Thermal Effect of the Back Radiation from Disk to Head after Laser Heating in HAMR

Abstract

In a heat-assisted magnetic recording, the thermal effect of the head/disk interface has an important influence on the stability of the recording data. In this paper, we will discuss the thermal radiation from the disk, more specifically, the magnetic recording layer, which is at high temperature after laser heating, to the magnetic head, which is at room temperature. The radiative heat flux can be represented by the Poynting vector. In the near-field band, an effective way to obtain the electromagnetic fields is to solve the Maxwell’s equations combined with fluctuational electrodynamics. The near-field back radiation between specific head and disk material is calculated by the fluctuation-volume-current method. The radiative heat energy will induce the thermal deformation of the magnetic head, which will be discussed by the simulation, laying the foundation for adjusting and controlling the flying status.

1. Introduction

One of the effective methods to increase the storage density in a magnetic recording is to decrease the size of the magnetic particle. When the magnetic particle gets smaller, the coercivity of which gets higher and causes difficulty when writing data [1]. Heat-assisted magnetic recording (HAMR) is a new technique in which the magnetic recording media is heated by a focused laser to reduce the coercivity; hence, a high storage density can be achieved. It has great potential to increase the areal density in HAMR. The main challenge of HAMR is how to quantify the nanoscale near-field heat transfer and increase the reliability of the head/disk interface due to the back heating of the disk [2,3].

As we know, the disk, more specifically the magnetic recording layer, is heated by the focused laser to above the Curie temperature during the writing process. When the writing process is over, the disk is at a high temperature; meanwhile, the magnetic head and the air gap between the head and disk, i.e., the air bearing, are at a lower temperature. Thermal energy will transfer back from the disk to the magnetic head and the air bearing by conduction, convection, and radiation, though mainly by radiation. After absorbing the thermal energy from the hot disk, the slider can generate thermal deformation, which can change the flying height as well as the flying stability of the slider. Hence, it is very important to investigate the back heating of the HAMR disk.

However, current thermal analysis in HAMR mostly focuses on the heating process [4]. Research on the energy transfer in the head/disk interface is relatively rare. There are three layers in the head/disk interface, which are the following: (a) the lubricant layer [5]; (b) the near-field transducer [6]; and (c) the air bearing [7,8,9]. As discussed before, the thermal effects on the magnetic head have a great influence on the air bearing indirectly. Research on the thermal effects of the slider on the head is mainly focused on the results considering multiple factors rather than a single factor [10]. The net temperature rise of the slider caused by the laser absorption and the heat radiation of the locally heated recording magnetic disk are studied in Ref. [9], where the influence of the laser absorption and the heat radiation is considered combined. A new model, together with a finite element (FE) model for an integrated HAMR slider and an optical absorption model for a temperature profile of media hot spot, is then used to solve for the temperature rise and thermal protrusion on the slider body in Ref. [10]. However, because of the limitations of the experimental technology, the accuracy of the theoretical calculation is difficult to verify. As the development of the high-precision micro-electromechanical system (MEMS), tip-enhanced Raman spectroscopy (TERS) temperature measurement and other experimental technology progressed in recent years, the theoretical calculation of the near-field radiative heat transfer is gradually revised and improved so that the particular research on the influence of the near-field radiative heat transfer on the slider is badly needed to be supplemented.

The flying height, i.e., the distance between the head and disk, has been reduced to be around 2 nm, which is much smaller than the characteristic wavelength of the electromagnetic waves. Due to the scale effect, such as the surface wave, evanescent wave, radiation tunneling, and so on, traditional radiation theories, such as Kirchhoff’s law and Planck’s law, cannot be used to predict the radiative heat transfer. Research on the near-field heat transfer is concentrated on the theory of heat transport by phonon tunneling, see Refs. [11,12,13]. In recent years [14], the theoretical and experimental studies on the near-field radiative heat transfer have obviously increased. For different theoretical models, the calculation methods of the near-field radiative heat transfer are different for different geometries, such as heat transfer between two close infinite bulks [15], a small spherical particle and a bulk [16,17], two close infinite bulks with coats on the surface [18,19], several arbitrary shape objects [20,21], nanoparticles [22,23], thin films [24,25], and so on. The main difference between the calculation methods is focused on the method to solve the electric field and magnetic field on the target position, i.e., the solving methods of the stochastic the Maxwell’s equations are different. The most common method to solve the Maxwell’s equations is the discrete approximation method [26], such as the finite-different time-domain (FDTD) method, finite-difference frequency-domain (FDFD) method, boundary finite element (BEM) method, and the thermal dipole discrete approximation (T-DDA) method [20,21]. The limitation of the existing discrete approximation method is that the calculation is of great complexity and time-consuming. Another mainstream method to solve the Maxwell’s equations is to bring in the fluctuation–dissipation theory [27,28], the main idea of which is to regard an object as a collection of many dipoles in constant motion and then to integrate over all sources in the volume of the object to obtain the electromagnetic field on the target position. For simple shape objects, the volume of the object is generally integrated directly, known as the fluctuate-volume-current (FVC) method [25,28,29]. And for complex shape objects, the volume integrals can be converted into surface integrals on the surface of objects to calculate the integral, known as the fluctuate-surface-current (FSC) method [28]. Back to the head/disk model discussed in this paper, the fluctuate-volume-current method can be chosen to calculate the near-field radiative heat transfer.

