Structural properties of self-organized semiconductor nanostructures

Instabilities in semiconductor heterostructure growth can be exploited for the self-organized formation of nanostructures, allowing for carrier confinement in all three spatial dimensions. Beside the description of various growth modes, the experimental characterization of structural properties, such as size and shape, chemical composition, and strain distribution is presented. The authors discuss the calculation of strain fields, which play an important role in the formation of such nanostructures and also influence their structural and optoelectronic properties. Several specific materials systems are surveyed together with important applications.

Figure 1

The $2\times 1$ surface reconstruction of $\mathrm{Si}\left(001\right)$: top panel, side view; lower panel, top view. In the lower panel, $S_{\mathrm{A}}$ and $S_{\mathrm{B}}$ steps are sketched parallel and perpendicular to the dimer rows, respectively. After 429 with permission from Elsevier. Copyright 2001.

Figure 2

Scanning tunneling microscope (STM) images of $\mathrm{Si}\left(001\right)$ with $4°$ miscut in the $\left[110\right]$ direction prior and after Ge deposition: (a) filled state of pure $\mathrm{Si}$; (b) filled state after deposition of $0.02\mathrm{ML}$’s of $\mathrm{Ge}$; (c) empty-state image of the area shown in (b). $\mathrm{Ge}$ incorporation leads to the formation of buckled dimer rows (indicated by boxes and ovals). From 287.

Figure 3

STM pictures $(350\times 350{\mathrm{nm}}^2)$ demonstrating the movement of steps during step-flow growth of $\mathrm{Si}\left(001\right)$: panel (a) the starting morphology before growth; panels (b) and (c) steps at $\mathrm{Si}$ coverages of 1.2 and 3.5 ML's, respectively. From 429, reprinted with permission from Elsevier. Copyright 2001.

Figure 4

Evolution of a stepped surface calculated using the strain model for step bunching: (a) time evolution of the average bunch size; (b) growth time expressed in arbitrary units. Adapted from 405.

Figure 5

Atomic force microscope (AFM) images of the surfaces of $10\times (3\text{−}\mathrm{nm}\mathrm{Si}∕30\mathrm{nm}{\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x)$ superlattices grown at various temperatures and with various $\mathrm{Ge}$ content. The direction of the substrate miscut is denoted by the arrow. The step bunch structure is not affected by the misfit stress (proportional to $x_{\mathrm{Ge}}$) but depends only on growth temperature. From 261.

Figure 6

$\mathrm{Si}∕\mathrm{Ge}$ layers: (a)–(c) STM images at different magnifications of a tensile strained $\mathrm{Si}$ layer deposited on a graded $\mathrm{SiGe}$ buffer layer. From 152. (d) AFM image of a $\mathrm{Ge}∕\mathrm{Si}$ surface (size $1600\times 1600{\mathrm{nm}}^2$) showing a triangular instability of $S_{\mathrm{A}}$ steps caused by the compressive stress in the $\mathrm{Ge}$ layer. The white spots are 3D $\mathrm{Ge}$ islands nucleated at the apexes of the $S_{\mathrm{A}}$ triangles. From 50.

Figure 7

STM images of the surface evolution during growth of a $\mathrm{Ge}$ layer on $\mathrm{Si}\left(001\right)$. In panels (a)–(d), the $\mathrm{Ge}$ coverage grows from 2.8 to $4.0\mathrm{ML}$’s. During growth (a)–(c) first mounds (prepyramids) develop, then these convert (d) to $\left\{105\right\}$ facetted pyramids. From 421.

Figure 8

Dependence of the island energy on volume (a) and the corresponding cross sections of $\mathrm{SiGe}$ islands calculated for various island volumes (b). For volumes between $V_1$ and $V_3$ two (meta)stable shapes exist—a shallow prepyramid without facets and a facetted pyramid. The pairs of solid circles in (b) denote the side facet. From 406.

Figure 9

The volume dependence of the energy necessary to create a pyramidal island, calculated for three different orientations of the facets. Energy and volume are normalized to the critical values of $\left\{105\right\}$ facetted pyramids. Adapted from 404; 33.

Figure 10

The size distribution of several island types occurring during deposition of $\mathrm{Ge}$ on $\mathrm{Si}\left(001\right)$. The inset shows the time evolution of the island density for different types. Adapted from 421.

