# Experimental investigation of the effect of topological insulator on the
magnetization dynamics of ferromagnetic metal: $BiSbTe_{1.5}Se_{1.5}$ and
$Ni_{80}Fe_{20}$ heterostructure

Sayani Pal, Soumik Aon, Subhadip Manna, Sambhu G Nath, Kanav Sharma $\&$
Chiranjib Mitra  Corresponding<EMAIL_ADDRESS>Indian
Institute of Science Education and Research Kolkata,
Mohanpur 741246, West Bengal, India

###### Abstract

We have studied the spin-pumping phenomenon in ferromagnetic
metal($Ni_{80}Fe_{20}$)/topological insulator($BiSbTe_{1.5}Se_{1.5}$) bilayer
system to understand magnetization dynamics of ferromagnetic metal (FM) in
contact with a topological insulator (TI). TIs embody a spin-momentum-locked
surface state that spans the bulk band gap. Due to this special spin texture
of the topological surface state, the spin-charge interconversion efficiency
of TI is even higher than that of heavy metals. We evaluated the parameters
like effective damping coefficient ($\alpha_{eff}$), spin-mixing conductance
($g_{eff}^{\uparrow\downarrow}$) and spin current density ($j_{S}^{0}$) to
demonstrate an efficient spin transfer in
$Ni_{80}Fe_{20}/BiSbTe_{1.5}Se_{1.5}$ heterostructure. To probe the effect of
the topological surface state, a systematic low-temperature study is crucial
as the surface state of TI dominates at lower temperatures. The exponential
increase of $\Delta H$ for all different thickness combinations of FM/TI
bilayers and the enhancement of effective damping coefficient ($\alpha_{eff}$)
with lowering temperature confirms that the spin chemical potential bias
generated from spin-pumping induces spin current into the TI surface state.
Furthermore, low-temperature measurements of effective magnetization ($4\pi
M_{eff}$) and magnetic anisotropy field ($H_{k}$) showed anomaly around the
same temperature region where the resistivity of TI starts showing metallic
behavior due to the dominance of conducting TI surface state. The anomaly in
$H_{k}$ can result from the emerging exchange coupling between the TI surface
state and the local moments of the FM layer at the interface without any long-
range ferromagnetic order in TI at the interface.

Spintronics, Ferromagnetic Resonance

###### pacs:

Valid PACS appear here

## Introduction

Spintronics is one of the emerging fields that has witnessed remarkable
progress on both fundamental and technological fronts over the past couple of
decades. Phenomena like spin-orbit torque [1], spin Hall effect[2], giant
magnetoresistance [3], tunnelling magnetoresistance [4], domain wall motion
[5] provide basics for applications in memory devices[6], storage
technology[7], logic gates [8] and magnetic sensors [9]. These devices utilize
the spin degrees of freedom of electrons and their interaction with orbital
moments through spin-orbit coupling. Complete knowledge of the process of
generation, manipulation, and detection of spin degrees of freedom or the spin
current is essential for widespread applications in this field. If one focuses
on the currently available spin current generation processes, spin-pumping
[10, 11] is one of the most efficient methods where the precessing
magnetization in the ferromagnet (FM) injects spin current into the adjacent
layer by transferring spin angular momentum. This raises a need to study the
effect of spin pumping with special emphasis on exploring new materials which
can give rise to significant spin-charge interconversion efficiency.
Topological insulators (TI) are a new class of materials that have an
interesting spin texture of the surface state, owing to spin-momentum
locking[14, 15, 16, 17]. The momentum direction of the electron in the surface
state of TI is perpendicularly locked to its spin polarization direction. Thus
the spin-charge interconversion for TI is even higher than the heavy metals
which makes TIs suitable for spintronics application[12]. As surface states
are robust against deposition of FM layers on top of TI [18], the topological
surface states remain intact and gapless [19]. TI/FM bilayers have been
successfully used for the spin current generation in spin-pumping
experiments[36, 37, 34, 35, 20]. The effect of spin pumping can be witnessed
in the enhanced damping coefficient ($\alpha_{eff}$) value of the ferromagnet
upon excitations of ferromagnetic resonance (FMR) because, in the spin pumping
process, the net transfer of spin angular momentum into TI layer brings about
an additional damping torque on the precessing magnetization in the FM. It is
difficult to fabricate a perfect TI thin film where the bulk state of TI is
completely insulating. Thus for a complete understanding of the effect of TI
surface state on FM magnetization dynamics, the low-temperature study is
necessary where the surface states of TI dominate.
In this paper, we present the study of the spin-pumping phenomenon in
ferromagnetic metal (FM)/ topological insulator (TI) bilayer system. We chose
$Ni_{80}Fe_{20}$ as the FM layer and $BiSbTe_{1.5}Se_{1.5}$ as the TI layer.
Currently, $BiSbTe_{1.5}Se_{1.5}$ is one of the best 3D TI materials in which
bulk conduction in thin films is negligible even at room temperature and the
dominance of surface state is very prominent at lower temperatures [21, 22,
23]. In our low-temperature measurements, we have witnessed exponential
enhancement of FMR linewidth ($\Delta H$) and effective damping coefficient
($\alpha_{eff}$) at lower temperatures. It supports the proposal of the spin
chemical potential bias induced spin current injection into the surface state
of TI given by Abdulahadet al.[50]. For further investigation of the effect of
the TI surface state on the FM magnetization, we have also studied low-
temperature variations of effective magnetization and anisotropy field. We
calculated the interfacial magnetic anisotropy of the bilayer to be in-plane
of the interface. At low temperatures, this magnetic anisotropy field shows a
hump-like feature concomitant with the resistivity behavior of
$BiSbTe_{1.5}Se_{1.5}$ with temperature. It predicts the existence of exchange
coupling between the surface states of TI and the local moments of the FM
layer which acts perpendicular to the TI/FM interface. We have also evaluated
the values of spin-transport parameters like spin-mixing conductance,
$g_{eff}^{\uparrow\downarrow}$ and spin current density, $j_{s}^{0}$ at room
temperature to ensure a successful spin injection into the TI layer from the
FM layer.

## SAMPLE PREPARATION and CHARACTERIZATION

For this particular work, we have prepared different thickness combinations of
topological insulator(TI)/ferromagnet(FM) bilayer heterostructure.
$BiSbTe_{1.5}Se_{1.5}(BSTS)$ has been taken as the TI material and
Permalloy($Ni_{80}Fe_{20}$) has been used as the ferromagnetic material. BSTS
thin films were grown on silicon (Si 111) substrate using pulsed laser
deposition(PLD) technique [24, 25]. The target material was prepared using
$99.999\%$ pure Bi, Sb, Te, and Se in a 1:1:1.5:1.5 stoichiometric ratio. The
films were deposited through ablation of the target by a KrF excimer laser
(248 nm, 25 ns pulse width) at a low repetition rate of 1Hz and $1.2Jcm^{-2}$
laser fluence keeping the substrate temperature fixed at $250^{0}C$ and the
chamber partial pressure at 0.5 mbar (base pressure $2\times 10^{-5}$ mbar)
with a continuous flow of Ar gas. After deposition, TI films were immediately
transferred into the thermal evaporation chamber for the deposition of the FM
layer. Commercially available $99.995\%$ pure permalloy ($Ni_{80}Fe_{20}$)
pallets were used for deposition. The Py film was deposited [26] on top of TI
film at a rate of 1.2$\AA$ (crystal monitor: Inficon SQM 160) keeping the
chamber pressure fixed at $1\times 10^{-6}$ torr (base pressure $1\times
10^{-7}$ torr). For the characterization of the films X-ray diffraction
analysis (XRD), field emission scanning electron microscope (FE-SEM) imaging,
and atomic force microscopy (AFM) facilities have been used. X-ray
reflectometry technique has been used for thickness measurements here. For
convenience we are defining the BSTS of different thicknesses as follows: 10nm
BSTS as BSTS1, 21nm BSTS as BSTS2, 28nm BSTS as BSTS3, and 37nm BSTS as BSTS4.

## RESULTS and DISCUSSION

Figure 1: (a) In the left diagram, a schematic illustration of the
experimental set-up has shown where the FM/TI bilayer is placed upside down on
top of a CPW, and in the right diagram, net injected spin current
($I_{S}^{pump}$) due to spin-pumping into the TI layer (BSTS) from the FM
layer (Py) has shown, it results faster magnetization relaxation in FM; (b)
Ferromagnetic Resonance spectra of absorption at different frequencies for
Py/BSTS bilayer system at room temperature after background subtraction.

