Imaging Strain in Nanostructures ESRF and ILL Summer School 2017 - - PDF document

Imaging Strain in Nanostructures

ESRF and ILL Summer School 2017 Report by Jan Bendix Hagedorn

Imaging Strain in Nanostructures Jan Bendix Hagedorn

Contents

1 Introduction 2 2 Project Summary 3 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Strain in Semiconductor Structures . . . . . . . . . . . . . . . . 3 2.1.2 The K-Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Building a Model in Comsol . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.3 Physics and Boundary Conditions . . . . . . . . . . . . . . . . . 8 2.2.4 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Computation and Visualization . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Sources 15 1

Imaging Strain in Nanostructures Jan Bendix Hagedorn

1 Introduction

During the four weeks of the ESRF and ILL Student Summer Programme 2017, I worked under the supervision of Ga´ etan Girard in the X-Ray Nanoprobe Group at the ESRF. The members of the Group at beamline ID01 have developed an imaging technique called Quick Mapping, or K-Map for short, that aims to resolve lattice strain and 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 references using the Comsol Multyphysics software to run finite element simulations of strain in semiconductor 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 an

• verview of the data acquired through the simulations and how it could be used in the
• future. As I had no prior experience in experimental diffraction, much of my time was

spend studying the concepts investigated and the methodology employed by the group at 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 of the workings of a large scale research facility such as the ESRF. The lectures held for

• ur group by scientists of both the ESRF and ILL provided me with an overview of the

numerous interesting fields of science present on the EPN Campus. These experiences combined 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. 2

Imaging Strain in Nanostructures Jan Bendix Hagedorn 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 because it can alter the electrical properties of a material. In a strained crystal the lattice parameters differ from those in an unstrained crystal. The subsequent change in the electric potential affects the electronic band structure. Strained crystals can be grown by epitaxy as illustrated in figure 1 [Berthelon et

• al. 2017]. The silicon-germanium layer in this example adopts the horizontal lattice

parameter of the silicon it is grown on, bringing the atoms closer together than they would be in unstrained silicon-germanium and resulting in an in plane strain (ǫxx, ǫyy) The material is stretched in the vertical direction to compensate, creating an out of plane strain (ǫzz). This surface based effect would be negligible for a bulk material but in a nanostructures such as epitaxial films it has significant effects. By purposely engineering semiconductor structures to be stressed, these effects can be used to increase the performance of certain elements for example metaloxidesemi- conductor field-effect transistors or MOSFETs for short [Maiti and Maiti, 2012]. With strain 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 technique has been developed by scientists working on microdiffraction at the ESRF’s beamline ID01. 3

Imaging Strain in Nanostructures Jan Bendix Hagedorn 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 or require 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 thus to the local strain. To produce a K-Map of a sample it is mounted on the piezo stage in the center of the diffractometer (see figure 2). The sample is positioned so that the incoming microbeam is diffracted in the area that is to be investigated. The motions of hexapod and piezo stage are remote controlled and a microscope is used for visual feedback. The detector is then moved to capture a single Bragg reflex, that is chosen based on the investigated strain component. Measurements of at least three different reflexes, corresponding to non 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 orientation

• f the piezo stage. For every point of this scan the intensity around the chosen Bragg

peak is measured with the two dimensional detector. Step size can be chosen depending

• n the size of the investigated area. Quasi continuous measurements, supported by a

software package developed at ID01, allow for much faster scanning times than an SXDM conducted in a step wise fashion. The procedure is repeated with the sample 4

Imaging Strain in Nanostructures Jan Bendix Hagedorn 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 of incidence 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 three dimensional Q-Space can be obtained. The position of the peak in Q-Space yields information 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 of scattering planes: dhkl = 2π | Q| = 2π Q2

x + Q2 y + Q2 z

tilt[◦] = 180 π arccos

• Qz

Q2

x + Q2 y + Q2 z

• 2.2

Building a Model in Comsol

2.2.1 Geometry The model geometry is determined directly by the investigated samples. Comsol allowed me to create geometries for simulations via the built in CAD software. All model dimensions were expressed in parametric form to allow for easy rescaling of the created

• bjects.

