current address: ]Lumileds Lighting Co., San Jose, CA 95131 USA.

# Crystal Truncation Rods from Miscut Surfaces with Alternating Terminations

Guangxu Ju<EMAIL_ADDRESS>[  Materials Science Division, Argonne
National Laboratory, Lemont, IL 60439 USA  Dongwei Xu  Materials Science
Division, Argonne National Laboratory, Lemont, IL 60439 USA  School of Energy
and Power Engineering, Huazhong University of Science and Technology, Wuhan,
Hubei 430074, China  Carol Thompson  Department of Physics, Northern Illinois
University, DeKalb, IL 60115 USA  Matthew J. Highland  X-ray Science Division,
Argonne National Laboratory, Lemont, IL 60439 USA  Jeffrey A. Eastman
Materials Science Division, Argonne National Laboratory, Lemont, IL 60439 USA
Weronika Walkosz  Department of Physics, Lake Forest College, Lake Forest, IL
60045 USA  Peter Zapol  Materials Science Division, Argonne National
Laboratory, Lemont, IL 60439 USA  G. Brian Stephenson<EMAIL_ADDRESS>Materials Science Division, Argonne National Laboratory, Lemont, IL 60439 USA

(revision resubmit: February 18, 2021)

###### Abstract

Miscut surfaces of layered crystals can exhibit a stair-like sequence of
terraces having periodic variation in their atomic structure. For hexagonal
close-packed and related crystal structures with an $\alpha\beta\alpha\beta$
stacking sequence, there have been long-standing questions regarding how the
differences in adatom attachment kinetics at the steps separating the terraces
affect the fractional coverage of $\alpha$ vs. $\beta$ termination during
crystal growth. To demonstrate how surface X-ray scattering can help address
these questions, we develop a general theory for the intensity distributions
along crystal truncation rods (CTRs) for miscut surfaces with a combination of
two terminations. We consider half-unit-cell-height steps, and variation of
the coverages of the terraces above each step. Example calculations are
presented for the GaN $(0001)$ surface with various reconstructions. These
show which CTR positions are most sensitive to the fractional coverage of the
two terminations. We compare the CTR profiles for exactly oriented surfaces to
those for vicinal surfaces having a small miscut angle, and investigate the
circumstances under which the CTR profile for an exactly oriented surface is
equal to the sum of the intensities of the corresponding family of CTRs for a
miscut surface.

##  I Introduction

Crystal truncation rods (CTRs) are features in the scattering patterns of
crystals that arise from the truncation of the bulk crystal lattice at a
surface [1]. They consist of streaks of scattering intensity extending away
from all Bragg peaks in the direction normal to the crystal surface. The
intensity distributions along the CTRs provide a sensitive measure of the
atomic structure of the surface. In particular, they provide information about
the termination plane of the crystal and the reconstruction of surface layers
[2], as well as the arrangement of atomic-scale steps on the surface [3]. In
situ X-ray measurements of CTRs have been used to determine the atomistic
mechanisms of crystal growth [4, 5, 6, 7, 8, 9, 10], such as the classical
homoepitaxial growth mechanisms of 1-dimensional step flow, 2-dimensional
island nucleation and coalescence, and 3-dimensional roughening [11, 12].
Because of their penetrating nature and atomic-scale sensitivity, X-ray
methods are uniquely suitable for in situ studies in non-vacuum environments.
Here we calculate the behavior of CTR intensities to address a long-standing
issue in crystal growth, the variation in the morphology of surfaces having
alternating terminations and step types, driven by differences in step
kinetics during step-flow growth.

Figure 1: Miscut surfaces with (a) identical terraces and (b) terraces with
alternating structures. Alternating structures often arise for crystal
structures with a screw axis normal to the surface. In this case half-unit-
cell-height steps separate terraces with differing orientations of the
terminating layer. The alternating structures and kinetic properties of the
steps can lead to unequal terrace fractions.

The simplest atomic model of a miscut (vicinal) crystal surface consists of a
stair-like sequence of identical terraces, separated by identical steps,
typically of full-unit-cell height, as shown in Fig. 1(a). The average step
spacing is determined by the miscut angle of the surface (i.e. the angle of
the surface away from the closest low-index crystallographic orientation).
During stable step flow growth, the terrace widths are typically all nearly
equal, since all steps have identical kinetic properties. However, miscut
surfaces can acquire more complex morphologies, with terraces separated by
fractional-unit-cell-height steps, and step structures and properties that
vary from step to step, as shown in Fig. 1(b). A typical case is a unit cell
with a multiple-layer structure [2]. In particular, when the space group of
the crystal includes a screw axis or a glide plane, the surface nearly
perpendicular to this symmetry element can have a sequence of terraces with
the same atomic arrangement, but different in-plane orientations. This gives
differences in the structure and kinetics of neighboring steps, leading to
fundamentally new crystal growth characteristics that produce fascinating
terrace morphologies [13].

A commonly encountered version of this effect occurs on the basal-plane
$\\{0001\\}$-type surfaces of crystals having hexagonal close-packed (HCP) or
related structures, which are normal to a $6_{3}$ screw axis. The close-packed
layers in HCP crystals have 3-fold symmetry alternating between
$180^{\circ}$-rotated orientations from layer to layer, as shown by the
$\alpha$ and $\beta$ terrace structures in Fig. 2. The
$\alpha\beta\alpha\beta$ stacking sequence typically results in half-unit-
cell-height steps on miscut surfaces. Often the lowest energy steps are normal
to $[01\overline{1}0]$-type directions. The alternating structures of these
steps are conventionally labelled $A$ and $B$ [14, 12] as shown on Fig. 3.
When the in-plane azimuth of an $A$ step changes by $60^{\circ}$, e.g. from
$[01\overline{1}0]$ to $[10\overline{1}0]$, its structure changes to $B$, and
vice versa. Differences in the dynamics of adatom attachment at $A$ and $B$
steps have strong effects on the surface morphology produced during growth.

Figure 2: Structure of $\alpha$ and $\beta$ terraces of the (0001) surface of
an HCP-type crystal, e.g. the Ga sites in wurtzite-structure GaN. Red triangle
of top-layer sites around $6_{3}$ screw axis shows difference between
alternating $\alpha$ and $\beta$ layers. Blue rhombus shows conventional HCP
unit cell; green rectangle shows orthohexagonal unit cell. Axes give
coordinates in terms of orthohexagonal lattice parameters $a$ and
$b=\sqrt{3}a$.  Figure 3: Terrace and step structure of an $(0001)$ surface of
an HCP-type crystal with regions miscut in two different azimuths. Circles
show in-plane positions of top-layer atoms on each terrace, with color
indicating height. Orthohexagonal lattice parameters are $a$, $b$, and $c$.
Steps of height $c/2$ typically have lowest edge energy when they are normal
to $[01\overline{1}0]$, $[10\overline{1}0]$, or $[1\overline{1}00]$. Steps of
a given azimuth have alternating structures, $A$ and $B$. The step structure
changes from $A$ to $B$ or $B$ to $A$ when they change azimuth by
$60^{\circ}$.

