Controlled formation of three-dimensional cavities during lateral epitaxial growth

Epitaxial growth is a fundamental step required to create devices for the semiconductor industry, enabling different materials to be combined in layers with precise control of strain and defect structure. Patterning the growth substrate with a mask before performing epitaxial growth offers additional degrees of freedom to engineer the structure and hence function of the semiconductor device. Here, we demonstrate that conditions exist where such epitaxial lateral overgrowth can produce complex, three-dimensional structures that incorporate cavities of deterministic size. We grow germanium on silicon substrates patterned with a dielectric mask and show that fully-enclosed cavities can be created through an unexpected self-assembly process that is controlled by surface diffusion and surface energy minimization. The result is confined cavities enclosed by single crystalline Ge, with size and position tunable through the initial mask pattern. We present a model to account for the observed cavity symmetry, pinch-off and subsequent evolution, reflecting the dominant role of surface energy. Since dielectric mask patterning and epitaxial growth are compatible with conventional device processing steps, we suggest that this mechanism provides a strategy for developing electronic and photonic functionalities.


Supplementary Notes
Supplementary Note 1: Thermodynamics and kinetics Fig. S9 illustrates the main idea of the phase-field model developed to understand the underlying mechanisms of the ELO and coalescence phenomenon.Two concurrent processes are modeled: the gas-solid phase transition of Ge and the transport of Ge atoms.Phase-field models are commonly used in simulating phase transition processes and the corresponding interfacial evolution.A continuous and conserved phase field is introduced to capture kinetics of quasi-immobile solidified Ge atoms, oversaturated gaseous Ge and Ge atoms on interfacial layers.
A continuous order parameter, (, ), is introduced as a function of the spatial coordinates  and time .We select the normalized concentration of Ge as the order parameter, (, ) = ((, ) −  )/( −  ) , where  is the local Ge concentration, and  and  are the equilibrium concentrations of the coexisting solid and gas phases, respectively.The order parameter varies continuously but abruptly between  = 0 in the saturated vapor and  = 1 in the solid phase; the narrow transition zone can be regarded as a representation of the surface with a layer of irregular atomic packing at the gas/solid interface.
Following the common practice of the Cahn-Hilliard models 1 , we write the free energy density into two terms.A simple double-well function Φ () =  ( − 1) , with minima at  = 1 and 0 , is introduced to separate the system into coexisting solid and gas phases.The parameter  characterizes the energy barrier between the two phases.The second contribution, a gradient term, Φ = |∇| is always positive.A comparison between the two energy contributions leads to a length scale  =  2 ⁄ , which characterizes the equilibrium thickness of the solid-gas interface.
In summary with repeated index representing a summation.
When the domain is connected to a reservoir of Ge atoms with chemical potential , we let  be the number of Ge inserted into the domain from the reservoir.The total potential energy of the system, including the reservoir, reaches a stationary point,  = 0 .The integration in Eq. ( 2) is carried out over the entire computational domain Ω .Conservation of Ge atoms requires that  = ∫  .Through variational calculations and integration by parts, we obtain the Euler equation: which can be solved for the equilibrium phase field (), when proper boundary conditions are imposed.
Now let us consider a general non-equilibrium process.We still assume local equilibrium, so that the inhomogeneous field of chemical potential (, ) is still given by Eq. ( S3).Writing the chemical potential in the form of Eq. (S3) also implies the choice of taking the equilibrium coexisting states as the reference, in which  = 0.The first term on the right-hand-side of Eq. ( S3) accounts for the mean-curvature contribution on an interface 2,3 , with a non-vanishing gradient of .The second term corresponds to the chemical potential in the bulk, far from the interface.The two states,  = 1 and  = 0, share the same chemical potential, corresponding to the coexisting phases.
To model the migration of Ge atoms, we adopt linear kinetics and assume that the diffusion flux is related to the gradient of chemical potential through: The positive-definite mobility tensor  captures both the differential diffusivities between the gas and solid phases, and the contribution from interface diffusion.Here, we write the mobility in the following anisotropic form: The first term on the right-hand-side captures bulk diffusion, with the phase-dependent isotropic mobility  () =  + (1 − )  interpolated between  (0) =  in gas phase and a much lower level  (1) =  in solid phase.The second term represents the surface diffusion.
The resulting transversely isotropic mobility tensor has a principal value  in the through-thickness direction of the interface (along ∇ ) and two equal principal values  + |∇|  along the other two directions perpendicular to the interface normal.Note that |∇| takes non-zero values only within the transition zone between two phases, where the surface diffusion contribution is effective.It is also possible to extend the current model to include anisotropic diffusion (e.g. that due to Ehrlich-Schwoebel barriers 4 ), by introducing orientation dependence in the already anisotropic mobility tensor M in Eq. (S4).Anisotropic diffusion changes the kinetics by directing adatom flux along preferred directions, but does not alter the thermodynamics.Therefore, it may delay some phenomena (e.g. the coalescence), but cannot prevent them from happening.
Finally, the conservation of Ge atoms dictates that in the bulk, and on a boundary of unit normal vector  and known outward flux of Ge, .Substitution of Eqs. ( S3) and (S5) into (S4), and then into (S6), we arrive at a partial differential equation for the evolution of the phase field (, ), which is then written into a weak form, and solved numerically by using a finite element method through the commercial package COMSOL Multiphysics.