5

Imaging Strain in Nanostructures Jan Bendix Hagedorn In the case of the silicon-germanium lines the longitudinal relaxation was considered negligible due to the high aspect ratio of the structure and only transverse strain and tilt components were of interest. A two-dimensional geometry representing the cross section of a single line was therefore sufficient. Such a geometry had the added benefit

• f 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 the Comsol model I used a ‘parametric sweep’ to increase the width of the silicon layer step wise until its boundary remained unaffected by the relaxation of the silicon-germanium and the simulated strain pattern was invariant under a further increase. This should result in a good approximation of isolated lines while again reducing the necessary computational resources. The height of the simulated silicon layer was chosen in a similar manner for all

• geometries. The silicon wafer in the actual experiment was several orders of magnitude

thicker than the silicon-germanium film. Its base was consequently not expected to have any impact on the deformation observed in the silicon-germanium layer. To model the square patterns a different approach was necessary. Due to the quadratic 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 their size, the assumption of isolated structures had to be dropped as well. Instead I emulated the 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 physical properties for the domains representing different materials in the geometry. As both silicon and silicon dioxide were assumed to be homogeneous, it was sufficient to use the associated 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 orientation rotated relative to the silicon substrate. This required a manual input of the elasticity matrix. Due to symmetries of the standard < 100 > orientation the matrix has a relatively simple initial shape: C =         c11 c12 c12 c12 c11 c12 c12 c12 c11 c44 c44 c44         6

Imaging Strain in Nanostructures Jan Bendix Hagedorn 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 of

• unity. In the modeled sample x = 0.24, resulting in the following numerical values of

the 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 by a basis transformation of the form L[001](45◦) =  

1 √ 2

− 1

√ 2 1 √ 2 1 √ 2

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, d respectively according to the key below: 1 → 11, 2 → 22, 3 → 33 4 → 23, 5 → 13, 6 → 12 7

Imaging Strain in Nanostructures Jan Bendix Hagedorn This yields the elements of the transformed elasticity matrix: C′ =         c11 − 1

2cc

c12 + 1

2cc

c12 c12 + 1

2cc

c11 − 1

2cc

c12 c12 c12 c11 c44 c44 c44 + 1

2cc

        ≈         18.5 3.2 6.0 3.2 18.5 6.0 6.0 6.0 15.7 7.7 7.7 4.8         GPa The only other parameter needed by Comsol is the density of the silicon-germanium

• heterocrystal. 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 mechanics

• package. This means the program chose all parameters according to the selected ma-

terials and attempted to numerically solve the equations governing the deformation of linear elastic materials on the user constructed mesh (see section 2.2.4). To accurately model the samples, Comsol needs information about boundary conditions and initial strain. All models are given a fixed constraint on their lower boundary, emulating the sample fixed on a silicon wafer. The models of silicon-germanium squares need an additional periodic boundary condition on all vertical faces of the silicon and silicon dioxide layers. This reflects the squares being arranged in a grid with distances between squares 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 lattice parameters divided by the ‘natural’ lattice parameter of bulk silicon-germanium. This follows directly from the relation ǫhkl = 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 8

Imaging Strain in Nanostructures Jan Bendix Hagedorn With aSi = 5.431˚ A the resulting equation for the initial in plane strain is ǫxx = ǫyy = 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 function for the density, based on the expression from the previous section, would provide an-

• ther layer of flexibility to the model. Different silicon-germanium alloys could then be

emulated 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 same concentration 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 the problem 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 to triangles 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 but the elements can have very high aspect ratios. The two options I considered for my models were bricks and tetrahedrons (tetragons and triangles for the two dimensional models). Bricks and tetragons promised to work well 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 models meant that a brick or tetragon based mesh would consist of fewer elements compared to a tetrahedral mesh with the same vertical ‘resolution’. The advantage of tetrahedral and triangular meshes was that they enabled me to use Comsol’s adaptive mesh refinement process. In this process Comsol adapts the size

• f mesh elements based on an initial solution (see figure 4). In areas where the solution

is more heterogeneous the the program attempts to refine the mesh in order to create a 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 step

• f customization was defining different domains for which mesh parameters could be

chosen individually in the second step. I chose domains coinciding with the different materials, as this was both simple and functional for our purposes. In the region of highest interest, the silicon germanium layer, the mesh size was chosen as small as 9

Imaging Strain in Nanostructures Jan Bendix Hagedorn Figure 4: User controlled tetrahedral mesh for a 250nm SiGe square (top) and an adaptive refinement of the same mesh (bottom) 10

