WO2006029604A1

WO2006029604A1 - Method for determining the crystal orientation of a crystallite of a multi-crystalline solid body and for measuring mechanical internal stresses in solid bodies by means of micro-raman-spectroscopy

Info

Publication number: WO2006029604A1
Application number: PCT/DE2005/001609
Authority: WO
Inventors: Horst Strunk; Michael Becker
Original assignee: Friedrich-Alexander- Universität Erlangen-Nürnberg
Priority date: 2004-09-17
Filing date: 2005-09-14
Publication date: 2006-03-23
Also published as: DE102004045175A1; DE112005002808A5

Abstract

In order to obtain detailed information on the expansion/stress tensors in the crystallite of a multi-cryistalline solid body, it is necessary to know the crystallite-orientation in relation to a fixed table co-ordinate system. The information on the crystal orientation is maintained in the intensities of the Raman-lines. According to a first step of the invention, said crystal orientation can be determined by evaluating the intensity path of the Raman strips according to the polarisation adjustments of the incident and scattered light. According to a second inventive step, the mechanical stresses can be determined by determining a level state of stresses in relation to the global table co-ordinate system for each crystallite. According to a third inventive step, mechanical stresses in a solid body having a know crystal orientation can be measured by measuring the frequency offsets of the Raman strips.

Description

description

Method for determining the crystal orientation of a crystallite of a multicrystalline solid and for measuring mechanical internal stresses in solids by means of micro-Raman spectroscopy

The present invention relates to a method for determining the crystal orientation of a crystallite of a multicrystalline solid by means of Raman spectroscopy

Method for determining mechanical stresses in multicrystalline solids by means of Raman spectroscopy using the crystal orientation determined beforehand according to the invention and a method for determining mechanical stresses in solids having known crystal orientation by means of Raman spectroscopy.

An important area of semiconductor technology relates to the production of solar cells from semiconductor materials, in particular the production of thin-film solar cells. An essential role for the market success of thin-film solar cells play the production costs and the efficiency in the application. To improve these criteria, new application-relevant manufacturing processes are constantly being developed and tested. For the production costs, the process yield and the process reliability and the reliability of the products produced are of importance. For this reason, suitable methods for process controls as well as for quality assurance and diagnostics are being researched. A distinction is made here between in-situ and ex-situ methods, ie methods which are carried out during a production process or after its completion. The semiconductor layer produced must fulfill certain quality criteria in order to be suitable for further processing into a solar cell. In general, however, will be Quality controls carried out only after the completion of the complete solar cell. Statements about their quality are not made directly after the preparation of the active semiconductor layer, but only at the end of the process, so that problems with the layer deposition or incorrectly set process parameters or the like are only ascertained at the end of the process. Under certain circumstances, this can lead to unnecessary production steps and make it more difficult to identify the problem that has arisen. All this leads to relatively high reject rates of the solar cells produced and to unnecessary use of materials and time. Especially for "in-line" production lines, however, it is advantageous to integrate a quality control into the production line as directly as possible after the production of the active semiconductor layer, in order to avoid unnecessary further processing steps on the semiconductor layer, if its inferior quality is at of quality control.

As a semiconductor material for photovoltaic applications in particular multi- or polycrystalline silicon has been found to be particularly relevant to application. However, it has been shown that structural defects in multicrystalline silicon wafers have an unfavorable effect on the production costs of solar cells in many respects. It is generally assumed that residual stresses remaining in the material are for the most part responsible for microcracks and for sudden wafer fractures. Furthermore, inhomogeneous stress fields occurring on specific microstructures induce defects (eg dislocations) which reduce the efficiency of silicon solar cells. It would be desirable to diagnose such internal stresses in the multicrystalline silicon material at an early stage during the production process of a solar cell, in order to avoid the production process due to exceeding of certain internal voltage values or already occurring microcracks to break expected wafer breakage or too poor efficiency of the solar cell.

One known method for determining internal voltages of semiconductors is Raman spectroscopy. In principle, it has been known for some time that the Raman bands are shifted in their frequency by the internal stresses, so that the magnitude of the frequency shifts can be regarded as a measure of the internal stresses. However, such measurements depend on the crystal orientation of the sample to be examined, which is initially unknown in microcrystalline silicon with respect to the individual crystallites. The crystallites adjoining the free surface of the multicrystalline silicon layer may have any desired crystal orientation. Therefore, in its application to the investigation of internal stresses in the past, Raman spectroscopy has been limited to monocrystalline semiconductor materials having a defined and known crystal orientation, whereby the methods known in the art are in need of improvement in this regard as well.

It is accordingly an object of the present invention to provide a method for determining the crystal orientation of a crystal of a multicrystalline solid. It is a further object of the present invention to specify a non-destructive method for determining mechanical stresses in multicrystalline solid-state materials. It is a further object of the present invention to specify a non-destructive method for determining mechanical stresses in solid materials with known crystal orientation, such as monocrystalline solid-state materials.

These objects are achieved by the features of the independent claims 1 to 3. Advantageous developments and refinements are the subject of dependent claims. The method according to the invention relates to a method for determining the crystal orientation of a crystallite of a multicrystalline solid by means of Raman spectroscopy. This method comprises in detail the following steps: a) measuring the intensity (I _φx ) of a Raman band at a defined first polarization direction (x) at variable polarization angle of the excitation radiation and b) measuring the intensity (I _φy ) of the Raman Band at a second polarization direction (y), which is orthogonal to the first polarization direction, with variable polarization angle (φ) of the excitation radiation and σ) determining the crystal orientation from the measured intensity gradients.

The inventive method according to claim 2 relates to a method for determining mechanical stresses in multicrystalline solids by means of Raman spectroscopy. This is an essential aspect of the invention

Method is first in a first step, the Kris¬ tallorientierung a specific area as a Kris¬ tallkorns pers on the surface of the multicrystalline Festkör¬ determine according to claim 1 and then in a second step to measure the frequency shift Ramanban¬ the and measured frequency shift and the previously determined crystal orientation to determine the interes¬ sierenden stress components of the multicrystalline Festkör pers in the selected area. Both steps can in principle be carried out by one and the same Raman measuring apparatus.

