Modified Stranski–Krastanov Growth

• Modified Stranski–Krastanov growth schemes are processes that adjust the classic layer-plus-island model by modifying the wetting layer behavior, critical thickness, and islanding mechanisms using interfacial chemistry and kinetic control.
• They employ techniques such as surfactant addition, topological templating, and near-critical kinetic steering to precisely manipulate island density, strain relaxation, and surface morphology during epitaxial deposition.
• These engineered growth strategies are pivotal for applications in superconducting trilayers, semiconductor photonics, and advanced device fabrication, offering tunable and reliable film characteristics.

Modified Stranski–Krastanov growth scheme denotes a family of growth behaviors that retain the canonical SK motif—an initial wetting layer followed by three-dimensional islanding or roughening—but alter its mechanism, sequence, or controllability through interfacial chemistry, epitaxial multiplicity, surfactants, topology, confinement, or near-critical kinetic control. In this expanded usage, the wetting layer need not remain coherently strained up to the morphological transition, the critical thickness need not be unique or purely thermodynamic, and the 3D stage may be externally triggered, kinetically delayed, redirected into step-flow, or built on an interfacial phase that already differs structurally from the depositing material (Welander, 2010, Berdnikov et al., 2023, Lewis et al., 2017, Rougemaille et al., 2019, Hückmann et al., 15 Jan 2026).

1. Classical baseline and the basis for modification

In its classical form, SK growth is a sequence of Frank–van der Merwe layer growth followed by Volmer–Weber-like three-dimensional islanding. The thermodynamic language used in the literature expresses this through the wetting function or wetting parameter $\Phi$, surface and interface energies, and the strain contribution to the film chemical potential. In the treatment of Prieto and Markov, the film chemical potential can be written as $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$, with $\varepsilon_e(f)$ the homogeneous strain energy per atom, so that complete chemical wetting can coexist with a later instability of the strained film; in that formulation, monolayer islands are necessary precursors of three-dimensional islands, and the positive wetting function required for the Volmer–Weber part of the sequence originates from vertical displacements of atoms near island edges during lattice-misfit relaxation (Prieto et al., 2017).

Subsequent work has shown that the textbook picture is incomplete even before any explicit “modification” is introduced. Kinetic Monte Carlo simulations of SK growth found that the wetting layer is thermodynamically stabilized by entropy, that a true wetting-layer thickness must be distinguished from an apparent critical thickness, and that the apparent critical thickness depends on deposition rate, temperature, and alloy intermixing (Baskaran et al., 2011). In a different formalism, dynamical simulations of strained solid droplets on rigid substrates produced stationary SK islands together with an extremely thin wetting layer and emphasized that the resulting morphology is a non-equilibrium steady state controlled by surface drift diffusion, mismatch stress, and triple-junction kinetics rather than by a prescribed equilibrium contact angle (Ogurtani et al., 2010).

A second refinement concerns criticality itself. In the near-critical InAs/InP literature, the relevant quantities are the equilibrium wetting-layer thickness $h_{eq}$, the islanding threshold $h_c$, and the wetting-layer superstress $\Delta h = h-h_{eq}$. In that framework, $h_c$ is explicitly not the dislocation critical thickness, and the experimentally accessible window $h_{eq} < h < h_c$ becomes a growth regime in its own right rather than a negligible prelude to abrupt islanding (Berdnikov et al., 2023). Modified SK schemes generally arise by exploiting one or more of these non-idealities.

2. Principal architectures of modification

The term covers several distinct but related departures from the idealized SK sequence.

Modification class	Representative system	Defining departure
Domain-structured wetting layer	Re(0001)/Nb(110) (Welander, 2010)	KS and NW variants coexist from the start; morphology changes near $150\,\text{\AA}$ while strain relaxes by $\sim 200\,\text{\AA}$
Near-critical kinetic steering	InAs/InP (Berdnikov et al., 2023)	As/P-exchange wetting layer plus $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$0–$\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$1 InAs enables density tuning from $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$2 to $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$3
Surfactant-induced or post-growth SK	InAs/GaAs(110) (Lewis et al., 2017, Lewis et al., 2019)	Bi converts 2D FvdM growth into SK and can trigger a 2D→3D transition after deposition
Confinement-mediated suppression of islanding	Cu/graphene/Ru(0001) (Rougemaille et al., 2019)	Bare Ru shows SK, whereas intercalated Cu under graphene grows by confined step-flow
Topology-directed bilayer wetting	Ti/Si(100) (Hückmann et al., 15 Jan 2026)	A TiSi bilayer acts as the wetting layer and nucleation template for C49-TiSi$\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$4