Figure 11

Typical $\mathrm{Ge}$ island shapes obtained by STM during $\mathrm{Si}$ capping of $\mathrm{Ge}$ domes grown on $\mathrm{Si}\left(001\right)$: (a),(b), domes; (c) pyramids; (d)–(f) prepyramids. The $\mathrm{Si}$ coverages are 0, 1, 2, 4, 8, and $16\mathrm{ML}$’s for panels (a)–(f). From 290.

Figure 12

(Color in online edition) Facetting of $\mathrm{InAs}$ islands (a) sketch of facets of an InAs islands on $\mathrm{GaAs}\left(001\right)$ obtained by STM; (b) and (c) zoom of the reconstructed $\left\{\overline{1}\overline{3}7\right\}$ facet; (d) structure model of facet reconstruction. From 237.

Figure 13

Size distributions of domes and pyramids calculated by means of equilibrium statistics (lines) for two different $\mathrm{Ge}$ coverages $\Theta$, and obtained from AFM (black dots). From 317, reproduced courtesy of University of California, Lawrence Livermore National Laboratory, and the U.S. Department of Energy.

Figure 14

Shape transition of $\mathrm{Ge}$ or $\mathrm{SiGe}$ islands grown on $\mathrm{Si}\left(001\right)$ during growth (solid arrow), postgrowth annealing (dotted arrow), and $\mathrm{Si}$ capping (dashed arrow). The solid curves represent the critical volumes for pyramids and domes. From Rastelli, Kummer, and von Kaenel, 2002, reprinted with permission from Elsevier. Copyright 2002.

Figure 15

AFM images of the surfaces of the ${\mathrm{Si}}_{0.25}{\mathrm{Ge}}_{0.75}$ layer in a $\mathrm{SiGe}\left(25\mathrm{\AA }\right)∕\mathrm{Si}\left(100\mathrm{\AA }\right)$ superlattice and their power spectra (insets in upper right corners): (a) the first period (at the substrate); (b) the 20th period of the superlattice. From 398. The inset in the upper left corner of panel (b) shows the distribution of the surface stress that affects the chemical adatom potential. Buried islands give rise to deep stress minima above them. In addition, shallow minima appear between the deep minima, if the island distance is large enough. From 407.

Figure 16

$\mathrm{InAs}∕\mathrm{GaAs}$ island layers: (a) cross-section TEM images of $\mathrm{InAs}∕\mathrm{GaAs}$ double layers with spacer thickness of $46\mathrm{ML}$’s; (b) spacer thickness of $92\mathrm{ML}$’s; (c) perfectly correlated $\mathrm{InAs}∕\mathrm{GaAs}$ superlattice with $36\text{−}\mathrm{ML}$-thick $\mathrm{GaAs}$ spacers. From 439.

Figure 17

Vertical pairing probability $P_P$ calculated by a kinetic Monte Carlo, method: (a) dependence on the number of deposited layers for constant spacer thickness of $15\mathrm{ML}$’s; (b) dependence on coverage for constant spacer thickness of $10\mathrm{ML}$’s; (c) dependence on spacer thickness for a coverage of 0.35 and 20 periods. From 247.

Figure 18

(Color in online edition) Surface distribution of the chemical potential of an adatom calculated for various materials and surface orientations. The insets show the calculated 2D distributions, the main graphs depict the calculated distributions along the arrows in the insets. Only the relative change in the chemical potential with respect to the value for a homogeneously strained epitaxial layer is plotted. The light color in the insets corresponds to a minimum of the chemical potential.

Figure 19

Cross-sectional TEM images of $\mathrm{PbSe}∕\mathrm{PbEuTe}$ superlattices with different thicknesses of the $\mathrm{PbEuTe}$ spacers: (a) $37\mathrm{nm}$; (b) $45\mathrm{nm}$; (c) $60\mathrm{nm}$. The inset in (b) shows the hexagonal in-plane arrangement of islands. From 371.

Figure 20

Dependence of the stacking type on the PbSe island size (a) and on PbEuTe spacer layer thickness (b). The insets (c), (d) show the energy distribution on the surface for two different spacer thicknesses. From 371.