In the first method step of claim 2 and in claim 1 is exploited that the information is contained in the intensities of the Raman lines through the spatial crystal orientation of the crystallite.

The inventive method for determining mechanical stresses according to claim 2 has in detail the following steps:

Determining the crystal orientation of a crystallite according to claim 1,

Determining the voltages by measuring the frequency shift (Δcüi, Δω ₂ , Δco ₃ ) of the Raman bands and calculating certain voltage components (α **, τ _xy , σ _yy ) of the voltage tensor from the frequency shift (ACO ₁ , Δω ₂ , Δω ₃ ) and the previously determined crystal orientation.

The crystallites of the multicrystalline solid can, as is the case with silicon, have sizes of a few to a few micrometers. For this reason, it is indicated in the method according to claim 1 and in both method steps of claim 2 to use the so-called micro-Raman spectroscopy, which is known per se, in which the excitation beam is directed onto the top side by means of a microscope objective Surface of the solid sample to be examined is directed. The excitation beam can thus be focused on a focal spot with a diameter of a few micrometers, in particular less than 5 μm, or even less than 1 μm. Thus, it is possible to limit the measurement to a single crystallite lying on the free surface of the solid sample to be examined.

In each solid, there are a limited number of phonon modes that appear as Raman bands, whose frequency shifts due to the internal stresses of the solid can be measured in the second step and used to determine the voltage components. In the semiconductor silicon there are, for example, three optical phonon modes, namely two transversal optical (TO) phonon modes and one longitudinal optical (LO) phonon mode.

For reasons of the limited number of Raman bands and thus measurable frequency shifts or for reasons of general simplification, it is advantageous to assume a plane stress state of the solid in the xy plane, in which no stress components exist in the z-direction and thus only the three in the xy plane lying

Stress components σ _xx , τ _xy , σy _{y of} the stress tensor are ermit¬ determined. This is expedient and appropriate inasmuch as a wavelength is usually used for the excitation light, which in silicon has a penetration depth of only a few micrometers, about 3 μm, so that only a thin surface layer is excited, as assumed can be that there is no z-voltage component in it.

If the solid to be examined has cubic crystal symmetry, as will be explained in detail below, there are only three independent elastic compliances or elastic constants Sn, S ₂ , S ₄₄ which can be used for the calculation of the stress components.

However, the method according to the invention can in principle also be applied to solids with non-cubic crystal symmetry, such as solids with tetragonal or hexagonal crystal symmetry. The necessary criterion, however, is the Raman activity of the solid.

At too low stresses or strains in the solid, the differences between the frequency shifts of the Raman bands are usually too small to be visualized. It can then be the ramangestreute light polar- are independently analyzed and the determined frequency shift of the observed Raman band used in the calculation of the voltage components uniformly for all frequency shifts appearing in the equations.

The inventive method according to claim 3 relates to a method for determining mechanical stresses in solids with known crystal orientation by Raman spectroscopy. This method therefore assumes that the crystal orientation is known in advance and does not have to be determined separately. This method can thus be used, for example, in a monocrystalline solid. In the method, the frequency shifts (Δ (θ, Δω ₂ , Δω ₃ ) of the Raman bands are measured and the voltage components (σ _xx , τ _xy , σ _y y) of the stress tensor from the Fre¬ quenzverschiebung (Δα> i, Δω ₂ , Δω ₃ ) and the known crystal orientation, whereby the individual frequency shifts of the Raman bands belonging to different phonon modes are measured, by varying the polarization direction of the excitation light and measuring the Raman scattered light at the first (x) or the second polarization direction (y) three settings are found, in each of which one of the three optical phonons is dominant.

In the method steps of claims 1 to 3, the polarization of the excitation light can be varied by a λ / 2 plate. Furthermore, in these cases an analyzer can be positioned in the beam path of the light Raman scattered by the solid sample to be examined, which analyzer can be set to the first or the second polarization direction. When carrying out the method according to the invention, the excitation light is preferably focused on the sample by means of an objective such as a microscope objective so that the excitation light can be focused to a focal spot with a diameter of one or a few microns or less than one micrometre ,

In the following the invention will be explained in more detail with reference to drawing figures in conjunction with exemplary embodiments. Show it:

Fig. 1 Euler angle and rotation of the table coordinate system in the crystal coordinate system according to Glei chung (11) (s.u.);

FIG. 2 shows a two-dimensional model of a grain twisted in relation to the table coordinate system and planar stress state in the table coordinate system; FIG.

3 schematic structure and beam path of the micro-Raman spectrometer;

Fig. 4 rotation of the plane of oscillation linear-polarized

Light through a λ / 2 plate;

FIG. 5 shows normalized intensities of the individual phonons as a function of the polarization direction φ of the incident beam (analyzer position x); FIG.

FIG. 6 shows normalized intensities of the individual phonons as a function of the polarization direction φ of the incident beam (analyzer position y); FIG. 7 shows intensity ratios of the individual phonons as a function of the polarization direction φ of the incident beam (analyzer position x);

FIG. 8 shows intensity ratios of the individual phonons as a function of the direction of polarization φ of the imaginary beam (analyzer position y); FIG.

FIG. 9 shows normalized intensities of the individual phonons as a function of the polarization direction φ of the incident beam (analyzer position x); FIG.

FIG. 10 shows normalized intensities of the individual phonons as a function of the polarization direction φ of the incident beam (analyzer position y); FIG.

FIG. 11 shows intensity ratios of the individual phonons as a function of the polarization direction φ of the incident beam (analyzer position x); FIG.

FIG. 12 shows intensity ratios of the individual phonons as a function of the direction of polarization φ of the imaginary beam (analyzer position y); FIG.

FIG. 13 Raman spectrum of the crack tip, polarization setting 27 ° / y _/

14 shows excerpts of Raman spectra of the crack tip and of the unstretched region, polarization setting 27 ° / y, a): Si peak, b): plasmaline at ca.