The paper is constructed as follows: The theoretical analysis of the fluctuate-volume-current method to calculate the near-field radiative heat flux by the Poynting’s theory from disk to head is presented in Section 2. The properties of the imitating material are considered, and then the numerical results and discussion of the near-field radiative heat flux from disk to head in HAMR are given in Section 3. At last, the simulation results and discussion of the thermal deformation of the magnetic head are given in Section 4.

2. Numerical Analysis

2.1. Theoretical Model

The HAMR disk is consisted of the following several layers: lubricant film (~1 nm), carbon overcoat (~2 nm), magnetic recording layer (~15 nm), interlayer (~30 nm), heat sink (~15 μm), and substrate. A schematic diagram of an HAMR recording system is shown in Figure 1. The thickness of the magnetic recording layer is around 15 nm, much smaller than the horizontal size of the disk so that it can be simplified as a nanoscale film while the head/disk interface system is simplified as a multilayered structure, shown in Figure 2. Note that to analyze the thermal effect of radiation individually, in order to make targeted corrections to the effects of radiation in the future, the conduction and convection heat transfer is ignored here. Meanwhile, due to the neglect of the conduction heat transfer, assume that the lower surface of the magnetic recording layer is thermal insulation material. In Figure 2, there are 4 layers, numbered from 0 to 3 from bottom to top. The half space 0 under the magnetic recording film is the dielectric substrate, which is in thermal equilibrium, at a temperature of $T_0$, the permittivity is ${\epsilon }_0$. The film layer 1 represents for the magnetic recording film, which is assumed to be in thermal equilibrium and at a temperature of $T_1$. The magnetic recording layer is made of magnetic material, the permittivity and permeability of which is represented as ${\epsilon }_1$ and ${\mu }_1$, respectively. The thickness of film layer 1 is assumed to be $d_1$. The film 2 between the head and disk is vacuum, in a thickness of $d_2$. Similarly, the half space 3 represents for the head, which is in thermal equilibrium, and the temperature, which is $T_3$, the permittivity and permeability is represented as ${\epsilon }_3$ and ${\mu }_3$, respectively.

2.2. Spectral Heat Flux

The numerical problem is to discuss the radiative heat flux flow into half space 3 from film 1, as shown in Figure 2. In fluctuational electrodynamics, the source of thermal radiation is modeled as stochastic electric and magnetic current densities $j^e$ and $j^m$, respectively. The electric and magnetic fields at point $r$ due to electric source $j^e$ located at $r^{\prime }$ inside a body of volume $V$ are given by the following:

$$E^e\left(r,\omega \right)=i{\mu }_v{\mu }_s\omega {\displaystyle {\int }_V{\overline{\overline{G}}}^{Ee}\left(r,r^{\prime },\omega \right)·j^e\left(r^{\prime },\omega \right)d{V}^{\prime }}$$

(1)

$${\mathbf{H}}^e\left(r,\omega \right)={\displaystyle {\int }_V{\overline{\overline{G}}}^{He}\left(r,r^{\prime },\omega \right)·j^e\left(r^{\prime },\omega \right)d{V}^{\prime }}$$

(2)

Similarly, the electric and magnetic fields at point $r$ due to magnetic source $j^m$ located at $r^{\prime }$ inside a body of volume $V$ are given by the following:

$$E^m\left(r,\omega \right)=-{\displaystyle {\int }_V{\overline{\overline{G}}}^{Em}\left(r,r^{\prime },\omega \right)·j^m\left(r^{\prime },\omega \right)d{V}^{\prime }}$$

(3)

$${\mathbf{H}}^m\left(r,\omega \right)=i{\epsilon }_v{\epsilon }_s\omega {\displaystyle {\int }_V{\overline{\overline{G}}}^{Hm}\left(r,r^{\prime },\omega \right)·j^m\left(r^{\prime },\omega \right)d{V}^{\prime }}$$