For a systematic study of the FM/TI bilayer system, we have done in-plane FMR
measurements in reflection mode geometry using a short-circuited CPW as shown
in fig.1a. We obtained typical FMR signal at different microwave frequencies
for Py(15nm)/BSTS2 sample in fig.1b. From the Lorentz formula fitting [53] of
the FMR signal we extracted the frequency dependence of the field linewidth
($\Delta H$ vs. $f$) and the resonance frequency vs. resonance field ($f$ vs
$H$) data as shown in fig.2a and fig.2b respectively. These give us valuable
information about the magnetization dynamics in ferromagnet which can be
described within the framework proposed by Landau, Lifshitz, and Gilbert [30],

$\frac{d\vec{M}}{dt}=-\gamma\vec{M}\times\vec{H}_{eff}+\frac{\alpha_{eff}}{M_{S}}\vec{M}\times\frac{d\vec{M}}{dt}$
(1)

where, $\gamma$ is the gyromagnetic ratio, $\vec{M}$ is the magnetization
vector, $M_{S}$ is the saturation magnetization, $H_{eff}$ is the effective
magnetic field which includes the external field, demagnetization and
crystalline anisotropy field and $\alpha_{eff}$ is the effective damping
coefficient of the system.

Figure 2: (a) Field linewidth ($\Delta H$) variation with resonance
frequencies ($f$) at 300K for Py/BSTS bilayer samples with different Py
thicknesses. Eq.4 has been used for fitting the curve and to determine the
damping coefficient ;(b) Resonance field ($H$) vs. resonance frequency ($f$)
for Py(20nm)/BSTS2 system at different temperatures . Eq.2 has been used for
fitting the curve and to determine the effective magnetization; (c) Effective
magnetization ($4\pi M_{eff}$) variation with thickness of Py(t), Py(t)/BSTS2
and Py(15nm/BSTS(t) at room temperature. Eq3 has been used for fitting the
curve and to evaluate saturation magnetization ($4\pi M_{S}$) and magnetic
anisotropy field($H_{k}$).

Figure 3: (a) $\alpha_{eff}$ variation with Py thickness for Py(t)/BSTS2
heterostructure at room temperature which fits in Eq.5; (b) $\alpha_{eff}$ as
a function of BSTS thickness for Py(15nm)/BSTS(t) heterostructure at room
temperature.

For a given magnetic material at ferromagnetic resonance, the resonance field
and frequency follow Kittel equation[27] given by,

$f=\frac{\gamma}{2\pi}\sqrt{(H+H_{k})(H+H_{k}+4\pi M_{eff})}$ (2)

Figure 4: Temperature dependence of resistivity of the BSTS sample of
thickness 21nm deposited on Si(111) substrate.

Figure 5: (a)Temperature dependence of the field linewidth ($\Delta H$) for
different thickness combinations of Py/BSTS bilayer systems and for a bare Py
thin film. The solid lines are the fits in the expression $exp(-T/T_{0})$;
(b)Temperature dependence of effective damping coefficient, $\alpha_{eff}$ of
Py(20nm)/BSTS2 and bare Py(20nm) film.

Figure 6: (a)Temperature dependence of effective magnetization of
Py(20nm/BSTS2); (b)Temperature dependence of the anisotropy field of
Py(20nm)/BSTS2.

where $H$, $H_{k}$, and $4\pi M_{eff}$ are the externally applied field,
magnetic anisotropy field, and effective magnetization respectively. We have
obtained $H_{k}$ and $4\pi M_{eff}$ for different FM/TI bilayer systems by
fitting the Kittel equation to the $f$ vs. $H$ curve as shown in fig.2b. The
obtained $4\pi M_{eff}$ value contains saturation magnetization($4\pi M_{s}$)
and other anisotropic contributions. We can evaluate $4\pi M_{s}$ value by
analyzing the thickness dependent measurement of $4\pi M_{eff}$ of the FM
layer. In the lower thickness region of the ferromagnetic thin films, $4\pi
M_{eff}$ is inversely proportional to the film thickness and follows the
equation[28],

$4\pi M_{eff}=4\pi M_{s}-\frac{2K_{s}}{M_{s}d}$ (3)

where $K_{s}$ is the surface anisotropy constant and $d$ is the thickness of
the FM film. The slope of the linear fit gives the anisotropy field
contribution to $4\pi M_{eff}$ and the intercept gives the $4\pi M_{s}$ value
as shown in fig.2c. The $4\pi M_{eff}$ does not depend on the thickness
variation of BSTS at room temperature but $4\pi M_{eff}$ for Py(t) monolayer
samples and for Py(t)/BSTS2 bilayer samples vary linearly with the inverse Py
thickness as shown in Fig.2c. From the linear fitting (Eq.3) of $4\pi M_{eff}$
vs. $1/t_{Py}$ data for the Py(t) and Py(t)/BSTS2 samples, we evaluated the
saturation magnetization, $M_{s}$ of the Py/BSTS bilayer that has been
decreased from that of the bare Py sample by an amount of 183$emu/cc^{3}$. It
is a result of the loss of ferromagnetic order in the Permalloy layer. Due to
the intermixing of the Py and BSTS at the interface, a magnetic dead layer
could have formed at the interface which resulted in the reduction of
saturation magnetization value as suggested by some previous studies [44, 43,
42] also. The $K_{s}$ value has decreased from $0.092\pm 0.008erg/cm^{2}$ in
bare Py film to $0.091\pm 0.015erg/cm^{2}$ in Py/BSTS2 bilayer. So interfacial
anisotropy constant, $K_{i}$($=K_{s}^{Py/TI}-K_{s}^{Py}$) for the Py/BSTS2
sample is $-0.001erg/cm^{2}$. From the negative value of $K_{i}$, we can
ensure an in-plane magnetic anisotropy in the Py/BSTS interface at room
temperature. A detailed discussion of magnetic anisotropy has been provided in
the last section where the temperature variation of $H_{k}$ is discussed.
$\alpha_{eff}$ can be determined by analysing $\Delta H$ at different
frequencies. $\Delta H$ contains both the intrinsic and extrinsic
contributions to the damping. Linewidth due to intrinsic damping is directly
proportional to the resonance frequency($f$) and follows the equation[29],

$\Delta H=\Delta H_{0}+(\frac{2\pi\alpha_{eff}}{\gamma})f$ (4)

where $\Delta H_{0}$ describes inhomogeneous linewidth broadening [38, 39] due
to different extrinsic contributions like magnetic inhomogeneities [40, 41],
surface roughness, and defects in the sample. We have evaluated the
$\alpha_{eff}$ values by fitting the $\Delta H$ vs $f$ curve for FM/TI
bilayers as shown in fig.2a. This $\alpha_{eff}$ consists of Gilbert damping
in the bulk ferromagnet($\alpha_{FM}$) and the enhanced damping($\alpha_{SP}$)
resulting from spin pumping into the adjacent TI layer [31, 32, 33],
$\alpha_{eff}=\alpha_{FM}+\alpha_{SP}$. The $\alpha_{FM}$ value for bare Py
film of thickness 15nm was calculated to be 0.0074 and for the FM/TI bilayer
system there has been significant enhancement in the $\alpha_{eff}$ value over
the bare Py value due to spin pumping, $\alpha_{SP}$. In this heterostructure,
$\alpha_{eff}$ increases gradually as the thickness of Py decreases both for
Py(t) and Py(t)/BSTS2 samples as shown in fig.3a. From the linear fit of
$\alpha_{eff}$ vs. $1/t_{Py}$ data we have obtained the spin-mixing
coefficient, $g_{eff}^{\uparrow\downarrow}$ for the BSTS/Py interface to be
$5.26\times 10^{18}\pm 0.71\times 10^{18}m^{-2}$ by using the equation[34,
37],

$\alpha_{eff}-\alpha_{FM}=\frac{g\mu_{B}}{4\pi
M_{s}t_{FM}}g_{eff}^{\uparrow\downarrow}$ (5)

where, $g$ and $\mu_{B}$ are the $g$-factor and Bohr magneton respectively. We
have also calculated the spin current density($j_{s}^{0}$) for the FM/TI
heterostructure using the $g_{eff}^{\uparrow\downarrow}$ value in the
following equation[36, 20],

