Morphological Relaxation of Strained Epitaxial Films for Stripe‐Geometry Devices

One of the major issues when reducing the channel length in strained transistors is the stress relaxation that significantly degrades the carriers mobility. A new morphological evolution is reported here, describing the relaxation of a strained epitaxial film deposited on a pattern characterized by stripe geometry, both paradigmatic of two‐dimensional (2D) like systems, and characteristic of many devices. The thermodynamic surface diffusion framework accounting for elasticity and capillarity is investigated. The former is solved in two dimensions thanks to the Airy formalism. The resulting dynamical Schrödinger‐like equation governing the film shape evolution is then solved thanks to a decomposition on eigenmodes. It reveals different developments depending upon the system's geometric parameters and the time scale. Mass transfer occurs toward the relaxed areas and creates a ridge at the nanolayers edges, controlled by the geometry and the scale of the structure. These results allow to refine the potential control of electronic properties via the geometry of a system.


Introduction
The "Internet of Things" is growing at an amazing rate and is expected to continue to accelerate.It should connect in the future years tens of billions of physical devices embedded with tiny computing devices that can sense and communicate.These achievements are based on the pursuit of the miniaturization that is DOI: 10.1002/admt.202301655achievable thanks to the transistor downscaling.In this respect, strain engineering has been used over the past ten years in the microelectronic industry for modifying the band-structure, and improving the carriers mobility.For example, epitaxial strain can be tuned by the lattice mismatch engineering, such as in uniaxial strained SiGe channels in ptype MOSFET. [1]SiGe (Si 1−x Ge x ) is also used as a higher mobility channel material for CMOS applications. [2]In recent and future technology nodes starting from and beyond 28nm, epi SiGe layers are integrated in all MOSFETs [3] and are the central part and the active areas of the transistors: drain/source (D/S) recessed or raised, or channel material.Since the morphology of the SiGe channels will impact the function of the final device, it is crucial to characterize their morphological evolution when confined in small areas during deposition, as well as during post-deposition thermal budgets.
[6][7][8][9] For example, the development of the fully-depleted complementary metal oxide semiconductors (FD-CMOS) technology [10] is based on ultra-thin silicon-on-insulator (SOI) [11] for which strain engineering in Si and SiGe layers of gate and source/drain areas respectively is a major technological challenge. [12,13]The unique features of nanogates can offer dramatic changes in growth modes, as well as matter and strain redistribution that are conceptually new as compared to bulk substrates given that finite-size effects come significantly into play.Strain may be used to change the band structure, effective mass and mobility [14,15] and boost the device performance. [16,17]f special interest for different technological developments is the strain and morphological evolution of SiGe epitaxial systems when confined on such reduced nanogates.It raises fundamental questions about the role of finite size effects and inhomogeneities on out-of-equilibrium dynamics, especially for systems with long-range (here elastic) interactions. [18,19]We consider here the morphological evolution of SiGe layers on top of strippatterned Si, that is both a paradigmatic 2D like geometry, and characteristic of the short-channel of a device.Strain distribution in similar patterned films has been investigated, see e.g., [20][21][22] and revealed the inhomogeneous strain that has an impact on the A film of thickness h is coherently deposited on a stripe of thickness e and width w.The system is deposited on a semi-infinite substrate (not displayed).electronic properties.We reveal here a new morphological evolution precisely due to the strain inhomogeneity, that describes the film relaxation thanks to surface diffusion.To do so, we first compute analytically the elastic relaxation in strained nanolayers deposited on a strip-patterned substrate thanks to the exact Airy formalism in two dimensions.We show that the relaxation is concentrated at the film free frontiers over a relaxation zone whose extension and amplitude depend on the film and stripe geometric characteristics (lateral size, film thickness, depth of the pattern, etc).We then investigate the evolution of the morphology of such a constrained nanofilm, by deriving a dynamic model of the surface diffusion induced by the inhomogeneous relaxation.Thanks to an analytic solution of the dynamical model, we characterize the relaxation dynamics that drives mass transfer toward the relaxed edges, and the resulting morphologies that may affect system properties.

