View Online PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics A solid-state NMR and DFT study of compositional modulations in AlxGa1ÀxAsw Paulus J. Knijn,a P. Jan M. van Bentum,a Ernst R. H. van Eck,a Changming Fang,a Dennis L. A. G. Grimminck,a Robert A. de Groot,a Remco W. A. Havenith,a Martijn Marsman,b W. Leo Meerts,a Gilles A. de WijsaDownloaded by RADBOUND UNIV. NIJMEGEN UNIVERSITEITSBIBLIOTHEK on 04 August 2010 and Arno P. M. Kentgens*a Received 25th February 2010, Accepted 17th May 2010 DOI: 10.1039/c003624b We have conducted 75As and 69Ga Nuclear Magnetic Resonance (NMR) experiments to Published on 02 August 2010 on http://pubs.rsc.org | doi:10.1039/C003624B investigate order/disorder in AlxGa1ÀxAs lift-oﬀ ﬁlms with x B 0.297 and 0.489. We were able to identify all possible As(AlnGa4Àn) sites with n = 0–4 coordinations in 75As NMR spectra using spin-echo experiments at 18.8 Tesla. This was achieved by employing high rf ﬁeld strengths using a small solenoid coil and an NMR probe speciﬁcally designed for this purpose. Spectral deconvolution, using an evolutionary algorithm, complies with the absence of long-range order if a CuAu based order parameter is imposed. An unconstrained ﬁt shows a deviation of the statistics imposed by this type of ordering. The occupational disorder in the Ga and Al positions is reﬂected in a distribution of the Electric Field Gradients (EFGs) experienced at the diﬀerent arsenic sites. We established that this can be modelled by summing the eﬀects of the ﬁrst coordination sphere and a Czjzek type distribution resulting from the compositional variation in the Al/Ga sub-lattice in the higher coordination spheres. 69Ga 3QMAS and nutation data exclude the presence of highly symmetric sites and also show a distribution in EFG. The experimentally obtained quadrupolar interactions are in good agreement with calculations based on Density Functional Theory (DFT). Using additivity of EFG tensors arising from distant charge perturbations, we could use DFT to model the EFG distributions of the n = 0–4 sites, reproducing the Czjzek and extended Czjzek distributions that were found experimentally. On the basis of these calculations we conclude that the 75As quadrupolar interaction is sensitive to compositional modulations up to the 7th coordination shell in these systems. Introduction positive excess enthalpy, these structures can order metastably in bulk and can become stable in epitaxially grown ﬁlms. Other Thin ﬁlm semiconductors are of great importance for electronic crystallographic arrangements such as the CuPt structure do not and photonic devices. The development of new or improved possess enough geometrical degrees of freedom to permit all devices relies on a detailed knowledge of the structural properties chemical bonds to attain their ideal length, and are therefore of the materials. In this respect compositional modulation and unstable. Nevertheless, ordering of alternating diagonal planes ordering play an important role. As was predicted by Zunger in the h111i direction was found experimentally which was and coworkers1 in the mid eighties, long-range order can occur explained by assuming atomic dimerization at the surface, in semiconductor alloys with size-mismatched constituents, inducing CuPt ordering in the subsurface layers.4 despite the fact that the mixing enthalpy is positive. This can As there is hardly any size-mismatch in AlGaAs, long-range be explained by the fact that the statistical average over the order has not been widely observed. Nevertheless, Krahmer many diﬀerent local environments is positive, because some of et al.5 reported evidence of a direct–indirect conduction band these clusters are strained. Periodically repeating a three- transition for x = 0.4 in AlxGa1ÀxAs by using reﬂectance dimensional geometric re-arrangement can minimize strain anisotropy spectrometry, indicating an increase in order. They and leads to long-range order. These predictions were soon report that doping, temperature and substrate orientation all veriﬁed by experimental observations of ordering in semi- aﬀect the anisotropy of the lattice, but did not quantify this in conductors by Stringfellow and coworkers2 and Gomyo and terms of an order parameter. X-ray measurements from Kuan6 Suzuki.3 For many III–V compounds the ordered phase has et al. reported a low long-range order parameter S o 0.5 for a Al0.75Ga0.25As grown by MOCVD, depending on growth Institute for Molecules and Materials, Radboud University Nijmegen, Toernooiveld 1, NL-6525 ED Nijmegen, The Netherlands conditions. More recent X-ray measurements by van Niftrik b Institut fu¨r Materialphysik and Centre for Computational Materials et al.7 indicated a lower long-range ordering of S r 0.05 for Science, Universita Wien, A 1090, Wien, Austria ¨t Al0.5Ga0.5As and S r 0.06 for Al0.25Ga0.75As. Despite the w Electronic supplementary information (ESI) available: Additional prediction of almost no long-range ordering with X-ray, information including an experimental NMR section concerning the QCPMG data and a theoretical DFT part. See DOI: 10.1039/ studies of the local interface for dilute AlGaAs with cross- c003624b sectional STM (XSTM) indicate the appearance of short This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
