# Doped two-dimensional diamond: properties and potential applications

Bruno Ipaves<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo,
CEP 05508-090 São Paulo, SP, Brazil  João F. Justo<EMAIL_ADDRESS>Escola
Politécnica, Universidade de São Paulo, CEP 05508-010, São Paulo, SP, Brazil
Biplab Sanyal<EMAIL_ADDRESS>Department of Physics and
Astronomy, Uppsala University, 75120 Uppsala, Sweden  Lucy V. C. Assali
<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo, CEP
05508-090 São Paulo, SP, Brazil

###### Abstract

This paper examines the structural, thermodynamic, dynamic, elastic, and
electronic properties of doped 2D diamond C4X2 (X = B or N) nanosheets in both
AA′A′′ and ABC stacking configurations, by first-principles calculations.
Those systems consist of three diamond-like graphene sheets, with an undoped
graphene layer between two 50% doped ones. Our results, based on the analysis
of ab-initio molecular dynamics simulations, phonon dispersion spectra, and
Born’s criteria for mechanical stability, revealed that all four structures
are stable. Additionally, their standard enthalpy of formation values are
similar to the one of pristine 2D diamonds, recently synthesized by
compressing three graphene layers together. The C4X2 (X = B or N) systems
exhibit high elastic constant values and stiffness comparable to the diamond.
The C4N2 nanosheets present wide indirect band gaps that could be advantageous
for applications similar to the ones of the hexagonal boron nitride (h-BN),
such as a substrate for high-mobility 2D devices. On the other hand, the C4B2
systems are semiconductors with direct band gaps, in the 1.6 - 2.0 eV range,
and small effective masses, which are favorable characteristics to high
carrier mobility and optoelectronics applications.

##  I INTRODUCTION

Graphene is the most popular two-dimensional (2D) material, being a zero-gap
semimetal with a honeycomb carbon structure and $sp^{2}$ hybridization. It
carries a unique combination of physical properties in nature, such as high
electrical conductivity, tensile strength, and optical transparency.
Additionally, it is the elementary structure for several other nanomaterials,
such as fullerenes, nanotubes, graphite, and the single-layer diamond (2D
diamond) [1, 2, 3]. As a result of recent developments in the synthesis and
characterization of 2D materials, the 2D diamond has received great attention,
with promising applications in several fields, such as batteries, quantum
computing, nano-optics, and nanoelectronics [4].

The stabilization of 2D diamond often requires surface functionalization,
leading to a variety of structures, which have received different labels, such
as diamane, diamene, diamondol, and diamondene [4, 5]. 2D diamonds can also be
built out of bilayer graphene (BLG) or few-layer graphene (FLG) through
different techniques. For example, the hydrogenated (HD) and fluorinated (FD)
2D diamonds can be synthesized at ambient pressure without a substrate, in
which the HD can be produced using hot filament chemical vapor deposition
(CVD) [6], while FD by combining FLG and gaseous ClF3 [7].

The pristine 2D diamond (PD) is hard to synthesize as high pressures are
required to transform $sp^{2}$ bonds from graphene layers into interlayer
$sp^{3}$ ones [5]. Nevertheless, the PD has recently been synthesized without
a substrate, by compressing three graphene layers [8]. Additionally, a
theoretical investigation has shown that it is possible to stabilize the 2D
diamond made of two graphene layers with nitrogen substitution [9]. For
example, the NCCN 2D structure, composed of two carbon layers functionalized
with nitrogen ones on both sides, has also been investigated, suggesting it
could be used as a selective ammonia sensor [10, 11, 12].

The physical properties of 2D diamonds may vary considerably, depending on the
synthesis methods, leading to structures with different configurations,
functional groups, and heteroatoms [5, 4]. At room temperature, the thermal
conductivity of HD is high and the heat transport arises from the acoustic
phonon modes. On the other hand, under the same conditions, the thermal
conductivity of FD is lower than that of HD and the heat transport is
controlled by the optical phonon modes [13]. 2D diamonds also present
remarkable mechanical properties, with stiffness and Young’s modulus similar
to the ones of graphene and bulk diamond [14]. Furthermore, unlike graphene,
2D diamonds have band gap features that depend on the stacking arrangement,
the number of layers, and the functional groups present in the structures [15,
4]. Despite several recent experimental and theoretical investigations on 2D
diamonds, the origin of all these peculiar properties has been the subject of
debate [5, 4].

In this paper, we present a study of the physical properties of 2D diamonds
doped with substitutional N or B atoms. The reference systems consist of three
graphene sheets: an undoped graphene layer between two 50% doped ones, where
the C-C bonds between neighboring layers are strong covalent bonds. Here, we
considered four structure configurations labeled AA′A′′-C4N2, ABC-C4N2,
AA′A′′-C4B2, and ABC-C4B2. Their structural, thermodynamic, dynamic, elastic,
and electronic properties and potential applications are explored and
discussed in depth.

##  II COMPUTATIONAL METHODS

This investigation was performed using first-principles calculations based on
the Density Functional Theory (DFT) [16], using the plane-wave basis set and
projector augmented-wave (PAW) method [17], as implemented in the Quantum
ESPRESSO computational package [18, 19]. We utilized the generalized gradient
approximation of Perdew–Burke–Ernzerhof (GGA-PBE) exchange–correlation
functional [20] and the Dion et al. scheme [21] optimized by Klimeš et al.
(optB88-vdW [22]) to properly describe the effects of the dispersive van der
Waals (vdW) interactions. For an accurate description of the energy band gap
values, we employed the hybrid Heyd-Scuseria-Ernzerhorf (HSE) functional [23]
at the relaxed structures obtained from the optB88-vdW approximation. The
plane-wave energy cutoff was set to 1100 eV with a convergence threshold of
0.1 meV/atom for the total energy. We used a $16\times 16\times 1$ $k$-point
mesh to describe the irreducible Brillouin zone [24], and the forces on atoms
were converged down to 1 meV/Å. To obtain the phonon dispersion curves, we
used the Density Functional Perturbation Theory (DFPT) [25] with an $8\times
8\times 1$ $q$-point mesh.

The primitive hexagonal cells of the 2D structures were constructed using 6
atoms. To determine the cell parameters in the $xy$-plane, a variable-cell
optimization was carried out with the BFGS quasi-newton algorithm. In order to
avoid interactions among cell images, the lattice parameter perpendicular to
the sheets ($z$-axis) was fixed at 25 Å. This approach has been successfully
applied to similar 2D systems in previous studies [26, 10, 27].

In order to determine the elastic properties of the systems, we built a
rectangular cell with 12 atoms and used the strain–energy method [28, 29].
Accordingly, for isotropic structures and small deformations $(\epsilon)$ near
their equilibrium configurations, the elastic energy, per unit area, was
approximated as

$E(\epsilon)-E(0)\approx\frac{1}{2}E^{(2)}\epsilon^{2},$ (1)

where $E(\epsilon)$ is the total energy of strained configurations, while
$E(0)$ is the total energy of the respective unstrained ones. We applied two
in-plane deformations, ranging from -1.2% to 1.2%, in order to obtain the
$E^{(2)}$, which allowed to obtain the elastic constants after fitting a
second-order polynomial to the data. Herein, $E^{(2)}=C_{11}$ elastic constant
for the zigzag axial deformation, while $E^{(2)}=2(C_{11}+C_{12})$ for the
biaxial planar deformation [28, 29].

The thermal stability was studied by computing the standard enthalpy of
formation, per atom, of the structures at 0 GPa ($\Delta H_{\\!f}^{0}$), by
using

$\Delta H_{\\!f}^{0}=\frac{E_{t}({\rm C}_{4}{\rm X}_{2})-4E_{t}({\rm
C})-2E_{t}({\rm X})}{6},$ (2)

where $E_{t}({\rm C}_{4}{\rm X}_{2})$ is the total energy of the 2D nanosheet,
with 4 C atoms and 2 X atoms (X = B or N) in the primitive cell. $E_{t}$(C)
and $E_{t}$(X) are the total energies, per atom, of the respective C and X
standard ground states, i.e., of graphite and the crystalline boron in the
trigonal structure ($\beta$-boron) or the isolated N2 molecule. This procedure
to determine enthalpies and/or energies of formation has been successfully
used to investigate several other systems [30, 31, 32, 33, 34].

Additionally, ab-initio molecular dynamics simulations (AIMD) were carried out
using the Vienna ab initio simulation package (VASP) [35], where a 6 $\times$
6 $\times$ 1 hexagonal 216-atom supercell was adopted to allow possible
structural reconstructions. A Nose–Hoover thermostat (NVT) ensemble was
employed, from 300 to 1000 K for 5 ps, with a simulation time step of 1 fs.

##  III RESULTS AND DISCUSSION

Initially, we explored the physical properties of pristine 2D diamond in
AA′A′′ and ABC stacking structural configurations, composed of three graphene
layers in which the C atoms between layers are covalently bonded with a near
$sp^{3}$ hybridization. Starting the simulations with the diamond-like
configuration, they converged, after the relaxation of atomic positions, to
trilayer graphene systems with vdW interactions between layers (graphite-
like). This behavior has also been found in a previous theoretical
investigation of 2D diamond, starting the simulations with two graphene layers
[36]. These results can be understood as a consequence of the absence of an
external pressure-induced and/or surface passivation to promote the $sp^{2}$
to $sp^{3}$ hybridization transformation [36]. Those pristine structures
represent the reference systems used here to study and understand the effects
of their functionalization.

