Chapter 4 Phase Diagrams
4.7 Bandgap Energy of Binary Nanosemiconductor Alloys
Now size-tunable properties are a hallmark of quantum dots and related nanostructures due to the potential applications to optoelectronics, high-density memory, quantum-dot lasers, vivo imaging, and lately biosensing and biolabeling. Size also plays a role when nanocrystals must be incorporated into larger superstructures such as mesoporous materials in photovoltaics [30]. However, the tuning of physical and chemical properties only by chang-ing r could cause problems in many applications, particularly, if unstable small particles (less than ∼ 2 nm) are used. One effective way to solve the problem of dual requirements is to employ alloy nanocrystals. As a valid method, alloying could tune the spectrum continuously since the interatomic interactions among different elements or compounds are different. Actually, tuning bandgap of nanocrystals by alloying could result in high lumines-cence and stability compared to the case for the component with narrower bandgap. An example to illustrate the advantages of alloying is the com-parison of (ZnSe)x(CdSe)1−xwith CdSe nanocrystals. A proven strategy for increasing the luminescence and stability of CdSe nanocrystals is to grow a thin inorganic layer of a wider bandgap semiconductor on the surface of the core nanocrystals, where r is very small. By contrast, r of (ZnSe)x(CdSe)1−x
nanocrystals increases and thus the nanocrystals have better thermal stabil-ity with high crystallinstabil-ity (or less defects). Note that subscript “x” shown in the alloy composition denotes the fraction of the first component in the alloy with 0 < x < 1. For example, blue-emitting (ZnSe)x(CdSe)1−x nanocrystals have particle sizes of over 3.5 nm, which is 3 times larger than that of the blue-emitting CdSe/ZnS nanocrystals. In addition, the stronger ZnSe bond stabilizes the weaker CdSe bond, and the shorter bond length of ZnSe intro-duces stiff struts into the system. Both lead to an increase of the dislocation energy. The addition of ZnSe into the CdSe lattice results in an increased covalency and a reduced ionicity, thus inhibiting plastic deformation and the generation of defects. Moreover, although inorganic-capped CdSe can provide a potential step for electrons and holes originating in the nanocrystals, and re-duce the probability of the carriers to migrate to the sample surface, spatial compositional fluctuations in (ZnSe)x(CdSe)1−x nanocrystals can produce atomically abrupt jumps in the chemical potential that can further localize free exciton states in the crystalline alloy.
After the above alloying, Eg(x, r) with a bowing behavior induced by in-teratomic interactions is however a nonlinear function of x [31], which leads to difficulty for materials design because the Eg(x, r) value could be determined only by experiments. If the Eg(x, r) function is a linear one, a certain spec-trum could be predicted only by that of two components with the additive rule. Note that for semiconductor alloys, the component could be compounds as shown in the above.
4.7 Bandgap Energy of Binary Nanosemiconductor Alloys 149
On the basis of assumption that a bulk ternary semiconductor alloy or psedo-binary semiconductor compound alloy is a regular solution of compo-nents, Eg(x,∞) has been given by [32],
Eg(x,∞) = xEg(0,∞) + (1 − x)Eg(1,∞) + ω(x, ∞)x(1 − x). (4.56) According to a virtual-crystal approximation model [33], ω(x,∞) essentially originates from the intersubstitutional crystalline structure with the normal CN . Up to now, although some experimental results have been brought about, the theoretical way to quantitatively determine ω(x,∞) values is still imma-ture. Theoretical Eg(x, r) functions of alloys cannot be determined either.