View Online strings. In 1994, an XSTM study by Smith et al.8 conﬁrmed order of nq = 3Cq/(2I(2I À 1)) whereas the shape of the that an interrupted growth process can create short period observed patterns reﬂects the asymmetry parameter Z of the super lattices in AlGaAs. In 1996, Smith et al.9 noticed a interaction. The central transition, h1/2;À1/2i is not aﬀected small deviation from random ordering in the Al–Al pair by the quadrupolar interaction in ﬁrst order but the second- distributions at the interface of GaAs and Al0.05Ga0.95As. order features are dispersed over a frequency range of the In 1999, Heinrich et al.10 clearly observed strings of Al order of n2 /n0, where the shape is again characteristic for the q atoms in the GaAs matrix, up to ﬁve atoms in MOVPE asymmetry of the EFG tensor. Even at high external ﬁelds thisDownloaded by RADBOUND UNIV. NIJMEGEN UNIVERSITEITSBIBLIOTHEK on 04 August 2010 grown Al0.15Ga0.85As along low indexed crystallographic second-order interaction can result in MHz bandwidths for directions. nuclei such as 75As. To excite such a bandwidth eﬃciently, In general, the composition of III–V semiconductors is suﬃciently large rf ﬁeld strengths are needed. In relation to diﬃcult to describe since various structural features can exist this one has to realize that the behavior of the spin system in (simultaneously) at diﬀerent length scales. For instance, the the rotating frame during rf-irradiation depends on the ratio presence of both ordered and disordered domains makes it of the quadrupolar frequency and the rf ﬁeld strength.12,13 Published on 02 August 2010 on http://pubs.rsc.org | doi:10.1039/C003624B hard to analyze these structures in detail. The degree of order Therefore, the eﬀective excitation of the spin systems has to be will aﬀect many physical and electronic properties such as modeled accurately, especially when quantitative results electronic transport, mobility, thermodynamic stability and of diﬀerent spin systems with widely diﬀering quadrupolar electro-optical properties. Many properties of AlGaAs are parameters are needed. Additionally, with such diﬀerent sites described in detail in the work of Adachi.11 Considering the present spectral overlap will often occur, demanding a robust limited number of reports on long-range order in AlGaAs thin approach to spectral deconvolution. Finally, to interpret ﬁlms as studied by diﬀraction techniques, whereas speciﬁc NMR spectra in relation to structures, a ﬁrst principle ordered structures are seen in a number of microscopic approach such as DFT is desirable. observations, it is interesting to study these materials with In the present contribution we address all of the issues a complimentary method such as NMR. As NMR probes described above. Using small rf-coils it is possible to generate short-range order, the presence of locally occurring structures appropriate rf-ﬁelds to excite a broad bandwidth. These coils can be detected if present in suﬃciently high concentrations. also help in overcoming the sensitivity problems related Since the NMR signal is composed out of contributions from to the study of thin ﬁlm materials leading to very low the entire sample, the possibility that one is studying a speciﬁc quantities of sample material.14 Here we compare spin-echo anomaly is reduced. Long-range ordered structures will result experiments and frequency-stepped Quadrupolar Carr–Purcell– in a speciﬁc signature which can be directly distinguished from Meiboom–Gill (QCPMG)15–18 experiments to obtain static 75 a disordered structure. As spectra of two AlGaAs ﬁlms with diﬀerent composition. NMR is a versatile technique that in principle can detect all Spectral deconvolution and quantiﬁcation of the spectra is nuclei which possess a spin through their interaction with an addressed with a dedicated program based on an evolutionary external magnetic ﬁeld (the Zeeman interaction). A number of algorithm. 69Ga 3QMAS19 and nutation12,13 NMR spectra interactions make NMR useful as they give the spectra a have been obtained to gain complementary information about speciﬁc appearance which can be related to local structure the local symmetry of the diﬀerent lattice sites. Although and dynamics of the material under study. In the present work this work focuses on AlGaAs thin ﬁlm materials, the two of these are important; the chemical shift and the approach presented can be applied for the investigation of quadrupolar interaction. The external magnetic ﬁeld used in compositional modulations in most III–V semiconductors NMR induces local ﬁelds in molecules and materials that and materials with related variations in the occupancy of modify the resonance frequencies and makes diﬀerent sites diﬀerent lattice positions, sometimes called substitutional distinguishable, this is referred to as the chemical shift. For a disorder. I 4 1/2 nucleus, the quadrupolar interaction arises from the Furthermore, we develop a modeling approach to accurately presence of an electric ﬁeld gradient (EFG), resulting from the describe the EFG distributions of various kinds of sites. To local charge distribution around the site of a nucleus which this end we extended the ﬁrst-principles DFT code of interacts with the non-spherically symmetric charge distribution VASP (Vienna Ab initio Simulation Program), to calculate of that nucleus. Although NMR is an appealing technique that EFGs and build a model wherein we can quickly calculate may form a bridge between diﬀraction studies and microscopic the EFGs for several millions of conﬁgurations based on observations in the study of III–V semiconductors, there are a just one DFT calculation. This model is used to estimate number of challenges that have to be addressed. Many nuclei the sensitivity of EFGs to variations in the order parameter. encountered in III–V semiconductors are quadrupolar nuclei, It has a wider scope, as it in principle provides the capability possessing a large quadrupolar moment eÁQ. The strength of to quickly obtain electronic density distributions for the interaction expressed through the quadrupole coupling any structural model for any ‘‘random’’ alloy with ﬁxed sub- constant Cq = e2qQ/h can be very large, easily in the tens of lattices. MHz regime. Although this interaction can still be treated as perturbation of the Zeeman Hamiltonian, if suﬃciently high Studying (dis-)order with NMR external ﬁelds are applied, the quadrupolar interaction has, in general, to be treated as a second-order perturbation. Based on theory used to analyze diﬀraction data, an order To ﬁrst-order the spectral features of the so-called satellite parameter S can be deﬁned to quantify the degree of transitions are dispersed over a wide frequency range of the long-range order related to the fractional occupancy of lattice Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
View Online sites by their preferred atom.6 S = 0 represents a completely pCuPt = 3(1 À rB)rA(1 À rA)2 + 3rA(1 À rB)rB2 2 2 2 random structure and S = 1 corresponds to a completely + 3(1 À rB)2rB(1 À rA) + 3rA2(1 À rA)rB ordered lattice: 2 2 S = |rA + rB À 1|, (1) pCuPt = 1rA3rB + 1(1 À rB)3(1 À rA) + 3rA2(1 À rB)(1 À rA) 3 2 2 2 where rA and rB are the fractions of A and B lattice sites that + 3(1 À rB)2rArB 2 are occupied by their preferred atom. For binary systems, as pCuPt = 1(1 À rB)rA3 + 1rA(1 À rB)3Downloaded by RADBOUND UNIV. NIJMEGEN UNIVERSITEITSBIBLIOTHEK on 04 August 2010 considered here, this means that 1 À rA constitutes the fraction 4 2 2 (5) of A lattice sites occupied by B atoms and 1 À rB the fraction of B sites occupied by A atoms. In case of random occupation of pCuAu = (1 À rA)2rB2 0 the lattice sites, i.e. the absence of ordering, rA = x and rB = 1 À x. For (partially) ordered structures the fractional pCuAu = 2rA(1 À rA)rB2+2(1 À rB)(1 À rA)2rB 1 occupancy of the A and B sites by their preferred atoms can be Published on 02 August 2010 on http://pubs.rsc.org | doi:10.1039/C003624B re-written as: pCuAu = rA2rB2+(1 À rB)2(1 À rA)2 +4rA(1 À rB)rB(1 À rA) 2 rA = x + S/2 (2) rB = (1 À x)+S/2 (3) pCuAu = 2rA2(1 À rB)rB+2rA(1 À rB)2(1 À rA) 3 Using this deﬁnition we should realize that the order para- pCuAu = (1 À rB)2rA2 4 (6) meter S can reach a value of 2x at most. A perfectly ordered structure can only be achieved for x = 0.5. S can be scaled, however, so that the maximum possible order As the probabilities correspond to the integrated peak for any given composition x gives an order parameter intensities of the diﬀerent sites, one can determine the long- S = 1.20 range order parameter by drawing the experimental peak The presence of (dis-)order in a structure is in most cases intensities in a plot of calculated intensities versus order manifested in quantiﬁable NMR parameters. Diﬀerent parameter S. Using the relative 31P peak intensities Tycko structural environments in a material can be distinguished et al.22 were able to set an upper limit of S o 0.6 for the CuPt based on the NMR interactions which are inﬂuenced by the type ordering in InGaP. Degen et al.21 observed the 75As diﬀerent coordinating atoms. For example, Tycko et al.22 resonances of the locally symmetric As[Al4] and As[Ga4] distinguished ﬁve 31P sites in the 31P MAS NMR spectrum coordinations in static single-pulse excitation experiments on of GaInP. These ﬁve resonances were assigned to the diﬀerent powdered AlGaAs ﬁlms. They were able to set an upper limit phosphorus nearest neighbors coordinations P[In4ÀnGan] with of S o 0.2 and S o 0.4 for CuAu and CuPt ordering n = 0–4. The sites are resolved in this spectrum due to their respectively. Although spin-echo experiments hinted at the diﬀerent isotropic chemical shifts. The presence of order in the presence of the other 75As coordinations, these proved to be sample is reﬂected in the statistics of the occurrence of the various too broad to be eﬃciently excited and resolved in the coordinations.21,22 The probabilities pn for the occupations of spectrum. the C[AnB4Àn] sites are Finding low values for the long range order parameter S indicates that the atomic distribution of A and B atoms p0 = pC[B4] becomes more or less random