Airy Function
We first compute elastic field in the strained stripe geometry, by considering a film of thickness h, epitaxially deposited on a stripe with a finite width w and height e, see Figure 1.The film is bound to evolve due to surface diffusion during annealing at constant temperature T. Given that mechanical equilibrium occurs under a timescale much shorter than that of thermodynamics, we first compute the elastic field within the body.We assume the stripe to be long enough as to neglect field variations along the y-direction, rendering our problem a 2D one effectively.We look for solutions of local equilibrium  ⋅  = 0, in which the stress  is linearly and isotropically related to strain  via  = Y 1+ [ +  1−2 Tr()]. [23]For the sake of simplicity, we assume that the Young modulus Y and the Poisson ratio  are equal in film and patterned substrate.In the 2D case, strain fields can be computed in an exact manner by utilizing the Airy function formulation (See, e.g.,ref.[24]).Within this framework, the components of the stress are derived from a scalar function ϕ (the Airy function) such that  xx = ∂ 2 ϕ/∂z 2 (and x ⇋ z),  xz = −∂ 2 ϕ/∂x∂z.Equilibrium is then satisfied when ∇ 4  ≡  4  x 4 + 2  4  x 2 z 2 +  4  z 4 = 0.In the stripe geometry considered here, an exact solution for ϕ was found using a double Fourier transform in x and z [25] (x, z) n sinh( n L) with  n = n/H,  n = n/L.Here, a coordinate system with the origin in the center of the body has been considered, with x ranging from −L to L (where L = w/2) and z ranging from −H to H [with H = (e + h)/2].The full solution to the elastic problem is subject to appropriate boundary conditions; i) the top and side facets are stress-free, i. e.,  ⋅ n z = 0 when z = H and  ⋅ n x = 0 when z = ±L (where n  is the unit vector in the direction  = x or z); ii) there is no displacement on the z = −H lower surface since the stripe is deposited on a rigid Si substrate [26] ; iii) the force perpendicular to the interface ( ⋅ n z ) and the displace- This offers the possibility to make use of the aforementioned Airy formalism and tackle the problem in an analytical fashion.Furthermore, one can show that u* is the equilibrium displacement for a single body with its side facets being subjected to a step-like load with form …∞, plus 2 equations relating J and K to the other coefficients, where the a p n,m , c p n,m , … i p n,m , j n , k n , L n are given as algebraic expressions involving tanh ( n L), tanh ( n H), sin ( n H), e/H.These linear matrix equations can be numerically solved by truncating series at a given order N and inverting the linear relationships.This gives the values of the different coefficients A n , C n , … and thence the solution for the Airy function.With this solution in hand, we compute stresses and derive the strain tensor.