$j_{s}^{0}=\frac{g_{eff}^{\uparrow\downarrow}\gamma^{2}h_{m}^{2}\hbar[4\pi
M_{s}\gamma+\sqrt{(4\pi
M_{s})^{2}\gamma^{2}+4\omega^{2}}]}{8\pi\alpha^{2}[(4\pi
M_{s})^{2}\gamma^{2}+4\gamma^{2}]}$ (6)

where $\gamma$, $h_{m}$, $\hbar$, $\omega$, and $\alpha$ are the gyromagnetic
ratio, microwave magnetic field, Planck’s constant, Larmour precession
frequency, and effective damping parameter respectively. Using Eq.6 the
$j_{s}^{0}$ value for Py/BSTS2 was obtained to be $0.901\times 10^{-10}\pm
0.122\times 10^{-10}Jm^{-2}$ in our experiment. The
$g_{eff}^{\uparrow\downarrow}$ and $j_{s}^{0}$ values obtained from Py
thickness-dependent study of $\alpha_{eff}$ are in a comparable range of the
previously reported values for other combinations of ferromagnet and TI
bilayer structures [37, 34, 35]. This gives evidence of successful spin
injection into the BSTS layer from the Py layer due to spin pumping [31, 32,
33, 50]. We also report the TI thickness-dependent study of $\alpha_{eff}$ as
shown in fig.3b. For bilayer structures of Py(15nm)/BSTS2(t) there is a sudden
jump in the $\alpha_{eff}$ value from that of the bare FM film
($\alpha_{FM}=0.0074$) because of spin pumping. Then with the thickness
variation of TI layer in the range of 10nm to 37nm, $\alpha_{eff}$ increases
slowly from 0.015 to 0.02. The TI thickness dependence of $\alpha_{eff}$ for
Py(15nm)/BSTS(t) bilayer is almost linear which certainly can not be described
by the conventional spin diffusion theory [48] for FM/NM proposed by
Tserkovnyak et al. [47]. For Py/BSTS heterostructure, $\alpha_{eff}$ vs.
$t_{BSTS}$ study suggests an efficient spin-sink nature of the TI bulk with
increasing thickness at room temperature [49]. From the room temperature study
we certainly can not distinguish the TI surface state contribution from the TI
bulk state contribution because growing a BSTS thin film with a perfectly
insulating bulk state is still very challenging. Thus it was imperative to
study the effect of topological surface state at low-temperature where bulk
states of TI get suppressed and surface states of TI starts to dominate.

In this section, we have focused on low-temperature measurements specifically
to understand the effect of topological surface states (TSS) on the
magnetization relaxation of FM. At higher temperatures, a significant amount
of bulk carriers are available to participate in the transport but with the
reduction of phonon scattering, surface carriers dominate at a lower
temperature. From the resistivity vs. temperature data of BSTS2 in fig.4, we
can see an insulating behavior of resistivity due to the enhanced insulating
nature of the bulk state of TI at higher temperatures and a metallic behavior
of resistivity below a certain temperature where the topological surface
states dominate. We measured temperature variation of FMR linewidth ($\Delta
H$), enhanced damping coefficient ($\alpha_{eff}$), anisotropy field ($H_{k}$)
and effective magnetization ($4\pi M_{eff}$). For different thickness
combinations of Py/BSTS bilayer, we obtained the $\Delta H$ variation with
temperature. It increases exponentially with decreasing temperature that fits
the expression, $exp(-T/T_{0})$ as shown in fig.5a. For bare Py(15nm) film, we
can note that there is no significant variation in $\Delta H$ at low
temperatures as can be seen from the curve at the bottom of fig.5a. To gain
further understanding, the temperature variation of $\alpha_{eff}$ has also
been studied for Py(20nm)/BSTS2 as shown in fig.5b and compared with
$\alpha_{eff}$ for bare Py film. From the enhancement of $\alpha_{eff}$ value
for Py(20nm)/ BSTS2 at room temperature we can ensure a successful spin
injection due to the spin pumping effect. But the exponential increase of
$\alpha_{eff}$ with decreasing temperature for the bilayer implies a huge
increment in the amount of spin angular momentum transfer into the TI layer at
lower temperatures. We attribute the origin of the exponential increase of
$\alpha_{eff}$ and $\Delta H$ at lower temperatures to the spin chemical
potential bias induced spin current into the surface state of TI as proposed
by Abdulahad et al. [50]. The induced spin current into the TI surface state
at lower temperatures corresponds to the rapid relaxation of magnetization
precession of FM which is reflected in the exponential increase of $\Delta H$
and $\alpha_{eff}$ of the ferromagnet.

To further investigate the effect of TI surface state on the magnetization of
FM, we studied the temperature variation of $4\pi M_{eff}$ and $H_{k}$ for
Py(20nm)/BSTS2. In our previous study [26] with bare Py thin films, we have
seen that $4\pi M_{eff}$ increases monotonically as saturation magnetization
increases with lowering the temperature. But from fig.6a, we can see that the
low-temperature dependence of $4\pi M_{eff}$ for Py/BSTS2 bilayer deviates
from the single layer Py film [Supplementary fig.S11(a)]. This anomaly in
$4\pi M_{eff}$ is related to the change of magnetic anisotropy energy of the
system as well as the other effects like spin chemical potential induced
current and exchange coupling between TSS and FM as mentioned by Abdulahad et
al. [50]. In a previous section, we evaluated the interfacial magnetic
anisotropy coefficient ($K_{i}=-0.001erg/cm^{2}$) to be in-plane of the
interface of the Py/BSTS2 bilayer. The anisotropy field associated with the
system anisotropy energy shows an interesting nature as we lower the
temperature. We can see from fig.6b that $H_{k}$ increases initially with
decreasing temperature until a certain value is reached and then the
anisotropy field weakens against a further decrease in temperature. Thus we
get a hump-like feature of $H_{K}$ for the same temperature region where $4\pi
M_{eff}$ shows the anomaly and it is concomitant with the resistivity vs
temperature behavior of the BSTS2 sample. The low-temperature behavior of
$H_{k}$ and $4\pi M_{eff}$ can be justified by the argument proposed by
Abdulahad et al. [50]. In their phenomenological model, they propose an
existence of exchange interaction between the surface states of TI and local
moments of the ferromagnetic layer. Several theoretical as well as
experimental predictions confirm the existence of gapless topological surface
states even after transition metal deposition on TI [51, 52]. These surface
states can couple with the local moments of the FM through exchange
interaction without any long-range ferromagnetic order. This exchange coupling
acts perpendicular to the TI surface and weakens the in-plane anisotropy at
lower temperatures where the surface states of TI dominate.

## CONCLUSIONS

In summary, we have carried out spin-pumping experiment in
$BiSbTe_{1.5}Se_{1.5}$(TI)/ $Ni_{80}Fe_{20}$(FM) bilayer system. From the
thickness-dependent measurements of FM/TI bilayers, we obtained the spin-
transport parameters like damping coefficient due to spin-pumping, spin mixing
conductance, and spin current density at room temperature. These results
demonstrate a successful spin transfer from the FM layer to the TI layer due
to spin-pumping. We have performed low-temperature measurements to
specifically understand the surface state contribution of TI on the FM
magnetization because the surface states of TI are more pronounced at lower
temperatures. We have confirmed the suppression of the insulating bulk state
of TI at lower temperatures from the resistivity vs. temperature data of TI.
In our low-temperature measurements of FMR linewidth and effective damping
coefficient, we have witnessed an exponential increase in both parameters with
the decrease in temperature. It suggests a spin chemical potential bias-
induced spin current injection into the surface states of TI that gets
enhanced at low temperatures [50]. We have also studied temperature variations
of the effective magnetization of the system. It showed a deviation from the
bare Py film [26] in the temperature regime where TI surface states dominate.
This deviation of effective magnetization results from the change in the
anisotropy energy of the system. At room temperature, we evaluated the
magnetic anisotropy energy coefficient which is found to be in-plane of the
interface. This in-plane anisotropy weakens when conducting surface state of
TI starts to dominate. It reflects from the hump-like feature in the magnetic
anisotropy field vs. temperature data of the bilayer system. The decrease in
in-plane magnetic anisotropy below a certain temperature can result from the
exchange coupling between the surface states of TI and the local moments of
the FM layer which act perpendicular to the interface [50]. Combining the
results of our low-temperature measurements we can conclude that there exists
an exchange coupling between the TI surface state and FM which does not create
any long-range ferromagnetic order in the TI and is unable to alter the
overall spin texture of the TI surface state at the interface[18]. However, it
affects the magnetization dynamics of the ferromagnetic metal quite
significantly. These added features of enhancing the damping coefficients
enables another fast control of magnetization dynamics in the FM layer.