(4)

where ${\mu }_v$ and ${\epsilon }_v$ represent for the permittivity and permeability in vacuum, generally ${\mu }_v=4\pi \times {10}^{-7}{\mathrm{NA}}^{-2}$ and ${\epsilon }_v=8.854187817\times {10}^{-12}\mathrm{F}/\mathrm{m}$, respectively. Similarly, ${\mu }_s$ and ${\epsilon }_s$ represent for the permittivity and permeability of media $s$, respectively. Terms ${\overline{\overline{G}}}^{Ee}\left(r,r^{\prime },\omega \right)$, ${\overline{\overline{G}}}^{Em}\left(r,r^{\prime },\omega \right)$, ${\overline{\overline{G}}}^{He}\left(r,r^{\prime },\omega \right)$, and ${\overline{\overline{G}}}^{Hm}\left(r,r^{\prime },\omega \right)$ are the electric and magnetic DGFs relating the fields with frequency $\omega$ at point $r$ due to a source located at $r^{\prime }$, respectively. The first $E/H$ in the superscript of DGFs represents for the type of the field at point $r$, and the second $e/m$ in the superscript of DGF represents for the type of the source at point $r^{\prime }$. The total electric field at point $r$ due to sources located at $r^{\prime }$ inside a body of volume $V$ is the sum of two types of sources, i.e., the sum of Equations (1) and (3), namely, the total magnetic field.

Generally, the heat flux of the electromagnetic fields can be calculated by the Poynting vector, which is the rate of energy flow per unit area in a plane electromagnetic wave. The electromagnetic fields radiated from a thermal object changes over time and can be represented by the Poynting vector. So that the instantaneous energy flow density, i.e., the instantaneous heat flux, is given by the following:

$$S=E\times H$$

(5)

The monochromatic radiative heat flux is given by the time-averaged Poynting vector as follows:

$$S\left(r,\omega \right)=\frac{1}{2}\mathrm{Re}\left[E\left(r,\omega \right)\times H^{\ast }\left(r,\omega \right)\right]$$

(6)

After ignoring the horizontal boundary, there is no energy transfer along the $x$ and $y$ directions in Cartesian coordinates, so that only the z-direction heat flux is considered. Since only positive frequencies are considered in the Fourier decomposition of the time-dependent fields into frequency-dependent quantities, the spectral heat flux is given by the following:

$$q\left(z,\omega \right)=\langle S_z\left(r,\omega \right)\rangle =4\times \frac{1}{2}\mathrm{Re}\left[\langle E_x{H}_y^{\ast }-E_y{H}_x^{\ast }\rangle \right]$$

(7)

The ensemble average of the spatial correlation functions of the stochastic electric and magnetic current densities are given by the following [18,25]:

$$\langle j_{\alpha }^e\left(r^{\prime },\omega \right)j_{\beta }^{e*}\left(r^{″},{\omega }^{\prime }\right)\rangle =\frac{\omega {\epsilon }_v\mathrm{Im}\left({\epsilon }_s\right)\Theta \left(\omega ,T\right)}{\pi }\delta \left(r^{\prime }-r^{″}\right)\delta \left(\omega -{\omega }^{\prime }\right){\delta }_{\alpha \beta }$$

(8)

$$\langle j_{\alpha }^m\left(r^{\prime },\omega \right)j_{\beta }^{m*}\left(r^{″},{\omega }^{\prime }\right)\rangle =\frac{\omega {\mu }_v\mathrm{Im}\left({\mu }_s\right)\Theta \left(\omega ,T\right)}{\pi }\delta \left(r^{\prime }-r^{″}\right)\delta \left(\omega -{\omega }^{\prime }\right){\delta }_{\alpha \beta }$$

(9)

$$\langle j_{\alpha }^e\left(r^{\prime },\omega \right)j_{\beta }^{m*}\left(r^{″},{\omega }^{\prime }\right)\rangle =0$$

(10)

$$\langle j_{\alpha }^m\left(r^{\prime },\omega \right)j_{\beta }^{e*}\left(r^{″},{\omega }^{\prime }\right)\rangle =0$$

(11)

where the mean energy of a Planck’s oscillator in thermal equilibrium at frequency $\omega$ and temperature $T$ is [27,28,30]

$$\Theta \left(\omega ,T\right)=\frac{\hslash \omega }{\exp \left(\hslash \omega /k_{B}T\right)-1}+\frac{1}{2}\hslash \omega$$

(12)

where $\hslash \omega /2$ represents for the term of vacuum fluctuation, i.e., the zero-point energy, the zero-point energy of two objects can be cancelled out; therefore, it can be ignored during the calculation.

