Stress control of tensilestrained InGaP nanomechanical string resonators
Abstract
We investigate the mechanical properties of freely suspended nanostrings fabricated from tensilestressed, crystalline InGaP. The intrinsic strain arises during epitaxial growth as a consequence of the lattice mismatch between the thin film and the substrate, and is confirmed by xray diffraction measurements. The flexural eigenfrequencies of the nanomechanical string resonators reveal an orientation dependent stress with a maximum value of 650 MPa. The angular dependence is explained by a combination of anisotropic Young’s modulus and a change of elastic properties caused by defects. As a function of the crystal orientation a stress variation of up to 50 % is observed. This enables fine tuning of the tensile stress for any given Ga content , which implies interesting prospects for the study of high Q nanomechanical systems.
Introducing strain in material systems enables the control of various physical properties. Examples include the improved performance of semiconductor lasers,Ghiti, O’Reilly, and Adams (1989); Lau et al. (1989) enhanced carrier mobility in transistors,People (1986); Welser, Hoyt, and Gibbons (1994) direct formation of quantum dotsLeonard et al. (1993); Landin et al. (1998) and increased mechanical quality factors (Q) in micro and nanomechanical systems (M/NEMS). Verbridge et al. (2006); Faust et al. (2012) In particular, tensilestrained amorphous silicon nitride has evolved to a standard material in nanomechanics in recent years. The dissipation dilutionYu, Purdy, and Regal (2012); González and Saulson (1994) arising from the inherent tensile prestress of the silicon nitride film gives rise to room temperature Q factors of several 100 000 at 10 MHz resonance frequencies,Verbridge et al. (2006); Norte, Moura, and Gröblacher (2016); Faust et al. (2012); Ghadimi, Wilson, and Kippenberg (2017); Villanueva and Schmid (2014) while additional stress engineering has been shown to increase Q by a few orders of magnitude.Tsaturyan et al. (2017); Norte, Moura, and Gröblacher (2016); Ghadimi et al. (2018) However, defectsPohl, Liu, and Thompson (2002) set a bound on the attainable dissipation and hence Q in amorphous materials,Rivière et al. (2011); Faust et al. (2014); Villanueva and Schmid (2014) provided that other dissipation channels can be evaded.Villanueva and Schmid (2014); Imboden and Mohanty (2014) Stressfree single crystal resonators, on the other hand, feature lower room temperature Q factors but exhibit a strong enhancement of Q when cooled down to millikelvin temperatures,Tao et al. (2014) as a result of the high intrinsic Q of single crystal materials.Hamoumi et al. (2018) Combining dissipation dilution via tensile stress with high intrinsic Q of single crystal materials could open a way to reach ultimate mechanical Q at room temperature.
In the recent years, a few possible candidates for tensilestrained crystalline nanomechanical resonators have emerged. Those include, for example, heterostructures of the silicon based 3CSiCKermany et al. (2014) and the IIIV semiconductors GaAsWatanabe, Onomitsu, and Yamaguchi (2010), GaNAsOnomitsu et al. (2013), and InGaP.Cole et al. (2014) Advantages of ternary InGaP (InGaP) are the direct bandgap (for ) and the broad strain tunability. When grown on GaAs wafers, this alloy system may be compressively strained, strainfree or tensile strained, with possible tensile stress values exceeding 1 GPa, by varying the groupIII composition . The prospects of InGaP in nanomechanics range from possible applications in cavity optomechanicsCole et al. (2014); Guha et al. (2017) to coupling with quantumelectronic systems, such as quantum wellsSete and Eleuch (2012) and quantum dots.WilsonRae, Zoller, and Imamoǧlu (2004)
Here we explore freely suspended nanostrings fabricated from InGaP as nanomechanical systems. Our analysis reveals that even for fixed the tensile stress state of the resonator can be controlled by varying the resonator orientation on the chip. This implies that unlike for the case of silicon nitride NEMS, resonator orientation will be an important design parameter allowing to finetune the tensile stress for any given Ga content .
