# Superconductivity in Bi based Bi2PdPt

A. Kataria  Department of Physics, Indian Institute of Science Education and
Research Bhopal, Bhopal, 462066, India  T. Agarwal  Department of Physics,
Indian Institute of Science Education and Research Bhopal, Bhopal, 462066,
India  S. Sharma  Department of Physics, Indian Institute of Science Education
and Research Bhopal, Bhopal, 462066, India  D. Singh  Department of Physics,
Indian Institute of Science Education and Research Bhopal, Bhopal, 462066,
India  S. Marik  Department of Physics, Indian Institute of Science Education
and Research Bhopal, Bhopal, 462066, India  R. P. Singh<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science Education and Research
Bhopal, Bhopal, 462066, India

###### Abstract

We report synthesis and properties of superconducting Bi2PdPt, a new member of
Bi-Pd based compounds known for their simultaneous existence of topological
surfaces states and superconductivity. It crystallizes in a hexagonal
structure having space group $P6_{3}/mmc$. A detailed investigation of the
properties via transport, magnetization, and specific heat measurements
confirm bulk superconductivity with transition temperature, TC = 4.0(1) K in
moderate coupling limit.

##  I Introduction

The topological aspects of electronic states provide a thrust in the latest
research of condensed matter physics. With the realization of spin
polarization in these electronic states, various exotic quantum phases emerge,
such as topological insulator, topological semimetals, and topological
superconductor (TSC) Ts ; ts1 ; ts2 . Particularly, topological
superconductors attracted immense attention due to the potential appearance of
exotic quasiparticles called Majorana fermions and their application in
topological quantum computing. A topological superconductor is characterized
by a bulk superconducting gap and Majorana zero modes at the symmetry
protected topological surface states (TSSs) ts2 . The various proposed ways to
realize TSC include carrier doped topological insulator and superconductor-
topological insulator heterostructures MF ; Mf1  which introduce the challenge
to prepare high quality samples and observe interfacial phenomenons. Bulk
compounds where non-trivial band structure and superconductivity intrinsically
coexists emerge as a promising route to realize topological superconductivity.

Bi-based materials are known for the presence of TSSs Bi-sb ; Bi-se ; TBS ;
TBS1 ; TBT ; BPST ; PBT ; HH . The Bi-based binary systems $\alpha$-PdBi,
$\alpha$-PdBi2, and $\beta$-PdBi2 are among the promising bulk topological
superconductivity candidates. Notably, in $\alpha$-PdBi2 and non-
centrosymmetric $\alpha$-PdBi, Rashba surface states near the Fermi level,
$E_{\mathrm{F}}$ were theoretically predicted and experimentally verified via
angle-resolved photoemission spectroscopy (ARPES) with intrinsic
superconductivity a-PB2 ; a-PB2_1 ; a-PB2_2 ; a-PB2_3 ; BP1 ; BP2 . Moreover,
the observed non-helical spin texture near $E_{\mathrm{F}}$ for $\beta$-PdBi2
combined with the various reports of fully gapped multiband superconductivity
in bulk, suggests the realization of Majorana fermions at the surface states
b-PB2_A ; b_PB2 ; b-PB2_s ; Msc ; Msc1 .

Recently, PtBi2 displays a large anisotropy for the in-plane and inter-plane
transport and is stated as a bulk topological metal with pressure-induced
superconductivity at 2.0 K Ptbi ; Ptbi2 . It indicates that the vast family of
Bi compounds is associated with various aspects of topological electronic
states and accompanied by superconductivity, which demands detailed
investigation in this family. Further, the spin-orbit coupling (SOC) strength
affect the Fermi surface to capture the non-trivial band topology Bi-se ; TI
and also impact the low-energy properties like superconductivity socsc . Thus,
the superconductivity with high SOC strength and the possible non-trivial band
structure may direct unconventionality in the superconducting ground state. As
SOC is directly proportional to the fourth power of atomic number $Z$,
introducing a heavier element will increase the SOC strength of the system.

We have synthesized a new member of the Bi compound family, Bi2PdPt, where a
Pt atom partially replaces a Pd atom from PdBi. Powder XRD and Laue
diffraction were performed to investigate the crystal structure and
orientation. The superconducting and normal state properties of Bi2PdPt
studied using transport, magnetization, and specific heat measurements. Type-
II superconductivity in moderate coupling limit is established in Bi2PdPt.

Figure 1:  The NiAs hexagonal crystal structure of Bi2PdPt is shown with solid
orange ball as Bi atoms and half green or grey solid ball as Pd/Pt atoms. (a)
Shows the side view of the crystal structure and (b) shows the top view. (c)
Rietveld refined pattern, represented by black line, of XRD recorded at room
temperature (orange open circle) for Bi2PdPt is shown. Inset shows the Laue
diffraction pattern of the single-crystal.

##  II Experimental Details

Single-crystal of Bi2PdPt was prepared by the modified Bridgman method.
Firstly, the polycrystalline sample was prepared using the solid-state
reaction method by taking a stoichiometric mixture of required elemental
powder Bi (99.99%), Pd (99.95%) and Pt (99.9%). The palletized form of the
mixture was subsequently heated at 700$\degree$C for four days in a vacuum
sealed tube, which was then transferred in a conical end quartz ampoule for
single-crystal growth. The ampoule dwelled at 700$\degree$C for 12 h and
slowly cooled up to 650$\degree$C at the rate of 0.4$\degree$C/hr, followed by
the water quenching. The obtained single-crystal is extremely brittle with a
metallic lustre. Mechanically cleaved single-crystals were used for all the
measurements. Powder x-ray diffraction data for grinded single-crystal was
collected at room temperature using PANalytical X’Pert powder diffractometer
with Cu$K_{\alpha}$ ($\lambda$ = 1.5406 Å) radiation. The Laue diffraction
pattern was recorded using the Photonic-Science Laue camera system for single-
crystal orientation and quality determination. The magnetic measurements were
performed on the vibrating sample magnetometer (VSM) option of the Quantum
Design Magnetic Property Measurement System (MPMS-3). AC transport and
specific heat measurements with two-$\tau$ relaxation method were performed
using Quantum Design Physical Property Measurement System (PPMS) 9 T with an
applied magnetic field parallel to the crystallographic [001] axis.

##  III Results and Discussion

###  a Sample characterization

Rietveld refinement of powder x-ray diffraction pattern of Bi2PdPt was
performed using FullProf software (shown in Fig. 1) fp . A small elemental
impurity peak of Pt is observed shown by an asterisk in the Fig. 1. The
refinement confirms the crystallization of Bi2PdPt in NiAs hexagonal crystal
structure having space group $P6_{3}/mmc$, which is different from the
structures reported for other superconducting compounds of the family,
$\alpha$-BiPd (non-centrosymmetric) and $\alpha$-, $\beta$-PdBi2, PtBi2
(layered structure) a-PB2 ; b_PB2 ; BP1 ; Ptbi .

The structure of Bi2PdPt consists of Bi atoms that lie inside the hexagonal
geometry while Pd and Pt atoms share the edges sites with equal probability,
as shown in the Fig. 1(a) and (b). Detailed refinement results with the
associated Wyckoff position of each atom are summarized in Table I. The Laue
diffraction pattern of different crystals of the same batch depicts the
orientation of the crystal along [001] direction, and bright spots tell the
quality of the single-crystal as shown in the inset of Fig. 1(c).

Table 1: Structure parameters of Bi2PdPt obtained from the Rietveld refinement
of XRD

Structure | Hexagonal
---|---
Space group | $P6_{3}/mmc$
Lattice parameters |
a = b(Å) | 4.294(5)
c (Å) | 5.567(1)

Atom | Wyckoff position | x | y | z | Occupancy
---|---|---|---|---|---
Bi | 2c | 0.333 | 0.666 | 0.250 | 0.0839
Pd | 2a | 0 | 0 | 0 | 0.0417
Pt | 2a | 0 | 0 | 0 | 0.0416

###  b Superconducting and normal state properties

In-plane AC resistivity at zero magnetic field in temperature range, 1.8 K to
250 K, was measured for single-crystal of Bi2PdPt, as shown in Fig. 2. A sharp
resistivity drop from the normal state resistivity at $T_{C,onset}$ = 4.0(1) K
with the width of 0.1 K is registered as a superconducting transition. Above
transition temperature resistivity behaviour of Bi2PdPt shows slight deviation
from linear behaviour, which is described by Wilson’s theory WT . However, the
high value of absolute resistivity with a saturated trend is allied with
$\alpha$-BiPd and other A-15 compounds where a parallel resistor model is
implanted SCbp . The resistivity behaviour for Bi2PdPt is characterised using
parallel resistor model described by Wisemann as Wm ,

$\frac{1}{\rho(T)}=\frac{1}{\rho_{s}}+\frac{1}{\rho_{i}(T)}$ (1)

Figure 2:  Temperature dependence of resistivity in zero field fitted using
the parallel resistor model above transition temperature is shown in main
panel. The inset shows the limited temperature region showing superconducting
transition at $T_{C,onset}$ = 4.0(1) K.

where $\rho_{s}$ is saturated resistivity independent of the temperature and
$\rho_{i}(T)$ is the ideal contribution consisting two terms which expressed
as,

$\rho_{i}(T)=\rho_{i,0}+C\left(\frac{T}{\Theta_{R}}\right)^{n}\int_{0}^{\Theta_{R}/T}\frac{x^{n}}{(e^{x}-1)(1-e^{-x})}dx\\\
$ (2)

here the first term $\rho_{i,0}$ is temperature-independent residual
resistivity, and the second term represents the temperature dependent
contribution including the effect of electron-phonon scattering stated by the
generalized Bloch-Grüneisen expression bg ; WT . $\Theta_{R}$ is the Debye
temperature from resistivity measurement, and $C$ is a material-dependent
quantity. Best fitting of Bi2PdPt for $n$ = 3 yields Debye temperature,
$\Theta_{R}$ = 123(3) K, $C$ = 5.70(2) $\mu\ohm$-cm, $\rho_{s}$ = 47(3)
$\mu\ohm$-cm and $\rho_{i,0}$ = 3.32(1) $\mu\ohm$-cm. The estimated residual
resistivity ratio, $\rho(250K)/\rho(5K)$ = 2.5 is much smaller than the
reported values for the other Bi-Pd compounds, which can be possibly due to
the increase in disorder of the system by Pt presence SCbp ; scpb2 .

Figure 3:  a) Shows the superconducting transition in 1 mT magnetic field at
$T_{c,onset}$ = 3.98(1) K. (b) Magnetization hysteresis loop at 1.8 K. (c)
Magnetic moment variation under magnetic field at different temperature upto
transition temperature. (d) Lower critical field with temperature fitting
using GL equation Eq. (4).