As r decreases from bulk, surface/volume ratio of the compound increases while the component and the structure are retained. Under this condition, Eg(0, r) function varies continuously as r drops. This continuity of Eg(0, r) function terminates at the lowest limit value of r0(0), as stated early. Thus, r can vary from ∞ to r0(0) continuously. The cases for Eg(1, r) and ω(x, r) functions should be similar. Under the above considerations, Eg(x, r) function can be obtained from Eq. (4.56) by simply substituting∞ by r, namely,
Eg(x, r) = xEg(0, r) + (1− x)Eg(1, r) + ω(x, r)x(1− x). (4.57) Eg(0, r) function of unary nanosemiconductor in terms of Ec(0, r) function has been described in Sec. 3.5 without any adjustable parameter, which is shown in a little modified form as
Eg(0, r) In the original work, the calculation is deduced for the solid-vapor transi-tion of particles where d = 0 is defined. To suit a more general case for all low dimensional materials, r0(0) = (3− d)h(0) is used in Eq. (4.58). To balance this modification, a constant of 12 is added in Eq. (4.58). At the smallest crystal size 2r0(0) = 2(3− d)h(0), Eg(0, r)/Eg(0,∞) → 2. Thus, Eg(0, r) is an ascending function with r. Equation (4.58) is simple since only two variables of ΔSb(0,∞) and h(0) are needed in order to predict Eg(0, r)/Eg(0,∞) or Eg(1, r)/Eg(1,∞) value when 0 in the function is sub-stituted by 1. Moreover, ω(x, r) function has been determined in Eq. (4.27), namely, ω(x, r)/ω(x,∞) = 1 − 2r0(x)/r where r0(x) = (3− d)h(x) and h(x) = xh(0) + (1− x)h(1). h is less dependent on composition concerned here with data given in Table 4.4 since both components are in the same group in the Periodic Table of Elements.
Substituting Eqs. (4.58) and (4.27) into Eq. (4.57) gives rise to an analytic Eg(x, r) function,
Eg(x, r) = xEg(0, r)+(1−x)Eg(1, r)+[1−2r0(x)/r]ω(x,∞)x(1−x). (4.59) In order to determine ω(x,∞) values of IIB–VIB semiconductors, Eg(x,∞) functions of some bulk pseudo-binary IIB–VIB chalcogenide semiconductors
150 Chapter 4 Phase Diagrams
are plotted in Fig. 4.13. The unknown ω(x,∞) values of WZ-(CdS)x(CdSe)1−x, WZ-(ZnS)x(CdS)1−x, and WZ-(ZnSe)x(CdSe)1−x used in Eq. (4.59) are de-termined from Fig. 4.13 (WZ denotes the wurtzite structure).
Fig. 4.13 Eg(x,∞) functions of some bulk pseudo-binary IIB–VIB chalco-genide semiconductors of experimental (symbols) and fitting results (curves). (a) WZ-(ZnSe)x(CdSe)1−x, (b) WZ-(ZnS)x(CdS)1−x, (c) WZ-(CdS)x(CdSe)1−x. The mean ω(x,∞) values for WZ-(ZnSe)x(CdSe)1−x, WZ-(ZnS)x(CdS)1−x, and WZ-(CdS)x(CdSe)1−x are respectively identified as -0.92 eV, -1.01 eV and -0.50 eV by quadratic curve fitting. The curves are shown in terms of Eq. (4.27) using the mean ω(x,∞) values derived from quadratic curve fitting and Eg(0,∞) and Eg(1,∞) val-ues listed in Table 4.4. (Reproduced from Ref. [29] with permission of Wiley-VCH)
Figure 4.14 shows comparisons of Eg(x, r) functions of some homoge-neously alloyed pseudo-binary IIB–VIB chalcogenide semiconductor nanopar-ticles with r < 10 nm (d = 0) in terms of Eq. (4.59) and available experimental results while Fig. 4.15 presents the related cases for the nanocrystals at r 20 nm (d = 0 or d = 1).
It is shown that Eg(x, r) functions increase on deceasing r, similar to the cases of Eg(0, r) and Eg(1, r). An obvious increase in Eg(x, r) is observed for
4.7 Bandgap Energy of Binary Nanosemiconductor Alloys 151 Table 4.4 The relevant data used in the calculations of Eq. (4.59)
ΔSb(∞)/
Structure (J·g-atom−1 Lattice constant/nm h/nm Eg(∞)/eV
·K−1)
ZnSe ZB 68.53 a= 0.567 0.245 2.72
GaN ZB
the size range of r < 10 nm shown respectively in Fig. 4.14 while this increase is less obvious for r 10 nm shown in Fig. 4.15. For a given r, Eg(x, r) shifts higher from the narrower bandgap sideα1(x = 0) to the wider bandgap side β1(x = 1) with the increase of x. This blue shift [or increase in Eg(x, r)] is ascribed to the formation of alloyed nanocrystals via the intermixing of the wider bandgapβ1with the narrower bandgapα1. Moreover, bowing shape of Eg(x, r) curves drops with r; in contrast to the prediction curves with r 10 nm in Fig. 4.15, an almost linear relationship as x varies is interestingly observed for the curves shown in Fig. 4.14 especially when r < 2.5 nm.
