imaging strain in nanostructures

Imaging Strain in Nanostructures

ESRF and ILL Summer School 2017Report by Jan Bendix Hagedorn

Imaging Strain in Nanostructures Jan Bendix Hagedorn

Contents

1 Introduction 2

2 Project Summary 32.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Strain in Semiconductor Structures . . . . . . . . . . . . . . . . 32.1.2 The K-Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Building a Model in Comsol . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Physics and Boundary Conditions . . . . . . . . . . . . . . . . . 82.2.4 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Computation and Visualization . . . . . . . . . . . . . . . . . . . . . . 112.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Sources 15

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1 Introduction

During the four weeks of the ESRF and ILL Student Summer Programme 2017, Iworked under the supervision of Gaetan Girard in the X-Ray Nanoprobe Group atthe ESRF. The members of the Group at beamline ID01 have developed an imagingtechnique called Quick Mapping, or K-Map for short, that aims to resolve lattice strainand tilt in small crystalline samples. The purpose of my work was to provide a refer-ence for the measurements obtained through the K-Map. I obtained these referencesusing the Comsol Multyphysics software to run finite element simulations of strain insemiconductor nanostructures modeled after the samples investigated by my supervisor.

With this report I aim to provide a summary of my work at the ESRF as well as anoverview of the data acquired through the simulations and how it could be used in thefuture. As I had no prior experience in experimental diffraction, much of my time wasspend studying the concepts investigated and the methodology employed by the groupat beamline ID01. I will attempt to reflect this in my report.

Beyond the work described in this report, the Summer School gave me a glimpse ofthe workings of a large scale research facility such as the ESRF. The lectures held forour group by scientists of both the ESRF and ILL provided me with an overview of thenumerous interesting fields of science present on the EPN Campus. These experiencescombined with the opportunity to meet like minded students from all over Europe,made me consider my stay in Grenoble as a valuable addition to my education.

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Figure 1: Epitaxial growth of a strained SiGe layer

2 Project Summary

2.1 Motivation

2.1.1 Strain in Semiconductor Structures

Strain plays an important role in semiconductor structures on the nanoscale becauseit can alter the electrical properties of a material. In a strained crystal the latticeparameters differ from those in an unstrained crystal. The subsequent change in theelectric potential affects the electronic band structure.

Strained crystals can be grown by epitaxy as illustrated in figure 1 [Berthelon etal. 2017]. The silicon-germanium layer in this example adopts the horizontal latticeparameter of the silicon it is grown on, bringing the atoms closer together than theywould be in unstrained silicon-germanium and resulting in an in plane strain (εxx, εyy)The material is stretched in the vertical direction to compensate, creating an out ofplane strain (εzz). This surface based effect would be negligible for a bulk material butin a nanostructures such as epitaxial films it has significant effects.

By purposely engineering semiconductor structures to be stressed, these effects canbe used to increase the performance of certain elements for example metaloxidesemi-conductor field-effect transistors or MOSFETs for short [Maiti and Maiti, 2012]. Withstrain thus being an important property of modern electronic devices, imaging tech-niques that are able to resolve it on small scales become interesting. One such techniquehas been developed by scientists working on microdiffraction at the ESRF’s beamlineID01.

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Figure 2: Diffractometer used at ID01 [ID01 Homepage]

2.1.2 The K-Map

Quick mapping is an imaging method based on scanning x-ray diffraction microscopy(SXDM). Where techniques such as atomic force microscopy, scanning electron mi-croscopy or transmission electron microscopy can only probe the sample surface orrequire the sample to be specifically prepared for measurements (e.g. in thin slices),SXDM is able to probe a sample volume without limitations to its shape or situation.The method is sensitive to the crystal lattice parameter [Evans et al. 2012] and thusto the local strain.

To produce a K-Map of a sample it is mounted on the piezo stage in the center of thediffractometer (see figure 2). The sample is positioned so that the incoming microbeamis diffracted in the area that is to be investigated. The motions of hexapod and piezostage are remote controlled and a microscope is used for visual feedback. The detectoris then moved to capture a single Bragg reflex, that is chosen based on the investigatedstrain component. Measurements of at least three different reflexes, corresponding tonon parallel crystal planes, are needed to obtain full information about the strain.