## ACKNOWLEDGEMENTS

The authors sincerely acknowledge the Ministry of Education, Government of
India and Science and Engineering Research Board (SERB) (grant no:
EMR/2016/007950), and Department of Science and Technology (grant no.
DST/ICPS/Quest/2019/22) for financial support. S.P. acknowledges the
Department of Science and Technology(DST)-INSPIRE fellowship India, S. A.
acknowledges the Ministry of Education of the Government of India, S.M.
acknowledges the Council Of Scientific and Industrial Research(CSIR), India,
S.G.N and K.S acknowledges the University Grant Commission, India for research
fellowship. The authors would like to thank Dr. Partha Mitra of the Department
of Physics, Indian Institute of Science Education and Research Kolkata, for
providing the lab facilities for sample deposition. The authors would like to
acknowledge Prof. Anjan Barman and Mr. Pratap Kumar Pal of the Department of
Physics, SN Bose National Centre for Basic Sciences for helping with thickness
measurements using the XRR facility in their institute.

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$q$) in the counterclockwise order. Then the string of $M_{2}$ corresponds to
the permissible curve $c_{\mathrm{proj}}^{\gamma}$ which is projective, and
the strings of $M_{1}$ and $M_{3}$ correspond to $c^{\prime}$ and
$\tilde{c}^{\prime}$, respectively. ∎

######  Remark 6.7.

If $\gamma$, $\gamma^{0}_{1}$ and $\tilde{\gamma}^{0}_{1}$ satisfy the
P-condition (we have $i<m$ and $j<n$ in this case), then in
$\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$,
$c_{\mathrm{inj}}^{\gamma}$ and $c_{\mathrm{proj}}^{\gamma_{1}^{0}}$ have an
intersection $x$, and $c_{\mathrm{inj}}^{\gamma}$ and
$c_{\mathrm{proj}}^{\tilde{\gamma}_{1}^{0}}$ have an intersection $y$. The
permissible curves $c^{\prime}$, $\tilde{c}^{\prime}$ and
$c_{\mathrm{proj}}^{\gamma}$ can be understood as obtained by “negative
rotating respect to $c_{\mathrm{proj}}^{\gamma^{1}_{1}}\oplus
c_{\mathrm{proj}}^{\tilde{\gamma}^{1}_{1}}$” of $c_{\mathrm{inj}}^{\gamma}$.
To be precise, $c_{\mathrm{inj}}^{\gamma}$ is divided into three parts
$c_{\mathrm{I}}$, $c_{\mathrm{II}}$ and $c_{\mathrm{III}}$, $x$ is the common
endpoint of $c_{\mathrm{I}}$ and $c_{\mathrm{II}}$, and $y$ is the common
endpoint of $c_{\mathrm{II}}$ and $c_{\mathrm{III}}$. Then $c^{\prime}$
(respectively, $\tilde{c}^{\prime}$) can be obtained as follows:

  * Step 1.

following the positive direction of boundary, move the endpoint of
$c_{\mathrm{I}}$ (respectively, $c_{\mathrm{III}}$) lying in $\mathcal{M}_{A}$
to the previous vertex of the elementary polygon $\Delta_{\gamma^{0}_{1}}$
(respectively, $\Delta_{\tilde{\gamma}^{0}_{1}}$), where $\gamma^{0}_{1}$ and
$\gamma^{1}_{1}$ (respectively, $\tilde{\gamma}^{0}_{1}$ and
$\tilde{\gamma}^{1}_{1}$) are sides of $\Delta_{\gamma^{0}_{1}}$
(respectively, $\Delta_{\tilde{\gamma}^{0}_{1}}$);

  * Step 2.

move $x$ (respectively, $y$) to the endpoint of
$c_{\mathrm{proj}}^{\gamma^{1}_{1}}$ (respectively,
$c_{\mathrm{proj}}^{\tilde{\gamma}^{1}_{1}}$) such that $c_{\mathrm{I}}$
(respectively, $c_{\mathrm{III}}$) makes a negative rotation.

######  Proposition 6.8.

Keeping the notations in Lemma 6.6, we have

$\mathrm{proj.dim}M(c_{\mathrm{inj}}^{\gamma})=\max\big{\\{}1,\mathfrak{C}(\Delta_{\gamma^{0}_{1}})-1,\mathfrak{C}(\Delta_{\tilde{\gamma}^{0}_{1}})-1\big{\\}},$

where $\Delta_{\gamma^{0}_{1}}$ $($respectively,
$\Delta_{\tilde{\gamma}^{0}_{1}}$$)$ is the elementary polygon with sides
$\gamma^{0}_{1}$ and $\gamma^{1}_{1}$ $($respectively,
$\tilde{\gamma}^{0}_{1}$ and $\tilde{\gamma}^{1}_{1}$$)$. To be more precise,

  * (1)

if $\gamma^{0}_{1}$ and $\gamma^{1}_{1}$ $($respectively,
$\tilde{\gamma}^{0}_{1}$ and $\tilde{\gamma}^{1}_{1}$$)$ exist, then
$\mathrm{proj.dim}M(c_{\mathrm{inj}}^{\gamma})=\max\\{\mathfrak{C}(\Delta_{\gamma^{0}_{1}}),\mathfrak{C}(\Delta_{\tilde{\gamma}^{0}_{1}})\\}-1$;

  * (2)

if at least one of $\gamma^{0}_{1}$ and $\gamma^{1}_{1}$ does not exists and
at least one of $\tilde{\gamma}^{0}_{1}$ and $\tilde{\gamma}^{1}_{1}$ does not
exists, then $\mathrm{proj.dim}M(c_{\mathrm{inj}}^{\gamma})=1$.

###### Proof.

Suppose that $\gamma^{0}_{1}$ and $\gamma^{1}_{1}$ (respectively,
$\tilde{\gamma}^{0}_{1}$ and $\tilde{\gamma}^{1}_{1}$) exist. Similarly to
Proposition 5.6, regard $\gamma^{0}_{1}$ and $\gamma^{1}_{1}$ as the functions
$\gamma^{0}_{1},\gamma^{1}_{1}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$,
where $\gamma^{0}_{1}(0^{+})=p$ and
$\gamma^{0}_{1}(1^{-})=\gamma^{1}_{1}(0^{+})$. If there are arcs
$\gamma^{2}_{1},\cdots,\gamma^{\theta}_{1}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$
such that
$\gamma^{0}_{1},\cdots,\gamma^{\theta}_{1}\in\mathfrak{C}(\Delta_{\gamma^{0}_{1}})$
satisfy (PD1)–(PD3), then the minimal projective resolution of $M(c^{\prime})$
is of the form:

$\displaystyle\cdots\to M(c_{\mathrm{proj}}^{\gamma^{\theta}_{1}})\to\cdots\to
M(c_{\mathrm{proj}}^{\gamma^{2}_{1}})\to
M(c_{\mathrm{proj}}^{\gamma^{1}_{1}})\to M(c^{\prime})\to 0.$

We claim that $\Delta_{\gamma_{1}^{0}}$ is not $\infty$-elementary. Otherwise,
there exists an arc $\gamma^{\vartheta}_{1}$ with $\vartheta\geq\theta$ such
that $\gamma^{\vartheta}_{1}(1^{-})=\gamma^{0}_{1}(0^{+})$, this is a
contradiction because $c_{\mathrm{inj}}^{\gamma}$ is injective. Since
$\sharp\mathcal{M}_{A}<\infty$, there is a sequence of arcs
$\gamma^{\theta}_{1},\gamma^{\theta+1}_{1},\cdots,\gamma^{d}_{1}$
($d\geq\theta$) such that
$\gamma^{0}_{1},\gamma^{1}_{1},\cdots,\gamma^{d}_{1}$ satisfy (PD) by
Proposition 5.6. Then the minimal projective resolution of $M(c^{\prime})$ is
as follows:

$\displaystyle 0\to M(c_{\mathrm{proj}}^{\gamma^{d}_{1}})\to\cdots\to
M(c_{\mathrm{proj}}^{\gamma^{2}_{1}})\to
M(c_{\mathrm{proj}}^{\gamma^{1}_{1}})\to M(c^{\prime})\to 0.$ (6.1)

Without loss of generality, suppose that

  * •

there are arcs $\gamma^{0}_{1},\gamma^{1}_{1},\cdots,\gamma^{d}_{1}$ with
$d=\mathfrak{C}(\Delta_{\gamma^{0}_{1}})$ satisfy (PD);

  * •

and dually, there are arcs
$\tilde{\gamma}^{0}_{1},\tilde{\gamma}^{1}_{1},\cdots,\tilde{\gamma}^{\tilde{d}}_{1}$
with $\tilde{d}=\mathfrak{C}(\Delta_{\tilde{\gamma}^{0}_{1}})$ satisfy (PD)′.