We investigate crystalline string resonators from two differently stressed, MBE grown IIIV heterostructures, illustrated in Fig. 1 (a) and (b). Both structures consist of two thick InGaP layers, each capped by of GaAs. Both InGaP layers are situated atop a sacrificial layer of high aluminum content AlGaAs (AlGaAs), with . Note that only the top InGaP and AlGaAs layers were employed as resonator and sacrificial layer, respectively in this work.
By varying the Ga content of InGaP, the latticeconstant changes by up to . Since the substrate lattice constant of AlGaAs changes by only , as a function of its Al content, we assume the lattice constant of AlGaAs to equal that of plain GaAs, . The difference in lattice constants results in a lattice mismatch between the InGaP and the GaAs lattice. This mismatch induces an inplane strain in the InGaP layer and is defined by the ratio:Pietsch, Holy, and Baumbach (2004)
(1) 
with the distorted inplane lattice constant of the strained InGaP layer, which in case of a 100 % pseudomorphic layer equals the lattice constant of the substrate . An InGaP layer grows strainfree (latticematched) on a GaAs substrate for Ga content, i.e. .Ozasa et al. (1990); Cole et al. (2014) The layer is grown tensile (compressive) strained for a higher (lower) Ga content. With this heterostructure it is thus possible to adjust and tailor the strain in a film up to a critical thickness determined by Matthews, Mader, and Light (1970); People and Bean (1985). In this work we investigate InGaP with Ga contents of (highstress) and (lowstress). The resulting strain values are and , for InGaP on GaAs, respectively.
String resonators were defined by electronbeamlithography followed by a SiCl inductively coupled plasma etch, using negative electronbeamresist maN 2403 as an etchmask, before releasing them with a buffered HF wet etch. The resonators are additionally cleaned via digital wet etching.DeSalvo et al. (1996) In the end we criticalpoint dried the samples, to avoid stiction and destruction of the structures.Kim and Kim (1997) Examples of free standing string resonators are shown in Figure 1 (c).
The samples are explored at room temperature and mounted inside a vacuum chamber (pressure mbar) to avoid degradation of the AlGaAs sacrificiallayer under ambient conditionsDallesasse and Holonyak (2013) as well as gas damping. We measured the fundamental resonance frequency of the outofplane flexural mode of resonators of different length and orientation on the substrate, using piezoactuation and interferometric detection. The InGaP resonators exhibit quality factors up to 70 000. Figure 2 presents the measured frequencies of several sets of resonators fabricated from the highstress InGaP epitaxial structure as a function of the resonator length for two different resonator orientations on the chip. Resonators with an angle of are oriented parallel to the cleaved chip edges, see inset of Fig. 2, which correspond to the 110 crystal directions for IIIV heterostructures on (001) GaAs substrate wafers. Hence, the strings point along a 110 direction. For comparison, we also discuss resonators which are rotated clockwise by , and hence are oriented along a 100 direction of the crystal.
Following EulerBernoulli beam theory,Weaver Jr., Timoshenko, and Young (1990); Cleland (2002) we can express the eigenfrequency of the th harmonic as
(2a)  
(2b) 
where is the Young’s modulus, is the area moment of inertia, is the mass density, is the crosssectional area, and the stress. For the case of sufficiently strong tensile stress Eq. 2a reduces to Eq. 2b. The resonance frequencies shown in Fig. 2 are fitted with Eq. 2b and clearly follow the expected dependence. Being in the high tensile stress regime, a change in frequency for a given resonator length can only originate from a different tensile stress . The frequency mismatch between the and data indicates that the stress depends on the resonator’s orientation. Solving Eq. 2a for and calculating the weighted mean from all data points yields MPa and MPa, indicating that the tensile stress varies by almost with crystal direction.
For anisotropic materials, stress and strain are related by the fourth rank compliance or stiffness tensors, and .Hopcroft, Nix, and Kenny (2010) For cubic crystals, those tensors simplify to matrices with three independent components, , and (see supplementary material). For InGaP, each component depends on the Ga content , values are taken from Ref. Shur, Levinshtein, and Rumyantsev, 1999.
