J\Omega^{2}+Lh^{2})/\gamma^{2}$. Thus, the FM magnon spectrum is given by,

Since $m=b_{s}(y)$, the positive spin-wave dispersion relation has two branches only when the net magnetic moment per site $m$ is finite. At a given transverse field, the SR from the out-of-plane to an in-plane direction where $m_{z}=0,\,m_{x}=m$ occurs at the SR temperature $\tau_{R}(\Omega_{0})$ derived from the solution of the equation,

with $\rm\beta^{-1}=\tau,\,y=\gamma/\tau$. Similarly, the Curie temperature is extracted from Eq.  (37) in the limit of zero TF,

In the presence of TF, the magnetic moment and the spin-wave energy follow different relations in certain temperature regions. Instead of Eq. (36), the energy spectrum of the FM magnon is explicitly expressed in different ranges of temperature which are

i/ Below the SR temperature ($0<\tau<\tau_{R}(\Omega_{0})$)

From that, the spin reorients from the out-of-plane to the in-plane direction at the SR field which is

The magnon spectrum is then described by

where the zero-momentum magnon mode simplifies to,

At finite temperatures, the gap of the low-energy magnon branch at the center of the BZ closes, $\epsilon^{-}_{0}=0$, when the transverse field reaches the SR field value in Eq. (40). In this case, the low-energy magnon branch is purely acoustic, which implies $\epsilon_{\bf k}^{-}\rightarrow 0$ when ${\bf k}\rightarrow 0$.

Near the Dirac point K’$\rm\Big{(}{2\pi\over 3a_{0}},-{\rm 2\pi\over 3\sqrt{3}a_{0}}\Big{)}$, the magnon energy has a linear wave-vector dependence,

The magnon speed $v_{m}$ is a function of transverse field and temperature,

ii/ Above the SR temperature ($\tau\geq\tau_{R}$),

The magnetization and the dispersion relation of magnon are correspondingly given by

$m_{z}=0$,  $m_{x}=m=b_{s}(y^{\prime})$,  $y^{\prime}=\gamma^{\prime}/\tau$;

$\gamma^{\prime}=\Omega_{0}+3(L+2J^{\prime})b_{s}(y^{\prime})$;

$\epsilon^{\pm}_{\bf k}=\Omega_{0}+\Big{[}3L+J^{\prime}(6-\xi_{\bf 2k})\pm J|\xi_{\bf 1k}|\Big{]}b_{s}(y^{\prime})$.

Near the Dirac point K’, $\epsilon^{\pm}_{\bf k}\approx\Omega_{0}+(3L+9J^{\prime})b_{s}(y^{\prime})\pm\hbar v^{\prime}_{m}|{\bf k-K^{\prime}}|$ where the magnon speed is non-explicitly temperature and field dependent,

iii/ At zero temperature

The spin reorientation is then interpreted as the quantum phase transition due to the TF. At zero temperature, the SR field equals the critical field $\rm\Omega_{0R}(0)=\Omega_{0C}$. In this case, the Brillouin function reaches a saturated value, $b_{s}(y)=s$ with the spin moment per site $s$. Consequently, the critical transverse field causing QPT is (see Eq. (40))

The spin wave energy behaves differently depending on the transverse field, which is

a) $\Omega_{0}\leq\Omega_{0C}$,

and the FM magnon follows a dispersion relation which is

where the zero-momentum magnon gaps at Brillouin zone center for two branches are

It is obviously seen from Eq. (50) that the low-energy branch magnon gap closes at the critical field $\Omega_{0C}$.

b) $\Omega_{0}\geq\Omega_{0C}$

The magnetization is simplified to $m_{z}=0$, $m_{x}=s$ and the magnon spectrum linearly varies on the TF, which is

For numerical calculations, energy-dependent quantities such as FM-magnon energy $\epsilon^{\pm}({\bf k})$, temperature $\tau$; parameters L, J’; field strength $\gamma$, h, $\Omega$ are measured in terms of NN exchange J and lengths are expressed in terms of $a_{0}$.

Fig. 2LABEL:sub@fig:2a illustrates the MF thermomagnetic behaviors by using Eqs. (37)-(39), when $\Omega_{0}$=0 and 0.8, $L=0.4$, $J’=0.1$, $s=3/2$. The reduction of the spin reorientation temperature $\tau_{R}$ with increasing transverse field is exhibited in Fig. 2LABEL:sub@fig:2b. The Curie temperature $\rm\tau_{C}=4.5$ is obtained as the spin reorientation temperature when $\Omega_{0}=0$ according to Eq. (38). At the critical value $\Omega_{0C}=2.7$, the SR temperature becomes zero.