Ge and Ge

Article pubs.acs.org/JPCC

Configuration-Interaction Excitonic Absorption in Small Si/Ge and Ge/Si Core/Shell Nanocrystals E. L. de Oliveira,† E. L. Albuquerque,‡ J. S. de Sousa,*,§ G. A. Farias,§ and F. M. Peeters§,⊥ Universidade Federal Rural do Semi-Á rido, Avenida Francisco Mota 572, 59625-900, Mossoró, Rio Grande do Norte, Brazil Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, 59072-970, Natal, Rio Grande do Norte, Brazil § Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, 60455-760 Fortaleza, Ceará, Brazil † ‡

ABSTRACT: The excitonic properties of Si(core)/Ge(shell) and Ge(core)/Si(shell) nanocrystals (NC’s) with diameters of ∼1.9 nm are investigated using a combination density functional ab initio method to obtain the single particle wave functions and a configuration interaction method to compute the exciton fine structure and absorption coefficient. These core/shell structures exhibit type II confinement, which is more pronounced for the Si/Ge NC as a consequence of strain. The absorption coefficients of these NC’s exhibit a single dominant peak, which has a much larger oscillator strength than the multipeaks found for pure Si and Ge NC’s. The exciton lifetime in Si, Ge, and Ge/Si shows a small temperature dependence in the range 10−300 K, whereas in Si/Ge, the exciton lifetime decreases more than an order of magnitude in the same temperature range.

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INTRODUCTION In recent years, Si and Ge nanostructures have become potentially important for many different applications because of their light-emitting/absorbing properties and their compatibility with Complimentary Metal Oxide Semiconductor technology.1−9 They can be synthesized by techniques as diverse as self-assembled growth over suitable substrates,10 ion implantation within oxide layers,11,12 and anodization and laserinduced crystallization of Si1−xGex films.13,14 Because of the high degree of solubility of both compounds, alloying is always expected. In fact, Si1−xGex nanocrystals can be grown with a reasonable degree of control of the Ge content.15,16 An interesting way of taking advantage of the properties of both Si and Ge at the nanoscale is by combining them in an ordered structure. Such multicomponent heterostructures of semiconductor NC’s are attractive because of the possibility to tune their band offsets at the interface through which one can engineer their electronic and optical properties. It was recently pointed out that Ge/Si core−shell nanowires/whiskers are promising materials for building blocks of future implementations of field effect transistors.17−19 Alternatively, core−shell nanoparticles are important because of the possibility of combining the rich physics of two-dimensional (2D) superlattices with the strong three-dimensional (3D) confinement of quantum dots (QDs).20 Much effort has been devoted to the synthesis, fabrication, and characterization of core−shell nanoparticles made of binary II−VI compounds such as CdSe/ZnS, ZnS/CdS, and CdSe/ CdS with tailored properties.21−24 Although the investigation of surface-grown Si/Ge capping-shell structures is a very active research field,25−27 the fabrication of isolated Si/Ge core−shell nanoparticles is still a challenge because of the very different © 2012 American Chemical Society

growth of Si on Ge and of Ge on Si. In the former, a high degree of intermixing is observed, and the latter is characterized by the formation of islands.28−30 However, with recently developed techniques to produce colloidal Si nanoparticles,31 we expect that Si/Ge core−shell nanoparticles will soon be synthesized in a manner similar to their II−VI counterparts. The electronic properties of Si/Ge core−shell nanocrystals (NC’s) are expected to be strongly affected by band alignment, quantum confinement, and strain. The band alignment is responsible for the relative localization of electrons and holes within the NC’s. There are two main types of band-edge alignment: (i) type I, in which the minimum of the conduction band and the maximum of the valence band are in the same material, and (ii) type II, in which the minimum of the conduction band and maximum of the valence band are in different materials. Type II confinement results in longer radiative lifetimes,32 lower exciton binding energies, and smaller exciton oscillator strengths33 as compared with type I systems. Such distinct characteristics are a consequence of the small overlap of the electron and hole wave functions, and they are useful for different types of applications: type I systems are important for light-emitting devices, whereas type II systems are of particular interest for photovoltaic devices.34 From a theoretical point of view, type I and type II confinement in Si/Ge heterostructures unleashed a strong interest in understanding their electronic properties. For instance, the growth conditions are crucial to defining the band alignment in such nanostructures. In addition, photoReceived: September 13, 2011 Revised: January 3, 2012 Published: January 4, 2012 4399

dx.doi.org/10.1021/jp2088516 | J. Phys. Chem. C 2012, 116, 4399−4407

The Journal of Physical Chemistry C

Article

Table 1. DFT and Modified Energy Spectrum near the Bandgap in Ascending Ordera Si NC h6 h5 h4 h3 h2 h1 e1 e2 e3 e4 e5 e6 a