over their lattice positions. It may well be, however, that there is still a preference for local p1 = pC[A1B3] p2 = pC[A2B2] p3 = pC[A3B1] p4 = pC[A4] (4) Here, two types of ordering are considered for AxB1ÀxC type materials, see Fig. 1. CuPt ordering, with diagonal planes, stacked in the h111i direction or a CuAu, with alternating Fig. 1 Order in the face centered cubic lattice structure of AlGaAs. planes in h001i. For each conﬁguration, we can quantify the The black atoms are Arsenic, the other atoms are the Al and Ga probability of their occurrence. The 16 possible atomic con- cations. left 2-fold symmetry of the As nucleus for a AsAl2Ga2 ﬁgurations in the ﬁrst shell contribute each with a diﬀerent conﬁguration, CuAu ordered with planes alternating along the h001i probability: direction. right 3-fold tetrahedral symmetry of the As nucleus for a AsAl3Ga and AsAlGa3 conﬁguration, also called CuPt ordered lattice pCuPt = 1rB(1 À rA)3 + 1(1 À rA)rB3 0 2 2 with planes extending along the h111i direction. Coordination numbers and distances are 0.40 and 0.245 nm for the ﬁrst coordination pCuPt = 1(1 À rB)(1 À rA)3 + 1rArB3 + 3rArB(1 À rA)2 sphere, 1.20 and 0.40 nm for the second coordination sphere and 1.20 1 2 2 2 and 0.469 nm for the third coordination sphere. The lattice constant is + 3(1 À rB)(1 À rA)rB2 2 Ax = 0.566140 À 0.000809Áx nm.21 This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
View Online ordering of Al and Ga in the sample, e.g., when A atoms prefer can be obtained but this approach is still not satisfactory as Cq B atoms in their vicinity and vice versa. Alternatively, atoms and Z are treated independently, needing a starting value for Z. can show a tendency to be close neighbors which is referred to Moreover, the distribution is generated numerically lacking an as clustering. Short-range ordering and clustering is very analytical description. diﬃcult to detect with diﬀraction techniques as they cause A physically more sound model for describing the distribution only very weak diﬀuse scattering. In NMR the statistics of the of electric ﬁeld gradients in amorphous solids has been developed diﬀerent coordinations will be modulated by either short range by Czjzek et al.25 They proposed a joint probability densityDownloaded by RADBOUND UNIV. NIJMEGEN UNIVERSITEITSBIBLIOTHEK on 04 August 2010 order or clustering, which should be expressed in increased function numbers for p0 or p4. The exact eﬀect and size of the variations will depend on the scale and abundance of the locally 2 ÀVzz ð1 þ Z2 =3Þ ordered structures and may be diﬃcult to interpret depending PðVzz ; ZÞ ¼A Á exp 2s2 on the accuracy with which the spectral line intensities can be ð7Þ determined. V dÀ1 Á Zð1 À Z2 =9Þ A ¼ zz pﬃﬃﬃﬃﬃﬃ ; Published on 02 August 2010 on http://pubs.rsc.org | doi:10.1039/C003624B Besides modulating the intensities of the various C[AnB4Àn] 2psd resonances in the spectra it should be realized, however, that the distribution of A and B atoms over the lattice also aﬀects in which the average quadrupolar coupling constant depends the NMR parameters such as the chemical shift and the on s and d, see Table 1. In this work, we show that for a quadrupolar interaction parameters nq and Z of these sites. randomly ordered lattice, the quadrupolar distributions Especially for the quadrupolar interaction it is important to caused by a random occupancy of the A and B sites over consider that the interaction is not fully determined locally. the lattice can be described by a Czjzek distribution with Although the electric ﬁeld gradient tensor is dominated by the d = 5 for the symmetric C[A4] and C[B4] sites. These ﬁrst shell bonds, further coordination shells still have a distinct distributions are isotropic and rotation invariant. Eqn (7) is inﬂuence. This is witnessed by the fact that the symmetric derived from the fact that the real terms Um(m = 0–4) (deﬁned ﬁrst shell As coordinations, As[Al4] and As[Ga4], observed in references)25,26 of the EFG tensor are independently, by Degen et al.21 in AlGaAs, show non-zero quadrupolar normally distributed variables for amorphous solids with interactions with quadrupolar coupling constants of 800 and random ionic coordinations. The ﬁve independent spherical 600 kHz respectively. tensor elements Vi,k of the EFG tensor are linear combinations Structural disorder leads to distributions in chemical shift of these Um. A similar approach was introduced by Stock- ¨ and quadrupolar interactions, which aﬀect the measured line mann27 describing the eﬀect of randomly distributed defects in shapes in a characteristic way. Therefore, a detailed study of cubic crystals. the line shapes in disordered solids can be interpreted in terms As was outlined by Massiot and coworkers,28 the Czjzek of structural variations in these materials if one can extract the distribution can only be used if the number of structural probability density function for the quadrupolar interaction elements contributing to the EFG is suﬃciently large, meaning parameters from the spectra and compute the EFG tensors for that for a given coordination shell the coordination number a given