Absorptivity of three kinds of surface morphology, i.e., V-type groove surface, sinusoidal surface, and random distribution, isinvestigated using a rigorous electromagnetic theory and a finite element method. Influences of surface contour parameters(span distance, intersection angle, and height) and light wave parameters (incident angle and wavelength) on absorptivity arenumerically simulated and analyzed for the three kinds of surfaces, respectively. Absorbing spectra about three silicon waferswith different surface roughness are recorded, and the results are coincident with simulated results.
1. Introduction
Improving absorptivity of light-absorbing materials is verysignificant in laser machining, solar cell manufacturing, andlight-sensitive detector fields, wherein surface morphologyand surface roughness become one major factor of influenc-ing material absorptivity. When heat treatment is with a laserbeam, a metallic material with a smooth surface has only lessthan ten percent absorptivity to laser [1], and absorptivity ofnonmetallic materials relates to the incident angle of laserbeam, which is affected by material surface roughness[2, 3]. For metallic and nonmetallic materials, its absorptivitycan be heightened through adjusting material surface rough-ness due to increased light reflected and absorbed a numberof times. Solar cell can transfer radiation energy to electricalenergy, and its absorptivity is one of the most important per-formance parameters. Many methods can be used to enhanceabsorptivity to solar light, e.g., overlaying a surface layer withhigh absorptivity [4–6], doping with metal nanoparticles tomotivate surface plasma polarization excimer [7–9], andcontrolling surface morphology [10–14]; here, improvingsurface roughness is a very effective method. Furthermore,to improve the performance of a solar radiometer, someblack coating material with high absorptivity is overplayedon the surface and its surface roughness is changed [15].
Therefore, investigating the relationship between materialabsorptivity and surface roughness of light-absorbing mate-rials is significant.
Li et al. [15] and Su et al. [16] have analyzed the lightabsorbing ability of several materials with different types ofa surface morphology-based light ray tracing method andachieved some common characteristics of improving lightabsorptivity through changing surface morphology andderived a relationship formula between absorptivity and sur-face morphology parameters. Chen et al. [17] have equiva-lently dealt with surface contour line and established acomputational model of influence of surface roughness onlaser absorptivity using a light ray tracing method and pro-vided a more convenient method for calculating laser absorp-tivity in a laser heat treatment field.
However, the light ray tracing method is based on ahypothesis that light propagates along a straight line, whichis only valid when the characteristic structure dimension ofsurface morphology of absorbing material, e.g., the periodof a microstructure and span of random fluctuation, is fargreater than incident light wavelength. But the majority ofsurface random fluctuations of a common material belongto a micrometer magnitude order and are close to visible lightwavelength; therefore, the light ray tracing method is notsuited to calculate the absorptivity of a rough surface.
HindawiInternational Journal of PhotoenergyVolume 2019, Article ID 1476217, 9 pageshttps://doi.org/10.1155/2019/1476217
In this thesis, a rigorous electromagnetic theory isadopted to analyze the influence of surface morphology andsurface roughness on material absorptivity. COMSOLMulti-physics, which is a commercial software and can be utilizedto numerically solve partial differential equations based ona finite element analysis method, is used to solve Maxwell’sequations and simulate the absorbing effect of the rough sur-face to light. The absorptivity of the three kinds of surfacecontour profiles such as the V-type groove structure, sinu-soidal wave structure, and random fluctuation structure isnumerically calculated, and relationship curves between lightabsorbing performance and surface morphology parame-ters are obtained. At last, absorbing spectra of three siliconwafers with different surface roughness are measured, andthe results are basically consistent with the numericalsimulation results.
2. Shortcoming of a Light Ray Tracing Method
Optical wave is a kind of electromagnetic wave in nature,and its propagation obeys Maxwell’s equations, which canbe expressed as
∇ ×H = J + ∂D∂t
,
∇ × E = −∂B∂t
,
∇·D = ρ,∇·B = 0,
1
where E and D represent the electric intensity vector andelectric displacement vector, respectively, and have a rela-tionship as D = εE, where ε is the dielectric constant. Hand B are the magnetic intensity vector and magneticinduction intensity vector, respectively, and have a rela-tionship as B = μH, where μ is the magnetic permittivity.Furthermore, J is the current density, and J = σE, whereσ represents the electrical conductivity.