Magnetization versus temperature measurement at 1 mT magnetic field in zero-
field-cooled warming (ZFCW) and field-cooled cooling (FCC) mode is shown in
Fig. 3(a). Curve shows a diamagnetic onset at temperature $T_{C}$ = 3.98(1) K
perpendicular to crystallographic [001] axis of single-crystalline Bi2PdPt.
The poor overlapping of the ZFCW and FCC curve depicts the presence of a
strong pinning centre in the sample with superconducting volume fraction close
to 96% certain the bulk property. Magnetization hysteresis loop under an
applied magnetic field of $\pm$ 1 T at temperature, 1.8 K is shown in Fig.
3(b). The loop represents the conventional type-II behaviour along with an
irreversible nature of magnetization below $H_{irr}$ = 0.45 T, above which
unpinning of the vortices start taking place.

Further, to estimate the lower critical field value $H_{C1}$, the
magnetization variation in the applied magnetic field at different isotherms
up to transition temperature, $T_{C}$, were analyzed. $H_{C1}$ for each
isotherm is considered as a magnetic field value where the curve starts to
deviate from the linear Meissner state, as shown in Fig. 3(c). The temperature
evolution of the lower critical field is fitted with the Ginzburg-Landau (GL)
equation,

$H_{C1}(T)=H_{C1}(0)\left[1-\left(\frac{T}{T_{C}}\right)^{2}\right]$ (3)

which gives $H_{C1}(0)$ = 2.96(4) mT for $T_{C}$ = 3.98(1) K (Fig. 3(d)). The
upper critical field $H_{C2}$ is evaluated by the observed shifts in
superconducting transition temperature with the increasing applied magnetic
field. The shift in the transition temperature, $T_{C}$ is marked from
magnetization at onset and for resistivity measurements at 90$\%$ drop from
normal state resistivity value ($T_{C,90\%}$), as shown in the inset of Fig.
4. The upper critical field $H_{C2}(0)$ is estimated from the expression,

$H_{C2}(T)=H_{C2}(0)\left[\frac{(1-t^{2})}{(1+t^{2})}\right].$ (4)

here $t=T/T_{C}$, reduced temperature. The fitted data yields $H_{C2,M}(0)$ =
0.92(1) T and $H_{C2,\rho}(0)$ = 1.08(1) T from magnetization, and resistivity
measurements, respectively displayed in Fig. 4.

Figure 4:  The upper critical field variation with temperature is determined
via magnetization and resistivity measurements and fitted by using the
equation Eq. (4) is shown. Inset shows the resistivity drop temperature
variation with applied magnetic field.

In addition to the resistivity and magnetization measurements, specific heat
is also measured at zero field and 2 T (well above the upper critical field)
under temperature variation, shown in Fig. 5. A small anomaly in 0 T specific
heat measurement is inscribed around 3.93(7) K, as shown in the inset of Fig.
5, which is in allied with the superconducting transition temperature recorded
from resistivity and magnetization measurements. Whereas the measurement
recorded at 2 T presents the normal state of Bi2PdPt. The normal state
specific data is fitted using the Debye relation,

$C=\gamma_{n}T+\beta_{3}T^{3}+\beta_{5}T^{5}$ (5)

where $\gamma_{n}$ is Sommerfeld coefficient corresponding to electronic
contribution, $\beta_{3}$ Debye constant represents phononic contribution, and
$\beta_{5}$ produces the anharmonic contribution to the specific heat. The
fitted data produces $\gamma_{n}$ = 5.1(6) mJmol-1K-2, $\beta_{3}$ = 1.72(6)
mJmol-1K-4 and $\beta_{5}$ = 11.2(1) $\mu$Jmol-1K-6 (Fig. 5). The Sommerfeld
coefficient value matches well with the $\alpha$-PdBi reported value, but a
large deviation from $\alpha$\- and $\beta$-PdBi2 is observed SCbp ; scpb2 ;
p-pb2_1 . According to the free-electron theory, the Sommerfeld coefficient is
used to evaluate the density of state at Fermi level as per the relation
$\gamma_{n}=\left(\frac{\pi^{2}k_{B}^{2}}{3}\right)D_{C}(E_{\mathrm{F}})$,
where $k_{B}$ = 1.38$\times$10-23 J K-1 is Boltzmann constant. The
$D_{C}(E_{\mathrm{F}})$ is calculated to be 2.1(2) states eV-1 f.u.-1.
Further, considering the Debye model, the Debye temperature $\Theta_{D}$ is
estimated to be 165(2) K for $\beta_{3}$ = 1.72(6) mJ mol-1 K-4 using the
expression
$\theta_{D}=\left(\frac{12\pi^{4}RN}{5\beta_{3}}\right)^{\frac{1}{3}}$, where
$R$ = 8.314 J mol-1K-1 is a gas constant. The calculated Debye temperature
value is consistent with the value obtained from parallel resistor model. The
average electron-phonon coupling constant, $\lambda_{e-ph}$ is calculated by
implanting McMillan’s theory given as mcm ,

$\lambda_{e-ph}=\frac{1.04+\mu^{*}\mathrm{ln}(\theta_{D}/1.45T_{C})}{(1-0.62\mu^{*})\mathrm{ln}(\theta_{D}/1.45T_{C})-1.04}$
(6)

where $\mu^{*}$ varies in the range of 0.1 $\leq\mu^{*}\leq$ 0.15, a material-
specific value accounts for screened Coulomb repulsion. Considering $T_{C}$ =
3.93(7) K, $\theta_{D}$ = 165(2) K and $\mu^{*}$ = 0.1 (as speculate for Bi-Pd
binary compounds) obtained $\lambda_{e-ph}$ = 0.64(1), which puts Bi2PdPt in
the moderately coupled superconductor regime respective to other Bi-Pd
compounds $\alpha$-, $\beta$-PdBi2, and $\alpha$-PdBi SCbp ; p-pb2_1 .

Figure 5:  The temperature dependence specific heat of the Bi2PdPt represented
in $C$ vs $T$ form at 2 T magnetic field is fitted with the equation Eq. (5).
Inset shows the $C$ vs $T$ curve at zero magnetic field curve where anomaly is
clearly observed.

###  c Superconducting parameters and Uemura Plot

The external applied magnetic field puts a limiting field on the
superconductor, portrayed independently by spin paramagnetic and orbital
limiting effects. According to the BCS theory, spin paramagnetic effect comes
from pairing breaking due to Zeeman effect defined as the Pauli limiting
field, $H_{C2}^{P}(0)$ = 1.86 $T_{C}$ Pauli_1 ; Pauli_2 . For Bi2PdPt with
$T_{C}$ = 3.98(1) K, yields $H_{C2}^{P}(0)$ = 7.40(2) T. The orbital limit for
an upper critical field refers to the critical field where vortex cores begin
to overlap. The orbital limiting field is given by Werthamer-Helfand-Hohenberg
(WHH) expression WHH_1 ; WHH_2 ,

$H_{C2}^{orbital}(0)=-\alpha
T_{C}\left.\frac{dH_{C2}(T)}{dT}\right|_{T=T_{C}}$ (7)

where $\alpha$ = 0.69 for dirty limit and 0.73 for clean limit
superconductors. The initial slope $\frac{-dH_{C2}(T)}{dT}$ at $T=T_{C}$ is
estimated to be 0.23(1) T/K which gives $H_{C2}^{orbital}(0)$ = 0.64(1) T
considering the dirty limit. Moreover, the relative strength of Pauli and
orbital limiting field is measured by Maki parameter defined by the relation
$\alpha_{M}=\sqrt{2}H_{C2}^{orb}(0)/H_{C2}^{P}(0)$, evaluates $\alpha_{M}$ =
0.12(2) maki . The small value of Maki parameter indicates the negligible
effect of Pauli limiting field for Bi2PdPt.

Further, the fundamental superconducting length parameters is calculated from
upper, $H_{C2}(0)$, and lower, $H_{C1}(0)$ critical field values. The
Ginzburg-Landau coherence length ($\xi_{GL}(0)$) is estimated from
$\xi_{GL}(0)=\sqrt{\frac{\Phi_{0}}{2\pi H_{C2}(0)}}$ Coh_Leng , here
$\Phi_{0}$ = 2.07 $\times$ 10-15 Tm2 is the magnetic flux quantum. Using
$H_{C2,M}(0)$ = 0.92(1) T, $\xi_{GL}(0)$ is evaluated to be 190(1) Å. The
equation MIB ,

$H_{C1}(0)=\frac{\Phi_{0}}{4\pi\lambda_{GL}^{2}(0)}\left(\mathrm{ln}\frac{\lambda_{GL}(0)}{\xi_{GL}(0)}+0.12\right)$
(8)

provides the Ginzburg-Landau penetration depth, $\lambda_{GL}(0)$ = 4231(76) Å
for $H_{C1}(0)$ = 2.96(4) mT and $\xi_{GL}$(0) = 190(1) Å. Moreover, the ratio
$\frac{\lambda_{GL}(0)}{\xi_{GL}(0)}$ gives the Ginzburg-Landau parameter,
$\kappa_{GL}$ = 22.2(5) which is much greater than $1/\sqrt{2}$ indicating a
strong type-II superconductivity in Bi2PdPt. The thermodynamic critical field
$H_{C}$, is evaluated by using the expression MIB ;
$H_{C1}(0)H_{C2}(0)=H_{C}^{2}\mathrm{ln}\kappa_{GL}$, yielding $H_{C}$ = 29(1)
mT.

For Bi2PdPt, in order to determine the rest of the superconducting parameters
including London penetration depth $\lambda_{L}$, electronic mean free path
$l_{e}$ and to verify dirty limit superconductivity, a set of equation is used
assuming the spherical Fermi surface. The electronic mean free path, $l_{e}$
is estimated from the quasiparticle number density, $n$ and Sommerfeld
coefficient, $\gamma_{n}$. The quasiparticle number density of Bi2PdPt has
been extracted from Hall measurement, $n=2.45(2)\times 10^{28}$ m-3 (more
details in appendix). The Fermi vector, $k_{\mathrm{F}}$ is evaluated using
$k_{\mathrm{F}}$ = (3$\pi^{2}n)^{1/3}$, which gives $k_{\mathrm{F}}$ = 0.90(1)
Å-1. The effective mass, $m^{*}$ is calculated from Sommerfeld coefficient,
$m^{*}={(\hbar k_{\mathrm{F}})^{2}\gamma_{n}/\pi^{2}nk_{B}^{2}}$. For $k_{B}$
is Boltzmann constant and $\gamma_{n}$ = 95.51 J m-3 K-2, $m^{*}$ = 2.0(3)
$m_{e}$. According to Drude Model Coh_Leng , Fermi velocity is determined from
the relation $v_{\mathrm{F}}=\hbar k_{\mathrm{F}}/m^{*}$ = 5.1(7) $\times$ 105
m s-1 and scattering time from $\tau^{-1}=ne^{2}\rho_{0}/m^{*}$, which further
used to calculate electronic mean free path, $l_{e}=v_{\mathrm{F}}\tau$
providing $l_{e}$ = 455(68) Å.