Images of $\\{0001\\}$ surfaces suggesting the alternating nature of the steps
have been obtained for several HCP-related systems, including SiC [15, 16,
17], GaN [14, 18, 19, 20, 21, 22, 23, 24, 25], AlN [26], and ZnO [27]. Such
images typically indicate a tendency for local pairing of steps (i.e.
alternating step spacings), consistent with predictions that $A$ and $B$ steps
can have significantly different energies and/or attachment kinetics [14, 21,
24, 28, 29, 30, 31, 32, 33]. In particular, different attachment kinetics at
$A$ and $B$ steps can produce a tendency to step pairing during growth and
thus to different local fractions of $\alpha$ and $\beta$ terraces. In
limiting cases, the $\alpha$ terrace fraction $f_{\alpha}$ can approach zero
or unity, when pairs of half-unit-cell-height steps join to form full-unit-
cell-height steps. However, for $\\{0001\\}$ surfaces of HCP-related systems
it has been difficult to distinguish experimentally the terrace orientation,
and thus to determine whether a given set of steps is of $A$ or $B$ type. Thus
it has not been possible to experimentally determine whether $A$ or $B$ steps
have faster adatom attachment kinetics. This can be solved by a method to
distinguish the fraction of $\alpha$ and $\beta$ terraces, especially an in
situ measurement in the relevant growth environment.

Streaks of scattering intensity extending away from Bragg peaks normal to the
surfaces of finite-sized crystals are inherent in exact treatments of X-ray
scattering, extending back to early work [34, 35]. As synchrotron sources
enabled detailed study of these surface-sensitive features, theoretical
treatments specific to CTRs from truncated perfect crystals were developed
[36, 1]. Subsequent work included the effect of overlayers or reconstructions
[37, 2] and miscut surfaces [38, 39, 40, 41, 3]. Previous general treatments
of miscut surfaces typically considered only full unit cell steps; half-unit
cell steps on Si $(001)$ have also been considered [39, 41].

Here we develop expressions for the CTR intensity distributions for surfaces
with two terrace types, $\alpha$ and $\beta$. We explicitly include the
effects of surface reconstructions. We consider straight steps, periodically
spaced. Surface roughness is accounted for by applying a roughness factor, as
derived in prior work [3]. New aspects of our calculations include a general
treatment of half-unit-cell-height steps, with fractional surface coverage of
two terrace types; generalization from a cubic to an orthorhombic bulk unit
cell, which allows us to consider HCP type crystals using orthohexagonal
coordinates; and the effects of absorption and extinction. We use a
mathematical formalism that emphasizes how exact results can be obtained
through the summation of geometric sequences. We compare the CTR profiles for
exactly-oriented and miscut surfaces, and consider the relationships between
the two cases.

To provide a concrete example, we make calculations for the GaN $(0001)$
surface, for which we have recently carried out a surface X-ray scattering
study [42]. We show how the CTRs vary as a function of the surface fraction of
$\alpha$ vs $\beta$ terraces. This demonstrates that surface X-ray scattering
can be used to distinguish the fraction of the surface covered by $\alpha$ or
$\beta$ terraces during growth, and thus unambiguously determine differences
in the attachment kinetics at $A$ and $B$ steps.

##  II Surface X-ray Scattering Theory

In this section we develop expressions for the intensity distributions along
CTRs and demonstrate their sensitivity to the surface termination
arrangements. The X-ray reflectivity along the CTRs can be calculated by
adding the complex amplitudes from the bulk crystal and the terminating
overlayers, with proper phase relationships. We start with the simple case of
an exactly-oriented surface, and then extend this to the case of a vicinal
surface having an array of straight steps. After developing general
expressions, we show calculations for the GaN $(0001)$ surface for both cases.

Figure 4: (a) Hexagonal pattern of CTRs at fixed $L=0.5$, for a GaN basal
plane surface. Axes give $HK$ in orthohexagonal units; Miller-Bravais indices
are listed by each CTR. (b) Reciprocal space map of CTRs for a GaN basal plane
surface. Color indicates variation in reflectivity as a function of $L$. Axes
give $HKL$ in orthohexagonal units.  Figure 5: Color indicates reflectivity of
CTRs as a function of $L$ and $\Delta K\equiv K-K_{0}$. Width of CTRs in $K$
is arbitrarily set to $0.001$ r.l.u. (a) CTRs for an exactly oriented GaN
$(001)$ surface with no reconstruction and $f_{\alpha}=0$, for various $H_{0}$
and $K_{0}$ shown at top. (b) CTRs for a miscut GaN $(001)$ surface with
$M=100$, no reconstruction, and $f_{\alpha}=0$, for various $H_{0}$, $K_{0}$,
and $L_{0}$. Orthohexagonal indices are used for $HKL$.

Although our motivation for this work is based on hexagonal crystals, it is
more convenient to use the Cartesian coordinates of an orthorhombic unit cell.
As shown in Figs. 2 and 4, the hexagonal lattice can be mapped to an
orthorhombic lattice using a doubled unit cell and an orthohexagonal
coordinate system [43]. The lattice parameters of the orthorhombic unit cell
are $a$, $b$, and $c$ in the $x$, $y$, and $z$ directions, respectively. The
corresponding Cartesian components of the scattering wavevector $\mathbf{Q}$
are given by $(Q_{x},Q_{y},Q_{z})$, which are related to reciprocal lattice
units $HKL$ by $Q_{x}=(2\pi/a)H$, $Q_{y}=(2\pi/b)K$, $Q_{z}=(2\pi/c)L$. Bragg
peaks locations have integer indices $H_{0}K_{0}L_{0}$ in reciprocal lattice
units.

For a hexagonal crystal, the orthohexagonal coordinate system has a lattice
parameter $b\equiv a\sqrt{3}$, where $a$ is the in-plane lattice parameter of
the conventional hexagonal unit cell, as shown in Fig. 2. The out-of-plane
lattice parameter $c$ is the same in both coordinate systems. This gives
Cartesian $x$, $y$, and $z$ axes parallel to the hexagonal
$[2\overline{1}\overline{1}0]$, $[01\overline{1}0]$, and $[0001]$ directions,
respectively. In reciprocal space, the orthohexagonal coordinates $HKL$ are
related to the hexagonal Miller-Bravais reciprocal space coordinates $hki\ell$
by $H=h$, $K=h+2k$, $L=\ell$, as shown in Fig. 4. In particular, the hexagonal
$(10\overline{1}0)$ and $(01\overline{1}0)$ correspond to the orthohexagonal
$(110)$ and $(020)$, respectively. Because of the doubled unit cell,
orthohexagonal indices $H$ and $K$ must both be even or odd integers at
allowed Bragg peaks. For the remainder of this paper, we will use 3-index
orthohexagonal indices $HKL$ rather than 4-index hexagonal Miller-Bravais
indices $hki\ell$.

###  II.1 Exactly oriented surface with two terminations

For an exactly oriented surface normal to the $z$ direction, the CTRs extend
continuously in the $Q_{z}$ direction at fixed $Q_{x}$ and $Q_{y}$ through
each Bragg peak. The CTRs from the Bragg peaks of different $L_{0}$ at the
same $H_{0}K_{0}$ all overlap, as shown in Fig. 5(a).