Substitution of the electric and magnetic fields Equations (1)–(4) and the ensemble average of the spatial correlation functions Equations (8)–(11) into heat flux Equation (7), Equation (7) can be reduced into the following:

$$q\left(z,\omega \right)=\frac{2k_v^2\Theta \left(\omega ,T\right)}{\pi }\mathrm{Re}\left[\begin{array}{l}i\mathrm{Im}\left({\epsilon }_s\right){\mu }_s{\displaystyle {\int }_{V}d{V}^{\prime }\left({\overline{\overline{G}}}_{x\alpha }^{Ee}{\overline{\overline{G}}}_{y\alpha }^{He\ast }-{\overline{\overline{G}}}_{y\alpha }^{Ee}{\overline{\overline{G}}}_{x\alpha }^{He\ast }\right)}\\ -i{\epsilon }_s\mathrm{Im}\left({\mu }_s\right){\displaystyle {\int }_{V}d{V}^{\prime }\left({\overline{\overline{G}}}_{x\alpha }^{Em}{\overline{\overline{G}}}_{y\alpha }^{Hm\ast }-{\overline{\overline{G}}}_{y\alpha }^{Em}{\overline{\overline{G}}}_{x\alpha }^{Hm\ast }\right)}\end{array}\right]$$

(13)

where the subscript $\alpha$ refers to all the orthogonal components, i.e., $\alpha =x,y,z$ in Cartesian coordinates. $k_v$ represents for the wavevector in vacuum, and there is a relationship $k_v^2={\omega }^2{\mu }_v{\epsilon }_v$.

To solve Equation (13), the dyadic correlation term ${\displaystyle {\int }_{V}d{V}^{\prime }{\overline{\overline{G}}}_{u\alpha }^{Ee}{\overline{\overline{G}}}_{v\alpha }^{He\ast }}$ and ${\displaystyle {\int }_{V}d{V}^{\prime }{\overline{\overline{G}}}_{u\alpha }^{Em}{\overline{\overline{G}}}_{v\alpha }^{Hm\ast }}$ need to be determined first. Here, the subscripts $u,v=x,y$, $u\ne v$ and $\alpha =x,y,z$ have been discussed before. Because of the transition of the Cartesian coordinate to the cylindrical coordinate, the subscripts hereafter stand for $u,v=\rho ,\theta$, $u\ne v$ and $\alpha =\rho ,\theta ,z$. Therefore, the dyadic correlation term in DGFs of fields generated by the electric sources leads to the following:

$${\displaystyle {\int }_{V}d{V}^{\prime }{\overline{\overline{G}}}_{u\alpha }^{Ee}{\overline{\overline{G}}}_{v\alpha }^{He\ast }}={\displaystyle {\int }_{-\infty }^{\infty }\frac{d{k}_{\rho }}{{\left(2\pi \right)}^2}}{\displaystyle {\int }_{z}d{z}^{\prime }{g}_{u\alpha }^{Ee}}{g}_{v\alpha }^{He\ast }=\frac{1}{2\pi }{\displaystyle {\int }_0^{\infty }{k}_{\rho }d{k}_{\rho }}{\displaystyle {\int }_{z}d{z}^{\prime }{g}_{u\alpha }^{Ee}}{g}_{v\alpha }^{He\ast }$$

(14)

Similarly, the dyadic correlation term in DGFs of fields generated by the magnetic sources leads to the following:

$${\displaystyle {\int }_{V}d{V}^{\prime }{\overline{\overline{G}}}_{u\alpha }^{Em}{\overline{\overline{G}}}_{v\alpha }^{Hm\ast }}=\frac{1}{2\pi }{\displaystyle {\int }_0^{\infty }{k}_{\rho }d{k}_{\rho }}{\displaystyle {\int }_{z}d{z}^{\prime }{g}_{u\alpha }^{Em}}{g}_{v\alpha }^{Hm\ast }$$

(15)

The source object in Figure 2 is the film 1, so that take $s=1$ into Equation (13). And then substitution of Equations (14) and (15) into Equation (13), the monochromatic radiative heat flux under cylindrical coordinate is given by the following:

$$q\left(z,\omega \right)=\frac{k_v^2\Theta \left(\omega ,T\right)}{{\pi }^2}\mathrm{Re}\left[\begin{array}{l}i\mathrm{Im}\left({\epsilon }_1\right){\mu }_1{\displaystyle {\int }_0^{\infty }{k}_{\rho }d{k}_{\rho }}{\displaystyle {\int }_{z}d{z}^{\prime }\left(\begin{array}{l}{g}_{\rho \rho }^{Ee}{g}_{\theta \rho }^{He\ast }\\ -g_{\theta \theta }^{Ee}{g}_{\rho \theta }^{He\ast }\\ +g_{\rho z}^{Ee}{g}_{\theta z}^{He\ast }\end{array}\right)}\\ -i{\epsilon }_1\mathrm{Im}\left({\mu }_1\right){\displaystyle {\int }_0^{\infty }{k}_{\rho }d{k}_{\rho }}{\displaystyle {\int }_{z}d{z}^{\prime }\left(\begin{array}{l}-g_{\theta \rho }^{Em}{g}_{\rho \rho }^{Hm\ast }\\ +g_{\rho \theta }^{Em}{g}_{\theta \theta }^{Hm\ast }\\ -g_{\theta z}^{Em}{g}_{\rho z}^{Hm\ast }\end{array}\right)}\end{array}\right]$$