By applying matrix rotations and transformations, one can calculate the angle dependent Young’s modulus of an ideal and defect free system (see supplementary material). Figure 3 shows for the two different Ga contents and . The Young’s modulus displays a similar behavior for both Ga contents, and varies between 80 GPa and 125 GPa, between the 100 and 110 crystal directions, respectively. In addition, Fig. 3 clearly reveals the rotation symmetry of . To calculate the tensile stress we multiply the Young’s modulus by the strain from Eq. 1 according to Hooke’s law:
(3) 
The resulting stress values for both angles, and , coincide well with the experimental results.
To further investigate the angular stress dependence of InGaP, we fabricated similar sets of resonators with angles changing in steps. For each orientation the tensile stress is extracted using Eq. 2a. The top plots of Fig. 4 show the resulting angular stress dependence for two different Ga contents. In both cases, local stress maxima are observed at and , i. e. along 110 crystal directions. Accordingly, the minima are found at and , which correspond to 100 directions.
The gray dashed line in Fig. 4 depicts the stress values obtained by using Eqs. 1 and 3, which does not completely coincide with the experimental data. While the model conforms with the data at and , there are deviations around for both and .
To elucidate the deviation of stress, we have performed high resolution xray diffraction (HRXRD) reciprocal space map measurementsPietsch, Holy, and Baumbach (2004) along two orthogonal 110 crystal directions as shown in Fig. 5 (a). The diffraction peak arising from the InGaP layer lies directly above the substrate peak (circles in Fig. 5(a)), i.e. at the same positions, and coincides with the expectation for a 100 % pseudomorphic film within an accuracy of . In particular the HRXRD measurements show the same outofplane strain for both the and sample orientations. However, the InGaP layer peaks show a different diffuse scattering which can be mainly attributed to point defects,Kaganer et al. (1997) indicating different defect densities along the orthogonal 110 crystal directions.
Additional cathodoluminescence (CL) measurements in Fig. 5 (b) are done to obtain further insight on the dislocation density in the epitaxial material. Dislocation lines have a higher density along the direction than along . These measurements confirm different defect densities along the orthogonal 110 crystal directions.
It has been shown, that defects can influence the elastic properties of crystalline materials, and can lead to a softening as well as a hardening of the elastic constants.Dai et al. (2015); Chen et al. (2016)
This change of elastic properties can be treated as an effective Young’s modulus . We extract the deviation from the experimentally obtained stress, the strain using Eq. 1, and the theoretically calculated Young’s modulus determined in Fig. 3. The extracted values are shown in the bottom plots of Fig. 4 and clearly reveal an angular deviation from the theoretical Young’s modulus. Both the softening and hardening of elastic constants can be seen for our two different InGaP compositions. One can see softening for the highstress sample, while the lowstress sample shows both softening and hardening. Fitting a phenomenological function to the data, leads to the deviation functions GPa and GPa, respectively. Adding those functions to the theoretical Young’s modulus to calculate the angular stress (Eq. 3), we obtain the solid blue lines in Fig. 4 which indicate the added effect.
In conclusion, we have explored tensilestrained nanomechanical string resonators fabricated from crystalline InGaP.
The initial InGaP thin film is pseudomorphically strained for a thickness of and a Ga content of .
For the given composition we extracted an angledependent tensile stress of up to 650 MPa.
InGaP with a Ga content of shows lower tensile stress around 100 MPa with a similar angledependence as the highstress InGaP.
The observed angular stress dependence with respect to the crystal orientation is explained by a combination of anisotropic Young’s modulus and a change of elastic properties caused by defects.
This enables control over the stress of a nanomechanical resonator for a given heterostructure with fixed Ga content, which in turn could be optimized to enable maximum tensile stress.
In addition, angular stress control opens a way to investigate the influence of tensile stress on the dissipation of nanomechanical systems.