In consideration of BCS theory, the coherence length $\xi_{0}$ is defined as
$0.18\hbar v_{\mathrm{F}}/k_{B}T_{c}$ providing $\xi_{0}$ = 1756(147) Å. The
ratio of $\xi_{0}/l_{e}$ = 3.9 is greater than 1, implying Bi2PdPt is a dirty
limit superconductor. Further in the dirty limit, the London penetration depth
$\lambda_{L}$ Umera_Ref , is written as,

$\lambda_{L}=\left(\frac{m^{*}}{\mu_{0}ne^{2}}\right)^{1/2}.\\\ $ (9)

From the above expression the estimated $\lambda_{L}$ is 483(36) Å. The order
of calculated parameters matches well with $\beta$-PdBi2 where dirty limit
superconductivity is observed Msc  whereas $\alpha$-PdBi lies in a clean type-
II superconducting limit SCbp .

Figure 6:  The Uemura plot showing the superconducting transition temperature
$T_{C}$ vs the effective Fermi temperature $T_{{\mathrm{F}}}$ for Bi2PdPt with
$\beta$-PdBi2, shown as solid green circle. Other data points represent the
different families of superconductors together with the shaded region
representing the families of unconventional superconductors. The dashed line
corresponds to the Bose-Einstein condensation temperature $T_{B}$.

Moreover, to classify the Bi2PdPt in the context of other superconductors,
Uemura plot Umera  of $T_{C}$ with effective Fermi temperature,
$T_{\mathrm{F}}$ has been plotted, which is used to define the character of
unconventionality of a superconductor based on the ratio of
$\frac{T_{C}}{T_{\mathrm{F}}}$. For 3D spherical Fermi surface, Fermi
temperature is calculated using Tf ,

$k_{B}T_{{\mathrm{F}}}=\frac{\hbar^{2}}{2}(3\pi^{2})^{2/3}\frac{n^{2/3}}{m^{*}},$
(10)

Table 2: Parameters in the superconducting and normal state of Bi2PdPt and $\alpha$-BiPd Parameters | Unit | Bi2PdPt |  $\alpha$-BiPd SCbp
---|---|---|---
$T_{C}$ | K | 4.0 | 3.8
$H_{C1}(0)$ | mT | 2.96 | 12.0
$H_{C2,M}(0)$ | T | 0.92 | 0.8
$H_{C2}^{P}(0)$ | T | 7.40 | 7.07
$H_{C2}^{Orb}(0)$ | T | 0.64 |
$\xi_{GL}(0)$ | Å | 190 | 170
$\lambda_{GL}(0)$ | Å | 4231 | 1792
$k_{GL}$ |  | 22.2 | 7.6
$\gamma_{n}$ | mJ mol-1 K-2 | 5.1 | 4.0
$\theta_{D}$ | K | 165.0 | 169.0
$\xi_{0}/l_{e}$ |  | 3.9 | 0.01
$v_{\mathrm{F}}$ | 105 m s-1 | 5.1 |
$n$ | 1028m-3 | 2.45 |
$T_{\mathrm{F}}$ | K | 17576 |
$m^{*}$/$m_{e}$ |  | 2.0 |

where $n$ is the quasiparticle number density per unit volume, and $m^{*}$ is
the effective mass of quasiparticles. The $T_{F}$ value is estimated to be
$T_{\mathrm{F}}$ = 17576(104) K, by considering the respective values of
parameters from Table II. The ratio $\frac{T_{C}}{T_{\mathrm{F}}}$ = 0.0002
places this material, Bi2PdPt close to other family member, $\beta$-PdBi2 and
outside the band of unconventional superconductor which includes iron based
superconductors, cuprates and Chevrel phase Unconv_1 ; Unconv_2 . The
conventional superconductors are considered to be on the right-hand side of
the Uemura plot. Bi2PdPt with the other member of the Bi-Pd compound family,
$\beta$-PdBi2 Msc  are shown by the solid green symbol in Fig. 6.

Among the Bi-Pd compound family, Bi2PdPt is the highest possible SOC strength
candidate, based on the presence of heavy constituent elements, establishes a
strong foundation to exhibit intriguing properties. The different crystal
structure NiAs type of Bi2PdPt eliminates all the possibilities of any
secondary superconducting phase presence of Bi-Pd/Pt compound, implementing
the intrinsic bulk superconductivity. Hall measurement (see appendix) suggests
dominated hole charge carries (1028m-3) same as for other Bi-Pd compounds,
which indicate a very low effect in the electronic state with the Pt presence.
The specific heat of Bi2PdPt gives Sommerfeld coefficient 5.1(6) mJmol-1K-2,
which is slightly greater than what is reported for $\alpha$-BiPd, suggests
only a minimal increase in density of states at $E_{\mathrm{F}}$. However, the
observed value is comparable to that of $\beta$-PdBi2 and Pt1.26Bi2 pb .
Concluding neither substitution at Pd site nor increasing the Bi concentration
predominately affects the density of states in Bi-Pd/Pt compounds. Despite the
presence of heavy 5d Pt element, no change in phonon frequency, extracted from
Debye temperature, is noted between Bi2PdPt and $\alpha$-BiPd SCbp .

Moreover, the bulk measurement of Bi2PdPt is not sufficient to establish the
gap symmetry of the sample. The possible explanation of low specific jump
value can be due to multiband superconductivity with a single superconducting
gap bi2pd . The layered transition metal dichalcogenide superconductors have
been well known to exhibit no anomaly in specific heat measurement tmc . In
addition, the low value of specific heat jump or no jump at all is also
observed in many Bi-based half Heusler alloys (RPdBi), where superconductivity
is attributed to the surface states BiHH . Further, the similar behaviour is
determined for the case of strongly or locally disordered systems such as
quasi-skutterudites sku .

No deviation or anisotropy has been observed in the upper critical field of
Bi2PdPt, directing towards the bulk nature of the superconducting state and
absence of multigap superconductivity. Apart from this, the variation of
superconducting volume fraction with the different cooling procedures of
sample preparation showed the grain boundaries effect in the sample (details
in appendix). To fully understand the effect of increased SOC strength, the
possible reason for the low value of specific heat jump and address any non-
trivial band structure in the normal state of Bi2PdPt detailed theoretical and
experimental analysis of the superconducting ground state with band topology
is required.

##  IV CONCLUSION

In summary, we synthesized the single-crystalline Bi2PdPt and transport,
magnetization, and specific heat measurements performed for detailed analysis
of superconducting properties. The superconducting transition of Bi2PdPt has
been reported at 4.0(1) K with upper and lower critical field value, $H_{C1}$
= 2.96(4) mT and $H_{C2,M}$ = 0.92(1) T, fitted using Ginzburg-Landau
equation. An anomaly was observed around transition temperature in specific
heat measurement confirms the bulk superconductivity. The comparison of
various normal and superconducting state parameters of Bi2PdPt and
$\alpha$-BiPd have been summarized in Table II. The Bi2PdPt is classified as a
moderately coupled type-II dirty limit superconductor. The estimated value of
Fermi temperature, $T_{\mathrm{F}}$ places Bi2PdPt close to the other member
of the family, $\beta$-PdBi2. Further microscopic measurements such as muon
spectroscopy, scanning tunnelling microscope (STM), and angle-resolved
photoemission spectroscopy (ARPES) are required to understand the
superconducting ground state and the possible topological nature of Bi2PdPt.

##  V Acknowledgments

A. Kataria acknowledges the funding agency Council of Scientific and
Industrial Research (CSIR), Government of India, for providing SRF fellowship
(Award No: 09/1020(0172)/2019-EMR-I). R. P. S. acknowledges Science and
Engineering Research Board, Government of India for the Core Research Grant
CRG/2019/001028.

##  VI Appendix

###  a Characterization

Superconducting volume fraction in Bi2PdPt is highly dependent on the sample
cooling procedure during preparation. Three different samples made, such as
S1, cooled at 1.0$\degree$C/hr, S2 cooled at 0.5$\degree$C/hr, and S3 cooled
at 0.4$\degree$C/hr while preparing the sample. XRD of the two samples S2 and
S3 is shown in Fig. 7(a). The composition of one of the samples extracted from
energy dispersive spectroscopy (EDS) is also shown in the inset of Fig. 7 (a),
clearly indicating the Pt presence. The results from the magnetic
characterization of S1 and S2 are presented in Fig. 7(b). The observed
superconducting transition temperature is the same for all three samples;
however superconducting shielding fraction 96$\%$ is the maximum for S3. S2
only shows 60$\%$ volume fraction with less than 10$\%$ fraction is observed
in S1, implying samples cooled only at a specific rate show a large volume
fraction with the same diamagnetic onset temperature in magnetization
measurement. Thus, reflecting the effect on grain boundaries and point defects
of samples due to different cooling procedures PST .

Figure 7:  (a) Shows the XRD comparison of powder sample having two different
cooling rate 0.5$\degree$C/hr (S2) and 0.4$\degree$C/hr (S3) while
preparation, with the inset shows the EDS pattern of S3 sample. (b) Shows the
magnetic measurement compilation of S2 and S1 in the inset, whereas S3 is
shown in Fig. 3(a).

###  b Normal state properties

Figure 8:  (a) Hall resistivity at $T$ = 30 K under field variation of $\pm$ 9
T. (b) Represent the MR data taken at $T$ = 5 K and $T$ = 10 K on Bi2PdPt
crystal.

To perceive the normal state properties of Bi2PdPt, magnetoresistance (MR) and
normal hall effect were also measured. The hall measurements were done on
Bi2PdPt under varying magnetic field of $\pm$ 9 T at 30 K to extract the
carrier density as shown in the left panel of Fig. 8(a), where a linear
variation with magnetic field represents the small contribution from the
longitudinal resistivity. The hall coefficient, $R_{H}$, was determined by the
slope of the variation $\rho_{xy}$ with both +$H$ and $-H$ fields. The
estimated $R_{H}$ = 2.52(2) $\times$ 10 -4 cm3/C, and the positive sign
indicate the hole as the dominant charge carrier in the sample. By simply
using the single band expression $n=1/R_{H}q$, the electron concentrations was
calculated, $n$ = 2.45(2) $\times$ 10${}^{28}\textit{m}^{-3}$. The carrier
density order matches with the other reported for $\alpha$-PdBi and PdBi2 n .
MR was measured in the longitudinal configuration under a varying magnetic
field of 9 T at different temperatures. MR with magnetic field dependence is
defined as $\frac{\rho_{xx}(H)-\rho_{xx}(0)}{\rho_{xx}(0)}$. A significant
small MR value is observed in our sample Fig. 8(b).