Supplementary Note 2: Contact angle
Ge crystals share the same lattice structure as Si, but differs from that of SiO2.The difference in coherency results in the preferential growth of Ge on Si surface, and the different wettability of Ge on Si and SiO2.To account for the different surface energy, an additional boundary condition should be imposed on the SiO2 surface 5 .To derive a consistent boundary condition, let us consider the simplest one-dimensional case, with a flat and infinitely large planar phase boundary dividing the entire space into two.
Let  be the one-dimensional coordinate, its axis normal to the solid-gas interface, pointing from the gas phase to the solid phase.The coexisting state of the two phases is still taken as the reference for chemical potential,  = 0.The equilibrium condition, Eq. ( S3), now becomes Multiplying both sides of Eq. (S8) by d d ⁄ , we arrive at Integration of both sides of Eq. (S9) from deep inside the bulk gas phase, where  = 0, d d ⁄ = 0, and Φ = 0, to a point near the interface leads to Thus, the slope profile of the equilibrium phase field near an interface takes the form Now let us consider the case of a Ge atom deposited on a SiO2 surface.Near the gas-solid interface, when the curvature effect to the local phase-field profile is negligible, we may assume the local phase-field profile to be close to that of the ideal one-dimensional case, as given by Eq. (S11).As sketched in Fig. S12a, in equilibrium, the gas-solid interface of Ge forms a specific angle  with the SiO2 surface, known as the static contact angle.Thus, the projection of the phase-field gradient ∇ onto the normal direction of the SiO2 surface can be evaluated as We therefore take Eq.(S12) as the boundary condition for a SiO2 surface.
To validate the effectiveness of the boundary condition, a series of numerical experiments have been carried out.With different values of  taken and boundary condition as Eq.(S12) imposed, finiteelement calculations all show that the corresponding contact angles are achieved when the system equilibrates after sufficient time, as shown by Fig. S12b.In the current study, the value  ≈ 120° is taken from experimental observation.

Supplementary Note 3: Growth front evolution process
To quantify the growth-front evolution process, we first compare the growth front morphology of anisotropic ELO on a square mask, as shown in Fig. S16(a-c).We take Si [110] and [100] as representative directions for observing the crystal geometry as epi-growth proceeds.During anisotropic growth, no overgrowth was observed at the edge of square mask along Si [110], and the morphology of the upper part of the growth front was defined by {105}, {111} and {113} facets.
Along Si [100], where Ge overgrowth started at the four corners of the mask, an obtuse contact angle of 100° was found.
There are three stages of growth front evolution after the anisotropic-isotropic transition, illustrated in Fig. S17.
Stage 1: contact angle of the growth front changing from <90° to ~100°, showing a uniform morphology of growth fronts from all directions towards the center of mask.The chemical potential gap between corner and edge vanishes, suggesting a uniform inward growth velocity.
Stage 2: contact angle between Ge growth front and mask is around 100°, suggesting the growth front moves forward with a uniform morphology.This stage only exists when the mask diameter is large enough that the growth fronts proceed independently of each other, as shown in Fig. S16(de).
Stage 3: contact angle increases sharply from ~100° to 135°, indicating the front morphology is strongly affected by the negative curvature-induced adatom migration.