Stress control and further characterization of strained crystalline resonators will help to gain a deeper understanding in pursuit of ultimate mechanical quality factors.Tsaturyan et al. (2017); Ghadimi et al. (2018)
Finally, InGaP is a promising material for cavity optomechanics, as two photon absorption is completely suppressed at telecom wavelengths.Cole et al. (2014); Guha et al. (2017)
Moreover, tensile strained InGaP could open a way to combine a high Q nanomechanical system with a quantum photonic integrated circuit on a single chip.Dietrich et al. (2016); Bogdanov et al. (2017)
See supplementary material for detailed descriptions of the fabrication process, calculations of the Young’s modulus, comments on the critical thickness of InGaP lattice matched to GaAs, and for more details on the HRXRD measurements.
G.D.C. would like to thank Chris Santana and the team at IQE NC for the growth of the epitaxial material used in this study. Financial support by the Deutsche Forschungsgemeinschaft via the collaborative research center SFB 767, the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement No 732894 (FET Proactive HOT), and the German Federal Ministry of Education and Research (contract no. 13N14777) within the European QuantERA cofund project QuaSeRT is gratefully acknowledged.
The data and analysis code used to produce the nanomechanical plots are available at http://dx.doi.org/10.5281/zenodo.1477912.
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Supplementary material:
Stress control of tensilestrained InGaP nanomechanical string resonators
S1 Fabrication
The employed heterostructures were grown by molecular beam epitaxy on diameter, thick (001) GaAs wafers.
Both InGaP (InGaP) layers on each wafer have a thickness of 86 nm, and are capped by 1 nm GaAs on both sides, respectively.
The two investigated Ga contents are (highstress) and (lowstress).
From bottom to top, the structure consists of a GaAs buffer layer followed by a GaAs/AlGaAs distributed Bragg reflector.
The bottom sacrificial AlGaAs (AlGaAs) layer has a thickness of 1065 nm.
The following two InGaP layers are separated by the 265 nm thick top sacrificial AlGaAs layer.
The aluminum content of all AlGaAs layers is .
We only use the top InGaP and sacrificial AlGaAs in this work.
String resonators are defined by electronbeamlithography. The negative electronbeamresist maN 2403 serves as etchmask. To avoid delamination, we apply an adhesion promoter TI Prime before spincoating the resist. The roughly 240 nm thick resist features good resistance against dry and wet etches. Etchresistance is further increased by an additional hard bake after development, for 10 min at 120C in a convection oven. We pattern the string resonators with an inductively coupled plasma (ICP) etch, using a SiCl:Ar (1:3) gas mixture. Etching is carried out at a pressure of 1.7 mTorr with 250 W ICP power and 60 W RF power and at a temperature of 30C. To remove possible chlorine residues of the ICP etch, we soak the samples for 10 min in DI water. With an oxygen plasma cleaner we remove the resist etchmask. The following buffered HF etch releases the string resonators, by etching the sacrificial AlGaAs with an etch rate of 50–90 nm/s, depending on the crystal direction. After thoroughly rinsing the samples, we use a digital wet etchDeSalvo et al. (1996) to remove the GaAs cap layers and possible etch residues. In the end we dry the samples via critical point drying. To avoid degradation of the AlGaAs sacrificiallayer under ambient conditions,Dallesasse and Holonyak (2013) the samples are quickly mounted inside the vacuum chamber of the measurement setup.
Under ambient conditions the AlGaAs sacrificial layer quickly degrades and swells, fracturing the suspended InGaP resonators, see Fig. S1 (a), (b). Alternatively the AlGaAs surface can be passivated by rapid thermal oxidation (RTO) as shown in Fig. S1 (c)(e). This is done by a 5 min long rapid thermal anneal at 550C in an oxygen atmosphere. The AlGaAs surface is stable for at least three weeks. To what extent this treatment changes the mechanical properties of InGaP string resonators, remains a topic of further investigation.