A harmonic plane wave can be expressed as
E r, t = E r ejωt ,H r, t =H r ejωt ,
E r = e r ejk0S r ,
E r = e r ejk0S r ,
2
where ω is the circular frequency, k0 depicts the wave vectorin vacuum, and S r represents the optical path length and isa real scalar function.
Supposing that light wavelength λ can be approximatedto an infinitely small quantity, an equation can be deducedas ∇S r = n, where n represents the refractive index ofpropagation medium. For an isotropic and homogeneousmedium, it can be derived from the equation ∇S r = n thatlight wave propagates along a straight line.
As mentioned above, the light ray tracing method is validonly when three conditions are satisfied: (1) a harmonicwave, (2) λ→ 0, and (3) n = constant.
If a monochromatic plane wave is incident on a materialwith a smooth and infinite surface, as shown in Figure 1,according to the Fresnel formula [18], the reflectance canbe expressed as
ρ = 12
n − cos α 2 + k2
n + cos α 2 + k2+ n − 1/cos α 2 + k2
n + 1/cos α 2 + k2, 3
where n is the material refractive index, k is the light extinc-tion coefficient, and α is the incident angle. If the thickness ofthe absorbing material is enough to absorb all the transmis-sion light, absorptivity of the absorption material to theincident light can be expressed as A α = 1 − ρ.
A V-type groove structure is always used to replace thesurface contour profile of a naturally formed and processedcoarse surface. According to the light ray tracing method,an incident light beam will endure many times the reflectionin a V-type groove, as shown in Figure 2(a). In Figure 2(a),L represents the opening width, H is the depth of a V-typegroove, and θ is an intersection angle, and there is a relation-ship formula: L = 2H∙sin θ/2 .Thewhole absorbing percent-age of light in a V-type groove can be obtain by consideringreflectance in every time according to equation (3) [15, 16].
As mentioned above, the light ray tracing method can beadopted only when the opening width L of the V-type grooveis far greater than incident light wavelength λ. If L is approx-imately equal to or less than λ, the diffraction of the V-typegroove and interference among reflecting wave must beconsidered. In this moment, the light field distribution willbe complex and must be calculated through a rigorouselectromagnetic theory.
In Figure 2(b), the electric intensity E 2 distributionfor a V-type groove structure with L = 2 μm, θ = 60°, andλ = 0 8 μm is shown. It can be seen from Figure 2(b) that,due to considering many time reflection, diffraction, andmultiple-beam interference in the rigorous electromagnetictheory, the last light field distribution is inhomogeneous,but the result cannot be obtained through the light raytracing method.
Absorption material
Incident light beam
Normal line
�훼
Figure 1: Schematic diagram of light incident on an absorptionmaterial.
2 International Journal of Photoenergy
3. Simulation Results and Analysis Based upon aRigorous Electromagnetic Theory
In order to analyze the influence of surface morphology onabsorptivity more comprehensively, two types of surface con-tour profiles are adopted, that is, regular surface morphologyand random surface morphology. Here, regular surface mor-phology includes the V-type groove structure and sinusoidalstructure, which are two-dimension symmetry and elemen-tary shape functions and often used to replace contourprofiles of coarse surfaces. A random surface has randomheight values with normal distribution and is more similarto realistic surface morphology.
Figure 3 shows three kinds of different surface morphol-ogies, which are the V-type groove structure, sinusoidalstructure, and random structure.
The absorbing performances of different surfacemorphol-ogies are numerically simulated and analyzed by adoptingthe wave optics module in commercial software COMSOLMultiphysics, which is based upon the finite element methodto solve Maxwell’s equations.
Simulation parameters are listed in Table 1. For conve-nience, a 2-dimension structure is considered and the simu-lation region sizes are 10 μm in the horizontal directionand 10μm in the vertical direction. The left and right bound-aries adopt the periodic boundary condition, and the up anddown boundaries use the “port” boundary condition. Thelight is incident from air to silicon substrate. In order toinvestigate the general absorption rule, the refractive indexand extinction coefficient of air and silicon subtracts arechangeless for all simulated light wavelength.
Figure 4 shows the mesh subdivision diagram for thethree kinds of surface structures, wherein the upper half isair and the lower half is crystalline silicon. In order to guar-antee the accuracy of the solution, the biggest mesh dimen-sion is less than λ0/6∙n, where λ0 represents the incidentlight wavelength in vacuum and n is the refractive index ofair or crystalline silicon.