The sample is then moved along the x and y directions given by the orientationof the piezo stage. For every point of this scan the intensity around the chosen Braggpeak is measured with the two dimensional detector. Step size can be chosen dependingon the size of the investigated area. Quasi continuous measurements, supported by asoftware package developed at ID01, allow for much faster scanning times than anSXDM conducted in a step wise fashion. The procedure is repeated with the sample

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Figure 3: Schematics of a K-Map measurement [Chahine et al. 2014]

tilted at different angles, producing measurements along a rocking curve at each point.This results in a five dimensional data set consisting of x- and y-coordinates, angle ofincidence ω and scattering angles 2Θ and ν (see figure 3). [Chahine et al. 2014]

From the three angles measured at each point, an image of the Bragg peak in threedimensional Q-Space can be obtained. The position of the peak in Q-Space yieldsinformation about strain and tilt of the lattice at the measured point in real space,since the scattering vector Q is connected to the orientation and relative distance ofscattering planes:

dhkl =2π

| ~Q|=

2π√Q2

x +Q2y +Q2

z

tilt[◦] =180

πarccos

(Qz√

Q2x +Q2

y +Q2z

)

2.2 Building a Model in Comsol

2.2.1 Geometry

The model geometry is determined directly by the investigated samples. Comsol allowedme to create geometries for simulations via the built in CAD software. All modeldimensions were expressed in parametric form to allow for easy rescaling of the createdobjects.

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In the case of the silicon-germanium lines the longitudinal relaxation was considerednegligible due to the high aspect ratio of the structure and only transverse strain andtilt components were of interest. A two-dimensional geometry representing the crosssection of a single line was therefore sufficient. Such a geometry had the added benefitof significantly reducing computation times.

As the spacing of the lines was much larger than the dimension of their cross section,it was assumed that relaxation in a single line is unaffected by its neighbors. In theComsol model I used a ‘parametric sweep’ to increase the width of the silicon layer stepwise until its boundary remained unaffected by the relaxation of the silicon-germaniumand the simulated strain pattern was invariant under a further increase. This shouldresult in a good approximation of isolated lines while again reducing the necessarycomputational resources.

The height of the simulated silicon layer was chosen in a similar manner for allgeometries. The silicon wafer in the actual experiment was several orders of magnitudethicker than the silicon-germanium film. Its base was consequently not expected tohave any impact on the deformation observed in the silicon-germanium layer.

To model the square patterns a different approach was necessary. Due to thequadratic shape in the XY-plane, no direction could be neglected and a three dimen-sional model had to be built. With the distance between squares being equal to theirsize, the assumption of isolated structures had to be dropped as well. Instead I emulatedthe periodicity of the array utilizing boundary conditions (see section 2.2.3).

2.2.2 Materials

In order to simulate the deformation of a sample I had to specify certain physicalproperties for the domains representing different materials in the geometry. As bothsilicon and silicon dioxide were assumed to be homogeneous, it was sufficient to use theassociated Young’s modulus, Poisson’s ratio and density for the corresponding domains.All these properties could be obtained directly from Comsol’s built-in materials library.

The silicon-germanium film was both inhomogeneous and grown in an orientationrotated relative to the silicon substrate. This required a manual input of the elasticitymatrix. Due to symmetries of the standard < 100 > orientation the matrix has arelatively simple initial shape:

C =

c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44

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The matrix elements were obtained using the relations [Schaffler et al. 2001]:

c11 = (165.8− 37.3x)GPa

c12 = (63.9− 15.6x)GPa

c44 = (79.6− 12.8x)GPa

at T = 300K

where x is the concentration of germanium in the silicon-germanium crystal in parts ofunity. In the modeled sample x = 0.24, resulting in the following numerical values ofthe matrix elements:

c11 ≈ 15.6848GPa

c12 ≈ 6.0156GPa

c44 ≈ 7.6528GPa

To obtain the matrix describing our rotated sample the above matrix had to be modified.If the rotation from the < 100 > orientation to the < 110 > orientation is described bya basis transformation of the form

L[001](45◦) =

1√2− 1√

20

1√2

1√2

0

0 0 1

=:

l1 m1 n1

l2 m2 n2

l3 m3 n3

The components of the elasticity matrix will transform according to the equations[Wortman and Evans, 1965]

cij′ = cij + cc(lalblcld +mambmcmd + nanbncnd − δij) for i, j ≤ 3

cij′ = cij + cc(lalblcld +mambmcmd + nanbncnd) for i, j > 3

where cc = c11 − c12 − 2c44 and the indices i and j map onto the indices a, b and c, drespectively according to the key below:

1 → 11, 2 → 22, 3 → 33

4 → 23, 5 → 13, 6 → 12

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This yields the elements of the transformed elasticity matrix:

C′ =

c11 − 1

2cc c12 + 1

2cc c12 0 0 0

c12 + 12cc c11 − 1

2cc c12 0 0 0

c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44 + 1

2cc

≈

18.5 3.2 6.0 0 0 03.2 18.5 6.0 0 0 06.0 6.0 15.7 0 0 00 0 0 7.7 0 00 0 0 0 7.7 00 0 0 0 0 4.8

GPa

The only other parameter needed by Comsol is the density of the silicon-germaniumheterocrystal. It is given by [Schaffler et al. 2001]:

ρSi1−xGex = (2.329 + 3.493x− 0.499x2)g/cm3, T = 300K

where, again, x is the concentration of germanium in the crystal.

2.2.3 Physics and Boundary Conditions

In order to simulate strain in the modeled structure, I used Comsol’s solid mechanicspackage. This means the program chose all parameters according to the selected ma-terials and attempted to numerically solve the equations governing the deformation oflinear elastic materials on the user constructed mesh (see section 2.2.4). To accuratelymodel the samples, Comsol needs information about boundary conditions and initialstrain.

All models are given a fixed constraint on their lower boundary, emulating thesample fixed on a silicon wafer. The models of silicon-germanium squares need anadditional periodic boundary condition on all vertical faces of the silicon and silicondioxide layers. This reflects the squares being arranged in a grid with distances betweensquares being equal to the length of their sides.

To simulate the initial strain I calculated the lattice mismatch between silicon-germanium and pure silicon. The in plane strain is given by the difference in latticeparameters divided by the ‘natural’ lattice parameter of bulk silicon-germanium. Thisfollows directly from the relation

εhkl =dhkl,unstrained − dhkl,strained

dhkl,unstrained

The lattice parameter of silicon-germanium is given by [Dismukes et al. 1964b]

aSi1−xGex = (5.431 + 0.20x+ 0.027x2)A, T = 300K

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With aSi = 5.431A the resulting equation for the initial in plane strain is

εxx = εyy =0.20x+ 0.027x2

5.431 + 0.20x+ 0.027x2

In our case this meant an initial strain of approximately 0.9%.It was possible to create custom functions for lattice parameter (asige(x)) and in

plane strain (ips(x)) in Comsol and define a global parameter for the germanium con-centration (xGe) as an argument for these functions. Adding a third custom functionfor the density, based on the expression from the previous section, would provide an-other layer of flexibility to the model. Different silicon-germanium alloys could then beemulated with Comsol’s built-in parametric sweep functionality or by manually alter-ing the parameter xGe. While this was not done, because all samples had the sameconcentration of germanium, it would only be a minor change to the existing model.

2.2.4 Mesh

At the core of the finite element method is the partition of the region on which theproblem is to be solved into the eponymous finite elements. Comsol offers four differ-ent shapes of mesh elements for three dimensional models: tetrahedrons, hexahedrons(“bricks”), prisms and pyramids. For two dimensional models this choice reduces totriangles and tetragons. Since any model can be subdivided in tetrahedrons or tri-angles, they are the default choice for three and two dimensional cases respectively.Meshes consisting of bricks, prisms or tetragons are limited to certain geometries butthe elements can have very high aspect ratios.

The two options I considered for my models were bricks and tetrahedrons (tetragonsand triangles for the two dimensional models). Bricks and tetragons promised to workwell with the model geometries since both the two and three dimensional models con-sisted entirely of rectangular surfaces. In addition, the high aspect ratio of the modelsmeant that a brick or tetragon based mesh would consist of fewer elements comparedto a tetrahedral mesh with the same vertical ‘resolution’.

The advantage of tetrahedral and triangular meshes was that they enabled me touse Comsol’s adaptive mesh refinement process. In this process Comsol adapts the sizeof mesh elements based on an initial solution (see figure 4). In areas where the solutionis more heterogeneous the the program attempts to refine the mesh in order to createa more accurate solution. This led me to choose tetrahedral over brick based meshes(and triangular over tetragonal ones).