S2 Calculating Young’s modulus
The Young’s modulus relates stress and strain in the onedimensional case of isotropic, uniaxial materials via Hooke’s law . For anisotropic materials, stress and strain are related by the fourth rank compliance or stiffness tensors, and .Hopcroft, Nix, and Kenny (2010) Those tensors are simplified to matrices with three independent components, for the case of the cubic symmetry of e.g. the zincblende crystal structure. For the [100] crystal direction, Young’s modulus simply equals the inverse first component of the compliance matrix :
(S1) 
However, since we know the elastic constants of InGaP,Shur, Levinshtein, and Rumyantsev (1999) we start with the stiffness matrix to calculate the Young’s modulus for any desired crystal direction:
(S2) 
Using the rotationmatrices Eqs. S7, we can perform clockwise rotations through an angle about a desired major crystal axis, initially =[100], =[010] and =[001]:
(S3) 
For example the application of to a [100] direction, produces a vector pointing along the [110] direction. By inverting the rotated stiffness matrix, we calculate the compliance matrix:
(S4) 
As in Eq. S1, the Young’s modulus is the inverted first component of the compliance matrix. But now it is along the rotated [100] direction, thus pointing in any desired direction:
(S5) 
For the Young’s modulus in the (001) plane, it simplifies to the following equation:
(S6) 
which is used to calculate the contour lines in Fig. 3 of the main paper. A full three dimensional plot of the Young’s modulus is shown in Fig. S2. The red line represents Eq. S6, the area cut through the (001) plane.
(S7a)  
(S7b)  
(S7c) 
S3 Critical thickness
This section provides an overview of the existing models to calculate the critical thickness of strained epilayers, and is a summary of Refs. Matthews, Mader, and Light, 1970; People and Bean, 1985.
There are two different approaches to treat the generation of dislocations, and thus yield a different thickness when a strained epilayer starts to relax. The first approach is the forcebalancing model by Matthews.Matthews, Mader, and Light (1970) This model considers the forces on dislocation lines. Those are misfit strain (as driving force), which is opposed by the tension in the misfit dislocation line. An epilayer starts to relax when the force exerted by the misfit strain becomes larger than the tension in a dislocation line. The other approach to calculate the critical thickness is the energybalancing model by People and Bean.People and Bean (1985) It compares the homogeneous strain energy density with the energy density associated with the generation of dislocations. If the surface strain energy density exceeds the selfenergy of an isolated dislocation, dislocations are introduced which lead to a relaxation.
The formulas to calculate the critical thickness are as follows:

Matthews:
(S8) 
People & Bean:
(S9)

[leftmargin=*]

: lattice constant of InGaP

: magnitude of Burgers vector of dislocation

: Poisson ratio of InGaP; elastic constants.

: inplane strain, distorted lattice constant due to lattice mismatch, as in Eq. 1.

: angle between dislocation line and Burgers vector (60 for most IIIV semiconductors)

: angle between slip direction and direction in epilayer plane which is perpendicular to the line of intersection of the slip plane and the interface (60)
By solving these formulas we obtain the curves of Fig. S3, which demonstrate that the model of Matthews gives a more conservative estimate of the critical thickness than the one by People & Bean. The horizontal line marks the thickness of the investigated InGaP film, whereas the vertical lines indicate the two Ga contents. Clearly, the model of Matthews is not able to describe the pseudomorphic high stress material which would greatly exceed the critical thickness. We thus conclude that the MBE growth process is better described by the model of People & Bean.
S4 High resolution xray diffraction measurements
High resolution xray diffraction (HRXRD) (using CuK radiation) is implemented for the characterization of the structural properties as a nondestructive method with a very high sensitivity to lattice parameter changes.Bauer, Li, and Holy (1996); Pietsch, Holy, and Baumbach (2004) The xrays scatter from the electronic density of the crystal, reproducing the lattice planes of the crystal. Measuring the angle of the scattered xrays and using Bragg’s law it is possible to calculate the distance between lattice planes, which can be related to the lattice constant of the crystal. Thin, mismatched layers distort tetragonally and therefore the lattice parameter parallel to the wafer surface differs from the perpendicular one, .
