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Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures
Superlattices and Microstructures xxx (2017) 1e8
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Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures Delong Han a, Han Ye a, *, Zhongyuan Yu a, **, Yunzhen Zhang a, Yumin Liu a, Yinfeng Li b a
State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China b Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 July 2017 Received in revised form 29 August 2017 Accepted 2 September 2017 Available online xxx
In this paper, the influences of composition on Raman scattering from Ge/Si-GeSi coreshell nanowire heterostructures standing along [011] and [111] crystal directions are numerically investigated. Uniform, linear and spontaneous nonlinear composition profiles (CPs) in GeSi alloy shell are taken into consideration. In uniform CP case, clear double peaks in Raman spectra contributed by core and shell are observed. The strain-induced shift follows linear relation with Ge concentration and nonlinear relation with shell thickness. Larger strain-induced shifts are obtained in nanowires along [111] direction. In linear CP case, the peaks contributed by shell cannot be distinguished in the total spectra and plateaus are formed on the low frequency side. Moreover, the nonlinear CP accounts for the spontaneous composition transition near heterointerface during lateral epitaxy of GeSi shell. Due to the rapid Ge concentration transition, Raman spectra are shown nearly identical to uniform CP cases. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Raman spectrum Core-shell nanowire heterostructure GeSi alloy Composition profile
1. Introduction Due to its unique characteristics, semiconductor heterostructure has been essential building block in modern electronic and optoelectronic devices such as transistors, laser diodes, light emitting diodes, solar cells and photo detectors [1e3]. As an important branch of nano materials, one-dimensional nanowire based heterostructures can be divided into axial and coreshell type by the direction of heterostructure. For both types, due to the stronger elastic strain relaxation induced by lateral free surfaces, the critical thickness of generating misfit dislocations is larger than epilayer grown on the flat substrate. There also exist critical diameters of nanowire below which coherency will be maintained regardless of the thickness of epilayer [4e8]. This attractive property is an important basis for high-quality heterostructure and provides design flexibility. Unlike axial type based on vapor-liquid-solid (VLS) mechanism, the core-shell nanowire heterostructures have been grown based on vapor-solid (VS) mechanism, which can even introduce Stranski-Krastanov growth mode for quantum dots (QDs) on nanowire sidewall. In the last two decades, fabrication of group-IV [9e12] and IIIV compound [13e16] core-shell nanowire heterostructures have been realized by metal organic chemical vapor deposition (MOCVD) and molecular beam epitaxy
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (H. Ye), [email protected] (Z. Yu). http://dx.doi.org/10.1016/j.spmi.2017.09.002 0749-6036/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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(MBE). As a typical lattice mismatch system, GeSi core-shell heterostructure has drawn considerable and increasing attention due to its potential applications in next generation field effect transistor [17,18], high-performance thermoelectric device [19] and platform for spin quantum bits [20,21]. Recently, the micro-Raman spectroscopy has been used to analyze the strain in GeSi alloy core-shell nanowire heterostructures in experiments [22e24], taking the advantages of nondestructive test and no special sample preparation. Through inelastic scattering, the analysis is performed based on the frequency difference between scattered light and incident light. Two transverse optical (TO) phonon modes and one longitudinal optical (LO) phonon mode are degenerate in the absence of strain at Brillouin-zone center. These three degenerate optical phonon modes would be shifted and split by the strain perturbation induced by lattice mismatch in alloy epilayer [22,25]. Qualitatively, the tensile strain and compressive strain lead to red shift and blue shift respectively. D. C. Dillen et al. [23] presented the measured Raman spectra of Si-Ge0.61Si0.39 coreshell nanowire, in which the Si-Si mode of Si core shifted to 516 cm1 from the unstrained value 520.0 cm1. As for GeGe0.5Si0.5 core-shell nanowire, the Ge-Ge mode of Ge core shifted to 305.9 cm1 from the unstrained value 300.5 cm1 [22]. The absolute Raman shift induced by the strain was observed diameter dependent and decreased with increasing diameter of ndez et al. demonstrated a linear relationship between the core while thickness of shell was kept constant [22,23]. J. Mene strained Raman shift of core and the axial-strain in core through analytical model [25]. The analytical expressions for strain distribution in core-shell nanowire heterostructures were also validated by comparing the Raman spectra calculated from them and numerical simulations [26]. In previous theoretic calculations, the distribution of composition was simply assumed to be uniform within the GeSi alloy shell. In the VS growth mechanism, the heterointerface is not always sharp due to the processes like atom exchange between bulk and surface [27]. Since the non-uniform composition profile was observed and required for some applications [28e30], it behooves us to explore the influences of concentration and composition profile on Raman spectra in alloy core-shell nanowire heterostructures. In this paper, we present systematic calculations of the Raman spectra of GeSi alloy core-shell nanowire heterostructures. The growth directions of the nanowire, [111] and [011], are modeled. Pure Ge, Si and GeSi alloy cores with uniform, linear and nonlinear composition profiles (CPs) in shell are taken into consideration. The strain field is calculated by finite element method (FEM) with anisotropic continuum elasticity theory. The Raman shift induced by strain is calculated though the lattice dynamical theory [23,26,31] at each mesh point, and the final spectra is superposition of these points after Lorentzian fitting. The spectra of Si-Si (Ge-Ge) mode in Si (Ge) core nanowire heterostructure are presented. The nonlinear CP is obtained from the transient thermodynamically consistent model [32,33].