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# Integrated Access and Backhaul in Cell-free Massive MIMO Systems

Ali Hosseinalipour Jazi, S. Mohammad Razavizadeh, and Tommy Svensson Ali
Hosseinalipour ans S. Mohammad Razavizadeh are with the School of Electrical
Engineering, Iran University of Science and Technology (IUST), Tehran
16846-13114, Iran (e-mail: {alihosseinalipour, smrazavi}@iust.ac.ir)Tommy
Svensson is with the Electrical Engineering Department, Chalmers University of
Technology, 412 96 Gothenburg, Sweden, (e-mail<EMAIL_ADDRESS>
###### Abstract

One of the major challenges with cell-free (CF) massive multiple-input
multiple-output (MIMO) networks is providing backhaul links for a large number
of distributed access points (APs). In general, providing fiber optics
backhaul for these APs is not cost-effective and also reduces network
scalability. Wireless backhauling can be a promising solution that can be
integrated with wireless access links to increase spectrum efficiency. In this
paper, the application of integrated access and backhaul (IAB) technique in
millimeter-wave (mmWave) CF massive MIMO systems is investigated. The access
and backhaul links share a frequency spectrum in the mmWave bands, and in
both, hybrid beamforming techniques are adopted for signal transmission. The
bandwidth allocation (division) parameter between the two link types as well
as the beamforming matrices are optimized to maximize the end-to-end data-
rate. This leads to a non-convex optimization problem for which an efficient
solution method is proposed. The simulation results show the effectiveness of
the IAB technique and our proposed scheme in CF massive MIMO systems. These
simulations also compare the proposed hybrid beamforming method with a fully
digital solution in terms of the number of radio frequency (RF) chains and the
volume of backhaul traffic. Finally, the effect of increasing the number of
APs on the users’ data rates in terms of wireless access and backhaul links
constraints is also examined.

###### Index Terms:

Integrated Access and Backhaul (IAB), massive MIMO, millimeter-wave (mmWave),
cell-free (CF), hybrid beamforming.

##  I Introduction

During past years, there have been increasing demands for new wireless
services like enhanced mobile broadband (eMBB) and ultra-reliable and low
latency communication (URLLC) that motivate researchers to develop new
technologies for efficient and reliable transmission of more data in minimal
time and frequency resources. Massive multiple-input multiple-output (MIMO) is
one of the promising techniques for accommodating these demands[1]. However,
implementation of a large number of antennas of massive MIMO on a limited
space base station (BS) is not easy in practice and leads to a degradation in
the expected performance. In addition, in the centralized implementation
scenarios of the massive MIMO arrays, there is a significant difference in the
signal powers of cell-edge and cell-center users. This motivates distributed
or cell-free (CF) massive MIMO systems in which the antennas of the massive
MIMO are distributed among a number of access points (APs) in a wide area [2].
CF massive MIMO systems can provide uniform quality of service (QoS) over a
cell region and reduce multi-user interference [3]. It is also a good choice
for implementing new services like massive machine type communication (mMTC)
and internet of things (IoT), in which devices are distributed in a wide
service area [4].

In spite of these advantages, CF networks face many challenges, including
synchronization among the APs, user association, and backhaul link
provisioning for the APs [5]. In general, existing cable and optical fiber-
based backhaul links are less suitable for future cellular networks due to
implementation costs and low flexibility. Hence, the wireless backhaul has
attained much attention due to its lower implementation complexity and higher
flexibility [6]. Microwave backhaul links that operate in line-of-sight (LOS)
conditions have been used for a long time, utilizing dedicated international
telecommunications union (ITU) frequency bands [7]. However, in 5G,
millimeter-wave (mmWave) frequency bands are a potential candidate to meet the
growing bandwidth demand in the future wireless backhaul link. In addition,
due to the wide bandwidth in the mmWave access spectrum, there is a new
interest to share radio resources between the wireless backhaul and access
links, leading to the concept of integrated access and backhaul (IAB) [8].

There are many papers that study wireless backhauling and IAB techniques in
wireless cellular networks. For example, in [9], the authors consider a
cellular heterogeneous network (HetNet) with wireless backhaul links and
design the beamforming matrices for this network. The authors in [10] maximize
the wireless backhaul link rate in the mmWave bands and then use this rate as
a constraint in maximizing the rate of users in the access link that operate
in the sub-6 GHz frequency band. The design of IAB networks by multiplexing
the access and backhaul links in time domain is studied in [11]. The authors
in [12] maximize end-to-end sum rate of users (i.e., from the central
processing unit (CPU) in the network core to the users) in a mm-wave cellular
network by optimizing the dedicated bandwidth for the access and backhaul
links and power allocation coefficients in the macro-cell BS (MBS). In
addition, the authors in [7] and [13] evaluate the coverage probability of a
cellular network equipped with IAB by considering different backhaul scenarios
(fiber, wireless/fiber, and IAB). In the case of wireless backhauling, the
small-cell BSs (SBSs) in the backhaul link and users in the access link are
served by different carrier frequencies. In most of the references, the
authors do not optimize the bandwidth allocation coefficient between the
access and backhaul connections and use a frequency division multiplexing in
the access and backhaul links.

There are also a few papers that assume wireless backhauling in the CF
networks. For example, in [14] and [15], the wireless backhaul links
parameters are considered as the constraints for the access link optimization
in the CF networks. In [16] the authors optimize end-to-end rate in a user-
centric CF massive MIMO network by jointly optimizing the beamforming matrix
in the backhaul link and the power allocation coefficients in the access link.
This paper considers different frequency bands (mm-wave band and sub-6 GHz)
for the access and backhaul links. However, to the best of our knowledge, the
IAB technique and specially frequency multiplexing between backhaul and access
links in the CF networks has not been studied before in the literature.

In this paper, we study the use of IAB in the downlink of a CF massive MIMO
network in which the wireless backhaul and access links are multiplexed in the
frequency domain. In the considered network, there is one CPU (IAB-donor) and
multiple APs (IAB-nodes) that serve a large number of users at the same time
and frequency resource. In this paper, we use the terms CPU/APs and IAB-
donor/IAB-nodes interchangeably. Both wireless access and backhaul links
operate at the same mm-wave frequency band, and hybrid beamforming techniques
are used for signal transmission at both of them. For optimal design of the
IAB scheme, the bandwidth allocation coefficient between the access and
backhaul links is optimized to maximize the minimum end-to-end rate over them.
At the same time, the hybrid beamforming matrices at the CPU and APs are also
optimized, which finally leads to a non-convex optimization problem that
cannot be solved efficiently. Hence, we propose a solution method that
optimizes the above variables for access and backhaul links alternatively. We
also derive a closed-form expression for dividing the mm-wave frequency band
between the access and backhaul links. We verify the performance of the
proposed scheme through computer simulations. The results show the
effectiveness of using IAB in the CF massive MIMO systems. We also evaluate
the performance of the proposed hybrid beamforming optimization scheme by
comparing it with fully digital beamforming and centralized beamforming at the
CPU, which illustrates the effectiveness of using this technique in
conjunction with IAB. Then, we investigate the impact of the number of APs on
coverage enhancement of the CF network in the access link and also on the rate
of the backhaul link. Finally, by considering the effects of both the access
and backhaul rates on the end-to-end rate of the networks, we show that in the
CF massive MIMO systems with wireless backhaul, there is an optimal number of
APs achieving the best performance.

The rest of this paper is organized as follows. The system model is described
in Section II. Following the definition of the main problem in Section III,
the parameters of the backhaul and access links are optimized in Sections IV
and V, respectively. Section VI specifies a closed-form equation for the
bandwidth allocation coefficient. Section VII presents the numerical results
of the proposed algorithms. In Section VIII, the paper is summarized.

Throughout this paper, the following notations are used: $a$, a, and A stand
for a scalar, a column vector, and a matrix, respectively;
${\left[{\bf{A}}\right]_{i,j}}$ denotes the $(i;j)$-th element of matrix A and
the $i$-th element of vector a is denoted by ${\left[{\bf{a}}\right]_{i}}$;
${\rm rank}({\bf{A}})$ is the rank of A; $({\bf{A}})^{*}$, $({\bf{A}})^{T}$,
and $({\bf{A}})^{H}$, denote conjugate, transpose, and Hermitian transpose of
A, respectively. The Euclidean and Frobenius norms of A are denoted by
$\left\|.\right\|$ and ${\left\|.\right\|_{F}}$, respectively. Furthermore, we
use $Tr\left\\{{}\right\\}$, $\Re\left\\{{}\right\\}$, and
$\mathbb{E}\left\\{{}\right\\}$ to respectively represent the trace, real part
taking, and expectation operators. ${\rm diag}\left\\{{{\bf{a}}}\right\\}$
forms a diagonal matrix of the vector ${\bf{a}}$, and ${{\bf{I}}_{{N}}}$
denotes the $N\times N$ identity matrix.
${z}\sim\,\mathbb{C}\mathbb{N}(0,\sigma^{2})$ denotes a circularly symmetric
complex Gaussian random variable $z$ with zero mean and variance $\sigma^{2}$.
Further, $\mathbb{C}$ and $\mathbb{C}^{m\times n}$ describe a complex value
and a complex matrix of dimension $m\times n$, respectively. The amplitude and
phase of a complex value $z$ are denoted by $\left|.\right|$ and $\angle$,
respectively.

##  II System Model

As depicted in Fig. 1, we consider a CF massive MIMO system consisting of $M$
$N_{A}$-antenna APs (i.e., IAB-nodes) that serve $K$ single-antenna users. All
the IAB-nodes are connected to an $N_{C}$-antenna CPU (i.e., IAB-donor). Both
the access and backhaul links operate at the same frequency band in the mmWave
frequencies. Assuming that $B$ and $\eta\in(0,1]$ indicate the total available
bandwidth in the network and the bandwidth allocation coefficient between
access and backhaul links, respectively, dedicated bandwidth to the access and
backhaul links will be $\eta B$ and $(1-\eta)\,B$, respectively. In the
following, we discuss signal transmission and reception in the access and
backhaul links.

Figure 1: System Model.

###  II-A Access link

Let $x_{k}\in\,\mathbb{C}$ denotes the $k$-th user’s signal in the access
link, which is transmitted from all the APs and
$\mathbb{E}\\{|x_{k}|^{2}\\}=1$. The received signals at the $k$-th user is

$\displaystyle\begin{split}{y_{k}}&=\sum\limits_{m=1}^{M}{\sum\limits_{j=1}^{K}{{{\bf{h}}_{k,m}}{\bf{W}}_{m}^{RF}{\bf{w}}_{j,m}^{BB}{x_{j}}}}+{n_{k}}\\\
&=\underbrace{\sum\limits_{m=1}^{M}{{{\bf{h}}_{k,m}}{\bf{W}}_{m}^{RF}{\bf{w}}_{k,m}^{BB}{x_{k}}}}_{{\rm{Desired\,signal}}}+\underbrace{\sum\limits_{m=1}^{M}{\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{K}{{{\bf{h}}_{k,m}}{\bf{W}}_{m}^{RF}{\bf{w}}_{j,m}^{BB}{x_{j}}}}}_{{\rm{Interference}}}\\\
&+{n_{k}},\end{split}$ (1)

where
${{\bf{W}}_{m}}={\bf{W}}_{m}^{RF}{\bf{W}}_{m}^{BB}\in{\mathbb{C}^{{N_{A}}\times
K}}$ denotes the hybrid beamforming matrix of the $m$-th AP which is product
of an analog beamforming matrix
${\bf{W}}_{m}^{RF}\in{\mathbb{C}^{{N_{A}}\times N_{RF}^{A}}}$ and a baseband
digital beamforming matrix ${\bf{W}}_{m}^{BB}\in{\mathbb{C}^{N_{RF}^{A}\times
K}}$ (where ${\bf{w}}_{k,m}^{BB}$ is the $k$-th column of
${\bf{W}}_{m}^{BB}$). $N_{RF}^{A}$ is the number of RF chains at each AP. The
analog beamforming matrix has a unit-module constraint (i.e.,
$\left|{{{\left[{{\bf{W}}_{m}^{RF}}\right]}_{i,j}}}\right|=\frac{1}{{\sqrt{{N_{A}}}}}$)
and the digital beamforming matrix must satisfy AP power constraint
$\left\|{{\bf{W}}_{m}^{BB}}\right\|_{F}^{2}\leq{P^{A}}$ where ${P^{A}}$ is the
total AP’s power constraint for the access link. In addition,
${n_{k}}\sim\,\mathbb{C}\mathbb{N}(0,\sigma_{k}^{2})$ is the additive white
Gaussian noise at the $k$-th user.