(16)

2.3. Dyadic Green’s Function

The sources in the film generate two kinds of electromagnetic waves, emitting to the forward and backward directions. Meanwhile, resulting from multiple reflections between the upper and lower surfaces of the film, the electromagnetic waves travel both forward and backward in the end. So that the electromagnetic waves consist of four parts, i.e., the forward travelling waves arising from a source emitting in the forward direction, the backward travelling waves arising from a source emitting in the forward direction, the forward travelling waves arising from a source emitting in the backward direction, and the backward travelling waves arising from a source emitting in the backward direction, as shown in Figure 3.

DGF between source $r^{\prime }$ located in the magnetic recording layer film 1 and the target position $r$ located in magnetic head half space 3 is given by the following:

$${\overline{\overline{G}}}^{Ee}\left(r,r^{\prime },\omega \right)=\frac{i}{8{\pi }^2}{\displaystyle \int \frac{d{k}_{\rho }}{k_{z1}}{e}^{i{k}_{\rho }\cdot \left(R-R^{\prime }\right)}\left[\begin{array}{l}\left(t_{13}^{s++}\widehat{s}\widehat{s}+t_{13}^{p++}{\widehat{p}}_3^{+}{\widehat{p}}_1^{+}\right)e^{i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}\\ +\left(t_{13}^{s+-}\widehat{s}\widehat{s}+t_{13}^{p+-}{\widehat{p}}_3^{-}{\widehat{p}}_1^{+}\right)e^{-i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}\\ +\left(t_{13}^{s-+}\widehat{s}\widehat{s}+t_{13}^{p-+}{\widehat{p}}_3^{+}{\widehat{p}}_1^{-}\right)e^{i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}\\ +\left(t_{13}^{s--}\widehat{s}\widehat{s}+t_{13}^{p--}{\widehat{p}}_3^{-}{\widehat{p}}_1^{-}\right)e^{-i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}\end{array}\right]}$$

(17)

where the term $t_{sl}^{p++}{\widehat{p}}_l^{+}{\widehat{p}}_s^{+}$ stands for the direction changes of the polarized waves, for example, the term $t_{13}^{p++}{\widehat{p}}_3^{+}{\widehat{p}}_1^{+}$ means the direction of p-polarized waves in film 1 is ${\widehat{p}}_1^{+}$, and it changes to ${\widehat{p}}_3^{+}$ in bulk 3, meanwhile the amplitude has a coefficient $t_{13}^{p++}$, the superscript refers to the type of the electromagnetic, i.e., $p++$ stands for the forward travelling waves arising from a source emitting in the forward direction. To make it clear, the amplitude coefficient $t_{12}^{s,p++},t_{12}^{s,p+-},t_{12}^{s,p-+},t_{12}^{s,p--}$ can be represented as $A^{s,p},B^{s,p},C^{s,p},D^{s,p}$. And substitution of the unit vectors of s-polarized and p-polarized into Equation (17), DGFs of the electric field due to electric sources are given by the following:

$$\begin{array}{ll}{\overline{\overline{G}}}^{Ee}\left(r,r^{\prime },\omega \right)& =\frac{i}{8{\pi }^2}{\displaystyle \int \frac{d{k}_{\rho }}{k_{z1}}{e}^{i{k}_{\rho }\cdot \left(R-R^{\prime }\right)}}\\ & \times \left[\begin{array}{c}\frac{k_{z1}{k}_{z3}}{k_1{k}_3}\left(\begin{array}{l}{A}^p{e}^{i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}-B^p{e}^{-i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}\\ -C^p{e}^{i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}+D^p{e}^{-i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}\end{array}\right)\widehat{\rho }\widehat{\rho }\\ +\frac{k_{\rho }{k}_{z3}}{k_1{k}_3}\left(\begin{array}{l}-A^p{e}^{i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}+B^p{e}^{-i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}\\ -C^p{e}^{i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}+D^p{e}^{-i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}\end{array}\right)\widehat{\rho }\widehat{z}\\ +\left(\begin{array}{l}{A}^s{e}^{i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}+B^s{e}^{-i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}\\ +C^s{e}^{i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}+D^s{e}^{-i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}\end{array}\right)\widehat{\theta }\widehat{\theta }\\ +\frac{k_{\rho }{k}_{z1}}{k_1{k}_3}\left(\begin{array}{l}-A^p{e}^{i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}-B^p{e}^{-i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}\\ +C^p{e}^{i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}+D^p{e}^{-i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}\end{array}\right)\widehat{z}\widehat{\rho }\\ +\frac{k_{\rho }^2}{k_1{k}_3}\left(\begin{array}{l}{A}^p{e}^{i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}+B^p{e}^{-i{k}_{z3}\left(z-z_3\right)-i{k}_{z1}{z}^{\prime }}\\ +C^p{e}^{i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}+D^p{e}^{-i{k}_{z3}\left(z-z_3\right)+i{k}_{z1}{z}^{\prime }}\end{array}\right)\widehat{z}\widehat{z}\end{array}\right]\end{array}$$

