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[2022-12-05]


# Exploring the double-slit interference with linearly polarized photons

Wangmei Zha  University of Science and Technology of China, Hefei, China
James Daniel Brandenburg<EMAIL_ADDRESS>Shandong University,
Jinan, China  Brookhaven National Laboratory, New York, USA  Lijuan Ruan
Brookhaven National Laboratory, New York, USA  Zebo Tang<EMAIL_ADDRESS>University of Science and Technology of China, Hefei, China  Zhangbu Xu
Brookhaven National Laboratory, New York, USA  Shandong University, Jinan,
China

###### Abstract

The linearly polarized quasi-real photons from the highly Lorentz-contracted
Coulomb fields of relativistic heavy ions can fluctuate to quark-antiquark
pairs, scatter off a target nucleus and emerge as vector mesons. In the
process, the two colliding nuclei can switch roles to act as photon emitter or
target, forming a double-slit interference pattern. The product from
photoproduction inherits the photon polarization states, leading to the
asymmetries of the decay angular distributions. In this letter, we study the
interference effect in polarization dimension from the asymmetries of the
decay angular distributions for photoprodution in heavy-ion collisions and
find a periodic oscillation with the transverse momentum of vector meson,
which could reasonably explain the transverse momentum dependence of the 2nd-
order modulation in azimuth for the $\rho^{0}$ decay observed by the STAR
collaboration.

In relativistic heavy-ion collisions, the giant electromagnetic field
accompanied by the heavy nuclei can be viewed as a spectrum of quasi-real
photons Krauss  _et al._ (1997). The photon emitted from one nucleus can
fluctuate to quark-antiquark pairs which then scatters off the other nucleus,
emerging as a real vector meson. The photoproduction of vector meson is
copious in relativistic heavy-ion collisions due to the large electric charge
and the highly Lorentz-contracted electric field. In the production process,
the elastic scattering occurs via Pomeron exchange, which impose a restriction
on the production site within one of the two colliding nuclei. This indicates
that the production source consists of two well separated nuclei (two slits).
There are two possibilities: either nucleus 1 emits a photon, whereas nucleus
2 acts as a target or vice versa. The two possibilities are indistinguishable,
which present the production process as a perfectly double-slit experiment of
individual particle. Due to the negative parity of vector meson, the
amplitudes from the two nuclei are assigned with opposite signs, which leads
to a destructive interference. The destructive interference of $\rho^{0}$ for
ultra-peripheral collisions (UPC) has been proposed by Klein and Nystrand
Klein and Nystrand (2000) and verified by the STAR Collaboration Abelev  _et
al._ (2009). In Refs. Zha  _et al._ (2018, 2019), we extended the scenario to
hadronic heavy-ion collisions (HHIC) and proposed to exploit the strong
interactions in overlap region to test the observation effect on the
interference.

The photons generated from the highly Lorentz-contacted electromagnetic field
are expected to be fully linearly polarized. It has been suggested by C. Li et
al. Li  _et al._ (2019) that the collisions of the linearly polarized photons
($\gamma+\gamma\rightarrow l^{+}+l^{-}$) result in the 2nd-order and 4th-order
modulations in azimuth (in the plane perpendicular to the beam direction)
between the pair momentum and the lepton momentum for diletpon production. The
experimental signature was confirmed by STAR Collaboration for the dielectron
measurements Adam  _et al._ (2019). The vector meson production from linearly
polarized photons also possesses distinctive signature in the asymmetries of
the decay angular distributions. Continuous efforts Schilling and Storim
(1968); Criegee  _et al._ (1968); Ballam  _et al._ (1970); Eisenberg  _et al._
(1976); Derrick  _et al._ (1996); Abelev  _et al._ (2008); Hiraiwa  _et al._
(2018) has been made for more than fifty years to utilize the linearly
polarized photons in vector meson photoproduction as a parity filter for the
exchange of particles in $t$ channel, which is an effective tool to separate
the natural from unnatural parity exchange in $t$ channel. In this letter, we
exploit the linearly polarized photons to investigate the interference effect
for the vector meson photoproduction in heavy-ion collisions and demonstrate a
periodic oscillation with transverse momentum for the asymmetries of the decay
angular distributions.

Hereinafter, we take the process of
$\gamma+A\rightarrow\rho^{0}+A\rightarrow\pi^{+}+\pi^{-}+A$ in heavy-ion
collisions to illustrate the interference effect on the asymmetries of the
decay. The decay distribution is discussed in its helicity system: the
$z$-direction is chosen to the direction of flight of vector meson in the
photon-nucleon center of mass frame. The decay angles $\theta$ and $\phi$ are
the polar and azimuthal angles of the decay daughters in the vector meson rest
frame, respectively. In experiment, the direction of $z$-axis is approximated
by that of incoming beams. It has been verified by Monte Carlo calculations
that this is a good approximation. Here, we adopt the approximation in our
calculation for direct comparisons with experimental results. Following Ref.
Schilling  _et al._ (1970), the decay angular distribution of $\rho^{0}$ with
the assumption of no helicity flip is:

$\frac{d^{2}N}{d\cos\theta
d\phi}=\frac{3}{8\pi}\sin^{2}\theta[1+P_{\gamma}\cos 2(\phi-\Phi)],$ (1)

where $P_{\gamma}$ is the degree of linear polarization, and $\Phi$ is the
angle between the photon polarization plane and the $\rho^{0}$ production
plane. The helicity nonflip assumption is consistent with the spin density
matrix element measurements for $\rho^{0}$ from STAR Abelev  _et al._ (2008)
and ZEUS Derrick  _et al._ (1996). As revealed in Eq. 1, the linearly
polarized photons result in the 2nd-order modulations in azimuth between the
vector meson momentum and the $\pi^{\pm}$ momentum. The magnitude of 2nd-order
modulation gives:

$2\langle\cos(2\phi)\rangle=P_{\gamma}\cos(2\Phi).$ (2)

The strength of modulation relies on the degree and direction of linear
polarization. The photons generated from the nuclei are fully linearly
polarized, which gives $P_{\gamma}=1$. The orientation of linearly polarized
photons is determined by the direction of the electric field vector. As
demonstrated in Fig. 1, the electric vector for the photons, which interact
with the nuclei, is parallel to the impact parameter vector. This suggests
that the polarization plane is completely coincidence with the reaction plane.
Here, we employ the point-like assumption for the nuclei, which is appropriate
in ultra-peripheral collisions. We can redefine the $x$-axis as the impact
parameter vector, then $\Phi$ is the azimuth angle of $\rho^{0}$ transverse
momentum. For event by event average, the modulation strength can be written
as:

$2\langle\cos(2\phi)\rangle=P_{\gamma}\langle\cos(2\Phi)\rangle.$ (3)

That is to say that the modulations of the decay distribution for $\rho^{0}$
are determined by the asymmetries in its two-dimensional transverse momentum
distribution.

Figure 1: (Color online) Schematic diagram for the direction of electric
vector in ultra-peripheral heavy-ion collisions.

Figure 2: (Color online) Panel (a): two-dimensional transverse momentum
distribution pattern with point-like assumption at mid-rapidity. Panel (b):
the modulation strength, 2$\langle\cos\phi\rangle$, as a function of
transverse momentum at mid-rapidity. Panel (c): 2$\langle\cos\phi\rangle$ as a
function of transverse momentum at forward rapidity ($A_{1}\neq A_{2}$).
Figure 3: (Color online) The modulation strength,
$2\langle\cos(2\phi)\rangle$, as a function of transverse momentum in ultra-
peripheral Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV for mutual dissociation
mode XnXn.

The probability distribution of $\rho^{0}$ photoproduciton in two-dimensional
transverse momentum space can be expressed by performing a Fourier
transformation to the amplitude in coordinate space:

$\frac{d^{2}P}{dp_{x}dp_{y}}=\frac{1}{2\pi}|\int
d^{2}x_{\perp}A(x_{\perp})e^{ip_{\perp}\cdot x_{\perp}}|^{2},$ (4)

where $A(x_{\perp})$ is the amplitude distribution in transverse plane. As a
start, we can approximate the production over the two nuclei as two point
sources. If there is no interference between the two source, the probability
distribution in transverse momentum space will be uniform in azimuth, which
would give no modulation for the decay distribution. With the interference
effect, the amplitude distribution from the two colliding nuclei for point-
like assumption can be written as:

$A(x_{\perp})=A_{1}\delta(x_{\perp}-\frac{1}{2}b)-A_{2}\delta(x_{\perp}+\frac{1}{2}b),$
(5)

where $A_{1}$ and $A_{2}$ are the relative amplitudes for the two colliding
nuclei, $\delta$ is the Dirac $\delta$ function, $b$ is the impact parameter,
and the minus sign in the equation corresponds to the negative parity of
$\rho^{0}$. The corresponding two-dimensional transverse momentum distribution
with $b=$ 20 fm at mid-rapidity ($A_{1}=A_{2}$) is shown in Fig. 2 panel (a).
The two-dimensional transverse momentum distribution reveals a typical Young’s
double-slit interference pattern with a series of alternating light and dark
fringes, which possesses significant asymmetries in azimuth. According to Eq.
3, the modulation strength factor, $2\langle\cos(2\phi)\rangle$, versus
transverse momentum ($p_{T}$) can be extracted from the two-dimensional
distribution and the result is shown in panel (b) of Fig. 2. The modulation
strength shows a periodic oscillation like the hyperbolic Bessel Functions
with the transverse momentum. The oscillation cycle is determined by the
distant between the two source (impact parameter). For the production at
forward rapidity in heavy-ion collisions, the relative amplitudes from the two
nuclei are not equal. Panel (c) of Fig. 2 shows the modulation strength as a
function of transverse momentum with unequal amplitudes from the two sources.
Since the interference is not complete in this case, the oscillation level is
weaker than that at mid-rapidity. Furthermore, the modulation strength is 0 at
$p_{T}=$ 0, while it is maximum for the case at mid-rapidity. This is due to
that the interference contribution is close to 0 for $p_{T}=$ 0.