structure, e.g., based on ﬁrst principles DFT based should be large. This will in general not be the case for the ﬁrst calculations. coordination shell. Therefore the Czjzek distribution can in Extraction of the probability density function from NMR principle only be used if the ﬁrst coordination sphere hardly spectra of randomly oriented powder samples is not trivial contributes to the EFG as is the case for the symmetric as the distribution in quadrupolar NMR parameters is As[Al4] and As[Ga4] sites in AlGaAs or the octahedrally and convoluted with the powder averaging over all possible tensor tetrahedrally coordinated aluminium sites discussed by orientations in the ensemble. In the NMR literature several Massiot et al.28 Clearly, the ﬁrst coordination sphere dominates approaches to model the probability density function(s) for the EFG for the asymmetric As[AlnGa4Àn] sites with n = 1,2,3 the quadrupolar interaction parameters can be found. The in AlGaAs. To deal with this situation an extension of the most basic approach is to assume Gaussian distributions Czjzek approach is needed, considering the case of an EFG in Cq and Z. Although such distributions are not physically contribution of a well-deﬁned local coordination of a given justiﬁed they regularly do produce line shapes similar to the atomic species and adding to that the eﬀect of disorder of more experimental ones and can therefore be used in spectral remote atomic shells and possible other random contributions deconvolutions in order to properly quantify the relative line to the total EFG. Due to the good lattice matching of the intensities. One should be careful with interpreting these results in terms of structural disorder, however, especially where the asymmetry parameter Z is concerned. First of all, Table 1 The ratio of the average hVzzi to the s in a Czjzek the structural parameters are not additively described by distribution as a function of the power factor d obtained by integrating eqn (7) over all Z for a given s quadrupolar parameters and are therefore not expected to be normally distributed and moreover nq and Z are coupled as d hVzzi/s they both depend on the local electronic coordination around 1 0.74 a nucleus. Jager et al.23 and Meinhold et al.24 have improved ¨ 2 1.17 on this basic model by assuming normal distributions in the 3 1.49 eigenvalues of the EFG tensor and applying constraints to 4 1.76 5 2.00 warrant the tracelessness of this tensor. Again good line shapes Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
View Online AlGaAs lattice, we neglect any stress in the lattice and propose case of a strong ﬁrst shell tensor Vzz(0) 4 40 Â 1020 V mÀ2. to use additivity for the local and remote contributions to these EFGs. This is implemented numerically by adding the ZZ ~0 ~0 U Á U ð0Þ (ﬁxed) contribution of the ﬁrst coordination shell to the Czjzek f ðVzz ; ZÞ / A Á exp 2s2 distribution resulting from (random) occupancy of Al and Ga ! lattice sites in the further coordination shells. First, the Vzz ð0Þ2 Á ð1 þ Zð0Þ2 =3Þ À Vzz Á ð1 þ Z2 =3Þ 2 principal EFG tensor values are generated from the Czjzek À dasinðbÞdbdg 2s2Downloaded by RADBOUND UNIV. NIJMEGEN UNIVERSITEITSBIBLIOTHEK on 04 August 2010 distribution using Vxx = ÀVzz(Z+1)/2 and Vyy = Vzz(Z À 1)/2. The resulting tensor V is then rotated over a set of Euler angles ! ! Zð0ÞVzz ð0Þa15 using Vrot = R(a, b, g) V R(a, b, g)T, with RT the transposed U 0 Á U 0 ð0Þ ¼ 2 Á Vzz Á Vzz ð0Þa11 þ pﬃﬃﬃ 3 rotation matrix. The relative orientation is randomized by taking all possible orientations into account. The ﬁrst shell 2 Á ZVzz Zð0ÞVzz ð0Þa55 þ pﬃﬃﬃ Vzz ð0Þa51 þ pﬃﬃﬃ ; EFG is added to this tensor after which it is diagonalized and 3 3 Published on 02 August 2010 on http://pubs.rsc.org | doi:10.1039/C003624B sorted using Vzz Z Vxx Z Vyy. The ﬁnal distribution is ð10Þ generated by integration over all possible angles on a sphere ensuring the isotropic character of the distribution. The resulting Vzz and Z values are placed on a new grid with with Aij(a, b, g) as deﬁned in ref. 26 and A as in eqn (7). For the weights of the original Czjzek distribution. For Al2Ga2, Z(0) = 0 this equation can be written into a modiﬁed Bessel the ﬁrst shell tensor V(0) is deﬁned by a single scalar value VZ1 function form (see eqn 61 in ref. 26). Practically, the (Vzz, Z) and Z(0) = 1 distribution for Z(0) = 0 showed good overlap with the previous equation in a simpliﬁed, approximated form without 2 3 integral ÀV Z1 0 0 V¼4 0 0 0 5 ð8Þ 0 0 V Z1 Vzz Á Vzz ð0Þ f ðVzz ; ZÞ $ A Á exp 2s2 The ﬁrst shell tensor of a Al1Ga3/Al3Ga1 site is deﬁned by VZ0 ! ÀVzz ð0Þ2 Á ð1 þ Zð0Þ2 =3Þ À Vzz Á ð1 þ Z2 =3Þ 2 and Z(0) = 0 2s2 2 3 ð11Þ ÀV Z0 =2 0 0 V ¼4 0 ÀV Z0 =2 0 5 ð9Þ 0 0 V Z0 allowing a much faster calculation. The only diﬀerence with the extended Czjzek distribution that was introduced The ‘‘extended Czjzek’’ distribution generated in this way previously, is the scaling of the tensor terms and the extra y coincides with the extended Gaussian Isotropic Model parameter. Although Le Caer and Brand26 use y as a ¨ (GIM) introduced by Le Caer and Brand.26 These authors ¨ parameter for local