Figure 21

AFM images of the surface of a $\mathrm{Ge}$ layer grown on lithographically prepatterned $\mathrm{Si}\left(001\right)$ substrates. In the sample shown at top left, the islands are arranged into a regular array along two orthogonal $<110>$ directions. In the sample shown at bottom right, the unit vectors of the 2D array of pits are oriented along $<100>\text{and}<110>$ directions, leading to a $45°$ island alignment. Reprinted with permission from 455, © 2004 American Institute of Physics.

Figure 22

$\mathrm{PbSe}$ islands with $\left\{001\right\}$ facets: (a) AFM image of the top surface of a $\mathrm{PbSe}∕\mathrm{PbEuTe}$ island multilayer grown on ${\mathrm{BaF}}_2\left(111\right)$; (b) polar plot, showing the angular distribution of surface normals.

Figure 23

Lateral island correlation: (a) AFM image $\left(3\times 3\mu {\mathrm{m}}^2\right)$ of the top surface of a $\mathrm{PbSe}∕\mathrm{PbEuTe}$ island multilayer; (b) Fourier transform of the image; the white line indicates the $\left[11\overline{2}\right]$ direction; (c) the corresponding line profile with maxima due to island position correlation; (d) the autocorrelation function. Islands are arranged in a regular array up to the sixth-nearest neighbor.

Figure 24

Conventional topography (a) and ultrasonic force microscopy (c) images of $\mathrm{SiGe}$ islands on $\mathrm{Si}\left(001\right)$. (b) and (d) are the respective line profiles. The island height in (a), and (c) is about $15\mathrm{nm}$. From 179.

Figure 25

Cross-sectional scanning tunneling microscopy: (a) XSTM image of a stack of $\mathrm{InAs}$ islands in $\mathrm{GaAs}$; (b) the thickness of the $\mathrm{GaAs}$ spacers as a function of lateral position; (c) comparison between a measured and simulated height profile for a similar sample; (d) the electronic wave function measured at two different tip biases, compared to simulations for the ground state and the first exited state. Panels (a) and (b) from 29. Panels (c) and (d) from 105.

Figure 26

Lattice parameter in growth direction in an $\mathrm{InAs}$ island: $∎$, obtained from cross-sectional STM; solid line, obtained from a finite-element simulation assuming an In content increasing from island base to island apex. From 30.

Figure 27

(Color in online edition) Strain distributions ${\epsilon }_{xx}$ obtained from TEM images of $\mathrm{InGaAs}$ islands in $\mathrm{GaAs}$ using the method of digital analysis of lattice images (DALI). From 189.

Figure 28

Lattice fringe analysis of a high-resolution TEM image of an $\mathrm{InGaAs}$ layer embedded in $\mathrm{GaAs}$. From 310, reprinted with permission of Elsevier. Copyright 1998.

Figure 29

Energy-selective imaging (ESI): (a) conventional TEM bright-field image of $\mathrm{InAs}$ islands in $\mathrm{GaAs}$; (b) In concentration map obtained from ESI; (c) line profiles of In concentration along the growth direction averaged over the stripes indicated in (b). From 431.

Figure 30

Sequence of low-energy electron microscope (LEEM) images during capping of $\mathrm{SiGe}$ islands with $\mathrm{Si}$: prior to $\mathrm{Si}$ deposition, and at 4.6-, 9.2-, and $13.8\text{−}\mathrm{ML}$ $\mathrm{Si}$ cap. The image size is $1.5\times 1.5\mu {\mathrm{m}}^2$. From 391.

Figure 31

Reciprocal-space sketch of a general scattering experiment. For a detailed description see the text.

Figure 32

Scattering geometry in grazing-incidence small-angle x-ray scattering (GISAXS): ${\mathbf{k}}_i$ is the incident beam with an angle ${\alpha }_i$ with respect to the sample surface, ${\mathbf{k}}_f$ is a (diffusely) scattered beam in the direction to the detector, ${\mathbf{k}}_s$ denotes the specularly reflected beam. The in-plane scattering angle is $2\theta$ and ${\alpha }_f$ is the exit angle of ${\mathbf{k}}_f$ with respect to the surface. Note that ${\mathbf{k}}_f$ does not lie in the plane spanned by ${\mathbf{k}}_i$, ${\mathbf{k}}_s$, and the surface normal. The upper left inset depicts the accessible range in reciprocal space.

