J. Phys. Chem. C 2008, 112, 2851-2856
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Size, Dimensionality, and Constituent Stoichiometry Dependence of Bandgap Energies in Semiconductor Quantum Dots and Wires C. C. Yang* and S. Li School of Materials Science and Engineering, The UniVersity of New South Wales, NSW 2052 Australia ReceiVed: August 20, 2007; In Final Form: December 9, 2007
The bandgap energies of semiconductor quantum dots and wires are investigated with respect to the effects of size, dimensionality, and also composition using a nanothermodynamic model. The results indicate: (1) that the bandgap energy increases with decreasing the nanocrystal size for groups IV, III-V, and II-VI semiconductors; (2) that the influence of crystal size on the bandgap energy of quantum wires is weaker than that in the case of quantum dots; (3) that the ratio of ∆Eg(D, d)QW/∆Eg(D, d)QD is size-dependent, where ∆Eg(D, d) is the size- and dimensionality-dependent change in bandgap energy; (4) that the category of crystallographic structure (i.e., zinc-blende and wurtzite) appears to have a limited influence on the bandgap energy of semiconductors; and (5) that irrespective of whether or not it occurs in bulk or as nanosized semiconductor alloys, the composition effects on the bandgap energy are substantial, having a common nonlinear (bowing) relationship. These calculated results are consistent with experimental findings and may provide new insights into the influence of size, dimensionality, and composition effects on the bandgap energy of semiconductors.
1. Introduction In last two decades, a diverse range of semiconductor nanocrystals have been vigorously developed in order to exploit their physical properties for high-performance optoelectronic devices. These semiconductors include covalent Si(IV),1-4 III-V semiconductors (GaN,5,6 GaP,6-8 GaAs,9,10 InN,11 InP,12-14 InAs,13,15-19 etc.), and II-VI semiconductors (ZnS,20-24 ZnSe,25-27 ZnTe,28 CdS,13,29-35 CdSe,32,36-41 CdTe,32,42,43 etc.), which have zinc-blende (ZB) or wurtzite (WZ) crystallographic lattice structures. With properties different from their bulk properties, these nanocrystalline semiconductors possess unique electronic, magnetic, optic, catalytic, and thermodynamic properties. When the size of semiconductor nanocrystals approaches 1-10 nm and hence becomes comparable to that of their exciton diameter, quantum confinement becomes a significant determinant of their properties.44,45 Both zero-dimensional quantum dots (QDs)6,16,18 and one-dimensional quantum wires (QWs)14,19,41 have demonstrated quantization effects under these conditions. Semiconductor nanocrystalline materials have been used in many applications as advanced functional materials, especially in optoelectronic devices, e.g., laser diodes, high-speed field-effect transistors, solar cells, UV photodetectors, biomedical tags, etc.44,45 Many of these applications are related to the most identifiable aspect of quantum confinement of semiconductor QDs and QWs: the valence-conduction bandgap energy Eg increases as the diameter (D) of QDs and QWs decreases.44,45 As a result, an understanding of the size-dependence of bandgap energy is one of the most important topics in the QD and QW research community. Aside from the change of size, differences in dimensionality also results in different bandgap energies of nanocrystals due to the various dimensionalities of confinement.14,19,40,41,45 Moreover, it is believed that QWs exhibit more potential technological * Corresponding author. Fax: +61-2-93855956. E-mail: ccyang@ unsw.edu.au.