(18)

It is easy to deduce the Wyle components respectively by Equation (18). And the Wyle components of other DGFs can be obtained similarly. Substituting the DGFs into the heat flux Equation (16) and noting that the integral interval of the magnetic recording layer is $z=\left[z_1,z_2\right]=\left[0,d\right]$. Finally, the spectral heat flux is given by the following:

$$q\left(z,\omega \right)=\frac{k_v^2\Theta \left(\omega ,T\right)}{{\pi }^2}\mathrm{Re}\left[\begin{array}{l}i\mathrm{Im}\left({\epsilon }_1\right){\mu }_1{\displaystyle {\int }_0^{\infty }{k}_{\rho }d{k}_{\rho }}{\displaystyle {\int }_0^{d}d{z}^{\prime }\left(\begin{array}{l}{g}_{\rho \rho }^{Ee}{g}_{\theta \rho }^{He\ast }\\ -g_{\theta \theta }^{Ee}{g}_{\rho \theta }^{He\ast }\\ +g_{\rho z}^{Ee}{g}_{\theta z}^{He\ast }\end{array}\right)}\\ -i{\epsilon }_1\mathrm{Im}\left({\mu }_1\right){\displaystyle {\int }_0^{\infty }{k}_{\rho }d{k}_{\rho }}{\displaystyle {\int }_0^{d}d{z}^{\prime }\left(\begin{array}{l}-g_{\theta \rho }^{Em}{g}_{\rho \rho }^{Hm\ast }\\ +g_{\rho \theta }^{Em}{g}_{\theta \theta }^{Hm\ast }\\ -g_{\theta z}^{Em}{g}_{\rho z}^{Hm\ast }\end{array}\right)}\end{array}\right]$$

(19)

3. Results and Discussion

3.1. Materials in HAMR

The magnetic recording layer in HAMR is made out of magnetic materials, which is in progress along with the development of the magnetic recording technology. Until now, FePt has become the next generation of high-density magnetic recording media material. FePt has significant advantages in magnetic recording because it has several excellent characteristics, such as high anisotropy, similar saturation magnetization, small superparamagnetic critical size, and the ability to keep high thermal stability when the size of the particle is reduced to 3–4 nm [31,32,33,34]. To meet the practical requirement of a magnetic recording medium, the alloy needs to be modified by doping. But, in theoretical calculations, the modifying of the material can be ignored [32,35]. Several studies on the properties of alloys show that the dielectric function of thin alloy films of different alloy ratios basically fluctuates in a similar trend and within a small range [36,37]. To simplify the calculation, the parameters of a single metal material can be used instead of a complex alloy material.

$$\epsilon =1-\frac{{\omega }_p^2}{{\omega }^2+i{\gamma }_e\omega }$$

(20)

where ${\omega }_p$ and ${\gamma }_e$ stand for the equivalent plasma frequency and damping factor of the material, respectively. The two parameters of metal Fe are ${\omega }_p=0.62204\times {10}^{16}\mathrm{rad}/\mathrm{s}$ and ${\gamma }_e=0.27709\times 2\pi \times {10}^{14}\mathrm{rad}/\mathrm{s}$. In the radiation band, the susceptibility of the metal is very small so that the permeability can be regarded as constant by ignoring the influence on the radiative heat flux contributed by the magnetism.

The permittivity of the material of the magnetic head for calculation, ${\mathrm{Al}}_2{\mathrm{O}}_3$, is approximated by the following:

$$\epsilon ={\epsilon }_{\infty }{\displaystyle \prod _j\frac{{\Omega }_{jLO}^2-{\omega }^2+i{\gamma }_{jLO}\omega }{{\Omega }_{jTO}^2-{\omega }^2+i{\gamma }_{jTO}\omega }}$$

(21)

where ${\Omega }_{jLO}$ and ${\Omega }_{jTO}$ are longitudinal and transverse resonance frequency respectively, ${\gamma }_{jLO}$ and ${\gamma }_{jTO}$ are longitudinal and transverse damping factor respectively. These parameters are related to the temperature, as discussed in Ref. [38].