Figure 33

GISAXS intensity distribution for two different sample orientations. Satellite peaks due to the positional correlation are visible as well as facet streaks due to island shape. From 331.

Figure 34

GISAXS spectrum of a sample with triangular $\mathrm{Ge}$ pyramids on a $\mathrm{Si}\left(111\right)$ surface: dots; -, kinematical approximation; solid line, full DWBA simulation; dotted line, contribution to the DWBA from the wave reflected at the sample surface prior to scattering in the islands. From 297.

Figure 35

Lateral island correlation: (a) in-plane GISAXS intensity distribution of a $\mathrm{Si}∕\mathrm{SiGe}$ island multilayer, recorded at constant ${\alpha }_i=0.69°$ and ${\alpha }_f=0.26°$, resulting in a penetration depth of about $900\mathrm{nm}$, i.e., illuminating all multilayer periods; (b) the Fourier transform of (a), showing different degrees of spatial island order in different azimuths; (c) and (d) line profiles through (b) for small and large distances $r$, respectively. A fit to the former yields information on island size, while the latter reveals positional correlation.

Figure 36

Scattering geometry in x-ray reflectivity (XRR) in real space (top) and reciprocal space (bottom). ${\mathbf{k}}_i$ is the incident beam with an angle $\omega$ with respect to the sample surface, ${\mathbf{k}}_f$ is a (diffusely) scattered beam in the direction to the detector, ${\mathbf{k}}_s$ denotes the specularly reflected beam. The scattering angle is $2\theta$, ${\mathbf{k}}_i$, ${\mathbf{k}}_f$, ${\mathbf{k}}_s$, and the surface normal lie in a common plane. The insets at top left depict the accessible range in reciprocal space for XRR and x-ray diffraction (XRD). In XRD the same geometry is used, only the angles $\omega$ and $2\theta$ are larger. In XRR, instead of $\omega$ and $2\theta$ often the incidence angle ${\alpha }_i=\omega$ and the exit angle ${\alpha }_f=2\theta -\omega$ are used in the description.

Figure 37

XRR reciprocal-space map from a rough $\mathrm{Si}∕\mathrm{SiGe}$ multilayer exhibiting resonant diffuse scattering sheets: (a) measured; (b) after correction for refraction. From 128.

Figure 38

$\mathrm{SiGe}$ island layer grown by liquid-phase epitaxy on $\mathrm{Si}\left(001\right)$: (a) measured (004) reciprocal-space map; (b) and (c) simulations with different assumptions on the vertical $\mathrm{Ge}$ distribution. From 433.

Figure 39

(Color in online edition) $\mathrm{Ge}$ content and strain profile obtained from x-ray diffraction reciprocal-space maps for uncapped dome-shaped islands and after capping with $\mathrm{Si}$ at $T=550°\mathrm{C}$. The insets sketch the different island shapes. From 126.

Figure 40

(Color in online edition) Comparison of $\mathrm{Ge}$ contents along island height obtained from x-ray diffraction experiments (solid line, data from 126) with results from TEM using digital analysis of lattice images obtained for the island center (dashed line, data from 332.

Figure 41

Reciprocal-space maps (symmetric diffraction 111) measured for $\mathrm{PbSe}∕\mathrm{PbEuTe}$ multilayers with three different arrangements of $\mathrm{PbSe}$ islands. From 132.

Figure 42

(Color) Four scattering processes from first-order perturbation theory sketched in side view (top panels) as well as in 3D (center panel). The resulting intensity distribution is sketched in the lower panels. The interference of beams of the four scattering paths leads to a characteristic shift of the primary maximum along $Q_z$ for each isostrain volume at a certain height above the substrate surface (left); with increasing distance from the reciprocal lattice point of the substrate along $Q_r$, the width of the intensity distribution along $Q_a$ increases (right). From 165, 166.

Figure 43

(Color) In-plane strain with respect to the substrate (left panels) and $\mathrm{Ga}$ content in the center of $\mathrm{InGaAs}$ islands as obtained from the isostrain-scattering method, for islands grown at two different temperatures. From 166.

