Historically, the growth of thin films has been categorized into three types：(1) Frank-Van der Merwe (FM) (2D layer-by-layer), (ii) Volmer-Weber (VW) (3D islands), and (3) Stranski-Krastanow (SK) (2D layer followed by 3D islands). In Fig.
2-1, we show three fundamental growth features in growing thin films materials.
These modes are deduced from equilibrium considerations of the surface and interface energies of lattice matched systems. In most practical applications the epitaxy of semiconductors profits from the existence of conditions where the layer-by-layer deposition mechanism (FM) occurs. This mechanism is typically realized for nearly lattice-matched combinations (i.e. <1%) with high interfacial bond energies, low supersaturation to suppress 3D nucleation, such a deposition manner are particularly favorite at high temperature on a slightly lattice-mismatched substrate. On the other hand, the SK growth mode operates in relatively higher mismatched systems (i.e.
~2-10%) in which strained films can grow in registry with the substrate until reaching a critical thickness, tc. At this thickness the accumulated elastic strain energy initiates the formation of dislocations and the strain can be partially relaxed by the formation of a dislocation network and/or relieved partially through the formation of 3D islands.
Combinations of highly mismatched (>10%) and dissimilar materials, Au/NaCl, on the other hand, preferentially crystallize in the Volmer-Weber mode, forming islands or clusters on the bare unwetted surface.
2-1 Quantum Dots Growth Mechanisms
The primary factors that determine the island growth manner of deposition of epitaxial film on substrate are the surface free energy of the substrate (σsubstrate), the surface free energy of the deposited film (σfilm), and the interface strain energy (σinterface). The interface strain energy includes the interface energy (σif) due to lattice mismatch between substrate and deposited film and the strain energy (σst) due to induced strain caused by wetting layer and island film itself, which is increased with the increasing thickness of wetting layer. Table 2-1 list the required surface energy conditions for the SK and the VW growth modes. For both island growth modes, the sum of the surface free energy of the deposited film and their interface strain energy has to be greater than the surface free energy of substrate.
Table 2-1: Surface energy conditions of VW and SK growth modes
Surface energy conditions
VW mode σsubstrate<σfilm + σinterface, (σinterface=σif+σst(t), t<one lattice layer) SK mode σsubstrate<σfilm + σinterface, (σinterface=σif+σst(t), t>one lattice layer)
As shown in Fig. 2-2, the primary difference between these two modes is the thickness of wetting layer. If the required wetting layer thickness to produce island growth is greater than one lattice layer, the associated island growth mode is categorized into SK growth mode, if not, VW mode.
Surface energy
On the surface between the vapor and solid phases, the surface energy per unit area γs can be given approximately by：
(
1 w u)
HvoN023s = − Δ
γ (2-1) where u is the number of nearest neighbors of an atom in the bulk of the crystal and w is the number of nearest neighbors in the crystal of an atom on the surface in question.
So, w/u means the number of bonds which connect a surface atom to atoms in the crystal, and (1-w/u) means the number of dangling bonds of an atom on the surface.
v0
ΔH is the enthalpy of evaporation of the material, and N0 is the number of atoms per unit volume. The argument used is that the surface energy is the energy to break all of the nearest-neighbor bonds across a given plane. The number of atoms per unit surface area Ns can be related to N0 as follows:
3 2
N0
Ns = (2-2) for a III-V wurtzite-type compound, Ns is given by:
3 2
2
Ns = a (2-3) for the (0001) face, where a is the lattice constant of the III-V binary compound. For growth from vapor, ΔHv0is given by the enthalpy of evaporation per mole Δ as H Stringfellow’s model written by :
5 .
−2
=
ΔH Ka (2-5) a is the lattice constant of the binary compound and the value of K is 1.15×107
cal/mole-Å2.5. Therefore, the surface energy σ can be written as follows: QD), and α is the reconstruction ratio of dangling bonds on the surface. A part of dangling bonds on the surface makes bonds with each other on the surface. The number of dangling bonds decreases as α increases.
Interface energy
In order to calculate roughly the energy σif of the interface between the film and the substrate (FM and SK modes) or that between the cluster and the substrate (the VW mode), the bonding ratio at the interface was calculated. When the lattice constant of the film and cluster a is larger than that of the substrate asub (a>asub), a is given by
where Δa /a means the lattice misfit between the film (or cluster) and the substrate.
In this calculation, as shown in Fig. 2-3, we assumed that at the interface all dangling bonds on the film and cluster side are bonding with dangling bond on the substrate side. At the interface, assuming the bonding ratio β1 on the upper film or cluster side is given by
where this means that the fraction 1/k of the dangling bonds are satisfied. The
where γsub is the surface energy per unit are of the substrate. Therefore, the interface energy σif can be written as follows:
For the calculation of the strain energy, the method developed by Nakajima et al., is used in order to calculate the precise stress distribution of an island on top of a substrate. In this method, the island is divided into many imaginary thin layers with coherent interfaces. Shear lag analysis is then used to calculate the longitudinal stress distribution over the imaginary thin layers in the FM, SK, and VW mode structures.
The total strain energy of each structure σst which includes the strain energy of the layer, cluster, buffer layer, and substrate is given by
∑
where Ui is the elastic strain energy in the ith imaginary thin layer, m is total number of thin layers which constitute each structure, and σi, E , i A and i d are the i stress (N/m2), Young modulus (GPa), surface area (m2), and thickness (m) of the ith thin layer.