Ge NC

Ge/Si NC

Si/Ge NC

DFT

modified

DFT

modified

DFT

modified

DFT

modified

−5.680 −5.680 −5.674 −5.556 −5.551 −5.550 −3.183 −3.180 −3.180 −3.087 −3.087 −3.069

−0.030 −0.030 −0.024 −0.006 −0.001 0.000 3.012 3.015 3.015 3.108 3.108 3.126

−5.320 −5.319 −5.317 −5.244 −5.240 −5.235 −3.068 −3.065 −3.058 −3.035 −2.984 −2.978

−0.085 −0.084 −0.082 −0.009 −0.005 0.000 2.687 2.690 2.697 2.720 2.771 2.777

−5.574 −5.534 −5.534 −5.329 −5.329 −5.323 −3.089 −3.089 −3.084 −2.991 −2.991 −2.974

−0.051 −0.011 −0.011 −0.006 −0.006 0.000 2.816 2.816 2.821 2.914 2.914 2.931

−5.436 −5.436 −5.432 −5.378 −5.378 −5.374 −3.087 −3.087 −3.086 −3.045 −3.005 −3.005

−0.062 −0.062 −0.058 −0.004 −0.004 0.000 2.891 2.891 2.892 2.933 2.973 2.973

The energy states are given in eV.

functional.49 DMOL3 uses numerical orbitals for the basis functions, and a cutoff radius of 0.47 nm was used to evaluate the matrix elements. The type of basis set used is DNP (double numerical with polarization). The core electrons are treated by means of norm-conserving pseudopotentials fitted from an allelectron DFT calculation. The adopted displacement (force) convergence criteria in the atomic relaxation procedure was 5 × 10 −3 Å (2 × 10 −3 Hartree/Å). The average bond lengths are the following: For the Si−Si bond, we obtained 2.3678, 2.3973, and 2.3707 Å in Si, Si/Ge, and Ge/Si NC’s, respectively. For the Ge−Ge bond, we obtained 2.4617, 2.4469, and 2.4557 Å in Ge, Si/Ge, and Ge/Si NC’s, respectively. For the Si−Ge bond, we found 2.4223 and 2.409 Å in Si/Ge and Ge/Si NC’s, respectively. The Si−H bond length is 1.4978, and 1.4983 Å in Si and Ge/Si NC’s, respectively. Finally, for the Ge−H bond, we obtained 1.5402 and 1.5393 Å in Ge and Si/Ge NC’s, respectively. We note that these values are average bond lengths, and the actual lengths depend on the distance to the NC center. Our values are in good agreement with the bond lengths recently obtained by Ramos et al.40 The excitonic properties are obtained by means of a configuration interaction formalism41,42 in which the exciton wave functions are expressed as linear combinations of two particle basis set Φv,c Slater determinants

luminescence measurements have shown evidence of temperature-dependent band alignment. For example, the band alignment of the dots changed from type II to type I with increasing temperature.35 In type II Si/Ge heterostructures, holes (electrons) are mainly localized in the Ge (Si) layers.36,37 These systems have been investigated using theoretical approaches as different as single-band effective mass approximation,38 k·p formalism,39 and ab initio methods.40 The exciton fine structure of spherical (e.g., InP, CdSe, Si, Ge, PbSe)41−44 and self-assembled (InGaAs/GaAs) quantum dots45 has been theoretically investigated by a few groups using an atomistic methodology that can handle structures containing up to millions of atoms (by comparison, density functionalbased methods can handle up to a thousand atoms). This methodology is flexible enough to easily include subtle effects such as spin−orbit coupling, leading to a spectacular ability to preview and explain the fine structure of many types of quantum dots.46,47 In particular, it was shown that that spin− orbit coupling leads to symmetry changes in the excitonic structure of self-assembled quantum dots.45 The aim of this work is to perform an ab initio investigation of the electronic structure of Si(core)/Ge(shell) and Ge(core)/ Si(shell) core−shell NC’s, focusing on their excitonic properties. Our method is based on the density functional theory (DFT) to obtain the single-particle states, which are used to calculate the excitonic properties within a configuration interaction framework. We also estimate the strain distribution in the interior of the NC.