The channel gain of the $m$-th AP to the $k$-th user is denoted by
${{\bf{h}}_{k,m}}\in{\mathbb{C}^{1\times{N_{A}}}}$ that has Saleh-Valenzuela
(SV) channel model [17] and is defined as

$\begin{split}{{\bf{h}}_{k,m}}=&\sqrt{{N_{A}}\,}\vartheta_{k,m}^{L}\,\,{\bf{a}}_{L}(\varphi_{k,m}^{AoD})\\\
&+\sum\limits_{l=1}^{{L_{k,m}}}{\sqrt{\frac{{{N_{A}}}}{{{L_{k,m}}}}\,}\vartheta_{l,k,m}^{N}\,\,{\bf{a}}_{L}(\varphi_{l,k,m}^{AoD})\,}.\end{split}$
(2)

In this equation, $L_{k,m}$, $\varphi_{k,m}^{AoD}(\varphi_{l,k,m}^{AoD})$, and
${\bf{a}}_{L}(.)$ represent the number of non-line-of-sight (NLOS) paths, the
angle of departure, and the normalized array response vector, respectively.
Without loss of generality, we consider a uniform linear array (ULA) at each
AP and CPU. For an N-element ULA, the array response can be expressed as

$\begin{split}{{\bf{a}}_{L}}(\varphi)\buildrel\Delta\over{=}&\frac{1}{{\sqrt{{N}}}}\left[{1,\,{e^{j\frac{{2\pi}}{\lambda}{d_{A}}\sin(\varphi)}},...,{e^{j\frac{{2\pi}}{\lambda}{d_{A}}({N}-1)\sin(\varphi)}}}\right]\\\
&\in{\mathbb{C}^{1\times{N}}},\end{split}$ (3)

where $\lambda$ is the wavelength, and $d_{A}$ indicates inter-element
spacing. Also,
${\vartheta_{k,m}^{L}}\in\mathbb{C}\mathbb{N}\left({0,I(d)\,{{10}^{-0.1\kappa}}}\right)$
indicates path loss of the channel between the $m$-th AP and the $k$-th user
in the LOS link in which $I(d)$ is a parameter that shows the existence of a
LOS link between APs and users, where $I(d)=1$ means there is a LOS link in
the access link by probability $\rho$, which for Urban Microcellular (UMi)
scenario can be defined as [18]

$\rho=\min\left({\frac{{20}}{d},1}\right)\left({1-{e^{\frac{{-d}}{{39}}}}}\right)+{e^{\frac{{-d}}{{39}}}}.$
(4)

Similarly,
${\vartheta_{l,k,m}^{N}}\in\mathbb{C}\mathbb{N}\left({0,{{10}^{-0.1\kappa^{N}}}}\right)$
determines the NLOS link path loss model. $\kappa$ is also obtained as below
[19]

$\kappa=32.4+20\,{\log_{10}}({f_{c}})+10\,{\log_{10}}({d^{\alpha}}),$ (5)

where $f_{c}$, $d$, and $\alpha$ represent the carrier frequency in GHz,
distance between transmitter and receiver in meter, and the channel path loss
exponent, respectively.

The signal-to-interference-plus-noise-ratio (SINR) at each user (in the access
link) can be expressed as

$SINR_{k}^{A}=\frac{{\sum\limits_{m=1}^{M}{{{\left|{{{\bf{h}}_{k,m}}{\bf{W}}_{m}^{RF}{\bf{w}}_{k,m}^{BB}}\right|}^{2}}}}}{{\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{K}{\sum\limits_{m=1}^{M}{{{\left|{{{\bf{h}}_{k,m}}{\bf{W}}_{m}^{RF}{\bf{w}}_{j,m}^{BB}}\right|}^{2}}}}+\sigma_{k}^{2}}}.$
(6)

Thus, the sum rate of all users in the access link is

${R^{A}}=\eta
B\,\,\sum\limits_{k=1}^{K}{\log\left({1+{SINR_{k}^{A}}}\right)}.$ (7)

###  II-B Backhaul link

In the backhaul link, the IAB-donor with $N_{C}$ antennas and $N_{RF}^{C}$ RF
chains transmits a single stream to each IAB-node through the backhaul link
[20]. If ${s_{m}}\in\mathbb{C}$ ($\mathbb{E}\\{|s_{m}|^{2}\\}=1$) denotes the
transmitted signal for the $m$-th IAB-node, the received signal at the $m$-th
IAB-node is obtained as

$\begin{split}y_{m}^{AP}=&{{\bf{\tilde{w}}}_{m}}{\bf{H}}_{m,C}^{B}\sum\limits_{n=1}^{M}{{{\bf{F}}^{RF}}\,{\bf{f}}_{n}^{BB}{s_{n}}}+{{\bf{\tilde{w}}}_{m}}n_{m}^{AP}\\\
=&\,{{\bf{\tilde{w}}}_{m}}\underbrace{{\bf{H}}_{m,C}^{B}\,{{\bf{F}}^{RF}}\,{\bf{f}}_{m}^{BB}{s_{m}}}_{{\rm{desired\,signal}}}+{{\bf{\tilde{w}}}_{m}}\underbrace{{\bf{H}}_{m,C}^{B}\sum\limits_{\begin{subarray}{c}n=1\\\
n\neq
m\end{subarray}}^{M}{{{\bf{F}}^{RF}}\,{\bf{f}}_{n}^{BB}{s_{n}}}}_{{\rm{Interference}}}\\\
&+{{\bf{\tilde{w}}}_{m}}n_{m}^{AP},\end{split}$ (8)

where ${\bf{f}}_{m}^{BB}$ denotes the $m$-th column of digital beamforming
matrix ${{\bf{F}}^{BB}}\in{\mathbb{C}^{N_{RF}^{C}\times M}}$ and
${{\bf{F}}^{RF}}\in{\mathbb{C}^{{N_{C}}\times N_{RF}^{C}}}$ is the analog
beamforming matrix at the IAB-donor. Also,
${{\bf{\tilde{w}}}_{m}}\in{\mathbb{C}^{1\times N_{A}}}$ is the analog
beamforming vector the $m$-th IAB-node in the backhaul link.
$n_{m}^{AP}\sim\mathbb{C}\mathbb{N}(0,\sigma_{m}^{2})$ represents the additive
white Gaussian noise at each AP in the backhaul link. For fixed IAB-node
locations, we only consider the LOS channel
${\bf{H}}_{m,C}^{B}\in{\mathbb{C}^{N_{A}\times N_{C}}}$ between the CPU and
the $m$-th IAB-node that can be modeled as

$\begin{split}{\bf{H}}_{m,C}^{B}=&\sqrt{{N_{C}\,N_{A}}\,}\,\,{\zeta_{m,C}^{L}}\,\,{\bf{a}}_{L}^{H}(\varphi_{m,C}^{AoA})\,\,{\bf{a}}_{L}(\varphi_{m,C}^{AoD})\\\
&\in{\mathbb{C}^{N_{A}\times{N_{C}}}},\end{split}$ (9)

where
${\zeta_{m,C}^{L}}\in\mathbb{C}\mathbb{N}\left({0,{{10}^{-0.1\kappa}}}\right)$
is the path loss of the LOS channel, and $\varphi_{m,C}^{AoA}$ determines the
angle of arrival.

The received SINR at the $m$-th IAB-node in the backhaul link can be shown as

$SINR_{m}^{B}=\frac{{{{\left|{{\bf{\tilde{w}}}_{m}}{{\bf{H}}_{m,C}^{B}\,{{\bf{F}}^{RF}}\,{\bf{f}}_{m}^{BB}}\right|}^{2}}}}{{\sum\limits_{\begin{subarray}{c}n=1\\\
n\neq
m\end{subarray}}^{M}{{{\left|{{\bf{\tilde{w}}}_{m}}{{\bf{H}}_{m,C}^{B}\,{{\bf{F}}^{RF}}\,{\bf{f}}_{n}^{BB}}\right|}^{2}}}+{\left\|{{{{\bf{\tilde{w}}}}_{m}}}\right\|^{2}}\sigma_{m}^{2}}}.$
(10)

Thus, the achievable rate of the $m$-th IAB-node in the backhaul link can be
obtained as

$R_{m}^{B}={(1-\eta)B}\log(1+SINR_{m}^{B}).$ (11)

Since in the CF systems, all APs simultaneously transmit users’ signals in the
access link, therefore it is important for all APs to fully receive data
streams in the backhaul link before transmitting them. Thus, the effective
rate of the backhaul link will be

${R^{B}}=\mathop{\min}\limits_{m}(R_{m}^{B}).$ (12)

Hence, the end-to-end network rate can be expressed as

$R=\min({R^{B}},{R^{A}}).$ (13)

##  III Problem formulation

As stated before, our target in this paper is to maximize the end-to-end rate
of an IAB-assisted CF massive MIMO system by optimizing the system parameters
consisting of the hybrid beamforming matrices at CPU and APs and also the
bandwidth allocation parameter. For maximizing the system’s end-to-end rate,
the minimum backhaul link rate must be greater than or equal to the access
link sum rate to ensure that each IAB-node can provide the access link sum
rate. Therefore, (13) equivalents to ${R^{A}}$, and the end-to-end rate
optimization problem can be formulated as

$\displaystyle\max_{\begin{subarray}{c}\scriptstyle\eta,\,\left\\{{{{{\bf{\tilde{w}}}}_{m}}}\right\\}_{m=1}^{M},\,{{\bf{F}}^{BB}},{{\bf{F}}^{RF}},\,\\\
\left\\{{{\bf{W}}_{m}^{RF},{\bf{W}}_{m}^{BB}}\right\\}_{m=1}^{M}\end{subarray}}\,{R^{A}}$
(14a) $\displaystyle\,\,s.t.$ $\displaystyle\qquad\,\,{R^{B}}\geq\,{R^{A}},$
(14b)
$\displaystyle\qquad\sum\limits_{m=1}^{M}{{{\left\|{\bf{f}_{m}^{BB}}\right\|}^{2}}}\leq\,{P^{B}},$
(14c)
$\displaystyle\qquad{\left|{{{\left[{{{\bf{F}}^{RF}}}\right]}_{i,j}}}\right|^{2}}=\frac{1}{{{N_{C}}}}\,\,,\forall
i,j,$ (14d)
$\displaystyle\qquad\sum\limits_{k=1}^{K}{{{\left\|{\bf{w}}_{k,m}^{BB}\right\|}^{2}}}\leq\,{P^{A}}\,\,,m=1,...,M,$
(14e)
$\displaystyle\qquad\,{\left|{{{\left[{{\bf{W}}_{m}^{RF}}\right]}_{i,j}}}\right|^{2}}=\frac{1}{{{N_{A}}}}\,\,,\forall
i,j\,\,,\,m=1,...,M,$ (14f)
$\displaystyle\qquad\,{\left|{{{\left[{{\bf{\tilde{w}}}_{m}}\right]}_{i}}}\right|^{2}}=\frac{1}{{{N_{A}}}}\,\,,\forall
i\,\,,\,m=1,...,M,$ (14g) $\displaystyle\qquad 0<\eta\leq 1,$ (14h)

where $P^{B}$ is the CPU total transmitted power constraint.