advantage over QDs, and this has been experimentally demonstrated by the observation of linearly polarized emission and lasing from QWs.19 However, in contrast to studies of QDs, studies of QWs are still in their infancy and related research publications are limited.45 However, with rapid developments in nanocrystals synthesis technologies, the size and dimensionality of nanocrystals can be manipulated in a controlled fashion. This provides the opportunity to study the dimensionality dependence of the quantum confinement effects and also the difference in bandgap structures between QDs and QWs. Recently, efforts toward new materials development have focused primarily on the fabrication of semiconductor nanocrystals of various sizes and dimensionalities.45 However, the modification of physical and chemical properties through variations in nanocrystal size may also affect the performance of optoelectronic devices on account of the variation of stability induced by the size reduction (e.g., D < 2-3 nm). Recent advances have led to the exploration of tunable optical properties in nanosized alloys by changing the constituent stoichiometries in materials such as ZnxCd1-xS and ZnxCd1-xSe (0 e x e 1 is the atomic fraction) etc., rather than by altering the nanocrystal size.46-51 It is reported that these nanoalloys have luminescent properties comparable to or even better than the best-reported binary semiconductor nanocrystals and that they are promising materials for optoelectronic devices.48 On the other hand, theoretical and experimental efforts have been primarily focused on investigating the variation in bandgap energy as a function of size.44,45,52 However, reports on the dimensionality, constituent stoichiometry, and crystallographic lattice structure (ZB and WZ structures) effects on the bandgap energies of QDs and QWs are very limited. This has become an impediment to the rapid development of advanced optoelectronic devices. Therefore, a general quantitative model needs to be developed for: (1) revealing the intrinsic factor that dominates the variation of bandgap energy in the strong quantum confinement regime and (2) calculating the bandgap energies
10.1021/jp076694g CCC: $40.75 © 2008 American Chemical Society Published on Web 02/06/2008
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of IV, III-V, and II-VI semiconductor QDs and QWs with respect to size, dimensionality, and composition effects. In this work, a simple and unified model is established to calculate the size-, dimensionality-, and composition-dependent bandgap energy of semiconductor QDs and QWs based on the cohesive energy of nanocrystals. An expanded understanding of the bandgap energy of semiconductor QDs and QWs and their related quantum confinement effects will advance the development of their applications in optoelectronic devices. 2. Methodology According to the nearly free-electron approximation, Eg originates from the crystal potential V. The width of the bandgap is dependent on the integration of V and the Bloch wave of the nearly free electron. As a result, the bandgap energy is determined by the first Fourier coefficient of the crystalline field V1, namely, Eg ) 2|V1|.4,36,52,53 This relationship is applicable for both bulk and nanosized materials. Therefore, ∆Eg(D,d)/ Eg(∞) ) |∆V(D,d)/V(∞)|, where (1) Eg(∞) is the bandgap energy of bulk materials (∞ denotes the bulk), (2) V(∞) is bulk crystalline field, (3) ∆Eg(D, d) ) Eg(D, d) - Eg(∞) expresses the size- and dimensionality-dependent energy change with the dimensionality d ) 0 for QDs and d ) 1 for QWs, and (4) ∆V(D, d) ) V(D, d) - V(∞) is the corresponding crystalline field change. Recently, size-dependent properties of nanocrystals were investigated by studying the relationship between the cohesive energy Ec and physical-chemical properties. The results show that the variation of the potential profile for nanocrystals, which is related to the crystallographic structures and the corresponding transition functions, can be determined by the change of Ec.54 Moreover, since Ec is related to the total coordination number of a particular atom and also the interatomic interaction, it provides the crystal potential. Hence, Ec(∞) ∝ V(∞) and Ec(D, d) ∝ V(D, d).4,36,52,53 With above considerations, we have: |∆V(D,d)/V(∞)| ) |∆Ec(D,d)/Ec(∞)| ) 1 - Ec(D,d)/Ec(∞) and the size- and dimensionality-dependent Eg(D, d) function can be expressed as follows:
Eg(D, d)/Eg(∞) ) 2 - Ec(D, d)/Ec(∞)
(1)
Combining the developed Ec(D) function54,55 and the consideration of dimensionality effects, the Ec(D, d)/E(∞) function in eq 1 can be written as
Ec(D,d) Ec(∞)
{
) 1-
}
1 × 6D/[(3 - d)h] - 1 2Sb 1 exp 3R 6D/[(3 - d)h] - 1
{
}
(2)
where Sb ) Eb/Tb is the bulk solid-vapor transition entropy of the crystals as determined by the bulk solid-vapor transition enthalpy Eb and the solid-vapor transition temperature Tb, R is the ideal gas constant, and h denotes the atomic or molecular diameter. From eqs 1 and 2, we have
Eg(D,d) Eg(∞)
{
)2- 1-
}
1 × 6D/[(3 - d)h] - 1 2Sb 1 exp 3R 6D/[(3 - d)h] - 1
{
}
(3)
For the bulk semiconductor alloys, the Eg(x, ∞) value is located within the range from Eg(0, ∞) to Eg(1, ∞). The
Figure 1. Eg(D, d) of Si QDs. The solid line denotes the model prediction from eq 3. The symbols b,1 4,2 3,3 ×,4 and [4 are experimental results.