Imaging Strain in Nanostructures Jan Bendix Hagedorn possible without running out of memory during computations. For the silicon dioxide and silicon layers I build an increasingly coarse mesh, as these regions were expected to deform less dramatically and not of particular interest. In order for mesh elements to grow gradually between domain boundaries, I applied a ‘growth factor’ greater than

• ne 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 of the according domain. This meant the mesh could easily be reused for any model size and would adapt during a ‘parametric sweep’. Taking into account the model’s high aspect ratio, I applied a separate scaling factor to the z-direction for the thin film of silicon-germanium. A second much coarser mesh was created with the same technique. This was then used 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 check for incorrect expressions in the initial conditions or material parameters before wasting computation 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 ranged from 30 minutes to several hours depending on model size and mesh refinement. I gradually refined the meshes until further refinement produced no noticeable changes in 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 Comsol corresponded 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 pure silicon. ǫplane,comsol = aSi − a aSi To be able to compare the simulated strain to the K-Map results, which measured strain relative to the lattice parameter of unstrained silicon-germanium, ǫplane,kmap = aSiGe − a aSiGe I had to plot a custom expression based on parameters defined in the Comsol model (see figure 6b) ǫplane,kmap = ǫplane,comsol · aSi − aSi + aSiGe aSiGe The variable a in the first two expressions denotes the computed and measured lattice parameter of the strained material respectively. For the out of plane strain Comsol’s 11

Imaging Strain in Nanostructures Jan Bendix Hagedorn Figure 5: Z-Displacement in a 130nm wide line of 20nm thick Si0.76Ge0.24 separated from 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 square

• f 20nm thick Si0.76Ge0.24 separated from the Si wafer by 20nm of SiO2). The shown

XZ-plane corresponds to the (-110) crystal plane through the center of the square. 12

Imaging Strain in Nanostructures Jan Bendix Hagedorn Figure 7: Out of plane strain in a 250nm square of 20nm thick Si0.76Ge0.24 separated from the Si wafer by 20nm of SiO2). The shown diagonal cross section corresponds to the (100) crystal plane through the center of the square. definition coincided with the one used for K-Map results and the plots could be used directly. 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. The average strain along that plane would be equivalent to the strain calculated from a K-Map. Since the necessary integration was not possible within the Comsol software itself, the data needed to be extracted. I extracted data from the Comsol simulations into text files, creating a lists of points in a three dimensional grid spanning the entire volume of the silicon-germanium

• sample. Grid spacing was based on the step size of K-Maps made of the samples. The

lists 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 the extracted data into three dimensional arrays. From the extracted data I calculated the surface tilt of the modeled samples using the 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 to that obtained via K-Map (see figure 8). There was however a quantitative discrepancy in form of a factor of about 10 between the simulations and the measurements. Due to the 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 above but the data extraction method used for the tilt should be suitable for this with minor

• adjustments. The same grid could be used for the calculation of out of plane strain,

13

Imaging Strain in Nanostructures Jan Bendix Hagedorn Figure 8: Tilt [◦] of the (001) plane on the surface of a 250nm SiGe square (left) and a 500nm square (right) around [110]. Based on the displacement field simulated in Comsol. either extracting the strain component directly from Comsol or calculating it from the displacement field. For other strain components and according Bragg peaks, the grid might have to be altered. With the help of a specialized Python package it was possible to construct the diffraction patterns the strained sample was expected to produce. Again, the patterns

• btained matched the actual measurements qualitatively but there was no time for a

quantitative 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 as a reference for a range of samples with different geometries and, provided some minor changes 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 existing Python code could be built upon in order to obtain a function providing the averaged strain components perpendicular to different crystal planes. These would enable a direct comparison to the K-Map measurements, thus making the model fully function as intended. 14

Imaging Strain in Nanostructures Jan Bendix Hagedorn

3 Sources

Berthelon et al. 2017: Berthelon R., Andrieu F., Ortolland S., Nicolas R., Poiroux T., Baylac E., Dutartre D., Josse E., Claverie A., Haond, M., “Characterization and modelling of layout effects in SiGe channel pMOSFETs from 14 nm UTBB FDSOI technology” 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 Sch¨ ulli T. U., “Imaging of strain and lat- tice orientation by quick scanning X-ray microscopy combined with three-dimensional reciprocal 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 up to 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 Sch¨ ulli T. U., “Nanoscale Distortions 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., Shur M.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 Applied Physics 36, 153-156 (1965); doi: 10.1063/1.1713863 15