## Abstract

The electromagnetic field enhancement (FE) at the end of the probe of an Apertureless Scanning Near-field Optical Microscope (ASNOM) is used to write nanometric dots in a phase-change medium. The FE acts as a heat source that allows the transition from amorphous to crystalline phase in a Ge_{2}Sb_{2}Te_{5} layer. Through the 2D Finite Element Method (FEM) we predict the size of the dot as a function of both the illumination duration and the incoming power density. Numerical results are found to be in good agreement with preliminary experimental data.

©2004 Optical Society of America

## 1. Introduction

The use of phase change alloys for optical data storage is now well acquired and remains promising for higher density data storage (DDS). Nowadays, rewritable CD or DVD are commonly used and several research centers are developing the so-called Blu-ray Disc to increase the DDS. Also, techniques using probes [1], near field (such as Solid Immersion Lens (SIL) [2]) or Scanning Near Field Optical Microscopy (SNOM) using a stretched optical fiber tip [3, 4] are under investigation at the present time. Another promising approach consists to use the electric field enhancement (FE) at the end of the metallic scatter-probe. This effect occurs in an apertureless SNOM, when the illuminating field is a p-polarized laser beam and has been studied previously [5, 6, 7]. This phenomenon is localized within an area of a few nanometers around the probe end and it has been reported to be ten times the intensity of the illuminating laser beam [8]. The FE has been recently used to induce nanometer-sized domains in a photosensitive sample [8, 9, 10]. This FE at the end of the probe of an apertureless SNOM [11, 12] may be considered as a nanometric energy source and thus can be used to induce an amorphous to crystalline transition in a phase change material.

The purpose of this work is the numerical study of the bit size in the phase change material. The dot size is supposed to be directly related to the absorbed optical power density that depends on the temporal variation of the permittivity. The results of the simulations are in good agreement with the experimental data. Therefore, the model is used to predict theoretically the size of the phase-change dot as a function of time and illumination power.

The paper is organized as follows. Sec. 2, is devoted to the description of the experimental setup. In Sec. 3, we describe the method used to predict the bit size. In Sec. 4, numerical results are presented, discussed and compared to experimental data, before concluding in Sec. 5.

## 2. Description of the experiment

#### 2.1. Presentation of the medium

The sample is a typical rewritable disc with a multilayered structure as shown in Fig. 1. In the following, the thickness (*d*) and the relative permittivity of the media (*ε*_{r}
) are indicated. These parameters have been used in the model.

A protective very thin layer (*d*=2 nm, *ε*_{r}
=2.89) covers the 30 nm thick Ge_{2}Sb_{2}Te_{5} phase change layer to prevent oxidation and mechanical contact. The crystalline phase of Ge_{2}Sb_{2}Te_{5} (*ε*_{r}
=-2.37+31.16 *j*) is produced from the amorphous phase (*ε*_{r}
=11.13+11.84 *j*) by heating. A crystalline dot into the amorphous layer may be used as a bit of information. The active layer is separated from the glass substrate by a ZnS-SiO_{2} dielectric layer (*d*=30 nm, *ε*_{r}
=4.53+0.04 *j*). These layers have been deposited by sputtering technique on a thick glass substrate (*ε*_{r}
=2.28). For an easier readout process in transmission configuration, no reflective layer is deposited.