Supplementary Note 4: Orientation-dependent surface energy
To arrive at the facetted surface of a single crystal, following Siem and Carter 6 , we can replace the interfacial energy coefficient  with an orientation-dependent parameter in the form of where  = /|| is the unit normal vector of the solid-gas interface,  is the unit normal vector of one of the  facet surfaces,  and  are positive parameters controlling the width and depth of the corresponding energy well 6 , and () is the Heaviside function.The values of  and  along particular directions are obtained from refs.(7,8), as listed in Table S1.         the mask after full coalescence.f, Illustration of how a plan-view TEM lamella is fabricated above the cavity, with plane parallel to Si (001).The FIB milling process was from the Si to Ge, so that the position of the mask can be marked in the final TEM lamella.g, Plan-view TEM image of the Ge on top of mask at the coalescence point for the cavity, showing dislocation-free Ge at the area above the mask (dashed square) including the coalescence point, while threading dislocations are observed from the Ge grown on Si (white arrows).h-i, from the same position as illustrated in (f), Plan-view TEM lamella from a pentagon and round oxide pad were fabricated above the cavity, no threading dislocation was observed at the coalescence point (the center of dielectric mask).The scale bars are 500 nm.

Fig. S2
Fig. S2 Low temperature growth of Ge on patterned Si (001).a, the mask was in 20×20 μm 2 square shape with oxide width of 1.0 μm.b, Top view SEM image of this pattern after epi-growth.The epigrowth Ge was in a trapezoid shape with dominant facets of {311}.Moreover, the profile of the epi-Ge was rather rough due to the low mobility of Ge adatoms during growth.c, tilted SEM view of the low temperature epi-Ge after carbon deposition for FIB milling.Ravines can be clearly seen on low-temperature Ge. d, tilted view of the oxide stripe, showing oxide is covered by Ge without forming tunnel or cavity.e, lithography defined oxide margins, imperfections can be clearly observed which led to the Ge ravines after growth due to low mobility.

Fig. S4
Fig. S4 Ge ELO on mask with different geometries.ELO first starts from the corner of masks and the growth velocity at concave corners is faster than at the edges.Anisotropic growth transitions to isotropic when the straight edges vanish, then the front grows inward with equal velocity.The scale bar is 2 m.

Fig. S5
Fig. S5 a-b, Ge ELO on square and triangular mask with edges rotated 22.5° from Si [110], showing that the anisotropic-isotropic transition occurs irrespective of the angle between mask and Si [110].c, enlarged view of (b2) showing stripy contrast indicating that step bunches and surface facets are formed during growth.The scale bar is 2 m.

Fig
Fig. S6 (a) Cross-section SEM image and (b) Plan-view SEM image of a cavity formed after Ge full coalescence on mask.The samples are fabricated by FIB across the center of mask and at the Ge-Si interface, respectively.SEM images are taken at a tilt angle of 52°.

Fig. S7 FIB
Fig. S7 FIB procedure for plan-view analysis of cavities on oxide.(a-c) Lift-out of a specimen containing Ge on oxide-patterned Si; (d-e): Top-view of specimen transfer to a grid; (f-g): rotated 90 ° for FIB milling; (h-j): Si substrate was milled step by step, and the cavity gradually appears at the Geoxide interface.

Fig. S9 a ,
Fig. S9 a, Illustration of how a TEM specimen with thickness ~100 nm is fabricated by FIB to image the Ge/Si interface.b, Plan-view TEM image of the specimen at Ge/Si interface.c, is the higher magnification image, where the misfit dislocation network is observed at Ge/Si interface while dislocation-free Ge is found over the oxide.d-e, Dark field TEM image of cross-section view across

Fig
Fig. S10 A simplified illustration of concentration field in the growth model.Free diffusion in the gas atmosphere is considered.For the Ge adatoms at gas-crystal interface, there are three pathways for motion: adsorption, desorption, and diffusion.Additionally, Ge adatom diffusion at interface is mediated by curvature in the model.

