Al Ga As Pccp Paulus

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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 Wijsa
Downloaded 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

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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 this

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

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

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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)3

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

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


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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 density

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 ;

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


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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
2s2

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

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.


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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.

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

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


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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 conditions

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

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


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