The reciprocal space maps (RSM) were performed using an  Empyrean (panalytical) diffractometer equipped with a combination of a Goebel mirror and a single channel cut (Ge220) monochromator on the source side. The scattered intensity was collected using a Pixcel 2D detector (514514 pixels at 55 m). The HRXRD line scans (Fig. S7) were done with a SeifertGE XRD3003HR diffractometer using a point focus equipped with an spherical 2D Goebel mirror allowing a beam size in the order of 1 mm horizontally and vertically. A Bartels monochromator and a tripleaxis analyzer in front of a scintillation counter were installed to achieve the highest resolution in reciprocal space.
a HRXRD measurements on highstress InGaP wafer
From symmetric RSMs, i.e. the 002 and 004 reflections of Fig. S4 and S5, one can extract the perpendicular lattice constant. The following symmetric RSMs show a strong GaAs substrate peak at wavevectors of about Å and Å, for the 002 and 004 reflection respectively. This corresponds to the substrate lattice constant of Å. The layer peak of InGaP can be seen at Å and Å and corresponds to the strained, perpendicular lattice constant Å.
When additionally measuring asymmetric RSMs, the 224 reflections of Fig. S4 and S5 and the 404 reflections of Fig. S6, it is possible to extract the parallel lattice parameters from the peak position on the axis. Since both the substrate and layer peak have the same component, their parallel lattice constants equal: Å, this is true for all the scan directions ([110], , [010], ).
We can clearly see with these findings, , that the highstress InGaP wafer is 100 % pseudomorphic.
In addition to the RSMs, to get further insight on the strain, one can look at a reflection curve as a function of scattering angle to extract structural information from the epitaxial structure. Figure S7 shows a HRXRD curve of the 002 reflection in blue. The substrate peak can be seen at a scattering angle of about , and the InGaP layer peak at about . The smaller, regularly distributed peaks arise from the GaAs/AlGaAs superlattice. By simulating and fitting the slow and rapid oscillations of the curve, it is possible to extract the thicknesses and compositions of the different layers in a heterostructure. The simulation, red line in Fig. S7, is done with the heterostructure of Tab. S1. The nominal heterostructure composition is shown in Tab. S2. From the measurement one can see, that the two InGaP layers have different compositions and also point to an In gradient along the growth direction. We have not taken into account the compositional gradients in our calculations in the main paper. This will be subject to followon work.
Layer  Repeat  Material  Al content  In content  Thickness (m) 

0  1  GaAs substrate  —  —  — 
1  1  AlGaAs  0.99536  —  0.01449 
2  41  GaAs  —  —  0.07801 
AlGaAs  0.96612 (top)  —  0.09002  
0.98612 (bottom)  
3  1  GaAs  —  —  0.07670 
4  1  AlGaAs  0.96032  —  1.05490 
5  1  GaAs  —  —  0.00100 
6  1  InGaP  —  0.46721 (top)  0.07762 
0.42233 (bottom)  
7  1  AlGaAs  0.98550  —  0.26441 
8  1  GaAs  —  —  0.00100 
9  1  InGaP  —  0.42227 (top)  0.08643 
0.38401 (bottom)  
10  1  GaAs  —  —  0.00100 
Layer  Repeat  Material  Al content  In content  Thickness (m) 

0  1  GaAs substrate  —  —  — 
1  1  AlGaAs  0.92  —  0.2720 
2  41  GaAs  —  —  0.0780 
AlGaAs  0.92  —  0.0906  
3  1  GaAs  —  —  0.0780 
4  1  AlGaAs  0.92  —  1.0653 
5  1  GaAs  —  —  0.0010 
6  1  InGaP  —  0.41  0.0859 
7  1  GaAs  —  —  0.0010 
8  1  AlGaAs  0.92  —  0.2646 
9  1  GaAs  —  —  0.0010 
10  1  InGaP  —  0.41  0.0859 
11  1  GaAs  —  —  0.0010 
b HRXRD measurements on lowstress InGaP wafer and comparison to highstress InGaP
For the lowstress wafer we performed the same RSM measurements as in subsection a. Due to the lower lattice mismatch the InGaP layer peak, at about Å, is located close to the underlying substrate and AlGaAs superlattice peak. Similar to the highstress wafer, the InGaP layer peaks show a different diffuse scattering in the asymmetric 224 reflections. We clearly see the enhanced diffuse scattering along the [110] crystal direction.