The contribution to the complex amplitude $r_{\mathrm{bulk}}$ of the
reflectivity from the truncated bulk crystal below the terminating overlayers
can be obtained by coherently summing the scattering from each unit cell in a
column normal to the surface, taking into account a factor $Z$ difference
between the scattering amplitude from each unit cell [37, 44]. The sum of this
geometrical sequence is given by

$r_{\mathrm{bulk}}=r_{\mathrm{f}}\,F_{\mathrm{bulk}}\sum_{m=-\infty}^{0}Z^{m}=r_{\mathrm{f}}\,F_{\mathrm{bulk}}\,\frac{Z}{Z-1},$
(1)

where $r_{\mathrm{f}}\equiv 4\pi ir_{0}/AQ$, $r_{0}=2.817\times 10^{-13}$ cm
is the Thomson radius of the electron, $A$ is the in-plane area of the unit
cell, and $Q$ is the magnitude of the wavevector. The bulk structure factor
$F_{\mathrm{bulk}}$ is

$F_{\mathrm{bulk}}=\sum_{k}g_{k}(Q)\sum_{n}\exp(i\mathbf{Q}\cdot\mathbf{r}_{kn}^{\mathrm{bulk}}).$
(2)

Here the first sum is over the $k$ chemical elements present in the crystal,
$g_{k}(Q)\equiv f_{k}(Q)\exp(-u_{k}^{2}Q^{2}/2)$ is the atomic form factor
$f_{k}(Q)$ modified by the Debye-Waller thermal vibration length $u_{k}$ for
element $k$, the second sum is over the $n$ bulk atoms of type $k$ in a unit
cell, and $\mathbf{r}_{kn}^{\mathrm{bulk}}$ is the position of bulk atom $n$
of type $k$. Unlike Bragg peak intensities for bulk crystals, CTR profiles
depend upon the choice of the boundaries of the unit cell. Shifting some atoms
between the top and bottom of the unit cell by a lattice translation does not
change $F_{\mathrm{bulk}}$ at the $\mathbf{Q}$ of Bragg peaks, but it does
change $F_{\mathrm{bulk}}$ at $\mathbf{Q}$ values along the CTR between the
Bragg peaks. The unit cell boundaries used for
$\mathbf{r}_{kn}^{\mathrm{bulk}}$ determine the surface termination, which
generally affects the CTR profile [2].

The quantity $Z$ in Eq. (1) is the ratio of the scattering amplitude from one
unit cell to that from the unit cell at $\Delta z=-c$ below it. It consists of
a phase factor and an absorption factor,

$\displaystyle Z$ $\displaystyle\equiv\exp(iQ_{z}c)\,\exp(\epsilon c/Q_{z})$
$\displaystyle=\exp(2\pi iL)\,\exp(\epsilon c^{2}/2\pi L),$ (3)

where $\epsilon=4\pi/(\lambda\ell_{\mathrm{abs}})$ is related to the photon
wavelength $\lambda$ and absorption length $\ell_{\mathrm{abs}}$. Typically
the absorption factor is very close to unity and only becomes important in
$Z-1$ when the phase factor is unity, e.g. at Bragg peaks. One can see from
Eq. (1) that the reflectivity amplitude is built up by summing the scattering
from each layer of the semi-infinite crystal in the $z$ direction from
$m=-\infty$ to $m=0$.

Figure 6: Bulk unit cells (black) and reconstructed unit cells (red), for an
exactly oriented surface with an island of $\alpha$ termination in a $\beta$
matrix. The extra half unit cells producing the shift from the $\beta$ to the
$\alpha$ termination are shown in green. Blue shade indicates index $m$ of sum
in Eq. (1), with final term $m=0$ darkest.

To provide a direct comparison with the results for miscut surfaces obtained
below, we allow the surface to have a mixture of two terminations, $\alpha$
and $\beta$. For an exactly oriented surface, this consists of islands of
$\alpha$ in a matrix of $\beta$, or vice versa, as shown in Fig. 6. We also
allow reconstructions in which the atoms in the top layer of unit cells at the
surface are relaxed from their bulk crystal positions, and there can be extra
atoms bonded to the surface. The reconstructed reflectivity amplitude per unit
area of the $x=\alpha$ or $\beta$ overlayer is

$r_{\mathrm{rec}}^{x}=r_{\mathrm{f}}\,\,F_{\mathrm{rec}}^{x}\,\,Z,$ (4)

where the structure factor of the reconstruction $F_{\mathrm{rec}}^{x}$ is

$F_{\mathrm{rec}}^{x}=\sum_{j}\theta_{xj}\sum_{k}g_{k}(Q)\sum_{n}\exp(i\mathbf{Q}\cdot\mathbf{r}_{jkn}^{x}).$
(5)

Here the first sum is over the $j$ possible domain orientations of the
reconstruction, $\theta_{xj}$ is the fraction of domain $j$ for the $x=\alpha$
or $\beta$ component, the second sum is over the $k$ chemical elements present
in the reconstruction, the third sum is over the $n$ atoms of type $k$ in a
unit cell, and $\mathbf{r}_{jkn}^{x}$ is the position of atom $n$ of type $k$
in domain orientation $j$ for the $x=\alpha$ or $\beta$ component. The total
reflectivity amplitude is the sum of the complex amplitudes from the bulk and
both reconstructed components,

$r_{\mathrm{tot}}=r_{\mathrm{bulk}}+f_{\alpha}r_{\mathrm{rec}}^{\alpha}+(1-f_{\alpha})r_{\mathrm{rec}}^{\beta},$
(6)

where $f_{\alpha}$ is the fraction of the surface covered by the $\alpha$
termination.

The total reflectivity amplitudes calculated above are for the kinematic limit
in which they are much smaller than unity. Near the Bragg peaks, where the
reflectivity amplitude approaches unity, the kinematic expressions can be
corrected using [44]

$r_{\mathrm{tot}}^{dyn}=\frac{2r_{\mathrm{tot}}}{1+\sqrt{1+4|r_{\mathrm{tot}}|^{2}}},$
(7)

which insures that the amplitude of the reflectivity does not exceed unity.
The intensity reflectivity is the square of the modulus of the amplitude
reflectivity,

$R=|r_{\mathrm{tot}}^{dyn}|^{2}\,S,$ (8)

where the final factor $S$ has been introduced to account for surface
roughness. Here we use a form for the roughness factor as a function of $L$
for the exactly oriented surface adopted from that obtained for a miscut
surface [3],

$\displaystyle S$ $\displaystyle=\sin^{2}(\pi
L)\sum_{L_{0}=-\infty}^{\infty}\frac{\exp\big{[}-\sigma_{R}^{2}(2\pi/c)^{2}(L-L_{0})^{2}\big{]}}{\pi^{2}(L-L_{0})^{2}}$
$\displaystyle=1-\sin^{2}(\pi
L)\left(\frac{4\sigma_{R}}{\pi^{1/2}c}-\sum_{n=1}^{\infty}\cos(2\pi
nL)\times\right.$
$\displaystyle\left.\left[4n\,\mathrm{erfc}\left(\frac{nc}{2\sigma_{R}}\right)-\frac{8\sigma_{R}}{\pi^{1/2}c}\exp\left(-\frac{n^{2}c^{2}}{4\sigma_{R}^{2}}\right)\right]\right)$
$\displaystyle\approx 1-(4\sigma_{R}/\pi^{1/2}c)\sin^{2}(\pi L),$ (9)

where $\sigma_{R}$ is the surface roughness (related to the total step
roughness given in [3] by $\sigma_{R}=c\,\tilde{\sigma}_{tot}$), and the final
approximation holds for $\sigma_{R}<0.3c$. Figure 7 shows plots of this
function for various values of $\sigma_{R}/c$.