433 cm ^"1 for reference.

FIG. 15 a): CCD image of a microcrack in multicrystalline silicon, X marks the location of the wavenumber measurement solution of the unstretched material; b), c), d): Raman shifts in the marked area of the image a); used polarization settings: b) 85 ° / x, c) 36 ° / x, d) 27 ° / y corresponding to each of a dominant line of the three optical phonons.

The vibrations of the atoms in a solid can be described as collective movements of all atoms in the form of plane waves. Every possible lattice vibration can be represented as a superposition of natural vibrations. Each natural oscillation j has a frequency Cθ _j , a Wellenvek¬ tor ^ y and a normal coordinate Q _j . This is given at location r at time t by the following expression [2]:

Q _j = Q _j0 ■ exp (/ (g _y • f ± ω _i • t)) (D

The movement of the atoms can be detected by establishing the Newton equations of motion [2], which contain the coupling of the atoms to each other as the sum of inertial force and spring forces. With the above approach for plane waves (1) this leads to the following eigenvalue problem:

The quantities D ^ (q) form the dynamic matrix [2].

This contains the coupling constants between the atoms a, b when they are deflected in the directions m and n, respectively.

The linear homogeneous equation system (2) has non-trivial solutions only when the determinant disappears. This results in the dispersion relation. For every q there are 3n (n atoms per unit cell) different solutions ω (q) which forms the individual branches of the dispersion relation. Depending on the direction of oscillation, a distinction is made between transverse (perpendicular to the direction of propagation) and longitudinal (in the direction of propagation).

Silicon has three optical phonon modes, two transversal optical (TO) modes, and one longitudinal optical (LO) mode. These oscillate near the center of the Brillouin zone (q «0) with a frequency of COJ ≠ 0.

Here, Co ₁ = ω ₂ - co ₃ , ie the three optical phonons are triply degenerate in the silicon in the unstressed state. This is due to the cubic symmetry of the diamond structure.

The Raman effect is caused by the modulation of the polarizability of valence electrons by lattice vibrations.

An incident light wave is generated by its electrical

Field E = E ₀ Cθs (/ c _E • r -ω _E ^■ t) a polarization P via the polarizability χ (3x3 matrix) in the crystal lattice:

P {f, t) = χE = χE ₀ cos (k _E -r-ω _E -t) (3)

A periodic change of P, in turn, results in the emission of a wave, namely the scattered wave. Their frequency and wave number coincide with those of the irradiated wave. This is the so-called Rayleigh scattering. If, at temperatures T> OK, lattice vibrations occur in the crystal, the polarizability χ changed periodically. The polarizability χ is thus one

Function of the nuclear deflections Qy.

The Taylor series expansion from χ to Qy is:

χ ₀ denotes the polarizability of the crystal lattice in the absence of lattice vibrations. The Taylor series development can be aborted after the second term. If this expression is used for χ and expression (1) in (3), the following relationship results for the polarization in the presence of the lattice vibrations [3]:

p {r, t, Q _j ) = P ₀ {r, t) + P _ind {r, t, Q _j ) (5)

Equation (5) thus contains terms that describe different polarizations.

The first term represents a wave oscillating in phase with the incident wave:

P ₀ = χ ₀ E ₀ cos (k _E -r-ω _E 't), (6)

whereas the second term describes a phonon-induced polarization:

Path + ξ _j ) • f - (ω _E + COJ) t] + cos [(£ _E - q _} ) ■? - (ω _E - ω _} ) t \)

(7) The scattered radiation thus contains in addition to the elastic portion

(Rayleigh radiation) (6) two further components with the frequencies ω _E + COji the so-called Raman bands.

Raman scattering occurs only when ≠ Oist.

Plus and minus sign indicate scattering of the photon with annihilation (+) or generation (-) of a phonon. Light of lower frequency than ω _E is referred to as Stokes scattered light, that with higher frequency is the anti-Stokes scattered light. The phonon frequency is the difference frequency of the incident and the inelastically scattered photon and is therefore referred to as Raman shift or frequency. Raman spectra are plots of the light intensity over the frequency, or the calculated from the frequency re¬ lative wavenumber. In silicon, the Raman bands of the three optical phonons are ± 520.7 cm ^-1 .

The intensities of the Raman bands depend on the polarization direction θ of the incident e _E and the scattered light θ _s and are given for triply degenerate phonons (unstrained Si) by the following expression [3]:

Strictly speaking, this term only applies if no microscope objectives are used in the Raman measurements. When using lenses with a small numeric aperture (for example, 10x), GIg. (8) can be used as a very good approximation.

R _j is the Raman tensor of phoneme j, a symmetric second-order tensor. Rj is defined as: )

ie, if the derivative of the polarizability χ disappears after Qy, or the normal coordinate Q, - itself, then no scattering intensity is present, independently of e _£ and e _s .

Each crystal class has characteristic Raman tensors. For crystals possessing the diamond structure, there are three Raman tensors, which in the crystal coordinate system x = [100], y = [010], z = [001] are as follows [4]:

The index of each tensor indicates the polarization direction of the phonon. The division into longitudinal and transversal optical phonons depends on the experimental geometry. The simplest scattering geometry is backscatter, e.g. from one

(001) surface. The wave vector of the incident k _e as well as the scattered radiation k _s is orien¬ along (001). In this case, R _x and R _{y are} transverse and R _{2 are} longitudinal optical phonons.

For certain polarization combinations of incident and scattered light, the scattered intensity disappears. In the above-mentioned scattering geometry, only the longitudinal phonon is active, ie the scattered intensity (8) consists only of the term with the Raman tensor R ₂ . Tab.l shows which of the three phonons can be observed as a function of the scattering geometry.

Tab.l: Raman Selection Rules for Coordinate Systems: x = (100), y = (010), z = (001) and x ^f = 1 / ^ 2 (110), y '=

Because of the dependence of the intensities of the Raman bands on the polarization direction of the incident and scattered radiation, information about the relative orientation of the crystal with respect. of a defined coordinate system (xy sample stage).

