ARTICLE pubs.acs.org/JPCC
Band Gap Tunability in Semiconductor Nanocrystals by Strain: Size and Temperature Effect Ziming Zhu,† Ai Zhang,† Gang Ouyang,*,†,‡ and Guowei Yang§ †
Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of the Ministry of Education, Department of Physics, Hunan Normal University, Changsha 410081, Hunan, People's Republic of China ‡ School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore § State Key Laboratory of Optoelectronic Materials and Technologies, Nanotechnology Research Center, School of Physics & Engineering, Sun Yat-sen University, Guangzhou 510275, Guangdong, People's Republic of China ABSTRACT: We put forward a theoretical model to elucidate the impact of strain on the electronic band gap tunability of semiconductor nanocrystals both in self-equilibrium state and stimulated by thermal approach from the perspective of atomistic origin based on the recently developed relationship between bond length and bond energy. Theoretical analyses pointed out the band gaps are significantly modified with changing solid size and temperature due to the development of a large lattice strain and found that the strains originated from thermal approach and that in a selfequilibrium state play the competitive role for the band offsets. Agreement between our theoretical predictions and experimental observations and simulations provides a new insight into the physical understanding on the electronic and optical properties of semiconductor nanocrystals.
1. INTRODUCTION Strain is of crucial impact on properties of semiconductor nanocrystals such as electronics, optics, structural phase transition, and so on, which is not only of great interest but also attractive technologically for applications such as light emitting diodes and photovoltaic and laser diodes.13 Essentially, strain is a consequence of changing the intrinsic bond length and modifying the energy levels of the bonding electrons,1,2,4 and its result can lead to the densification and localization of charge, energy, and mass in the area of surface or interface of nanostructures,57 which would create a perturbation to the total Hamiltonian and further influence on the electronic band structure.79 In general, with shrinking the solid size of a material to nanoscale, the strain will be spontaneous taken place due to high ratio of uncoordinated atoms located at the surface and associated with higher energy state in comparison with the bulk counterpart.1015 Thus, the nanosolids will be in self-equilibrium state, and the strain can be determined on the minimal total strain energy principle.12 Importantly, the strain in nanosolids has a great influence on their properties, which has been verified by a large number of investigations both experimentally and theoretically. For example, Bazhanov et al.16 indicated that the surface relaxation, stability, and electric performance of transition metal carbide films can be strongly affected by both compression and tensile strain. Smith and co-workers2 reported the coherence strain resulting from the lattice mismatch would effectively regulate the opticalelectronic properties in the colloidal nanostructures. Yang et al.17 theoretically studied the CdSe/CdTe r 2011 American Chemical Society
core/shell nanowires and found that the interface strain has a great contribution on the modulation of band gap using firstprinciples calculations. Also, we have recently established an analytic model to address the band gap shift in SnO2 nanodots and nanowires in self-equilibrium state based on the relationship between bond length and bond energy correlations.18 Therefore, it is clearly seen that the energy band gap of semiconductor nanocrystals would expand as a function of size reduction because of the imperfect coordination and strain in the selfequilibrium state. In addition, apart from the nanosolids in the self-equilibrium state, the strain can also be induced by external fields approach such as thermal approach, pressure, the epitaxy layer, and so on,2,1923 and it plays a significant role in the related electronic and optical performances.18,2429 For instance, Li et al.22 found that the optical properties of CdSe nanocrystals can be effectively tuned with hydrostatic pressure. Also, Zhang and co-workers23 experimentally investigated the large effects of interface strain on near-band-gap photoluminescence of GaN epilayers grown on different substrates. In particular, it can be obtained that the band gap of nanocrystals and corresponding bulk demonstrate a similar variation trend with temperature, and both of them exhibit obvious red shift with increasing the temperature in various experiments. Theoretically, there are a variety of models
Received: January 29, 2011 Revised: February 28, 2011 Published: March 18, 2011 6462
dx.doi.org/10.1021/jp2009644 | J. Phys. Chem. C 2011, 115, 6462–6466
The Journal of Physical Chemistry C
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to describe the temperature effect of band gap in semiconductor nanostructures,19,2729 as summarized in eq 1. 8 > E0 RT 2 =ðT þ βÞ > > < E 2R ðexpðΘ =TÞ 1Þ 0 B B Eg ðD, TÞ ¼ E0 Shpwi½cothðhpwi=2kTÞ 1 > > > : E0 RT 2 =ðT þ βÞ þ ΔEð0Þ þ γðTECfilm TECsub ÞT where E0 is the band gap at 0 K, and R and β are both the given coefficient for a special solid. RB refers to the interaction strength between electron and average phonon; ΘB corresponds to the average phonon temperature. S and Æpwæ represent a dimensionless coupling constant and an average phonon energy, γ and ΔE(0) are both strain related terms, and TECfilm and TECsub denote the thermal expansion coefficient (TEC) of film and substrate. However, the theoretical basis of Varshni’s equation is rather weak because parameter β related to Debye temperature may be negative in some cases.27 Logothetidis et al.28 suggested a BoseEinstein-type expression based on the temperature dependence of shift in band gap induced by electronphonon interaction.30,31 In fact, Logothetidis’s equation fits somewhat better than Varshni’s under low temperatures. Moreover, O’Donnell et al.19 addressed an extended expression from Varshni’s model, which shows a reasonable basis for the phonon influence on band gap. Furthermore, taking the thermal effect into consideration between film and substrate, Lee et al.29 recently proposed a new resolution to describe the temperature effect of band gap, which is also an empirical equation although it has given better fit than the conventional Varshni approach. Although these studies reported significant strain in the lowdimensional systems, few explanations were provided for how strain influences the band gap shift from the perspective of atomistic origin that it is connected with bond identifies (bond length and bond strength). In order to better understand potentially useful strain-induced electronic property tuning by size and temperature effect, in this paper we focus on the response of band gap shift to strain in two parts: one is the selfequilibrium strain and the other is from thermal strain.