As mentioned herein before, we have qualitatively demonstrated the
interference effect on the asymmetries of the decay angular distributions for
$\rho^{0}$ from photoproduction. The quantitative calculations is described
herein after. Following Ref. Zha  _et al._ (2018), the production amplitude
distribution for $\rho^{0}$ photoproduction is determined by the spatial
photon flux and the corresponding $\gamma A$ scattering amplitude
$\Gamma_{\gamma A\rightarrow\rho^{0}A}$. According to equivalent photon
approximation, the spatial photon flux generated by the heavy nuclei can be
written as:

$\displaystyle\frac{d^{3}N_{\gamma}(\omega_{\gamma},\vec{x}_{\perp})}{d\omega_{\gamma}d\vec{x}_{\perp}}$
$\displaystyle=\frac{4Z^{2}\alpha}{\omega_{\gamma}}|\int\frac{d^{2}\vec{k}_{\gamma\perp}}{(2\pi)^{2}}\vec{k}_{\gamma\perp}\frac{F_{\gamma}(\vec{k}_{\gamma})}{|\vec{k}_{\gamma}|^{2}}e^{i\vec{x}_{\perp}\cdot\vec{k}_{\gamma\perp}}|^{2}$
(6)
$\displaystyle\vec{k}_{\gamma}=(\vec{k}_{\gamma\perp},\frac{\omega_{\gamma}}{\gamma_{c}}),\omega_{\gamma}=\frac{1}{2}M_{\rho^{0}}e^{\pm
y},$

where $\vec{x}_{\perp}$ and $\vec{k}_{\gamma\perp}$ are two-dimensional photon
position and momentum vectors perpendicular to the beam direction,
$\omega_{\gamma}$ is the energy of emitted photon, $Z$ is the electric charge
of nucleus, $\alpha$ is the electromagnetic coupling constant, $\gamma_{c}$ is
the Lorentz factor of the beam, $M_{\rho^{0}}$ and $y$ are the mass and
rapidity of $\rho^{0}$, and $F_{\gamma}(\vec{k}_{\gamma})$ is the nuclear
electromagnetic form factor. The form factor can be obtained via the Fourier
transformation of the charge density in the nucleus. The charge density
profile of the nucleus can be parametrized by the Woods-Saxon distribution:

$\rho_{A}(r)=\frac{a^{0}}{1+\rm{exp}[(r-R_{WS})/d]},$ (7)

where the radius $R_{WS}$ and skin depth $d$ are based on fits to electron-
scattering data Barrett and Jackson (1977) and $a^{0}$ is the normalization
factor.

The scattering amplitude $\Gamma_{\gamma A\rightarrow\rho^{0}A}$ with the
shadowing effect can be estimated via GlauberMiller  _et al._ (2007a) \+
vector meson dominance (VMD) approachBauer  _et al._ (1978):

$\Gamma_{\gamma A\rightarrow\rho^{0}A}(\vec{x}_{\perp})=\frac{f_{\gamma
N\rightarrow\rho^{0}N}(0)}{\sigma_{\rho^{0}N}}[1-\rm{exp}(-\sigma_{\rho^{0}N}T(\vec{x}_{\perp}))],$
(8)

where $f_{\gamma N\rightarrow\rho^{0}N}(0)$ is the forward-scattering
amplitude for $\gamma+N\rightarrow\rho^{0}+N$, $\sigma_{\rho^{0}N}$ is the
total $\rho^{0}N$ cross section, and $T(\vec{x}_{\perp})$ is the thickness
function for nucleus defined as
$T(\vec{x}_{\perp})=\int_{-\infty}^{+\infty}\rho(\sqrt{\vec{x}^{2}_{\perp}+z^{2}})dz$.
The $f_{\gamma N\rightarrow\rho^{0}N}(0)$ can be determined from the
measurements of forward scattering cross section, $\frac{d\sigma_{\gamma
N\rightarrow\rho^{0}N}}{dt}|_{t=0}$, which is well parametrized in Ref. Klein
_et al._ (2017). Using the optical theorem and VMD relation, the total cross
for $\rho^{0}N$ scattering can be given by

$\sigma_{\rho^{0}N}=\frac{f_{\rho^{0}}}{4\sqrt{\alpha}}f_{\gamma\rho^{0}\rightarrow\rho^{0}N},$
(9)

where $f_{\rho^{0}}$ is the $\rho^{0}$-photon coupling. To account for the
coherent length effect, the thickness function $T(\vec{x}_{\perp})$ is
redefined as

$T(\vec{x}_{\perp})=\int_{-\infty}^{+\infty}\rho(\sqrt{\vec{x}^{2}_{\perp}+z^{2}})e^{iq_{L}z}dz,q_{L}=\frac{M_{\rho^{0}}e^{y}}{2\gamma_{c}},$
(10)

where $q_{L}$ is the longitudinal momentum transfer required to produce a real
$\rho^{0}$. Finally, the production amplitude distribution is given by

$A(\vec{x_{\perp}})=\Gamma_{\gamma
A\rightarrow\rho^{0}A}\sqrt{\frac{d^{2}N_{\gamma}}{d^{2}\vec{x_{\perp}}}}.$
(11)

In addition to the coherent production with interference, there are also
incoherent contribution for $\rho^{0}$ photoproduction in heavy-ion
collisions, which is significant at relative high transverse momentum. The
incoherent production should have no contribution to the asymmetries of decay
distribution, which would weaken the overall modulation strength. Similarly to
the case of coherent $\rho^{0}$ photoproduction on nuclei, the incoherent
cross section, $\sigma_{\gamma A\rightarrow\rho^{0}A^{{}^{\prime}}}$, could be
related to the exclusive $\gamma p$ cross section $\sigma_{\gamma
p\rightarrow\rho^{0}p}$ via Glauber + VMD approach. Here, $A^{{}^{\prime}}$ is
the final nuclear state containing products of the nuclear disintegration. The
approach gives:

$\displaystyle\sigma_{\gamma
A\rightarrow\rho^{0}A^{{}^{\prime}}}=\sigma_{\gamma p\rightarrow\rho^{0}p}\int
d^{2}{\vec{x}_{\perp}}T(\vec{x}_{\perp})e^{-\sigma^{in}_{\rho^{0}N}T(\vec{x}_{\perp})},$
(12)

where
$\sigma^{in}_{\rho^{0}N}=\sigma_{\rho^{0}N}-\sigma^{2}_{\rho^{0}N}/(16\pi
B_{\rho^{0}})$ is the inelastic $\rho^{0}$-nucleon cross section, and
B${}_{\rho^{0}}$ is the slope of the $t$ dependence of the $\gamma
p\rightarrow\rho^{0}p$ scatteringKlein  _et al._ (2017).

Here, we focus on the ultra-peripheral collisions, where there is no nuclear
overlap to reject hadronic background. According to the optical Glauber model
Miller  _et al._ (2007b), the mean number of projectile nucleons that interact
at least once in $A+A$ collisions with impact parameter $b$ is:

$P_{H}(b)=\int
d^{2}\vec{x_{\perp}}T(\vec{x_{\perp}}-\vec{b})\\{1-exp[-\sigma_{NN}T(\vec{x_{\perp}})]\\},$
(13)

where $\sigma_{NN}$ is the total nucleon-nucleon cross section. Then, the
probability of having no hadronic interaction is $exp[-P_{H}(b)]$. In
experiment, the UPC events are usually triggered or selected by the mutual
Coulomb dissociation of the nuclei, in which the number of emitted forward
neutrons can be detected by Zero Degree Calorimeters. Following the EPA
approach, the Coulomb excitation probability of an ultra-relativistic nucleus
can be determined by the photon flux companied with the nucleus and the
appropriate photon-absorption cross section of nuclei. The lowest-order
probability for an excitation of nucleus which emits at least one neutron
($X_{n}$) is

$m_{Xn}(b)=\int dkn(b,E)\sigma_{\gamma A\rightarrow A^{*}}(E),$ (14)

where $E$ is the photon energy,b is the impact parameter, $n(b,E)$ is photon
flux described by Eq. 6, and $\sigma_{\gamma A\rightarrow A^{*}}(E)$ is the
photoexcitation cross section with incident energy $E$. The photoexcitation
cross section $\sigma_{\gamma A\rightarrow A^{*}}(E)$ can be extracted from
the experimental measurements A  _et al._ (1970); Leprêtre  _et al._ (1981);
Carlos  _et al._ (1984); Armstrong  _et al._ (1972a); Caldwell  _et al._
(1973); Michalowski  _et al._ (1977); Armstrong  _et al._ (1972b). The
probability of mutual dissociation for the two nuclei with at least one
neutron emission for each beam (XnXn)is then given by:

$P_{XnXn}(b)=(1-\rm{exp}[-m_{Xn}(b)])^{2}.$ (15)

As described in our recent work Brandenburg  _et al._ (2020), the mutual
dissociation probability with any number of neutron emission can be estimated
in similar way.

With the inputs described herein before, the cross section and differential
distributions for $\rho^{0}$ photoproduction can be calculated, which could
reasonably described the measurements from STAR Abelev  _et al._ (2008);
Adamczyk  _et al._ (2017) and ALICE Adam  _et al._ (2015, 2020). The estimated
modulation strength, $2\langle\cos(2\phi)\rangle$, of $\rho^{0}$ as a function
of transverse momentum in ultra-peripheral Au+Au collisions at
$\sqrt{s_{NN}}=200$ GeV for mutual dissociation mode XnXn is shown in Fig. 3.
Similar to the case for point-like assumption, the modulation strength shows a
periodic oscillation with the transverse momentum. The amplitude of the second
oscillation period is weaker than that of the first oscillation period, which
is due to more fraction of incoherent production at higher $p_{T}$. The
oscillation feature disappears for $p_{T}>$ 0.16 GeV/c, where the incoherent
production plays dominate role in the photoproduction. The results from
calculations can reasonably describe the transverse momentum dependence of the
2nd-order modulation in azimuth for the $\rho^{0}$ decay observed by the STAR
collaboration Brandenburg (2019).

In Summary, the double slit interference is explored in polarization dimension
with the linearly polarized photons accompanied by relativistic nuclei in
heavy-ion collisions. We demonstrate how the interference between the two
colliding nuclei affect the asymmetries of the decay angular distributions for
vector meson photoproduction from linearly polarized photons. Using the vector
meson dominance with Glauber approach, the 2nd-order modulation in azimuth for
the $\rho^{0}$ decay from photoproduction in ultra-peripheral Au+Au collision
at $\sqrt{s_{NN}}=$ 200 GeV is estimated, and reveals a periodic oscillation
with the transverse momentum, which could reasonably described the decay
asymmetries observed by the STAR Collaboration. The calculation can be easily
extended to other vector mesons, e.g. $\omega$ and $\phi$, which could be
studied by future experiments.

This work was funded by the National Natural Science Foundation of China under
Grant Nos. 11775213, 11505180 and 11375172, the U.S. DOE Office of Science
under contract No. DE-SC0012704, and MOST under Grant No. 2014CB845400.

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# A Pulse-Shape Binary Multiplex Modulation

Pavel Loskot The author is with ZJU-UIUC Institute, Haining, China (e-mail:
pavelloskot@intl.zju.edu.cn).This work was supported by a research grant from
Zhejiang University.