Figure 44

Vertical island correlation: (a) sketch of the diffuse scattering sheets in grazing-incidence diffraction due to a $\mathrm{SiGe}$ island multilayer with vertically correlated island positions, together with a measured intensity distribution; (b) increase of the half-width along $q_z$ as a function of lateral momentum transfer $q_{\Vert }$ [relative to the (220) Bragg point] of a resonant diffuse scattering sheet due to the imperfect vertical stacking of $\mathrm{SiGe}$ islands. From 168.

Figure 45

Decrease of the degree of vertical correlation $P$ of island positions in a $\mathrm{Si}∕\mathrm{SiGe}$ multilayer as a function of $\mathrm{Si}$ spacer thickness, calculated from the rms deviation of island positions in subsequent layers ${\sigma }_{\perp }$. The solid symbols are results from a grazing-incidence diffraction experiment, open symbols are results from TEM. From 379.

Figure 46

Anomalous x-ray scattering: (a) anomalous corrections to the atomic scattering factor $f^{\prime }$ and $f^{″}$ as a function of energy around the $\mathrm{Ge}$ $K$ edge; (b) momentum-transfer dependence of the real part of the atomic scattering factor for $\mathrm{Si}$ and for $\mathrm{Ge}$ for two energies indicated in (a); (c) intensity ratio for scattering at the two energies—solid line, pure $\mathrm{Ge}$; dashed line, ${\mathrm{Si}}_{0.1}{\mathrm{Ge}}_{0.9}$; dotted line, ${\mathrm{Si}}_{0.3}{\mathrm{Ge}}_{0.7}$. From 342.

Figure 47

Anomalous diffraction from $\mathrm{SiGe}$ islands: (a) radial scans in grazing-incidence diffraction geometry around the (800) Bragg reflection: $\times$, recorded at $11043\mathrm{eV}$; $●$, recorded at $11103\mathrm{eV}$; $∎$, intensity ratio; (b) lateral dimension; and (c) $\mathrm{Ge}$ content of $\mathrm{SiGe}$ islands and lattice parameter as a function of island height, obtained from anomalous x-ray scattering. From 342.

Figure 48

Photocurrent spectra of $\mathrm{InAs}$ islands embedded in a $p\text{−}i\text{−}n$ structure at different reverse biases, showing a strong quantum confined Stark effect. Peaks due to the interband transitions in the dots are visible. The inset shows the polarization dependence of the $0-\mathrm{V}$ spectrum at $300\mathrm{K}$ ($E_{\Vert }$ is parallel to the growth plane, $E_{\perp }$ is along the growth direction). From 90.

Figure 49

(Color in online edition) Photoluminescence (PL) spectra from $\mathrm{SiGe}$ hut clusters overgrown at different energies. Inset: the PL line position saturates for $T_{\mathrm{growth}}<360°\mathrm{C}$. From 65.

Figure 50

Photoluminescence from dome-shaped $\mathrm{SiGe}$ islands as a function of growth temperature during $\mathrm{Si}$ capping. From 333.

Figure 51

(Color in online edition) Strain tensor components ${\epsilon }_{xx}$ (left) and ${\epsilon }_{xx}$ (right) in and around a $\mathrm{Ge}$ pyramid buried below a $\mathrm{Si}$ surface. The strain distributions in the upper row have been calculated assuming an infinite isotopic $\mathrm{Si}$ matrix [Eqs. (27, 29)]. In the middle row an anisotropic infinite continuum was assumed [Eqs. (25, 29)]. In the lower row the strain calculation in a semi-infinite anisotropic continuum was performed using Eqs. (32, 35). The step of the contours is $\Delta \epsilon =0.002$. The upper edges of the graphs correspond to the free surface.

Figure 52

Strain tensor components ${\epsilon }_{xx}$ and ${\epsilon }_{zz}$ calculated along the growth axis of a tenfold stack of $\mathrm{Ge}$ islands in a multilayer.

Figure 53

Strain tensor components along the axis of a square $\mathrm{InAs}$ pyramid with $\left\{110\right\}$ side facets buried in $\mathrm{GaAs}$: solid lines, obtained by valence force field approach; dashed lines, obtained by finite-element method. Panels (a)–(c) show the strain distributions, panels (d)–(f) the differences of both curves. The strains are defined with respect to unstrained $\mathrm{InAs}$. From 285.