Similarly, we get the minimal projective resolution of $M(\tilde{c}^{\prime})$
as follows:

$\displaystyle 0\to
M(c_{\mathrm{proj}}^{\tilde{\gamma}^{\tilde{d}}_{1}})\to\cdots\to
M(c_{\mathrm{proj}}^{\tilde{\gamma}^{2}_{1}})\to
M(c_{\mathrm{proj}}^{\tilde{\gamma}^{1}_{1}})\to M(\tilde{c}^{\prime})\to 0.$
(6.2)

Thus

$\mathrm{proj.dim}M(c_{\mathrm{inj}}^{\gamma})=\max\\{\mathrm{proj.dim}M(c^{\prime}),\mathrm{proj.dim}M(\tilde{c}^{\prime})\\}-1=\max\\{d,\tilde{d}\\}-1.$

Now suppose that at least one of $\gamma^{0}_{1}$ and $\gamma^{1}_{1}$ does
not exists and at least one of $\tilde{\gamma}^{0}_{1}$ and
$\tilde{\gamma}^{1}_{1}$ does not exists. Then
$M(c_{\mathrm{proj}}^{\gamma^{1}_{1}})=0=M(c_{\mathrm{proj}}^{\tilde{\gamma}^{1}_{1}})$
in (6.1) and (6.2), and hence $M(c^{\prime})=0=M(\tilde{c}^{\prime})$. By
Lemma 6.6(2), we have that $\mathrm{Ker}\ p(M(c_{\mathrm{inj}}^{\gamma}))\cong
M(c_{\mathrm{proj}}^{\gamma})$ is projective and
$\mathrm{proj.dim}M(c_{\mathrm{inj}}^{\gamma})=1$. ∎

For any basic finite dimensional $k$-algebra $A$, we know that
$A\cong\bigoplus_{i\in I}e_{i}A$ and $D(A)\cong\bigoplus_{i\in I}D(Ae_{i})$
where $D=\mathrm{Hom}_{k}(-,k)$ and $\\{e_{i}\mid i\in I\\}$ is a complete set
of primitive orthogonal idempotents of $A$, each $e_{i}A$ is an indecomposable
projective right $A$-module, and each $D(Ae_{i})$ is an indecomposable
injective right $A$-module. Thus Proposition 6.8 provides a method for
calculating the projective dimension of $D(A)$. Similarly, we can establish
the dual of Proposition 6.8 and calculate the injective dimension of $A$.

######  Theorem 6.9.

Let $A=kQ/I$ be a non-simple gentle algebra and
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ its marked
ribbon surface. If the quiver $(Q,I)$ of $A$ is not an oriented cycle with
full relations, then

$\displaystyle\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\max\limits_{\Delta_{j}\in\mathsf{f}\text{-}\Delta}\big{\\{}1,{\mathfrak{C}(\Delta_{j})}-1\big{\\}}=\max\limits_{\mathit{\Pi}\in\mathsf{f}\text{-}\mathcal{F}_{A}}l(\mathit{\Pi}).$
(6.3)

###### Proof.

Let $A=kQ/I$ be a gentle algebra. Then
$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\mathrm{proj.dim}D(A)$. Similarly
to the proofs of Proposition 5.6 and Theorem 5.10, we get

$\mathrm{proj.dim}D(A)=\max\big{\\{}1,\max\limits_{\Delta_{i}\in\mathsf{f}\text{-}\Delta}{\mathfrak{C}(\Delta_{i})}-1\big{\\}}=\max\limits_{\Delta_{j}\in\mathsf{f}\text{-}\Delta}\big{\\{}1,{\mathfrak{C}(\Delta_{j})}-1\big{\\}}=\max\limits_{\mathit{\Pi}\in\mathsf{f}\text{-}\mathcal{F}_{A}}l(\mathit{\Pi})$

by Proposition 6.8. ∎

Note that the number of all elements in $\mathfrak{C}(\Delta_{i})$ is equal to
that of all sides of $\Delta_{i}$ minus $1$. Thus, Theorem 6.9 shows that the
self-injective dimension of $A$ is equal to the number of sides of element
polygon(s) with the largest number of sides minus $2$. Moreover, by [14,
Theorem 3.4], any gentle algebra is Gorenstein. Thus, the finitistic dimension
of a gentle algebra equals to its left and right self-injective dimensions
(see [16]). Thus Theorem 6.9 provides a description of the finitistic
dimension of a gentle algebra.

###  6.3 The number of indecomposable Gorenstein projective modules

Recall that an algebra is said to be CM-finite if the number of indecomposable
Gorenstein projective modules (up to isomorphism) is finite. It is known that
gentle algebras are CM-finite [10]. We use
$\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))$ to denote the
subcategory of $\bmod A$ consisting of all indecomposable non-projective
Gorenstein projective modules. In this subsection, we obtain a formula for
calculating $\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))$ by AG-
invariants.

######  Lemma 6.10.

Let $A=kQ/I$ be a gentle algebra, and let
$c_{1}=\alpha_{1}\alpha_{2}\cdots\alpha_{m}$ and
$c_{2}=\beta_{1}\beta_{2}\cdots\beta_{n}$ be two oriented cycles with full
relations of $A$. If $c_{1}$ and $c_{2}$ have a common vertex $v_{0}$, that
is, there are arrows $\alpha_{h}$, $\alpha_{h+1}$, $\beta_{\hbar}$ and
$\beta_{\hbar+1}$ such that
$v_{0}=t(\alpha_{h})=t(\beta_{\hbar})=s(\alpha_{h+1})=s(\beta_{\hbar+1})$,
then $\alpha_{h}A$ and $\beta_{\hbar}A$ are indecomposable Gorenstein
projective modules satisfying
$\mathrm{top}(\alpha_{h}A)=\mathrm{top}(\beta_{\hbar}A)\cong
M(c_{\mathrm{simp}}^{\gamma_{0}})=S(v_{0})$.

###### Proof.

We know that $\alpha_{h}A$ and $\beta_{\hbar}A$ are indecomposable non-
projective Gorenstein projective by Theorem 6.1, Now we show that
$\mathrm{top}(\alpha_{h}A)=\mathrm{top}(\beta_{\hbar}A)\cong
M(c_{\mathrm{simp}}^{\gamma_{0}})=S(v_{0})$ by using the marked ribbon surface
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ of $A$. Let
$\gamma_{0}$ be the arc in $\Gamma_{A}$ corresponding to $v_{0}$ by the
bijection $\mathfrak{v}:Q_{0}\to\Gamma_{A}$. Then $\gamma_{0}$ is a common
side of two $\infty$-elementary polygons $\Delta_{1}$ and $\Delta_{2}$, that
is, there are arcs $\gamma_{-1},\gamma_{1},\gamma_{2},\cdots,\gamma_{a}$ and
$\tilde{\gamma}_{-1},\tilde{\gamma}_{1},\tilde{\gamma}_{2},\cdots,\tilde{\gamma}_{b}$
such that