###### Abstract

The root raised-cosine pulse commonly used in linear digital modulations
yields exactly two intersymbol interference components from the preceding and
the subsequent data symbols, provided that the roll-off factor is $100\%$ and
the modulation packing factor is set to $50\%$. This can be exploited to
symmetrically multiplex two data streams of transmitted symbols. Hence, the
proposed scheme is referred to as pulse-shape binary multiplex modulation. The
demodulation of the two multiplexed data streams at the receiver can be aided
by making the streams mutually orthogonal. It can be achieved by superposition
modulation with symbol-by-symbol interference cancellation, proper design of
transmission sequences interleaving pilot and data symbols in order to also
enable channel estimation, and using orthogonal spreading sequences. The
presented numerical results indicate that the proposed modulation scheme can
outperform Nyquist signaling in terms of transmission reliability or the time
required for transmitting the whole sequence of data symbols. For instance,
differentially encoded modulation symbols can be transmitted twice as fast by
the proposed modulation scheme with a 3 dB penalty in signal-to-noise ratio
over additive white Gaussian noise channels.

###### Index Terms:

Intersymbol-interference; linear modulation; Nyquist signaling; partial
response signaling; root raise cosine pulse; sequence multiplexing.

##  I Introduction

The spectrum scarcity necessities the use of spectrally efficient modulations.
The Nyquist signaling is a well established and robust technique for
constructing linear digital modulations which are employed in a vast majority
of today’s communication systems. These modulation schemes are often combined
with channel encoding to improve the transmission reliability and even
approach the channel capacity. An alternative strategy is to assume
modulations having a controlled level of intersymbol interference (ISI), which
can increase the rate of information transmission as well as act as a form of
information encoding for improving the transmission reliability [1], albeit at
the cost of increased detection complexity at the receiver. Such so-called
faster-than-Nyquist (FTN) schemes are linear modulations that can be used over
band-limited channels [2].

The renewed interest in FTN signaling schemes goes back to the early 2000’s
[3]. However, a closely related idea of partial response linear modulations
with controlled ISI appeared much earlier [4]. The achievable spectral
efficiency of coded and uncoded FTN schemes is evaluated in [2], [5], and [6].
The observation that up to $25\%$ increase in the transmission rate is
possible without deteriorating the error performance is known as the Mazo
limit [3, 2, 7, 6]. The energy and complexity costs of FTN signaling are
reviewed in [2].

The FTN schemes can be implemented both in time and in frequency domains [2,
8]. An orthogonal FTN scheme based on OFDM was designed in [9]. Alternatively,
Nyquist signaling with dual root raised-cosine (RRC) pulses akin to duobinary
modulation has been investigated in [10]. This scheme was further refined for
the RRC pulses with zero roll-off in [11]. The link between duobinary
modulation and FTN signaling has been pointed out in [1].

An important issue is how to efficiently perform the detection of transmitted
symbols at the receiver. Unlike the ISI due to multipath propagation, the ISI
created by FTN signaling also correlates samples of additive noise. The
optimum detection necessitates the use of whitening matched filter (WMF) prior
to symbol decisions. The ISI at the detector input can be equivalently
represented as an auxiliary channel [5, 6]. The output signal of such channel
has a trellis-like structure, which can be optimally equalized by the Viterbi,
BCJR and other such algorithms with varying complexity [2, 7, 6]. These
decoding methods can approach the performance of zero-ISI (Nyquist)
modulations over additive white Gaussian noise (AWGN) channels [2]. The
symbol-by-symbol detector for FTN signals was devised in [6] and [12]. The
detection of FTN signals with oversampling and one-bit quantization was
developed in [5]. A low complexity linear equalization for FTN signaling was
designed in [13]. The joint channel estimation and decoding of FTN signals was
studied in [14] and in [15].

Nearly all investigations of FTN signaling schemes in the literature assume
the RRC modulation pulse. The RRC pulse is parameterized by a time period,
$T_{\mathrm{p}}$, and a roll-off factor, $\alpha$. Linear modulations combine
the RRC pulses weighted by data symbols, which are then transmitted once every
symbol period, $T_{\mathrm{s}}$. The packing factor defines the relationship
between $T_{\mathrm{p}}$ and $T_{\mathrm{s}}$, i.e.,
$\tau=1-T_{\mathrm{s}}/Tp$. The design and analysis of FTN signaling in the
literature usually assumes arbitrary values of $0\leq\alpha\leq 1$ and
$0\leq\tau<1$. The search for good values of $\alpha$ and $\tau$ over an
entire $\alpha-\tau$ plane to allow symbol-by-symbol decisions was carried out
in [12]. However and importantly, the case of $100\%$ bandwidth roll-off is
rarely explicitly considered in the literature [6]. The authors in [5] noticed
that, for $\alpha=1$ and an arbitrary value of $\tau$, the ISI is
approximately limited to the two previous and the two subsequent symbol
samples.

In this paper, we show that the RRC pulse with $100\%$ roll-off and $50\%$
packing has a well-defined ISI, which is exactly and symmetrically constrained
to one previous and one subsequent symbol. Such a unique property of the RRC
pulse appears to remain unnoticed in the literature. Interestingly, reference
[1] states that ISI with only two components can be obtained with $100\%$
roll-off and $50\%$ packing assuming prolate spheroidal wave pulses, but not
RRC pulses. Although such a modulation scheme can be assumed to be a special
case of FTN signaling, it is argued that RRC pulses having $100\%$ roll-off
and $50\%$ packing offers symmetric multiplexing of the two transmitted data
streams. For this reason, such a partial response signaling is referred to in
this paper as a pulse-shape binary multiplexing (PSBM) modulation. The main
task then is how to separate the two multiplexed data streams at the receiver
with acceptable reliability and complexity. As with other partial response
signalings, the modulation constellation and the dependency between
transmitted symbols must be carefully selected in order to trade-off the
performance and the decoding complexity. We design several transmission
sequences interleaving pilot and data symbols, discuss superposition
modulation with symbol-by-symbol sequential interference cancellation (SIC),
and also consider orthogonal spreading sequences to aid separation of the data
streams at the receiver. In addition, the performance of multiplexed
differentially encoded phase-shift keying (PSK) modulation symbols is
evaluated numerically. The numerical results identify several cases when the
proposed PSBM modulation outperforms the Nyquist signaling in terms of either
transmission reliability or the time required to transmit a given number of
data symbols.

The rest of this paper is organized as follows. Linear modulation schemes that
are related to the proposed pulse-shape multiplexing signaling are outlined in
Section II. System model and the received signal structure are described in
Section III. The proposed pulse-shape multiplexing modulation is defined in
Section IV including the design of transmitted symbol sequences. Numerical
results are presented in Section V. Section VI concludes the paper.

We adopt the following notations: ${\operatorname{E}}\\!\left[\cdot\right]$ is
expectation, $\circledast$ is convolution, $|\cdot|$ is absolute value,
$(\cdot)^{\ast}$ is complex conjugate, $\real{\cdot}$ and
${\operatorname{Im}}\\!\left\\{\cdot\right\\}$, respectively, denote the real
and imaginary part of a complex number, $\mathrm{Card}\\{\cdot\\}$ is
cardinality of a set, $(\cdot)^{T}$ is matrix transpose, $(\cdot)^{-1}$ is
matrix inverse, and $\left\|\cdot\right\|$ is the Euclidean norm of a matrix
or vector.

##  II Related Linear Modulation Schemes

A linearly modulation signal is constructed as,

$x(t)=\sum_{k}s_{k}\,p(t-kT_{\mathrm{s}})$ (1)

where $s_{k}$ are $M$-ary modulation symbols transmitted every symbol period,
$T_{\mathrm{s}}$, and $p(t)$ denotes a deterministic pulse-shape, which is
also known at the receiver. The stationary sequence of transmitted symbols,
$s_{k}$, has zero-mean, and the variance,
${\operatorname{E}}\\!\left[|s_{k}|^{2}\right]=E_{\mathrm{s}}$. The symbols
are usually obtained as output of a finite-state modulator, i.e.,

$s_{k}=s(q_{k},c_{k})$ (2)

where the states, $q_{k}$, represent modulation memory, and the data symbols,
$c_{k}$, each carry, $\log_{2}M$, bits of input information. In this paper,
$p(t)$ is assumed to be the unit-energy RRC pulse, [4]

$p(t)=\frac{\mathrm{rrc}_{\alpha}(t/T_{\mathrm{s}})}{\sqrt{T_{\mathrm{s}}}}$
(3)

where

$\mathrm{rrc}_{\alpha}(t)=\frac{1}{1-16\alpha^{2}t^{2}}\left(\frac{\sin\left((1-\alpha)\pi
t\right)}{\pi t}+\frac{4\alpha\cos\left((1+\alpha)\pi t\right)}{\pi}\right).$
(4)

The roll-off factor, $0\leq\alpha\leq 1$, however, it is possible to also
consider pulse shapes having a roll-off greater than $100\%$.

Since the sequence of symbols, $s_{k}$, is stationary, the auto-correlation,
$R_{s}(i-j)={\operatorname{E}}\\!\left[s_{i}s_{j}^{\ast}\right]$. The
corresponding power-spectrum density (PSD) of signal (1) is computed as, [4]

$S_{x}(f)=\frac{1}{T_{\mathrm{s}}}|P(f)|^{2}\
\sum_{k}R_{s}(k)\,{\operatorname{e}}^{{\operatorname{j}}2\pi
fkT_{\mathrm{s}}}$ (5)

where $P(f)$ denotes the Fourier transform of $p(t)$.

Correlative coding assumes the discrete modulator (2) to be a finite impulse
response (FIR) filter, i.e.,

$s_{k}=\sum_{i=0}^{K-1}v_{i}c_{k-i}.$ (6)

The filter weights, $v_{i}$, are normalized, so that, $\sum_{i}|v_{i}|^{2}=1$.
More importantly, with a change in indices, modulated signal (1) with symbols
(6) can be rewritten as,

$\begin{split}x(t)=&\sum_{k}\sum_{i=0}^{K-1}v_{i}\,c_{k-i}p(t-kT_{\mathrm{s}})\\\
=&\sum_{k}c_{k}\sum_{i=0}^{K-1}v_{i}\,p(t-(k+i)T_{\mathrm{s}})=\sum_{k}c_{k}\,\tilde{p}(t-kT_{\mathrm{s}})\end{split}$
(7)

where the compound pulse,
$\tilde{p}(t)=\sum_{i=0}^{K-1}v_{i}\,p(t-iT_{\mathrm{s}})$.

Duobinary modulation is a special case of correlative coding, such that the
FIR filter has only two non-zero weights, $v_{o}=v_{1}=1/\sqrt{2}$, the
modulation symbols are binary, i.e.,
$c_{k}\in\\{-\sqrt{E_{\mathrm{s}}},+\sqrt{E_{\mathrm{s}}}\\}$, and the RRC
pulse has the smallest possible roll-off, $\alpha=0$. Modified duobinary
modulation assumes instead the weights, $v_{0}=1/\sqrt{2}$, $v_{1}=0$, and
$v_{2}=-1/\sqrt{2}$.