corresponding Eg(x, ∞) function can be determined by the Fox equation:56
1/Eg(x, ∞) ) (1 - x)/Eg(0, ∞) + x/Eg(1, ∞)
(4)
For semiconductor nanoalloys, since the structure and optoelectronic properties of the binary semiconductors in III-V or II-VI are similar, the Eg(x, D, d) function is given as
1/Eg(x, D, d) ) (1 - x)/Eg(0, D, d) + x/Eg(1, D, d) (5) 3. Results and Discussions Figures 1-3 plot the calculation and experimental results of Eg(D, d) for Si, III-V, and II-VI semiconductors with the configurations of QDs and QWs. The related parameters used in the modeling are listed in Table 1. It is discernible that the calculated Eg(D, d) values are in good agreement with the experimental data even when D < 3 nm, thereby confirming the accuracy of the developed model. As shown in the figures, the Eg(D, d) function increases with a decreasing D value and the quantum confinement effect is pronounced when D ( 10 nm, R approaches 0.67, which is consistent with the above experimental and other theoretical results. 3.3. Bandgap Energies of ZB and WZ Structures. In order to compare the influence of crystallographic structure on the bandgap energy of semiconductor nanocrystals, we plot the β ) ∆Eg(D, d)WZ/∆Eg(D, d)ZB values of CdSe QDs and QWs in Figure 5. It demonstrates that the nanocrystal size effects on β can be divided into two segments, namely, the size-dependent and size-independent regions. From the figure, we can see that β decreases with increasing D sharply and subsequently approaches a saturation value of 0.83 when D > 10 nm. In this case, the bandgap energies determined by eq 3 for these two crystallographic structures are similar despite differences in the
wave function symmetry and the fine structures of the energy spectrum for ZB and WZ structures. This implies that the crystallographic structure effects on the bandgap energy of CdSe QDs and QWs are negligible, and that the bandgap is dominated by the parameters of size and dimensionality. 3.4. Bandgap Energies of Bulk and Nanosized Alloys. Experimental findings in the field of semiconductor alloys have led to the conclusion that the Eg(x, ∞) function varies smoothly and monotonically (but not linearly) with x changing over the whole range of composition.46-51,60 The Eg(x, ∞) function is often described by an empirical formula, Eg(x, ∞) ) Eg(0, ∞) +[Eg(1, ∞) - Eg(0, ∞) - b]x + bx2, where b is named the bowing parameter.45,46,60 The value of b is a measure of the fluctuation magnitude of the crystal field or the nonlinear effect caused by the anisotropic nature of binding. According to Hill and Richardson theory,61 b is composition-dependent when the alloy components have different lattice constants. Moreover, the value of b is difficult to measure experimentally, with limited means of calculation using theoretical methods.45,46,60 Except for the above empirical formula, the Vegard formula has also been used to determine the Eg(x, ∞) function, where Eg(x, ∞) ) xEg(1, ∞) + (1 - x)Eg(0, ∞).48,50,51 However, the calculated results did not match the experimental data.45,46,60 Figure 6 plots the results calculated using eq 4 and the experimental data for the Eg(x, ∞) function of bulk ZnxCd1-xS and ZnxCd1-xSe alloys.
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Figure 4. ∆Eg(D, d)QW/∆Eg(D, d)QD of CdSe. The solid and dashed lines denote the model predictions from eq 3 for ZB and WZ structures, respectively.
Figure 3. Eg(D, d) of II-VI semiconductor QDs and QWs. The solid lines denote the model predictions from eq 3, and the symbols are experimental results. (a) ZnS QDs: [,20 b,21 3,22 4,23 O.24 (b) ZnSe QDs: [,25 O,26 4.27 (c) ZnTe QDs: [.28 (d) CdS QDs: 2,13 [,29 +,30 3,31 4,32 O,33 b,34 ×.35 (e) CdSe QDs: ×,32 [,36 3,37 O,38 4.39 (f) CdSe QWs: [,40 b.41 (g) CdTe QDs: [,32 4,42 O.43
Figure 5. ∆Eg(D, d)WZ/∆Eg(D, d)ZB of CdSe. The solid and dashed lines denote the model predictions from eq 3 for QDs and QWs, respectively.
TABLE 1: Parameters Used in the Model Predictions Si GaN (WZ) GaP (ZB)c GaP (ZB)c GsAs (ZB) InN (WZ) InP (ZB) InAs (ZB) ZnS (ZB) ZnSe (ZB) ZnTe (ZB) CdS (ZB) CdSe (ZB) CdTe (ZB) CdS (WZ) CdSe (WZ)
Eg(∞) (eV)
h (nm)a
Sb (J mol-1 K-1)b
1.124 3.45 2.78(direct)8 2.22(indirect)8 1.63345 2.044 1.4552 0.952 3.1623 2.5858 2.2658 2.4258 1.7458 1.6143 2.3046 1.6747
0.235258 0.162445 0.23737 0.23737 0.244745 0.180145 0.254245 0.262445 0.234359 0.245559 0.264159 0.252059 0.262059 0.280659 0.210345 0.219245
144 13R 13R 13R 13R 13R 13R 13R 13R 13R 13R 13R 13R 13R 13R 13R
a h ) a2/(6c) + c/4 and h ) ((3)1/2/4)a for WZ and ZB structures, respectively, with a and c being the corresponding lattice constants. b For Si, Sb ) Eb/Tb with Eb ) 456 kJ mol-1 and Tb ) 3173 K.57 For III-V and II-VI semiconductor compounds, Sb ≈ 13R as a first-order approximation, which is equal to that of the mean value of most elements (70-150 J mol-1 K-1).57 c Note that the direct and indirect Eg(∞) values of GaP are different.