If a crystal is twisted in relation to the table coordinate system x = [100], y = [010], z = [001], the table coordinate system can be moved into the crystal coordinate system by three successive rotations around the three Euler angles et, ß, γ Transfer X, y, Z. These rotations are shown schematically in FIG. A rotation about an axis is a linear image that can be represented in the form of a matrix. The rotation around the three Euler angles mathematically corresponds to the multiplication of the three rotary matrices, which In turn, this results in a matrix, the rotational matrix D or also the orientation matrix.

The orientation matrix D has the following shape [5]:

cos (α) -sin (or) 0 1 0 0 cos (^) -sin (^) 0

D = sin (α) cos (α) 0x0 cos (/?) -Sin (^) xsin (χ) COS (^) 0 (11)

0 0 1 0 sin (/?) Cos (/?) 0 0 1

Rotation around the z-axis Rotation around the x-axis Rotation around the z-axis

The γ-dependent matrix describes the rotation about the z-axis, the / ^ -dependent matrix revolves around the already rotated x-axis (denoted by x _γ in FIG. 1) and the α-dependent matrix rotates around it rotated z-axis.

Twisted crystals have twisted Raman tensors R. For the Raman intensity of a by a, ß, γ respect. of the table coordinate system twisted crystal thus results:

^• D ^τ (α, β, γ) ^> R _r D (α, β, γ) -e _i * (12)

The polarization directions of the incident and scattered beam can be adjusted as follows by means of a λ / 2 plate and an analyzer, which can be aligned in the x and y directions with respect to the table coordinate system.

The correction factor k takes into account all depolarization effects in the incident beam path. Calibration measurements give k: k = 1.0507. That is to say that the incident laser beam polarized in the x-direction has on the specimen the Tc ² C = 1.104) -fold intensity than the laser beam polarized in the y-direction.

Similarly, the correction factor m takes into account the depolarization effects occurring in the scattered beam (inter alia, influence of the optical grating on the polarization direction). The value for m is 1.8626.

If the above polarization vectors are used in the general relationship (12), two intensity curves of the Raman bands that depend on the angle .phi. (.Lambda. / 2 platelets) result. The first equation describes the intensity profile for an analyzer position in the x-direction, the second the intensity curve for an analyzer position in the y-direction:

I _.phi.x ∞ [k ² - f ₁ (a, ß _l γ) -f ₃ (a, ß, γ) \ ² cos (φ) + k f ₂ (a, ß, γ) - cos (φ) - sm (φ) +

+ f ₃ (a, ß, γ)

l _φy oc

+ g ₃ (a>ß> r)

(13)

On the basis of the fitted experimental intensity curves of I _φx and I _φy , the orientation _{- dependent} values of the

Functions f ₁ (a, ß, γ), f ₂ (<x, ß, γ), etc. are determined. From the value relations of these trigonometric functions, the three Euler angles a, ß, γ xm.ά can be used

Kornorientierungen numerically (Gauss-Newton method) determine. Strains / stresses in the material change the "spring constants" of the atomic bonds.Thus, the strains are large enough that the degeneracy of the phonon frequencies of the silicon can be canceled because of the resulting break in the crystal symmetry ACOJ = cθ _j - ω _{0 of} the Raman peaks, the stress state can be determined (the index 0 refers to the unstrained state).

The relation between the spring constants and the frequencies of the three optical phonons [8] applies in equation (2):

K is the spring constant tensor, V is the eigenvector, and w is the frequency of a phonon. By solving the eigenvalue problem, the phonon frequencies can be determined accordingly.

If the material is stretched, the components of the spring constant sensor thereby change. For small strains, a linear dependence on the components of the strain tensor S _k i can be assumed [8], [9]:

/ C _p / are the phonon deformation potentials and W ₀ are the phonon frequencies in the unstrained state. δ ± j is the crown icon. For cubic symmetry there are only three independent ones

Phonon deformation potentials [8]:

P ⁱ _{1 1 1 1} - l \ ₂ 222 ^* ^ 3333 ~ H

K ₁₁₂₂ - ^ 1133 ⁼ ^ 2233 ⁼ Q ( ^{1 6} ) ^f \ ₁₂₁₂ - ^rx 1313 ~ ^ 2323 ~ '(The values for silicon are given in Tab.

If the components of the spring constant sensor (15) are calculated with the deformation potentials (16) and inserted into the intrinsic value problem (14), this leads to the following secular equation (coated strain / stress components are now part of the crystal coordinate system, uncoated to the table coordinate system ) [9]:

= 0

ε _y ' _y ) -Ä (17)

The eigenvalues λj represent the Raman frequency shifts in the strained state:

λ _j = [ωf -

-co ₀ y [a) _j + co ₀ ) ∞ Δω _j - 2 - ω ₀ (iδ)

The measurements result in a maximum of three frequency shifts, since three optical phonon modes exist. However, equation (17) contains six different expansion components. A clear solution is therefore only possible if simplifying assumptions about the state of stress / strain are made. The penetration depth of the laser beam used is about 3 microns. In this area, it can be assumed that the voltage level is even. This means there are no stress components in the z-direction of the table coordinate system. The stress tensor with respect to the table coordinate system looks under this assumption as follows:

'xx τ _xy 0 σ = "xy _σyy 0 (19)

0 0 0

A grain which has the orientation matrix D in relation to the table coordinate system has a stress tensor rotated as follows in the crystal coordinate system:

σ '= D ^{~ 1} -σ-D (20)

The components of this tensor are used in the general Hooke's law for calculating the expansion components.

ε '= S'-σ' (21)

The matrix S 'contains the elastic compliances of the material. The shape of this matrix depends on the symmetry of the crystal.

For cubic symmetry there are only three independent elastic compliances [10]:

(The values for silicon are in Tab.2.)