Elastic fields
In Figure 2, we show the resulting lateral strain map ϵ xx , the Poisson dilatation ϵ zz and the total strain Tr[] (where ϵ yy = −m) in different typical geometries.One can see that for small aspect ratios, e/w = h/e = 1/10, see Figure 2a, away from the close vicinity of the facets, the film is nearly fully strained according to the substrate, with the flat and infinite film displacement vector ū.The latter corresponds to a biaxial compression in x and y (ϵ xx = −m) and a Poisson dilatation in z.On the contrary, the vicinity of the side facets is characterized by "relaxation zones" where the film is no longer fully strained, and where the lateral strain evolves from some value depending on z at the vertical free facet, to −m, away from the facet.This relaxation takes place over some length  that is found to be a function of the aspect ratios e/w and h/e, see Figure 2. At the edges, ϵ xx vanishes and the film is relaxed in the x direction, as was experimentally measured in other similar geometries. [27]The relaxation zone near the edges characterizes the typical inhomogeneity of strain enforced by finite size effects.For aspect ratios e/w = h/e = 1, see Figure 2d, finite size effects are stronger, with the overlap of the two relaxation zones leading to an almost fully relaxed film.In these situations, the system, even in its midst, is far from resembling the fully biaxially strained limit: both strains ϵ xx and ϵ zz are very small within the film, leading to a reduced full strain Tr[] (yet ϵ yy is still −m).This kind of geometry is thus very suitable to effectively and efficiently relax the epitaxial stress and to obtain highly relaxed SiGe channels, that are appropriate for instance as templates for the growth of Si tensile strain channels.However, it is totally unsuitable for fully strained MOSFET devices.These two cases (full strain with clearly separated relaxation zones versus relaxed with overlapping relaxation zones, Figure 2a,d) are the two extremes when varying the geometric parameters (e/w and h/e).For intermediate values of the different aspect ratios (e/w and h/e), one finds that the size of the frelaxation zone increases when either h/e increases for a given e/w width ratio, or when e/w decreases for a given h/e, see Figure 2b,c.It is noticeable that the full strain case is already achieved for values of h/e and e/w below typically 0.1, while the relaxed case occurs for values of these parameters of the order or larger than 1.Hence, we can argue that relax-ation has rather very localized effects on the edges of the system as soon as the aspect ratios e/w and h/e are both smaller than typically 0.1, whereas relaxation starts to significantly penetrate the system as soon as one of the aspect ratios is larger than typically 0.1.If one seeks to relax the epitaxial stress almost completely and uniformly, one should note that this is the case as soon as h/e and e/w are equal to or greater than 1, for example as in Figure 2d.On the other hand, the relaxation remains globally very low when the Si stripe is ten times thicker than the epitaxial SiGe layer.
In order to characterize the inhomogeneity at the surface, and for subsequent use in the dynamical analysis, we consider the elastic energy density  = 1 2  i j  i j computed on the surface z = H.In Figure 3, we plot the typical evolution of this energy inhomogeneity for two aspect ratios e/w but varying the film thickness h.The energy is maximum in the middle of the film where strain is higher, and it decreases towards the free facet, where relaxation is maximum.For geometries where the film is strongly strained (i.e. for either thin enough films or sufficiently wide stripes with small aspect ratio e/w and h/e), the energy density in the middle of the film is close to its value for a thin infinite film,  = Ym 2 ∕(1 − ), see Figure 3a.In this situation, the film is clearly relaxed not only at its edges, but also in its middle in case the aspect ratios e/w or h/e are no longer small, see Figure 3b.Naturally, we can see from the elastic energy curves that the relaxation is greater as the ratio h/e increases for a given e/w shape, but also it is greater as e/w increases for a given h/e, see the comparison between Figure 3a 2 Ym 2 = 1 2 (1 − ) , which is ≃ 0.36  for SiGe.This value is indeed well found by the numerical calculation at or near the corners, except in some conditions where the usual Gibbs effect can be detected. [26,28]Utilizing the boundary conditions similarly, we are able to deduce that the energy density profile at the corners satisfies dϵ/dx(x = ± L,z = H) = 0 which is also verified by the full solution (except for the Gibbs effect at the corners).For later use in the dynamical analysis, we fit this energy profile with the following form with with two fitting parameters, the amplitude A( e w , h e ) and the relaxation zone width *.This fit is plotted as a dotted line in Figure 3, and is relatively close to the Fourier series calculation except at the corners when the Gibbs effect is non negligible.At the corners, we impose on the fit to tend toward the exact analytical value  c ∕  = 1 2 (1 − ). Figure 4 gives the typical width of the relaxation zone * as a function of the system geometric parameters, h/e and e/w.The results show that the relaxation zone width * naturally increases when the film height increases, with a typical algebraic behavior.It is noticeable that its exponent is similar for e/w = 0.1 and 1, but departs from the fully strain case when e/w = 0.01.