Nv Nc

Ψλ(re, rh) =

∑ ∑ Cvλ, c Φv , c v

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c

(1)

where two Slater determinants Φv1,c1 and Φv2,c2 belong to the same configuration if the single-particle states are degenerate; that is, εc1 = εc2 and εv1 = εv2. The excitonic spectrum can then be obtained by solving the following set of algebraic equations:

THEORETICAL MODEL AND METHODOLOGY Our NC’s are modeled as nearly spherical by starting from one atom and adding its nearest neighbors, assuming tetrahedral coordination. A core−shell structure is obtained by defining an inner radius measured from the central atom and replacing the atoms inside this radius with the core species, resulting in an overall Td symmetry. The dangling bonds in the surface are saturated by hydrogen atoms. Our NC’s have 35 atoms in the core, 124 atoms in the shell, and 124 hydrogen atoms on the surface. This represents NC’s with a diameter of ∼1.9 nm, whereas the core region has a diameter of 1.2 nm, and the shell has a thickness of 0.35 nm. The one-electron states are determined from first principles using DFT in the generalized gradient approximation (GGA), as implemented in the DMOL3 code.48 The electron−electron interaction is described by the Perdew−Burke−Ernzerhof

Nv Nc

∑ ∑ ⎡⎣(εc − εv − Eλ)δv , v ′δc , c ′ − Jvc , v ′ c ′ + K vc , v ′ c ′⎤⎦ v′ c′

Cvλ′ , c ′ = 0

(2)

where εc and εv represents the energy states obtained with the DFT calculation. Because of the well-known DFT problem of band gap underestimation, the excitonic spectrum as calculated by eq 2 would also contain errors. The strategy to reduce these errors and still be able to provide qualitative and quantitative results is the following: Within the DFT framework, the energy 4400



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of a noninteracting electron−hole pair (also known as quasiparticle gap) Eqp is related to the Kohn−Sham HOMO− LUMO gap EHL through Eqp = EHL + Σ, where Σ represents the self-energy correction. Eqp can also be calculated by using Eqp = E(N + 1) + E(N − 1) − 2E(N), where E(N) represents the total energy of the system containing N electrons.50,51 It was recently shown that the use of the GGA-PBE functional leads to improved quasi-particle gaps in comparison to LDA functionals for pure and alloyed SiGe NC’s,52 exhibiting very good agreement with GW-corrected LDA calculations for the same systems. In an accurate model, one would obtain Eqp = EHL. Then, we rescale our DFT energy spectrum to match Eqp without changing the energy difference between adjacent states in both conduction and valence bands. The DFT and modified energy spectrum of the NC species investigated here are listed in Table 1. The quantities Jvc,v′c′ and Kvc,v′c′ in eq 2 represent the direct Coulomb and exchange energies, which are given by Jvc , v ′ c ′ = e2

∫∫

ψ*v ′(r1) ψ*c (r2) ψv(r1) ψc ′(r2) ε(r1, r2) r1 − r2

which can be conveniently evaluated in the reciprocal space as E(Ω) =

K vc , v ′ c ′ = e2

∫∫

ε(r1, r2)|r1 − r2|

where ρ1(G), ρ2(G), and g(G), are the Fourier transforms of ρ1*(r), ρ2(r′) and g(r − r′), respectively. This sum is carried out over the reciprocal lattice vectors G of the supercell with volume Ω. In particular, g(G) has the following analytical form: g (G) =

ε−1

(G) =

−

a mad =

(10)

q1q2 4π a D1·D2 + NCC(R ) mad 3Ωε0NC(R ) ε0

2π 3Ωε0NC(R )

∑ R≠0

−

(q1Q 2 + q2Q1)

(11)

erfc(R β1/2) 4π + R Ω

∑ G≠0

exp( − G2 /4β) G2

⎛ 4β ⎞1/2 π ⎜ ⎟ − ⎝π⎠ βΩ

(12)

Here, erfc is the complementary error function, β is the arbitrary Ewald parameter, and the sums are carried out over the nonvanishing real space lattice vectors, R, and reciprocal space lattice vectors, G. The first-order radiative recombination lifetime of the excitonic states, Ψλ, is obtained by using Fermi’s golden rule:

In the above equation, q = 2π (3π n0) is the Thomas− Fermi wave vector, n0 is the average valence band electron density, and ρ∞ is obtained by solving sinh(qρ∞)/(qρ∞) = NC NC ε∞ (R). The macroscopic dielectric constant, ε∞ (R), for Si and Ge NC’s was obtained by interpolating the data of Tsolakidis et al.55 with the following expression:42