2. PRINCIPLE It is generally known that the atoms in surface shell of nanocrystals with coordination imperfection and different energy state compared with the core interior.5 The surface atomic distance will shrink spontaneously and the bond strength becomes stronger.7,8 Thus, it makes the total energy be optimized and the nanocrystals be in the self-equilibrium state.12 In this case the self-equilibrium state can be calculated from the condition of extreme value of total strain energy. Conventionally, the bond strain of a nanocrystal can be written as ε ¼ d=d0 1
ð2Þ
where d* and d0 are the mean bond length of a nanocrystal and that of the bulk, respectively. Actually, under the condition of thermal approach, the strain is from two contributions, including ε(ΔD) from self-equilibrium strain and ε(ΔT) from thermal strain. In accordance with our previous considerations,4,18 the ε(ΔD) has been deduced by taking into account a shellcore configuration for nanocrystals. Thus
ðVarshniÞ ðLogothetidis et al:Þ ðO0 Donnell et al:Þ ðLee et al:Þ
εðΔD, i Þ ¼ ci 1 εðΔD Þ ¼
specific atom layer 1
1þ
ð1Þ
2d0 ðns D
∑
ci Þ
1
total strain
i < ns
ð3Þ where ci, zi, and ns are the coefficient of bond contraction, the effective CNs of the specific ith atom, and the number of surface atom layers, respectively. In light of the bond length and bond strength correlations,7,8 the bond contraction coefficient ci = 2/(1 þ exp((12 zi)/8zi)). In addition, in the case of free nanocrystals under thermal approach, the thermal strain is Z T ð4Þ RðD, TÞ dT εðΔT Þ ¼ 0
Here R(D,T) is the TEC determined by solid size and ambient temperature. Physically, the TEC is defined by R = (1/l0)(∂l/ ∂T), where l0 is the bond length in bulk at 0 K. Considering the joint effects from size effect and thermal approach, the TEC is given by32 RðD, TÞ ¼
1 Dl 1 DU=DT ¼ ¼ ÆηðD, TÞæ=½ςðlÞEb l0 DT l0 DU=Dl
ð5Þ
with
Z CV kB T 3 ΘD =T x4 ex ÆηðD, TÞæ ¼ ¼9 dx ÆzæNa Æzæ ΘD ðex 1Þ2 0
where U, Æηæ, Æzæ, Na, and kB denote the potential function between two atoms, the mean heat capacity per bond of an atom, the mean coordination numbers (CNs) of an atom, Avogadro’s number, and Boltzmann's constant, respectively. Because the restoring force F(l) = (∂U/∂l), we set F(l)l0 = ς(l)Eb. Note that ζ(l) is a prefactor dependence on bond length and Eb is single bond energy in equilibrium state. Importantly, it should be noted that the Debye temperature is a significant parameter to describe the thermal effect process in the above discussion. According to the Debye temperature consideration under the applied stimuli,33 the thermal strain could be determined. Additionally, due to the CNs of surface shell of a nanocrystal imperfection, in this case, we can assume the shellcore configuration and define the mean CNs expressed as hzi ¼
1 Nc ð zc, i þ N i¼1
∑
Ns
∑ zs, j Þ j¼1
where Nc and Ns are the summation of interior atoms and the surface atoms. zc and zs are the CNs of lattice and surface shell, respectively. Thus hzi ¼ ð1 γi Þzb þ γi hzs i 6463
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where γi = τhci/D is surface-to-volume ratio, h is an atomic diameter in the bulk, τ is shape factor of a spherical nanodot (τ = 6), a nanowire (τ = 4), and a thin film (τ = 2).9 On the other hand, in terms of band theory,9 the band gap (Eg) is approximately determined by the first Fourier coefficient of the crystal field and in proportion to the mean cohesive energy (ECOH) per bondÆE0æ Eg ÆE0 æ ¼ ECOH =ðNÆzæÞ
ð6Þ
Meanwhile, it is worthwhile to point out that the single bond energy is from the minimum of potential energy in the equilibrium state, any disturbance will cause the atomic bond strengthening or weakening and lead to the total band structure modification. Evidently, in the case of a nanocrystal under the condition of ambient temperature, the perturbations can be divided into two contributions: one is the size effect on potential energy; the other is the thermal energy. Considering the discrepancies between the surface shell and the core interior, the cohesive energy of a spherical nanocrystal can be deduced as ECOH ðD, TÞ ¼
Ni zi Ei s þ zb EC ðN ∑ Ni Þ ∑ i