#### 2.2. The writing setup

The ASNOM experimental setup used for the recording process is depicted in Fig. 1(a). A commercial silicon (*ε*_{r}
=14.98+0.08 j) AFM probe (CSC17/PT/50, NTMDT) coated with a 25 nm thick platinum layer (*ε*_{r}
=-11.27+18.72 *j*) is used. The radius of curvature of the probe is 20 nm and the half-angle of its vertex is 10°. This metallic probe is mounted on a commercial AFM (“M5” model, Park Scientific Instrument) working in contact mode. The scan of the sample in x and z direction is ensured by a holder including a piezotranslator (NPS XY100A, Queensgate). The illumination is performed by using an AlGaInP semiconductor laser diode @λ=660 nm. The laser source is collimated and polarized into the incident plane (p-polarization), focused within an area of 6 *µ*m^{2} by an objective (20 X, Numerical Aperture NA=0.42). The angle between the axis of the probe (*y*) and the laser beam is 65°. The writing process is carried out by applying light pulses cycles and it needs a laser power *P*_{W}
=7.4 mW, that is chosen just below the threshold ${P}_{W}^{0}$ of the phase-change in Ge_{2}Sb_{2}Te_{5}. The FE at the end of the metallic probe is expected to produce the amorphous to crystalline transition. The duration of the writing pulse *τ*=100*µ*s is chosen to be long enough to produce the phase-change and its minimum value is related to experimental constraints.

## 3. Numerical modeling of the bit size in the phase-change material

In this paper, we use FEM to model the writing process. The main advantage of the FEM approach is its ability to treat any type of geometry and material inhomogeneity (with complex permittivity). An iterative and adaptive process is used to control the accuracy of the computed solution. Figure 2 shows a schematic of the modeled structure.

#### 3.1. Modeling the phase-change writing process

In the present case, the phase-change dot is supposed to be produced by the FE occurring at the end of the ASNOM probe. The actual phase-change process is very complex as it involves a complicated interaction between multiple physical phenomena. Indeed, the optical source induces heat transfer, specific heat, thermal phase-change, solid state transformations (nucleation, crystal growth) and mechanical constraints. Thus, the energy balance may be written:

where Cristal(t) concerns the energy description of the media in solid state physics, Heat(t) its thermal behavior and *Q*
_{em}(*ε*_{r}
(*t*)) is the absorbed optical power density in the material. To avoid complexity in the model, in this approach, we focus our attention only on the evolution of *Q*_{em}
(*ε*_{r}
(*t*)) considering it as the “source of crystallization” or “source of phase-change”. Therefore, no further physical parameters on the nucleation, crystal growth, thermal or mechanical effects are needed in this model. We assume that the size of the crystalline dot is directly related to the zone where the energy is highly absorbed in the material.

Let us note *Q*_{em}
(*ε*_{r}
(*t*)) the optical power density absorbed by the materials during the illumination [13]:

where *ω* is the angular frequency of the illumination field, *ε*
_{0} is the vacuum permittivity, ℑ(*ε*_{r}
) is the imaginary part of the relative complex permittivity *ε*_{r}
for each material and **E** is the electric field vector. Let us note that *ε*_{r}
is a function of the illumination duration *t* and of the spatial coordinates. The growth of the crystallized dot at a certain time *t*_{i}
results from the absorbed optical power density at time *t*_{i}
and from the previous material phase-change (*t*<*t*_{i}
) which produces a strong variation of the relative permittivity. Indeed, the amorphous state of Ge_{2}Sb_{2}Te_{5} is an absorbing dielectric *ε*_{r}
=11.13+11.84 *j* and the crystalline one is a metallic medium: *ε*_{r}
=-2.37+31.16 *j*.

The characteristic time of the crystallization process *τ*_{pc}
(~50 ns for Ge_{2}Sb_{2}Te_{5} [14]) is 2000 times smaller than the illumination duration *τ*(~100 *µ*s). Therefore, the temporal evolution of *Q*_{em}
(*t*) may be considered as the discrete sum of absorbed optical power density terms *Q*_{k}
(*ε*_{r}
(*t*_{k}
)) for each time step of duration *τ*_{pc}
:

where *θ*[*t*] is the Heavyside step function, *Q*
_{0} is the absorbed power density before crystallization, *Q*_{k}
is the *k*^{th}
absorbed optical power density taking into account the crystallized zone during the previous time step (see Fig. 3) and *N* is the integer part of the ratio *t/τ*_{pc}
. The nanometric dot is written only if the following conditions are fulfilled:

• The absorbed optical power density below the probe is sufficient *i.e*. the probe end is small and it is close to the PC layer [8],

• the power of the illumination *P*_{W}
is greater than a given threshold ${P}_{W}^{0}$
which is characteristic from the material. Experimentally, ${P}_{W}^{0}$ is fixed to be at the limit of phase change of the whole illuminated region (without the probe).