2. Raman spectrum simulation The three-dimensional models of cylindrical GeSi alloy core-shell nanowire heterostructures standing along [011] and [111] directions are built as shown in Fig. 1. Both pure silicon or germanium cores and GeSi alloy shells with uniform, linear and nonlinear composition profiles are taken into consideration. Cartesian coordinates, whose z axis is h set along growth direction h of nanowire, are adopted. When nanowire is along [011] ð½111Þ, the x and y axes stand for ½100 110 and ½011 112 respectively. The elastic strain distribution induced by lattice mismatch is simulated by finite element method within framework of anisotropic elasticity. The original stiffness matrix is rotated to meet the adopted direction of nanowire by below matrix
1 cos4cosq sin4cosq sinq @ R¼ sin4 cos4 0 A; cos4sinq sin4sinq cosq 0
(1)
Fig. 1. Schematic illustration of GeSi alloy core-shell nanowire model and composition profiles considered.
Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
D. Han et al. / Superlattices and Microstructures xxx (2017) 1e8
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where 4 and q are the angles rotated around z and y axis respectively. Only three independent constants C11, C12 and C44 exist in original stiffness matrix for Si and Ge. As the initial state, the shell strictly matches the lattice of core with strain εini ¼ ðalocal acore Þ=alocal implemented, where acore is lattice constant of core and alocal is determined by local composition in shell following Vegard's law. The displacements of equilibrium state and the corresponding strain fields are obtained by the FEM solver after relaxation. The lateral and top surfaces of nanowire are set free while the bottom surfaces are set fixed to account for stiff substrate effect. The length of nanowire is chosen 300 nm, and core diameter is acquiescently set 40 nm which is comparable with experimental observations [22,23,26]. To build a clear relationship between the strain and the optical phonon frequencies, we use the secular equation of lattice vibrational dynamics [22]:
pεxx þ q εyy þ εzz l 2rεxy 2rεxz ¼ 0; 2rεxy pεyy þ qðεxx þ εzz Þ l 2rε yz 2rεxz 2rεyz pεzz þ q εxx þ εyy l
(2)
where p, q and r are phonon deformation potentials (PDPs) which are obtained experimentally for Ge and Si. The eigenvalue l represents the strain-induced shift of mode. It should be noted that the strain εij in Eq. (2) should be given in the crystaloriented coordinate system with axis [100], [010] and [001] directions. The strain fields ε0pq obtained from FEM is then transformed by relation
εij ¼
X
aip ajp ε0pq :
(3)
pq
The transformation matrix for ½011 and ½ 111 growth direction can be expressed as
0
a½011
1
B B0 B ¼B B @ 0
1 0 1 1 1 1 pffiffiffi pffiffiffi C 0 0 B pffiffiffi 6 3C B 2 1 1 C C B pffiffiffi pffiffiffi C B 1 1 1 C C C: p ffiffiffi p ffiffiffi p ffiffiffi 2 2 C and a½111 ¼ B C B 2 C 6 3 C B C B 1 1 A @ pffiffiffi pffiffiffi 2 1 A p ffiffiffi p ffiffiffi 0 2 2 6 3
(4)
The Raman shift frequency can be estimated by the simple relation Du ¼ u u0 ¼ l=ð2u0 Þ, where u and u0 are frequencies for strained and unstrained cases. The intensities of phonon modes are evaluated by Refs. [22,25]
0 1 2 X 0 Ii fei $@ vj $vi $Rj A$es ; j
(5)
where ei and es are polarization vectors of incident and scattered light respectively. Experiments have shown much larger local fields exist in the setup that electromagnetic waves polarize parallel to nanowire axis compared with transverse polarization. To ensure the highest intensity of Raman spectra, we here only focus on the setup that both ei and es parallel to the nanowire axis [011] or [111]. The unstrained phonon wave vectors nj are given by n1 ¼ ½100, n2 ¼ ½010 and n3 ¼ ½001. The 0 strained wave vectors vi are eigenvectors obtained in solutions of Eq. (2). The unstrained Raman tensor Rj can be described by Rkl ðjÞ ¼ dð1 dkl Þð1 djk Þð1 djl Þ, in which k, l ¼ 1,2,3 correspond to the coordinates and d is the non-zero Raman tensor elements. The intensities are computed at each mesh point (0.1 nm mesh spacing) inside both core and shell, and finally integrated over entire cross section of nanowire and over all possible modes to draw the total Raman spectra. For GeSi alloy, the number of SieSi (GeeGe) bond and subsequent intensity of corresponding mode is assumed linearly proportional to Si (Ge) fraction [34]. When considering composition profile in GeSi alloy shell, parameters would be related to the local composition. The frequencies of Si-Si mode and Ge-Ge mode follow linear empirical relations with the concentration of Ge [35]:
u0
SiSi
¼ 520:7 66:9x;
u0
GeGe
¼ 280:3 þ 19:4x;
(6)
where x is the Ge concentration in alloy. Besides the shift of center frequency, larger full width at half maximum (FWHM) of Raman spectra in alloy would be expected compared with pure Si or Ge crystal. The dependence of FWHM on the concentration of minority species xm in GeSi alloy is fitted based on the experimental data in Ref. [36]:
W ¼ 3:74 8:05xm þ 37:55x2m
(7)
This second-order polynomial relation is adopted for both Si-Si and Ge-Ge modes. The PDP values also strongly depend on the Ge concentration, which are fitted based on the measured data in Ref. [37] Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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. . p u20 ¼ q u20 ¼ h0 þ h1 x þ h2 x2 þ h3 x3 þ h4 x4 . rSiSi u20 ¼ 1:00½23;
(8)
. rGeGe u20 ¼ 1:11½22
The fitting parameters hi are given in Table 1. The Ge concentration x ranges from 0 to 0.5 for Si-Si mode and from 0.5 to 1.0 for Ge-Ge mode. The Raman frequencies of Ge-Ge, Ge-Si and Si-Si modes in unstrained Ge0.5Si0.5 shell obtained from our calculation (experiment [38]) are 290.0 cm1 (289.2 cm1), 407.0 cm1 (406.1 cm1) and 487.2 cm1 (487.2 cm1) respectively. The good agreement between experimental data and our calculation indicates that the fitting equations of unstrained values are reliable. For non-uniform CPs, both linear and nonlinear cases are taken into consideration. The linear CP is simply assumed that Ge concentration linearly increases from core to shell edge. The nonlinear CP is more complicated and accounts for the deposited rates of Si and Ge, the atom exchange between bulk and growing surface, and the surface segregation. To simulate this spontaneous CP, we resort to the theory proposed by G. Vastola [32]. Based on the mass conservation, the time-related bulk and surface compositions (xb and xs) can be calculated by Ref. [32].