Figure 1: Schematic illustration of the C4X2 (X = B or N) systems. (a) Top and (b) side views of AA′A′′-C4X2, (c) top and (d) side views of ABC-C4X2. The black and gray spheres represent the C and X atoms, respectively. The red lines denote the simulation unit cell limits, with the rectangle cells used to determine the elastic properties. The graphs also indicate the labels given to the intralayer ($d_{\rm C-X}$) and interlayer ($h_{\rm C-X}$) distances, structure thickness ($\Delta h$), and the intraplanar bond angle ($\theta$). Table 1: Structural properties of C4X2 (X = B or N): lattice parameter ($a$), intralayer ($d$) and interlayer ($h$) distances, thickness ($\Delta h$), and the intraplanar bond angle ($\theta$), labeled according to figure 1. The distances are given in Å and angles in degrees. The standard enthalpies of formation ($\Delta H_{\\!f}^{0}$) at 0 GPa are given in meV/atom. For PD, HD, and FD, X = C. System | $\\!\\!a$ | $\\!\\!d_{\mathrm{C-X}}$ | $d_{\mathrm{C-C}}$ | $\\!\\!h_{\mathrm{C-C}}$ | $\\!\\!\Delta h$ | $\\!\\!\theta$ | $\Delta H_{\\!f}^{0}$
---|---|---|---|---|---|---|---
AA′A′′-C4N2 | 2.42 | 1.49 | 1.49 | 1.60 | 4.74 | 108.9 | 348
ABC-C4N2 | 2.44 | 1.50 | 1.50 | 1.57 | 4.66 | 109.0 | 333
AA′A′′-C4B2 | 2.66 | 1.55 | 1.62 | 1.66 | 4.25 | 118.4 | 424
ABC-C4B2 | 2.67 | 1.55 | 1.63 | 1.65 | 4.18 | 118.5 | 365
PD111Reference [8]. The $(\overline{2}110)$-oriented h-diamane exhibits two $d_{\mathrm{C-X}}$ and $\theta$ values. Y = bond lengths are 1.35 and 1.54 Å with the angles presenting $sp^{3}$ and $sp^{2}$ hybridizations. | 2.43 | Y${}^{\text{a}}$ | 1.54 | 1.65 | — | Y${}^{\text{a}}$ | 300
HD222Reference [15] | 2.53 | 1.56 | — | 1.56 | — | — | —
FD${}^{\text{b}}$ | 2.56 | 1.55 | — | 1.55 | — | — | —
NCCN | 2.39333Reference [10] | 1.47${}^{\text{c}}$ | — | 1.58${}^{\text{c}}$ | 2.59444Reference [9] | 108.8${}^{\text{c}}$ | 211${}^{\text{d}}$
Figure 2: Systems’ total energy variation during 5 ps, computed with AIMD
simulations. At 300 K, results of (a) AA′A′′-C4N2, (b) ABC-C4N2, (c)
AA′A′′-C4B2, and (d) ABC-C4B2. At 1000 K, results of (e) AA′A′′-C4N2, (f)
ABC-C4N2, (g) AA′A′′-C4B2, and (h) ABC-C4B2.

Then, we explored the properties of C4X2 (X = B or N) systems, which can be
described as three graphene sheets in which four C atoms are bonded covalently
(2D diamond-like) in each unit cell. The two external layers are 50% doped
with substitutional X atoms, hence, each X atom is bonded to three C atoms.
Figure 1 presents a schematic representation of the optimized and relaxed C4X2
(X = B or N) systems, in both AA′A′′ and ABC stacking configurations, as well
as the respective labels given to the intraplanar bond angle ($\theta$),
intralayer ($d_{\rm C-X}$) and interlayer ($h_{\rm C-X}$) distances, and
systems’ thickness ($\Delta h$).

The optimized structural parameters of C4X2 (X = B or N) nanosheets are shown
in Table 1, where the distance labels are consistent with the ones defined in
figure 1. It can be observed that all the nanosystems functionalized with N
atoms keep the lattice constants almost unchanged when compared to the PD
ones. Additionally, for both stacking configurations of the C4N2, the
intraplanar bond angle ($\theta$) values are close to the $sp^{3}$
hybridization ones (109.47∘), leading to a thickness of $\approx$ 4.7 Å.
Nevertheless, the C4B2 nanosheet lattice parameters are slightly greater than
those of HD and FD systems, with $\theta$ close to the value of 120∘, i.e.,
the B atoms bonded to three adjacent C atoms present a $sp^{2}$-type
hybridization and hence we observed a smaller thickness of $\approx$ 4.2 Å as
compared to the N-functionalized structures.

We now discuss the stability of the C4X2 nanosheets. To study the thermal
stability of those structures, we computed the standard enthalpy of formation
$\Delta H_{f}^{0}$ using equation (2). Herein, we found positive $\Delta
H_{f}^{0}$ of 424, 365, 348, and 333 meV for AA′A′′-C4B2, ABC-C4B2,
AA′A′′-C4N2, and ABC-C4N2, respectively, displayed in Table 1, indicating
possible thermodynamic instability. However, the literature has reported 2D
materials synthesized by endothermic processes ($\Delta H_{f}^{0}>0$), such as
graphene, germanene, and silicene [33, 29]. Also, the $\Delta H_{f}^{0}$
values of C4X2 nanosheets are similar to the 300 meV/atom of PD with three
graphene layers at 0 GPa (Table 1), which was recently synthesized [8], and
slightly higher than NCCN that was theoretically studied to stabilize the 2D
diamond without any passivation [9].

The thermodynamic stability of the systems was also investigated by AIMD
simulations. The results exhibited a total energy small variation during 5 ps
at 300 K, as shown in figure 2, indicating that the structural integrity of
those systems is maintained at those conditions. At 1000 K, the same behavior
is observed for the C4N2 systems, while the C4B2 nanosheets presented some
broken bonds, suggesting some structural degradation.

Furthermore, the dynamic stability of the systems was investigated using the
phonon theory, in which a system is considered stable when its vibrational
spectrum contains only positive frequencies. The phonon dispersion curves of
C4B2 and C4N2 compounds, in both AA′A′′ and ABC stacking configurations, are
presented in figure 3. All spectra show 18 phonon branches, related to the 6
atoms present in the primitive cell. All those systems are dynamically stable
since there are only positive frequencies.

Figure 3: Phonon dispersion branches of (a) AA′A′′-C4B2, (b) ABC-C4B2, (c)
AA′A′′-C4N2, and (d) ABC-C4N2 along the main high-symmetry directions in the
BZ of the hexagonal lattice. Figure 4: Electronic band structures along the
main high-symmetry directions of the BZ and PDOS, obtained with the optB88-vdW
approach for the exchange-correlation energy: (a) AA′A′′-C4N2, (b) ABC-C4N2,
(c) AA′A′′-C4B2, and (d) ABC-C4B2. The PDOS on the C and X $s$-orbitals are
given in purple and blue, respectively, and on the C and X $p$-orbitals are
given in red and green, respectively. Ev represents the VBM. Energies and PDPS
are given in eV and states/eV, respectively.

Next, we computed the elastic constants of the systems using equation (1) to
verify their mechanical stability, according to the Born stability criteria
($C_{11}>0$ and $C_{12}<C_{11}$) [37]. Table 2 presents the elastic constants
$C_{11}$, $C_{12}$, and $C_{44}=\left(C_{11}-C_{12}\right)/2$, Young’s modulus
$\displaystyle Y^{\rm 2D}={\left(C_{11}^{2}-C_{12}^{2}\right)/C_{11}}$, and
the Poisson ratio $\displaystyle\nu={C_{12}/C_{11}}$ of C4X2 trilayers (X = B
or N), as well as those of several other 2D materials for comparison.

Table 2: Elastic constants $C_{11}$, $C_{12}$, and $C_{44}$, Young’s modulus $Y^{\rm 2D}$, Poisson ratio $\sigma$, formal density $\rho_{\rm 2D}$, longitudinal $v_{\rm LA}$ and transverse $v_{\rm TA}$ acoustic velocities of C4X2 (X = B or N), graphene, and other 2D diamonds. Elastic constants and Young’s modulus are given in N/m, Poisson ratio is dimensionless, formal density and velocities are given in $10^{-7}$ kg/m2 and km/s, respectively. The results with the ∗ symbols were obtained using the data from the table and the equations described in this paper.  System | $\\!\\!C_{11}$ | $\\!\\!C_{12}$ | $C_{44}$ | $Y^{\rm 2D}$ | $\sigma$ | $\rho_{\rm 2D}$ | $v_{\rm LA}$ | $v_{\rm TA}$
---|---|---|---|---|---|---|---|---
AA′A′′-C4N2 | 816 | 85 | 366 | 808 | 0.10 | 24.8 | 18.1 | 12.1
ABC-C4N2 | 777 | 82 | 348 | 769 | 0.11 | 24.5 | 17.8 | 11.9
AA′A′′-C4B2 | 627 | 88 | 270 | 615 | 0.14 | 18.9 | 18.2 | 11.9
ABC-C4B2 | 609 | 92 | 259 | 595 | 0.15 | 18.7 | 18.0 | 11.7
Graphene | 354555Reference [28] | 60${}^{\text{a}}$ | 147∗ | 340666Reference [38] \- experiments | 0.18777Reference [39] | 7.55${}^{\text{c}}$ | 21.6∗ | 13.9∗
HD | 474${}^{\text{c}}$ | 36${}^{\text{c}}$ | 219∗ | 471∗ | 0.08${}^{\text{c}}$ | 14.9${}^{\text{c}}$ | 17.8${}^{\text{c}}$ | 12.2${}^{\text{c}}$
ABC-HD | 718${}^{\text{c}}$ | 58${}^{\text{c}}$ | 330 | 713∗ | 0.08${}^{\text{c}}$ | 22.2${}^{\text{c}}$ | 18.0${}^{\text{c}}$ | 12.2${}^{\text{c}}$
FD | 485888Reference [15] | 49∗ | 218∗ | 480${}^{\text{d}}$ | 0.10∗ |  | 14.0999Reference [4] | 9.3${}^{\text{e}}$
NCCN | 568101010Reference [9] | 66${}^{\text{f}}$ | 243${}^{\text{f}}$ | 560∗ | 0.12∗ |  |  |
Diamond | 1079${}^{\text{f}}$ | 124${}^{\text{f}}$ | 578${}^{\text{f}}$ |  |  |  | 18.3${}^{\text{c}}$ | 12.4${}^{\text{c}}$

Accordingly, the C4X2 structures are mechanically stable since they satisfy
the Born criteria, agreeing with the phonon dispersion spectra shown in figure
3. The C4X2 nanosheets present high Young’s modulus values and characteristics
of isotropic systems since their Poisson ratio $\sigma$ values are lower than
0.5 [40, 41].