In the above optimization problem, the objective function and some constraints
are non-convex due to highly coupled optimization variables and integer
constraints. Thus, finding the optimal solution to this problem is
intractable. However, by taking a close look at the problem, we observe that
the backhaul and the access links have different transmitters (CPU or APs) and
receivers (APs or users). Therefore, we can separately optimize each pair of
the transmitters and receivers’ beamforming matrices. This motivates us to
propose an alternating solution to find the optimal $\eta$ and then the
beamforming matrices. The resulting sub-problems will be presented in the
following sections.

##  IV Minimum Backhaul Rate Maximization

For any given $\eta$ and access link’s parameters, it is evident that the
performance of the access link is upper bounded with the minimum rate of the
backhaul link. Therefore, in the end-to-end rate maximization problem we have
to maximize the minimum backhaul link rate of APs. In this sense, (14) can be
reformulated as

$\displaystyle\max_{\begin{subarray}{c}\left\\{{{{{\bf{\tilde{w}}}}_{m}}}\right\\}_{m=1}^{M},\\\
{{\bf{F}}^{BB}},{{\bf{F}}^{RF}}\end{subarray}}\mathop{\min}\limits_{m}\,\,\,\log\left({1+SINR_{m}^{B}}\right)$
(15a) $\displaystyle\,\,s.t.$ $\displaystyle(\ref{cons:
P_B}),\,(\ref{F_RF}),\,(\ref{w_tilda}).$

By considering perfect CSI and setting $N_{RF}^{C}=M$, the analog beamforming
matrix of CPU can be chosen as [21]

$\begin{split}{{\bf{F}}^{RF}}=&\left[{\bf{a}}_{L}^{H}(\varphi_{1,C}^{AoD}),{\bf{a}}_{L}^{H}(\varphi_{2,C}^{AoD}),\ldots,{\bf{a}}_{L}^{H}(\varphi_{M,C}^{AoD})\right].\end{split}$
(16)

In the same way, the analog beamforming vector at the $m$-th AP can be fixed
as ${{{\bf{\tilde{w}}}}_{m}}={\bf{a}}_{L}(\varphi_{m,C}^{AoA})$ for
$m=1,2,...,M$.

For the specified ${\bf{F}}^{RF}$ and
$\left\\{{{{{\bf{\tilde{w}}}}_{m}}}\right\\}_{m=1}^{M}$, the backhaul max-min
rate problem (15) can be written as

$\displaystyle\mathop{\max}\limits_{{{\bf{F}}^{BB}}}\,\,\,\mathop{\min}\limits_{m}\,\,\,\log\left\\{{1+\frac{{{{\left|{{{\bf{b}}_{m}}\,{\bf{f}}_{m}^{BB}}\right|}^{2}}}}{{\sum\limits_{\begin{subarray}{c}n=1\\\
n\neq
m\end{subarray}}^{M}{{{\left|{{{\bf{b}}_{m}}\,{\bf{f}}_{n}^{BB}}\right|}^{2}}}+\sigma_{m}^{2}}}}\right\\}$
(17a) $\displaystyle\,\,s.t.$ $\displaystyle(\ref{cons: P_B}),$

where
${{\bf{b}}_{m}}={{{\bf{\tilde{w}}}}_{m}}\,{\bf{H}}_{m,C}^{B}\,{{\bf{F}}^{RF}},m=1,2,...,M$.
By using auxiliary variable $t$, the above problem can be rewritten as

$\displaystyle\mathop{\max}\limits_{{{\bf{F}}^{BB}},t}\,\,\,t$ (18a)
$\displaystyle\,\,s.t.$
$\displaystyle\frac{{{{\left|{{{\bf{b}}_{m}}\,{\bf{f}}_{m}^{BB}}\right|}^{2}}}}{{\sum\limits_{\begin{subarray}{c}n=1\\\
n\neq
m\end{subarray}}^{M}{{{\left|{{{\bf{b}}_{m}}\,{\bf{f}}_{n}^{BB}}\right|}^{2}}}+\sigma_{m}^{2}}}\geq\,\,{2^{t}}-1\,\,,\forall
m,$ (18b) $\displaystyle(\ref{cons: P_B}),$

which due to (18b), this problem is non-convex. For a fixed $t>0$, we have the
following feasibility problem

$\displaystyle{\rm{Find\,\,\\{}}{{\bf{F}}^{BB}}{\rm{\\}}}$ (19a)
$\displaystyle\,\,s.t.$
$\displaystyle\left({1+\frac{1}{{{2^{t}}-1}}}\right){\left|{{{\bf{b}}_{m}}\,{\bf{f}}_{m}^{BB}}\right|^{2}}\geq\,\,\sum\limits_{n=1}^{M}{{{\left|{{{\bf{b}}_{m}}\,{\bf{f}}_{n}^{BB}}\right|}^{2}}}+\sigma_{m}^{2}\,\,,\forall
m,$ (19b) $\displaystyle(\ref{cons: P_B}).$

In particular, if $t^{\circledast}$ represents the optimal value of $t$, we
can conclude that, while the previous problem is feasible for a given $t$, we
have $t^{\circledast}\geq t$; on the contrary, if it is infeasible, we can
find that $t^{\circledast}<t$. Thus, the optimal value of $t$ can be found by
applying the bisection algorithm over $t$ and solving some optimization
problems[2]. Furthermore, when ${{\bf{F}}^{BB}}^{\circledast}$ is the optimal
solution of (19), we can find arbitrary phase shifts
$\left\\{{{\phi_{m}}}\right\\}_{m=1}^{M}$ for which
${{\bf{F}}^{BB}}^{\circledast}{\rm{diag}}\left\\{{{e^{j{\phi_{m}}}}}\right\\}$
is also optimal because the absolute value of ${e^{j{\phi_{m}}}}$ equals $1$
and it will not affect the SINR value [22]. Hence, by considering
${\bf{b}}_{m}\,{\bf{f}}_{m}^{BB}\geq 0$ for $\forall m$ (i.e. non-negative
real valued with zero imaginary part), we can express (19b) and (14c) as the
Second-order cone (SOC) constraints shown below

$\sqrt{1+\frac{1}{{{2^{t}}-1}}}\,{{\bf{b}}_{m}}\,{\bf{f}}_{m}^{BB}\,\geq\,\,\left\|\begin{array}[]{l}{{\bf{b}}_{m}}\,{\bf{f}}_{1}^{BB}\\\
{{\bf{b}}_{m}}\,{\bf{f}}_{2}^{BB}\\\ \,\,\,\,\,\,\vdots\\\
{{\bf{b}}_{m}}\,{\bf{f}}_{M}^{BB}\\\
\,\,\,\,\,{\sigma_{m}}\end{array}\right\|\,\,\,\,,\,\forall m,$ (20)
$\left\|\begin{array}[]{l}\,\,\,{{\bf{f}}_{1}}\\\ \,\,\,{{\bf{f}}_{2}}\\\
\,\,\,\,\vdots\\\
\,\,{{\bf{f}}_{\bf{M}}}\end{array}\right\|\leq\sqrt{{P^{B}}}.$ (21)

Therefore, (19) can be reformulated as

$\displaystyle{\rm{Find\,\,\\{}}{{\bf{F}}^{BB}}{\rm{\\}}}$ (22a)
$\displaystyle\,\,s.t.$ $\displaystyle(\ref{eq:
SINR_k_Acons}),(\ref{eq_BHpower}),$

which is a convex second-order cone programming (SOCP) problem and can be
solved by using CVX [23]. The above procedures are summarized in Algorithm
$1$.

Algorithm 1 Hybrid Beamforming Optimization in the Backhaul Link.

1: Input: ${\bf{H}}_{m,C}^{B}$ for $\,\,m=1,2,...,M\,$;

2: Initialize: $\varepsilon\,{\rm{>0,}}\,{t_{\min}},\,{t_{\max}}$;

3: Compute: ${{\bf{F}}^{RF}}$ based on (16);

4: Set: ${{{\bf{\tilde{w}}}}_{m}}={\bf{a}}_{L}(\varphi_{m,C}^{AoA})$ for
$m=1,2,...,M$;

5: Set: $t:=\frac{{{t_{\min}}+{t_{\max}}}}{2}$;

6: Find $\\{{{\bf{F}}^{BB}}\\}:$ for given $t$, solve (22);

7: if problem (22) is feasible then

8: ${t_{\min}}:=t;$

9: else

10: ${t_{\max}}:=t;$

11: end if;

12: if ${t_{\max}}-{t_{\min}}<\,\varepsilon$ then

13: ${{\bf{F}}^{BB}}:={{\bf{F}}^{BB}};$

14: else

15: go to step (5);

16: end if;

17: Output: ${{\bf{F}}^{BB}}$, ${{\bf{F}}^{RF}}$, and
$\left\\{{{{{\bf{\tilde{w}}}}_{m}}}\right\\}_{m=1}^{M}$.

##  V Access Link sum rate Maximization

In this section, we desire to maximize the sum rate of users in the access
link by optimizing the hybrid beamforming matrix of each AP. For given $\eta$
and the backhaul link’s parameters, (14) can be written as

$\displaystyle\mathop{\max}\limits_{\\{{{\bf{W}}_{m}^{RF},{\bf{W}}_{m}^{BB}}\\}_{m=1}^{M}}\,{R^{A}}$
(23a) $\displaystyle\,\,s.t.$
$\displaystyle\sum\limits_{k=1}^{K}{{{\left\|{\bf{w}}_{k,m}^{BB}\right\|}^{2}}}\leq\,{P^{A}}\,\,\,\,,m=1,...,M\,,$
(23b)
$\displaystyle\,{\left|{{{\left[{{\bf{W}}_{m}^{RF}}\right]}_{i,j}}}\right|^{2}}=\frac{1}{{{N_{A}}}}\,\,\,\,,\forall
i,j\,\,m=1,...,M.$ (23c)

Due to the non-convex objective function and integer constraints, we introduce
a two-stage optimization strategy that optimizes the ${\bf{W}}_{m}^{RF}$ and
${\bf{W}}_{m}^{BB}$ in the different stages. As the first step, we optimize
the analog beamforming matrix of each AP without considering inter-user
interference. In the next step, we optimize the digital baseband beamforming
matrix of each AP to eliminate interference among users in the access link.