  * •

$\gamma_{-1},\gamma_{0}\gamma_{1},\cdots,\gamma_{a}$ (respectively,
$\tilde{\gamma}_{-1},\tilde{\gamma}_{0}=\gamma_{0},\tilde{\gamma}_{1},\cdots,\tilde{\gamma}_{b}$)
have a common endpoint $p$ (respectively, $q$) and $\gamma_{i+1}$
(respectively, $\tilde{\gamma}_{j+1}$) appears after $\gamma_{i}$
(respectively, $\tilde{\gamma}_{j}$) for any $-1\leq i\leq a-1$ (respectively,
$-1\leq j\leq b-1$);

  * •

$\gamma_{-1},\gamma_{0}\in\mathfrak{S}(\Delta_{2})$,
$\gamma_{0},\gamma_{1}\in\mathfrak{S}(\Delta_{1})$ and
$\gamma_{i}\notin\mathfrak{S}(\Delta_{1})$ for any $3\leq i\leq a$;

  * •

$\tilde{\gamma}_{-1},\tilde{\gamma}_{0}\in\mathfrak{S}(\Delta_{1})$,
$\tilde{\gamma}_{0},\tilde{\gamma}_{1}\in\mathfrak{S}(\Delta_{2})$ and
$\gamma_{j}\notin\mathfrak{S}(\Delta_{2})$ for any $3\leq j\leq b$;

  * •

$\mathfrak{v}^{-1}(\gamma_{1})=t(\alpha_{h+1})$ and
$\mathfrak{v}^{-1}(\tilde{\gamma}_{1})=t(\beta_{\hbar+1})$.

For the permissible curve $c$ consecutively crossing
$\gamma_{0},\gamma_{1},\cdots,\gamma_{a}$ and $\tilde{c}$ consecutively
crossing $\tilde{\gamma}_{0},\tilde{\gamma}_{1},\cdots,\tilde{\gamma}_{b}$
such that

$\displaystyle
p=p(\gamma_{0},\gamma_{1},c)=p(\gamma_{1},\gamma_{2},c)=\cdots=p(\gamma_{a-1},\gamma_{a},c)\text{
and }$ $\displaystyle
q=p(\tilde{\gamma}_{0},\tilde{\gamma}_{1},\tilde{c})=p(\tilde{\gamma}_{1},\tilde{\gamma}_{2},\tilde{c})=\cdots=p(\tilde{\gamma}_{b-1},\tilde{\gamma}_{b},\tilde{c}),$

we have $M(c)\cong\alpha_{h}A$, $M(\tilde{c})\cong\beta_{\hbar}A$, and $c$ and
$\tilde{c}$ correspond to the strings

$\displaystyle\mathfrak{v}^{-1}(\gamma_{a})\longleftarrow\cdots\mathop{\longleftarrow}\limits^{\alpha_{h+2}}\mathfrak{v}^{-1}(\gamma_{1})\mathop{\longleftarrow}\limits^{\alpha_{h+1}}\mathfrak{v}^{-1}(\gamma_{0})\text{
and }$
$\displaystyle\mathfrak{v}^{-1}(\tilde{\gamma}_{0})\mathop{\longrightarrow}\limits^{\beta_{\hbar+1}}\mathfrak{v}^{-1}(\tilde{\gamma}_{1})\mathop{\longrightarrow}\limits^{\beta_{\hbar+2}}\cdots\longrightarrow\mathfrak{v}^{-1}(\tilde{\gamma}_{b}),\
\text{respectively,}$

where
$\mathfrak{v}^{-1}(\gamma_{0})=\mathfrak{v}^{-1}(\tilde{\gamma}_{0})=v_{0}$.
So $\mathrm{top}(\alpha_{h}A)\cong\mathrm{top}M(c)\cong
M(c_{\mathrm{simp}}^{\gamma_{0}})$ and
$\mathrm{top}(\beta_{\hbar}A)\cong\mathrm{top}M(\tilde{c})\cong
M(c_{\mathrm{simp}}^{\tilde{\gamma}_{0}})$. Notice that
$\gamma_{0}=\tilde{\gamma}_{0}$, thus
$M(c_{\mathrm{simp}}^{\gamma_{0}})=M(c_{\mathrm{simp}}^{\tilde{\gamma}_{0}})$
is isomorphic to the simple module
$S(\mathfrak{v}^{-1}(\gamma_{0}))=S(v_{0})$. ∎

######  Lemma 6.11.

Let $A=kQ/I$ be a gentle algebra, and let $Q_{0}^{\mathrm{ocf}}$ be the set of
all vertices on the oriented cycle with full relations. Then the map
$f:\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))\to
Q_{0}^{\mathrm{ocf}}$ via $\alpha A\mapsto t(\alpha)$ is surjective;
furthermore, if $v\in Q_{0}^{\mathrm{ocf}}$ is a common vertex of two oriented
cycles with full relations, then $\sharp f^{-1}(v)=\sharp\\{\alpha
A\in\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))|t(\alpha)=v\\}=2$; and
if otherwise, then $\sharp f^{-1}(v)=1$.

###### Proof.

For any $v\in Q_{0}^{\mathrm{ocf}}$, there exists at least one oriented cycle
with full relations $\alpha_{0}\cdots\alpha_{l-1}$ such that
$v=t(\alpha_{\overline{i}})=s(\alpha_{\overline{i+1}})$ for some $0\leq i\leq
l$ where $\overline{i}$ is equal to $i$ modulo $l$. Then by Theorem 6.1, we
have that $\alpha_{\overline{i}}A$ is an indecomposable non-projective
Gorenstein projective module and $f$ is surjective.

If $v$ is a common vertex of two oriented cycles with full relations
$\alpha_{0}\alpha_{1}\cdots\alpha_{l-1}$ and
$\beta_{0}\beta_{1}\cdots\beta_{\ell-1}$, then there are arrows
$\alpha_{\overline{i}}$, $\alpha_{\overline{i+1}}$, $\beta_{\underline{j}}$
and $\beta_{\underline{j+1}}$ such that
$v=t(\alpha_{\overline{i}})=s(\alpha_{\overline{i+1}})=t(\beta_{\underline{j}})=s(\beta_{\underline{j+1}})$
where $\overline{i}$ is equal to $i$ modulo $l$ and $\underline{j}$ is equal
to $j$ modulo $\ell$. Thus we have
$\\{\alpha_{\overline{i}}A,\beta_{\underline{j}}A\\}\subseteq f^{-1}(v)$ by
Lemma 6.10; and furthermore $\sharp f^{-1}(v)\geq 2$. Notice that $A$ is
gentle, thus there is no other arrow whose source or sink is $v$. Therefore
$\sharp f^{-1}(v)\leq 2$ and $\sharp f^{-1}(v)=2$.

It is obvious that if there exists a unique oriented cycle
$\mathsf{s}=\alpha_{0}\alpha_{1}\cdots\alpha_{l}$ with full relations of $A$
such that $v\in Q_{0}^{\mathrm{ocf}}$ is a vertex of $\mathsf{s}$, then
$\sharp f^{-1}(v)=1$. Indeed, suppose $t(\alpha_{i})=v$ with $0\leq i\leq l$.
Then
$f^{-1}(v)=\\{\alpha_{i}A\\}\subseteq\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))$.
∎

On the marked ribbon surface
$\mathcal{S}_{A}=(\mathcal{S}_{A},\mathcal{M}_{A},\Gamma_{A})$ of a gentle
algebra $A=kQ/I$, for each $\infty$-elementary polygon, its each side provides
an indecomposable non-projective Gorenstein projective module by the
permissible curve satisfying Proposition 6.3 (GP2), and the arc which is a
common side of two $\infty$-elementary polygons provides two indecomposable
non-projective Gorenstein projective modules. From this observation, we can
also prove Lemma 6.11. Moreover, we have the following proposition.

######  Proposition 6.12.

Let $A=kQ/I$ be a gentle algebra with the AG-invariant $\phi_{A}$, and let
$L\subseteq\mathbb{N}^{+}$ be a finite set such that $\phi_{A}(0,\ell)>0$ if
$\ell\in L$, otherwise, $\phi_{A}(0,\ell)=0$. Then we have

  1. (1)

$\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))=\sum\limits_{\ell\in
L}\ell\cdot\phi_{A}(0,\ell),$

that is, $\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))$ is the
number of arrows on oriented cycles with full relations.