If *P*_{W}
>${P}_{W}^{0}$, the whole illuminated layer is transformed into crystalline phase; otherwise if *P*_{W}
<${P}_{W}^{0}$ only the zone below the probe is expected to change.

In the model, the corresponding threshold ${Q}_{k}^{0}$ is the maximum value of the absorbed power density computed without the probe. Therefore, the zone where *Q*_{k}
(*ε*_{r}
(*t*_{k}
)) overpass this threshold is supposed to be crystallized and, for computation facilities, the resulting change of permittivity is supposed to be homogeneous in the 30 nm thick Ge_{2}Sb_{2}Te_{5} layer (along *y*-direction, see Fig. 2).

To study the temporal evolution of the dot size, we have to compute the electromagnetic field (Eq. (2)) by taking into account the temporal evolution of the layer permittivity (Eq. (3)) and the probe characteristics.

#### 3.2. Modeling the electromagnetic problem

In classical electromagnetics, the partial differential equations (PDE) are derived from Maxwell’s equations. The problem can be reduced to the wave equation, or Helmholtz’s equation if an harmonic time dependance of the form exp(*jωt*) is assumed in the electromagnetic field [15]. In the two-dimensional model, the fields and the media have no variation with respect to one cartesian coordinate (z). Assuming that the geometry is defined as a function of (x,y)-coordinates and that the illumination field is p-polarized, we compute the *z* component of the magnetic field *H*_{z}
and we deduce the electric field **E**(*x,y*) from Maxwell-Ampère’s equation. The z-component *H*_{z}
of the harmonic magnetic field in a domain Ω verifies:

where the different layers are supposed to be non magnetic [15] and *c* is the velocity of light in vacuum. Equation (4) is called the homogeneous scalar wave equation for the p-polarization.

To find the physical solution for the electromagnetic energy absorbed by the system (probe+media), a set of boundary conditions on the contour Γ=Γ_{0}+Γ_{1} of the computational domain Ω has to be established. The approximate boundary conditions have been extensively used in problems of wave propagation, radiation and guidance to simulate the material and geometric properties of surfaces [16]. In our approach, we consider the continuity of the tangential components of the electromagnetic field *H*_{z}
(impedance condition [16]). The artificial outer boundary, which limits the region of computation, is characterized by the incoming wave amplitude and the first order Sommerfeld’s (or running waves) boundary conditions [16]:

where *∂/∂ n* is the outgoing normal derivative operator and *H*_{i}
the illumination field.

To solve the problem, the FEM makes use of a variational formulation (or weak formulation):

where *ν* is a test function defined on *L*
^{2}(Ω) (the linear space of the scalar functions *ν*, being 2-integrable on Ω). This integral formulation enables us to consider the domain of computations as the union of *N* sub-domains, thus Ω=${\cup}_{i=1}^{N}$Ω
_{i}
. The problem is solved on each sub-domain and the global solution is then the sum of the solutions in each element. The Eq. (6) is the projection of the solution on a basis of test functions *ν*. The solution verifies exactly the PDE on each node for the given boundary conditions. The basis of polynomial functions ν gives an approximation of the solution into the element. Numerical implementation involves a finite basis of functions and consequently, a linear system can be solved by classical numerical methods such as Gaussian elimination, Cholesky, LU decompositions or by iterative procedures such as bi-conjugate gradient, Lanczos’ algorithm, for large systems.

We have developed a home-made code [17, 18] with self-adaptive mesh refinement, which is particularly adapted to the description of complex geometry. The validity of our method by comparison with other ones has been established in previous papers [17, 18].