The following modulations assume the RRC pulse-shape with an arbitrary roll-
off value. Differential PSK constructs the transmitted symbols as,

$s_{k}=c_{k}\,s_{k-1}$ (8)

where the data symbols,
$c_{k}\in\\{\sqrt{E_{\mathrm{s}}}\,{\operatorname{e}}^{{\operatorname{j}}2\pi(i-1)/M}\\}$,
$i=1,2,\ldots,M$. Generalized shift-keying extends the modulation alphabet of
amplitude or phase shift-keying modulations with a zero symbol [16]. Offset-
quadrature ($M=4$) PSK delays the imaginary part of the modulated signal by
half a symbol period, i.e.,

$x(t)=\sum_{k}\real{c_{k}}p(t-kT_{\mathrm{s}})+{\operatorname{j}}\,{\operatorname{Im}}\\!\left\\{c_{k}\right\\}p(t-kT_{\mathrm{s}}-T_{\mathrm{s}}/2).$
(9)

Finally, FTN signaling is a linear modulation described by eq. (1). More
importantly, the RRC pulse-shape in (3) can now be scaled by,
$T_{\mathrm{p}}=T_{\mathrm{s}}/(1-\tau)$, instead of $T_{\mathrm{s}}$, where
$0\leq\tau<1$ is so-called the packing factor, i.e.,

$x(t)=\sum_{k}s_{k}p(t-k(1-\tau)T_{\mathrm{p}})=\sum_{k}s_{k}p(t-kT_{\mathrm{s}})$
(10)

so that $T_{\mathrm{p}}$ is a design parameter of the pulse, $p(t)$, whereas,
$T_{\mathrm{s}}=(1-\tau)T_{\mathrm{p}}$, denotes the symbol period. Thus,
$\tau=0$ packing corresponds to a conventional Nyquist signaling, whereas
$\tau=1$ packing would completely overlap the transmitted symbols. More
importantly, the PSD of (10) is still given by eq. (5), and it is otherwise
completely independent of the packing factor, $\tau$.

##  III Received Signal

The standard wireless channel model with $L$ propagation paths is an FIR
filter with the impulse response,

$\tilde{h}(t)=\sum_{l=1}^{L}h_{l}(t)\delta(t-\tau_{l}).$ (11)

The signal delays, $\tau_{l}$, are assumed to be constant. The path
attenuations, $h_{l}(t)$, are zero-mean circularly symmetric Gaussian
processes. These processes are stationary, and generally mutually correlated.
They have a defined auto-correlation, $R_{h}(\Delta t)$, which determines the
coherence bandwidth.

For narrow-band signals, the number of paths, $L$, is small. For $L=1$, the
channel model (11) becomes frequency non-selective. In low-mobility scenarios,
the channel attenuations, $h_{l}(t)$, are often assumed to be constant over
blocks of transmitted symbols, and independent between the successive blocks,
which is often referred to as a block fading model.

The received signal corresponding to multi-path propagation model (11) is
written as,

$\begin{split}y(t)&=\tilde{h}(t)\circledast x(t)+w(t)\\\
&=\sum_{l=1}^{L}h_{l}(t)x(t-\tau_{l})+w(t)\end{split}$ (12)

where $w(t)$ is a zero-mean stationary circularly symmetric AWGN with the
variance, $\sigma_{w}^{2}={\operatorname{E}}\\!\left[|w(t)|^{2}\right]$.

The received signal is filtered through a filter matched to the transmitted
pulse, $p(t)$, and synchronously sampled at a rate, $1/T_{\mathrm{s}}$. In
particular, assuming RRC pulses, the matched filter, $p^{\ast}(-t)=p(t)$, and
provided that the channel attenuations are constant over blocks of transmitted
symbols, the received samples are modeled as,

$\begin{split}r_{n}=&y(t)\circledast
p^{\ast}(-t)\Big{|}_{t=nT_{\mathrm{s}}+\tau_{0}}\\\
=&\sum_{l=1}^{L}h_{l}\,x(t-\tau_{l})\circledast
p(t)\Big{|}_{t=nT_{\mathrm{s}}+\tau_{0}}+w(t)\circledast
p(t)\Big{|}_{t=nT_{\mathrm{s}}+\tau_{0}}\\\
=&\sum_{l=1}^{L}h_{l}\sum_{k}s_{k}\int_{-\infty}^{\infty}p(\zeta+(n-k)T_{\mathrm{s}}-\tau_{l})p(\zeta-\tau_{0}){\,\operatorname{d}}\zeta\\\
&+\int_{-\infty}^{\infty}w(\zeta+nT_{\mathrm{s}})p(\zeta-\tau_{0}){\,\operatorname{d}}\zeta\\\
=&\sum_{k}s_{k}\sum_{l=1}^{L}h_{l}\,p_{n-k,l}+w_{n}=\sum_{k}s_{k}\tilde{p}_{n-k}+w_{n}\\\
=&s_{k}\tilde{p}_{0}+\underbrace{\sum_{k\atop n\neq
k}s_{k}\tilde{p}_{n-k}}_{\mathrm{ISI}}+w_{n}.\end{split}$ (13)

The timing offset, $\tau_{0}$, at the receiver can be optimized to minimize
the ISI term (in some sense) in (13) defined as,

$\tilde{p}_{n-k}=\sum_{l=1}^{L}h_{l}\int_{-\infty}^{\infty}p(\zeta+(n-k)T_{\mathrm{s}}-\tau_{l})p(\zeta-\tau_{0}){\,\operatorname{d}}\zeta,\
n\neq k.$ (14)

Thus, the ISI arises when the orthogonality between the transmitter and the
receiver pulses is violated, for example, due to multi-path propagation, time-
synchronization errors between transmitter and receiver, and also due to
symbol-period compression in FTN signaling schemes [4].

An interesting question is how much ISI is produced for different combinations
of parameters $\alpha$ and $\tau$ in FTN signaling schemes using RRC pulses.
Hence, define the function,

$\mathrm{ISI}\\!\left({\mu}\right)=\mathrm{Card}\\{|\tilde{p}_{k}|>\mu,\ k\neq
0\\}$ (15)

to be the number of ISI components that are greater than a given threshold,
$\mu$. Note that, $\mathrm{ISI}\\!\left({\mu}\right)\in\\{0,2,4,\ldots\\}$,
due to even symmetry of the RRC pulses. Assuming different thresholds, $\mu$,
the roll-off, $0\leq\alpha\leq 2$, and the RRC pulses truncated to
$(-4T_{\mathrm{s}},+4T_{\mathrm{s}})$, the values
$\mathrm{ISI}\\!\left({\mu}\right)=0$ (red points) and
$\mathrm{ISI}\\!\left({\mu}\right)=2$ (blue points) in the $\alpha-\tau$ plane
are shown in Fig. 1. The empty (white) spaces in Fig. 1 indicate the values,
$\mathrm{ISI}\\!\left({\mu}\right)>2$. It can be observed that by decreasing
the threshold, $\mu$, several cases of interest for designing FTN signaling
schemes start to emerge. In particular, exactly two ISI components can be
obtained for these parameters: $\alpha=1.0$ and $\tau=0.5$, $\alpha=1.07$ and
$\tau\in(0.70,0.71)$, and $\alpha\in(1.65,1.85)$ and $\tau\in(0.47,0.50)$.

Figure 1: The $\mathrm{ISI}\\!\left({\mu}\right)=0$ components (red points)
and $\mathrm{ISI}\\!\left({\mu}\right)=2$ components (blue points) for four
different thresholds, $\mu$.

##  IV Pulse-Shape Binary Multiplex Modulation

As indicated in Fig. 1, the RRC pulse with $100\%$ roll-off and $50\%$ packing
has well-defined and finite ISI components. In particular, the RRC pulse (4)
for $\alpha=1$ becomes,

$\mathrm{rrc}_{1}(t)=\frac{4\cos(2\pi t)}{\pi(1-16t^{2})}.$ (16)

This pulse has the following ISI components in an AWGN channel without multi-
path. Such a fundamental property appears to remain unnoticed in the
literature.

###### Lemma 1

Let $n$ be a non-negative integer. The ISI integral involving the RRC pulse,
$\mathrm{rrc}_{1}(t)$, with $100\%$ roll-off has the exact solution,

$\begin{split}\int_{-\infty}^{\infty}&\mathrm{rrc}_{1}(t)\times\mathrm{rrc}_{1}(t-n/4){\,\operatorname{d}}t\\\
=&\left\\{\begin{array}[]{cc}8/(3\pi)&n=1\\\
\frac{8}{\pi(n-2)n(n+2)}&n-\mathrm{odd},\ n>1\\\ 1&n=0\\\ 1/2&n=2\\\
0&n-\mathrm{even},\ n>2.\end{array}\right.\end{split}$ (17)

Lemma 1 can be proved by solving the integral for the first few values of $n$
(for example, using Mathematica software), and then using induction.

However, the result (17) is exact only when the integration is performed over
an infinite interval. In practice, the pulse shapes must be truncated to a
finite interval. The numerically computed values of integral (17) when the
interval of integration is truncated to $(-d,+d)$ are shown in Fig. 2. It can
be observed that the RRC pulse shape, $\mathrm{rrc}_{1}(t)$, should not be
truncated to the intervals shorter than, $(-4,+4)$, in order to achieve the
RRC property given in Lemma 1 with at least $99.9\%$ accuracy.

Figure 2: Numerically computed integral (17) truncated to interval, $(-d,+d)$,
as a function of $d$ (solid lines). The exact values for an infinite interval
are shifted to be all equal to unity in order to enable comparison. The dashed
lines are mirrored solid lines about the unit value.

###### Definition 2

The modulated signal of pulse-shape binary multiplex modulation is written as,

$x(t)=\sum_{k}s_{k}\frac{\mathrm{rrc}_{1}\left(\frac{t-kT_{\mathrm{s}}}{2T_{\mathrm{s}}}\right)}{\sqrt{2T_{\mathrm{s}}}}.$
(18)

The synchronously sampled matched filter output of modulated signal (18)
received in AWGN, $w(t)$, is,

$\begin{split}r_{n}=&(x(t)+w(t))\circledast\frac{\mathrm{rrc}_{1}\left(\frac{t}{2T_{\mathrm{s}}}\right)}{\sqrt{2T_{\mathrm{s}}}}\Big{|}_{t=nT_{\mathrm{s}}}\\\
=&\sum_{k}s_{k}\int_{-\infty}^{\infty}\mathrm{rrc}_{1}(\zeta+(n-k)/2)\
\mathrm{rrc}_{1}(\zeta){\,\operatorname{d}}\zeta+w_{n}\\\
=&\left(\frac{1}{2}s_{n-1}+s_{n}+\frac{1}{2}s_{n+1}\right)+w_{n}.\end{split}$
(19)

The noise samples, $w_{n}$, in (19) are zero-mean, have the variance,
${\operatorname{E}}\\!\left[|w_{n}|^{2}\right]={\operatorname{E}}\\!\left[|w(t)|^{2}\right]=\sigma_{w}^{2}$,
and their stationary auto-correlation is,

$R_{w}(n-m)={\operatorname{E}}\\!\left[w_{n}w_{m}^{\ast}\right]=\left\\{\begin{array}[]{cc}\sigma_{w}^{2}&n=m\\\
\sigma_{w}^{2}/2&|n-m|=1\\\ 0&|n-m|>1.\end{array}\right.$ (20)

Such noise samples can be equivalently modeled by a simple FIR filter,

$w_{n}=\frac{u_{n}+u_{n-1}}{\sqrt{2}}$ (21)

where $u_{n}$ are the samples of a zero-mean, circularly symmetric Gaussian
process having the variance,
${\operatorname{E}}\\!\left[|u_{n}|^{2}\right]=\sigma_{w}^{2}$. In addition,
it is straightforward to show that the variance of the sum of $N$ noise
samples having the correlations (20) is,

${\operatorname{var}}\\!\left[\sum_{n=1}^{N}w_{n}\right]=(2N-1)\sigma_{w}^{2}$
(22)

which is greater than the variance, $N\sigma_{w}^{2}$, of the sum of $N$
uncorrelated samples.