ordering (crystalline, mixed or random give an in-depth and general discussion of models for the domains), we consider this eﬀect simply by adjusting the distributions of electric ﬁeld gradients in disordered solids. magnitude of the ﬁrst shell tensor V(0), omitting the need Their extended Gaussian Isotropic Model (GIM) considers the for an extra variable in this approach. Substituting V(0) = 0 case of an EFG tensor V0 added to a random tensor back- corresponds to the original Czjzek distribution. Very recently ˜ ˜ ground contribution VGI with a ratio y, V = (1 À y)ÁV0+yÁ Le Caer and coworkers have further developed their extension ¨ ˜ VGI. Here we assign y = 0.5, since there is an equal contribu- of the Czjzek model in view of an application to 71Ga NMR tion of the EFG tensor from the higher coordination spheres spectra of chalcogenide glasses.29 and the ﬁrst coordination shell. Since a normalization is A ﬁnal important step for interpreting distributions in ˜ missing, it is more convenient to consider V = V0 + VGI. ˜ quadrupolar NMR parameters is being able to predict the Le Caer and Brand describe the quadrupolar distribution by ¨ quadrupolar parameters on the basis of a given structural an exponential expression of three terms, integrated over a model. The calculation of EFGs using Density Functional sphere, see ref. 26, using the Euler angles a, b, g. This equation theory (DFT) has been implemented in the Full Potential is slightly modiﬁed here into eqn (10). The nominator of the Linearised Augumented Plane Wave (FLAPW) method by ~ ~ exponent consists of a mix term U 0 ÁU(0), an oﬀset term which Herzig et al.30 and in the PAW (Projector Augmented Wave) is constant Vzz(0)2Á(1 + Z(0)2/3) and merely scales the method by Petrilli et al.31 Following the latter we implemented intensities and a Czjzek term Vzz2Á(1 + Z2/3). The ‘‘ordered’’ the EFG calculation in the ﬁrst-principles electronic structure ﬁrst shell tensor has magnitude Vzz(0) and an asymmetry program VASP (Vienna Ab initio Simulation Package).32–35 parameter Z(0). In comparison with Z(0) = 0, there are two An additional challenge in this context is to appropriately ~ ~ additional cross-terms in U 0 ÁU(0), when Z(0) = 1. Even model the eﬀects of the structural variations present in the though the oﬀset term is constant, as the ratio Vzz/s can material. In case of AlGaAs, however, the eﬀects of local become large, all exponential terms need to be placed inside charge (re)distributions arising from single lattice modiﬁcations the integral and cannot be placed in front, since individually are additive, so that various disordered conﬁgurations can be the exponential terms can become extremely large 410700 in modeled with relatively low computational cost. This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.
View Online Experimental acquisition time was 125 ms with 7.5 ms experimental delay before and after the 1801 pulse of each echo. The individual Sample preparation 500 point whole-echos were T2 weighted before co-adding, and The AlGaAs samples were grown by Metal Organic Vapor subsequently swapped around the echo maximum and Fourier Phase Epitaxy. Undoped 2 inch GaAs wafers with crystal transformed. A more detailed description of acquisition and orientation h100i, 15 degrees oﬀ towards h111i were used as processing of the QCPMG data is given in the supplementary substrates. A typical sample consisted of a 15 nm Si-doped material.Downloaded by RADBOUND UNIV. NIJMEGEN UNIVERSITEITSBIBLIOTHEK on 04 August 2010 AlAs buﬀer and a 5 mm undoped AlxGa1ÀxAs layer. A growth series with nominal aluminium fractions of x = 0, 0.297 was II. 75As Hahn-echo. Hahn-Echo experiments were produced. A sample with x = 0.489 was grown on a substrate obtained at 18.8 T, corresponding to a 75As frequency of 2 degrees oﬀ towards h110i. An epitaxial lift-oﬀ process36 was 136.93 MHz, using a Varian InﬁnityPlus Console Àp x;Àx Á applied to separate the AlxGa1ÀxAs layer from the substrate 2 À t1 À pÀy À t2 À ACQx;Àx . A recycle delay of 1 s by selectively etching the intermediate Si-doped AlAs layer was employed. Whole echoes were recorded which were Published on 02 August 2010 on http://pubs.rsc.org | doi:10.1039/C003624B with a hydrogen ﬂuoride solution. The AlxGa1ÀxAs thin ﬁlms processed by swapping the data around the echo maximum (Emg quantities) were crushed into a powder with a typical followed by apodization (exponential) and zero-ﬁlling to grain size of a few micrometres and subsequently transferred 8192 points before Fourier transformation. The spectral width to plastic sample holders for NMR measurements. The sample was 5 MHz. The x E 0.297 Hahn-echo spectrum was acquired holders were made of PolyEtherEtherKeton (PEEK). with 75.000 scans and echo times of t1 = 200 ms, t2 = 75 ms and 25 ms delay from the acquisition and receiver delay. The NMR experiments 901 and the 1801 pulse times were 0.2 ms respectively 0.4 ms. 166.000 scans were averaged in the x E 0.489 Hahn-echo data A home-built, static XH probe operating at 137 MHz for with t1 = 175 ms, t2 = 5 ms and 20 ms delay from the arsenic (18.8 T) was designed for achieving high