Additionally, we estimated the longitudinal and the transversal acoustic
velocities, given respectively by

$v_{\rm LA}=\left(\frac{C_{11}}{\rho_{\rm
2D}}\right)^{1/2}\\!\\!\quad\mbox{and}\quad v_{\rm
TA}=\left(\frac{C_{44}}{2\rho_{\rm 2D}}\right)^{1/2}\\!\\!,$ (3)

where $\rho_{\rm 2D}$ is the formal density, allowing comparison among
systems, independent of their thickness [4]. The velocity values, listed in
Table 2, suggest that the stiffness of the C4X2 systems is comparable with
that of the diamond.

Following, we studied the electronic band structures and the projected density
of states (PDOS) of the C4X2 systems, displayed in figure 4 (a)-(d). The
electronic band structures and PDOS of AA′A′′-C4N2 and ABC-C4N2 nanosheets,
presented in figures 4 (a) and (b), exhibit some differences, despite their
analogous PDOS, showing that the N $p$-orbitals dominate at valence band
maximum (VBM) while the conduction band minimum (CBM) are mostly characterized
by a mixture of $s$\- and $p$-orbitals of N and C atoms. Both systems present
the VBM around the $\Gamma$-point with a Mexican-hat dispersion, in which the
two peaks lie on the $\Gamma$-K and $\Gamma$-M lines. The height of the
Mexican-hat band at $\Gamma$ point is 0.01 and 0.001 eV for AA′A′′-C4N2 and
ABC-C4N2, respectively. However, the CBM of ABC-C4N2 is well defined at the
M-valley, while in AA′A′′-C4N2 although the CBM is located at the M-valley,
the energy of the K-point is very close to the M-point one. On the other hand,
the AA′A′′-C4B2 and ABC-C4B2 nanosheets have very similar band structures, as
shown in figures 4 (c) and (d). Both systems present direct band gaps, where
the doubly degenerated VBM and the CBM are located at the $\Gamma$-point, in
which the B $p$-orbitals dominate at the CBM and the VBM is described by a
combination of B $p$-orbitals and C $p$-orbitals.

Table 3: Electronic band gap values of C4X2 (X = B or N) nanosheets (in eV), obtained with two different approximations for the exchange-correlation functional: optB88-vdW ($E_{g}^{\rm wdW}$) [22] and HSE ($E_{g}^{\rm HSE}$) [23]. The PD band gap value is also displayed. The electron ($m^{*}_{e}$) and hole ($m^{*}_{h}$) effective masses, in $m_{0}$ units, obtained with the optB88-vdW approach are also shown. The VBM of C4B2 systems are doubly degenerated at $\Gamma$ and have two values for hole carrier. The CBM of C4N2 systems displays two effective masses around the M-valley, $m_{e}^{*M\rightarrow\Gamma}$ and $m_{e}^{*M\rightarrow K}$.  System | $E_{g}^{\rm vdW}$ | $E_{g}^{\rm PBE}$ | $E_{g}^{\rm HSE}$ | Band gap | $m^{*}_{h}$ | $m^{*}_{e}$ | $m_{e}^{*M\rightarrow\Gamma}$ | $m_{e}^{*M\rightarrow K}$
---|---|---|---|---|---|---|---|---
AA′A′′-C4N2 | 4.40 |  | 5.56 | indirect | 6.23 |  | 2.94 | 0.30
ABC-C4N2 | 4.13 |  | 5.42 | indirect | 16.77 |  | 2.61 | 0.47
AA′A′′-C4B2 | 0.53 |  | 1.64 | direct | 0.34 (0.68) | 1.22 |  |
ABC-C4B2 | 0.84 |  | 1.97 | direct | 0.36 (0.80) | 1.30 |  |
HD111111Reference [15] |  | 3.32 |  | direct | 0.21 (0.58) | 1.11 |  |
FD${}^{\text{a}}$ |  | 4.04 |  | direct | 0.37 (1.13) | 0.55 |  |
PD121212Reference [8]. The ($\overline{2}110$)-oriented h-diamane with 3 graphene layers. |  |  | 2.70 | indirect |  |  |  |

As discussed in the introduction, the 2D diamond systems present non-zero band
gaps with characteristics that depend on several factors, such as doping with
different functional groups. Herein, we are working with B and N atoms as
X-doping elements, which the B atom belongs to group-III elements of the
periodic table, with a $2s^{2}2p^{1}$ valence electronic configuration, and
the N atom belongs to group-V elements, with a $2s^{2}2p^{3}$ valence
electronic configuration. As a result, we found a wider indirect band gap for
the C4N2 nanosheets, and a narrower direct band gap for the C4B2 systems when
compared to PD. Table 3 displays the band gap values of the C4X2 nanosheets
obtained with the optB88-vdW [22] ($E_{g}^{\rm vdW}$) and the hybrid HSE [23]
($E_{g}^{\rm HSE}$) functional approaches for the exchange-correlation energy.
For comparison, we also included the HD and FD band gap values acquired with
PBE functional [15] and of the three-layer graphene PD obtained with hybrid
HSE functional [8]. The band gap width of the latter is 2.70 eV, while for the
AA′A′′-C4N2, ABC-C4N2, AA′A′′-C4B2, and ABC-C4B2 functionalized compounds they
are 5.56, 5.42, 1.64 eV, and 1.97 eV, respectively, using the same exchange-
correlation functional. The C4N2 nanosheets have band gap width values,
obtained with the optB88-vdW approach, similar to those of the HD and FD
systems obtained with the PBE approximation.

Furthermore, since effective masses ($m^{*}$) can be used to investigate
electronic transport under the influence of electric fields or carrier
gradients, we estimated them by fitting parabolic functions to the CBM and VBM
via the formula as follows:

$\frac{1}{m^{*}}=\frac{1}{\hbar^{2}}\Bigg{|}\frac{\partial^{2}E(k)}{\partial
k^{2}}\Bigg{|}$ (4)

where $E(k)$ and $k$ are the energy and the wave vector of the CBM or VBM. The
values of the effective masses depend on the curvature radius of the
electronic band around the band-edge position, i.e., a larger curvature radius
implies a heavier effective mass.

Table 4: Summary of the qualitative properties and potential applications of C4X2 (X = B or N) systems. The column indicates dynamic (DS), mechanic (MS), and thermodynamic (TS) stability, Young’s modulus $Y^{\rm 2D}$, and effective masses ($m^{*}$). System | DS | MS | TS (300 K) | TS (1000 K) | $Y^{\rm 2D}$ | Stiffness | Band gap | $m^{*}$ | Potential applications
---|---|---|---|---|---|---|---|---|---
C4N2 | Yes | Yes | Yes | Yes | high | high | indirect | heavier | host material for
|  |  |  |  |  |  |  |  | single photon emission
|  |  |  |  |  |  |  |  | quantum computing/biosensors
C4B2 | Yes | Yes | Yes | No | high | high | direct | lighter | nano-optics/nanoelectronics

The electron ($m^{*}_{e}$) and hole ($m^{*}_{h}$) effective masses, in $m_{0}$
units, calculated with the optB88-vdW approach, are presented in table 3. In
both stacking structures of the C4B2 nanosheets, the effective masses of
electrons and holes are comparable to the ones of HD and FD systems, which
present extraordinary carriers mobility [15]. Furthermore, these estimated
effective masses are similar to the $m_{e}^{*}=1.06\,m_{0}$ and
$m_{h}^{*}=0.59\,m_{0}$ of silicon at a temperature of 4K [42]. Regarding C4N2
nanosheets, the $m^{*}_{h}$ is the magnitude of the effective mass at
$\Gamma$-point, i.e., we fitted a parabolic function considering the minimum
located at the $\Gamma$-point [43]. Accordingly, the $m^{*}_{h}$ depends on
the height of the Mexican-hat band and the radius centered at $\Gamma$-point
around band edges [43]. On the other hand, the $m^{*}_{e}$ displays two
effective masses around the M-valley. $m_{e}^{*M\rightarrow\Gamma}$ shows a
high electron effective mass, much higher than $m_{e}^{*M\rightarrow K}$.
$m_{e}^{*M\rightarrow\Gamma}$ is five times larger than $m_{e}^{*M\rightarrow
K}$ for ABC-C4N2 and nine times larger for AA′A′′-C4N2.