The modulated signal (18) in Definition 2 can be visualized as shown in Fig.
3. In particular, the transmitted data symbols can be viewed as consisting of
two multiplexed streams of data symbols, $a_{k}$, and, $b_{k}$, which are each
transmitted with a period $2T_{\mathrm{s}}$, but mutually shifted by
$T_{\mathrm{s}}$. The corresponding received symbol samples after the matched
filtering are,

$r_{n}=\left\\{\begin{array}[]{cc}a_{n}+\frac{b_{n-1}+b_{n}}{2}+w_{n}&n-\mbox{odd}\\\
b_{n}+\frac{a_{n}+a_{n+1}}{2}+w_{n}&n-\mbox{even}.\end{array}\right.$ (23)

Figure 3: A visualization of pulse-shape multiplex modulated signal.

Using (5), the PSD of modulated signal (18) is computed as,

$S_{x}(f)=2|\mathrm{RRC}_{1}(2T_{\mathrm{s}}f)|^{2}\,\sum_{k}{\operatorname{E}}\\!\left[s_{0}s_{k}^{\ast}\right]\,{\operatorname{e}}^{{\operatorname{j}}2\pi
fkT_{\mathrm{s}}}$ (24)

where the Fourier transform of the pulse, $\mathrm{rrc}_{1}(t)$, is,

$\mathrm{RRC}_{1}(f)=\left\\{\begin{array}[]{cc}\cos(\pi f/2)&|f|\leq 1\\\
0&\mbox{otherwise}.\end{array}\right.$ (25)

###  IV-A Transmitted Sequence Design

The optimum detection of transmitted symbols in the presence of ISI must
consider complete sequences of received samples. However, in the absence of
multi-path, the received samples have structure (19), and the transmitted
sequences can be designed, so that the complexity of detection at the receiver
can be reduced.

The key strategy for reducing the detection complexity is to exploit
orthogonality among sub-sequences of transmitted symbols. Offset-quadrature
PSK modulation (9) alternates one-dimensional modulation symbols along the in-
phase and quadrature components, which allows the optimum symbol-by-symbol
decisions.

Multiplexing two data streams as described by (23) can exploit the design
principles of superposition modulation and multiuser detection. In such a
case, symbol-by-symbol decisions can be performed by SIC. Specifically,
provided that symbols, $b_{n}$, can be reliably detected, even if the symbols,
$(a_{n}+a_{n+1})/2$, are not yet known, then the symbol, $a_{n}$, can be
reliably detected after canceling the ISI term, $(b_{n-1}+b_{n})/2$.

In the sequel, three other sequence design strategies are discussed in more
detail. The first strategy combines pilot and data symbols to aid the data
detection and channel estimation. The second strategy employs orthogonal
spreading codes in order to separate the two multiplexed data sequences. The
third strategy adopts the differential encoding of transmitted symbols.

###  IV-B Sequences with Interleaved Pilot Symbols

In general, pilot symbols for channel estimation can be interleaved with data
symbols or superimposed onto data symbols [17]. Here, the more common former
approach is adopted. Thus, consider a transmitted sequence consisting of
alternating groups of $L_{\mathrm{d}}$ data symbols and $L_{\mathrm{p}}\ll
L_{\mathrm{d}}$ pilot symbols, which are separated by a single zero-symbol as
shown in Fig. 4.

For instance, the following sub-sequences with reduced or no ISI can be
considered with pilot symbol, $\mathsf{p}$, and arbitrary data symbols,
$\mathsf{d}_{1}$, and, $\mathsf{d}_{2}$: $(0,\mathsf{p},0)$,
$(\mathsf{d}_{1},\mathsf{p},-\mathsf{d}_{1})$,
$(\mathsf{d}_{1},\mathsf{p},-\mathsf{d}_{1},-\mathsf{p},\mathsf{d}_{1})$, and
$(\mathsf{d}_{1},\mathsf{p},-\mathsf{d}_{1},-\mathsf{p},\mathsf{d}_{2},\mathsf{p},-\mathsf{d}_{2})$.
These sub-sequences enable ISI-free data detection and channel estimation, as
can be deduced from eq. (23) and Fig. 3. Recall also that the noise samples,
$w_{n}$, and, $w_{n\pm 2}$, are uncorrelated, i.e., independent.

Figure 4: The transmitted sequence with interleaved sub-sequences of data and
pilot symbols and a single zero-symbol separator.

In order to illustrate the ISI-free channel estimation, consider the sequence,
$(\mathsf{d}_{1},\mathsf{p},\mathsf{p},-\mathsf{d}_{1})$. The received samples
corresponding to the two pilot symbols in the middle are,

$\begin{split}r_{n}=&h\frac{3}{2}\mathsf{p}+h\frac{1}{2}\mathsf{d}_{1}+w_{n}\\\
r_{n+1}=&h\frac{3}{2}\mathsf{p}-h\frac{1}{2}\mathsf{d}_{1}+w_{n+1}\end{split}$
(26)

where $h$ denotes the complex-valued channel attenuation (i.e., frequency non-
selective slow fading). The samples, $r_{n}$, and, $r_{n+1}$, can be simply
combined as,

$r_{n}+r_{n+1}=3h\mathsf{p}+w_{n}+w_{n+1}$ (27)

where the total variance of the additive noise samples is equal to
$3\sigma_{w}^{2}$ due to correlations (20).

More generally, the transmitted sequence,

$(-\mathsf{p},\mathsf{d}_{1},\mathsf{p},\mathsf{d}_{2},-\mathsf{p},\mathsf{d}_{3},\mathsf{p},\mathsf{d}_{4},-\mathsf{p},\mathsf{d}_{5},\ldots,\mathsf{d}_{N},\pm\mathsf{p})$
(28)

where the last pilot symbol is $\mathsf{p}$, if $N$ is odd, and $-\mathsf{p}$,
if $N$ is even, allows the ISI-free symbol-by-symbol decisions of all data
symbols. Moreover, assuming again a slow fading channel, the received samples
corresponding to the pilot symbols can be summed up to obtain,

$\sum_{n=1}^{N}(-1)^{n}\,r_{2n-1}=N\,h\,\mathsf{p}+\sqrt{N}w$

where the noise sample, $w$, has the variance, $\sigma_{w}^{2}$, so the
signal-to-noise ratio (SNR) for estimating the channel coefficient, $h$, has
been improved $N$-times. Note also that once the channel has been estimated,
the pilot symbols can be subtracted from the received samples in order to aid
decisions of the remaining data symbols.

Finally, consider the case of a symbol repetition diversity. The transmitted
sequence, $(\mathsf{d},0,\mathsf{d},0,\ldots,0,\mathsf{d})$, of a data symbol,
$\mathsf{d}$, repeated $(N\geq 2)$-times corresponds to the canonical Nyquist
signaling. The pulse-shape multiplex modulation instead transmits the
sequence, $(\mathsf{d},\mathsf{d},\ldots,\mathsf{d})$, of $N_{1}$-times
repeated data symbol, $\mathsf{d}$. For the same sequence length,
$N_{1}=2N-1$. Assuming slowly fading channel, the detector combines the
received samples for the two modulation schemes, respectively, as,

$\begin{split}r=&N\,h\,d+\sqrt{N}w\\\ r=&(2\cdot
3/2+2(N_{1}-2))\,h\,d/\sqrt{2}+\sqrt{2N_{1}-1}w\end{split}$ (29)

where the scaling by $\sqrt{2}$ was introduced for the second modulation in
order to account for the larger number of symbols in its transmitted sequence.
The resulting SNR of these two schemes is proportional to, $\gamma\propto N$,
and, $\gamma\propto 2N-3/2$, respectively. Consequently, for symbol repetition
diversity, the SNR gain of the pulse-shape binary multiplexing is
asymptotically $3$ dB larger than for the Nyquist signaling.

###  IV-C Sequences with Orthogonal Spreading

Another strategy for transmitting interleaved, but orthogonal symbols in
modulated signal (18) is to use orthogonal spreading codes. In particular,
assume transmitted symbols,

$a_{n}=\mathsf{d}_{1}c_{n}^{(1)},\qquad b_{n}=\mathsf{d}_{2}c_{n}^{(2)}$ (30)

where $\mathsf{d}_{1}$ and $\mathsf{d}_{2}$ are two data symbols, and,
$c_{n}^{(1)}$ and $c_{n}^{(2)}$, $n=1,2,\ldots,N$, are generally complex-
valued, orthogonal spreading sequences, so that,

$\sum_{n=1}^{N}c_{n}^{(i)}c_{n}^{\ast(j)}=\left\\{\begin{array}[]{cc}N,&i=j\\\
0,&i\neq j.\end{array}\right.$ (31)

Then, the sequences of received samples (23) are linearly combined as,

$\begin{split}\sum_{n=1}^{N}r_{2n-1}c_{n}^{\ast(1)}=&\mathsf{d}_{1}+\mathsf{d}_{2}\sum_{n=1}^{N}\frac{c_{n}^{(2)}+c_{n+1}^{(2)}}{2}\,c_{n}^{\ast(1)}+\sum_{n=1}^{N}w_{2n-1}c_{n}^{\ast(1)}\\\
=&\mathsf{d}_{1}+\tilde{w}_{1}\\\
\sum_{n=1}^{N}r_{2n}c_{n}^{\ast(2)}=&\mathsf{d}_{2}+\mathsf{d}_{1}\sum_{n=1}^{N}\frac{c_{n}^{(1)}+c_{n+1}^{(1)}}{2}\,c_{n}^{\ast(2)}+\sum_{n=1}^{N}w_{2n}c_{n}^{\ast(2)}\\\
=&\mathsf{d}_{2}+\tilde{w}_{2}\end{split}$ (32)

provided that the spreading sequences, $c_{n}^{(1)}$, and, $c_{n}^{(2)}$, are
exactly orthogonal. In such a case, the SNR improvement for transmitting two
data symbols with orthogonal spreading sequences using the pulse-shape binary
multiplex modulation (18) is proportional to,