The expansion components of (21) are used in the Säkularglei¬ Chung (17). Forming and summarizing provides the following form of the secular equation for calculating the stress components with respect to the table coordinate system:

111 Ω 121 "• oil Ω-M2 Ω ₁₂₂ Ωi ₃₂ « 113 Q ₁₂₃ Q133 1 0 0

Det 121 Ω 221 Ω ₂₃₁ ^{• σ} xx + Ω ₁₂₂ Ω ₂₂₂ Ω ₂₃₂ ^{■ τ} xy + Q ₁₂₃ Q ₂₂₃ Q ₂₃₃ ^■ σ _yy - λ - 0 1 0 = 0

131 Ω 231 Ω331 Ω-I32 Ω ₂₃₂ "332 ^ 133 Q ₂₃₃ Q 323 0 0 1

(23)

The parameters Ωi _jk contain the elastic compliances and deformation potentials of the material. In addition, they depend on the components of the orientation matrix D and, consequently, are grain-specific constants in the further calculation. The determinant provides the following characteristic polynomial:

P (λ) = λ ^z + a ² + bλ + c (24)

The coefficients a, b and c are functions of the three voltages CTxx, Tχy _f <Tyy.

The characteristic polynomial can be expressed via the zeros λ which represent the measured values as follows: P (λ) = (λ-Aω _r 2ω ₀ ) - (λ-Aω ₂ -2ω ₀ ) - (λ-Aω ₃ -2ω ₀ ) (25)

By multiplying and by a coefficient comparison with equation (24), the following relationships result:

a = -2ω _Q • (AO) ₁ + Aω ₂ + Aω ₃ )

b = Aωl • (AO) ₁ • Aω ₂ + AO) ₁ • Aω ₃ + Am ₂ • Aω ₃ ) (26)

c = -8 #> o • (AO) ₁ • Aω ₂ • Aω ₃ )

It concerns a nonlinear system of equations regarding. of the three stresses σ _xx , x ^ y, σ _yy . Again, the Gauss-Newton numerical solving method is used. The method described for the determination of grain orientations and plane stress states is in principle suitable for any material of cubic symmetry, provided that it has Raman-active modes. In the following, only multicrystalline silicon will be investigated. The required material constants are in Tab.2.

Tab.2: Material constants of silicon used. In order to be able to carry out Raman measurements, a monochromatic light source (laser), a microscope optics which focuses the light on the sample, an optical grating for the spectral analysis of the scattered light and a corresponding detection and amplification system are required.

Our measurements are performed on a Jobin-Yvon Labram HR800 spectrometer. Due to the long focal length of 800 mm, this device has a very high spectral resolution (0.04 cm ^"1 ) and is equipped with two lasers: a HeNe laser (633 nm) and an argon ion laser (514 The structure provides for the use of a laser-specific interference filter which removes the plasma lines of the laser from the excitation light, but the plasma lines are of great use as a tool for voltage measurements Use All measurements are therefore carried out without the interference filter.

Fig. 3 shows schematically the structure used for the measurements and the beam path within the spectrometer.

An important component is the notch filter which, in terms of wavelength, totally reflects the excitation light in the direction of the specimen, but transmits the light leaked by the specimen Ramangestre. The exact position of the notch filter, matching the mirror S2, is controlled by a spacer. The totally reflected laser beam is focused on the sample via the microscope objective. There are four lenses available: 100x, 50x, 10x and a macro lens. The orientation measurements were carried out with the 10x objective and the voltage measurements with the 5Ox objective. The sample is located on a table which can be moved in the xy direction in 0.2 μm increments. If the beam splitter S3 is inserted, the sample surface can be viewed and the laser beam focused become. The intensity of the laser must be weakened for this purpose. This takes over the filter wheel. The software can be used to set the appropriate damping level. The D4 filter (ie attenuation by the factor IGT ⁴ ) is used for foaming. If the beam splitter is not used, the light backscattered by the sample will return the same way back to the notch filter. While the excitation light can not pass the notch filter, inter alia the Raman scattered light is transmitted and passes through the confocal aperture whose diameter is adjustable. The confocal

Aperture determines the effective diameter of the laser beam on the sample surface and thus the spatial resolution. The scattered light is spectrally decomposed by means of a grating (1800 lines / mm) and subsequently detected by a silicon CCD. For the orientation measurements, exact polarization settings of the excitation light as well as of the scattered light are required. For this purpose, a λ / 2 plate is positioned in front of the filter wheel, which rotates the polarization direction of the incident laser light by a fixed angle. In front of the confocal aperture is an analyzer. Die¬ ser leaves, depending on the position, the respect. of the xy Tischkoordi¬ natensystems x- or y-polarized portions of rückgestreu¬ th light.

The light emitted by the laser has a certain polarization direction. This differs by a certain angle from the x or y axis of the table coordinate system. The polarization direction of the exciter light is rotated in the x or y direction via a λ / 2 plate. A λ / 2 plate consists of a birefringent material which decomposes linearly polarized light into two mutually perpendicularly polarized partial waves (o-, e-wave) which, if the light is perpendicular to the optical axis of the λ / 2 plate comes in the same direction, but with different speeds to move forward. If the λ / 2 plate has a thickness corresponding to the wavelength used, the transition difference between the two partial waves is λ / 2. For the E-vector of one of the two partial waves (in FIG. 4 e-wave), this only means the reversal of the direction. Accordingly, a rotation of the λ / 2 plate by θ corresponds to a rotation of the polarization direction by 2θ. In the available spectrometer, when using the HeNe laser, with a position of the λ / 2 plate at 6 °, the polarization direction is rotated in the x-axis, at a position of 51 ° in the y-axis.

The use of the objectives results in the formations of the polarization vectors due to the shaped beam cones (aperture angle about 3 ° for the 10 × objective), which generate z components in addition to x and y components. These in turn produce light intensities in the Raman signal that would be zero if ideally polarized. That Polarization settings, where according to Tab. 1 actually no Raman signal should be present, show minimal residual intensities. The modified polarization vectors lead to a softening of the sharp selection rules and thus to an inaccuracy in orientation measurements. Since the polarization directions are smeared more strongly the greater the aperture angle of the beam cone, orientation measurements based on exact polarization settings are carried out with the 10 × objective.