Considering the stable structures presented previously and their physical
properties, it is interesting to explore their potential applications. Further
investigations could explore the applicability of C4N2 and C4B2 structures as
building blocks to build 2D/3D systems, such as van der Waals
heterostructures, with different properties [44, 10, 45]. Moreover, wide band
gap materials, such as hexagonal boron nitride (h-BN), serve as a substrate
for high-mobility 2D devices [46], a host material for single-photon emitter
defect-centers for quantum computing and biosensors [4], etc. Therefore, the
C4N2 nanosheets seem appropriate for these kinds of applications. Finally, the
C4B2 nanosheets presented direct band gaps, in the 1.6 - 2.0 eV range, being
more favorable for optoelectronics applications than C4N2 ones, which have
indirect band gaps in the 5.4 - 5.6 eV range [46]. In particular, the small
effective masses and high elastic modulus of C4B2 systems may contribute to
high electron mobility [15], being suitable to be applied in photovoltaic
cells. Table 4 summarizes their properties and the respective potential
applications.

In conclusion, we performed an ab-initio investigation on the structural,
thermodynamic, dynamic, elastic, and electronic properties of C4X2 (X = B or
N) systems. According to AIMD simulations, phonon calculations, and the Born
stability criteria, all the nanosheets are thermodynamically, dynamically, and
mechanically stable. Furthermore, the systems presented standard enthalpy of
formation close to the recently synthesized pristine 2D diamond composed of
three graphene layers. Elastic properties indicated that those nanosheets
possess a high Young’s modulus values and characteristics of isotropic
systems, and the estimated longitudinal and transversal acoustic velocities
revealed that their stiffness is comparable with that of the diamond. Finally,
the systems’ electronic properties presented some differences, in which C4N2
structures exhibited wide indirect band gaps and heavier effective masses,
while the C4B2 ones had narrow direct band gaps and lighter effective masses.
These results provide chemical routes to tune the electronic properties of 2D
diamonds by doping them for specific applications, such as optoelectronic
devices.

###### Acknowledgements.

Brazilian Federal Government Agencies CAPES (Grants 88882.332907/2019-01 and
88887.371193/2019-00), CNPq (Grants 314884/2021-1, 302800/2022-0, and
150595/2023-9) and FAPESP (Grant 22/10095-8) partially supported this
investigation. The authors acknowledge the National Laboratory for Scientific
Computing (LNCC/MCTI, Brazil) for providing HPC resources of the Santos Dumont
supercomputer, Centro Nacional de Processamento de Alto Desempenho em São
Paulo (CENAPAD-SP, Brazil) and SNIC-UPPMAX, SNIC-HPC2N, and SNIC-NSC centers
under the Swedish National Infrastructure for Computing (SNIC) resources.

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# Emergence of quasiperiodic behavior in transport and hybridization
properties of clean lattice systems

Cecilie Glittum  Antonio Štrkalj  Claudio Castelnovo  T.C.M. Group, Cavendish
Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### Abstract

Quasiperiodic behaviour is mostly known to occur in systems with enforced
quasiperiodicity or randomness, in either the lattice structure or the
potential, as well as in periodically driven systems. Here, we present instead
a rarer setting where quasiperiodic behaviour emerges in clean, non-driven
lattice systems. We illustrate this through two examples of experimental
relevance, namely an infinite tight-binding chain with a gated segment, and a
hopping particle coupled to static Ising degrees of freedom. We show how the
quasiperiodic behaviour manifests in the number of states that are localised
by the geometry of the system, with corresponding effects on transport and
hybridisation properties.

##  I  Introduction

Some of the most striking discoveries in physics occur when simple systems are
found to exhibit unexpectedly complex behaviour. These are often prime
examples of the beauty and elegance of mathematical modelling that succeeds in
describing the world around us. Examples include the chaotic motion of the
double pendulum [1, 2] and the quantum Hall effect of a 2D electron gas in a
strong applied field [3].

Without claim of drawing a comparison in importance, we present here a simple
result that fits well in this category. In one of its simplest forms, our
finding shows that the number of states localised 111The reader should bear in
mind here, for reference, the concept of Wannier-Stark localisation [35, 36,
37]. on a gated portion of an infinite tight-binding chain forms a
_quasiperiodic_ sequence in the number of gated sites. Equivalently, the
number of resonance peaks in the conduction along the chain also forms a
quasiperiodic sequence. This happens for both negative and positive gate
potential, within the bounds of the tight-binding band. We provide both an
exact solution to the problem as well as a simple, elegant and intuitive
(albeit only effective) derivation of the same solution.

Our results are of direct relevance to several experimental settings –
thinking for example about cold atoms [5, 6], trapped ions [7, 8], quantum dot
arrays [9], superconducting cubits [10] and other quantum simulators [11] –
and can be readily verified in a laboratory. Moreover, in addition to this
simplest formulation of the problem, we show that a similar phenomenology is
expected to play a role in other settings, for instance when particles hopping
on a lattice are coupled to underlying spin degrees of freedom, as in the
Falicov-Kimball model [12] and in systems that exhibit disorder-free
localisation [13].

Quasiperiodic systems have received significant attention of late (see for
instance Refs. 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28). In
most cases however the quasiperiodicity is built into the system, e.g., via
quasiperiodic modulation of on-site potentials and hopping amplitudes, or
induced by incommensurate periodic driving; then, properties deriving from it
are studied. Examples where quasiperiodicity is not seeded but rather emerges
from periodic constituents are rare (most notably this was uncovered to result
from the interplay of lattice filling fractions and interactions in Refs. 29,
30; however, see also Ref. 31 for an example of a quasicrystalline potential
emerging from collective light scattering). In our setting, the quasiperiodic
behaviour emerges in a clean system made of non-driven periodic components –
something that does not happen often in nature. We envision that the simple
mechanism discussed in this work is likely to operate in interesting ways in
other condensed matter systems, and possibly in higher dimensions as well.

##  II  Gated chain

Figure 1:  (a) Illustration of the gated chain. The solid black dots
illustrate the infinite leads, at potential $V=0$. The solid pink dots
illustrate the gated segment of length $L$ at potential $V_{0}$, here for
$L=5$. (b) Number of states localised in the gated region $N_{\mathrm{loc}}$
as a function of its length $L$. We subtracted the linear slope $(1-\alpha)L$
to emphasise the quasiperiodic nature of the fluctuations following from
sampling at integer $L$. Numerically obtained data from conductance
simulations (black dots) and IPR (blue squares) overlap and agree perfectly
with the analytical prediction in Eq. (4) (black line).  Figure 2:  (a)
Dispersion of the infinite leads (black solid line) and of the gated segment
in the $L\rightarrow\infty$ limit (pink dashed line). The two dispersions
overlap for $k\in[-\alpha\pi,\alpha\pi]$. We also show the energies of an
isolated gated segment with $L=9$ plotted as a function of $q_{n}=\pi n/L$;
$n=1,2,\ldots,L$ (pink solid dots). Counting the number of discretised
energies falling outside of the bandwidth of the leads gives the number of
states localised on the gated segment, which in this case is $4$ states. (b)
IPR for a gated segment of length $L=50$ with $V_{0}=0.33$ (left) and
$V_{0}=1.45$ (right), comparing two different lead sizes,
$L_{\mathrm{leads}}=5\cdot 10^{3}$ and $L_{\mathrm{leads}}=10^{4}$. States
with $E>2$ (highlighted by a green rectangle) are localised inside the gated
segment and their IPR is independent of the lead size. (c) Conductance through
the whole system as a function of energy of the incident particle. Here we
used $L=9$ and $V_{0}=1.45$. We remind that potentials and energies are
expressed in units of the hopping amplitude $t=1$.

We study a tight-binding chain of lattice spacing $a=1$ and hopping amplitude
$t=1$, with a segment of length $L$ gated at potential, i.e., on-site energy,
$V_{0}$. The rest of the system (i.e., the left and right leads) is kept at
the reference potential $V=0$. The Hamiltonian of the system is given by

$H=-t\sum_{i}\left[c^{\dagger}_{i+1}c_{i}+\mathrm{h.c.}\right]+\sum_{i}V_{i}c^{\dagger}_{i}c_{i},$
(1)

where $V_{i}$ is finite on the gated segment and vanishes on the leads. The
system is illustrated in Fig. 1(a).

Due to the mismatch in on-site potential between the gated segment and the
leads, some states localise on the gated segment, while other states are
extended over the whole system. Our goal is to investigate the nature of these
localised states, and more precisely how their number changes with the system
parameters $L$ and $V_{0}$. As we detail below, the number of states Wannier-
Stark-localised on the gated segment forms a _quasiperiodic_ sequence as a
function of the discrete length $L$ of the segment; see Fig. 1(b).

By choosing an appropriate ansatz for the wave function, we look for
eigenstates of the Schrödinger equation that are exponentially decaying on the
leads (i.e., localised on the gated segment); see App. A for details and App.
F for examples of wave function profiles. We find that the number of such
states corresponds to the number of solutions of either of the following
equations:

$\displaystyle-e^{\operatorname{arccosh}\left[V_{0}/2-\cos k\right]}$
$\displaystyle=$
$\displaystyle\frac{\cos\left[k-k\frac{L+1}{2}\right]}{\cos\left[-k\frac{L+1}{2}\right]},$
(2) $\displaystyle-e^{\operatorname{arccosh}\left[V_{0}/2-\cos k\right]}$
$\displaystyle=$
$\displaystyle\frac{\sin\left[k-k\frac{L+1}{2}\right]}{\sin\left[-k\frac{L+1}{2}\right]}\,.$
(3)

The task can be conveniently re-cast into counting the number of
$q_{n}=n\pi/L$ with $n\in\\{1,\dots,L\\}$ larger than
$\alpha\pi\equiv\arccos{\left(\frac{V_{0}}{2}-1\right)}$, which is the point
in the Brillouin zone where the dispersion of the isolated infinite gated
segment intersects the band edge of the isolated leads; see Fig. 2(a). Knowing
$q_{n}$ and $\alpha$, one can show that the number of states localised on the
gated segment is given by

$N_{\rm loc}=L-\mathrm{floor}\left(\alpha L\right)\,,$ (4)

which is a linearly growing function of $L$ with slope $1-\alpha$, and it
exhibits quasiperiodic fluctuations whenever $\alpha$ is irrational.