Surface Diffusion and Eigenmodes
We now turn to the dynamical evolution of the crystal shape in the presence of such an inhomogeneous strain.Morphological evolution takes place through the diffusion of matter on the crystal surface, driven by elastic energy gradients.We consider the usual thermodynamic framework in which surface diffusion occurs only on the upper surface described by z = h(x, t), [29] neglecting mass transport across the corners.Mass conservation thence requires ∂h/∂ t = −∇ • J where J is the surface current density.This relates to the surface chemical potential via J = −D∇, with some diffusion coefficient D. [30] The chemical potential at the surface relates to the capillary and elastic energies as  =−Δh + (x, t) in the small-slope approximation.In the following, we assume that the energy profile (x, t) is set by the finite-size effects that we calculated above, and that it does not change throughout the dynamical evolution, i.e., (x, t) = (x).This assumption is supported by the fact that the extra mechanical relaxation that would be enforced by the morphological change of the free surface is rather small compared to the inhomogeneous strain field (see e.g.ref. [31]).The morphological evolution thence occurs in a given energetic profile.It is a noticeable difference with the strain-induced Asaro Tiller Grinfel'd (ATG) instability [31][32][33] or quantum dot nucleation [34,35] where on the contrary, the energetic profile is precisely given by the corrugation of the surface and Figure 4. Typical size of the relaxation zone at the corners */L as a function of the ratio between the film and the patterned substrate thicknesses, h/e, for different stripe aspect ratio e/w = 1 (upper curve), 0.1 (middle curve) and 0.01 (lower curve).The dotted line is an algebraic fit */L∝(h/e)  , with  = 0.42 (upper curve), 0.41 (middle curve), and 0.27 (lower curve).(Values of * close to or greater than unity, which are characterized by large uncertainties are not represented.)Table 1.Examples of the characteristic time and space scales for the elastic relaxation for a Si 0.8 Ge 0.2 film on Si at the typical temperature T = 550 °C of a continuous nucleationless dynamics. [37]For comparison, the ATG time scale is in this case  ATG = 780 s, with a wavelength  ATG = 43 nm. [37]w = 1/100 ; proportional to h(x, t).[36] In the present framework, we use the so that the evolution equation takes the form with the elastic inhomogeneity f(x) given in Equation ( 2).It is noteworthy scales of the surface relaxation (4) are again different to those of the ATG instability that are  ATG = ∕  and  ATG =  4 ATG ∕D.Contrarily to  ATG ∝ 1/m 2 , here * does not depend on m and is just a function of the geometric parameters h, e, and w.As an example, and in order to give some orders of magnitude relevant to the experiments, we display in Table 1 the values of the time and space scales for a Si 0.8 Ge 0.2 film on Si at T = 550 °C (ensuring a continuous nucleationless dynamics [37] ).We first notice that these scales explicitly depend on the geometry of the system.We also note that the wider the system, the bigger the time scale * is.As a consequence, the evolution occurs much faster on a small stripe compared to a big one, for equal aspect ratios.The scales * and l* also decrease proportionally to w, so that the amplitude of the morphological evolution at the edges is expected to be smaller on a narrow strip.
It is possible to solve the evolution Equation ( 5) with the initial condition h(x, 0) = H utilizing standard methods of functional analysis in presence of a forcing term. [38]We assume the continuity of  (neglecting lateral corrugation) and neglect mass flow at the corners, so that we enforce  2 h∕ x2 = 0 and  3 h∕ x3 = 0 when x =± L where L = L∕ * .One may first compute the stationary solution hs ( x) of Equation ( 5) given by where Having this long-time stationary solution, one can find h thanks to the decomposition h = hs ( x) − h1 ( x, t) where h1 satisfies the Schrödinger equation The eigenfunctions ϕ n of  such as [ n ] = E n  n , n = 1, 2…, are merely where satisfies the characteristic equation originating from the boundary conditions tan(k n L) + tanh(k n L) = 0 ( 9 )   whose roots can only be found numerically.Yet, a very good approximation of them is k n L ≃ (n + 3∕4).Decomposing h 1 in terms of these eigenfunctions, one eventually finds the solution to the surface dynamics as where the coefficients c n are given by the initial condition

