X-ray reflectometry is a powerful tool for investigations on rough surface and interface structures of multilayered thin film materials. The X-ray reflectivity has been calculated based on the Parratt formalism, accounting for the effect of roughness by the theory of Nevot-Croce conventionally. However, in previous studies, the calculations of the X-ray reflectivity often show a strange effect where interference effects would increase at a rough surface. And estimated surface and interface roughnesses from the X-ray reflectivity measurements did not correspond to the TEM image observation results. The strange result had its origin in a used equation due to a serious mistake in which the Fresnel transmission coefficient in the reflectivity equation is increased at a rough interface because of a lack of consideration of diffuse scattering. In this review, a new accurate formalism that corrects this mistake is presented. The new accurate formalism derives an accurate analysis of the X-ray reflectivity from a multilayer surface of thin film materials, taking into account the effect of roughness-induced diffuse scattering. The calculated reflectivity by this accurate reflectivity equation should enable the structure of buried interfaces to be analyzed more accurately.

X-rays scattered from a material surface at a glancing angle of incidence provide a wealth of information on the structure of the surface layer of materials. X-ray scattering spectroscopy is a powerful tool for investigations on rough surface and interface structures of multilayered thin film materials [

Because the X-ray scattering vector in a specular reflectivity measurement is normal to the surface, it provides the density profile solely in the direction perpendicular to surface. Specular reflectivity measurements can yield the magnitude of the average roughness perpendicular to surface and interfaces but cannot give information about the lateral extent of the roughness. In previous studies, the effect of roughness on the calculation of the X-ray reflectivity only took into account the effect of the density changes of the medium in a direction normal to the surface and interface. On the other hand, diffuse scattering can provide information about the lateral extent of the roughness. In contrast to previous calculations of the X-ray reflectivity, in the present analysis, we consider the effect of a decrease in the intensity of penetrated X-rays due to diffuse scattering at a rough surface and rough interface.

In this review, we show that the strange result has its origin in a currently used equation due to a serious mistake in which the Fresnel transmission coefficient in the reflectivity equation is increased at a rough interface, and the increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient because of a lack of consideration of diffuse scattering. The mistake in Nevot and Croce’s treatment originates in the fact that the modified Fresnel coefficients were calculated based on the theory which contains the X-ray energy conservation rule at surface and interface. In their discussion, the transmission coefficients were replaced approximately by the reflection coefficients by the ignoring diffuse scattering term at the rough interface and according to the principle of conservation energy at the rough interface also. The errors of transmittance without the modification cannot be ignored. It is meaningless to try to precisely match the numerical result based on a wrong calculating formula even to details of the reflectivity profile of the experimental result. Thus, because Nevot and Croce’s treatment of the Parratt formalism contains a fundamental mistake regardless of the size of roughness, this approach needs to be corrected. In the present study, we present a new accurate formalism that corrects this mistake and thereby derive an accurate analysis of the X-ray reflectivity from a multilayer surface, taking into account the effect of roughness-induced diffuse scattering. The calculated reflectivity obtained by the use of this accurate reflectivity equation gives a physically reasonable result and should enable the structure of buried interfaces to be analyzed more accurately. This paper is the review article that is edited based on the two research articles of IOP Science [

The surface and interfacial roughness of the same sample of multilayered thin film material was measured by transmission electron microscopy (TEM) and compared them with those from X-ray reflectivity measurements. The surface sample for examination was prepared as follows; a GaAs layer was grown on

Cross-section image of

X-ray reflectivity measurements were performed using a Cu-K

Measured X-ray reflectivity from a silicon wafer covered with a thin (48 nm) GaAs layer.

In Section

The intensity of X-rays propagating in the surface layers of a material, that is, the electric and magnetic fields, can be obtained from Maxwell’s equations [

The reflectance of an

Following that approach, let

For X-rays of wavelength

We take the vertical direction to the surface as the

The electric field of X-ray radiation at a glancing angle of incidence

The incident radiation is usually decomposed into two geometries to simplify the analysis, one with the incident electric field

Because an X-ray is a transverse wave, the amplitude and the wave vector are orthogonal as follows:

We consider the relation of the electric field

Reflected and transmitted X-rays.