These categories share the formal SK signature of an initial wetting stage and a later change in growth dimensionality, but they differ in what actually controls that change. In some systems the principal perturbation is crystallographic, in others kinetic or surfactant-mediated, and in still others the nominal 3D-islanding step is replaced by another strain-relief pathway. This suggests that “modified SK” is best understood as a class of engineered layer-plus-instability schemes rather than a single alternative mechanism.

3. Interfacial registry, domain structure, and topology

A clear non-textbook example is Re(0001) on Nb(110). The Re overlayer follows a Stranski–Krastanov mode in the sense that growth is initially layer-by-layer and later becomes granular, but the wetting layer is never a single coherently registered orientation. XRD $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$5-scans show six in-plane orientations, dominated by Kurdjumov–Sachs and Nishiyama–Wassermann variants that account for $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$6 of the Re diffraction intensity, while the measured KS separation angle is $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$7 rather than the theoretical $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$8. AFM shows that Re layers thinner than $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$9 retain the smooth Nb morphology with rms roughness $\varepsilon_e(f)$0, whereas thicker films become granular and reach roughness $\varepsilon_e(f)$1 at $\varepsilon_e(f)$2; yet RHEED indicates that the in-plane strain is already only $\varepsilon_e(f)$3 at $\varepsilon_e(f)$4 and fully relaxed by $\varepsilon_e(f)$5. TEM further reveals a mixed Nb–Re interfacial region about 5 monolayers thick. The result is a modified SK sequence in which roughening occurs on top of a domain-structured, early-relaxed, chemically mixed interfacial state rather than after a long coherent elastic regime (Welander, 2010).

The Ti/Si(100) case radicalizes this point by redefining the wetting layer itself. Extensive DFT calculations show that Ti adsorption is directed by the reconstructed Si(100) dimer topology toward a surface-plus-subsurface $\varepsilon_e(f)$6–$\varepsilon_e(f)$7 motif, producing a TiSi bilayer at about $\varepsilon_e(f)$8. That bilayer reproduces the zigzag bonding motifs of C49-TiSi$\varepsilon_e(f)$9, providing an epitaxial C49(010)//Si(100) template even though bulk C54-TiSi$h_{eq}$0 is more stable by $h_{eq}$1. In this interpretation, the SK-like transition occurs because Ti wets Si until the TiSi bilayer is complete, but additional Ti does not wet the TiSi-terminated surface: single-Ti adsorption on TiSi is computed as $h_{eq}$2, whereas two Ti atoms on TiSi yield $h_{eq}$3, favoring clustering. The paper therefore recasts the “critical thickness” as a topology-directed chemical transition rather than a gradual strain-driven loss of planarity (Hückmann et al., 15 Jan 2026).

These cases correct a common simplification. Modified SK does not merely shift the thickness at which islands appear; it can alter the identity, registry, and relaxation state of the wetting layer itself.

4. Near-critical wetting-layer engineering

Near-critical semiconductor protocols convert SK from a spontaneous instability into a tunable operating window. In InAs/InP, the wetting layer is first prepared by As/P exchange during InP annealing in AsH$h_{eq}$4, which at $h_{eq}$5 and AsH$h_{eq}$6 flow $h_{eq}$7 produces $h_{eq}$8 InAs$h_{eq}$9P$h_c$0 after $h_c$1 and $h_c$2 InAs$h_c$3P$h_c$4 after $h_c$5; no QDs are observed even after $h_c$6, where the composition saturates near $h_c$7 InAs$h_c$8P$h_c$9. A second, much smaller InAs deposition of $\Delta h = h-h_{eq}$0–$\Delta h = h-h_{eq}$1 then places the system in a near-critical regime, while $\Delta h = h-h_{eq}$2 drives it into the supercritical regime with $\Delta h = h-h_{eq}$3. In this window, the model $\Delta h = h-h_{eq}$4, with a double-exponential $\Delta h = h-h_{eq}$5, captures the sharp but controllable onset of nucleation near $\Delta h = h-h_{eq}$6, and a subsequent growth interruption under As flux changes island height strongly while leaving density nearly unchanged (Berdnikov et al., 2023).