  2. (2)

Furthermore, if two gentle algebras $A$ and $A^{\prime}$ are AG-equivalent,
then

$\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))=\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A^{\prime})).$

###### Proof.

Recall from the proof of Proposition 5.12 that for each $\ell\in L$, the
number of oriented cycles with full relations of length $\ell$ is
$\phi_{A}(0,\ell)$. Thus we can suppose that $A$ has $t=\sum\limits_{\ell\in
L}\phi_{A}(0,\ell)$ oriented cycles with full relations
$\mathsf{s}^{i}=\alpha_{1}^{i}\alpha_{2}^{i}\cdots\alpha_{\ell_{i}}^{i}$ with
$1\leq i\leq t$. Then there exist are the following maps:

$Q_{1}^{\mathrm{ocf}}\mathop{\longrightarrow}\limits^{g}_{\sim}\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))\mathop{\longrightarrow}\limits^{f}Q_{0}^{\mathrm{ocf}},$

where $Q_{1}^{\mathrm{ocf}}=\bigcup\limits_{1\leq i\leq
t}\\{\alpha_{j}^{i}|1\leq j\leq\ell_{i}\\}$ and $f$ is surjective (see Lemma
6.11), and $g$ defined by $\alpha\mapsto\alpha A$ is bijective (see Theorem
6.1). Thus
$\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A^{\prime}))$ is equal
to the number $\sum_{\ell\in L}\ell\cdot\phi_{A}(0,\ell)$ of arrows on
oriented cycles with full relations.

Let $A^{\prime}$ be a gentle algebra such that $\phi_{A^{\prime}}=\phi_{A}$.
Then

$\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A^{\prime}))=\sum\limits_{\ell\in
L}\ell\cdot\phi_{A^{\prime}}(0,\ell)=\sum\limits_{\ell\in
L}\ell\cdot\phi_{A}(0,\ell)=\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A)).$

The second equality is obtained by $\phi_{A^{\prime}}=\phi_{A}$. ∎

######  Corollary 6.13.

[10, 20] Gentle algebras are CM-finite.

###### Proof.

Let $A=kQ/I$ be a gentle algebra. Since $(Q,I)$ is a finite quiver, the number
of oriented cycles with full relations is finite, and so there exists a finite
set $L\subseteq\mathbb{N}^{+}$ such that $0<\phi_{A}(0,\ell)<\infty$ for any
$\ell\in L$, and $\phi_{A}(0,\ell)=0$ for any $\ell\notin L$. By Proposition
6.12, we have
$\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))=\sum\limits_{\ell\in
L}\ell\cdot\phi_{A}(0,\ell)<\infty$. ∎

##  7 Examples

###  7.1 Some examples for calculating global dimension of gentle algebras

In this subsection, we provide some examples to calculate the global dimension
of gentle algebras and the projective dimension of simple modules.

######  Example 7.1.

For any $n\geq 1$, there exists a gentle one-cycle algebra $A$, that is, its
quiver has exactly one cycle, such that $\mathrm{gl.dim}A=n$. Indeed, if $A$
is such a gentle one-cycle algebra whose marked ribbon surface
$\mathcal{S}_{A}$ is shown in the FIGURE 21 subfigure 1, then $A\cong kQ/I$,
where $Q_{0}=\\{\gamma^{0},\gamma^{1},\cdots,\gamma^{n-1}\\}$, $Q$ is shown in
the FIGURE 21 subfigure 2 and
$I=\langle\alpha_{1}\alpha_{2},\alpha_{2}\alpha_{3},\cdots,\alpha_{n-1}\alpha_{n}\rangle$.

$q^{3}$$q^{2}$$q^{1}$$q^{0}=q^{n}$$q^{n-2}$$q^{n-3}$$p$$\iddots$$\gamma^{1}$$\gamma^{2}$$\gamma^{3}$$\iddots$$\gamma^{n-2}$$\gamma^{n-1}$$\gamma^{0}$subfigure
1

$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma^{1}}}$$\scriptstyle{\alpha_{2}}$$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma^{2}}}$$\scriptstyle{\alpha_{3}}$$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma^{0}}}$$\scriptstyle{\alpha_{1}}$$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma^{n-1}}}$$\scriptstyle{\alpha_{n}}$$\scriptstyle{\alpha_{n-1}}$subfigure
2

Figure 21: The subfigure 2 is the gentle algebra whose marked ribbon surface
is shown in subfigure 1.

Thus $\mathrm{gl.dim}A=n$ by Theorem 5.10. Let $\gamma^{0}$ be the arc
$\gamma^{0}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$ such
that $\gamma^{0}(0^{+})=p$ and $\gamma^{0}(1^{-})=q^{0}$, and let $\gamma^{i}$
be the arc
$\gamma^{i}:(0,1)\to\mathcal{S}_{A}\backslash\partial\mathcal{S}_{A}$ such
that $\gamma^{i}(0^{+})=q^{i-1}$ and $\gamma^{i}(1^{-})=q^{i}$ for any $1\leq
i\leq n-1$, and let $\gamma^{n}$ be the arc $\gamma^{0}(1-x)$ (note:
$\gamma^{n}=\gamma^{0}(1-x)\simeq\gamma^{0}(x)=\gamma^{0}$). Then the arcs
$\gamma^{0},\gamma^{1},\cdots,\gamma^{n-1},\gamma^{n}$ satisfy the condition
(PD), and the minimal projective resolution of the simple module
$M(c_{\mathrm{simp}}^{\gamma^{0}})$ can be determined by Proposition 5.6 as
shown in FIGURE 22, where $c_{i}\simeq c_{\mathrm{proj}}^{\gamma^{i}}$;
$c_{i+1}^{\prime}$ is the permissible curve corresponding to the kernel of
$\delta_{i}$ ($1\leq i\leq n-1$) and $c_{n}\simeq c_{n}^{\prime}\simeq
c_{\mathrm{proj}}^{\gamma^{n}}$.

$q^{3}$$q^{2}$$q^{1}$$q^{0}=q^{n}$$q^{n-2}$$q^{n-3}$$p$$\iddots$$c_{\mathrm{simp}}^{\gamma^{0}}$$c_{0}$$c_{1}^{\prime}$$c_{1}$$c_{2}^{\prime}$$c_{2}$$c_{n-3}^{\prime}$$c_{n-3}$$c_{n-2}^{\prime}$$c_{n-2}$$c_{n-1}^{\prime}$$c_{n-1}$$c_{n}$$\simeq$$c_{n}^{\prime}$
Figure 22: The minimal projective resolution of
$M(c_{\mathrm{simp}}^{\gamma^{0}})$.

Furthermore, by formula (5.1), we obtain

$0\longrightarrow
M(c_{\mathrm{proj}}^{\gamma^{n}})\mathop{\longrightarrow}\limits^{\delta_{n}}M(c_{\mathrm{proj}}^{\gamma^{n-1}})\mathop{\longrightarrow}\limits^{\delta_{n-1}}\cdots\mathop{\longrightarrow}\limits^{\delta_{2}}M(c_{\mathrm{proj}}^{\gamma^{1}})\mathop{\longrightarrow}\limits^{\delta_{1}}M(c_{\mathrm{proj}}^{\gamma^{0}})\mathop{\longrightarrow}\limits^{\delta_{0}}M(c_{\mathrm{simp}}^{\gamma^{0}})\longrightarrow
0,$

where
$M(c_{\mathrm{proj}}^{\gamma^{n-1}})\cong{{}_{\gamma^{1}}^{{}_{\gamma^{0}}^{\gamma^{n-1}}}}$
and $\mathrm{Ker}\delta\cong{{}_{\gamma^{1}}^{\gamma^{0}}}\cong
M(c_{\mathrm{proj}}^{\gamma^{n}}).$ Thus
$\mathrm{proj.dim}M(c_{\mathrm{simp}}^{\gamma^{0}})=n$. Similarly,
$\mathrm{proj.dim}M(c_{\mathrm{simp}}^{\gamma^{i}})=n-i$ for any $0\leq i\leq
n-1$.