Away from the Bragg peaks, so that the dynamical scattering correction can be
neglected, the intensity reflectivity can be written as

$R=|r_{\mathrm{f}}|^{2}\,|Z|^{2}\,\left|\frac{F_{\mathrm{bulk}}}{Z-1}+f_{\alpha}F_{\mathrm{rec}}^{\alpha}+(1-f_{\alpha})F_{\mathrm{rec}}^{\beta}\right|^{2}S.$
(10)

Figure 7: Roughness factor $S$ as a function of $L$, for various values of
$\sigma_{R}/c$ given in the legend.  Figure 8: Calculated reflectivity as a
function of $L$ of CTRs for an exactly oriented GaN $(001)$ surface with no
reconstruction. Left column: (0 0 L); middle column: $(02L)$; right column:
$(11L)$. Values of $f_{\alpha}$ are given for each row.  Figure 9: Calculated
reflectivity of CTRs for an exactly oriented GaN $(001)$ surface with various
reconstructions, for $f_{\alpha}=0$.

###  II.2 Exactly oriented GaN $(001)$ surface

To illustrate the behavior for the basal plane of an HCP-type system, we
consider Ga-face GaN $(001)$ surfaces with a Ga termination for the bulk, so
that the top-layer atoms shown in Fig. 3 are Ga. Table 1 in Appendix A lists
the atomic coordinates used for the bulk. We use atomic coordinates for the
reconstructed overlayers calculated previously [45]. Since these were
calculated using a $2\times 2$ unit cell, for consistency the unit cell sums
used in calculating the structure factors here are carried out over two
adjacent orthohexagonal unit cells having an area $A=2ab$, which is normalized
out in the denominator of $r_{\mathrm{f}}$. To compute the scattering from
surfaces terminated at $\alpha$ and $\beta$ terraces, we terminate the bulk at
a $\beta$ terrace, and incorporate an extra half unit cell of atoms in their
bulk positions into the bottom of the reconstructed overlayer for the $\alpha$
terrace regions. We also reverse the relaxation amounts in the $y$ direction
for the $\alpha$ terraces, relative to those for the $\beta$ terraces. Figure
6 illustrates these arrangements. Appendix A and Supplemental Tables I-IX [46]
give values of atomic coordinates $\mathbf{r}_{jkn}^{x}$ used for the
$x=\alpha$ and $\beta$ structure factors for each of the reconstructions
considered. For all calculations presented in this paper, we use a photon
energy of $25.78$ keV ($\lambda=0.4809$ Å) to correspond with recent
experiments [42]. We use atomic form factors for each type of atom [47] with
resonant corrections for this energy [48], an absorption length of
$\ell_{\mathrm{abs}}=103$ $\mu$m for GaN at this energy, a Debye-Waller length
of $u_{k}=0.16$ Å for all atoms, and a surface roughness of $\sigma_{R}=1$ Å.

Figure 8 shows the calculated reflectivity as a function of $L$ for different
integer $H_{0}K_{0}$ values and three $\alpha$ fractions, $f_{\alpha}=0$,
$0.5$ and $1$. Here we show calculations for an unreconstructed surface, with
all atoms in their bulk positions. The $(00L)$ CTR is insensitive to the
difference between the $\alpha$ and $\beta$ terminations ($f_{\alpha}=1$ and
$f_{\alpha}=0$, respectively); both give the same intensity distribution.
However, it is very sensitive to whether the surface has a single termination
($\alpha$ or $\beta$), or a mixture of the two. Deep minima form near the
forbidden Bragg positions (odd $L_{0}$) for $f_{\alpha}=0.5$ owing to
destructive interference between the scattering from the $\alpha$ and $\beta$
regions. We see sharp features exactly at forbidden Bragg peaks such as $L=1$
on the $(00L)$ CTR for all values of $f_{\alpha}$. These arise from the effect
of the absorption factor in $Z$ with a non-zero $\epsilon$. In contrast to
$(00L)$, the $(02L)$ and $(11L)$ CTRs show very different intensity
distributions for $\alpha$ and $\beta$ terminations. There are alternating
deep and shallow minima between the Bragg peaks, with the alternation being
opposite for the two terminations. Furthermore, the $(02L)$ scattering from
the $\alpha$ terrace is identical to the $(11L)$ scattering from the $\beta$
terrace, and vice versa, as required by symmetry. For surfaces with mixed
terrace coverages, the $(02L)$ CTR for $f_{\alpha}=X$ and the $(11L)$ CTR for
$f_{\alpha}=1-X$ are similar but not identical. This can be seen for the
$f_{\alpha}=0.5$ case shown, where the minima between the Bragg peaks all have
about the same depth. The small difference between the two CTRs is determined
by whether the $\alpha$ terraces or $\beta$ terraces are the top layer. The
calculations shown here are for $\alpha$ islands on top. By symmetry, the
$(02L)$ CTR for $f_{\alpha}=X$ with $\alpha$ islands is identical to the
$(11L)$ CTR for $f_{\alpha}=1-X$ with $\beta$ islands.

We have performed calculations using atomic coordinates for all of the stable
GaN $(001)$ reconstructions predicted previously [45]. On this surface, there
are 6 reconstruction domain orientations, related by 3-fold rotation about the
$6_{3}$ axis, and/or reflection about a plane passing through the axis at
$x=0$. Figure 9 shows calculations for $f_{\alpha}=0$ with equal fractions
$\theta_{xj}=1/6$ of all six domains for each reconstruction. The label “bulk”
refers to no reconstruction with all atoms in their bulk atomic positions;
“bare” has relaxed atomic positions in the surface layers with no extra atoms
(see Supplemental Table I [46]); other reconstructions have relaxed positions
and extra atoms at locations indicated (see Appendix A Table 3 and 2 for the
atomic coordinates of the 3H(T1) reconstruction and Supplemental Tables II to
IX [46] for the other 8 reconstructions). All show the same qualitative
behavior as the unreconstructed surface, with small quantitative differences.
Furthermore, because the X-ray scattering is dominated by the Ga atoms, which
occupy an HCP lattice, the same qualitative behavior found for GaN is also
obtained for an elemental HCP crystal.

###  II.3 Miscut surface with alternating terminations

Figure 10: Bulk unit cells (black), extra half unit cells (green) producing
shift between $\alpha$ and $\beta$ terminations of neighboring terraces, and
reconstructed unit cells (red), for a miscut surface with $M=6$, $N=2$, and
$f_{\alpha}=1/3$. Blue shade indicates index $m$ of sum in Eq. (11), with
final term $m=0$ darkest.

We now consider a miscut surface, with an array of straight steps parallel to
the $x$ axis, periodically distributed along the $y$ axis, as shown in Fig. 1.
We assume that the miscut corresponds to a decrease of the surface height by a
full unit cell $c$ every $M$ unit cells in $y$, so that the period of the step
array is $Mb$. The surface miscut angle $\gamma$ relative to $(001)$ is given
by $\tan{\gamma}=c/(Mb)$, and the surface is parallel to $(01M)$ planes. As
shown in Fig. 5(b), the CTRs from this surface are tilted in the $Q_{y}$
direction at an angle $\gamma$ from $(001)$. Because of the tilt, there are
$M$ times as many CTRs as in the exactly oriented case, indexed not just by
$H_{0}K_{0}$ but also by values of $L_{0}$ from $0$ to $M-1$. The $Q_{y}$
value varies with $L$ along the CTR according to
$Q_{y}=(2\pi/b)[K_{0}+(L-L_{0})/M]$, where $H_{0}K_{0}L_{0}$ is the primary
Bragg peak associated with the CTR. The spacing in $L$ along a given CTR
between Bragg peak positions is $M$, rather than unity as in the exactly
oriented surface. (This neglects systematic absences in the Bragg peaks; for
the orthohexagonal case, the spacing between allowed peaks is $2M$.)