The orientation matrix (11) can be represented by the direction cosines of the angles between the table coordinate axes x, y, z and the crystal axes X, y, Z and then has the following form: cos (zxx) cos (Zxy) cos (Zx.z? Y) \

D = (27)

The lines of the orientation matrix contain the cosines of the angles, each including a table coordinate axis with the three respective crystal axes. Accordingly, the columns represent the three angles, each forming a crystal axis with the table coordinate axes.

Since, for a cubic crystal, the three coordinate axes are physically equivalent, a set of Raman intensity ratios has a number of different orientation matrices whose components are not different in magnitude, but in which lines of D are reversed. If a fixed table coordinate system is assumed, interchanging the lines means interchanging the respective crystal axes. The number of possible orientation matrices based on pure line permutations is 3! = 6. If the signs of a line change, this means the direction reversal of the respective crystal axis.

(A column change or a sign change of a column has the same consequences, however, for the respective table coordinate axis (s).) Since the set polarization directions are fixed at the table coordinate system, column exchanges logically lead to reversed intensity ratios, but row permutations do not - ments of table coordinate axes, that is, sign changes of columns, are harmless, since the polarization directions are preserved.) Line permutations and sign reversals of lines and / or columns correspond to the following rules for the Euler angles [6]:

γ - »γ + π

H -P (28) α - » ^■ α + π / 2 (β, α) → (π - β, ^" - α)

These transposition processes generate physically equivalent orientations, which therefore can not be distinguished on the basis of Raman polarization measurements (intensity ratios do not change).

In order to make a uniform choice, it is advisable to make sure that the three Euler angles are in the following angular ranges [6]:

0 ° <β <45 ° 0 ° <a ≤ 180 ° (29)

0 ° <γ ≤ 90 °

If the stresses / strains in the material are too low (<1.5 GPa) in order to bring about a visible degeneracy of the optical phonons, the following assumption had to be made hitherto:

If polarization settings are used in which a mode (dominant mode) is involved in the scattering intensity, then the measured frequency shift can be quasi assigned to this one dominant phonon. By deliberately twisting the λ / 2 plate and possibly exchanging the analyzer direction, three polarization directions can generally be found, in each of which one of the three optical phonons is dominant.

The analyzer can be placed either in the x or in the y position. The λ / 2 plate, however, can be rotated by a be¬ any angle θ. When the λ / 2 plate is rotated by θ, the polarization direction of the incident light is rotated by 2θ = φ with respect to the table coordinate system and then reads as follows (due to the small deviation, the correction factor k can be 1) be ignored) :

For each phonon j, the scattered intensity according to (12) (while the three sum terms are considered individually) is simulated as a function of the angle φ of the λ / 2 plate and the analyzer position (x or y).

For each phonon there are thus two intensity functions, in total six functions:

U = / 'φ "x ^»

= / 'φ ⁽² x ⁾ b ^u φ = /' φ ⁽ y ²⁾ (30)

c - / ⁽³⁾ The subscript indicates the polarization setting, the subscript indicates the phonon mode.

For the intensity e.g. of phonon 1 at an x-analyzer position, the following expression applies according to (12):

l $ ∞e _E ^τ _φ -D ^τ (a, ß, y) -R, .D (a, ß, r) 'e Sx (31)

The other intensities can be represented analogously. These six functions can be simulated with known crystal orientation of the considered crystallite (α, β, γ known from orientation measurement) and plotted over the rotation angle j.

The intensities simulated in FIGS. 5 and 6 belong, for example, to the following measured grain orientation:

_a = 57 ° ß = 43 ° γ = 51 °

FIG. 5 contains those intensities (phonons 1-3) in which the analyzer is in the x-position, FIG. 6 shows the intensities belonging to the y-position. These graphs show areas in which one mode in each case dominates over the other two modes.

In order to more accurately determine the desired polarization settings for the dominant modes, the intensity ratios of in each case one mode are plotted to the sum of the two other modes. This results in six functions:

y φ

(32)

/; + / ?, + 0.01 a _φ + c ^ + 0.01

g _p + /; +0.01 a _φ + b _φ + 0W

The quantities f _{f /} gr 1, h _φ , a _φ , b _φ , c _φ represent the individual intensities defined in (30). The term 0.01 in the denominator ensures that the denominator never becomes 0 under any circumstances , These intensity ratios are shown for the specified Euler angles in FIGS. 7 and 8.

Measurements have shown that, as a rule, a peak can be recognized for each mode in which the intensities of the two other phonons are negligible. An exception is Si grains with (001) surface, where generally only the longitudinal phonon is active. In addition, the peaks are usually clearly demarcated so that, despite the conditional accuracy with which the λ / 2 plate can be tuned, a signal is obtained which can be primarily attributed to a single phonon.

The method is now demonstrated by measuring the stress field around a microcrack. The investigated material is multicrystalline solar silicon which has been mechanically and chemically polished and subsequently subjected to a Secco etch in order to visualize the grain boundaries. Imprints were artificially produced with a diamond tip, with microcracks forming on the edges. The investigated crack tip can be seen in FIG. 15 a). Before the actual voltage measurement, an orientation measurement must be carried out in order to determine the three grain-specific polarization settings for the mapping.

The orientation measurement yields the following three Euler angles:

α = 43 ° β = 42 ° γ = 15 °

The angles are thus in the interval ranges indicated in (29). The magnitude of the error vector becomes minimal at these echo angles, i. The intensity ratios calculated on the basis of these angles agree very well with the measured intensities.