$\gamma\propto\frac{N^{2}}{2N-1}.$ (33)

For instance, if the spreading symbols, $c_{n}$, are generated independently
at random and with an equal probability from the set, $\\{-1,+1\\}$, the
probability that two such sequences are orthogonal is,

$\begin{split}\operatorname{Pr}\\!\left(\sum_{n=1}^{N}c_{n}^{(1)}c_{n}^{\ast(2)}=0\right)=&\binom{N}{N/2}\left(\frac{1}{2}\right)^{N/2}\left(\frac{1}{2}\right)^{N-N/2}\\\
=&\binom{N}{N/2}2^{-N}.\end{split}$ (34)

Since the probability (34) of exact orthogonality asymptotically goes to zero
with large $N$, consider instead the probability,

$\begin{split}\operatorname{Pr}\\!\left(-\lceil\kappa\,N/2\rfloor\leq\sum_{n=1}^{N}c_{n}^{(1)}c_{n}^{\ast(2)}\leq\lceil\kappa\,N/2\rfloor\right)\\\
=\sum_{n=-\lceil\kappa\,N/2\rfloor}^{\lceil\kappa\,N/2\rfloor}\binom{N}{n}\left(\frac{1}{2}\right)^{N}\end{split}$
(35)

for some small $\kappa\geq 0$. The probabilities (35) as a function of $N$ for
two different values of factor, $\kappa$, are shown in Fig. 5. These
probabilities are indicative of how many random spreading sequences need to be
generated in order to select the required number of such sequences having an
acceptable level of mutual orthogonality.

Figure 5: The probability (35) vs. the spreading sequence length, $N$,
assuming $\kappa=5\%$ and $\kappa=10\%$, respectively.

###  IV-D Sequences with Differential Encoding

Differential PSK is a popular modulation scheme for fast fading channels,
since it alleviates the need for recovering the absolute phase reference. Fig.
6 shows differentially encoded $M$-ary PSK symbols (8) transmitted via pulse-
shape binary multiplex modulation. In particular, the $n$-th transmitted
symbol is,

$\begin{split}s_{n}=&\left(\prod_{k=0}^{n-2}c_{k}\right)\frac{1+c_{n-1}c_{n}+2c_{n-1}}{2}\\\
=&\frac{1}{2}\left(\prod_{k=0}^{n-2}c_{k}\right)+\frac{1}{2}\left(\prod_{k=0}^{n-1}c_{k}\right)c_{n}+\left(\prod_{k=0}^{n-1}c_{k}\right).\end{split}$
(36)

Consequently, the differential decoding can be performed as,

$\begin{split}c_{n}=&\left(2s_{n}-\left(\prod_{k=0}^{n-2}c_{k}\right)-2\left(\prod_{k=0}^{n-1}c_{k}\right)\right)\left(\prod_{k=0}^{n-1}c_{k}\right)^{\ast}\\\
=&2s_{n}\left(\prod_{k=0}^{n-1}c_{k}^{\ast}\right)-c_{n-1}^{\ast}-2.\end{split}$
(37)

The performance of this modulation scheme is evaluated in the next section.

Figure 6: Differentially encoded $M$-ary PSK symbols transmitted via pulse-
shape binary multiplex modulation.

##  V Numerical Examples

It is convenient to use a vector notation to generate samples of pulse-shape
binary multiplex modulation (18) received over a frequency non-selective
fading channel. The vector, $\bm{r}$, of $N$ received samples corresponding to
the vector, $\bm{s}$, of $N$ transmitted symbols can be obtained as,

$\bm{r}=\bm{s}\,\bm{A}\,{\operatorname{diag}}\\!\left(\bm{h}\right)+\bm{w}\,\bm{A}_{0}^{T}$

where $\bm{h}$ is a vector of fading channel coefficients, $\bm{w}$ are
samples of AWGN, and the $(N\times N)$ ISI matrix,

$\bm{A}=\left[\begin{matrix}1&1/2&&\\\ 1/2&1&1/2&\\\ &\ddots&\ddots&\\\
&&1/2&1\end{matrix}\right]=\bm{A}_{0}\,\bm{A}_{0}^{T}.$

The optimum detection requires that the additive noise is first whitened as,
[4]

$\bm{r}\,\bm{A}_{0}^{-T}=\bm{s}\,\bm{A}\,{\operatorname{diag}}\\!\left(\bm{h}\right)\,\bm{A}_{0}^{-T}+\bm{w}.$

Then the maximum likelihood (ML) detection of sequence $\bm{s}$ is,

$\hat{\bm{s}}=\arg\min\limits_{\bm{s}}\left\|\bm{r}\,\bm{A}_{0}^{-T}-\bm{s}\,\bm{A}\,{\operatorname{diag}}\\!\left(\hat{\bm{h}}\right)\,\bm{A}_{0}^{-T}\right\|^{2}$
(38)

where $\hat{\bm{h}}$ is the estimate of $\bm{h}$ representing channel state
information (CSI).

An uncoded binary phase shift keying (BPSK) modulation and Rayleigh-
distributed fading amplitudes, $\bm{h}$, are assumed for simplicity. The
transmitted sequence interleaves pilot symbols and data symbols as shown in
Fig. 4. The pilot symbols are used to estimate the channel coefficients,
$\bm{h}$, by linear minimum mean-square error (LMMSE) algorithm. The spectral
efficiency of pulse-shape binary multiplexing is, $2$, which is always larger
than the spectral efficiency of the Nyquist signaling being equal to,
$2/(1+\alpha)$.

The BER curves, $P_{\mathrm{e}}$, for short data sequences of
$L_{\mathrm{d}}=4$ and $L_{\mathrm{d}}=8$ binary symbols, respectively,
separated by a single zero-symbol are shown in Fig. 7 and Fig. 8. The SNR is
defined as, $\gamma_{b}=1/(2\sigma_{w}^{2})$. Both cases of perfect and
estimated CSI are considered. The Nyquist signaling (no ISI) with symbol-by-
symbol decisions is assumed as a reference. The ML data detector (38) is used
for pulse-shape multiplexing signaling.

It can be observed that the performance penalty due to channel estimation is
much larger for pulse-shape multiplexing than for the Nyquist signaling, which
is to be expected. The WMF improves the performance by several dB’s for both
signaling schemes. More importantly, the performance of pulse-shape
multiplexing improves with the data block length by exploiting the time
diversity over a fading channel, so it can significantly outperform the
Nyquist signaling at medium to large SNR values. It is likely that by
employing more sophisticated channel estimation and equalization techniques,
the performance of pulse-shape multiplexing can be further improved. In order
to demonstrate the effect of time diversity, Fig. 9 shows that, over an AWGN
channel, the performance of pulse-shape multiplexing is worse than that of
Nyquist signaling, even though some performance loss can be recovered by WMF.

Figure 7: The BER of BPSK vs. SNR over Rayleigh fading channel for sequences
of $4$ binary symbols. Figure 8: The BER of BPSK vs. SNR over Rayleigh fading
channel for sequences of $8$ binary symbols. Figure 9: The BER of BPSK vs. SNR
over AWGN channel for sequences of $2$ and $4$ binary symbols, respectively.

Lastly, the BER performance of Nyquist modulation and pulse-shape multiplex
modulation transmitting differentially encoded quadrature PSK (QPSK) symbols
over an AWGN channel is compared in Fig. 10. It can be observed that even
though the pulse-shape multiplexing suffers asymptotically a 3 dB penalty in
SNR, it reduces the time required for transmitting the whole symbol sequence
to one half.

Figure 10: The BER comparison of differentially encoded QPSK with Nyquist and
pulse-shape binary multiplexing (PSBM) modulation transmitted over an AWGN
channel.

##  VI Conclusion

The paper introduced a pulse-shape binary multiplex modulation. Such a
modulation scheme is akin to partial-response signaling, correlative coding,
offset-QPSK modulation and FTN signaling. It combines two data streams under
controlled ISI created by the RRC pulses having $100\%$ roll-off, and
transmitted at twice the Nyquist rate. The ISI analysis showed that this is
unique property among all the roll-off factors being at most $100\%$ and the
packing factors greater than $5\%$. However, the successive samples of
additive noises at the output of matched filter at the receiver become
correlated, which incurs a SNR performance penalty. This penalty could be
reduced or even removed by using more complex sequence-based detection schemes
as shown elsewhere in the literature. The BER performance as well as decoding
complexity of the proposed pulse-shape binary multiplexing modulation scheme
is critically affected by the choice of transmitted sequences. One can
consider superposition modulation with SIC decoding, interleave data symbols
with pilot and zero-symbols to aid channel estimation and data decoding, and
also employ orthogonal spreading sequences to separate the multiplexed data
streams. The numerical results indicate that pulse-shape binary multiplexing
can exploit time-diversity in fading channels to outperform the Nyquist
signaling. In addition, it has been shown numerically that a sequence of
differentially encoded PSK symbols can be transmitted twice as fast by the
proposed modulation scheme compared to canonical Nyquist signaling, although
with a 3 dB SNR penalty over AWGN channels.

## References

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††thanks: D. van Treeck and J. Lähnemann contributed equally to the
work††thanks: D. van Treeck and J. Lähnemann contributed equally to the work

Present address: ]Department of Materials Science and NanoEngineering, Rice
University, Houston, Texas, USA Present address: ]Institute for Optoelectronic
Systems and Microtechnology (ISOM) and Materials Science Department,
Universidad Politécnica de Madrid, Avenida Complutense 30, 28040 Madrid, Spain

# Sequential directional deposition of one-sided (In,Ga)N shells on GaN
nanowires by molecular beam epitaxy

David van Treeck  Jonas Lähnemann<EMAIL_ADDRESS>Guanhui Gao  [
Sergio Fernández Garrido  [  Oliver Brandt  Lutz Geelhaar  Paul-Drude-Institut
für Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V.,
Hausvogteiplatz 5–7, D-10117 Berlin, Germany

###### Abstract

Capitalizing on the directed nature of the atomic fluxes in molecular beam
epitaxy, we propose and demonstrate the sequential directional deposition of
lateral (In,Ga)N shells on GaN nanowires. In this approach, a sub-monolayer
thickness of each constituent atomic species, i. e. Ga, In, and N, is
deposited subsequently from the same direction by rotating the sample and
operating the shutters accordingly. Using multiple iterations of this process,
we achieve the growth of homogeneous shells on a single side facet of the
nanowires. For higher In content and thus lattice mismatch, we observe a
strain-induced bending of the nanowire heterostructures. The incorporation of
In and the resulting emission spectra are systematically investigated as a
function of both the growth temperature and the In/Ga flux ratio.