We assume that the surface has alternating terminations of $\alpha$ and
$\beta$ terraces along the $y$ axis. Figure 10 shows the bulk and
reconstructed unit cells used to calculate the CTRs for the vicinal surface.
The width of the $\alpha$ terraces is $N$ unit cells, and the width of the
$\beta$ terraces is $M-N$ unit cells. The $\alpha$ terrace fraction is given
by $f_{\alpha}=N/M$.

To calculate the scattering from a miscut crystal, it is useful to take
advantage of the new periodicity in the surface plane produced by the step
array [40]. The bulk reflectivity amplitude can be built up by summing the
scattering from each unit cell in the $y$ direction from $m=-\infty$ to $m=0$,
according to

$r_{\mathrm{bulk}}=\frac{r_{\mathrm{f}}\,F_{\mathrm{bulk}}}{M}\sum_{m=-\infty}^{0}Y^{m}=\frac{r_{\mathrm{f}}\,F_{\mathrm{bulk}}}{M}\frac{Y}{Y-1},$
(11)

where the quantity $Y$ is the ratio of the scattering amplitude from one unit
cell to that from the unit cell at $\Delta y=-b$ beside it, made up of a phase
factor and an absorption factor

$Y\equiv\exp(iQ_{y}b)\,\exp(\epsilon\,b\tan\gamma/Q_{z}).$ (12)

This is similar to the geometric sequence for the exactly oriented surface,
Eq. (1), except that the summation is carried out in the $y$ direction rather
than the $z$ direction, as indicated in Fig. 10. Substituting the above
expression for $Q_{y}$ along the CTR into the phase factor, the term involving
$K_{0}$ drops out. This gives

$\displaystyle Y$ $\displaystyle=\exp[i(Q_{z}c-2\pi L_{0})/M]\exp(\epsilon
c/MQ_{z})$ $\displaystyle=\exp[2\pi i(L-L_{0})/M]\exp(\epsilon c^{2}/2\pi
ML).$ (13)

Comparison with Eq. (3) shows that $Y^{M}=Z$.

The reflectivity amplitude per unit area of the reconstructed overlayer on the
$\alpha$ terraces can be written as

$r_{\mathrm{rec}}^{\alpha}=\frac{r_{\mathrm{f}}\,F_{\mathrm{rec}}^{\alpha}}{N}\sum_{m=1}^{N}Y^{m}=\frac{r_{\mathrm{f}}\,F_{\mathrm{rec}}^{\alpha}}{N}\frac{Y(Y^{N}-1)}{Y-1}.$
(14)

A similar expression applies to the $\beta$ terraces,

$r_{\mathrm{rec}}^{\beta}=\frac{r_{\mathrm{f}}\,F_{\mathrm{rec}}^{\beta}}{M-N}\sum_{m=N+1}^{M}Y^{m}=\frac{r_{\mathrm{f}}\,F_{\mathrm{rec}}^{\beta}}{M-N}\frac{Y(Y^{M}-Y^{N})}{Y-1}.$
(15)

In these equations the sums run over the fraction of the surface periodicity
covered by each termination.

As in the calculation for the exactly oriented surface, the total reflectivity
amplitude $r_{\mathrm{tot}}$ is the sum of the complex amplitudes from the
bulk and the reconstructed layers on the $\alpha$ and $\beta$ terraces.
Expressions similar to Eqs. (6-8) give the intensity reflectivity $R_{L_{0}}$
for the CTR from the $H_{0}K_{0}L_{0}$ Bragg peak, with the roughness factor
for each CTR on a vicinal surface now given by a simple Gaussian form [36, 49,
3],

$S_{L_{0}}=\exp\big{[}-\sigma_{R}^{2}(2\pi/c)^{2}(L-L_{0})^{2}\big{]},$ (16)

where we assume small miscut ($M$ large) so that only the Bragg peak at
$L=L_{0}$ need be considered for each CTR. Away from Bragg peaks, so that the
dynamical scattering correction can be neglected, the intensity reflectivity
can be written as

$\displaystyle R_{L_{0}}$
$\displaystyle=\left|\frac{r_{\mathrm{f}}}{M}\right|^{2}\,|Y|^{2}\,\,S_{L_{0}}$
(17)
$\displaystyle\times\left|\frac{F_{\mathrm{bulk}}+F_{\mathrm{rec}}^{\alpha}(Y^{N}-1)+F_{\mathrm{rec}}^{\beta}(Y^{M}-Y^{N})}{Y-1}\right|^{2}.$

###  II.4 Miscut GaN $(001)$ surface

Figure 11: Calculated reflectivities of CTRs for a miscut GaN $(001)$ surface
with $M=100$ and no reconstruction. Left column: $(00L)$; middle column:
$(02L)$; right column: $(11L)$. Black, red, green, blue, cyan, and magenta
curves are for $L_{0}=-1$ to $4$, respectively. Values of $f_{\alpha}$ are
given for each row.  Figure 12: Calculated reflectivity of CTRs for a miscut
GaN $(001)$ surface with $M=100$ and various reconstructions, for
$f_{\alpha}=0$. .  Figure 13: Relationship of CTR profiles for an exactly
oriented surface (solid blue curves) to those for miscut surfaces, for GaN
$(001)$ with no reconstruction. Red and green dashed curves show sum over
$L_{0}$ of intensities and square of sum of amplitudes, respectively, of all
CTRs at the same $H_{0}K_{0}$ from a miscut surface. (a) and (b) correspond to
single and mixed terminations, $f_{\alpha}=0$ and $0.5$, respectively.

Figure 11 shows the calculated reflectivity of various CTRs as a function of
$L$ for a miscut GaN $(001)$ surface. Three families of CTRs are shown,
$(00L_{0})$, $(02L_{0})$, and $(11L_{0})$, for $L_{0}=-1$ to $4$, with three
$f_{\alpha}$ values, 0.0, 0.5, and 1.0. These calculations were done for a
step period of $M=100$ and an unreconstructed surface. The result is
insensitive to changes in $M$; any value above $M=10$ gives reflectivities
that agree within $\sim 10\%$ at all $L$. As in the case of an exactly
oriented surface, the $(00L_{0})$ CTRs are identical for $f_{\alpha}=0$ and
$f_{\alpha}=1$, and they are very different for $f_{\alpha}=0.5$, with the
CTRs for even $L_{0}$ becoming stronger and the CTRs for odd $L_{0}$ becoming
very weak. The $(02L_{0})$ and $(11L_{0})$ CTRs have a more monotonic
dependence on $f_{\alpha}$. For $f_{\alpha}=0$ and $f_{\alpha}=1$, there are
alternating stronger and weaker intensities between the Bragg peaks, with the
alternation being opposite for $(02L_{0})$ and $(11L_{0})$. For
$f_{\alpha}=0.5$, the intensities between the Bragg peaks are about the same,
and there is no difference between the $(02L_{0})$ and $(11L_{0})$ CTRs. The
$(02L_{0})$ CTRs with $f_{\alpha}=X$ are identical to the $(11L_{0})$ CTRs
with $f_{\alpha}=1-X$, for any value $X$. As with the exactly oriented
surface, other GaN reconstructions have the same qualitative behavior, as
shown in Fig. 12, as do elemental HCP structures. The calculations for Fig. 12
use the same atomic coordinates for the reconstructions as in Fig. 9.