We note in passing that the same result can be obtained using Levinson’s
theorem [32], appropriately modified for a lattice model, from the phase
difference between the states at the bottom and top of the band (see App. B).

The quasiperiodicity can be distinctly seen in Fig. 1(b), where we also verify
the perfect agreement between the analytical result, Eq. (4), and the
numerical calculation of $N_{\rm loc}$ using both the inverse participation
ratio (IPR) and transport measurements. Albeit simple, we find this result
surprising since the leads and the gated segment are periodic, and there is no
additional quasiperiodicity or randomness present in the system.

###  II.1 Simple counting argument

It is tempting to interpret the above $q_{n}$ as the wave vector of a tight-
binding chain of length $L$ with open boundary conditions. The argument
appears then to be counting the number of discrete eigenenergies of the gated
segment, whose dispersion in the limit of infinite length is
$E_{\mathrm{gated}}(k)=V_{0}-2\cos k$, that fall outside the continuous band
of the leads, of dispersion $E_{\mathrm{leads}}(k)=-2\cos k$.

The two dispersions intersect at $k=\pm\alpha\pi$, as illustrated in Fig.
2(a), and one can see that the discrete energies of the gated segment fall
outside the band of the leads if $q_{n}>\alpha\pi$. Recalling that
$n=1,2,\ldots,L$, one can straightforwardly count such states and obtain
$L-\mathrm{floor}\left(\alpha L\right)$ for the number of states localised on
the gated segment, which coincides with the exact solution, Eq. (4).

This is however only an intuitive, and somewhat incorrect, picture which
happens coincidentally to give the correct answer. In the exact calculation,
the $q_{n}$’s are simply a tool to count the number of solutions and have no
direct physical interpretation.

###  II.2 Probing localised states

One way to probe the localised states is by computing the IPR of the
eigenstates of the Hamiltonian (1),

$\mathrm{IPR}(E_{n})=\frac{\sum_{j}|\psi_{j}(E_{n})|^{4}}{\sum_{j}|\psi_{j}(E_{n})|^{2}}\,.$
(5)

The IPR is known to scale as (i) $\mathcal{O}(1/L_{\mathrm{tot}})$ for states
extended over the whole system of length
$L_{\mathrm{tot}}=2L_{\mathrm{leads}}+L$, where $L_{\mathrm{leads}}$ is the
number of sites in a single lead; (ii) $\mathcal{O}(1/(2L_{\mathrm{leads}}))$
for states localised on the leads (when one allows for a small matrix element
due to tunnelling across the gated segment); and (iii) $\mathcal{O}(1/L)$ for
states localised only on the gated segment. We use exact diagonalisation and
compare the results for two different sizes of leads, $5\cdot 10^{3}$ and
$10^{4}$ sites, and for different gate potentials. The results for a gated
segment with $L=50$ sites are shown in Fig. 2(b), for $V_{0}=0.33$ and
$V_{0}=1.45$ (recall that we work in units where $t=1$). We see that all the
states with energy $E>2$ have an IPR that is independent of the size of the
leads, meaning that these states are localised on the gated segment. The
aforementioned states cannot hybridise with the states belonging to the leads
as their energy falls outside their bandwidth. On the other hand, states
falling within the bandwidth of the leads hybridise and delocalise.

The steplike behaviour seen in Fig. 2(b) is a numerical artefact of the
diagonalisation algorithm. When the energy of the eigenstates is smaller than
$-2+V_{0}$, the particle cannot propagate across the barrier; however, a small
tunnelling amplitude exists for finite values of $L$ such that the eigenstates
found by diagonalisation are even and odd superpositions of the states on the
left and on the right lead, and the IPR is approximately $1/(2L_{\rm leads})$
(which in the figure is indistinguishable from $1/(2L_{\rm leads}+L)$, the IPR
of delocalised states). As we continue to lower the energy of the eigenstates
however, the tunnelling amplitude is correspondingly suppressed; when it falls
below numerical accuracy, the diagonalisation routine is no longer able to
find even and odd superpositions of states on the left and on the right lead,
but rather sees the left and right lead states as eigenstates of the system.
This results in an IPR $\sim 1/L_{\rm leads}$ and in the step increase
observed when we consider the lowest energy eigenstates.

Another way to directly probe the aforementioned localised states is via
energy-dependent transport, i.e., via the conductance

$G(E)=\frac{e^{2}}{h}\mathcal{T}(E)\mathcal{T}^{*}(E)\,,$ (6)

where $e$ is the charge of the particle, $E$ is its incident energy, and
$\mathcal{T}$ is the transmission coefficient. The above conductance is
observed to have an oscillatory behavior in energy, with maxima reaching the
value $e^{2}/h$, and the distance between them determined by the length $L$
(see App. D). In Fig. 2(c) we show the numerical result from the simulation of
a system consisting of two identical and infinite leads and a gated segment of
$L=9$ sites (see App. C for a plot of the conductance as a function of $V_{0}$
for fixed $L$). The peaks in conductance occur once $E$ is in resonance with
the energies of the gated segment that hybridise with the leads, i.e., the
energies that lie within the bandwidth of the leads. By counting the number of
peaks $N_{\rm peaks}$, whose maxima reach the value $e^{2}/h$, we can extract
the number of states that localise on the gated segment as $N_{\rm
loc}=L-N_{\rm peaks}$. The result for the gated chain discussed before is
shown in Fig. 1(b). We observe perfect agreement with $N_{\rm loc}$ obtained
by IPR, as well as with the analytical prediction (4). Interestingly, a simple
analytical calculation of conductance which assumes that the leads do not
affect the dispersion of the gated segment gives the same expression for
$N_{\rm loc}$ as in Eq. (4); see App. D.

##  III  Ferromagnetic segment in an antiferromagnetic chain

Similar quasiperiodic behaviour occurs also in related systems of relevance to
other experimental settings. Consider for instance the case of an Ising chain
where $+$ and $-$ correspond to an on-site energy $\pm W$ for a tight-binding
particle along the same chain, with hopping amplitude $t=1$ (where without
loss of generality we set $W>0$). Specifically, take an infinite
antiferromagnetic (AFM) chain with a ferromagnetic (FM) insertion $++\ldots+$
of length $L$ (see Fig. 3(a) for a precise illustration of how we define the
inserted FM segment).

Figure 3:  (a) Left: Illustration of the FM segment in an AFM chain. Black
points and arrows illustrate the infinite AFM leads and pink ones illustrate
the FM segment of length $L$, here for $L=3$. Right: The AFM and FM
dispersions for $W=0.5$. The FM band overlaps with the lower AFM band in the
region $k\in[-\gamma\pi,\gamma\pi]$ (for $W\leq 1$) and with the upper AFM
band in the region $k\in[-\beta\pi,-\pi/2]$ and $k\in[\pi/2,\beta\pi]$. (b)
Number of states localized in the gated region as a function of its length for
$W=0.1$. We subtracted the linear slope $(1-\beta+1/2-\gamma)L$ to emphasise
the quasiperiodic nature of the fluctuations following from sampling at
integer $L$. We observe a perfect agreement between numerical data obtained
from conductance (black dots), IPR (blue squares) and the analytical
prediction in Eq. (9).

The dispersion of the AFM leads is given by

$E_{\mathrm{AFM}}(k)=\pm\sqrt{W^{2}+2\left(1+\cos k\right)},$ (7)

which exhibits a gap near zero energy, in contrast with the case considered
earlier. The dispersion of the FM segment is instead given by the usual
relation

$E_{\mathrm{FM}}(k)=W-2\cos k.$ (8)

Both dispersions are illustrated in Fig. 3(a). Note that the FM segment
overlaps with both bands of the leads for $W\leq 1$ and only with the upper
band for $W>1$. The FM band overlaps with the lower AFM band for
$k\in[-\gamma\pi,\gamma\pi]$ (for $W\leq 1$), where
$\gamma\pi\equiv\arccos(W)$, and with the upper AFM band for
$k\in\left\\{[-\beta\pi,\pi/2],[\pi/2,\beta\pi]\right\\}$ with
$\beta\pi\equiv\arccos\left[\frac{1}{2}\left(W-\sqrt{W^{2}+4}\right)\right]$
(the values of $\beta$ and $\gamma$ are visually represented in the right
panel of Fig. 3(a) for added clarity).

All states falling outside of the AFM bands correspond to states localised on
the FM domain, falling off exponentially when moving into the AFM leads (see
App. F for examples of wave function profiles). Using a corresponding ansatz
for the lattice wave function, as we did for the gated segment earlier, one
can explicitly find the localised eigenstates of the system (see App. E) by
solving equations similar to Eqs. (2) and (3). As shown in App. E, the number
of states localised on the FM segment with energies above the upper AFM band
can be found by counting the number of $q_{n}=n\pi/L$ with
$n\in\\{1,\dots,L\\}$ larger than $\beta\pi$, i.e., $L-\mathrm{floor}(\beta
L)$. Similarly, the number of states localised on the FM segment with energies
in the gap of the AFM dispersion can be found by finding the number of
$q^{\infty}_{n}=n\pi/(L+2)<\pi/2$ and if $W\leq 1$ then subtracting the number
of $q_{n}=n\pi/L\geq\gamma\pi$. This leads to
$\mathrm{ceil}\left(L/2\right)-\mathrm{floor}\left(\gamma L\right)$ localised
states in the gap for $W\leq 1$ and $\mathrm{ceil}\left(L/2\right)$ for $W>1$.
The total number of states localised on the FM domain is thus given by (App.
E) 222A similar simple counting argument can be used here as we did in the
case of the gated segment. However, it seems only to be able to reproduce the
exact expression in Eq. (9) up to an additive unit constant.