The simulated intensities and intensity ratios of the individual phonon modes are as follows:

From FIGS. 11 and 12, the following polarization settings result for the three modes:

Phonon 1: φ = 70 °, analyzer: x (-> λ / 2 plate: x + 70 ° / 2 = 85 °)

Phonon 2: φ = 152 °, analyzer: x (- »λ / 2-plate: x -

(180 ° -152 °) / 2 = 36 °)

Phonon 3: φ = 133 °, analyzer: y (-> λ / 2-plate: x -

(180 ° -133 °) / 2. 27 °)

The λ / 2 plate must be rotated by half the angle φ. The position φ = 0 ° in the diagram is the x-position of the λ / 2 plate which, at a 51 ° rotation of the λ / 2 Lies platelet. For this purpose, the respective half-rotation angle must then be added, or in the case of phonons 2 and 3, if the rotation angle is in the second quadrant, the difference to the negative x-axis {φ = 180 °) is subtracted.

If the sets Euler angles (28) are used, which lead to the same intensity ratios and therefore can not be distinguished, only the phonon designations interchange.

For example, the substitution leads to: a -> a + π / 2 causing phonon 1 to be reversed with phonon 2 in the diagrams. (The curve shape is retained, only the color assignment is exchanged).

An exchange of the phonon notation, however, has no consequences for the further calculation of the voltages, since the indices of the frequency shifts of the three phonons in the equations (26) are likewise arbitrarily interchangeable.

Now, three mappings of the Raman peak shifts of the region marked in FIG. 15 a) can be carried out with the three polarization settings determined.

After the measurement, the silicon peak and a plasma line (approximately 433 cm ^-1 ) of the laser are fitted with a Gauss / Lorentz function in all single spectra Polarization setting 27 ° / y (λ / 2 plate rotated by 27 °, analyzer in the y direction) was recorded.

In the course of a measurement, the opt. Lattices drift through temperature fluctuations and other influences, which leads to unwanted drifting of the wavenumber scale. such Influences Can be checked and corrected by using a reference line of be¬ known and constant wave number. Suitable for this are plasma lines of the laser used (in our case, the line at about 433 cm ^-1 ) This line is physically determined exactly.

In order to eliminate the effects of grid drift, the difference Δω ^PL from the wave number of the (dominant) silicon phonon and the wave number of the plasma line is determined from each individual spectrum. Since both peaks are equally affected by a grating drift, the differences now measured in the wavenumber difference are purely stress-induced.

The wavenumber difference is shown in color coded in the three mappings Fig. 15 b), c), d), which were carried out in the area marked in FIG. 15 a). The red color indicates areas that have a higher wavenumber difference compared to the unstressed material, the blue / black color indicates a lower wavenumber difference. (Unfortunately the existing evaluation program does not work

Color bars with wavenumber mapping too.) The mappings are similar, but do not match perfectly. This is to be expected since the three phonon modes and thus the three stress components a _** ,

σyy can generally show different location dependencies. The (absolute) maximum

Wavelength differences are in the same position in all three mappings.

The maximum differences in the two ranges to be read from the pictures are shown in the following table: Phonon 1 Phonon 2 Phonon 3

Δω ^PL 9 86.87 87.08 87.39

Δω ^PL S 87.32 87.34 87.65 unstrained Δω _Q ^FL 87.26 87.32 87.64

Table 3: Measured wave number differences Δω ^PL and Δω ₀ ^PL .

Table 3 also contains the wavenumber differences Δω ₀ ^{Ph of} the unstressed material. These were determined by individual measurements at the point marked X in FIG. Here it can be assumed that the material is no longer affected by the stress field around the crack. It should actually be the same for all polarization settings

Difference frequencies Δω ₀ ^PL result. This is not the case. The reason for this is a slight dependence of the shape of the plasma line on the polarization setting. Fitting the plasmaline then gives different plasma line peak positions.

The stress state is determined from the wavenumber shifts at the crack tip (Figure 15d). The voltage-dependent wave number shifts result to:

Δω = Δω _o ^PL- Δω ^PL

and assume the following values:

Δω _x = 0.39 cm ^"1! 0.04 ^{cm" 1} Δω ₂ = 0.24 Cm ^"1 + 0.04 ^{cm" 1} Δω ₃ = 0.25 cm ^"1! 0.04 ^{cm" 1} FIG. 14 contains two sections of spectra which were marked X in the polarization setting of phonon 3 (27 ° / y) in the unstressed region (in FIG. 15 a) and in the strained region at the crack tip. While the plasma line shows a minimal shift to higher frequencies at about 433 cm ^-1 , the frequency of the silicon peak in the span range is shifted towards lower values (generally tensile stress).

Subsequently, the conversion from the wavenumber cm ^"1 to the frequency s ^{" 1 takes place} .

The values are inserted in (26). The three nonlinear equations in (26) for a, b, c, are solved by specifying starting values for σ ^, τ _xy , σy _y using the Gauss-Newton method. With regard to the table coordinate system, the following stress condition results (with error indication):

The method described uses micro-Raman spectroscopy for the orientation-dependent determination of mechanical stress tensors in multicrystalline materials with diamond structure, such as, for example, multicrystalline silicon (an extension of the process to further Raman-active, multicrystalline materials with other crystal structures is planned).

For accurate determination of stress tensors in multicrystalline silicon, the following measuring steps are necessary:

1.) Data on the crystallite orientations are indispensable for the stress measurements. The crystallite orientations can be calculated from the sum intensities of the Raman bands, sufficiently accurately and quickly determined wer¬. This is crucial because the same device and thus the same reference coordinate system can be used for voltage and orientation measurements. This circumstance leads to a significant reduction in the measuring time and the measuring effort. 2.) To determine the three stress components of a given stress state, the three Raman peak shifts of the three optical phonons of the silicon must be measured. With small mechanical stresses (<1.5 GPa), there is no visible degeneracy of the three optical phonons, ie the three associated Raman peaks fuse into a single peak, which makes a direct measurement of the three displacements impossible.

On the basis of the data on the crystallite orientation, however, the Raman intensities of the three optical phonons can be simulated separately as a function of the polarization settings (λ / 2 plate and analyzer position). It has been shown that for the majority of crystallite orientations there is a special polarization adjustment for each phonon, in which the Raman intensity of this phonon is clearly dominant over the sum intensity of the other two phonons. An occurring peak shift can then be assigned to this one Pho¬ non. Accordingly, the three Raman peak shifts of the three optical phonons can be measured separately on the basis of three specific polarization settings. 3.) In order to determine the stress tensor of a plane stress state (three stress components with respect to the table coordinate system) in a crystallite of arbitrary orientation, the calculation methods used for stress measurements in crystallites of simplest orientations (eg surfaces in (100) Direction) have been significantly modified and further developed. The method described contains the solutions of the resulting mathematical For¬ malismen.