3.1. Absorptivity vs. Surface Morphology Parameters of aV-Type Groove Structure. Figure 5 displays the absorbingperformance of the V-type groove surface structure. Inthis case, light wave with wavelength λ = 0 8 μm is nor-mally incident on the crystalline silicon surface with theV-type groove as two electromagnetic polarization modes:TE mode (E vector is perpendicular to the water plane) andTM mode (H vector is perpendicular to the water plane). InFigure 5, the x-axis represents the intersection angle of theV-type groove and the y-axis represents absorptivity.
It can be seen from Figure 5 that, for two opening widthsL = 0 5 μm and L = 1 μm, absorptivity corresponding to theTM mode is larger than absorptivity to the TE mode, whichis because in this simulation a two-dimension structure isadopted, i.e., the V-type groove structure is changeless alongthe y-axis, so that there exists polarization selectivity. How-ever, for the three-dimension V-type groove structure, asshown in Figure 3, the absorbing performance should bemixed results by the TE mode and TM mode. Therefore, insubsequent discussion, the total absorptivity will be an aver-age value of the TE mode and TM mode.
Furthermore, it is obvious in Figure 5 that four absorptiv-ity curves roughly tend to decrease along with the increasingangle θ, which is similar to results according to the light raytracing method [15, 16]. But when the angle θ is less, absorp-tivity curves have some fluctuation. The appearance can beexplained as the light wave incident on the V-type groovesurface with less angle θ will endure more reflecting timesand there exist more complex multiple-beam interferenceresults, but for the case θ > 120°, light wave is only reflectedonce, so that absorptivity monotonously decreases accordingto the analysis of equation (3).
In addition, it can also be seen from Figure 5 that, forthe identical intersection angle θ, absorptivity correspond-ing to opening width L = 1 μm is larger than absorptivity toL = 0 5 μm, which is because when θ is a constant quantity,a larger opening width means a deeper V-type groove andmore light reflection times.
�휃�휃
L L
H
(a) Light ray tracing (b) E 2 calculated through the finite element method
Figure 2: Surface structure of V-type groove and light field distribution.
3International Journal of Photoenergy
In the light ray tracing method, suppose λ→ 0, lightabsorptivity is independent of light wavelength. But accord-ing to the rigorous electromagnetic theory, light propagationperformance in medium is influenced by light wavelengthdeeply. Figure 6 shows the absorptivity varying curve ofthe V-type groove structure along with light wavelengthwhen θ = 60°, L = 1 μm, and α = 0°. Within the wavelength
scope from 0.4 μm to 1μm, absorptivity has only weakvariety, but when light wavelength continues to increase,absorptivity will tend to decrease, and the absorptivitycurve has more and more fluctuation. This result can beexplained as follows: when the opening width L of theV-type groove is larger than light wavelength, the lightwave can be treated approximately as light beam with astraight propagation path, but when L < λ and H < λ, theV-type groove surface can be taken as an equivalenthomogeneous medium layer [19, 20]; the reflectivity willfluctuate along with increasing light wavelength due tomultiple-beam interference of the equivalent film, and atthe same time, larger light wavelength means thinner filmthickness and weaker absorbing ability.
Figure 7 shows the absorptivity varying curve alongwith the light incident angle α when L = 1 μm and λ =0 8 μm. It is obvious that in three curves, absorptivityroughly reduces with the incident angle α increasing, buta larger intersection angle θ corresponds to a larger fluctu-ation of the absorptivity curve. This appearance can beexplained as follows: when the intersection angle θ issmall, light wave will be reflected many times in theV-type groove, so that reflecting times reducing caused bythe light incident angle α changing can be neglected; butwhen θ is larger, light wave only endures less reflecting timesin the V-type groove, reflecting times changing with theincident angle will be very dominant.
3.2. Absorptivity vs. Surface Morphology Parameters of aSinusoidal Surface Structure. The sinusoidal surface structurecan be looked as a passivated V-type groove structure, andits equivalent intersection angle θ can be expressed as θ = 2
arctan T/4A , where T means the period of the sinusoidalsurface structure and A is the amplitude, as shown in Figure 8.
Figure 9 shows the absorptivity varying curve along withlight wavelength supposing that T= 1μm, θ = 60°, and α = 0°.
Figure 4: Mesh subdivision diagram of three surface structures.
160
Abso
rptio
n ra
tio
TM-mode L = 1 �휇m
TM-mode L = 0.5 �휇m
TE-mode L = 1 �휇m
TE-mode L = 0.5 �휇m
Angle of groove �휃 (°)14012010080
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.6040 60
Figure 5: Varying curve of absorptivity along with the intersection angle of the V-type groove structure.