For analog beamforming matrix optimization, we set $N_{RF}^{A}=K$ and assume
that each user’s signal is directly transmitted through one of the RF chains.
Also, assuming a fully connected hybrid beamforming, each RF chain is
connected to all APs’ antennas with $N_{A}$ phase shifters. By determining the
singular value decomposition of the $\bf{h}_{k,m}$ as
${{\bf{h}}_{k,m}}=\,{{\bf{U}}_{k,m}}{\bf{\sum}}_{k,m}\left[{{\bf{V}}_{k,m}^{1},{\bf{V}}_{k,m}^{0}}\right]^{H}$,
the optimal fully digital beamforming vector for the $k$-th user at the $m$-th
AP will be
${\bf{w}}_{k,m}^{opt}={\bf{V}}_{k,m}^{1}\in{\mathbb{C}^{{N_{A}}\times 1}}$
which is the right singular vector of the channel between the $k$-th user and
the $m$-th AP. If we define
${\bf{W}}_{m}^{RF}=[{\bf{w}}_{1,m}^{RF},{\bf{w}}_{2,m}^{RF},\,...,\,{\bf{w}}_{K,m}^{RF}]$,
we can not express ${\bf{w}}_{k,m}^{RF}$ as ${\bf{w}}_{k,m}^{opt}$ for
$k=1,...,K$ and $m=1,...,M$ due to the constant module constraint of the
elements of the analog beamforming matrix. Thus, for maximizing the sum rate
of the access link users, we minimize the distance between
${\bf{w}}_{k,m}^{RF}$ and ${\bf{w}}_{k,m}^{opt}$ [24, 25] by considering the
constraints of the ${\bf{w}}_{k,m}^{RF}$ elements that can be formulated as

$\displaystyle\mathop{\min}\limits_{{\bf{w}}_{k,m}^{RF}}\,\left\|{{\bf{w}}_{k,m}^{opt}-\,{\bf{w}}_{k,m}^{RF}}\right\|^{2}$
(24a) $\displaystyle\,\,s.t.$
$\displaystyle{\left|{{{\left[{{\bf{w}}_{k,m}^{RF}}\right]}_{i}}}\right|^{2}}=\frac{1}{{{N_{A}}}}\,\,\,,i=1,...,N_{A}.$
(24b)

We can assume
${\left({{\bf{w}}_{k,m}^{RF}}\right)^{H}}{\bf{w}}_{k,m}^{RF}\approx{{\bf{I}}_{{N_{A}}}}$
since non-diagonal elements of
${\left({{\bf{w}}_{k,m}^{RF}}\right)^{H}}{\bf{w}}_{k,m}^{RF}$ are the sum of
random numbers that with high portability are lower than constant diagonal
elements. Hence, by considering the orthogonal property of
${\bf{w}}_{k,m}^{opt}$, the objective function of (24) can be written as

$\begin{split}\\\
\left\|{{\bf{w}}_{k,m}^{opt}-{\bf{w}}_{k,m}^{RF}}\right\|^{2}&=Tr\left\\{{{{({\bf{w}}_{k,m}^{opt}-\,{\bf{w}}_{k,m}^{RF})}^{H}}({\bf{w}}_{k,m}^{opt}-\,{\bf{w}}_{k,m}^{RF})}\right\\}\\\
&\propto\,-Tr\left\\{{\Re\left({{{\left({{\bf{w}}_{k,m}^{opt}}\right)}^{H}}\left({{\bf{w}}_{k,m}^{RF}}\right)}\right)}\right\\}.\end{split}$
(25)

Thus, for minimizing the objective function of (24), we can maximize

$\begin{split}\\\
&Tr\left\\{{\Re\left({{{\left({{\bf{w}}_{k,m}^{opt}}\right)}^{H}}\left({{\bf{w}}_{k,m}^{RF}}\right)}\right)}\right\\}=\\\
&\sum\limits_{i=1}^{{N_{A}}}{\Re\left\\{{{\left({\left[{\bf{w}}_{k,m}^{opt}\right]_{i}}\right)^{*}}{\left[{\bf{w}}_{k,m}^{RF}\right]_{i}}}\right\\}}.\end{split}$
(26)

Also, we have

$\begin{split}&\Re\left\\{{{\left({\left[{\bf{w}}_{k,m}^{opt}\right]_{i}}\right)^{*}}{\left[{\bf{w}}_{k,m}^{RF}\right]_{i}}}\right\\}=\\\
&\left|{\left[{\bf{w}}_{k,m}^{opt}\right]_{i}}\right|\left|{\left[{\bf{w}}_{k,m}^{RF}\right]_{i}}\right|\,\cos\left({\angle{\left[{\bf{w}}_{k,m}^{RF}\right]_{i}}-\angle{\left[{\bf{w}}_{k,m}^{opt}\right]_{i}}}\right).\end{split}$
(27)

Obviously, for minimizing the objective function of (24), we set

$\angle{\left[{\bf{w}}_{k,m}^{RF}\right]_{i}}=\angle{\left[{\bf{w}}_{k,m}^{opt}\right]_{i}}\,,\,\,i=1,2,...,{N_{A}}.$
(28)

To eliminate inter-user interference in the access link, we propose the Block
Diagonalization (BD) algorithm that removes interference among all users by
optimizing the digital beamforming matrix at each AP[26].

Based on the given analog beamforming matrix, the effective channel between
the $k$-th user and the $m$-th AP in the access link can be expressed as

${{\bf{\bar{h}}}_{k,m}}={{\bf{h}}_{k,m}}\,{\bf{W}}_{m}^{RF}\in{\mathbb{C}^{1\times
N_{RF}^{A}}}.$ (29)

Thus, for inter-user interference cancellation, based on the given
${\bf{\bar{h}}}_{k,m}$, the baseband digital precoder of each user should be
orthogonal to the effective channel of other users (i.e.
${{\bf{\bar{h}}}_{j,m}}\,{\bf{w}}_{k,m}^{BB}=0\,\,\,;\,k\neq j$ where
${\bf{w}}_{k,m}^{BB}$ indicates the $k$-th column of ${\bf{W}}_{m}^{BB}$). To
be more specific, if we define
${{\bf{\tilde{H}}}_{k,m}}={\left[{{\bf{\bar{h}}}_{1,m}^{T}|{\bf{\bar{h}}}_{2,m}^{T}|...|{\bf{\bar{h}}}_{k-1,m}^{T}|{\bf{\bar{h}}}_{k+1,m}^{T}|...|{\bf{\bar{h}}}_{K,m}^{T}}\right]^{T}}\in{\mathbb{C}^{K-1\times
N_{RF}^{A}}}$, the ${\bf{w}}_{k,m}^{BB}$ should be lain in the null space of
${\bf{\tilde{H}}}_{k,m}$. In this regard, by introducing $r={\rm
rank}({{\bf{\tilde{H}}}_{k,m}})$ , which has to satisfy
${r_{k,m}}\leq\,\,K-1$, the singular value decomposition (SVD) of
${\bf{\tilde{H}}}_{k,m}$ can be written as

$\begin{split}{{\bf{\tilde{H}}}_{k,m}}&=\,{{\bf{\tilde{U}}}_{k,m}}{{\bf{\tilde{\Sigma}}}_{k,m}}{\bf{\tilde{V}}}_{k,m}^{H}\\\
&=\,{{\bf{\tilde{U}}}_{k,m}}{{\bf{\tilde{\Sigma}}}_{k,m}}{\left[{{\bf{\tilde{V}}}_{k,m}^{1}\,\,{\bf{\tilde{V}}}_{k,m}^{0}}\right]^{H}},\end{split}$
(30)

where ${\bf{\tilde{V}}}_{k,m}^{1}$ and ${\bf{\tilde{V}}}_{k,m}^{0}$ represent
${{\bf{\tilde{V}}}_{k,m}}(:,1:{r_{k,m}})$ and
${{\bf{\tilde{V}}}_{k,m}}(:,{r_{k,m}}+1:end)$, respectively. It is clear that
${\bf{\tilde{V}}}_{k,m}^{0}$ includes the orthogonal basis of the null space
of ${\bf{\tilde{H}}}_{k,m}$, so
${{\bf{\bar{h}}}_{j,m}}{\bf{\tilde{V}}}_{k,m}^{0}$ can be formulated as

${{\bf{\bar{h}}}_{j,m}}{\bf{\tilde{V}}}_{k,m}^{0}=\left\\{\begin{array}[]{l}0\,;\,\,k\neq
j\\\
{{\bf{U}}_{k,m}}{{\bf{\Sigma}}_{k,m}}{\bf{V}}_{k,m}^{H}\,;k=j\end{array}\right.\,.$
(31)

Let us define ${\bf{V}}_{k,m}$ as
${{\bf{V}}_{k,m}}{\bf{=}}\left[{{\bf{V}}_{k,m}^{1}\,{\bf{V}}_{k,m}^{0}}\right]$,
where ${\bf{V}}_{k,m}^{1}$ is the first column of ${\bf{V}}_{k,m}^{H}$. As a
result, for jointly suppressing the inter-user interference and enhancing the
spectrum efficiency, the digital beamforming matrix of the $m$-th AP can be
expressed as

${\bf{W}}_{m}^{BB}=\left[{{\bf{\tilde{V}}}_{1,m}^{0}\,{\bf{V}}_{1,m}^{1},{\bf{\tilde{V}}}_{2,m}^{0}\,{\bf{V}}_{2,m}^{1},...,{\bf{\tilde{V}}}_{K,m}^{0}\,{\bf{V}}_{K,m}^{1}}\right].$
(32)

The above procedure is summarized in Algorithm $2$, where $(*)$ is defined to
satisfy the APs power constraints in the access link[27].

Algorithm 2 Optimal Hybrid Beamforming Design in the Access Link.