The electric fields

In general, when X-rays that are linearly polarized at an angle

Next, we consider the reflection from a flat surface of a multilayer with flat interfaces. We consider the electric field

Reflection and transmission of X-rays in the

The electric fields

The amplitudes

The reflection coefficient is defined as the ratio

When the surface and interface have roughness, the Fresnel coefficient for reflection is reduced by the roughness [

Figure

Calculated (dots) and measured (line) reflectivity from a GaAs layer with a thickness of 48 nm on a Si substrate. The surface roughness

Calculated (dotted, dashed, and thin lines) and measured (thick line) reflectivity from a GaAs layer with a thickness of 48 nm on a Si substrate. In the calculation, the interface roughness

Although the calculated results did more closely approach those from experiment, they still showed poor agreement. The ratio of the oscillation amplitude to the value of the reflectivity near an angle of incidence of 0.36° in the calculated reflectivity for the GaAs surface of 4 nm roughness in Figure

Figure

Calculated reflectivity from a GaAs layer with a thickness of 48 nm on a Si substrate. The solid curve is for a flat surface and a flat interface. The dashed curve is for a surface roughness

In the reflectivity curve (dashed line) for a surface roughness of 4 nm and with a flat interface, the ratio of the oscillation amplitude to the size of the reflectivity near an angle of incidence of 0.36° is much larger than the reflectivity of the flat surface in Figure

To probe these effects further, we then recalculated the reflectivity for surface roughness of 3.5 nm, 4 nm, and 4.5 nm, and with a flat interface. Those calculated reflectivity results are shown in Figure

Calculated reflectivity from a GaAs layer with a thickness of 48 nm on a Si substrate. In the calculation, the interface roughness

For most angles of incidence within this range, the reflectivity of the surface with a roughness of 4 nm is near the mean value of the reflectivity of surfaces with roughnesses of 3.5 nm and 4.5 nm. However, near an angle of incidence of 0.36°, the reflectivity of the surface with a roughness of 4 nm is very much attenuated compared to that same average. It seems very strange that the reflectivity of the average roughness has a value quite different from the mean value of the reflectivity of each roughness, because the value of the roughness is not the value of the amplitude of a rough surface but the standard deviation value of various amplitudes of rough surface.

Figure

X-ray reflectivity from a silicon wafer covered with a thin (10 nm) tungsten film calculated by the theory in use prior to this work. Solid line shows the case of a flat surface. Dashed line shows the case of a surface with an rms surface roughness of 0.3 nm.

We now consider the previous strange result of the X-ray reflectivity which was calculated based on the Parratt formalism [

The Fresnel coefficients for refraction at the rough interface are derived using the Fresnel reflection coefficient

Here, we consider the energy flow of the X-ray. In electromagnetic radiation,

The amplitudes

This conservation expression should not apply any longer when the Fresnel reflection coefficient is replaced by the reduced coefficient

The penetration of X-rays should decrease at a rough interface because of diffuse scattering. Therefore, the identity equation for the Fresnel coefficients become

Initially, we consider the reduced Fresnel coefficient, which is known as the Croce-Nevot factor. When the

The modified Fresnel refraction coefficients

The derived Fresnel refraction coefficients

Moreover, if the deformation modulus of

In Nevot and Croce’s treatment of the Parratt formalism for the reflectivity calculation including surface and interface roughness [

The resulting increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient at the rough interface. Thus, because Nevot and Croce’s treatment of the Parratt formalism contains a fundamental mistake regardless of the size of the roughness, results using this approach cannot be correct. The size of the modification of the transmission coefficient is one-order smaller than that of reflection coefficient, but the size of transmission coefficient is one-order larger than the reflection coefficient at angles larger than critical angle. Thus, the errors of transmittance without the modification cannot be ignored.