5International Journal of Photoenergy
It can be seen from Figure 9 that in the whole scope from0.4μm to 4μm, there exists a distinct fluctuation butabsorptivity decreases with increasing light wavelength,approximately.
Figure 10 shows the absorptivity varying curve of the sinu-soidal surface structure along with the light incident angle αwhen T = 1 μm and λ = 0 8 μm. It is very similar to Figure 7that absorptivity approximately reduces with the increasingincident angle α, but the larger intersection angle θ corre-sponds to the larger fluctuation of the absorptivity curve.
3.3. Absorptivity vs. Surface Morphology Parameters of aRandom Surface with Normal Distribution. The height of acommon realistic structure of a natural or treated surface is
Abs
orpt
ion
ratio
Wavelength (�휇m)
0.75
0.80
0.85
0.90
0.95
1.00
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Figure 6: Absorptivity varying curve of V-type groove structurealong with wavelength.
Abs
orpt
ion
ratio
Incident angle �훼 (°)
0.6
0.7
0.8
0.9
1.0
0 10 20 30
�휃 = 30°�휃 = 60°�휃 = 90°
40 50 60 70 80
Figure 7: Absorptivity varying curve along with incident angle.
T
2A
�휃
Figure 8: Sinusoidal surface structure.
Abs
orpt
ion
ratio
Wavelength (�휇m)
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Figure 9: Absorptivity varying curve of sinusoidal surface structurealong with wavelength.
Abs
orpt
ion
ratio
Incident angle �훼 (°)
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30
�휃 = 30°�휃 = 60°�휃 = 90°
40 50 60 70 80
Figure 10: Absorptivity varying curve of sinusoidal surfacestructure along with incident angle.
6 International Journal of Photoenergy
often normal random distribution, in which height probabil-ity density function can be expressed as
p z = pk2πσ
exp −z − μ 2
2σ2 , 4
where p z represents the probability density of the surfaceheight, pk is the height scale coefficient, and μ and σ arethe mean value and root mean square value (RMS) of the sur-face height, respectively. In this thesis, a normal-distributionrandom surface with μ = 0 and σ = 1 μm is adopted.
Figure 11 illustrates the absorptivity varying curve of therandom surface structure with different pk values, where T isdefined as the span distance between two wave peaks in therandom distribution surface and T = 2 μm. It is clearly seenthat when pk = 0 25 and pk = 0 5, there are obvious fluctua-tions in the absorptivity varying curve, but when pk = 1,absorptivity is close to 1 and there are only small fluctuations.A larger pk value means a higher groove depth and a smallintersection angle, similar to cases in Figures 7 and 10, so thatwhen roughness is high enough, the absorptivity of the coarsesurface will be independent to the light incident angle.
The absorptivity varying curve of the random surfacestructure with T taking different values is shown inFigure 12. When the light incident angle is less than 60°, itis very clear that a smaller span distance T value correspondsto a higher absorption ratio, but for an incident angle biggerthan 60°, there are larger fluctuations in absorptivity varyingcurve and there exists no obvious tendency with a changing Tvalue. In most cases, light wave is incident with an incidentangle less than 60°; an identical pk value and a smaller Tvalue, corresponding to higher surface roughness, will havea higher absorption ratio.
Figure 13 shows the absorptivity varying curve of the ran-dom surface structure along with light wavelength supposingthat T = 2 μm, pk = 1, and α = 0°. It can be seen that in thewhole scope from 0.4μm to 4μm, absorptivity decreases with
light wave increasing approximately but with some fluctua-tion, the same as in Figure 9.
According to the above analysis, for improving theabsorption ability of silicon subtract, the random surfacestructure should have a larger pk value and a smaller T value,the suitable regions of the ratio pk/T are from 0.5 to 10,because a larger pk/T value is difficult to realize at the presenttechnological level. Moreover, in the above simulation, theextinction coefficient of silicon substrate is changeless for dif-ferent light wavelengths, but in fact, crystalline silicon mate-rial has larger absorption ability in a visible light region butlower absorption in a near-infrared light region. In consider-ation of the above simulation results and absorption charac-teristic of silicon material, the suitable wavelength region isfrom 400nm to 600nm.