$\displaystyle N^{\mathrm{FM}}_{\mathrm{loc}}=$
$\displaystyle\begin{cases}L-\mathrm{floor}\left(\beta
L\right)+\mathrm{ceil}\left(L/2\right)-\mathrm{floor}\left(\gamma
L\right),&W\leq 1\\\ L-\mathrm{floor}\left(\beta
L\right)+\mathrm{ceil}\left(L/2\right),&W>1\,.\end{cases}$ (9)

To confirm the above formula, we numerically extract the number of states
localised on the FM domain, $N^{\mathrm{FM}}_{\rm loc}$, using both IPR and
conductance. The results for $N^{\mathrm{FM}}_{\rm loc}$ as a function of size
of the FM domain $L$ are shown in Fig. 3 (b), where we observe perfect
agreement with the analytical prediction, Eq. (9).

As an aside, we note that it is possible to numerically compute the number of
localised states on the FM domain more efficiently by counting the number of
sign changes in the Sturm sequence obtained by relating the subdeterminants of
the Hamiltonian [34]. This enables one to treat larger system sizes at a
relatively low computational cost compared to full diagonalisation.

##  IV Conclusions

In this paper, we demonstrated how quasiperiodic behaviour can emerge in clean
periodic systems without driving, borne out of a mechanism similar to the one
that gives rise to Wannier-Stark localisation [35, 36, 37]. For instance, we
showed that the number of localised states on a gated segment of a tight-
binding chain (equivalently, the number of peaks in the conductivity along the
chain) is a quasiperiodic function of the length of the segment. A similar
behaviour arises when the tight-binding particle is instead subject to on-site
coupling to static Ising degrees of freedom, which form an antiferromagnetic
chain with a ferromagnetic segment of finite length inserted into it. We
provide both an exact solution as well as an intuitive effective description
of the phenomenon, and we discuss how it is directly relevant to various
settings, from gated quantum dot arrays to systems described by the Falicov-
Kimball model. Similar behaviour can be expected in general if a finite
segment is inserted into spatially translationally invariant leads.
Quasiperiodicity should arise when the dispersion of the inserted segment
intersects the band edge of the leads at some $k=\alpha\pi$, with $\alpha$
irrational.

(We note in passing that when both gating and coupling to an underlying spin
pattern are present, additional features may appear in the form of states
localised at the boundaries of the segment.)

The mechanism discussed in our work is not inherently limited to 1D
noninteracting systems and the experimental platforms mentioned in the
introduction. Similar mechanisms are expected to operate for example in
strongly interacting systems, e.g., quantum wires described by Luttinger
liquid theory, that are coupled to leads of finite bandwidth [38, 39, 40, 41].
They should also be relevant to two- and higher-dimensional gated systems. One
can trivially see it if one uses a separable potential:
$V(x,y)=V_{0}[\Theta(x)+\Theta(y)]$, where $\Theta$ is a function that takes
the value $1$ on a segment of length $L$ and vanishes elsewhere in the system.
This potential results in an infinite cross pattern in 2D (of potential
$V_{0}$), with a raised square at its core (of potential $2V_{0}$ and size
$L\times L$). The exact analytical solution follows directly from the 1D case,
and we verified numerically that the agreement is excellent already for $L$
and $L_{\rm leads}$ of the order of a few tens of sites. The case of a simple
square potential, where $V=V_{0}$ only on a region of size $L\times L$ and
$V=0$ elsewhere, is less straightforward and attempting an analytical solution
is beyond the scope of the present work. However, one would expect – and
indeed numerical simulations suggest – that the quasiperiodic behaviour of the
number of localised states survives in that case too; the numerical results
are unfortunately too limited in accessible system sizes to be conclusive on
their own.

On a more speculative note, one may wonder what happens when several FM
insertions of different lengths are present in the AFM Ising chain that
determines the on-site potential in the Falicov-Kimball-inspired model
presented in the main text. Based on our results, one would expect the system
to exhibit a quasiperiodically distributed number of quasilocalised states
that weakly interact with one another via the exponentially decaying tails of
their wave functions. The effects of such states on transport as well as on
the many-body properties of the system are interesting directions for future
work.

## Acknowledgements

We would like to thank Olav F. Syljuåsen and Attila Szabó for helpful
discussions, and John Chalker for suggesting the connection with Levinson’s
theorem. C.G. was supported by the Aker Scholarship. A.Š. acknowledges
financial support from the Swiss National Science Foundation (Grant No.
199969). This work was supported in part by the Engineering and Physical
Sciences Research Council (EPSRC) grants No. EP/P034616/1, No. EP/T028580/1
and No. EP/V062654/1.

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##  Appendix A  Analytical derivation of $N_{\mathrm{loc}}$
(direct calculation)

We study an infinite tight-binding chain with constant hopping amplitude $t=1$
and on-site potential

$V_{n}=\begin{cases}0,&\quad n\leq 0\\\ V_{0},&\quad 1\leq n\leq L\\\ 0,&\quad
n\geq L+1,\end{cases}$ (10)

where $n$ labels the sites and goes from $-\infty$ to $\infty$.

By solving the Schrödinger equation analytically, we derive an expression for
the number of states localised on the gated segment. There are two bulk
equations, one for the leads and one for the gated segment, given by

$\displaystyle-\psi_{n-1}-\psi_{n+1}$ $\displaystyle=$ $\displaystyle
E\psi_{n},$ (11) $\displaystyle-\psi_{n-1}+V_{0}\psi_{n}-\psi_{n+1}$
$\displaystyle=$ $\displaystyle E\psi_{n}.$ (12)

Using the ansatz $\psi_{n}=Ae^{ikn}$ separately for the leads and the gated
segment, we obtain the dispersions

$\displaystyle\begin{split}E_{\mathrm{gated}}(k)&=V_{0}-2\cos k,\\\
E_{\mathrm{leads}}(k)&=-2\cos k\,.\end{split}$ (13)

Being interested in studying the states which localise on the gated segment
and decay exponentially into the leads, we construct the following ansatz,

$\psi_{n}=\begin{cases}\hskip 8.0ptA\,e^{-(p+i\pi)(L+1)/2}e^{(p+i\pi)n},&\quad
n\leq 0\\\ \hskip
8.0ptB\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[kn-k(L+1)/2\right],&\quad 1\leq
n\leq L\\\ \pm A\,e^{(p+i\pi)(L+1)/2}e^{-(p+i\pi)n},&\quad n\geq
L+1,\end{cases}$ (14)

where we have taken advantage of the fact that the potential is symmetric
about the middle of the gated region, meaning that the eigenstates must be
symmetric or antisymmetric about $(L+1)/2$. Note that $(L+1)/2$ is not
restricted to integer values and the following derivation therefore holds
irrespective of $L$ being even or odd. Note also that there are only two
possible choices for the oscillatory phase $\varphi$ of the wave function in
the leads that give real energies: $0$ and $\pi$. In our case, all the
possible localised states are captured by $\varphi=\pi$ and we will thus only
consider this case here ($\varphi=0$ would only give $E<-2$). Having
explicitly accounted for the oscillating part of the wave function in the
leads, we consider only real $p>0$. The state in Eq. (14) has two unknowns:
$k$ and the relation between $A$ and $B$ ($p$ is given by requiring
$E_{\mathrm{leads}}(p)=2\cosh p=E_{\mathrm{gated}}(k)$; namely,
$p(k)=\operatorname{arccosh}\left[V_{0}/2-\cos k\right]$). To find these
unknowns, we have to impose the boundary equations which couple the leads and
the gated segment. These are in general four equations (two equations for each
boundary), but since we have already imposed the exchange symmetry of the wave
function, we only need to consider the equations for one of the boundaries.
The boundary equations for the left boundary read

$\displaystyle-\psi_{-1}-\psi_{1}$ $\displaystyle=E\psi_{0},$ (15)
$\displaystyle-\psi_{0}+V_{0}\psi_{1}-\psi_{2}$ $\displaystyle=E\psi_{1}\,.$
(16)

Inserting the ansatz Eq. (14) into Eqs. (15) and (16), we arrive at the
following equation for $k$:

$-e^{p(k)}=\frac{\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[k-k\frac{L+1}{2}\right]}{\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[-k\frac{L+1}{2}\right]},$
(17)

with $p(k)=\operatorname{arccosh}\left[V_{0}/2-\cos k\right]$.

This equation in general has several discrete solutions $k_{i}$, and each
$k_{i}$ corresponds to a state localised on the gated segment as long as
$p(k_{i})>0$. Finding an explicit expression for the solutions $k_{i}$ is
difficult, but we can count the number of solutions by considering the
behaviour of the functions on the two sides of the equation.

The left-hand side

$\mathrm{LHS}(k)=-e^{\operatorname{arccosh}\left[V_{0}/2-\cos k\right]}$ (18)

is a function starting at $\mathrm{LHS}(k)=-1$, for
$k=\arccos{\left(V_{0}/2-1\right)}\equiv\alpha\pi$, and decreasing
monotonically as $k\to\pi$; see Fig. 4. For convenience, we define the
starting point of $\mathrm{LHS}$ as a constant function

$C(k)=-1.$ (19)

The right-hand side (of which we have one for the symmetric eigenstates and
another for the antisymmetric eigenstates)

$\mathrm{RHS}(k)=\frac{\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[k-k\frac{L+1}{2}\right]}{\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[-k\frac{L+1}{2}\right]}$
(20)

is a function with derivative $\mathrm{RHS}^{\prime}(k)\geq 0$ for $k\geq 0$
(and $\mathrm{RHS}^{\prime}(k)=0$ only for $k=0,\pi$).