vxb =vt þ V$JGe ¼ 0
(9)
dvxs =vt ¼ vGe ðtÞ xb vtot ðtÞ þ JGe ; Where d is surface thickness, vtot and vGe are total growth rate and Ge contribution respectively. The normal exchanging flux between surface and bulk is expressed as JGe ¼ Kðxeq s xs Þ ðxs xb Þvtot , where K is exchanging rate and the surface equixb ek librium composition xeq s ¼ 1þðek 1Þxb . The parameter k is defined to measure the surface segregation of Ge. The nonlinear revolution of composition in Ge0.5Si0.5 alloy shell shown in Fig. 1 is obtained by solving Eq. (9) in FEM. The parameters adopted are d ¼ aGeSi =2, K ¼ 0.002 nm/s, vtot ¼ 0.02 nm/s and k ¼ 5, which can reproduce the composition profile of GeSi thin film grown on flat substrate. 3. Results and discussion We first consider the uniform CP in GeSi alloy shell. The Raman spectra of Si-Ge0.5Si0.5 and Ge-Ge0.5Si0.5 core-shell nanowire heterostructures standing along [011] and [111] directions are calculated separately. For pure Si (Ge) core nanowire, the Si-Si (Ge-Ge) mode is presented. The core diameter and shell thickness are set 40 nm and 5 nm respectively. As shown in Fig. 2, all Raman spectra have two explicit peaks. The right peak which can be attributed to the Si core (Ge core) has higher intensity since volume of core is larger than shell. The left additional peak belongs to the Si-Si mode (Ge-Ge mode) in Ge0.5Si0.5 shell. Because of the smaller FWHM of pure material compared to alloy, the peak of core is sharper. It is obvious that the intensity of nanowire along [111] direction is higher than that of [011] direction, deriving from the strain tensors and specific polarizations of incident and scattered light. It should be noted that only the contrast of peak intensities of core and shell in single model is quantitatively sound. This double peaks shape from our simulation shows good similarity with experimentally obtained Raman spectrum [23]. The black solid line in Fig. 2(a) demonstrates the Raman spectrum of Si-Ge0.5Si0.5 core-shell nanowire along [111] direction. The higher peak of Si-Si mode in Si core shifts from unstrained value 520.7 cm1 to 513.2 cm1 due to the tensile strain induced by lattice mismatch. The additional peak attributed to alloy shell locates at 501.5 cm1, where the unstrained value is 487.25 cm1. The blue shift is due to compressive strain in shell. The shape of spectrum of Si core nanowire along [011] direction (red dashed line in Fig. 2(a)) is similar. The strained Si-Si modes of Si core and alloy shell are 514.9 cm1 and 498.5 cm1 respectively. Nanowire along [111] direction shows a larger strain induced shift compared to [011] direction. Fig. 2(b) shows the spectra of Ge-Ge0.5Si0.5 core-shell nanowire heterostructures. The Ge-Ge mode of Ge core is 304.7 cm1 (303.5 cm1) along [111] ([011]) direction, as the unstrained value is 299.7 cm1. Blue shift takes place due to compressive strain in Ge core. Because of the tensile strain, the Ge-Ge mode of alloy shell along [111] ([011]) direction shifts to 281.4 cm1 (283.2 cm1) from the unstrained value of 290 cm1. Moreover, when simulating with identical geometry, Raman parameters and concentration in alloy shells, our code provides good consistency with peak values of Si core and Ge core reported in Ref. [22] and [23], where only 0.1 cm1 deviation is observed.
Table 1 The fitting parameters of PDP values. pSiSi =u20 qSiSi =u20 pGeGe =u20 qGeGe =u20
h0
h1
h2
h3
h4
1.87 2.34 3.89 3.31
11.18 3.98 31.83 33.00
74.65 20.41 65.45 69.22
171.55 43.46 57.67 61.65
131.92 32.38 18.49 19.93
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Fig. 2. (a) Raman shift of Si-Si mode in Si-Ge0.5Si0.5 core-shell nanowire along [011] and [111] direction. (b) Raman shift of Ge-Ge mode in Ge-Ge0.5Si0.5 core-shell nanowire along [011] and [111] direction.
To interpret the influences of average Ge concentration and shell thickness on Raman shift, we consider the uniform CP in shell. The strain induced shift Du of peaks core and shell are evaluated separately. Diameter of core is set 40 nm throughout all calculations. In Fig. 3(a) for Si core nanowire, Tshell is set 5 nm and Ge concentration ranges from 0.1 to 0.5. It is obviously that Du follows a linear function of concentration. The Du of Si core is negative due to the tensile strain while the values are positive GeSi shell due to the compressive strain. The absolute value of Du increases with concentration, because of the larger lattice mismatch. Although the Du of two growth directions are close at the beginning, the discrepancy of them becomes larger due to the larger strain-induced shift when nanowire is along [111] direction. The dependence of Du on Tshell is monotonous but nonlinear as shown in Fig. 3(b), where the Ge concentration x ¼ 0.5 is fixed and Tshell ranges from 5 nm to
Fig. 3. Strain-induced Raman shift of Si-GeSi and Ge-GeSi core-shell nanowires with uniform composition profile in shells. (a) Si core, shell thickness Tshell ¼ 5 nm and Ge concentration in shell ranges from 0.1 to 0.5. (b) Si-Ge0.5Si0.5 core-shell heterostructure with Tshell ranges from 5 nm to 20 nm. (c) Ge core, Tshell ¼ 5 nm and Ge concentration in shell ranges from 0.9 to 0.5. (d) Ge-Ge0.5Si0.5 core-shell heterostructure with Tshell ranges from 5 nm to 20 nm.
Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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Fig. 4. Raman spectra of Si-Si mode of Si core nanowire along (a) [011] and (b) [111] direction respectively. Raman spectra of Ge-Ge mode of Ge core nanowire along (c) [011] and (d) [111] direction respectively. The composition profiles in GeSi alloy shell are uniform, linear and nonlinear. The marked values are for linear and nonlinear CPs.
20 nm. The Du of Si-Si mode in shell decreases with the increase of Tshell due to smaller deformation and strain accommodated in shell. This phenomenon is induced by elastic compliance between core and shell. On the contrary, because of more mismatch accommodated in core which leads to a larger strain-induced shift, the absolute Du of Si core increases with Tshell. For Ge-GeSi core-shell nanowire, there also exists a linear relation between Ge concentration (ranges from 0.9 to 0.5) and Du of Ge-Ge mode, as shown in Fig. 3(c). Moreover, Fig. 3(d) demonstrates similar nonlinear relations between Du of Ge-Ge mode and Tshell. However, the changing trend is inverse compare to Fig. 3(a) and (b), due to the opposite polarity of strain. Next, we start to investigate the influences of non-uniform CPs on Raman scattering. Dcore and Tshell are set 40 nm and 5 nm respectively. The linear CP in GeSi shell is firstly taken as the simplest example. The linearly graded increase of concentration from core can observably reduce the lattice mismatch in heterostructures and avoid generation of misfit dislocations. For Si core cases, the Raman peak of alloy shell with linear CP cannot be distinguished in the total spectra, as shown in Fig. 4(a) and (b). The spectra are broadened and asymmetric on the low frequency side. The marked peak values of shell in Fig. 4 are obtained by computing only shell contribution. The Si-Si mode of Si core has a value of 517.5 cm1 (516.6 cm1) along [011] ([111]) direction, which is closer to unstrained value of silicon 520.7 cm1 compared with uniform CP. Next, we take the spontaneous nonlinear CP, which is induced by atom exchange between bulk and growing surface, into consideration. The heterointerface is not sharp and a transition area exists. The Ge concentration rises rapidly from 0 to 0.5 (deposited alloy concentration) and transition length is below 2 nm. The corresponding elastic strain energy density (Estrain/Vshell) of nonlinear CP case is 4:86 107 J=m3 ð4:97 107 J=m3 Þ for nanowire along [011] ([111]), about 6% smaller than value of heterostructure with uniform CP. The Raman peak of nonlinear CP shell has a value of 499.0 cm1 (502.2 cm1) along [011] ([111]), which is 0.5 cm1 (0.7 cm1) larger than uniform CP. Qualitatively, smaller compressive strain should lead to a smaller Raman shift, while on the contrary, lower Ge concentration means larger u0. Therefore, the little blue shift of Raman peak from shell with nonlinear CP indicates that the influence of concentration prevails. However, it can be seen the discrepancy between Raman spectra of nonlinear and uniform CP is too trivial to be distinguished in experiment. This attributes to the small spontaneous transition length of Ge concentration. As shown in Fig. 4(c) and (d), similar tendencies can be found for Ge core heterostructures, while linear CP introduces a more obvious plateau on the low frequency side of spectrum. Finally, Raman spectra of Ge-Ge mode in Ge0.5Si0.5-Ge0.7Si0.3 core-shell nanowire heterostructure are calculated. The Ge composition is kept uniform in core. For shell with linear CP, the peak of shell is merged into the strong peak of core as shown in Fig. 5. This can be attributed to little difference between Ge concentration of core and shell. The individual calculation of shell shows the value of peak should be 294.0 cm1 (294.5 cm1) along [011] ([111]) direction, while the value of core is Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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Fig. 5. Raman spectra of Ge-Ge mode of nanowires with Ge0.5Si0.5 core along (a) [011] and (b) [111]. Three composition profiles in GeSi alloy shell: linearly (black, solid) and nonlinearly (red, dash) increase from 0.5 to 0.7, and uniform 0.7 (blue, dash-dot). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
289.3 cm1 (289.1 cm1). For alloy core, the nonlinear CP is quite different. At the beginning of shell growth, core is at thermodynamic equilibrium, where surface composition is also equilibrium accounting for Ge segregation. A localized peak near heterointerface originates from the redistribution of atoms between bulk and surface. With nonlinear CP and growth direction of [011] ([111]), the Raman peak of core is 288.7 cm1 (288.2 cm1), while the value of shell is 296.2 cm1 (297.1 cm1). Although higher Ge concentration appears near the heterointerface, the Raman spectra of nonlinear CP are nearly identical to those of uniform CP. This is because the area in which Ge concentration keeps larger than 0.75, occupies only 7.7% of the whole shell. Its contribution to the intensity of Raman spectra is too small to be distinguished. 4. Conclusion In summary, we systematically analyze the influence of composition on Raman spectra of Ge/Si-GeSi core-shell nanowire heterostructures. Compared with [011] direction, larger strain-induced shifts are obtained in nanowires along [111] crystal direction. For uniform CP in alloy shell, the Raman spectra have explicit double peaks, and the strain-induced shift follows linear relation with Ge concentration and nonlinear relation with shell thickness. For linear CP, the peaks contributed by alloy shell cannot be distinguished and plateau is formed on low frequency side of spectra. Moreover, we take the nonlinear CP into consideration which accounts for spontaneous composition transition at heterointerface during epitaxy. Due to the small transition length, the observed spectra are nearly identical to the uniform CP cases. Acknowledgement This work was supported by the National Natural Science Foundation of China (Grants No.61401035 and No. 61372037), Fundamental Research Funds for the Central Universities (2016RCGD19) and the National Basic Research Program of China “973” (Grant No. 2014CB643901). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures Delong Han a, Han Ye a, *, Zhongyuan Yu a, **, Yunzhen Zhang a, Yumin Liu a, Yinfeng Li b a
State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China b Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 July 2017 Received in revised form 29 August 2017 Accepted 2 September 2017 Available online xxx