4nαω3λ 1 = |Mλ|2 τλ 3c 2

(13)

where the refractive index is n = (ε0)1/2, α is the fine structure constant, c is the speed of light in vacuum, ωλ = Eλ/h̵, and Mλ represents the dipole matrix elements:

ε bulk − 1

(6)

From there, we obtained R0(Si) = 5.428, η(Si) = 3.149, R0(Ge) = NC 7.27, η(Ge) = 2.113. For core/shell NC’s, we used ε∞ = NC(core) NC(shell) [ε∞ Vcore + ε∞ Vshell]/VNC, where Vcore(shell) represents the core (shell) volume, and VNC = Vcore + Vshell. The Coulomb and exchange integrals can be cast in the following general form:

∫ ∫ ρ1*(r) g(r − r′) ρ2(r′) dr dr′

)

In the above equation, qi = ∫ ρi(r) dr, Di = ∫ ρi(r)r dr, and Qi = ∫ ρi(r)r2 dr represent the monopole, dipole, and quadrupole moments, respectively. The Madelung parameter amad is given by

(5)

E=

(

G2 + q2

dr1 dr2

1/3

1 + (R 0/R )η

NC (R ) Gρ G2 + q2 sin(Gρ∞)/ ε∞ ∞

E(̃ Ω) = E(Ω) −

⎧ εNC(R )q/[sinh q(ρ − r ) + qr ] r ≤ ρ ⎪ ∞ ∞ ∞ ε ̅ (r , R ) = ⎨ ⎪ εNC(R ) r > ρ∞ ⎩ ∞

NC(R ) = 1 + ε∞

(9)

To overcome the slow convergence of eq 8 with respect to the supercell volume Ω, we used the method proposed by Franceschetti et al.,41 in which

dr1 dr2

The Coulomb and exchange integrals involve a screening dielectric function, ε(r1, r2), that depends on the particle's position. Pseudopotential calculations have shown that the dielectric constant is significantly reduced for decreasing NC sizes.53 Moreover, it is well-known that the exchange interaction exhibits both short- and long-range regimes.54 To take into account those effects, we make use of the screening model proposed by Franceschetti et al.,41,42 in which ε(r1,r2) ≈ ε(|r ̅ 1, r2|, R), and

2

e 2 −1 4π ε (G) 2 Ω G

The analytical form of the dielectric function in reciprocal space is

(4)

1/2

(8)

G

(3)

ψ*v ′(r1) ψ*c (r2) ψv(r2) ψc ′(r1)

∑ ρ1*(G) g(G) ρ2(G)

Mλ =

∑ Cvλ, c⟨ψc|r|ψv⟩ v ,c

(14)

Finally, the average exciton lifetime is obtained with 1 −(Eλ − E0)/ kBT ∑ τ− 1 λ e = λ τ ∑λ e−(Eλ − E0)/ kBT

(7) 4401

(15)



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where E0 is the lowest exciton energy. The absorption cross section is calculated using Fermi’s golden rule as I(ω) ∝

1 VNC

moderate hole confinement in the valence band of Ge with a large barrier ΔEV. As for the Si/Ge NC, the LUMO orbital is mostly localized in the core, with an appreciable tail in the shell, and the HOMO orbital is strongly localized in the shell region. This is consistent with an electron confinement in the Si core with small barrier ΔEC in the conduction band, and hole confinement in the Ge shell with a large barrier ΔEV in the valence band. These characteristics are confirmed in Figure 2, where the radial distribution of the HOMO and LUMO orbitals are shown.

∑ |Mλ|2 δ(ℏω − Eλ) λ

(16)

where VNC is the nanocrystal volume. A line-broadening of 10 meV is considered in the calculation of the absorption coefficient.

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SINGLE PARTICLE WAVE FUNCTIONS AND STRAIN EFFECTS Figure 1a and b shows schematically the band alignment of Si(core)/Ge(shell) and Ge(core)/Si(shell) core/shell NC’s,

Figure 2. Radial distribution of the HOMO (black) and LUMO (red) states of Si, Ge, Si/Ge, and Ge/Si NC’s with 1.9 nm of diameter. The vertical dashed line indicates the position of the Si/Ge interface.