## 4. Results and discussions

In Fig. 4, we present the temporal evolution of the optical intensity (square modulus of the electric field) and of the absorbed optical energy density in the phase-change material computed with a 2D FEM for the experimental parameters given in Sec. 2. Let us note that no scan is used neither in the experiment nor in the simulations. The temporal evolution in Fig. 4 shows the simulation of the modifications due to the material permittivities evolution during the illumination process. Figure 4(a) shows that the optical intensity is located below the probe end. A logarithmic scale of the intensity is used otherwise only the FE is visible. With reference to Fig. 4(b), let us note that the lateral size of the absorbed power density in Ge_{2}Sb_{2}Te_{5} is increasing with time due to the change of permittivity. The final size of the crystallized dot is larger than both the probe end and the associated field enhancement (see Fig. 4(a–b)). In the first steps of the illumination, the absorbed optical power density is located below the probe end. Then, the lateral size of the absorbed optical energy zone increases in time before reaching a constant value (see also Fig. 6). Moreover, the edges of the crystalline domain are visibly due to the FE of this phase-changed zone that is embedded in a dielectric material. The field response of the edges of the phase-change dot may be compared to the FE phenomenon at the end of the probe.

In Fig. 5, we present an experimental result of a bit written considering the conditions described above. Figure 5(a) is an Atomic Force Microscopy (AFM) data and Fig. 5(b) is the corresponding ASNOM image. The AFM map exhibits no topography despite the fact that the crystalline phase is denser than the amorphous one. The topography of the written bit is smaller than 1 nm, which is the maximum noise level in this recording data.

Figure 5(c) is the optical intensity profile extracted along the dashed line in Fig. 5(b). The lateral size of the dot along the direction of illumination (*x*) may be evaluated at *d*_{x}
=145 nm as predicted by the computations. The dot size along the *z* direction is twice as *d*_{x}
. This enlargement is due to the symmetry of the system with respect to the plane of incidence. The FE is the same on both sides of the probe. This effect cannot be predicted by a 2D model.

Let us focus now on the size of the dot as a function of time for various incoming laser power *P*_{W}
. The results are given as a function of the threshold power ${P}_{W}^{0}$, which is characteristic of the material. ${P}_{W}^{0}$ corresponds to an absorbed power density ${Q}_{k}^{0}$ equal to the maximum value of the *Q*_{k}
computed without the probe. Figure 6 shows the temporal evolution of the size of the written dot for five ratios of ${P}_{W}^{0}$ (between 0.5, and 0.9). We observe that the lateral size of the dot increases in time before reaching a constant value (around 145 nm as showed in the experimental image in Fig. (5)).

The left part of Fig. (6)) shows the smallest dot size that could be reached for different laser powers, if the pulse durations were of the same order as the characteristic crystallization time *τ*_{pc}
. This first plateau is only related to the FE and the size of the dot may be smaller than the FE if the power *P*_{W}
is correctly tuned. If *P*_{W}
is small enough, the useful zone of intensity (that could produce phase-change) decreases and may be smaller than the probe end size but it may be more complicated to be controlled experimentally.

If the duration of the illumination is greater than 1 *µ*s the size of the dot varies strongly and then reaches a limit value after 1.5 *µ*s. The problem exhibits a characteristic stabilization time *τ*_{st}
(~2 *µ*s) for the variation of the permittivity in the PC material. This second plateau corresponds to an increment of the dot size, that is induced by both the laser spot and the variation of the permittivity.

## 5. Conclusion

In this paper, we have presented a model of probe monitored phase-change writing process and we have shown a good agreement with the first experimental data. The critical parameters are the illumination duration and the power density. We have shown that the illumination duration should be lower than 1 *µ*s to generate small phase-change dots. Moreover, decreasing the power density leads to small dots but this configuration is more difficult to handle experimentally. In principle, a 3D approach should be introduced to discuss on the shape of the dot. Furthermore a more specific crystallization model should be introduced. However, due to the complex geometry of the problem, it would require very large memory resources and computation time. At this stage, a complete 3D model for the electro-thermal writing process would be prohibitive.

## Acknowledgments

Th. Grosges was supported by Grant #35935E from the “Conseil Régional de Champagne-Ardenne”. Authors thank Ludovic Poupinet for fruitful discussions and the members of LETICEA for providing the phase-change sample.

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