3.2. Spectral Heat Flux of the Back Radiation in HAMR

In Figure 2, assume that the thickness of film 1 is 15 nm, the flying height, i.e., the thickness of air film 2 between head and disk is 2 nm, the radiative heat flux from film 1 to bulk 3 is calculated at full frequency after ignoring the frequencies where the heat flux is several orders of magnitude below the maximum value. The frequency–flux graph at an effective frequency band, from $\omega =1\times {10}^{11}\mathrm{rad}/\mathrm{s}$ to $\omega =1\times {10}^{15}\mathrm{rad}/\mathrm{s}$, is plotted shown in Figure 4. Overall, the contribution of the near-field radiative heat flux (the red line) is several orders of magnitude larger than that of the far-field radiative heat flux (the blue line), related to the frequency band; thus, the contribution of far-field for radiative heat flux can be ignored here. Moreover, the near-field heat transfer is enhanced at the resonant frequency, which is related to the properties of the material. Notice the near-field part, there is a remarkable fluctuation of the heat flux during the frequencies from ${\omega }_1=1.10\times {10}^{13}\mathrm{rad}/\mathrm{s}$ to ${\omega }_2=2.04\times {10}^{13}\mathrm{rad}/\mathrm{s}$, which is because of the electromagnetic properties of the materials. The abrupt change in the heat flux at frequencies ${\omega }_1=1.10\times {10}^{13}\mathrm{rad}/\mathrm{s}$ (Figure 5a) and ${\omega }_2=2.04\times {10}^{13}\mathrm{rad}/\mathrm{s}$ (Figure 5b) is because the sign of the imaginary part of the electrical permittivity changes at these two frequencies, i.e., when $\omega <{\omega }_1=1.10\times {10}^{13}\mathrm{rad}/\mathrm{s}$, the imaginary part of the electrical permittivity is negative, and it changes to positive when ${\omega }_1=1.10\times {10}^{13}\mathrm{rad}/\mathrm{s}<\omega <{\omega }_2=2.04\times {10}^{13}\mathrm{rad}/\mathrm{s}$, and changes again to negative when $\omega >{\omega }_2=2.04\times {10}^{13}\mathrm{rad}/\mathrm{s}$, as shown in Figure 5. Noticed that there is a deviation between the peaks of the heat flux and the peaks of the electrical permittivity of ${\mathrm{Al}}_2{\mathrm{O}}_3$ during the frequency band from ${\omega }_1=1.10\times {10}^{13}\mathrm{rad}/\mathrm{s}$ to ${\omega }_2=2.04\times {10}^{13}\mathrm{rad}/\mathrm{s}$, which is because of the influence of the imaginary part of the electrical permittivity of Fe, as shown in Figure 6. Generally, the electromagnetic properties of the materials, particularly the electrical permittivity of the materials, have great influence on the spectral heat flux. In addition, the surface phonon–polariton resonance is visible during the frequencies ${\omega }_2=2.04\times {10}^{13}\mathrm{rad}/\mathrm{s}$ to ${\omega }_4=4.40\times {10}^{13}\mathrm{rad}/\mathrm{s}$, and the large value of the near-field heat flux is at the frequency ${\omega }_3=3.42\times {10}^{13}\mathrm{rad}/\mathrm{s}$.

The spectral heat flux at an effective frequency band under varying flying heights from 1 nm to 10 nm is plotted in Figure 7 to discuss the influence of the flying height on the spectral heat flux. As shown in Figure 7, the surface phonon–polariton resonance frequency band of the heat flux in the extreme near-field distance, i.e., $d=$ 1 nm and 2 nm, is narrower than that in the far-field band, i.e., $d=$ 5 nm and 10 nm. In addition, there is a remarkable end of the resonance frequency for the extreme near-field radiative heat transfer.

3.3. The Influence of the Flying Height on the Back Radiation in HAMR

As shown in Figure 7, obviously, the radiative heat transfer is decreased when the flying height is increased, and the peaks appear in the same frequencies, but the peaks become smoother. By integrating over the whole frequencies, the total near-field back radiation heat flux can be calculated. Taking the varying flying height from 1 nm to 10 nm, the graph of heat flux varying with flying height is plotted in Figure 8. Obviously, when the flying height is small enough, the contribution of the far-field heat flux can be ignored, and the contribution of the near-field heat flux plays a major role in the total flux. As shown in Figure 8, the near-field heat flux (the red line) is almost overlapped with the total heat flux (the black line). Even under the extremely small flying height, when the flying height is $d=1\mathrm{nm}$, the heat flux is almost two orders of magnitude larger than that when the flying height is $d=10\mathrm{nm}$. So, because the near-field radiative heat transfer is very sensitive to the changes in the flying height, slight fluctuations in the flying height can lead to huge near-field radiative heat transfer changes, resulting in thermal deformation of the head, the failure of the near-field transducer, the instability of the flying status, and so on.