The calculated results obtained using the empirical formula (b ) 0.2246 and 0.4547 eV for ZnxCd1-xS and ZnxCd1-xSe, respectively) and those derived from the Vegard formula are also plotted in Figure 6 for comparison. It can clearly be seen that the variation of Eg(x, ∞) deviates slightly from a linear dependence, displaying a downward bowing relationship. This
Figure 6. Eg(x, ∞) of bulk ZnxCd1-xS and ZnxCd1-xSe alloys. The solid, dashed, and dotted lines denote the model predictions from eq 4, empirical formula, and Vegard formula, respectively. The symbols [46 and O47 are experimental results of bulk ZnxCd1-xS and ZnxCd1-xSe alloys, respectively.
demonstrates that the results calculated with the developed model (eq 4) are in good agreement with the experimental data and also the empirical formula. However, the unique advantage of eq 4 is that it does not have any adjustable parameters in the equation, thereby substantially simplifying the calculation of Eg(x, ∞) for semiconductor alloys. Figures 7 and 8 plot Eg(x, D, d) as a function of constituent stoichiometry x for ZnxCd1-xS and ZnxCd1-xSe QDs, as calculated using eq 5 for different nanocrystal sizes. Experimental data are also included in Figures 7 and 8 for comparison. From these figures, it can be observed that the Eg(x, D, d) plots have downward shifts as D increases and that the bandgap energy increases with increasing x. It can hence be seen that
Semiconductor Quantum Dots and Wires
J. Phys. Chem. C, Vol. 112, No. 8, 2008 2855 energies of other II-VI and III-V semiconductors if the relevant parameters are available. The parameters used in the developed model have distinctive physical meanings, which makes it possible to reveal the physical and chemical nature behind the properties. 4. Conclusions
Figure 7. Eg(x, D, d) of ZnxCd1-xS nanoalloy QDs. The solid lines denote the model predictions from eq 5. The symbols 4 (4.8 nm),48 O (5.8 nm),48 b (6.4 nm),48 [ (8.0 nm),48 and 3 (20 nm)49 are experimental results.
A thermodynamic quantitative model was developed to calculate the bandgap energies of semiconductor quantum dots and wires on the basis of the size and dimensionality dependence of the cohesive energy. It was found that (1) the bandgap energy increases with decreasing nanocrystal size; (2) the nanocrystal size effects on the bandgap energy in quantum wires are weaker than those in quantum dots; (3) the ratio of ∆Eg(D, d)QW/∆Eg(D, d)QD increases from 0.67 to 1 with decreasing nanocrystal size; (4) the structure effects on bandgap energy are negligible while the composition effect in semiconductor alloys is substantial; and (5) a similar nonlinear (bowing) relationship between bandgap energies and the composition is discernible in both bulk and nanosized alloy systems. The developed model is in good agreement with experimental data. It reveals that the cohesive energy change is an intrinsic factor in determining the size, dimensionality, and composition effects on the bandgap energy of semiconductors. Acknowledgment. This project is financially supported by Australia Research Council Discovery Program (Grant No. DP0666412). References and Notes
Figure 8. Eg(x, D, d) of ZnxCd1-xSe nanoalloy QDs. The solid lines denote the model predictions from eq 5. The symbols [,50 O,50 4,50 3,50 b,50 and ×51 are experimental results.
the results obtained from eq 5 are consistent with experimental findings. As shown in Figures 7 and 8, the composition effects on the bandgap energy of nanoalloys are much stronger than that of the size effects, in particular for the mixture of narrowgapped and wide-gapped semiconductors. A comparison of the results in Figure 6 with the data in Figures 7 and 8 demonstrates a similar nonlinear (bowing) behavior in both bulk and nanosized alloy systems. This is because ZnS and CdS, as well as ZnSe and CdSe, have similar lattice constants and properties. These findings are in agreement with recent experimental observations.48,50 The size-, dimensionality-, and composition-induced variation of bandgap energy are common phenomena in semiconductor nanoalloys. The findings of this study indicate that the bandgap structure has the potential for quantitative modification through the manipulation of the size, dimensionality, and composition of QDs and QWs precisely. Such an understanding shall prove essential for the future application of QD and QW nanostructures of these materials in optoelectronic devices. Moreover, the developed model could also be used to calculate the bandgap
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