HOMO and LUMO states exhibit almost identical decay near the boundaries in Si and Ge NC’s. In addition, the HOMO states of Si and Ge NC’s are similar in the whole NC volume, but the LUMO states are different. This difference possibly arises from symmetry issues. Reboredo et al. have shown that the LUMO state of Si NC is predominantly dominated by X-valley Bloch states, whereas the LUMO state in Ge NC’s can be dominated by X, L, or mixed valleys Bloch states depending on the NC size.56 On the other hand, the orbitals in the core/shell NC’s clearly exhibit an inversion of behavior between NC center and edge. In the Si/Ge case, the LUMO state is visibly localized in the core and the HOMO state in the shell, which is consistent with a strong type II confinement. In the Ge/Si NC, type II confinement is still observed, but not as strong as in the Si/Ge one, because there is a significant leakage of the LUMO (HOMO) orbital in the core (shell) region. The strength of the type II confinement can be quantified by the term |⟨rh⟩ − ⟨re⟩|, which gives the radial dipole moment of the electron−hole pair, yielding values of 0.244, 0.517, 0.7, and 1.33 Å for Si, Ge, Ge/Si, and Si/Ge NC’s, respectively. The above orbital characteristics are in good qualitative agreement with the electron affinity confinement model which predicts ΔEC = χSi − χGe = 0.05 eV and ΔEV = χSi − χGe + EGSi − EGGe = 0.51 eV. Thus, we can conclude that strain effects are not large enough to modify qualitatively the band alignment as predicted by the electron affinity model. In fact, it was recently shown that the low lattice mismatch requirement for capping NC’s with a shell layer is not as stringent as in two-dimensional systems because the total area over which the strain accumulates is small; that is, there is much more room for strain relaxation. It is expected that strain becomes more important for nonspherical shaped NC’s that exhibit average curvatures that are intermediate between the surface of a spherical NC and a flat film.57

Figure 1. (Panels a and b) Schematics of the band edges of Ge/Si and Si/Ge NC’s estimated with the electron affinity model, respectively. (Panels c and e) Isosurface plots of the peak values of the LUMO and HOMO orbitals of Ge/Si NC’s. (Panels d and f) Isosurface plots of the peak values of the LUMO and HOMO orbitals of Si/Ge NC’s. The atomic species Si, Ge, and H are represented in yellow, green, and white, respectively.

respectively. These band edge profiles were estimated by aligning the electron affinity of bulk Si (χSi = 4.05 eV) and Ge (χGe = 4.00 eV) with respect to the vacuum level and disregarding strain effects. Within this simple picture, Ge/Si and Si/Ge NC’s exhibit type II band alignment in which the holes are localized in the Ge region, and electrons, in the Si region. However, there is a lattice mismatch of 4% between the Si and Ge lattice parameters, and consequently, this picture will be modified by strain effects. The HOMO and LUMO orbitals of Ge/Si and Si/Ge NC’s are displayed in Figure 1. In the Ge/Si NC, one can see that the LUMO state is located in the shell region but with a nonnegligible penetration in the core region, whereas the HOMO state is localized in the Ge core, with a small tail in the shell region. This is consistent with the picture of weak electron confinement in the Si shell with a small barrier ΔEC, and a 4402



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The role of strain can be calculated as follows: at k = 0, and the effective energy gap between the conduction and heavyhole bands is given by58 EG(r) = EG0 (r) + (ac(r) − av(r)) Tr[ε̂(r)]

(17)

EG (EG0 )

where is the strained (unstrained) band gap, and ac and av are the hydrostatic deformation potentials for conduction and valence bands, respectively. In addition, ε̂ is the strain tensor whose components are εij (i, j = x, y, z), and Tr[ε̂] = εxx + εyy + εzz represents its trace. In this expression, the shear component, Qε = −b(εxx + εyy − 2εzz)/2, where b is the shear deformation potential, is disregarded because of the nearly spherical symmetry of the NC’s investigated. Because the strain-induced modification in the conduction band can be well described by Ve(r) = V e(0)(r) + ac(r) Tr[ε̂(r)]

(18)

the modified valence band profile is obtained from Vh(r) = Ve(r) − EG(r)

(19)

Ve(0)