The presented measurement method is partly based on known facts, but some new approaches are pursued. For example, it has long been known in the literature that crystallite orientations can be determined based on the intensity relationships of the Raman bands. The orientation determination method we have outlined builds on this existing knowledge, but contains modifications which make the calculation of the crystallite orientations more accurate and faster.

The facts described under the above 2.) and 3.) represent a fundamental extension of the voltage measurement in silicon with micro-Raman spectroscopy. On the one hand, independent measurements of the Raman peak shifts of the three optical phonons are now possible , although due to the (low) voltages (<1.5 GPa) no visible degeneration takes place. This allows a much more accurate determination of the three sought stress components, if they are in the range 20 MPa (resolution limit) to 1.5 GPa.

On the other hand, stress tensors of plane stress states can now be measured in crystallites of any orientation. References:

[1] I. De Wolf, Semicond. Sci. Technol. 11, 139 (1996) [2] H. Ibach, H. Lüth, Solid State Physics (Introduction to the

Basics) (Springer, Berlin, 1995) p. 66

[3] M. Cardona, Light Scattering in Solids II (Topics in Applied Physics) (Springer, Berlin, 1982) p. 378 [4] R. Loudon, Adv. Phys. 13, 423 (1964) [5] H. Margenau, G.M. Murphy, The Mathematics of Physics and

Chemistry (Van Nostrand, New York, 1956) p. 288 [6] J.B. Hopkins, L.A. Farrow, J. Appl. Phys. 59 (4), 1103,

(1986)

[7] J.J. More, B.S. Garbow, K.E. Hillstrom, User's Guide to Minpack I, (Argonne National Laboratory Publication,

ANL-80-74, 1980 [8] S. Ganesan, A.A. Maradudin, J. Oitma, Ann. Phys. 56, 556

(1970)

[9] E. Anastassakis, A. Pinczuk, E. Burstein, Solid State Commun. 8, 133 (1970)

[10] F. Cerdeira, CJ. Buchenauer, F.H. Pollak, M. Cardona, Phys. Rev. B 5, 580 (1972)

Claims

claims

1. A method for determining the crystal orientation of a crystallite of a multicrystalline solid by Raman spectroscopy, comprising the steps of: a) measuring the intensity (I _φx ) of a Raman band at a defined first polarization direction with variable polarization angle (φ) of the excitation radiation, and b) measuring the intensity (I _φy ) of the Raman band in a second polarization direction which is orthogonal to the first polarization direction, with variable polarization angle (φ) of the excitation radiation, and c) determining the crystal orientation from the measured intensity profiles.

2. A method for determining mechanical stresses in multi¬ crystalline solids by Raman spectroscopy, comprising the steps of: - determining the crystal orientation of a crystallite according to

Claim 1,

Determining the voltages by measuring the frequency shift (Δcoi, Δω ₂ , Δω ₃ ) of the Raman bands and calculating certain voltage components (σ _xx , t _x y, σ yy) of the voltage tensor from the frequency shift (Δω i, Δω ₂ , Δω ₃ ) and the previously determined crystal orientation.

3. A method for determining mechanical stresses in Fest¬ bodies with known crystal orientation by Raman spectroscopy, by

Determining the voltages by measuring the frequency shifts (Δωi, Δω ₂ , Δω ₃ ) of the Raman bands and calculating specific voltage components (α 1, τ _xy , σ y y) of the span Frequency shift (ΔGÜ _I , Δω ₂ , Δω ₃ ) and the crystal orientation, wherein

- the individual frequency ^shifts' to be verschiede¬ NEN phonon modes belonging Raman bands are measured by the Anre¬ be found supply light, and by measuring the Raman-scattered light in the first (x) or the second direction of polarization (y) has three settings by varying the polarization direction, wherein each one of the three optical phonons is dominant

4. The method according to claim 2 or 3, characterized in that a

- When determining the voltages, a plane stress state is assumed in the xy plane, in which no voltage components z-direction exist, and thus only the three voltage components (σ _xx , _{x x} y, σ _yy ) determines the xy plane become.

5. The method of claim 2, wherein:

- The solid silicon and, when determining the Spannun¬ gen the frequency shift (Δωi, Δω ₂ , Δω ₃ ) of the three Ra¬ manbanden the three optical phonon modes of silicon are measured.

6. Method according to one of claims 2 to 5, characterized in that a

- The solid has cubic crystal symmetry and therefore has three independent elastic compliances or elasticity constants (Sn, Si ₂ , S ₄₄ ), which are used for the calculation of the stress components Be¬.

7. The method according to any one of claims 2, 4, 5 or 6, characterized in that - When the voltages are determined, the individual frequency shifts of the Raman bands belonging to different phonon modes are measured by finding three settings by varying the polarization direction of the excitation light and measuring the randomly scattered light in the first (x) or the second polarization direction (y) ¬ are those in each of which one of the three optical phonons is dominant.

8. The method according to any one of the preceding claims, d a d u r c h e c e n e c e in that e

- The polarization of the excitation light is varied by a λ / 2 plate.

9. The method according to one of the preceding claims, in which a

- In the beam path of the ramangestreuten from the sample to be examined ramangestreuten light analyzer is positioned, which is selectively set to the first (x) or the second polarization direction (y).

10. The method according to claim 1, wherein:

- The excitation light through a lens, in particular a microscope objective, is focused on the solid, wo¬ in particular the excitation light is focused to a focal spot with a diameter below 5 microns.

11. The method according to any one of claims 1, 2, 4 to 10, d a d u c h e c e n e c e s in that e

- When determining the crystal orientation, the crystal orientation is determined on the basis of the Euler angles (α, β, γ).