In this paper, the influences of composition on Raman scattering from Ge/Si-GeSi coreshell nanowire heterostructures standing along [011] and [111] crystal directions are numerically investigated. Uniform, linear and spontaneous nonlinear composition profiles (CPs) in GeSi alloy shell are taken into consideration. In uniform CP case, clear double peaks in Raman spectra contributed by core and shell are observed. The strain-induced shift follows linear relation with Ge concentration and nonlinear relation with shell thickness. Larger strain-induced shifts are obtained in nanowires along [111] direction. In linear CP case, the peaks contributed by shell cannot be distinguished in the total spectra and plateaus are formed on the low frequency side. Moreover, the nonlinear CP accounts for the spontaneous composition transition near heterointerface during lateral epitaxy of GeSi shell. Due to the rapid Ge concentration transition, Raman spectra are shown nearly identical to uniform CP cases. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Raman spectrum Core-shell nanowire heterostructure GeSi alloy Composition profile
1. Introduction Due to its unique characteristics, semiconductor heterostructure has been essential building block in modern electronic and optoelectronic devices such as transistors, laser diodes, light emitting diodes, solar cells and photo detectors [1e3]. As an important branch of nano materials, one-dimensional nanowire based heterostructures can be divided into axial and coreshell type by the direction of heterostructure. For both types, due to the stronger elastic strain relaxation induced by lateral free surfaces, the critical thickness of generating misfit dislocations is larger than epilayer grown on the flat substrate. There also exist critical diameters of nanowire below which coherency will be maintained regardless of the thickness of epilayer [4e8]. This attractive property is an important basis for high-quality heterostructure and provides design flexibility. Unlike axial type based on vapor-liquid-solid (VLS) mechanism, the core-shell nanowire heterostructures have been grown based on vapor-solid (VS) mechanism, which can even introduce Stranski-Krastanov growth mode for quantum dots (QDs) on nanowire sidewall. In the last two decades, fabrication of group-IV [9e12] and IIIV compound [13e16] core-shell nanowire heterostructures have been realized by metal organic chemical vapor deposition (MOCVD) and molecular beam epitaxy
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (H. Ye), [email protected] (Z. Yu). http://dx.doi.org/10.1016/j.spmi.2017.09.002 0749-6036/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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(MBE). As a typical lattice mismatch system, GeSi core-shell heterostructure has drawn considerable and increasing attention due to its potential applications in next generation field effect transistor [17,18], high-performance thermoelectric device [19] and platform for spin quantum bits [20,21]. Recently, the micro-Raman spectroscopy has been used to analyze the strain in GeSi alloy core-shell nanowire heterostructures in experiments [22e24], taking the advantages of nondestructive test and no special sample preparation. Through inelastic scattering, the analysis is performed based on the frequency difference between scattered light and incident light. Two transverse optical (TO) phonon modes and one longitudinal optical (LO) phonon mode are degenerate in the absence of strain at Brillouin-zone center. These three degenerate optical phonon modes would be shifted and split by the strain perturbation induced by lattice mismatch in alloy epilayer [22,25]. Qualitatively, the tensile strain and compressive strain lead to red shift and blue shift respectively. D. C. Dillen et al. [23] presented the measured Raman spectra of Si-Ge0.61Si0.39 coreshell nanowire, in which the Si-Si mode of Si core shifted to 516 cm1 from the unstrained value 520.0 cm1. As for GeGe0.5Si0.5 core-shell nanowire, the Ge-Ge mode of Ge core shifted to 305.9 cm1 from the unstrained value 300.5 cm1 [22]. The absolute Raman shift induced by the strain was observed diameter dependent and decreased with increasing diameter of ndez et al. demonstrated a linear relationship between the core while thickness of shell was kept constant [22,23]. J. Mene strained Raman shift of core and the axial-strain in core through analytical model [25]. The analytical expressions for strain distribution in core-shell nanowire heterostructures were also validated by comparing the Raman spectra calculated from them and numerical simulations [26]. In previous theoretic calculations, the distribution of composition was simply assumed to be uniform within the GeSi alloy shell. In the VS growth mechanism, the heterointerface is not always sharp due to the processes like atom exchange between bulk and surface [27]. Since the non-uniform composition profile was observed and required for some applications [28e30], it behooves us to explore the influences of concentration and composition profile on Raman spectra in alloy core-shell nanowire heterostructures. In this paper, we present systematic calculations of the Raman spectra of GeSi alloy core-shell nanowire heterostructures. The growth directions of the nanowire, [111] and [011], are modeled. Pure Ge, Si and GeSi alloy cores with uniform, linear and nonlinear composition profiles (CPs) in shell are taken into consideration. The strain field is calculated by finite element method (FEM) with anisotropic continuum elasticity theory. The Raman shift induced by strain is calculated though the lattice dynamical theory [23,26,31] at each mesh point, and the final spectra is superposition of these points after Lorentzian fitting. The spectra of Si-Si (Ge-Ge) mode in Si (Ge) core nanowire heterostructure are presented. The nonlinear CP is obtained from the transient thermodynamically consistent model [32,33].
2. Raman spectrum simulation The three-dimensional models of cylindrical GeSi alloy core-shell nanowire heterostructures standing along [011] and [111] directions are built as shown in Fig. 1. Both pure silicon or germanium cores and GeSi alloy shells with uniform, linear and nonlinear composition profiles are taken into consideration. Cartesian coordinates, whose z axis is h set along growth direction h of nanowire, are adopted. When nanowire is along [011] ð½111Þ, the x and y axes stand for ½100 110 and ½011 112 respectively. The elastic strain distribution induced by lattice mismatch is simulated by finite element method within framework of anisotropic elasticity. The original stiffness matrix is rotated to meet the adopted direction of nanowire by below matrix
1 cos4cosq sin4cosq sinq @ R¼ sin4 cos4 0 A; cos4sinq sin4sinq cosq 0
(1)
Fig. 1. Schematic illustration of GeSi alloy core-shell nanowire model and composition profiles considered.
Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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3
where 4 and q are the angles rotated around z and y axis respectively. Only three independent constants C11, C12 and C44 exist in original stiffness matrix for Si and Ge. As the initial state, the shell strictly matches the lattice of core with strain εini ¼ ðalocal acore Þ=alocal implemented, where acore is lattice constant of core and alocal is determined by local composition in shell following Vegard's law. The displacements of equilibrium state and the corresponding strain fields are obtained by the FEM solver after relaxation. The lateral and top surfaces of nanowire are set free while the bottom surfaces are set fixed to account for stiff substrate effect. The length of nanowire is chosen 300 nm, and core diameter is acquiescently set 40 nm which is comparable with experimental observations [22,23,26]. To build a clear relationship between the strain and the optical phonon frequencies, we use the secular equation of lattice vibrational dynamics [22]:
pεxx þ q εyy þ εzz l 2rεxy 2rεxz ¼ 0; 2rεxy pεyy þ qðεxx þ εzz Þ l 2rε yz 2rεxz 2rεyz pεzz þ q εxx þ εyy l
(2)
where p, q and r are phonon deformation potentials (PDPs) which are obtained experimentally for Ge and Si. The eigenvalue l represents the strain-induced shift of mode. It should be noted that the strain εij in Eq. (2) should be given in the crystaloriented coordinate system with axis [100], [010] and [001] directions. The strain fields ε0pq obtained from FEM is then transformed by relation
εij ¼
X
aip ajp ε0pq :
(3)
pq
The transformation matrix for ½011 and ½ 111 growth direction can be expressed as
0
a½011
1
B B0 B ¼B B @ 0
1 0 1 1 1 1 pffiffiffi pffiffiffi C 0 0 B pffiffiffi 6 3C B 2 1 1 C C B pffiffiffi pffiffiffi C B 1 1 1 C C C: p ffiffiffi p ffiffiffi p ffiffiffi 2 2 C and a½111 ¼ B C B 2 C 6 3 C B C B 1 1 A @ pffiffiffi pffiffiffi 2 1 A p ffiffiffi p ffiffiffi 0 2 2 6 3
(4)
The Raman shift frequency can be estimated by the simple relation Du ¼ u u0 ¼ l=ð2u0 Þ, where u and u0 are frequencies for strained and unstrained cases. The intensities of phonon modes are evaluated by Refs. [22,25]
0 1 2 X 0 Ii fei $@ vj $vi $Rj A$es ; j
(5)
where ei and es are polarization vectors of incident and scattered light respectively. Experiments have shown much larger local fields exist in the setup that electromagnetic waves polarize parallel to nanowire axis compared with transverse polarization. To ensure the highest intensity of Raman spectra, we here only focus on the setup that both ei and es parallel to the nanowire axis [011] or [111]. The unstrained phonon wave vectors nj are given by n1 ¼ ½100, n2 ¼ ½010 and n3 ¼ ½001. The 0 strained wave vectors vi are eigenvectors obtained in solutions of Eq. (2). The unstrained Raman tensor Rj can be described by Rkl ðjÞ ¼ dð1 dkl Þð1 djk Þð1 djl Þ, in which k, l ¼ 1,2,3 correspond to the coordinates and d is the non-zero Raman tensor elements. The intensities are computed at each mesh point (0.1 nm mesh spacing) inside both core and shell, and finally integrated over entire cross section of nanowire and over all possible modes to draw the total Raman spectra. For GeSi alloy, the number of SieSi (GeeGe) bond and subsequent intensity of corresponding mode is assumed linearly proportional to Si (Ge) fraction [34]. When considering composition profile in GeSi alloy shell, parameters would be related to the local composition. The frequencies of Si-Si mode and Ge-Ge mode follow linear empirical relations with the concentration of Ge [35]:
u0
SiSi
¼ 520:7 66:9x;
u0
GeGe
¼ 280:3 þ 19:4x;
(6)
where x is the Ge concentration in alloy. Besides the shift of center frequency, larger full width at half maximum (FWHM) of Raman spectra in alloy would be expected compared with pure Si or Ge crystal. The dependence of FWHM on the concentration of minority species xm in GeSi alloy is fitted based on the experimental data in Ref. [36]:
W ¼ 3:74 8:05xm þ 37:55x2m
(7)
This second-order polynomial relation is adopted for both Si-Si and Ge-Ge modes. The PDP values also strongly depend on the Ge concentration, which are fitted based on the measured data in Ref. [37] Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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. . p u20 ¼ q u20 ¼ h0 þ h1 x þ h2 x2 þ h3 x3 þ h4 x4 . rSiSi u20 ¼ 1:00½23;
(8)
. rGeGe u20 ¼ 1:11½22
The fitting parameters hi are given in Table 1. The Ge concentration x ranges from 0 to 0.5 for Si-Si mode and from 0.5 to 1.0 for Ge-Ge mode. The Raman frequencies of Ge-Ge, Ge-Si and Si-Si modes in unstrained Ge0.5Si0.5 shell obtained from our calculation (experiment [38]) are 290.0 cm1 (289.2 cm1), 407.0 cm1 (406.1 cm1) and 487.2 cm1 (487.2 cm1) respectively. The good agreement between experimental data and our calculation indicates that the fitting equations of unstrained values are reliable. For non-uniform CPs, both linear and nonlinear cases are taken into consideration. The linear CP is simply assumed that Ge concentration linearly increases from core to shell edge. The nonlinear CP is more complicated and accounts for the deposited rates of Si and Ge, the atom exchange between bulk and growing surface, and the surface segregation. To simulate this spontaneous CP, we resort to the theory proposed by G. Vastola [32]. Based on the mass conservation, the time-related bulk and surface compositions (xb and xs) can be calculated by Ref. [32].