where represents the conduction band profile obtained with the electron affinity model. In the above expression, one needs to evaluate the position-dependent trace of the strain tensor Tr[ε̂(r)]. This quantity can also be calculated from Tr[ε̂] = ΔV/V0, where ΔV = V − V0 is the strain-induced volume variation with respect to the unstrained volume V0. Within an atomistic approach, one can use the tetrahedron volume formed by the four chemical bonds of Si and Ge atoms to calculate the strain-induced volume modification on each atomic site. There are two possible reference volumes, V0, for Si and Ge atoms: (i) the tetrahedron volume of the respective bulk lattice and (ii) the tetrahedron volume of each corresponding atom in the relaxed NC that is used as reference. In particular, Si/Ge (Ge/Si) core−shell NC’s are compared with pure Si (Ge) ones with the same size. We remark that this model is not able to obtain the individual components, εij, which can be derived with models such as the one of Pryor et al.59 Figure 3a shows the radial distribution of Tr[ε̂(r)] on each atomic position as a function of the distance to the central atom for NC’s with 1.8 nm of diameter. Notice that the strain in both NC’s does not exhibit significant fluctuations, meaning a lack of angular dependence of Tr[ε̂(r)]. Thus, the strain profile is almost flat in both core and shell regions, exhibiting only small variations near the surface. In the Si/Ge NC (triangles), the tetrahedra surrounding of each atomic site is expanded with respect to the reference Si NC. The volume expansion in the core (shell) is ∼3% (10%). In the case of Ge/Si (circles), a volume contraction of the core (shell) of ∼3% (10%) is obtained in comparison with the reference Ge NC. Because of the flatness of the strain profile, the conduction band will also be approximately flat. Figure 3b displays the estimated conduction and valence band profiles in the NC’s. One can see that the band edges obtained with the electron affinity model are qualitatively correct, but the difference is in the strength of the confinement barriers. In the simple electron affinity model, the confinement barrier is 0.05 eV (0.51 eV) for electrons (holes) in both NC’s. When strain effects are included, the barriers for electrons and holes are 0.16 and 0.64 eV, respectively, in the Si/Ge NC. In the case of Ge/Si, the barriers for electrons and holes are 0.34 and 0.62 eV, respectively. Thus, as compared with the electron

Figure 3. (a) Radial distribution of the strain tensor of Si/Ge (triangle) and Ge/Si (circle) NC’s. The vertical dashed line indicates the border between core and shell regions. (b) Estimated band edge profiles in the NC’s.

affinity model, the confinement barriers are (i) enhanced and (ii) different for Si/Ge and Ge/Si NC’s.

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OPTICAL AND QUASI-PARTICLE GAPS

Table 2 gives the HOMO−LUMO and quasi-particle gaps and other excitonic properties of the NC species investigated in this work, which we compare with results available in the literature. In comparison with Eqp, EHL, and Σ calculated using LDA by Melnikov et al.,51 one can see that our EHL values are in good agreement, whereas their Eqp and Σ are larger than ours. The reason for this difference relies in the fact that GGA provides a better approximation for the quasi-particle gaps of either pure or mixed Si and Ge NC’s.52 Our Eqp are also in good agreement with the single particle gaps calculated by Reboredo et al.42 using the semiempirical pseudopotential method and the tightbinding calculations of Delerue et al.60 Our optical gaps, here defined as Eop = Eqp − Jeh, where Jeh represents the ground state exciton binding energy, are also comparable to other theoretical and experimental values. We note that there are other definitions for the optical gaps. For example, Garoufalis et al. defined Eop = EHL,61 where EHL was obtained within the DFT framework using a hybrid functional (B3LYP) which minimizes the self-energy correction problem. Tsolakidis et al. defined Eop as the energy of the first well-defined peak in the absorption spectrum calculated using a time-dependent LDA−DFT model.55 Finally, Reboredo et al. defined Eqp as the lowest excitation energy obtained with a configuration−interaction method, in which the single particle states were calculated within the framework of a semiempirical pseudopotential method. The experimental optical gaps of Wolkin et al. were obtained from photoluminescence measurements.4 The exciton binding and exchange energies (Jeh and Keh) (0) shown in Table 2 were defined using different models. Jeh (1) (0) represents the unscreened electron−hole interaction, Jeh = Jeh / NC (2) ε∞ , and Jeh was calculated using the screening model of eq 5. 4403



Article eh excitation energy, obtained as Epair = E(N, e + h) − E(N), was 63 compared with the optical gaps. Here E(N, e + h) represents the total energy of the N-electrons system with an excited electron−hole pair, obtained by means of the constraint that the highest occupied (unoccupied) single-particle state of the ground state system contains a hole (electron). Their results are also given in Table 2. However, the authors have shown eh (1) 52 that the Epair ≈ EHL − Jeh , and the self-energy correction Σ must be added so that the result is comparable with the optical eh gap. Thus, if we add their Epair values to Σ in Table 2, one recovers the agreement with the optical gap calculated using Eop = Eqp − Jeh. Concerning recombination lifetimes, Ramos et al.40 obtained values almost 3 orders of magnitude larger than ours for Si and Ge/Si NC’s, but their values are of the same magnitude for Ge and Si/Ge NC’s. They argued that the recombination lifetimes could be understood by means of the atomic bond lengths, which are larger for Ge. However, this is not necessarily true because we found bond lengths that differ by J(1). This is caused by the fact that the electron−hole interaction is unscreened when |re − rh| → 0. Ramos et al. also investigated Si, Ge, Si/Ge, and Ge/Si NC’s with varying shell thicknesses using a modified DFT model that allows the calculation of the total energy of the NC’s containing an excited electron−hole pair.40 In their method, the pair