The wetting layer itself may continue to evolve long after the first SK transition. Reflectance-difference spectroscopy studies of InAs/GaAs show that growth interruption can lower the effective critical thickness from $\Delta h = h-h_{eq}$7 to $\Delta h = h-h_{eq}$8 for $\Delta h = h-h_{eq}$9 GI and $h_c$0 for $h_c$1 GI. Without GI, the system follows a two-stage trajectory in which the wetting layer thickens and then saturates; with GI, it follows a three-stage trajectory in which the wetting layer continues to evolve while QDs form and later enters a ripening regime in which large dots can deplete the wetting layer (Zhang et al., 2022). Modified SK thus includes protocols in which the wetting layer is not a fixed residue beneath islands but a dynamically re-equilibrating reservoir.

A recent device-oriented implementation on InAs/InP makes this engineering explicit. A modified SK scheme in chemical beam epitaxy inserts a $h_c$2 GaP pseudomorphic interlayer between InP buffer and InAs, uses $h_c$3 growth and long growth interruption, and tunes InAs amount and interruption time to obtain highly symmetric dots with aspect ratios $h_c$4 and densities ranging from $h_c$5 to $h_c$6. The reported optical metrics include fine-structure splittings down to $h_c$7, $h_c$8, an upper-bound to the mean low-power linewidth of $h_c$9, and a best case of $h_{eq} < h < h_c$0 (Wakileh et al., 2 Sep 2025). Here, modified SK is a deliberate route to coherence engineering rather than merely a descriptive label.

5. Surfactant-induced, on-demand, and confinement-mediated variants

Surfactants provide the most direct route to externally switching SK behavior on and off. InAs on GaAs(110) is a canonical non-SK system under normal conditions, growing in a purely two-dimensional Frank–van der Merwe mode. Introducing Bi during molecular beam epitaxy changes that growth mode to SK-like: at $h_{eq} < h < h_c$1, $h_{eq} < h < h_c$2 InAs remains 2D, $h_{eq} < h < h_c$3 marks the onset of 3D islands with low density $h_{eq} < h < h_c$4, and $h_{eq} < h < h_c$5 yields an abrupt density increase to $h_{eq} < h < h_c$6. The effect can also be triggered after growth: a coherently strained 2D layer grown without Bi at $h_{eq} < h < h_c$7 transforms into 3D islands after $h_{eq} < h < h_c$8 exposure to $h_{eq} < h < h_c$9, with the critical thickness again between $150\,\text{\AA}$0 and $150\,\text{\AA}$1. Small islands are coherent, whereas larger islands become plastically relaxed by $150\,\text{\AA}$2 and $150\,\text{\AA}$3 misfit dislocations. The surfactant therefore creates both a Bi-induced SK mode during deposition and a post-growth SK transformation, demonstrating that the 2D→3D instability can be externally activated rather than merely exceeded by continued deposition (Lewis et al., 2017, Lewis et al., 2019).

The same logic extends to nanowire sidewalls. On GaAs nanowire $150\,\text{\AA}$4 facets, Bi induces InAs 3D islands by a process resembling SK, even though these facets otherwise support only 2D shell growth. Under Bi, no 3D islands are seen at $150\,\text{\AA}$5, islands become visible at $150\,\text{\AA}$6, and by $150\,\text{\AA}$7 many structures completely encircle the nanowire core as zig-zag nanorings. Nucleation occurs at the edges of the $150\,\text{\AA}$8 facets, and the islands elongate along $150\,\text{\AA}$9 directions in the sidewall plane. This is a facet-constrained, edge-biased variant of SK, in which the critical shell thickness and the sidewall topology jointly determine the resulting hierarchy of quantum dots, V-shaped islands, and nanorings (Lewis et al., 2017).