2-2 Photoluminescence (PL), Temperature Dependent of PL Spectra
The photoluminescence (PL) was used for measurement of the optical properties of GaN and InN nanodots in this thesis. It was known that the PL is a powerful and non-destructive technique to probe the optical emission properties of materials, especially in luminescent semiconductors. The auto mapping PL system is also widely used in the industry to monitor the quality od semi-finished devices on wafer. By analyzing the PL spectra, a set of characteristic spectral features can identify the impurity types, the band gap energy, one can estimate the contents in compound semiconductors. PL analysis can also survey buried interfaces of heterostructures which are difficult to be probed by direct physical and electrical contacts. However, it is difficult to find the correlation between the spectral line intensity and concentration of the specific impurity, due to variation of non-radiative recombination through deep-levels or surface recombination centers. The luminescence process typically involves three steps: excitation, thermalization, recombination. The electron-hole pairs generated by incident light, through quasi-thermal equilibrium distribution, will recombine and produce photos. The impurities and defects can form various energy levels in the band gap, and the corresponding energies will be estimated by radiative recombination process or absorbed by non-radiative recombination process. The transition rates of these impurities are different due to various matrix elements and density of states at respective energy levels. The luminescence of semiconductors can be divided into three regions: the excitonic edge emission, the donor-acceptor pair emission, and deep level relate emission.
A relation for the variation of the energy gap (Eg) with temperature (T) in semiconductors is proposed [28]:
)
Most of the variation in the energy gap with temperature is believed to arise from the following two mechanisms:
(1) A shift in the relative position of the conduction and valence bands due to the temperature-dependent dilatation of the lattice. Theoretical calculations show that the effect is linear with temperature at high temperatures. In that region this effect accounts for only a fraction (about 0.25) of the total variation of the energy gap with temperature. At low temperature the thermal expansion coefficient is nonlinear with T; indeed for a number of diamond structure solids it even becomes negative over a certain temperature interval. Correspondingly the dilatation effect on the energy gap is also nonlinear.
(2) The major contribution comes from a shift in the relative position of the conduction and valence bands due to a temperature-dependent electron lattice interaction. Theoretical treatments show that this leads to a temperature dependence of the following form:
T
Eq. (1) is consistent with the theoretical results if we assume that β ≈ . The θ constants in eq. (1) were evaluated from the experimental data for a member of semiconductors and are recorded in Table 2-2.
Table 2-2: Values of the parameters in Varshini’s equantion
Values of the parameters in Varshini’ equation
Substance E0 (eV) α (×10-4) β Debye θ
( )
0KDiamond 5.4125 -1.979 -1437 2220
Si 1.1557 7.021 1108 645
Generally, bandgap energies of semiconductor decreases with increasing temperature, namely, E0(T)= E0 −αT2/(β +T), where α and β are known as Varshini’s fitting parameters. For wurzite GaN, it has been reported that α=0.909 meV and β=830K for the temperature variation of A-exciton transition. This shows that the temperature-induced red shift of transition energies is about 60 meV between 0K and 300K.
The temperature dependence of the PL integrated intensity of films was generally sued to identify the mechanism of the PL quenching. In the bulk GaN (D0,X) process, the thermal dissociation of donor bound excitons involves two activation energies, namely the localization energy and exciton binding energy [29-30]. At low temperature, the free excitons tend to localize at the neutral donors (localization energy) so that there is interplay between the localization and ionization of neutral donors to reduce the number of available neutral donors. In this study, we also use the following formula to fit our results,
(2-12)
where I(T) and I(0) are the integrated intensity at temperatures T and 0 K. C1 and C2
are fitting parameters. Ea and Eloc are the activation energies at the high and low temperature regime, respectively.
The localization state may be one of below cases: (1) Interface fluctuations particularly in narrow QWs, produce localization states for excitons and that such localization states act like QD. (i.e. localization of excitons in either QDs or potential minima in QWs). (2) Localization of the free exciton at neutral donors. (3) Detrapping of excitons from interface roughness fluctuation. (4) Potential fluctuations can be realized, for example, by CdSe islands with size fluctuations etc.
For AlGaN/GaN MQWs case, From N Grandjean et al. report [31], taking into account the competitive non-radiative channel introduced by the dislocations, the PL intensity is given by Eq (2-12). The best fit of the experimental data is obtained with Ea1 = 125 meV and Ea2 = 12 meV. The 12 meV activation energy is consistent with the localization energy and accounts for the detrapping of excitons from interface roughness fluctuation. The activation energy of 125 meV corresponds to the carrier thermal escape out the well.
Fig. 2-1 Schematic diagram of typical films growth Frank-van-der Merwe (FvdM) mode, Volmer-Weber (VW) mode and Stranski-Krastanow (SK) mode.
Frank- van derMerwe d
Volmer-Weber Stranski-Krastanow
- Layer by layer growth - Lattice matched - GaN on AlGaN
- Direct island growth - Large lattice mismatched - GaN on sapphire
- Layer by layer followed by island nucleation - Dissimilar lattice spacing
Fig. 2-2 (a) schematic geometries of a strained film on substrate for (a) SK mode and (b) VW mode.
1 lattice layer
(a) (b)
Fig. 2.3 Model of the interface energy.