Morphological evolution under strain
The resulting evolution of the crystal surface is plotted in Figure 5 for a typical geometric configuration.The elastic density gradient clearly drives mass transfer toward the edges of the system, where the elastic energy is lower.At the beginning of the relaxation, we observe that mass transport initiates within the relaxation zones at both ends of the stripe, where there is a noticeable gradient in elastic energy.Meanwhile, the middle of the system remains mostly stationary and flat.The surface is then W-shaped with a flat central area that gradually narrows.In latter stages, however, the evolution progressively spreads in toward the middle of the system.This is the response of the surface capillarity, which tends to minimize the surface area facing the vacuum.
In the long run, the top surface corrugates due to mass transfer from regions close to the middle of the surface towards its edges.At this point, matter is transferred from the middle of the surface toward its edges, quickly giving rise to the equilibrium V-shape.It is noticeable that the evolution on the different geometries occurs on different time scales.If Figure 5 is characterized by dimensionless time scales following Equation (4), for large stripes (long channels), the relaxation occurs over a rather long time scale, while it occurs over short time scale for short stripes (short channels).In order to fix the orders of magnitude, we consider the typical temperature of continuous growth without nucleation T = 550 °C for a typical Si 0.8 Ge 0.2 film on Si.For the aspect ratios e/w = 1/100 and h/e = 1 of Figure 5, one finds * = 210 s on a Si stripe with w = 200 nm, while it is just 0.021s when w = 20 nm.Consequently, in the latter case, the final stationary V shape is quickly reached, while it is clearly not the case for the larger stripes.Note also that the counter-acting driving force of the capillarity is lessened on larger stripes due to lower surface-to-volume ratio on these.The equilibrium stationary solution (dashed line) corresponds to Equation ( 6).The typical relaxation zones (where strain is significantly different from the infinite-system limit) are shaded in blue.
The theoretical framework given here relies on different assumptions (static elastic field, small-slope approximation, absence of mass transfer toward the sides, corner rounding…), but we argue that the main physical picture is captured by this description : mass transfer to the surface edges where the elastic energy is lower, with a (transitory) W shape or (stationary) V shape and speed that are function of the stripe aspect ratios.This study therefore allows us to exhibit a new mode of relaxation on a minimal model including the crucial effects, and paves the way for further studies incorporating other effects that may come into play, such as anisotropy [39,40] or the competition between finite size relaxation and morphology-induced relaxation.

Conclusion
We have investigated the relaxation and morphological evolution of strained films epitaxially deposited on a stripe-patterned substrate (modeling the gate of a transistor).We have characterized a new relaxation instability that drives mass transfer toward the edges of the deposited layers and can result in significant morphological evolution.Its dynamical evolution depends on the geometrical parameters (film thickness, pattern length, and depth) that control the strain inhomogeneity.The evolution displays different shapes and speeds as a function of the stripegeometric parameters and eventually leads to a stationary profile when capillarity counterbalances elasticity.The work reveals and explains the general mechanisms expected to be at work in several gate and S/D geometries of microelectronic devices and also well beyond this framework.It opens the way for a better understanding and control of strain-engineering using finite-size effects experienced when using S/D stressors.It should help to optimize the design of new geometries intended to improve the materials and devices properties.We are currently investigating on this basis, the development of innovative systems for the pursuit of miniaturization in microelectronics, typically for node 28nm BiCMOS SiGe transistors and beyond.

Figure 1 .
Figure 1.Schematic of the strip-patterned system under consideration.A film of thickness h is coherently deposited on a stripe of thickness e and width w.The system is deposited on a semi-infinite substrate (not displayed).
ment u with respect to Si [accounting for the misfit m = (a SiGe − a Si )/a Si ] are continuous across the boundary since the SiGe/Si interface is coherent.To simplify the calculations, we use the decomposition u = ū + u * where ū corresponds to an infinitely large (w → ∞) flat film.In this special case, ū = 0 in the stripe, while in the film, ū = 2m∕(1 − ){0, 0, z − (e − h)∕2}.Decomposing u in this way produces a new unknown displacement vector u* that is effectively 2D, with the y-component identically 0.

Figure 2 .
Figure 2. Strain maps for different stripes with e/w and h/e = 1/10 and 1.For each geometry, we display ϵ xx , ϵ zz and Tr[] for a Si 0.8 Ge 0.2 film on a Si stripe.The maps correspond to the solution of (1) with a truncation number N = 15 000.
,b.At the corners (x = ±L, z = H), the value of the energy density is dictated by the boundary conditions : the elastic field must satisfy  xx =  zz =  xz = 0 so that ϵ xx = ϵ zz = m and  yy = −mY and eventually  c = 1