4. Experimental Verification
In order to verify the effectiveness of the above simulationand analyzed results, the three kinds of coarse surfaces ofsilicon wafer are considered to investigate absorbing ability.
The roughness of the three surfaces is measured by usinga Talyrond-365 type of a surface roughometer produced byTaylor Hobson Ltd. The measured roughness curves areshown in Figure 14.
In Figure 14, y-axis represents random fluctuation and itsunit is micrometer. Three surface roughness curves are allrandom distribution and belong to the three kinds of sur-faces, i.e., the smooth front surface, the rough rear surface,and the treated rear surface of a commercial silicon wafer,respectively. The contour arithmetic mean Ra is often usedto estimate the roughness level of the coarse surface, and Raof three surfaces shown in Figure 14 are 0.0153 μm,0.9994μm, and 0.9038μm, respectively.
Reflection of light on the coarse surface belongs to diffusereflection. Diffuse reflection spectra of three surfaces are
Figure 11: Absorptivity varying curve of random surface structure.
Abs
orpt
ion
ratio
Incident angle �훼 (°)
0.95
0.96
0.97
0.98
0.99
1.00
1.01
0 20
pk, = 1, T = 0.5 �휇mpk, = 1, T = 2 �휇mpk, = 1, T = 4 �휇m
40 60 80
Figure 12: Absorptivity varying curve of random surface structurewith different T values.
7International Journal of Photoenergy
measured by using a set-up shown in Figure 15, white opticalemitted from a halogen tungsten lamp (AvaLight-DHc) isguided into integrating sphere (AvaSphere-50) throughan optical fiber, a part of incident white light is absorbedby measured powder sample placed above a standardreflection whiteboard, and other incident white light isscattered in all directions. The scattered light will be scat-tered secondly or repeatedly when collided with the wall ofthe integrating sphere, so after many times of collision,remaining scattered light will have spatially uniform distri-bution in an integrating sphere. A fiber connector is used
to guide the remaining part of scattered light into a fiberspectrometer (AvaSpec-ULS2048-USB2) for measuring dif-fuse reflection spectrum. The absorptivity of the sample canbe expressed as A λ = 1 − I λ /I0 λ , where I λ representthe diffuse light intensity measured as the set-up shown inFigure 15 and I0 λ is the diffuse light intensity measuredonly for a standard reflection whiteboard.
Figure 16 shows the measured absorption spectrum ofthe three surfaces. For the smooth surface, there is a Ra valueof 0.0153 μm and the lowest roughness, so it should have thelowest absorption spectrum, which is in accordance with theresult shown in Figure 16. It can be seen in Figure 14 that therough surface and the treated rough surface have close Ravalues (0.9994μm and 0.9038μm), but the treated rough sur-face has more spacing distance T or intersection angle θ than
Abs
orpt
ion
ratio
Wavelength (�휇m)
0.850.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.90
0.95
1.00
Figure 13: Absorptivity varying curve of random surface structure along with wavelength.
−4
−2
0
(a) (b) (c)
2
4
Figure 14: Roughness measured from three different surfacepatterns.
Fiber connector(to optical source)
Fiber connector(to spectrometer)
Integrating sphereSample
Standard reflection whiteboard
Figure 15: Set-up diagram for measuring diffuse reflectionspectrum.
400 500 600 700 800Wavelength (nm)
90040
Abso
rptio
n ra
tio (%
)
50
60
70
80
90
1000
Rough surface
Treated rough surface
Smooth surface
Figure 16: Absorption spectrum for three surfaces.
8 International Journal of Photoenergy
the rough surface; therefore, the treated rough surface shouldhave lower absorptivity, which is also coincident with theresults in Figure 16.
5. Conclusion
In conclusion, the absorptivity of the three kinds of surfaceswith different morphologies is investigated by using the rig-orous electromagnetic theory and finite element method.The three surfaces are the V-type groove surface, sinusoidalsurface, and random distribution surface and are graduallyclose to a realistic coarse surface. For the three surface, influ-ences of surface contour parameters (span distance, intersec-tion angle, and height) and light wave parameters (incidentangle and wavelength) on absorptivity are numerically simu-lated and analyzed. Moreover, absorbing spectra about threesilicon wafers with different surface roughness are measured,and the results are coincident with simulated results.
Data Availability
The (figures and tables) data used to support the findings ofthis study are included within the article.
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper.
Acknowledgments
The thesis is funded by the “Project of Development andImprovement of Scientific Research of Beijing InformationScience and Technology University.”
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