rf ﬁelds. The acquisition and receiver delay. coil was a 5-turn solenoid with a 1.2 mm internal diameter, made of a silver-plated, 220 mm diameter wire. The sample III. 69Ga Nutation and 3QMAS. A series of 69Ga static space was conﬁned to a length of 2 mm and a diameter of nutation spectra using diﬀerent rf-ﬁeld strengths were obtained 0.6 mm. The Q factor of the probe was determined to be E38. for Al0.297Ga0.703 As at 14.1 T on a Chemagnetics Inﬁnty Two Allen-Bradley 1 kO high power resistors were put parallel 600 MHz spectrometer. The same probe was used for these to the tuning capacitor to reduce the Q factor to E 20 to allow experiments, tuned to 143.99 MHz with a Q factor of B50. A frequency-stepped experiments without retuning the probe. z-ﬁltered 69Ga 3QMAS spectrum, using hyper-complex The rf-ﬁeld strength was determined by obtaining 75As nutation States TPPI,37 was recorded at 9.4 T, 96.013087 MHz, on a spectra from a sample of GaAs. In this sample As occupies a Chemagnetics Inﬁnity 400 MHz spectrometer. Here a spinning cubically symmetric lattice and therefore the signal nutates speed of 15 kHz was employed, using a Chemagnetics 2.5 mm with frequency nrf. rf ﬁeld strength up to 800 kHz were HX probe. achieved using approximately 800 Watts of power. In the QCPMG and Hahn-echo pulse experiments very short pulses IV. Spin–lattice relaxation times. To ensure quantitative are used so one has to take into account that the maximum evaluations of the spectra, spin–lattice relaxation times were voltage is never reached due to the pulse rise time depending determined for Al0.297Ga0.703As using saturation recovery on the Q factor of the probe (see supplementary material). experiments, as summarized in Table 2. The values for this Therefore actual rf ﬁelds amounted to 390 kHz and 625 kHz in x E 0.297 composition do not diﬀer strongly from the the QCPMG and Hahn-echo experiment respectively. values determined earlier21 for a sample with composition Al0.498Ga0.511As, indicating no signiﬁcant changes in relaxation I. 75As QCPMG. A well-established experiment for mechanism for diﬀerent compositions. The T1 of 75As in bulk acquiring very wide-line spectra with high sensitivity is the GaAs was 0.33 Æ 0.05 s. frequency-stepped QCPMG experiment.15–18 For such an experiment it is useful to reduce the probes’ Q factor (here Spectral deconvolution to 20). The ﬂat response of the probe minimizes the phase variations as a function of frequency. This is advantageous An important objective of this study is to characterize the 75As since the probe does not need to be re-tuned and the coordinations in AlxGa1ÀxAs and determine their relative intensity loss at large frequency oﬀsets is almost negligible. intensities. The analysis can become complex due to the presence of several overlapping resonances with a distribution 24 respectively 22 experiments were performed at 18.8 T using in NMR interaction parameters. Here we resort to ﬁtting a Varian InﬁnityPlus console. The frequency step size was 125 kHz covering a spectral width of 3 MHz for Al0.297Ga0.703As Table 2 T1(s) relaxation times obtained using saturation recovery at and Al0.489Ga0.511As. The x E 0.297 spectrum was recorded 14.1 Tesla with 10.000 scans and the x E 0.489 with 5.400 scans at each frequency step. Because T2 is long, the number of echoes Al0.297Ga0.703As Al0.489Ga0.511As21 acquired (242) was limited by experimental limitations 69 Ga — 0.77Æ0.04 71 (121.000 data points in 30 ms). The 901 and 1801 pulse times Ga 0.11Æ0.04 0.15 Æ 0.08 27 were 0.32 ms and 0.64 ms respectively. A repetition delay of 1 s Al 18 Æ 1 16.1 Æ 0.4 75 As 0.27 Æ 0.02 0.28 Æ 0.03 was used, acquiring a spectral width of 4 MHz. Each echo Phys. Chem. Chem. Phys. This journal is c the Owner Societies 2010
View Online procedures based on an evolutionary algorithm to provide in negligible changes (o0.002 nm) in the atom positions. All knowledge about intensities, NMR parameters and their results below were obtained using the PBE functional.44 The distribution, and the associated error margins. The program PAW datasets have frozen [Ar], [Ne] and [Ar]3d10 cores for is based on the work by Meerts38 and will be described in Ga, Al and As respectively. On the basis of these results, we detail in a forthcoming publication. The algorithm uses a think that the obtained quadrupolar distributions are a good library database of time domain data sets which were generated representation for this lattice. using SIMPSON.39,40 In this way the experimental conditionsDownloaded by RADBOUND UNIV. NIJMEGEN UNIVERSITEITSBIBLIOTHEK on 04 August 2010 can be mimicked accurately for each experiment. For each site a group of simulated data sets are added to describe a Results and discussion distribution in interaction parameters. The weight of each 75 I. As QCPMG analysis set is determined by the selected distribution function, e.g., given by the work of Czjzek25 and Le Caer and Brand.26 For ¨ The high quadrupolar moment of 75As and its strategic anion the Hahn-echo spectra, approximately 1600 datasets were position in the AlGaAs lattice makes it the most interesting Published on 02 August 