The functions $\mathrm{LHS}(k)$ and $\mathrm{RHS}(k)$ are illustrated in Fig.
4, where we plot $\mathrm{RHS}(k)$, which is independent of $V_{0}$, for
$L=12$, together with $\mathrm{LHS}(k)$ for a range of different values of
$V_{0}$.

Figure 4: The pink lines illustrate $\mathrm{LHS}(k)$ for different $V_{0}$.
The blue lines represent $\mathrm{RHS}(k)$ for $L=12$. The black dashed line
shows $C(k)=-1$. The blue lines (RHS) cross this line for $q_{n}=\pi/Ln$. The
pink lines touch $C(k)$ for $\alpha\pi$.

To find an analytical expression for the number of solutions $k_{i}$ for an
arbitrary length $L$ of the gated segment, let us start from the limit
$V_{0}=0$ and increase $V_{0}$ continuously to see how the number of solutions
increases. For $V_{0}=0$ there are no solutions ($\alpha=1$ and
$k=\pi=\alpha\pi$ solves the equation, but this solution is delocalised as
$p(\alpha\pi)=0$ with $E_{\mathrm{gated}}(\alpha\pi)=2$). As one then lets
$V_{0}>0$, one immediately finds an allowed solution for the $\mathrm{RHS}$
line crossing $C(k)$ at $\pi$, but for a $k_{i}$ slightly smaller than $\pi$
($k_{i}$ decreases continuously from $\pi$ as $V_{0}$ increases, approaching
the divergence of the corresponding RHS line; see Fig. 4). We then have only
one solution until $V_{0}$ has increased enough for the next RHS line to
fulfil

$C(\alpha\pi)=\mathrm{RHS}(\alpha\pi)\,.$ (21)

In fact we first get a new allowed solution when $V_{0}$ is infinitesimally
larger than the value solving Eq. (21), as $k=\alpha\pi$ gives
$p(\alpha\pi)=0$. Then, upon further increasing $V_{0}$, the number of
solutions stays constant (but the values of $k_{i}$ change continuously) until
$V_{0}$ is larger than the next value solving Eq. (21). Consequently, the
number of solutions of Eq. (17) is the number of solutions
$C(q)=\mathrm{RHS}(q)$ with $q>\alpha\pi$. This follows from the fact that the
derivative of RHS is positive, as illustrated in Fig. 4. Here, we see that
once an RHS line crosses $-1$ for $q>\alpha\pi$, it must necessarily also
cross $\mathrm{LHS}(k)$ for some $k=k_{i}$ smaller than $q$ and it must give
us an allowed solution for this $k_{i}$.

It can easily be shown that $C(q)=\mathrm{RHS}(q)$ for $q_{n}=\pi n/L$ with
$n\in\\{-L,...,-1,1,...,L\\}$. Counting the number of solutions now reduces to
counting how many $q_{n}=\pi n/L>\alpha\pi$. This set of $q$’s is indeed not
the values $k_{i}$ solving Eq. (17), and thus not the proper $k$’s
characterising the localised eigenstates on the gated segment; they are simply
the values which solve $C(q)=\mathrm{RHS}(q)$. No physical significance should
be given to these $q$’s other than them being a tool to count the number of
solutions of Eq. (17). Negative $k$ gives the same physical state as positive
$k$ and Eq. (17) is symmetric for $k\to-k$; thus we only need to count the
positive solutions. In summary, we arrive at the condition for solutions of
the Schrödinger equation localised on the gated segment

$n>\alpha L.$ (22)

As $0<n\leq L$, the number of such solutions is given by

$N_{\mathrm{loc}}=L-\mathrm{floor}(\alpha L).$ (23)

##  Appendix B  Analytical derivation of $N_{\mathrm{loc}}$
(Levinson’s theorem)

Levinson’s theorem relates the difference in the phase shift of the extended
states at zero and infinite momentum to the number of bound states. We here
show that Levinson’s theorem, appropriately modified for our lattice model,
can be used to obtain the number of localised states, and it reproduces our
earlier result, Eq. (4). To show this, let us consider the same model as in
App. A, with finite size for convenience, and focus on the states that are
extended on the leads. We write one ansatz for the antisymmetric states
($\psi^{A}$) and one for the symmetric states ($\psi^{S}$):

$\psi^{A}_{n}=\begin{cases}A\sin\left[kn-k\frac{L+1}{2}+\delta^{A}(k)\right],&-L_{\mathrm{leads}}<n\leq
0\\\ B\sin\left[k^{\prime}n-k^{\prime}\frac{L+1}{2}\right],&1\leq n\leq
L,\end{cases}$ (24)
$\psi^{S}_{n}=\begin{cases}A\cos\left[kn-k\frac{L+1}{2}+\delta^{S}(k)\right],&-L_{\mathrm{leads}}<n\leq
0\\\ B\cos\left[k^{\prime}n-k^{\prime}\frac{L+1}{2}\right],&1\leq n\leq
L,\end{cases}$ (25)

where the form of the wave function on the right lead follows by symmetry.
$\delta^{A(S)}(k)$ is the scattering phase shift defined so that it vanishes
for $V_{0}=0$. $k^{\prime}$ is given by $k^{\prime}=\arccos(V_{0}/2+\cos k)$
and is imaginary for $-2\leq E<-2+V_{0}$ and real for $-2+V_{0}\leq E\leq 2$.

The open boundary conditions of the chain are imposed by assuming
$\psi(-L_{\mathrm{leads}})=\psi(L_{\mathrm{leads}})=0$. Consequently, the
allowed values of $k$ are those where either of the functions

$\displaystyle f^{A}(k)$
$\displaystyle=\left(-kL_{\mathrm{leads}}-k\frac{L+1}{2}+\delta^{A}(k)\right)/\pi,$
(26) $\displaystyle f^{S}(k)$
$\displaystyle=\left(-kL_{\mathrm{leads}}-k\frac{L+1}{2}+\delta^{S}(k)\right)/\pi-\frac{1}{2},$
(27)

take an integer value.

For sufficiently large $L_{\mathrm{leads}}$, $f^{A(S)}(k)$ is a monotonically
decreasing function of $k$ (we require $\frac{\partial\delta}{\partial
k}<L_{\mathrm{leads}}+\frac{L+1}{2}$, which includes the requirement that
$\delta(k)$ is continuous). If we consider $f^{A(S)}(k)$ at the minimum and
maximum momentum, i.e., $k=0$ and $k=\pi$, we can then count the number of
antisymmetric (symmetric) extended states as

$N^{A(S)}_{\mathrm{ext}}=\mathrm{ceil}[f^{A(S)}(k=0)]-\mathrm{floor}[f^{A(S)}(k=\pi)]-1.$
(28)

We have here paid extra caution to states with momenta $k=0$ and $k=\pi$, as
necessary. Their wave functions are zero everywhere and should thus not be
included in the counting. This explains the choices of $\mathrm{ceil}$ and
$\mathrm{floor}$ and why we subtract one in $N^{A(S)}_{\mathrm{ext}}$.

By imposing the boundary equations between the leads and the gated segment, we
find that the phase shift is given by

$\displaystyle\cot\left[\delta^{A}(k)-k\frac{L+1}{2}\right]$
$\displaystyle=\frac{\sin[k^{\prime}-k^{\prime}(L+1)/2]}{\sin[-k^{\prime}(L+1)/2]\sin
k}-\cot k,$ (29) $\displaystyle\tan\left[\delta^{S}(k)-k\frac{L+1}{2}\right]$
$\displaystyle=-\frac{\cos[k^{\prime}-k^{\prime}(L+1)/2]}{\cos[-k^{\prime}(L+1)/2]\sin
k}+\cot k,$ (30)

where an appropriate integer number of $\pi$ should be added to the phase
shift so that it is continuous.

For $V_{0}=0$, the phase shift is necessarily zero and all states are
extended. We then have

$\displaystyle\left.N^{A}_{\mathrm{ext}}\right|_{V_{0}=0}$
$\displaystyle=L_{\mathrm{leads}}-1+\mathrm{ceil}[(L+1)/2],$
$\displaystyle\left.N^{S}_{\mathrm{ext}}\right|_{V_{0}=0}$
$\displaystyle=L_{\mathrm{leads}}+\mathrm{ceil}[L/2],$

giving $2L_{\mathrm{leads}}+L$ states in total. As the total number of
antisymmetric and symmetric states both should be independent of $V_{0}$, we
find that the number of localised states is

$\displaystyle N^{A}_{\mathrm{loc}}=$
$\displaystyle\begin{cases}\frac{1}{\pi}\left[\delta^{A}(k=\pi)-\delta^{A}(k=0)\right]+\frac{1}{2},&\quad
L\;\mathrm{even}\\\
\frac{1}{\pi}\left[\delta^{A}(k=\pi)-\delta^{A}(k=0)\right],&\quad
L\;\mathrm{odd},\end{cases}$ (31) $\displaystyle N^{S}_{\mathrm{loc}}=$
$\displaystyle\begin{cases}\frac{1}{\pi}\left[\delta^{S}(k=\pi)-\delta^{S}(k=0)\right]+\frac{1}{2},&\quad
L\;\mathrm{even}\\\
\frac{1}{\pi}\left[\delta^{S}(k=\pi)-\delta^{S}(k=0)\right]+1,&\quad
L\;\mathrm{odd},\end{cases}$ (32)

Figure 5: The number of localised states found from Levinson’s theorem
compared to $L-\mathrm{floor}(\alpha L)$ for $V_{0}=0.33$. Figure 6: The
number of localised states found from Levinson’s theorem compared to
$L-\mathrm{floor}(\alpha L)$ for $V_{0}=1.45$.