vxb =vt þ V$JGe ¼ 0
(9)
dvxs =vt ¼ vGe ðtÞ xb vtot ðtÞ þ JGe ; Where d is surface thickness, vtot and vGe are total growth rate and Ge contribution respectively. The normal exchanging flux between surface and bulk is expressed as JGe ¼ Kðxeq s xs Þ ðxs xb Þvtot , where K is exchanging rate and the surface equixb ek librium composition xeq s ¼ 1þðek 1Þxb . The parameter k is defined to measure the surface segregation of Ge. The nonlinear revolution of composition in Ge0.5Si0.5 alloy shell shown in Fig. 1 is obtained by solving Eq. (9) in FEM. The parameters adopted are d ¼ aGeSi =2, K ¼ 0.002 nm/s, vtot ¼ 0.02 nm/s and k ¼ 5, which can reproduce the composition profile of GeSi thin film grown on flat substrate. 3. Results and discussion We first consider the uniform CP in GeSi alloy shell. The Raman spectra of Si-Ge0.5Si0.5 and Ge-Ge0.5Si0.5 core-shell nanowire heterostructures standing along [011] and [111] directions are calculated separately. For pure Si (Ge) core nanowire, the Si-Si (Ge-Ge) mode is presented. The core diameter and shell thickness are set 40 nm and 5 nm respectively. As shown in Fig. 2, all Raman spectra have two explicit peaks. The right peak which can be attributed to the Si core (Ge core) has higher intensity since volume of core is larger than shell. The left additional peak belongs to the Si-Si mode (Ge-Ge mode) in Ge0.5Si0.5 shell. Because of the smaller FWHM of pure material compared to alloy, the peak of core is sharper. It is obvious that the intensity of nanowire along [111] direction is higher than that of [011] direction, deriving from the strain tensors and specific polarizations of incident and scattered light. It should be noted that only the contrast of peak intensities of core and shell in single model is quantitatively sound. This double peaks shape from our simulation shows good similarity with experimentally obtained Raman spectrum [23]. The black solid line in Fig. 2(a) demonstrates the Raman spectrum of Si-Ge0.5Si0.5 core-shell nanowire along [111] direction. The higher peak of Si-Si mode in Si core shifts from unstrained value 520.7 cm1 to 513.2 cm1 due to the tensile strain induced by lattice mismatch. The additional peak attributed to alloy shell locates at 501.5 cm1, where the unstrained value is 487.25 cm1. The blue shift is due to compressive strain in shell. The shape of spectrum of Si core nanowire along [011] direction (red dashed line in Fig. 2(a)) is similar. The strained Si-Si modes of Si core and alloy shell are 514.9 cm1 and 498.5 cm1 respectively. Nanowire along [111] direction shows a larger strain induced shift compared to [011] direction. Fig. 2(b) shows the spectra of Ge-Ge0.5Si0.5 core-shell nanowire heterostructures. The Ge-Ge mode of Ge core is 304.7 cm1 (303.5 cm1) along [111] ([011]) direction, as the unstrained value is 299.7 cm1. Blue shift takes place due to compressive strain in Ge core. Because of the tensile strain, the Ge-Ge mode of alloy shell along [111] ([011]) direction shifts to 281.4 cm1 (283.2 cm1) from the unstrained value of 290 cm1. Moreover, when simulating with identical geometry, Raman parameters and concentration in alloy shells, our code provides good consistency with peak values of Si core and Ge core reported in Ref. [22] and [23], where only 0.1 cm1 deviation is observed.
Table 1 The fitting parameters of PDP values. pSiSi =u20 qSiSi =u20 pGeGe =u20 qGeGe =u20
h0
h1
h2
h3
h4
1.87 2.34 3.89 3.31
11.18 3.98 31.83 33.00
74.65 20.41 65.45 69.22
171.55 43.46 57.67 61.65
131.92 32.38 18.49 19.93
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Fig. 2. (a) Raman shift of Si-Si mode in Si-Ge0.5Si0.5 core-shell nanowire along [011] and [111] direction. (b) Raman shift of Ge-Ge mode in Ge-Ge0.5Si0.5 core-shell nanowire along [011] and [111] direction.
To interpret the influences of average Ge concentration and shell thickness on Raman shift, we consider the uniform CP in shell. The strain induced shift Du of peaks core and shell are evaluated separately. Diameter of core is set 40 nm throughout all calculations. In Fig. 3(a) for Si core nanowire, Tshell is set 5 nm and Ge concentration ranges from 0.1 to 0.5. It is obviously that Du follows a linear function of concentration. The Du of Si core is negative due to the tensile strain while the values are positive GeSi shell due to the compressive strain. The absolute value of Du increases with concentration, because of the larger lattice mismatch. Although the Du of two growth directions are close at the beginning, the discrepancy of them becomes larger due to the larger strain-induced shift when nanowire is along [111] direction. The dependence of Du on Tshell is monotonous but nonlinear as shown in Fig. 3(b), where the Ge concentration x ¼ 0.5 is fixed and Tshell ranges from 5 nm to
Fig. 3. Strain-induced Raman shift of Si-GeSi and Ge-GeSi core-shell nanowires with uniform composition profile in shells. (a) Si core, shell thickness Tshell ¼ 5 nm and Ge concentration in shell ranges from 0.1 to 0.5. (b) Si-Ge0.5Si0.5 core-shell heterostructure with Tshell ranges from 5 nm to 20 nm. (c) Ge core, Tshell ¼ 5 nm and Ge concentration in shell ranges from 0.9 to 0.5. (d) Ge-Ge0.5Si0.5 core-shell heterostructure with Tshell ranges from 5 nm to 20 nm.
Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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Fig. 4. Raman spectra of Si-Si mode of Si core nanowire along (a) [011] and (b) [111] direction respectively. Raman spectra of Ge-Ge mode of Ge core nanowire along (c) [011] and (d) [111] direction respectively. The composition profiles in GeSi alloy shell are uniform, linear and nonlinear. The marked values are for linear and nonlinear CPs.
20 nm. The Du of Si-Si mode in shell decreases with the increase of Tshell due to smaller deformation and strain accommodated in shell. This phenomenon is induced by elastic compliance between core and shell. On the contrary, because of more mismatch accommodated in core which leads to a larger strain-induced shift, the absolute Du of Si core increases with Tshell. For Ge-GeSi core-shell nanowire, there also exists a linear relation between Ge concentration (ranges from 0.9 to 0.5) and Du of Ge-Ge mode, as shown in Fig. 3(c). Moreover, Fig. 3(d) demonstrates similar nonlinear relations between Du of Ge-Ge mode and Tshell. However, the changing trend is inverse compare to Fig. 3(a) and (b), due to the opposite polarity of strain. Next, we start to investigate the influences of non-uniform CPs on Raman scattering. Dcore and Tshell are set 40 nm and 5 nm respectively. The linear CP in GeSi shell is firstly taken as the simplest example. The linearly graded increase of concentration from core can observably reduce the lattice mismatch in heterostructures and avoid generation of misfit dislocations. For Si core cases, the Raman peak of alloy shell with linear CP cannot be distinguished in the total spectra, as shown in Fig. 4(a) and (b). The spectra are broadened and asymmetric on the low frequency side. The marked peak values of shell in Fig. 4 are obtained by computing only shell contribution. The Si-Si mode of Si core has a value of 517.5 cm1 (516.6 cm1) along [011] ([111]) direction, which is closer to unstrained value of silicon 520.7 cm1 compared with uniform CP. Next, we take the spontaneous nonlinear CP, which is induced by atom exchange between bulk and growing surface, into consideration. The heterointerface is not sharp and a transition area exists. The Ge concentration rises rapidly from 0 to 0.5 (deposited alloy concentration) and transition length is below 2 nm. The corresponding elastic strain energy density (Estrain/Vshell) of nonlinear CP case is 4:86 107 J=m3 ð4:97 107 J=m3 Þ for nanowire along [011] ([111]), about 6% smaller than value of heterostructure with uniform CP. The Raman peak of nonlinear CP shell has a value of 499.0 cm1 (502.2 cm1) along [011] ([111]), which is 0.5 cm1 (0.7 cm1) larger than uniform CP. Qualitatively, smaller compressive strain should lead to a smaller Raman shift, while on the contrary, lower Ge concentration means larger u0. Therefore, the little blue shift of Raman peak from shell with nonlinear CP indicates that the influence of concentration prevails. However, it can be seen the discrepancy between Raman spectra of nonlinear and uniform CP is too trivial to be distinguished in experiment. This attributes to the small spontaneous transition length of Ge concentration. As shown in Fig. 4(c) and (d), similar tendencies can be found for Ge core heterostructures, while linear CP introduces a more obvious plateau on the low frequency side of spectrum. Finally, Raman spectra of Ge-Ge mode in Ge0.5Si0.5-Ge0.7Si0.3 core-shell nanowire heterostructure are calculated. The Ge composition is kept uniform in core. For shell with linear CP, the peak of shell is merged into the strong peak of core as shown in Fig. 5. This can be attributed to little difference between Ge concentration of core and shell. The individual calculation of shell shows the value of peak should be 294.0 cm1 (294.5 cm1) along [011] ([111]) direction, while the value of core is Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002
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Fig. 5. Raman spectra of Ge-Ge mode of nanowires with Ge0.5Si0.5 core along (a) [011] and (b) [111]. Three composition profiles in GeSi alloy shell: linearly (black, solid) and nonlinearly (red, dash) increase from 0.5 to 0.7, and uniform 0.7 (blue, dash-dot). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
289.3 cm1 (289.1 cm1). For alloy core, the nonlinear CP is quite different. At the beginning of shell growth, core is at thermodynamic equilibrium, where surface composition is also equilibrium accounting for Ge segregation. A localized peak near heterointerface originates from the redistribution of atoms between bulk and surface. With nonlinear CP and growth direction of [011] ([111]), the Raman peak of core is 288.7 cm1 (288.2 cm1), while the value of shell is 296.2 cm1 (297.1 cm1). Although higher Ge concentration appears near the heterointerface, the Raman spectra of nonlinear CP are nearly identical to those of uniform CP. This is because the area in which Ge concentration keeps larger than 0.75, occupies only 7.7% of the whole shell. Its contribution to the intensity of Raman spectra is too small to be distinguished. 4. Conclusion In summary, we systematically analyze the influence of composition on Raman spectra of Ge/Si-GeSi core-shell nanowire heterostructures. Compared with [011] direction, larger strain-induced shifts are obtained in nanowires along [111] crystal direction. For uniform CP in alloy shell, the Raman spectra have explicit double peaks, and the strain-induced shift follows linear relation with Ge concentration and nonlinear relation with shell thickness. For linear CP, the peaks contributed by alloy shell cannot be distinguished and plateau is formed on low frequency side of spectra. Moreover, we take the nonlinear CP into consideration which accounts for spontaneous composition transition at heterointerface during epitaxy. Due to the small transition length, the observed spectra are nearly identical to the uniform CP cases. Acknowledgement This work was supported by the National Natural Science Foundation of China (Grants No.61401035 and No. 61372037), Fundamental Research Funds for the Central Universities (2016RCGD19) and the National Basic Research Program of China “973” (Grant No. 2014CB643901). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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Please cite this article in press as: D. Han et al., Influences of composition on Raman scattering from GeSi alloy core-shell nanowire heterostructures, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.09.002