2010 on http://pubs.rsc.org | doi:10.1039/C003624B generated based on the experimental rf-ﬁeld strength and nucleus to probe for information about the symmetry and bandwidth. In the simulations, the maximum of the echo ordering of the crystal lattice. As detailed in the experimental was recorded to identify the intensity contribution for each section, frequency-stepped QCPMG experiments15–18 were site. For the QCPMG spectra a slightly diﬀerent approach was used to obtain good sensitivity 75As spectra with signals followed. Calculating all echoes in a QCPMG experiment for dispersed over 2 MHz. Fig. 2 displays the summation of the each quadrupolar interaction is too time consuming and experimental QCPMG subspectra for each frequency oﬀset for therefore only a single echo was calculated assuming ideal Al0.297Ga0.703As and Al0.489Ga0.511As. The spectra show two pulses. Each subset was Fourier transformed into ﬁve absolute narrow resonances separated by d = 188 Æ 5 ppm, in simulated spectra with ideal lineshapes but including accordance with previous work45 these lines are assigned to quadrupolar distributions, which were least square ﬁtted to the signals of As[Ga4] and As[Al4] sites having a symmetric the absolute, experimental spectrum. To cover all values ﬁrst coordination sphere. An arsenic nucleus surrounded appearing in the spectra, the quadrupolar interaction by four aluminium or gallium nuclei, has a tetrahedrally parameters of the As[Ga4] and As[Al4] site were varied over symmetric structure and should therefore display no quadrupolar a range of 0 r Z r 1, 0 r Cq r 2 MHz while the As[Al2Ga2] interaction. However, depending on the positions of the site had a range of 0.9 r Z r 1, 31 r Cq r 35 MHz and cations in the third and higher coordination spheres, there the As[Al1Ga3] and As[Al1Ga3] site 0 r Z r 0.1, 31 r Cq r can still exist a (relatively small) EFG. Indeed, in the work by 35 MHz. In all ﬁts, the cost function was calculated using a Degen et al.21 a Cq = 610 kHz and Z 4 0.97 was determined normalized, least square summation for As[Ga4] and Cq = 820 kHz, Z 4 0.88 for As[Al4]. Here we show that these sites can be properly ﬁtted to a Czjzek X fi n gi 2 w2 ¼ À ; ð12Þ distribution, with d = 5 and average quadrupolar values of i¼1 fmax gmax Cq = 610 and 820 kHz, as were obtained from 75As nutation NMR experiments.21 The average quadrupolar coupling here f is the experimental spectrum and g the calculated constants for these two sites were ﬁxed as the ﬁt was not very spectrum from the library and n the number of points. The sensitive to small changes in this parameter. calculations were performed using up to 20 nodes of a high The wide resonances, not directly observed in the previous speed cluster based on Sunﬁre X4100 computers. study, show distinct quadrupolar features of second-order central-transition powder patterns. A line shape with an Density functional theory asymmetry parameter Z close to zero is easily identiﬁed. The Electronic structure calculations were carried out to relate high signal to noise ratio allowed visualization of the small but quadrupolar interaction parameters to the diﬀerent cation distinctive features extending from both sides of the spectrum conﬁgurations. The local electric ﬁeld gradients have been of the spectrum indicating the presence of a species with a high calculated with the ﬁrst-principles molecular-dynamics asymmetry parameter. The broad resonances are assigned to programme VASP,32–35 using density functional theory and 75 As resonances with an asymmetric ﬁrst coordination shell. the PAW method41,42 following the approach by Petrilli Based on a simple point charge model46 we calculate a et al.31 several convergence tests were carried out, in particular quadrupolar coupling constant of approximately 33 MHz to estimate the eﬀect of the semi-core states and compared for these coordinations. The asymmetry parameter is predicted with state-of-the-art quantum chemical calculations using to be Z = 1 for the As[Al2Ga2] coordination and Z = 0 for the DALTON.43 A detailed summary of the convergence test, As[Al1Ga3] and As[Al1Ga3] coordinations. The enormous molecular simulations on small model systems for AlGaAs width of the spectrum makes it diﬃcult to determine the and the choice of DFT functional, can be found in the chemical shift of these resonances with any accuracy. supplementary material. Convergence of the results with The spectra were ﬁtted using an evolutionary algorithm38 to respect to k-point sampling, kinetic energy cut-oﬀ of the plane deconvolute the signals into the contributions from each wave basis set, and lattice relaxation was checked as well. AlAs individual 75As site as is shown in Fig. 2. The diﬀerent has an experimental lattice spacing of 0.56695 nm and GaAs quadrupolar subspectra used as library spectra for ﬁtting the of 0.56614 nm, see ref. 21, thus the lattice mismatch for QCPMG spectra were calculated assuming ideal non-selective Al0.5Ga0.5As is r 0.1%. Relaxation of the structure resulted excitation in SIMPSON.39 The ﬁt with the lowest average cost This journal is c the Owner Societies 2010 Phys. Chem. Chem. Phys.