1: Input: ${{\bf{h}}_{k,m}}$ for $\forall{k,m}$;

2: Repeat for each AP:

3: Compute: ${{\bf{W}}_{m}^{RF}}$ based on (28);

4: Compute:

5: for $k=1$ to $K$ do

6: ${{\bf{\bar{h}}}_{k,m}}={{\bf{h}}_{k,m}}\,{\bf{W}}_{m}^{RF}$;

7:
${{\bf{\tilde{H}}}_{k,m}}={\left[{{\bf{\bar{h}}}_{1,m}^{T}|{\bf{\bar{h}}}_{2,m}^{T}|...|{\bf{\bar{h}}}_{k-1,m}^{T}|{\bf{\bar{h}}}_{k+1,m}^{T}|...|{\bf{\bar{h}}}_{K,m}^{T}}\right]^{T}}$;

8:
${{\bf{\tilde{H}}}_{k,m}}=\,{{\bf{\tilde{U}}}_{k,m}}{\tilde{\sum}_{k,m}}{\left[{{\bf{\tilde{V}}}_{k,m}^{1}\,{\bf{\tilde{V}}}_{k,m}^{0}}\right]^{H}}$;

9:
${{\bf{\bar{h}}}_{k,m}}\,{\bf{\tilde{V}}}_{k,m}^{0}={{\bf{U}}_{k,m}}{\sum_{k,m}}{\bf{V}}_{k,m}^{H}$;

10:
${\bf{W}}_{m}^{BB}=\left[{{\bf{\tilde{V}}}_{1,m}^{0}\,{\bf{V}}_{1,m}^{1},{\bf{\tilde{V}}}_{2,m}^{0}\,{\bf{V}}_{2,m}^{1},...,{\bf{\tilde{V}}}_{K,m}^{0}\,{\bf{V}}_{K,m}^{1}}\right]$;

11:
${\bf{W}}_{m}^{BB}=\frac{{{P^{A}}\,{\bf{W}}_{m}^{BB}}}{{{{\left\|{{\bf{W}}_{m}^{BB}}\right\|}_{F}}}}\,\,\,\,(*)$;

12: end for;

13: ${\bf{Output}}:\,\,\\{{{\bf{W}}_{m}^{BB}},{\bf{W}}_{m}^{RF}\\}_{m=1}^{M}$.

##  VI Bandwidth allocation coefficient optimization

In the previous sections, the minimum backhaul rate and access link sum-rate
have been maximized (in bps/Hz). Now we try to find an optimal bandwidth
allocation coefficient that satisfies (14)’s constraints and maximize its
objective function. Hence, (14) can be rewritten as

$\displaystyle\mathop{\max}\limits_{\eta}\,\,\,\,\,\,\,\,\eta\,{C_{A}}$ (33a)
$\displaystyle\,\,s.t.$ $\displaystyle(1-\eta)\,{C_{B}}\geq\,\eta\,\,{C_{A}},$
(33b) $\displaystyle 0<\eta\leq 1,$ (33c)

where ${C_{A}}$ and ${C_{B}}$ are given as

${C_{A}}=B\,\sum\limits_{k=1}^{K}{\log\left({1+\frac{{\sum\limits_{m=1}^{M}{{{\left|{{{\bf{h}}_{k,m}}{\bf{W}}_{m}^{RF}{\bf{w}}_{k,m}^{BB}}\right|}^{2}}}}}{{\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{K}{\sum\limits_{m=1}^{M}{{{\left|{{{\bf{h}}_{k,m}}{\bf{W}}_{m}^{RF}{\bf{w}}_{j,m}^{BB}}\right|}^{2}}}}+\sigma_{k}^{2}}}}\right)},$
(34)
${C_{B}}=\mathop{\min}\limits_{m}\,\,B\,\log\left({1+\frac{{{{\left|{{\bf{\tilde{w}}}_{m}}{{\bf{H}}_{m,C}^{B}\,{{\bf{F}}^{RF}}\,{\bf{f}}_{m}^{BB}}\right|}^{2}}}}{{\sum\limits_{\begin{subarray}{c}n=1\\\
n\neq
m\end{subarray}}^{M}{{{\left|{{\bf{\tilde{w}}}_{m}}{{\bf{H}}_{m,C}^{B}\,{{\bf{F}}^{RF}}\,{\bf{f}}_{n}^{BB}}\right|}^{2}}}+\sigma_{m}^{2}}}}\right).$
(35)

If we rewrite (33b) as $\eta\leq\frac{{{C_{B}}}}{{{C_{A}}+{C_{B}}}}$ , it is
clear that $\eta$ is upper bounded, so the maximum value of $\eta$ (i.e.
optimal value of (33)) can be found as

${\eta}=\frac{{{C_{B}}}}{{{C_{A}}+{C_{B}}}}.$ (36)

Therefore, based on (36) and (34) (or 35), the end-to-end rate of the proposed
system determined in (13) can be calculated as

$R=\frac{{{C_{A}}\,{C_{B}}}}{{{C_{A}}+{C_{B}}}}.$ (37)

In the above sections, we optimize all optimization parameters. Algorithm $3$
summarizes the CF massive MIMO system end-to-end rate optimization. Using this
algorithm, the hybrid beamforming matrices of the access and backhaul links
can be optimized simultaneously. The bandwidth allocation coefficient is then
calculated using the $C_{A}$ and $C_{B}$ parameters. It should be noted that
the proposed hybrid beamforming algorithm in the access link is locally
calculated by each AP. Then, each user’s desired and interference signals
should be sent to the CPU by each AP. In the following, we evaluate the access
link performance for decentralized beamforming at each AP and centralized
beamforming at the CPU and evaluate the backhaul resources required for each
scenario.

Algorithm 3 End-to-end Rate Optimization in the CF massive MIMO Systems.

1: Input: ${{\bf{h}}_{k,m}},\,{\bf{H}}_{m,C}^{B}\,;\forall k,m$;

2: Optimize: the backhaul link parameters via Algorithm $1$ and optimize the
access link parameters via Algorithm $2$;

3: Find: optimum bandwidth allocation coefficient based on the (36).

##  VII simulation results

In this section, we evaluate performance of the proposed scheme by computer
simulation. The performance of the backhaul and access links are evaluated
based on the number of IAB-nodes, transmit power constraints, and different
bandwidth allocation coefficients. The simulation parameters are shown in
Table I.

TABLE I: CF massive MIMO system parameters PARAMETER | NOTATION | VALUE
---|---|---
LOS link path loss exponent | $\alpha$ | 2.1 [28]
NLOS link path loss exponent | $\alpha$ | 3.64 [28]
Number of users | K | 8
Number of CPU’s antenna | $N_{C}$ | 64
AP’s antennas for Access | $N_{A}$ | 64
Number of NLOS path | $L_{k,m}$ | 5
Career frequency | $f_{c}$ | 28 GHz
System bandwidth | B | 2 GHz
Noise variance in Access | $\sigma_{k}^{2}$ | -174 dBm
Noise variance in Backhaul | $\sigma_{m}^{2}$ | -174 dBm
AP-CPU distance | $d_{A-C}(m)$ | $30\leq d_{A-C}\leq 50$
User-AP distance | $d_{U-A}(m)$ | $150\leq d_{U-A}\leq 200$

Figure 2: Sum rate of Access link users versus different transmitter power constraint with $M=6,K=8$. TABLE II: Backhaul link resources for the access link beamforming optimization. Beamforming | UL | DL
---|---|---
Centralized-FD | $MN_{A}K$ | $1+MN_{A}K$
Decentralized-FD | $2MK$ | $1$
Decentralized-Hybrid | $2MK$ | $1$

Figure 3: Max-Min rate of APs in the Backhaul link versus different power
constraints and number of APs.

Figure 4: End-to-end rate versus different values of $\eta$ with $M=6,K=8$.

First, we compare our proposed hybrid beamforming BD-based algorithm with the
fully digital (shown by FD abbreviation) BD algorithm [26] (i.e., when the
number of RF chains equals to the number of the transmitter’s antennas) and
random analog beamforming matrices. Fig. 2 illustrates the sum rate of users
(in bps/Hz) in the access link versus the AP’s transmitter power constraints.
It can be seen that our proposed hybrid BD algorithm approximately achieves
the same performance as fully-digital BD. Also, it significantly outperforms
random beamforming. This figure also presents the performance of the
centralized FD beamforming algorithm, in which the CPU designs the beamforming
matrix of each AP in the access link using the CSI of all users. It is
observed that a centralized beamforming can increase the access link rate of
the users, but the instantaneous CSI of each user must be provided for the
CPU. Backhaul resources that should be used in the centralized and
decentralized scenarios are shown in Table II. We can see that the centralized
beamforming optimization requires considerable backhaul resources, especially
in large antenna cases, but does not provide a significant improvement.

Fig. 3 shows how increasing the number of IAB-nodes will reduce the max-min
rate of the IAB-nodes in the backhaul link due to the inter-IAB-node
interference. In fact, increasing the number of IAB-nodes can inversely affect
the max-min rate of the IAB-nodes in the backhaul link and directly affect the
sum rate of the access link’s users.

Fig. 4 depicts the end-to-end rate of the system using an exhaustive search
through all possible $\eta$ values. The end-to-end rate is also determined
based on the bandwidth allocation coefficient that has been defined in (36).
It can be seen that the proposed bandwidth allocation strategy maximizes the
end-to-end rate of the system for different power constraints in the access
and backhaul links. According to the figure, improving one link will result in
more bandwidth being allocated to another. For example, by increasing the
backhaul link power constraint from $0$ dBm to $30$ dBm, the system has to
allocate more bandwidth to the access link to maximize the end-to-end rate.

Figure 5: End-to-end rate versus the access and backhaul links power
constraints with $M=6,K=8$.

Fig. 5 shows the end-to-end rate of the system versus the access and backhaul
links power constraints. It can be observed that by decreasing the access link
APs power, the end-to-end rate is less affected by backhaul link power
constraints, indicating that the access link limits the system’s performance.
In the same way, the end-to-end rate is affected slowly by the access link
power variation as the backhaul link power decreases. As a result, we can
conclude that both the access and backhaul links must have minimum power in
order to satisfy the end-to-end rate constraint.

For coverage evaluation, we do not consider NLOS paths in (2). Fig. 6 shows
the end-to-end rate of the networks for different number of APs . It can be
seen by increasing the number of APs, the end-to-end rate of the system first
increases due to the access link limitation and then decreases as the max-min
rate of the backhaul link limits the end-to-end system’s performance. This
shows that an optimum number of APs can be found in the system, which is $9$
in this setting.

Figure 6: End-to-end rate versus the different number of APs with $P^{A}=30$
dBm, $P^{B}=30$ dBm.

##  VIII conclusion

In this paper, we considered a CF massive MIMO system and maximized the end-
to-end rate of the users by optimizing the hybrid beamforming matrices of the
CPU and the IAB-nodes in the backhaul and access links, respectively. For
integrating the two link types, we developed a close-form expression for the
bandwidth allocation coefficient between the access and backhaul links. In
comparison to fully digital beamforming, the simulation results demonstrate
the effectiveness of the proposed hybrid beamforming algorithm. Furthermore,
for different power constraints in the access and backhaul links, the proposed
bandwidth allocation strategy maximizes the end-to-end rate of the system. The
end-to-end rate of the system was analyzed by an increasing number of IAB-
nodes, demonstrating that the end-to-end rate of the users first increased and
then decreased due to the access and backhaul links’ limitations. Therefore,
due to the dependence of the end-to-end rate of the users in both access and
backhaul links, increasing the number of IAB-nodes in these systems may not
always be beneficial.

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