Of course, there are cases where that Nevot and Croce’s treatment can be applied. However, their method can be applied only to the case where there is no density distribution change at all in the direction parallel to the surface on the surface field side, and only when the scattering vector is normal to the surface. A typical example of surface medium to which this model can be applied is one where only the density distribution change in the vertical direction to the surface exists, as caused by diffusion, and so forth. In such a special case, Nevot and Croce’s treatment can be applied without any problem. However, because a general multilayer film always has structure in a direction parallel to the surface field side, Nevot and Croce’s expression fails even when the roughness is extremely small. The use of only Fresnel reflection coefficients by Nevot and Croce is a fundamental mistake that does not depend on the size of the roughness.

To proceed, we therefore reconsider the derivation of the average value of the matrix as the same derivation of (

When the

Then the amplitudes

The modified Fresnel refraction coefficients

Once again we consider process by which we derive the average value of the matrix. When the

Because X-rays that penetrate an interface reflect from the interface below and penetrate former interface again without fail, it is necessary to treat the refraction coefficients

The Fresnel refraction coefficients

New calculated reflectivities from a GaAs layer with a thickness of 48 nm on a Si substrate. The line is for a flat surface and a flat interface. The dashed curve is for a surface roughness

The ratio of the oscillation amplitude to the size of the reflectivity in the reflectivity curve (dot) in Figure

Figure

New calculated reflectivity from a GaAs layer with a thickness of 48 nm on a Si substrate. In the calculation, the interface roughness

Next, we again calculated the X-ray reflectivity for the W/Si system but now considered the effect of attenuation in the refracted X rays by diffuse scattering resulting from surface roughness. However, the reduced refraction coefficient in prior work varies [

X-ray reflectivity from a silicon wafer covered with a thin (10 nm) tungsten film calculated by the new calculation that considered diffuse scattering. Solid line shows the case of a flat surface. Dashed line shows the case of a surface with an rms surface roughness of 0.3 nm.

In this review, we investigated the fact that the calculated result of the X-ray reflectivity based on Parratt formalism [

Therefore, the Fresnel coefficients for refraction at the rough interface are necessarily larger than the Fresnel coefficient for refraction at the flat interface. The resulting increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient. These coefficients for refraction obviously contain a mistake because the penetration of X-rays should decrease at a rough interface because of diffuse scattering. We propose that the unnatural results in the previous calculation of the X-ray reflectivity originate from the fact that diffuse scattering was not considered. We found that the strange result originates in the currently used equation due to a serious mistake where the Fresnel refraction coefficient in the reflectivity equation is increased at a rough interface. The increase in the transmission coefficient completely overpowers any decrease in the value of the reflection coefficient because of a lack of consideration of diffuse scattering. The mistake in Nevot and Croce’s treatment originates in the fact that the modified Fresnel coefficients were calculated based on the theory, which contains the X-ray energy conservation rule at the surface and interface. In their discussion, the transmission coefficient was replaced by the reflection coefficient so as to conserve energy, and so diffuse scattering was ignored at the rough interface. It is meaningless to try to precisely match the numerical result based on a wrong calculating formula even to details of the reflectivity profile of the experimental result. Thus, because Nevot and Croce’s treatment of the Parratt formalism contains a fundamental mistake regardless of the size of roughness, results based on this approach are not correct.

We have developed a new formalism that corrects this mistake, producing more accurate estimates of the X-ray reflectivity for systems having surface and interfacial roughness, taking into account the effect of roughness-induced diffuse scattering.

The new, accurate formalism is completely described in detail. The X-ray reflectivity

The penetration of X-rays should decrease at a rough interface because of diffuse scattering. Therefore, the identity equation for the Fresnel coefficients become

Figure

X-ray reflectivity from a silicon wafer covered with a thin (10 nm) tungsten film with an rms surface roughness of 0.3 nm. Dashed line shows the calculated result by the theory in use prior to this work. Solid line shows the calculated result by the new calculation that considered the reduction in the sum intensity of reflective X-ray and refractive X-ray by diffuse scattering.

The reflectivity calculated with this new, accurate formalism (