4. Simulation Results and Discussion of the Thermal Deformation of Head

The back radiative heat flux from disk to head, which can be regarded as the thermal source on the disk heating the slider on the head, will cause a thermal deformation of the slider. The slider is modelled to find out the maximal deformation of the head. It makes no sense to consider the whole size of the magnetic head because, in comparison with the heated area, the magnetic head is too big; therefore, a calculation diameter of $4\mathsf{\mu }\mathrm{m}$ for the cylinder head is taken. The heated area is taken to a $200\mathrm{nm}\times 200\mathrm{nm}$ square in the center of the head. The thickness of the cylinder head is much larger than the size of the heated area too, so a thickness of 10 times the size of the heated area, $2\mathsf{\mu }\mathrm{m}$, is taken. After heating the square area by the radiative heat flux calculated in Section 2, the flying height of 1 nm was picked, under which distance the radiative heat flux is large enough to obviously observe the deformation, the distribution of temperature is plotted in Figure 9. The high temperature appears inside the heating square. The temperature is higher inside and lower outside the square. The heating has almost no influence on the other area in the head model.

The deformation caused by the radiative heat flux is shown in Figure 10. As expected, the largest deformation is in the center of the head model (Figure 10b), and the deformation in the square area is larger than the other area in the head model (Figure 10a). It shows that when the flying height is 1 nm, the maximum deformation is about 0.058 nm. It is going to make a big difference on the air bearing, i.e., the 1 nm-thick air film.

The thermal deformation and deformation rate, i.e., the rate of the deformation under different flying heights from 1 nm to 10 nm, is plotted in Figure 11. As shown in Figure 11, the deformation rate is increased when the flying height gets smaller, especially when the flying height is smaller than 5 nm, the deformation rate is too big to be ignored. It is because when the flying height is increased, the radiative heat flux is decreased, as discussed in Section 2. Ref. [10] shows that radiation at very small gap spacing could have a more significant effect on the thermal protrusions (thermal deformation) by comparing the thermal deformation with and without back radiation considered. The simulation results in Figure 11 basically coincide with the results in Ref. [10]. Now that the flying height reaching 2 nm, the discussion of the thermal deformation is meaningful.

5. Conclusions

In this paper, the fluctuate-volume-current (FVC) method is used to calculate the near-field back radiative heat transfer from the disk to the magnetic head in heat-assisted magnetic recording (HAMR). And the thermal deformation of the slider on the magnetic head caused by the back radiative heat from the hot magnetic recording layer is then analyzed.

The formulation for calculating the near-field radiative heat transfer by the fluctuate-surface-current method is based on the fluctuation–dissipation theory, which was brought in to solve Maxwell’s equations to obtain the electromagnetic fields on the target position in the magnetic head, and then the heat flux using the Poynting’s theory is obtained.

Materials Fe and ${\mathrm{Al}}_2{\mathrm{O}}_3$ are chosen to represent the magnetic recording film and the slider on the magnetic head, respectively. The radiative heat flux from the hot magnetic recording layer to the cold slider is calculated. It is shown in the spectral heat flux graph that the fluctuations of the heat flux are caused by the electromagnetic properties of the materials, and the frequencies reached the surface phonon–polariton resonance.

It is shown in the graph of heat flux varying with flying height that, under the small flying height band, the contribution of the far-field heat flux can be ignored. And even under the extremely small flying height from 1 nm to 10 nm, the heat flux has almost two orders of magnitude changes.

The thermal analysis and the structural analysis of the slider on the magnetic head are used to discuss the influence of the back radiation on the structure of the slider. It is shown that when the distance between the head and disk gets larger, the deformation rate decreases and becomes small enough to be ignored when the distance is larger than 5 nm. Now that the flying height is reaching 2 nm, the thermal deformation may cause an obvious decrease in the air bearing (i.e., the gap between slider and disk), leading to the fluctuation of the head flying. More research on eliminating the negative effects of the back radiation thermal effects should be conducted in the future, such as how to reduce the near-field radiative heat transfer, how to achieve ultra-fast heat dissipation, and how to maintain the dynamic balance of the flying height while the slider is deformed by the back heating from the disk. The analysis of the thermal effect of the back radiation in this paper will be a preparation for these studies.

6. Patents

This section is not mandatory but may be added if there are patents resulting from the work reported in this manuscript.