In addition to providing automatically created so called ‘physics controlled’ meshes,Comsol allowed me to build a customized or ‘user controlled’ mesh. The first stepof customization was defining different domains for which mesh parameters could bechosen individually in the second step. I chose domains coinciding with the differentmaterials, as this was both simple and functional for our purposes. In the region ofhighest interest, the silicon germanium layer, the mesh size was chosen as small as

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Figure 4: User controlled tetrahedral mesh for a 250nm SiGe square (top) and anadaptive refinement of the same mesh (bottom)

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possible without running out of memory during computations. For the silicon dioxideand silicon layers I build an increasingly coarse mesh, as these regions were expectedto deform less dramatically and not of particular interest. In order for mesh elementsto grow gradually between domain boundaries, I applied a ‘growth factor’ greater thanone to these two domains. The result can be seen in figure 4.

Minimum and maximum element sizes had to be specified manually for each region.Both were expressed in terms of the parameter specifying the x and y dimensions ofthe according domain. This meant the mesh could easily be reused for any model sizeand would adapt during a ‘parametric sweep’. Taking into account the model’s highaspect ratio, I applied a separate scaling factor to the z-direction for the thin film ofsilicon-germanium.

A second much coarser mesh was created with the same technique. This was thenused to run a fast simulation after every change to the model and before the actual,much more time consuming, simulation using the first mesh. In this way I could checkfor incorrect expressions in the initial conditions or material parameters before wastingcomputation time on an faulty model.

2.3 Computation and Visualization

With the finalized models the intended computations were run on the ESRF’s com-puter clusters accessed via the OAR resource manager. Computation times rangedfrom 30 minutes to several hours depending on model size and mesh refinement. Igradually refined the meshes until further refinement produced no noticeable changesin the solutions.

Strain and displacement fields could be visualized in Comsol as two or three di-mensional color plots (see examples in figures 5-7). The strain metric used by Comsolcorresponded to a variation in lattice parameter relative to the initial state of the ma-terial, and thus, in case of the in plane strain, relative to the lattice parameter of puresilicon.

εplane,comsol =aSi − aaSi

To be able to compare the simulated strain to the K-Map results, which measured strainrelative to the lattice parameter of unstrained silicon-germanium,

εplane,kmap =aSiGe − aaSiGe

I had to plot a custom expression based on parameters defined in the Comsol model(see figure 6b)

εplane,kmap =εplane,comsol · aSi − aSi + aSiGe

aSiGe

The variable a in the first two expressions denotes the computed and measured latticeparameter of the strained material respectively. For the out of plane strain Comsol’s

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Figure 5: Z-Displacement in a 130nm wide line of 20nm thick Si0.76Ge0.24 separatedfrom the Si wafer by 20nm of SiO2). The shown cross section corresponds to the (-110)crystal plane.

Figure 6: a) Out of plane strain (left) and b) in plane strain (right) in a 250nm squareof 20nm thick Si0.76Ge0.24 separated from the Si wafer by 20nm of SiO2). The shownXZ-plane corresponds to the (-110) crystal plane through the center of the square.

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Figure 7: Out of plane strain in a 250nm square of 20nm thick Si0.76Ge0.24 separatedfrom the Si wafer by 20nm of SiO2). The shown diagonal cross section corresponds tothe (100) crystal plane through the center of the square.

definition coincided with the one used for K-Map results and the plots could be useddirectly.

In order to compare the results of my simulations with K-Map measurements quan-titatively, the strain values needed to be integrated over the depth of the sample per-pendicular to the crystal plane corresponding to the investigated Bragg reflex. Theaverage strain along that plane would be equivalent to the strain calculated from aK-Map. Since the necessary integration was not possible within the Comsol softwareitself, the data needed to be extracted.

I extracted data from the Comsol simulations into text files, creating a lists ofpoints in a three dimensional grid spanning the entire volume of the silicon-germaniumsample. Grid spacing was based on the step size of K-Maps made of the samples. Thelists contained the three components of the displacement field (u, v, w) for each point(x, y, z). Using Python code, written by my supervisor for this purpose, I arranged theextracted data into three dimensional arrays.