Surfactants need not promote the 3D stage; they can also suppress or reroute it. Cu on bare Ru(0001) at $\sim 200\,\text{\AA}$0 exhibits conventional SK: a complete monolayer wetting layer forms at about $\sim 200\,\text{\AA}$1, second-layer islands appear near $\sim 200\,\text{\AA}$2, and 3D Cu islands appear above $\sim 200\,\text{\AA}$3. Under graphene, however, the same deposited Cu intercalates beneath graphene and grows by confined step-flow from graphene edges and holes, producing thickness-quantized rims instead of macroscopic 3D mounds. The authors explicitly classify this as a distinct confined step-flow mode rather than FvM, VW, or SK. This suggests an important limiting case of modified SK: the nominal strain-relief stage survives, but the islanding channel is kinetically bypassed by a surfactant-defined transport geometry (Rougemaille et al., 2019).

A related but device-oriented morphology control appears in InSbBi on InSb(001). There the Bi-containing alloy is interpreted as growing in a “2.5D growth regime, also known as the SK growth mode,” with strain-driven mounds promoted by significant Ehrlich–Schwoebel barriers. By dynamically adjusting Sb flux to keep V/III $\sim 200\,\text{\AA}$4 and controlling temperature and Bi flux, the morphology evolves from SK mounds to step-flow while achieving tunable Bi incorporation up to $\sim 200\,\text{\AA}$5. At the optimum $\sim 200\,\text{\AA}$6 and $\sim 200\,\text{\AA}$7 Bi flux, the film remains fully strained in-plane to InSb and shows clear mound morphology; by $\sim 200\,\text{\AA}$8, step-flow reappears but Bi incorporation is effectively zero (Edirisinghe et al., 14 Mar 2025). This is a modified SK scheme in which the 2.5D stage is used as a controllable intermediate between smooth homoepitaxy and droplet-dominated instability.

6. Applications, misconceptions, and broader scope

Modified SK schemes are technologically consequential because they decouple the benefits of a wetting layer from the liabilities of uncontrolled roughening. In the Re/Nb bilayer system, Re layers thinner than $\sim 200\,\text{\AA}$9 are atomically smooth with rms roughness less than $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$00, while the Re lattice is fully relaxed by about $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$01; the resulting strain-free and atomically smooth Re surface was identified as ideal for subsequent epitaxial ultra-thin oxide tunnel barriers and, more broadly, for all-epitaxial superconductor/insulator/superconductor trilayers (Welander, 2010). In Nb$\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$02Sn coating, the nucleation step with Sn/SnCl$\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$03 at $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$04 produces not only tin particles but also a tin film “resembling the surface obtained by Stranski-Krastanov growth mode,” and all nucleation attempts with SnCl$\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$05 yielded better uniformity of Nb$\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$06Sn coating than coating obtained without nucleation, which often included random patchy regions with irregular grain structure (Pudasaini et al., 2018). In telecom photonics, the modified InAs/InP schemes are already being used to suppress linewidth broadening and fine-structure splitting at source (Wakileh et al., 2 Sep 2025).

Several recurring misconceptions are corrected by this literature. Modified SK is not merely SK with a different critical thickness. The criticality may be thermodynamic, kinetic, morphology-specific, or only apparent, and $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$07, $\Delta\mu = 2\sigma a^2 \Phi + \varepsilon_e(f)$08, morphological roughening thresholds, and dislocation thresholds need not coincide (Berdnikov et al., 2023, Baskaran et al., 2011, Zhang et al., 2022). Nor does a modified scheme require a long coherent wetting layer: Re/Nb shows roughening on top of an already multi-variant, rapidly relaxed interfacial state, while Ti/Si(100) replaces the usual wetting layer with a TiSi bilayer that already encodes the preferred silicide polymorph (Welander, 2010, Hückmann et al., 15 Jan 2026). Finally, surfactants do not act in a single direction. Bi can induce SK on surfaces that otherwise remain 2D, whereas graphene can suppress the 3D-islanding stage of an SK system and replace it with confined step-flow (Lewis et al., 2017, Rougemaille et al., 2019).

The concept has also been extended far beyond conventional crystalline heteroepitaxy. At the water/graphite interface, AFM observations of a row-ordered epitaxial monolayer, followed by higher adlayers and three-dimensional dome-like structures, were explicitly described as resembling the layer-plus-island, or Stranski–Krastanov, growth mode in heteroepitaxy (Hwang et al., 2012). This suggests that the enduring value of modified SK is not a rigid taxonomy of epitaxial modes, but a transferable framework for systems in which wetting, strain or elastic frustration, interfacial energetics, and kinetic accessibility combine to produce a staged transition from planar adsorption to a higher-dimensional morphology.