Figure 4. Exciton fine structure of absorption spectra of Si, Ge, Si/Ge, and Ge/Si NC’s. In all panels, the first column is the single excitonic state calculated for NC = NV = 1. The other columns represent the eight lowest exciton states calculated with NC = NV = 6 considering single (second column) and mixed configuration (third column). The solid (open) circle represents the lowest bright (dark) exciton state. The exchange interaction energy ΔX is defined as the energy difference between the lowest dark/bright states.

displayed. In practice, the calculation with NC = NV = 1 leads to a single exciton state with energy Eex = Eqp − Jeh + Keh, which is nearly equal to the optical gap previously defined as Eop = Eqp − Jeh. One can see that the multiple exciton configurations lead to a rich set of excitonic states whose individual energies represent transitions involving more than a single pair of single-particle states. For example, in the Si NC case, the exciton energy calculated with NC = NV = 1 solely represents the transition h1−e1, whereas the lowest exciton state (NC = NV = 6, mixed configuration) represents the transitions h1−e1 (93%), h1−e3 (4%), and h2−e2 (3%). For the Ge NC, the lowest exciton state predominantly represents h1−e1 (89%), h2−e2 (7%) transitions. 4404



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For the Si/Ge NC, we have h1−e3 (62%), h2−e1 (18%), and h3−e2 (18%). For the Ge/Si NC, we found h1−e2 (99%). Concerning the exchange interaction energy ΔX, defined as the energy difference between the lowest dark/bright exciton states, we obtained ΔX(Si) = 2.2 meV, ΔX(Ge) = 1.3 meV, ΔX(Si/Ge) = 8.25 meV, and ΔX(Ge/Si) = 5 meV. Interestingly, the lowest exciton state in Si and Ge NC’s are bright (solid circle), and dark (open circles) in Si/Ge and Ge/Si ones. Here, we define a dark exciton when its dipole moment squared, |Mλ|2, is less than 3% of the highest squared dipole moment of the excitonic spectrum. Reboredo et al. also calculated the excitonic properties of Si using a configuration interaction formalism.42 Their lowest exciton state of Si NC’s (D = 1.78 nm) is 2.68 eV, whereas we obtained 2.78 eV. They also obtained ΔX(Si) ≈ 20 meV, much larger than ours. However, if one considers the exciton exchange energy calculated within the single-particle approach, (2) ΔX = 2Keh , we obtain ΔX = 15 meV, which is in much better agreement with Reboredo’s value. We suspect that this discrepancy arises from the fact that their NC’s exhibit full Td symmetry, whereas ours slightly deviates from Td symmetry. Another additional source of error is due to the truncation of the reciprocal space grid used in our calculations. It is wellknown that the number of grid points in the reciprocal space used to represent ρ(r) in eq 7 has to be larger than the grid points used to represent a single wave function. Figure 5 shows the absorption spectra of Si, Ge, Si/Ge, and Ge/Si NC’s considering single and mixed exciton configurations. Interestingly, the mixed configurations have little effect on the exciton states and dipole transition moments, resulting in negligible differences in the absorption spectra. In this calculation, we have used Ne = Nv = 6 and we disregarded the dielectric mismatch between the NC and its surroundings. Si and Ge NC’s exhibit three main absorption peaks. As expected, the absorption gap of Ge is smaller than the one of Si. In both cases, the separation between the highest and lowest absorption peak is ∼0.2 eV. For Si/Ge and Ge/Si, the absorption spectra are quite different from pure Si and Ge NC’s. Only a single absorption peak is observed with dipole transition moments that are much larger than those of Si and Ge; moreover, the absorption peaks are blue-shifted with respect to the lowest peak of Ge. The absorption peaks of the Si/Ge and Ge/Si are very close to the lowest absorption peak of the Si NC. One can also see that the lowest exciton states in Si/Ge (2.65 eV) and Ge/Si (2.70 eV) are nearly dark, which is a consequence of the type II character of the ground electron and hole states. Concerning the origin of these absorption peaks, we identified the following transitions. The first visible peak in Si and Ge NC’s are predominantly due to transitions between the lowest states of valence and conduction band. The last peaks are due to transitions between excited states of the valence and conduction bands. The intermediate peaks are due to crossed transitions between the lowest and highest states in both valence and conduction bands. For the Si/Ge NC, a single peak is observed. This peak is predominantly due to the transition from the lowest state of the valence band to the highest states of the valence band. For the Ge/Si NC, the highest absorption peak is due to transitions from the lowest valence band states to the highest states in the conduction band. We note that enumerating individually all the transitions involved in each excitonic absorption peak is quite a difficult task because of the