From the extracted data I calculated the surface tilt of the modeled samples usingthe equation

tilt[◦](xi, yj) =180

π· arctan

(∆wi,j

aSi + ∆ui,j

)where (xi, yi) denotes a point on the surface, ∆ui,j = u(xi+1, yj)−u(xi, yj) and ∆wi,j =w(xi+1, yj) − w(xi, yj). Plotting the obtained surface tilt yielded a pattern similar tothat obtained via K-Map (see figure 8). There was however a quantitative discrepancyin form of a factor of about 10 between the simulations and the measurements. Due tothe lack of time I was unable to find the cause of this.

I did not have the time to calculate any of the averaged strains mentioned abovebut the data extraction method used for the tilt should be suitable for this with minoradjustments. The same grid could be used for the calculation of out of plane strain,

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Figure 8: Tilt [◦] of the (001) plane on the surface of a 250nm SiGe square (left) anda 500nm square (right) around [110]. Based on the displacement field simulated inComsol.

either extracting the strain component directly from Comsol or calculating it from thedisplacement field. For other strain components and according Bragg peaks, the gridmight have to be altered.

With the help of a specialized Python package it was possible to construct thediffraction patterns the strained sample was expected to produce. Again, the patternsobtained matched the actual measurements qualitatively but there was no time for aquantitative comparison.

2.4 Conclusions

The Comsol software yielded a simple and flexible model of the investigated samples,meeting our expectations. Due to the models parameterized nature, it could be used asa reference for a range of samples with different geometries and, provided some minorchanges were made, different germanium concentrations.

Two steps could immediately be taken to further increase the models usefulness.First, the process of obtaining the tilt from the extracted data should be scrutinized,eliminating possible errors leading to the discrepancy in the results. Second, the existingPython code could be built upon in order to obtain a function providing the averagedstrain components perpendicular to different crystal planes. These would enable adirect comparison to the K-Map measurements, thus making the model fully functionas intended.

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3 Sources

Berthelon et al. 2017: Berthelon R., Andrieu F., Ortolland S., Nicolas R., PoirouxT., Baylac E., Dutartre D., Josse E., Claverie A., Haond, M., “Characterization andmodelling of layout effects in SiGe channel pMOSFETs from 14 nm UTBB FDSOItechnology” Solid State Electronics 128, 72-79 (2017); doi: 10.1016/j.sse.2016.10.011

Chahine et al. 2014: Chahine G. A., Richard M.-I., Homs-Regojo R. A., Tran-Caliste T. N., Carbone D., Jaques V. L. R., Grifone R., Boesecke P., Katzer J.,Costina I., Djazouli H., Schroeder T. and Schulli T. U., “Imaging of strain and lat-tice orientation by quick scanning X-ray microscopy combined with three-dimensionalreciprocal space mapping” Journal of Applied Crystallography 47, 762-769 (2014); doi:10.1107/S1600576714004506

Dismukes et al. 1964b: Dismukes J. P., Ekstrom L., Steigmeier E. F., Kudman I.and Beers D. S., “Thermal and Electrical Properties of Heavily Doped Ge-Si Alloys upto 1300K” Journal of Applied Physics 35, 2899-2907 (1964); doi: 10.1063/1.1713126

Evans et al. 2012: Evans P. G., Savage D. E., Prance J. R., Simmons C. B.,Lagally M. G., Coppersmith S. N., Eriksson M. A. and Schulli T. U., “NanoscaleDistortions of Si Quantum Wells in Si/SiGe Quantum-Electronic Heterostructures”Advanced Materials 24, 5217-5221 (2012); doi: 10.1002/adma.201201833

ID01 Homepage, 2017: www.esrf.eu/home/UsersAndScience/Experiments/XNP/

ID01/equipment/diffracto.html , as of 2017/10/28 13:15

Maiti and Maiti, 2012: Maiti C. K. and Maiti T. K., “Strain-Engineered MOS-FETs” CRC Press, 2012, 87-88; ISBN: 9781466500556

Schaffler et al. 2001: Schaffler F., in “Properties of Advanced Semiconductor Ma-terials GaN, AlN, InN, BN, SiC, SiGe” Eds. Levinshtein M.E., Rumyantsev S.L., ShurM.S., John Wiley & Sons, Inc., New York, 2001, 149-188.

Wortman and Evans, 1965: Wortman J. J. and Evans R. A., “Young’s Modulus,Shear Modulus, and Poisson’s Ratio in Silicon and Germanium” Journal of AppliedPhysics 36, 153-156 (1965); doi: 10.1063/1.1713863

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imaging strain in nanostructures

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