######  Example 7.2.

See FIGURE 23 subfigure 1, it is the marked ribbon surface of the gentle
algebra $A=kQ/I$, where $Q$ is shown subfigure 2 and

$I=\langle
a_{1}a_{3},a_{5}a_{4},a_{8}a_{7},a_{7}a_{6},a_{10}a_{2},a_{9}a_{10}^{\prime},a_{10}^{\prime}a_{14},a_{14}a_{13}\rangle.$

$1\\!3$$1\\!2$$1\\!1$$8$$7$$6$$5$$3$$4$$1$$2$$10$$9$$14$subfigure 1

$1\\!3$$1\\!2$$1\\!1$$8$$7$$6$$5$$3$$4$$1$$2$$10$$9$$1\\!4$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$a_{8}$$a_{9}$$a_{1\\!0}$$a_{1\\!1}$$a_{1\\!2}$$a_{1\\!3}$$a_{1\\!4}$$a_{1\\!0}^{\prime}$subfigure
2

Figure 23: The subfigure 2 is the gentle algebra whose marked ribbon surface
is shown in subfigure 1.

The elementary polygon of the marked ribbon surface gives the projective
dimension of each simple module. Such as the arc $S(8)$, by Proposition 5.6,
each term in the minimal projective resolution of $S(8)$ corresponds to an arc
of the elementary polygon $\Delta_{j}$ whose arc set
$\mathfrak{S}(\Delta_{j})=\\{6,7,8,9\\}$. Then the minimal projective
resolution of $S(8)$ is

$0\longrightarrow M(c_{\mathrm{proj}}^{6})\longrightarrow
M(c_{\mathrm{proj}}^{7})\longrightarrow
M(c_{\mathrm{proj}}^{8})\longrightarrow S(8)=M(c_{\mathrm{simp}}^{8})\to 0,$

thus $\mathrm{proj.dim}S(8)=2$. Similarly, we get

$\displaystyle\mathrm{proj.dim}S(1)=2,$
$\displaystyle\mathrm{proj.dim}S(2)=0,$
$\displaystyle\mathrm{proj.dim}S(4)=0,$
$\displaystyle\mathrm{proj.dim}S(5)=1,$
$\displaystyle\mathrm{proj.dim}S(6)=1,$
$\displaystyle\mathrm{proj.dim}S(7)=1,$
$\displaystyle\mathrm{proj.dim}S(11)=1,$
$\displaystyle\mathrm{proj.dim}S(12)=1,$
$\displaystyle\mathrm{proj.dim}S(13)=1,$
$\displaystyle\mathrm{proj.dim}S(14)=2.$

The simple module $S(9)=M(c_{\mathrm{simp}}^{9})$ corresponds to the arc $9$
which satisfies the P-condition, its minimal projective resolution is

$\displaystyle 0$ $\displaystyle\longrightarrow
M(c_{\mathrm{proj}}^{12})\longrightarrow M(c_{\mathrm{proj}}^{6})\oplus
M(c_{\mathrm{proj}}^{14})\longrightarrow M(c_{\mathrm{proj}}^{7})\oplus
M(c_{\mathrm{proj}}^{14})$ $\displaystyle\longrightarrow
M(c_{\mathrm{proj}}^{8})\oplus M(c_{\mathrm{proj}}^{10})\longrightarrow
M(c_{\mathrm{proj}}^{9})\longrightarrow
S(9)=M(c_{\mathrm{simp}}^{9})\longrightarrow 0,$

thus $\mathrm{proj.dim}S(9)=4$. Similarly, we get

$\mathrm{proj.dim}S(3)=1,\ \mathrm{proj.dim}S(10)=3.$

Consequently we conclude that $\mathrm{gl.dim}A=4$. It should be pointed out
that we can get $\mathrm{gl.dim}A=4$ by Theorem 5.10 directly.

######  Example 7.3.

Let $A$ be a gentle algebra whose marked ribbon surface $\mathcal{S}_{A}$ is
given in Example 6.4. Then $\mathrm{gl.dim}A=\infty$ by Theorem 6.9 because
$\mathcal{S}_{A}$ has an $\infty$-elementary polygon with the arc set
$\\{1,2,3\\}$. Indeed, the minimal projective resolution of $S(1)$ is

$\cdots\to P(3)\to P(2)\to P(1)\to P(3)\to P(2)\to P(1)\to S(1)\to 0$

by Proposition 5.6, and so $\mathrm{proj.dim}S(1)=\infty$.

###  7.2 Some examples for calculating self-injective dimension of gentle
algebras

In this subsection, we provide some examples to calculate projective
(respectively, injective) dimension of indecomposable injective (respectively,
projective) modules, and the self-injective dimension of gentle algebras.

######  Example 7.4.

Consider the gentle algebra $A$ given in Example 7.2. Then
$\mathrm{gl.dim}A=4$. Since
$I(8)={{}^{{}^{{}^{13}_{12}}_{11}}}{{}_{8}}{{}^{9}}=M(c_{\mathrm{inj}}^{8})$
is an indecomposable injective module with the top
$M(c_{\mathrm{simp}}^{13})\oplus M(c_{\mathrm{simp}}^{9})\cong S(13)\oplus
S(9)$, its projective cover is $M(c_{\mathrm{proj}}^{13}\oplus
c_{\mathrm{proj}}^{9})$. Consider the elementary polygons $\Delta_{9}$ with
sides $\\{9,10,14,13,12\\}$ and $\Delta_{12}$ with side $\\{12\\}$. We have
$\mathfrak{C}(\Delta_{9})=5$ and $\mathfrak{C}(\Delta_{12})=1$, thus
$\mathrm{proj.dim}I(8)=\max\\{5,1\\}-1=4$. Furthermore, by the proof of
Proposition 6.8, the minimal projective resolution of $I(8)$ is as follows:

$0\rightarrow M(c_{\mathrm{proj}}^{12})\longrightarrow
M(c_{\mathrm{proj}}^{13})\longrightarrow
M(c_{\mathrm{proj}}^{14})\longrightarrow M(c_{\mathrm{proj}}^{8})\oplus
M(c_{\mathrm{proj}}^{10})\longrightarrow M(c_{\mathrm{proj}}^{13})\oplus
M(c_{\mathrm{proj}}^{9})\longrightarrow I(8)\rightarrow 0.$

Thus we have
$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\mathrm{proj.dim}D(A)=4=\mathrm{gl.dim}A$.

In Example 7.4, for the gentle algebra $A$, its global and self-injective
dimensions are identical. Now we give two examples of gentle algebras with
infinite global dimension.

######  Example 7.5.

  1.   2. (1)

Let $A$ be the gentle algebra given in Example 6.4. Then
$\mathrm{gl.dim}A=\infty$ by Example 7.3. Since
$\max\limits_{\Delta_{j}\in\mathsf{f}\text{-}\Delta}\mathfrak{C}(\Delta_{j})=2$,
we have
$\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=\max\limits_{\Delta_{i}\in\mathsf{f}\text{-}\Delta}\\{1,\mathfrak{C}(\Delta_{i})-1\\}=1$
by Theorem 6.9.

  3. (2)

Let $A$ be the gentle algebra whose marked ribbon surface is shown in the
FIGURE 24 I.

$1$$2$$3$$4$$5$$7$$6$$8$$9$$10$I

$1$$2$$3$$4$$5$$7$$6$$8$$9$$10$II

Figure 24: The marked ribbon surface of gentle algebra given in Example 7.5
(subfigure I) and all permissible curves corresponding to indecomposable non-
projective Gorenstein projective modules (subfigure II).

Then $\mathrm{gl.dim}A=\infty$ by Theorem 5.10 (because the elementary polygon
with sides $\\{3,4,5,6\\}$ and the elementary polygon with sides $\\{5,7,8\\}$
are $\infty$-elementary), and $\mathrm{inj.dim}_{A}A=\mathrm{inj.dim}A_{A}=2$
by Theorem 6.9. Furthermore, since its AG-invariant
$\phi_{A}=[(9,4),(0,4),(0,3)]$, we have

$\sharp\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))=4\phi_{A}(0,4)+3\phi_{A}(0,3)=4\cdot
1+3\cdot 1=7$

by Proposition 6.12. Indeed, we have
$\mathrm{ind}(\mathrm{nproj}\text{-}\mathcal{GP}(A))=\\{3,\ 4,\
{{}^{{}_{5}}_{{}^{7}_{9}}},\ {{}^{7}_{9}},\ 8,\ {{}^{5}_{6}},\ 6\\}$ by
Proposition 6.3.

## Acknowledgement

The authors thank the referees for very useful and detailed suggestions.

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