###  II.5 Relation between CTR profiles for miscut and exactly oriented
surfaces

It has been previously discussed that under some circumstances the CTR
intensity profile for an exactly oriented surface can be expressed as the sum
of the CTR intensities from each Bragg peak along the rod [50, 49]. This is
analogous to the sum of the intensities of all the CTRs in the family having
the same $H_{0}K_{0}$ from a miscut surface. Here we make this comparison with
the expressions obtained above for the cases of surfaces with pure $\alpha$,
pure $\beta$, or mixed terminations.

For a miscut surface with a single termination, i.e. $f_{\alpha}=0$ or $1$,
giving full-unit-cell-height steps, one can show that the intensity
reflectivity $R$ for an exactly oriented surface is simply the sum of the
intensity reflectivities $R_{L_{0}}$ for all the CTRs with the same
$H_{0}K_{0}$, in the limit of small miscut angle and neglecting absorption,

$R=\lim_{M\rightarrow\infty}\sum_{L_{0}}R_{L_{0}}.$ (18)

For $N=0$ or $N=M$ ($x=\beta$ or $\alpha$ termination, respectively), using
$Y^{M}=Z$, Eq. (17) becomes

$R_{L_{0}}=\left|\frac{r_{\mathrm{f}}}{M}\right|^{2}\,|Y|^{2}\,\left|\frac{F_{\mathrm{bulk}}}{Z-1}+F_{\mathrm{rec}}^{x}\right|^{2}\,\left|\frac{Z-1}{Y-1}\right|^{2}S_{L_{0}}.$
(19)

Neglecting the absorption factors in $Z$ and $Y$, one obtains
$|Z-1|^{2}=4\sin^{2}(\pi L)$ and $|Y-1|^{2}=4\sin^{2}[\pi(L-L_{0})/M]$. For
small miscut angle (large $M$), the second expression becomes
$|Y-1|^{2}=4\pi^{2}(L-L_{0})^{2}/M^{2}$. Substituting these into Eq. (19)
gives

$\lim_{M\rightarrow\infty}R_{L_{0}}=|r_{\mathrm{f}}|^{2}\,|Y|^{2}\,\left|\frac{F_{\mathrm{bulk}}}{Z-1}+F_{\mathrm{rec}}^{x}\right|^{2}\frac{\sin^{2}(\pi
L)}{\pi^{2}(L-L_{0})^{2}}S_{L_{0}}.$ (20)

While the first two factors depend upon $Q$ and thus vary slightly with
$L_{0}$ at fixed $L$ due to the spacing of the CTRs in $K$, for small miscut
angle this is negligible, and these factors are the same as for an exactly
oriented surface. Comparison with Eqs. (8), (9), and (16) shows that the
summation Eq. (18) holds, if we neglect the factors of $|Z|^{2}$ and
$|Y|^{2}$, which can be set to unity when absorption is small.

This is illustrated in Fig. 13(a), where the summation over $L_{0}$ of the
miscut CTR intensities $R_{L_{0}}$ gives the same result as the calculation of
$R$ for the exactly oriented surface for $f_{\alpha}=0$. This equality holds
for all CTRs, even for those such as $(02L)$ and $(11L)$ which are sensitive
to the differences between the $\alpha$ and $\beta$ terminations. This result
differs from previous analyses [50, 49] because we include the $L$ dependence
of the structure factors $F_{\mathrm{bulk}}$ and $F_{\mathrm{rec}}$, which
properly accounts for the sensitivity of the CTRs to the surface termination
in the regions between the Bragg peaks. However, as shown in Fig. 13(b), the
summation Eq. (18) does not hold for mixed terminations, e.g.
$f_{\alpha}=0.5$.

It is interesting that, for the single termination surface, the exactly
oriented CTR is equal to the sum of the intensities of the miscut CTRs, not
the square of the sum of the amplitudes of the miscut CTRs. This is
demonstrated in Fig. 13(a). From an experimental perspective, this means that
there is no difference between the CTR profile measured from an exactly
oriented surface and that from a surface with a miscut so small that the
splitting of the CTRs cannot be resolved. For the mixed termination surface,
Fig. 13(b), the exactly oriented CTR is not equal to the miscut CTRs summed in
either intensity or amplitude. In this case the reflectivity of the exactly
oriented surface depends upon whether the $\alpha$ or $\beta$ islands are on
top, and the limit of zero miscut $M\rightarrow\infty$ does not reduce the
miscut surface structure to that of the exactly oriented surface.

##  III Discussion

The expressions developed above are an extension of those developed previously
[3] to the case of alternating terrace structures, while also including finite
absorption and orthorhombic symmetry. The kinematic expression for the CTR
intensity as a function of $L$ for a miscut surface, Eq. (17), is equivalent
to the second line of Eq. (12) in [3] if we neglect absorption so that
$Y=\exp[2\pi i(L-L_{0})/M]$ and $|Y-1|^{2}=4\sin^{2}[\pi(L-L_{0})/M]$,
consider a single termination ($N=0$ or $N=M$), and cubic symmetry $c=b=a$.
Likewise the kinematic expression for an exactly oriented surface, Eq. (10),
is equivalent to Eq. (17) in [3] under the same conditions.

The analysis presented above shows that CTR profiles are very sensitive to the
difference between $\alpha$ and $\beta$ terraces on vicinal basal plane
surfaces of HCP-type systems. This enables in situ X-ray measurements during
growth to determine the surface fraction covered by each type of terrace, and
thus to distinguish the dynamics of $A$ and $B$ steps. Measurement of the CTR
profiles as a function of $L$ can be carried out by running scans during
steady-state growth conditions. Recent measurements during step-flow growth by
organo-metallic vapor phase epitaxy of GaN $(0001)$ [42] show CTR profiles
that agree with the calculations presented here, allowing the steady-state
terrace fraction $f_{\alpha}$ to be determined as a function of growth
conditions. The observed increase in $f_{\alpha}$ with increasing growth rate
indicates that adatom attachment kinetics are faster at $A$ steps than $B$
steps for this system.

Figure 14: Calculated reflectivity of selected CTRs for a miscut GaN $(001)$
surface as a function of terrace fraction $f_{\alpha}$, for fixed values of
$L$ given.  Figure 15: Blue curves show calculated reflectivity of selected
CTRs for an exactly oriented GaN $(001)$ surface as a function of terrace
fraction $f_{\alpha}$, for fixed values of $L$ given. Magenta curves show sum
over $L_{0}$ of intensities of all CTRs at the same $H_{0}K_{0}$ from a miscut
surface.