Fig. S11
Fig. S11 Phase field model construction.a, Initial state of numerical simulation where a square mask was defined in the center of z = 0 plane of a cubic area.b, The Ge profile evolution is tracked through the order parameter C (volume fraction of solid Ge), with values 1 and 0 representing the solid phase (Ge crystal) and gas phase respectively, the interface between them being represented by the C =0.5 isosurface.

Fig. S12 a ,
Fig. S12 a, Schematic of interface (a contour of  ) at stasis. is the local direction normal to the interface from gas phase to solid phase. is the contact angle of solid Ge on mask.b, With different setting of the initial contact angle, the calculated morphology of the cavity inside the solid Ge at long time.

Fig. S13
Fig. S13 Phase field simulation (PFS) of ELO process and cavity formation on mask with varied geometry.a1-d1: Top view of the simulation of growth front changing from anisotropic to isotropic and maintaining isotropic growth until it coalesced, irrespective of the mask shape (circle, triangle, pentagon and hexagon are shown).a2-d2: Cavities are formed at the center of mask after growth front coalescence, corresponding to a1-d1.

Fig. S14
Fig. S14 Additional simulation of random masks.a1, b1: Illustrative scheme of the surface diffusion modeling, with the color map of the outline representing the curvature of mask while the curvature of the Ge growth front has the same value but opposite sign.a2, b2: Top view of the anisotropic to isotropic transition of ELO on the asymmetric masks.a3, b3: Side view of cavity formation after coalescence.

Fig. S15
Fig. S15 Simulation of surface diffusion at different choices of the value of the mobility of  : a, no surface diffusion,  = 0; b, moderate surface diffusion,  = 200 and (c) surface diffusion chosen in studying kinetics  = 400 .The Ge profile recedes from  to  in simulations where surface diffusion is set to be "strong" ( = 400) while recession is not observed when we set weaker surface diffusion  = 200 or zero surface diffusion  = 0.

Fig. S16
Fig. S16 Growth front morphology of anisotropic and isotropic ELO process.a, Top-view SEM image showing the accelerated rate of ELO at the corner of a quadrilateral mask.b and c are the crosssection views produced by FIB milling along Si[110] and Si[100] direction, respectively.C was deposited before FIB fabrication and the horizontal features are caused by the CxHy gas flow during deposition.The growth front at the mask edge is inhibited with contact angle smaller than 90° while the front at the mask corner moves forward with contact angle ~ 100°.d-f During isotropic-ELO process, the growth front is moving inward with uniform velocity and contact angle of ~ 100° along Si [110] and [100] directions.

Fig
Fig. S17 Illustration of growth front morphology evolution under anisotropic-and isotropic-ELO processes.The contact angle  is smaller than 90° when the growth front is inhibited at the mask edge (orientation Si <110>), while is ~100° when the growth front moves forward at the corner of the mask (Si <100>).The growth fronts become uniform when isotropic ELO begins;  remains ~100° until the diameter of the open cavity is smaller than d0.The d0 here denotes the critical diameter value of the transition zone where Ge adatom consumption cannot be supplied effectively thus causing accelerated growth of the front end and sluggish growth of the contact point.The contact angle then evolves from 100° to 135°, indicating a strong geometric shielding effect when the growth fronts are close enough.After full coalescence, the cavity is formed and the inner facets are reconfigured.

Fig. S18
Fig. S18 a, Plan-view TEM image of Ge-SiO2 interface showing cavity located at the center of a hexagonal mask.b, Enlarged image of cavity showing the cavity contact line with SiO2 has 12 edges.

Fig. S19
Fig. S19 a-b, Cross-section view of Ge ELO on round masks with different diameters.The mask diameters in a and b are 4.9 µm and 6.8 µm, respectively, and undergo the same growth condition and growth time.The epi-length (L) of each mask is shown on the images and indicate that smaller diameter masks undergo a faster growth rate with larger curvature.The contact angle of both is ~100°.c-d, Crosssection view of Ge ELO on square masks with different diameters.Viewed from Si [110] direction, the geometry of growth front is the same as a and b, indicating the contact angle between Ge and mask is determined by the interfacial energy between Ge and mask, and the curvature plays a minor effect at the early stages.The scale bars are 1 µm.

Table S1
Facet energy parameters for single-crystal Ge