Figure 5. Absorption spectra of Si, Ge, Si/Ge, and Ge/Si NC’s. Black (red) curves represents the calculation considering single (mixed) exciton configuration. The vertical lines indicate the squared dipole matrix elements, |Mλ|2. A line-broadening of 30 meV is considered in the calculation. The main absorption peaks are numbered, and the main transitions of the peak are displayed. For example, peak 1 in the Si case (transitions h1,2−e1,2) is due to the possible transitions h1−e1, h1−e2, h2−e1, h2−e2.

enormous number of possible transitions, each of them with its weight. Thus, we enumerate only the predominant transitions. For completeness, we also calculated the temperature dependence of the exciton lifetime τ. These results are depicted in Figure 6, and we found very different behavior for the different NC species. For Si, the exciton lifetime slightly increases with temperature, but it slightly decreases for Ge. For Si/Ge NC’s, the exciton lifetime decreases more than 1 order of magnitude when we pass from low to high temperatures. Interestingly, the exciton lifetime in the low temperature limit is very large, which is consistent with a type II confinement, but it decreases to the same order of magnitude of τSi at high temperatures. This behavior can be understood as follows: In the low temperature limit, the important transition for the exciton decay is LUMO → HOMO. These states are well confined in the core and shell regions, respectively, with small superposition of orbitals, thereby increasing τ. When the temperature increases, transitions from excited states in both 4405



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whereas the lifetime of Si/Ge NC decreases more than 1 order of magnitude in the same temperature range.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address ⊥

Department of Physics, University of Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium and also at ́ Departamento de Fisica, Universidade Federal do Ceará, Caixa Postal 6030, 60455-760 Fortaleza, Ceará, Brazil

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ACKNOWLEDGMENTS The authors acknowledge financial support from CNPq and the bilateral program between Flanders and Brazil and the Belgian Science Foundation (IAP).

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Figure 6. Temperature dependence of the exciton lifetime in Si, Ge, Si/Ge, and Ge/Si NC’s. Black (red) lines represents the calculated lifetimes considering single (mixed) exciton configuration.

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conduction and valence bands become important, as well. Since the conduction band confinement barrier between Si and Ge is small, the excited states are no longer confined in either core or shell and exhibit a large superposition of orbitals, thereby reducing τ. In the case of Ge/Si NC’s, we observed a negligible temperature dependence. This reinforces our previous results showing that the type II confinement in Ge/Si is weaker, resulting in exciton lifetimes of the same order of magnitude as for Ge. Finally, we found that the mixed exciton configurations are important only in the low temperature limit, whereas for higher temperatures, it is a good approximation to take the exciton states from single electron and hole states.

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REFERENCES

CONCLUSION

In conclusion, we calculated the quasi-particle gaps and excitonic properties of Si, Ge, Si/Ge, and Ge/Si NC’s by means of a DFT-based ab initio method. The radial distribution of the HOMO and LUMO orbitals show that both Ge/Si and Si/Ge NC’s exhibit type II confinement, whose oscillator strength is larger for Si/Ge. We estimated the confinement barriers using a model that can take into account strain effects. Although the electron affinity confinement model predicts the sign of the confinement barriers correctly, the quantitative values are underestimated because of strain effects that enhance the barriers in both core/shell cases. We also found the radial strain distribution (no angular dependence) and the bond lengths that are stretched in Si and compressed in Ge. This leads to different conduction and valence band edges for Si/Ge and Ge/Si, which explains the different strengths of type II confinement. Concerning simple models for the optical gaps, we found that the Eop's of Si/Ge and Ge/Si are smaller than the one for Si, and it is larger for Ge, while the absorption spectra calculated within a configuration interaction approach shows that the energy of the lowest absorption peak of Si, Si/Ge and Ge/Si are comparable. The difference relies on the fact that the dipole transition moments of Si/Ge and Ge/Si are much larger than that of Si. The temperature dependence of the exciton lifetime is also quite different between the investigated NC species. Moreover, the exciton lifetime of Si, Ge, and Ge/Si NC’s slightly changes in the temperature range of 10−300 K, 4406



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