In addition, the dynamics of the terrace fraction $f_{\alpha}(t)$ can be
observed by monitoring CTR intensities at particular $L$ values during changes
in growth conditions. Figure 14 shows calculations of the reflectivity as a
function of $f_{\alpha}$ at fixed $L$ positions on six CTRs, for a miscut
unreconstructed GaN $(001)$ surface. On the $(001)$ CTR at $L=0.99$, the
intensity is maximum for $f_{\alpha}=0$ or $f_{\alpha}=1$, and goes through a
deep minimum at $f_{\alpha}=0.5$. On the $(002)$ CTR at $L=0.99$, the
intensity is minimum for $f_{\alpha}=0$ or $f_{\alpha}=1$, and goes through a
maximum at $f_{\alpha}=0.5$. On the $(021)$ and $(022)$ CTRs at $L=1.6$, the
intensity increases almost monotonically as $f_{\alpha}$ increases from $0$ to
$1$. The $(111)$ and $(112)$ CTRs at $L=1.6$ show complementary behavior, with
an almost monotonic intensity decrease. These curves can be used to convert
observed intensity changes into terrace fraction dynamics $f_{\alpha}(t)$,
e.g. as net growth rate is changed during step-flow growth [42].

Figure 15 shows calculations of the reflectivity as a function of $f_{\alpha}$
at fixed $L$ positions on three CTRs for an exactly oriented unreconstructed
GaN $(001)$ surface. The $(00L)$ with $L=0.99$ position shows behavior
qualitatively similar to the $(001)$ CTR of a miscut surface, with a minimum
at $f_{\alpha}=0.5$. This position (or the equivalent $(13L)$ position) has
been used to monitor growth under island nucleation and coalescence
conditions, where oscillations in intensity occur with a period corresponding
to growth of half-unit-cell monolayers [7, 10]. Figure 15 also shows how the
sum of the CTR intensities for a miscut surface, which agree with the exactly
oriented surface at $f_{\alpha}=0$ and $1$, deviate at intermediate values of
$f_{\alpha}$.

##  IV Conclusions and Outlook

The above analysis extends previous calculations of CTR profiles for miscut
surfaces to include half-unit-cell-height steps and variation of the coverage
of the $\alpha$ and $\beta$ terraces above each step. We demonstrate that the
CTR intensity as a function of $L$ can be used to measure the terrace fraction
$f_{\alpha}$, and thus determine whether $A$ or $B$ steps have faster kinetics
during crystal growth on basal plane surfaces of HCP-type systems. In
particular the orthohexagonal $(02L)$ and $(11L)$ CTRs (equivalent to
$(01\overline{1}L)$ and $(10\overline{1}L)$, respectively, in Miller-Bravais
notation) show characteristic alternating high and low intensities between the
Bragg peaks that depend on $f_{\alpha}$. Intensities between Bragg peaks e.g.
at $L=1.6$ on the $(022)$ or $(112)$ CTRs change almost monotonically with
$f_{\alpha}$. This behavior is qualitatively similar for different GaN $(001)$
surface reconstructions, and for elemental HCP systems. A comparison of CTR
profiles for exactly oriented and miscut surfaces shows that a summation
relation Eq. (18) holds for all CTRs, but only when the surface has a single
termination, $f_{\alpha}=0$ or $1$.

While the example CTR calculations presented here are for wurtzite-structure
GaN, the ability to distinguish the terrace fraction $f_{\alpha}$ applies
directly to many other HCP-type systems with a $6_{3}$ screw axis, including
other compound semiconductors, as well as one third of the crystalline
elements and many more complex crystals. The treatment for miscut surfaces
developed above considers only two terminations that alternate, separated by
half-unit-cell-height steps, to provide a method to distinguish $A$ and $B$
step properties in HCP-type systems. For crystal structures with other types
of screw axes or multiple layers [2], the geometrical sequence formulas Eqs.
(14-15) are straightforward to extend to periodic sequences of more than two
terminations.

###### Acknowledgements.

Work supported by the U.S Department of Energy (DOE), Office of Science,
Office of Basic Energy Sciences, Materials Science and Engineering Division.
This work was inspired by and developed to analyze data from experiments at
beamline 12ID-D of the Advanced Photon Source, a DOE Office of Science user
facility operated by Argonne National Laboratory.