##  I Introduction

The three-dimensional geometry of nanowires (NWs) allows the growth of more
complex heterostructures than possible for conventional planar thin films,
which opens up new possibilities for device design.[1] For example, in NWs,
strain partitioning enables the accommodation of higher degrees of strain
arising from lattice mismatch without plastic relaxation.[2] Another benefit
is that core-shell NW heterostructures offers the possibility for an increased
volume per substrate area of the active region of semiconductor devices. This
concept is attractive for light-emitting diodes and has been pursued, e.g.,
for the material system (In,Ga)N/GaN.[3] Furthermore, the NW geometry can also
facilitate more advanced heterostructures, such as one-sided shells. For
classical III-V semiconductors, namely GaAs, GaP, InSb, and related alloys,
the intentional deposition of a heterostructure on only one lateral surface of
NWs has been demonstrated.[4, 5, 6, 7, 8] This selectivity is based on the
directionality of fluxes in molecular beam epitaxy (MBE) and utilizes either
tilted core NWs[4] or “digital alloy” deposition,[5] where the group-III
elements of a ternary III-V shell are deposited in an alternating fashion. In
contrast, for group-III nitrides, asymmetric NW shells have been reported only
as a transitional phenomenon caused by inhomogeneous shell nucleation.[9]

For this specific material class of group-III nitrides, the formation of core-
shell heterostructures in general is complicated by the peculiarities of the
NW growth mechanisms in MBE.[10, 11, 12] Whereas conventional III-V NWs are
typically grown by the vapor-liquid-solid mechanism, in which uniaxial growth
is induced by a droplet,[13] GaN and related NWs form spontaneously.[14] Thus,
the interplay between incorporation on the top and sidewall facets is
particularly pronounced, and the enhancement of radial growth needed for shell
formation is more difficult than for conventional III-V NWs. When samples are
grown with continuous rotation and fluxes, the incidence of the constituent
elemental fluxes from different directions together with the three-dimensional
nature of the NWs leads to a situation which in planar growth is known as
migration enhanced epitaxy:[15] on individual side facets, the different
elements are deposited in an alternating sequence. However, in the case of GaN
NWs, the permanent presence of N on the top facet causes this facet to
constitute a pronounced sink for the metal atoms.[11] Thus, during the growth
of (In,Ga)N shells the enhanced diffusion of In and Ga to the tip leads to the
formation of large parasitic top segments.[12] This competition between the
incorporation on the side and top facets of the NWs impedes the usability of
such core-shell heterostructures for potential devices. To shift the balance
of this competition toward incorporation on the side facets requires an
improved deposition sequence that allows us to control which atomic species is
present where and when.

In this paper, we present an innovative approach for the growth of (In,Ga)N
shells on GaN nanowires that exploits the directed nature of the material
beams in MBE: Only a single atomic species is deposited at a given time and
the sample is rotated accordingly to enable sequential directional deposition
(SDD) of either GaN or (In,Ga)N on only one side of GaN core NWs. This
approach allows to reduce the diffusion of metal adatoms to the NW tip by
avoiding the permanent presence of active N species on this facet. This
protocol enables the growth of uniform, one-sided shells on thin GaN NWs. From
a more general perspective, our study of one-sided shell growth as a
conceptionally simplified system helps to assess the potential of molecular
beam epitaxy for the growth of core-shell structures in the nitride material
system. In particular, in comparison to shell growth with continuous
fluxes[12] the parasitic growth on top of the NWs is substantially diminished.

##  II Results and Discussion

###  II.1 Sequential directional deposition

For these investigations, self-assembled GaN NWs on TiN grown at a V/III flux
ratio of 2.5 and a substrate temperature of 780 ∘C, as introduced in Refs. 16,
11, are used as core NWs. These NW ensembles have a mean diameter and length
of about 35 nm and 790 nm, respectively, at a density of about $10^{9}$ cm-2.
This low area density facilitates studies of shell growth since shadowing by
neighboring NWs is avoided.

Figure 1:  (a) Sketch of the SDD of a one-sided GaN shell. (b) Bird’s eye SE
micrograph of sample G, where one-sided GaN shells were grown on GaN core NWs
using the SDD approach. (c) Modeled shell thickness of a one-sided GaN shell
grown using the SDD approach (blue profile) and for constantly open N shutter
(green profile) at otherwise similar modeling parameters.

To introduce the SDD approach, we start with binary GaN shells (sample G).
Figure 1 (a) outlines the SDD of a one-sided GaN shell on a GaN core NW. In
the first step (i), Ga [typically $<1$ monolayer (ML)] is deposited on one
specific side of the NW by opening the corresponding shutter for a few
seconds. After closing the Ga shutter, the sample is rotated (ii) to align the
very same side with the N source. In the final step (iii), the shutter of the
N source is opened and GaN is grown. After closing the N shutter, the NW is
rotated back to position (i) for the next deposition cycle.

Figure 1 (b) shows a bird’s eye secondary electron (SE) micrograph of the
resulting sample G. The shell was grown at a substrate temperature of 490 ∘C
with a Ga-flux of 2.4 nm/min and a N-flux of 12.4 nm/min (direct fluxes in
equivalent growth rate units for two-dimensional GaN layers). During each
cycle, the NW side was exposed to the Ga beam for 4 s—corresponding to the
deposition of approximately 0.6 ML of Ga on the exposed side facets and 0.8 ML
of Ga on the top facet/substrate, with the difference being caused by the
different deposition angles in relation to the effusion cell. The wetted NW
side was then rotated toward the N source and exposed to N for 25 s. Both
shutters were closed during the 4 s long rotation steps. This cycle was
repeated 130 times. Thus, a 20–25 nm thick GaN shell was grown on the right
side of the NWs as shown in the top view sketch in the inset to Fig. 1 (b).
The shell thickness was estimated by analyzing the NW diameter along the
growth direction of the shell from SE micrographs. The one-sided shells are
very homogeneous along the entire NW length, as seen in the micrograph of Fig.
1 (b). This homogeneity is in contrast to the hourglass-shape obtained in Ref.
11 using continuous Ga and N fluxes both for radially symmetric GaN shells
obtained for continuous rotation and for one-sided shells grown without
substrate rotation. However, note that the growth temperature in this study
was reduced compared with Ref. 11, in order to decrease adatom diffusivity,
avoid Ga desorption and to subsequently enhance the incorporation of In in
ternary (In,Ga)N shells.

To better understand the SDD, we modeled the shell thickness of a one-sided
GaN shell on a core NW based on our work in Ref. 11. Instead of the shell
growth for continuous rotation investigated there, we here simulated the
different deposition and rotation sequences of the SDD mode. The model
consists of a system of coupled one-dimensional diffusion equations for the
different growth regions (top facet, sidewall facets, and substrate) that are
solved numerically. As input parameters, we used the growth conditions of
sample G described above and modeling parameters corresponding to those from
Ref. 11. The resulting profile (blue) is shown in Fig. 1 (c). For comparison,
the green profile corresponds to a shell for constantly open N shutter with
otherwise similar modeling parameters. In agreement with the experiment, the
modeled profile for the SDD approach (blue) shows a rather uniform shell
thickness along the NW with a slight barrel-shape.

During steps (i) and (ii), when the N shutter is still closed, the Ga adatom
concentration on the side, the top facet and the substrate reaches an
equilibrium state, where the surface is rather homogeneously wetted,
explaining the homogeneous shell thickness. Note that due to the slightly
higher adatom concentration on the top facet/substrate, compared to the side
facet, there may be small diffusion fluxes onto the side facets. Only when the
N-shutter is opened for step (iii), Ga adatoms diffuse from the NW side to the
NW top and the substrate due to the higher incorporation rates in these
regions (caused by the higher effective N flux related to the deposition
geometry[11]). Due to these diffusion effects, the barrel shape is much more
pronounced in case the N shutter is kept continuously open (green profile) and
the increased Ga diffusion, away from the side facet, results in a much
smaller mean shell thickness ($<10$ nm).

Figure 2:  Sketch of the SDD process of a one-sided (In,Ga)N shell with the
source arrangement corresponding to our MBE system. (i) Ga deposition on the
side of the NW. (ii) Rotation towards the In source and In deposition. (iii)
Rotation towards the In source and opening the N shutter for (In,Ga)N growth.
Note that all shutters are closed during the rotation phases.

###  II.2 Systematic analysis of ternary alloy shells

Figure 3:  SE micrographs of GaN NW ensembles with a one-sided (In,Ga)N shell
and the respective CL line scans (measured at 10 K) of single, dispersed NWs
as a function of the growth temperature and the nominal In/Ga flux ratio. The
CL intensities are color-coded in arbitrary units using a logarithmic scale.
The top-right inset displays ensemble PL spectra of the samples C1, C2, and D
acquired at room temperature. The growth conditions for the presented samples
are summarized in Tab. 1.

Turning to ternary alloy shells, Fig. 2 outlines the SDD process for a one-
sided (In,Ga)N shell. Compared to the binary shell, an intermediate step is
added after the Ga deposition, where the growth side faces the In cell and In
is deposited. The total amount of deposited metal was kept constant at 0.3 ML
(0.4 ML) on the side facet (top facet) per deposition cycle. Finally, the
facets were exposed to an N flux of 16.2 nm/min for approximately 25 s
seconds.

A series of six samples with different In/Ga flux ratios (between 1 and 5) and
at different growth temperatures (between 440 and 590 ∘C) were grown, as
summarized in Tab. 1. Only for sample D, with an In/Ga flux ratio of 5, the
metal flux had to be increased to 0.45 ML (0.6 ML) for the side facet (top
facet) to maintain a reasonable calibration of the Ga flux and shutter
response times. By adjusting the number of SDD cycles for each sample, we
aimed to maintain a shell thickness of 10–15 nm.

The morphology and emission properties of these GaN NWs with a one-sided
(In,Ga)N shell (samples A–D), plotted as a function of the growth temperature
and the nominal In/Ga flux ratio, are shown in Fig. 3. The presented line
scans are representative choices among 10–20 line scans on dispersed NWs
performed for each sample. To resolve the emission along the NW axis, the NWs
were dispersed on an Au-coated Si substrate and cathodoluminescence (CL)
spectral line scans were measured using a Gatan MonoCL4 system mounted on a
Zeiss Ultra55 field-emission scanning electron microscope (SEM) operated at 5
kV acceleration voltage with a probe current of 1–1.5 nA. The data was
analyzed and plotted using the python packages HyperSpy[17] and LumiSpy.[18]
In Fig. 3, representative CL line scans collected at a sample temperature of
10 K on individual NWs are plotted alongside bird’s eye scanning electron (SE)
micrographs of the corresponding NW ensemble. The SE micrographs of the single
NWs corresponding to each of the line scans are presented in the Supplementary
Material. With respect to the CL measurements, in addition to the broad
emission band of the (In,Ga)N shell at longer wavelengths, all samples also
exhibit a notable luminescence at around 357 nm (3.47 eV), which can be
attributed to the near band edge emission of the GaN cores. The contribution
of this band varies between individual line scans even from the same sample,
which might just be a geometric effect of the CL excitation depending on where
the shell is oriented on the dispersed NWs. Note that the streaky features in
the CL linescans are an experimental artefact due to the drift correction
employed during the measurements.