For clarity, Fig. 4.16 shows ω(x, r) of IIB-VIB chalcogenide semiconduc-tors as function of r in terms of Eq. (4.59), which determines the bowing shape of Eg(x, r) curves. Corresponding to the bowing shape of Eg(x, r) curves in Fig. 4.14 (r < 10 nm) and Fig. 4.15 (r 10 nm), ω(x, r) ≈ ω(x, ∞) when r 10 nm, but it lowers clearly as r < 5 nm. Especially, ω(x, r) → 0 when r → 2r0(x) ≈ 1.4 nm for d = 0 or r → 2r0(x) ≈ 1 nm for d = 1. Now Eg[x, 2r0(x)] function deteriorates into V´egard’s Law where about a half of atoms are located on the particle surface, which leads to evident enhance-ment of solubility due to the lack of elastic energy induced by solute atoms within a solution.
Although Eg(x, r) curves determined experimentally have a bowing shape, which leads to difficulty of materials design, the bowing shape of Eg(x, r) curves becomes weak and deteriorates into an almost linear function of x as r decreases. This is especially true when r < 2.5 nm at d = 0 or 1. Since the requirement of miniaturization of electronic and optic parts or devices is gradually enhanced, a smaller r value is asked in materials design where ω(x, r) → 0. This result suggests that when r → 2r0(x), even if we do not know the exact ω(x,∞) value, we can still determine Eg(x, r) function of the alloys in terms of V´egard’s Law. Note that at r > 10 nm (Fig. 4.15), the size effect on Eg(x, r) is much weak, and ω(x, r)≈ ω(x, ∞) in this regime. In this
152 Chapter 4 Phase Diagrams
case, Eq. (4.56) should be directly used to estimate Eg(x, r) value.
Fig. 4.14 Eg(x, r) functions of zinc blende ZB–(CdS)x(CdSe)1−xnanoparticles (d
= 0) of Eq. (4.59) (curves) and experimental results (symbols). α and β denote the compounds with narrower bandgap at x = 0 and wider bandgap at x = 1, respectively. r values are selected based on known experimental results. Different r values are shown respectively in (a)–(e). The mean ω(x,∞) = -0.53 eV is taken for simplicity. h(x) = xh(0) + (1− x)h (1). For ΔSb(0,∞), ΔSb(1,∞), Eg (0,∞), Eg (1, ∞), h (0) and h (1) values see Table 4.4. (Reproduced from Ref. [29] with permission of Wiley-VCH)
4.7 Bandgap Energy of Binary Nanosemiconductor Alloys 153
Fig. 4.15 A comparison of Eg(x, r) functions of nanostructured materi-als between Eq. (4.59) (curves) and experimental results (symbols) for (a) ZB-(CdSe)x(CdTe)1−x (d = 0), (b) WZ-(ZnS)x(CdS)1−x(d = 0), (c) ZB-(CdS)x(CdSe)1−x(d = 0), (d) WZ-(CdS)x(CdSe)1−x(d = 1) and (e) ZB-(ZnSe)x(CdSe)1−x (d = 1). r values are selected based on known experimental results. (Reproduced from Ref. [29] with permission of Wiley-VCH)
154 Chapter 4 Phase Diagrams
Fig. 4.16 ω(x, r) of some pseudo-binary IIB–VIB chalcogenide semiconductors as function of r in terms of Eq. (4.59). r0(x) is calculated by the relationship r0(x) = (3−d)h(x) with d = 0 or d = 1. The averaged h values of all cited IIB–VIB chalcogenides semiconductor compounds are about 0.24 nm as shown in Table 4.4.
For simplification, this value will be taken as a rough estimate of h to determine the corresponding 2r0(x). (Reproduced from Ref. [29] with permission of Wiley-VCH)
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