##  Appendix A Atomic coordinates

Table 1:  Fractional orthohexagonal coordinates of bulk GaN used to calculate the substrate contribution to the CTRs. Atom | Site | $x$ | $y$ | $z$
---|---|---|---|---
$k$ | $n$ |  |  |
Ga | 1 | 0.5000 | 0.1667 | -0.5000
Ga | 2 | 0.0000 | 0.6667 | -0.5000
Ga | 3 | 1.5000 | 0.1667 | -0.5000
Ga | 4 | 1.0000 | 0.6667 | -0.5000
Ga | 5 | 0.0000 | 0.0000 | 0.0000
Ga | 6 | 0.5000 | 0.5000 | 0.0000
Ga | 7 | 1.0000 | 0.0000 | 0.0000
Ga | 8 | 1.5000 | 0.5000 | 0.0000
N | 1 | 0.0000 | 0.0000 | -0.6232
N | 2 | 0.5000 | 0.5000 | -0.6232
N | 3 | 1.0000 | 0.0000 | -0.6232
N | 4 | 1.5000 | 0.5000 | -0.6232
N | 5 | 0.5000 | 0.1667 | -0.1232
N | 6 | 0.0000 | 0.6667 | -0.1232
N | 7 | 1.5000 | 0.1667 | -0.1232
N | 8 | 1.0000 | 0.6667 | -0.1232
Table 2:  Fractional orthohexagonal coordinates $x$, $y$, $z$ of atoms in domain $j=1$ of the 3H(T1) reconstruction used to calculate the $\alpha$ terrace contribution to the CTRs, as well as their differences $\Delta x$, $\Delta y$, $\Delta z$ relative to bulk lattice positions. The differences for H atoms are relative to N sites. The lowest four Ga and N sites are an extra half unit cell of bulk lattice to account for the difference in height of the $\alpha$ and $\beta$ terraces. Atom | Site | $x$ | $y$ | $z$ | $\Delta x$ | $\Delta y$ | $\Delta z$
---|---|---|---|---|---|---|---
$k$ | $n$ |  |  |  |  |  |
Ga | 1 | 0.5000 | 0.1667 | -0.5000 | 0.0000 | 0.0000 | 0.0000
Ga | 2 | 0.0000 | 0.6667 | -0.5000 | 0.0000 | 0.0000 | 0.0000
Ga | 3 | 1.5000 | 0.1667 | -0.5000 | 0.0000 | 0.0000 | 0.0000
Ga | 4 | 1.0000 | 0.6667 | -0.5000 | 0.0000 | 0.0000 | 0.0000
Ga | 5 | 0.0000 | 0.0000 | 0.0076 | 0.0000 | 0.0000 | 0.0076
Ga | 6 | 0.5075 | 0.4975 | -0.0015 | 0.0075 | -0.0025 | -0.0015
Ga | 7 | 1.0000 | 0.0050 | -0.0015 | 0.0000 | 0.0050 | -0.0015
Ga | 8 | 1.4925 | 0.4975 | -0.0015 | -0.0075 | -0.0025 | -0.0015
Ga | 9 | 0.4929 | 0.1643 | 0.5223 | -0.0071 | -0.0024 | 0.0223
Ga | 10 | 0.0000 | 0.6667 | 0.4294 | 0.0000 | 0.0000 | -0.0706
Ga | 11 | 1.5071 | 0.1643 | 0.5223 | 0.0071 | -0.0024 | 0.0223
Ga | 12 | 1.0000 | 0.6714 | 0.5223 | 0.0000 | 0.0047 | 0.0223
N | 1 | 0.0000 | 0.0000 | -0.6232 | 0.0000 | 0.0000 | 0.0000
N | 2 | 0.5000 | 0.5000 | -0.6232 | 0.0000 | 0.0000 | 0.0000
N | 3 | 1.0000 | 0.0000 | -0.6232 | 0.0000 | 0.0000 | 0.0000
N | 4 | 1.5000 | 0.5000 | -0.6232 | 0.0000 | 0.0000 | 0.0000
N | 5 | 0.4988 | 0.1663 | -0.1254 | -0.0012 | -0.0004 | -0.0022
N | 6 | 0.0000 | 0.6667 | -0.1201 | 0.0000 | 0.0000 | 0.0031
N | 7 | 1.5012 | 0.1663 | -0.1254 | 0.0012 | -0.0004 | -0.0022
N | 8 | 1.0000 | 0.6675 | -0.1254 | 0.0000 | 0.0008 | -0.0022
N | 9 | 0.0000 | 0.0000 | 0.3766 | 0.0000 | 0.0000 | -0.0002
N | 10 | 0.5064 | 0.4979 | 0.3775 | 0.0064 | -0.0021 | 0.0007
N | 11 | 1.0000 | 0.0043 | 0.3775 | 0.0000 | 0.0043 | 0.0007
N | 12 | 1.4936 | 0.4979 | 0.3775 | -0.0064 | -0.0021 | 0.0007
H | 13 | 0.5002 | 0.1667 | 0.8196 | 0.0002 | 0.0000 | -0.0572
H | 15 | 1.4998 | 0.1667 | 0.8196 | -0.0002 | 0.0001 | -0.0572
H | 16 | 1.0000 | 0.6666 | 0.8196 | 0.0000 | -0.0001 | -0.0572
Table 3:  Fractional orthohexagonal coordinates $x$, $y$, $z$ of atoms in domain $j=1$ of the 3H(T1) reconstruction used to calculate the $\beta$ terrace contribution to the CTRs, as well as their differences $\Delta x$, $\Delta y$, $\Delta z$ relative to bulk lattice positions. The differences for H atoms are relative to N sites. Atom | Site | $x$ | $y$ | $z$ | $\Delta x$ | $\Delta y$ | $\Delta z$
---|---|---|---|---|---|---|---
$k$ | $n$ |  |  |  |  |  |
Ga | 1 | 0.5075 | 0.1692 | -0.5015 | 0.0075 | 0.0025 | -0.0015
Ga | 2 | 0.0000 | 0.6667 | -0.4924 | 0.0000 | 0.0000 | 0.0076
Ga | 3 | 1.4925 | 0.1692 | -0.5015 | -0.0075 | 0.0025 | -0.0015
Ga | 4 | 1.0000 | 0.6617 | -0.5015 | 0.0000 | -0.0050 | -0.0015
Ga | 5 | 0.0000 | 0.0000 | -0.0706 | 0.0000 | 0.0000 | -0.0706
Ga | 6 | 0.4929 | 0.5024 | 0.0223 | -0.0071 | 0.0024 | 0.0223
Ga | 7 | 1.0000 | -0.0047 | 0.0223 | 0.0000 | -0.0047 | 0.0223
Ga | 8 | 1.5071 | 0.5024 | 0.0223 | 0.0071 | 0.0024 | 0.0223
N | 1 | 0.0000 | 0.0000 | -0.6201 | 0.0000 | 0.0000 | 0.0031
N | 2 | 0.4988 | 0.5004 | -0.6254 | -0.0012 | 0.0004 | -0.0022
N | 3 | 1.0000 | -0.0008 | -0.6254 | 0.0000 | -0.0008 | -0.0022
N | 4 | 1.5012 | 0.5004 | -0.6254 | 0.0012 | 0.0004 | -0.0022
N | 5 | 0.5064 | 0.1688 | -0.1225 | 0.0064 | 0.0021 | 0.0007
N | 6 | 0.0000 | 0.6667 | -0.1234 | 0.0000 | 0.0000 | -0.0002
N | 7 | 1.4936 | 0.1688 | -0.1225 | -0.0064 | 0.0021 | 0.0007
N | 8 | 1.0000 | 0.6624 | -0.1225 | 0.0000 | -0.0043 | 0.0007
H | 10 | 0.5002 | 0.5000 | 0.3196 | 0.0002 | -0.0000 | -0.0572
H | 11 | 1.0000 | 0.0001 | 0.3196 | 0.0000 | 0.0001 | -0.0572
H | 12 | 1.4998 | 0.4999 | 0.3196 | -0.0002 | -0.0001 | -0.0572

To provide a detailed example of how we calculate the CTR intensities
including the effects of reconstruction, we here provide an example of the
calculated atomic coordinates we use for a particular GaN (0001)
reconstruction, 3HT1. Atomic coordinates calculated for the other
reconstructions are given in Supplemental Tables I to IX [46]. All coordinates
were obtained by ab initio methods in [45]. The qualitative behavior we
observe does not depend upon the reconstruction chosen or the exact values of
the atomic coordinates used.

We first show the coordinates used for bulk positions in Table 1. The
fractional coordinates $x$, $y$, and $z$ given are the components of the
positions $\mathbf{r}_{kn}^{\mathrm{bulk}}$ used to calculate
$F_{\mathrm{bulk}}$, normalized to the respective orthohexagonal lattice
parameters $a$, $b$, and $c$, i.e. $\mathbf{r}=(ax,by,cz)$. A $2\times 2$
surface unit cell is used, equivalent to two orthohexagonal unit cells, so
there are 8 Ga and 8 N sites in each unit cell thickness. These coordinates
place a bulk Ga site on a $\beta$ layer at the origin; $\beta$ terraces occur
at integer values of $z$, while $\alpha$ terraces occur at half-integer
values. We use $u=0.3768$ for the internal lattice parameter of bulk GaN, i.e.
the fractional distance between Ga and N sites, which deviates slightly from
the ideal $3/8$ value, in agreement with ab initio calculations [45, 51] and
experiments [52].

Tables 2 and 3 give the atomic coordinates for the 3H(T1) reconstruction. The
fractional coordinates $x$, $y$, and $z$ given in the tables are the
normalized components of the positions $\mathbf{r}_{jkn}^{\alpha}$ and
$\mathbf{r}_{jkn}^{\beta}$ used to calculate $F_{\mathrm{rec}}^{\alpha}$ and
$F_{\mathrm{rec}}^{\beta}$. Relaxed positions were calculated for a one-unit-
cell thick layer at the surface. For the $\alpha$ terrace, an extra half unit
cell of bulk (unrelaxed) atoms is attached to the bottom to account for the
difference in height of the $\alpha$ and $\beta$ terraces, as shown in Fig.
10. Coordinates for only one domain are given. Those for other 5 domains are
obtained by 3-fold rotation about the $6_{3}$ axis and/or reflection of the
$y$ coordinate. One can see that the Ga atoms bonded to the three adsorbed
hydrogens of the 3H(T1) reconstruction relax to higher $z$ positions.

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