In terms of morphology, the growth map reveals that with decreasing growth
temperature from 540 ∘C to 440 ∘C and increasing In/Ga flux ratio from 1 to 5,
the NWs bend more and more to one side, while the CL emission of the (In,Ga)N
shell shifts from around 380 nm (3.26 eV) for sample B1 to about 480 nm (2.58
eV) for sample D. Only for sample A, grown at the highest temperature, no
(In,Ga)N shell emission can be detected. Note that the one-sided (In,Ga)N
shells show luminescence even without passivation by an outer GaN shell,
indicating that the localization of carriers in the ternary alloy prevents
them from recombining non-radiatively at the surface, as observed
previously.[12] However, such carrier localization and variations in its
effectiveness can lead to the observed variations in (In,Ga)N emission
intensity along the length of single NWs, most pronounced for the line scan
displayed for sample B2. Note that the apparent changes in the full width at
half maximum of the emission band, correlating with these intensity
variations, result from the logscale intensity colorscale chosen for the
plots. Additional experiments presented in the Supplementary Material show
that a conventional co-deposition of all constituents without rotation also
leads to one-sided shells. However, this approach results in very rough shells
with an inhomogeneous composition. Furthermore, for SDD the deposition
sequence (Ga-In-N) is important to incorporate In, as also discussed in the
Supplementary Material.

The bending of the NWs visible in the SE micrographs is a consequence of the
larger lattice constant of (In,Ga)N with respect to GaN. Hence, once an
(In,Ga)N shell is grown on one side of the GaN NWs (right side in the
micrographs of Fig. 3), they bend to the opposite (left) side. Thus, the
bending of the NWs confirms the presence of a one-sided (In,Ga)N shell, with a
larger In content and thus lattice mismatch resulting in a more pronounced
bending of the NWs. For sample A, the straight NWs indicate no or only little
In incorporation into the shell, consistent with the absence of a pronounced
(In,Ga)N emission. Correspondingly, the red-shift of the shell luminescence
with decreasing growth temperature can be attributed to an increase in In
incorporation resulting from a reduced desorption of In and a lower
dissociation rate of InN.[19, 20, 21, 22] Note that the bending of a NW
results in a non-uniform strain distribution in the core-shell structure,
which includes a gradient in the $\epsilon_{zz}$ component of the strain
tensor across the NW diameter, affecting the band structure[5] and hence the
emission wavelength of the (In,Ga)N shell.

The increasing In/Ga flux ratio from samples B1 to B2 (and C1 to C2) further
increases the In incorporation, corresponding to an additional red-shift and
broadening of the shell emission. This red-shift and broadening is also
visible when analyzing the ensemble photoluminescence (PL) spectra of the
samples C1 and C2 shown in the top-right inset of Fig. 3. The spectra were
acquired at 300 K using a Horiba Jobin Yvon (Labram HR 800 UV) µ-PL setup with
excitation from a HeCd laser (325 nm) with a spot diameter of about 3 µm, thus
probing about 70 NWs. The PL spectrum of sample D, grown at the lowest growth
temperature and highest In/Ga flux ratio, shows an even broader emission band
with rather constant emission intensities between 440 nm (2.82 eV) and 510 nm
(2.43 eV).

For planar (In,Ga)N layers,[23, 24] the observed red-shift from 380 nm (3.26
eV) to 480 nm (2.58 eV) between samples B1 and D would correspond to a change
of the In content from 6% to 22%. For our samples, this composition can only
be a rough estimate, since the different strain state of the one-sided
(In,Ga)N shells is not taken into account. With respect to the nominal In/Ga
flux ratios used during the growth, only a small fraction of the In is
incorporated. This discrepancy is not surprising, since even for sample D, In
desorption was detected by line-of-sight quadruple mass spectrometry. The
desorption being another loss channel in addition to the discussed diffusion
of In from the sidewalls to the top facet of the NW and the substrate.

Sample | $T_{\mathrm{sub}}$ | In/Ga | In per | Ga per | # cycles
---|---|---|---|---|---
| (∘C) | flux ratio | cycle (ML) | cycle (ML) |
A | 590 | 1 | 0.2 | 0.2 | 250
B1 | 540 | 1 | 0.2 | 0.2 | 250
B2 | 540 | 3 | 0.3 | 0.1 | 350
C1 | 490 | 1 | 0.2 | 0.2 | 250
C2 | 490 | 3 | 0.3 | 0.1 | 350
D | 440 | 5 | 0.5 | 0.1 | 250
Table 1:  Summary of the growth conditions for the investigated samples,
including the growth temperature, the In/Ga flux ratio, the amount of
deposited In and Ga per deposition cycle, as well as the total number of
cycles carried out for the respective sample.

For radially symmetric (In,Ga)N shells,[16] such a redshift of the shell
emission with increasing In/Ga flux ratio has not been observed. The higher In
incorporation for one-sided shells can be explained by a reduced In diffusion
away from the side facet during the SDD process due to the sequential and
directional N exposure. However, a broadening of the PL spectrum towards
higher In/Ga flux ratios was also observed for the radially symmetric
shells.[12] For the one-sided (In,Ga)N shells, this broadening may be
attributed to the increasing variation of the emission wavelength from NW to
NW observed between CL line scans, but also the emission band in individual
NWs is significantly broadened with increasing In content. Note that different
strain states along the axis of the one-sided (In,Ga)N shells due to the
bending of the NW may also contribute to the emission broadening.[5]

In Ref. 12, we observed significant axial growth on the top facet for NWs with
radially symmetric shells grown with continuous fluxes, which would complicate
the processing of devices out of such NW heterostructures. This parasitic
growth is notably reduced for our one-sided (In,Ga)N shells. The reduced
volume of the top segment can be attributed to the smaller amount of material
that is deposited, but also to the aforementioned reduction of adatom
diffusion towards the NW top for the SDD approach. Since the CL analysis of
the different samples focused on the emission properties of the shells, the
spectral region where the top segments might emit was not covered. Only for
samples B1 and C1, the onset of the top segment emission is visible at about
560 nm (2.21 eV), while for sample A, the NW tip is the only part showing
(In,Ga)N emission.

Figure 4:  (a) Cross-sectional, bright-field STEM micrograph of sample D and
(b) zoom into the mid segment of a representative NW. (c) STEM-EDX intensity
profiles of the In and Ga signal across the NW diameter (blue arrow) of the NW
presented in (b). (d) Exemplary high-resolution STEM micrograph of the
interface between the (In,Ga)N shell and the GaN core NW along the
[$10\bar{1}0$] zone axis.

To independently verify the presence of one-sided (In,Ga)N shells, Fig. 4
shows the results of an analysis of sample D by scanning transmission electron
microscopy (STEM) and STEM-based energy-dispersive X-ray spectroscopy (EDX) in
a JEOL JEM-3010 microscope. The STEM sample was prepared using a classic
polishing procedure where the NW ensemble was embedded in epoxy glue. The
bending of the NWs and the parasitic underlayer are highlighted in Fig. 4(a).
As already observed in SE micrographs, both the bending radius and the shape
of the top segments can vary from NW to NW. Figure 4(b) shows an enlarged
bright-field STEM micrograph of the middle segment of a representative NW. The
approximately 10-nm-thick shell is clearly visible on only one side of the NW,
while the opposite side shows no shell growth. A STEM-EDX line scan across the
NW diameter, displayed in Fig. 4(c), shows a clear increase in the In signal
for the position of the shell, proving that the shell does indeed contain a
significant amount of In, even if a quantification is not possible. Note that
the low intensity of the In-L lines results in a rather noisy profile with a
notable contribution from the background. Finally, Fig. 4(d) shows a
representative high-resolution micrograph of the interface between the
(In,Ga)N shell and the GaN core NW. Analysis of such high resolution images
along the NW showed that there is no evidence of plastic relaxation or
dislocations in the shells. Apparently, most of the strain resulting from the
different lattice constants of GaN and (In,Ga)N is released by the bending of
the NW.

##  III Conclusions

We have presented a sequential directional deposition method for GaN and
(In,Ga)N using MBE to grow one-sided shells on GaN NW cores, while minimizing
the parasitic growth on the top facet of the NW compared to shell growth
employing continuous fluxes. Using a dedicated sequence of shutter openings
and rotations, the individual metal fluxes and subsequently the nitrogen flux
are directed towards the same facet. For the growth of GaN shells, the SDD
mode allows not only the deposition of single shells on selected lateral sides
of a NW, but also enables the growth of a homogeneous GaN shell along the
entire NW length. For ternary, one-sided (In,Ga)N shells, the emission
wavelength can be shifted in a controlled manner from the UV to the green
range (510 nm) by lowering the growth temperature (down to 440 ∘C) and
increasing the In/Ga flux ratio (up to 5), while the shell thickness remains
homogeneous along the NW length. Using thin core NWs (average diameter 35 nm),
the one-sided (In,Ga)N shells can even induce a bending of the NWs.

Our new approach to the growth of ternary (In,Ga)N shells can be exploited for
device fabrication in two different ways. First, by applying the same
sequential directional deposition sequence to all six NW sidewalls, radially
symmetric homogeneous core-shell quantum wells can be grown with minimal
parasitic axial growth. This benefit compared to continuous deposition
simplifies the subsequent processing into devices such as light-emitting
diodes. Second, new opportunities for device design are opened up. More
specifically, one-sided shell quantum wells with different emission
wavelengths can be grown on different sidewall facets of one and the same NW,
thus potentially enabling multi-color light-emitting diodes within a single
NW.

## Supplementary Material

See Supplementary Material for corresponding SE micrographs for the single NW
CL line scans displayed in Fig. 3, as well as an investigation of one-sided
shell growth with continuous fluxes (co-deposition) and samples grown with an
inverted metal deposition sequence (In-Ga-N).

###### Acknowledgements.

The authors would like to thank Andrea Ardenghi for a critical reading of the
manuscript. The authors also thank Katrin Morgenroth, Carsten Stemmler and
Michael Höricke for the maintenance of the MBE system, as well as Anne-Kathrin
Bluhm for the SE micrographs. Sergio Fernández-Garrido acknowledges the
partial financial support from the Spanish program Ramón y Cajal (co-financed
by the European Social Fund) under Grant No. RYC-2016-19509 from the
Ministerio de Ciencia, Innovación y Universidades.

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

## Data Availability Statement

The data that support the findings of this study are available within the
article and its Supplementary Material. The raw data may be obtained from the
corresponding author upon reasonable request.

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