The total number of localised states is
$N_{\mathrm{loc}}=N^{A}_{\mathrm{loc}}+N^{S}_{\mathrm{loc}}$. Numerically
solving for $\delta(k)$, we end up with the number of localised states shown
in Figs. 5 and 6, which is in perfect agreement with Eq. (4).

##  Appendix C  Dependence on $V_{0}$

Figure 7 shows the dependence of peaks in the conductance, from which the
number of localised states can be extracted, on $V_{0}$. Notice that the
behaviour of the system depends continuously on the gated voltage, and
$N_{\mathrm{loc}}(V_{0})$ does not show quasiperiodic behaviour as a function
of the gate voltage $V_{0}$.

Figure 7: Conductance as a function of $V_{0}$ for $L=10$, in the gated chain
model in Sec. II.

##  Appendix D  Analytical calculation of the conductance

The conductance through a single-channel 1D system is given by the Landauer
formula

$G(E)=\frac{e^{2}}{h}\mathcal{T}(E)\mathcal{T}^{*}(E)\,,$ (33)

where $E$ is the energy of the incident particle and $\mathcal{T}$ is the
transmission coefficient through the gated region. To obtain the transmission
coefficient, we first write the wave function of the free particle in
continuum in the left and right leads (L, R) and in the gated region (M)

$\displaystyle\psi_{\mathrm{L}}(x)$
$\displaystyle=e^{ikx}+\mathcal{R}e^{-ikx}$ (34)
$\displaystyle\psi_{\mathrm{M}}(x)$
$\displaystyle=Ae^{ik^{\prime}x}+Be^{-ik^{\prime}x}$ (35)
$\displaystyle\psi_{\mathrm{R}}(x)$ $\displaystyle=\mathcal{T}e^{ikx}\,,$ (36)

where $k$ and $k^{\prime}$ are the momenta of the leads and the middle region,
respectively. Using the following boundary conditions,

$\displaystyle\begin{split}\psi_{\mathrm{L}}(0)=\psi_{\mathrm{M}}(0);&\qquad\psi_{\mathrm{M}}(L)=\psi_{\mathrm{R}}(L)\\\
\partial_{x}\psi_{\mathrm{L}}(0)=\partial_{x}\psi_{\mathrm{M}}(0);&\qquad\partial_{x}\psi_{\mathrm{M}}(L)=\partial_{x}\psi_{\mathrm{R}}(L)\,,\end{split}$
(37)

we obtain the expression for the transmission coefficient

$\mathcal{T}=\left[1-\left(\frac{1+\kappa}{1-\kappa}\right)^{2}\right]\frac{e^{-ikL}}{e^{ik^{\prime}L}-\left(\frac{1+\kappa}{1-\kappa}\right)^{2}e^{-ik^{\prime}L}}\,,$
(38)

where $\kappa\equiv k/k^{\prime}$. The energy dependence is obtained by
inverting Eqs. (13), which gives

$\displaystyle\begin{split}k(E)&=\arccos\left(-\frac{E}{2}\right)\\\
k^{\prime}(E)&=\arccos\left(\frac{V_{0}-E}{2}\right)\\\
\kappa(E)&=\frac{\arccos\left(-\frac{E}{2}\right)}{\arccos\left(\frac{V_{0}-E}{2}\right)}\,.\end{split}$
(39)

The energy-dependent conductance then follows,

$\displaystyle
G(E)=\frac{\frac{e^{2}}{h}\left[1-\left(\frac{1+\kappa}{1-\kappa}\right)^{2}\right]^{2}}{1+\left(\frac{1+\kappa}{1-\kappa}\right)^{4}-2\left(\frac{1+\kappa}{1-\kappa}\right)^{2}\cos\left[2\arccos\left(\frac{V_{0}-E}{2}\right)L\right]}\,.$
(40)

Note that $G(E)$ is an oscillatory function with maxima reaching
$\frac{e^{2}}{h}$ located at energies $E_{n}$ that maximise the cosine term in
the denominator, namely when

$\arccos\left(\frac{V_{0}-E_{n}}{2}\right)=\pi\frac{n}{L}\,,$ (41)

with $n\in\mathbb{N}$. To count the total number of oscillations in the
conductance for a given length $L$, we first need to restrict the energies
$E_{n}$ to an interval $[-2,2]$, which is the bandwidth of the leads. The
number of oscillations in conductance for a fixed length $L$ is

$\displaystyle N_{\rm peaks}(L)$ $\displaystyle=$
$\displaystyle\left\lfloor\frac{L}{\pi}\arccos\left(\frac{V_{0}}{2}-1\right)\right\rfloor$
(42) $\displaystyle=$ $\displaystyle\mathrm{floor}\left(\alpha L\right)\,,$
(43)

with the same definition of $\alpha$ as in the main text and in App. A.

##  Appendix E  Analytical derivation of $N^{\mathrm{FM}}_{\mathrm{loc}}$

The second example we consider in the main text is an infinite tight-binding
chain with constant hopping amplitude $t=1$ and on-site potential

$V_{i}=\begin{cases}(-1)^{i+1}W,&\quad i\leq 0\\\ +W&\quad 1\leq i\leq L\\\
(-1)^{i+L+1}W,&\quad i\geq L+1,\end{cases}\,$ (44)

where $i$ labels the sites and goes from $-\infty$ to $\infty$. This can be
viewed as a hopping particle interacting with classical Ising spins living on
the sites of the chain, arranged in an infinite antiferromagnetic (AFM)
pattern with a ferromagnetic (FM) insertion of length $L$. (Notice that, for
convenience, we adopted a labelling convention for the potential in Eq. (44)
such that $L=0$ corresponds to an infinite AFM pattern.)

We analytically derive the expression for the number of states localised on
the FM segment by solving the Schrödinger equation. In this case, the
potential is symmetric about $i=(L+2)/2$. Thus, we only need to solve for
$i\leq(L+2)/2$ and enforce a symmetric/antisymmetric wave function. Note that
$(L+2)/2$ is not restricted to integer values and the following derivation
therefore holds irrespective of $L$ being even or odd.

The AFM leads are divided into two sublattices. We therefore have three bulk
equations: Two for the leads and one for the FM segment, given by

$\displaystyle-\psi^{-}_{n-1}+(+W)\psi^{+}_{n}-\psi^{-}_{n}$
$\displaystyle=E\psi^{+}_{n},$ (45)
$\displaystyle-\psi^{+}_{n}+(-W)\psi^{-}_{n}-\psi^{+}_{n+1}$
$\displaystyle=E\psi^{-}_{n},$ (46)
$\displaystyle-\psi^{FM}_{n-1}+(+W)\psi^{FM}_{n}-\psi^{FM}_{n+1}$
$\displaystyle=E\psi^{FM}_{n}\,,$ (47)

where $\psi^{+}$ ($\psi^{-}$) is the wave function on the sublattice of
potential $+W$ ($-W$) in the left AFM lead and $\psi^{FM}$ is the wave
function in the FM segment. We have introduced a new index $n$ to label the
unit cells, implying $i=2n-1,2n$ in the left lead whereas $i=n$ in the FM
segment.

Using the ansatz $\psi^{+}_{n}=Ae^{ikn}$ for the leads, we get the dispersion

$E_{\mathrm{AFM}}(k)=\pm\sqrt{W^{2}+2\left(1+\cos k\right)}\,,$ (48)

and using the ansatz $\psi^{\mathrm{FM}}_{n}=Ae^{ikn}$ for the FM segment, we
get the dispersion

$E_{\mathrm{FM}}(k)=W-2\cos k\,.$ (49)

We are interested in finding the states which localise on the FM segment and
therefore decay exponentially into the leads. To search for such solutions, we
construct the following ansatz

$\displaystyle\psi^{+}_{n}$
$\displaystyle=Ae^{-(p+i\varphi)(L+2)/2}e^{(p+i\varphi)n},\quad n\leq 0$ (50)
$\displaystyle\psi^{\mathrm{FM}}_{n}$
$\displaystyle=B\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[kn-k\frac{L+2}{2}\right],\quad
1\leq n\leq L+1$ (51)

where the wave function on the right lead ($n>L$) follows from symmetry
considerations.

Inserting the ansatz into the bulk equations, we get

$E_{\mathrm{AFM}}(p,\varphi)=\pm\sqrt{W^{2}+2\left(1+\cosh(p+i\varphi)\right)}$
(52)

and

$\psi^{-}_{n}=-\frac{(1+e^{p+i\varphi})}{E_{\mathrm{AFM}}(p,\varphi)+W}\psi^{+}_{n}\,.$
(53)

To find the allowed values for $k$ in Eq. (51), we need to impose the boundary
equations which couple the leads and the FM segment ($p$ is given by requiring
$E_{\mathrm{AFM}}(p,\varphi)=E_{\mathrm{FM}}(k)$; namely,
$p(k)=\operatorname{arccosh}\left[2\cos^{2}k-2W\cos k-1\right]$). The boundary
equations for the left boundary read

$\displaystyle\begin{split}-\psi^{+}_{0}-W\psi^{-}_{0}-\psi^{\mathrm{FM}}_{1}&=E\psi^{-}_{0}\,,\\\
-\psi^{-}_{0}+W\psi^{\mathrm{FM}}_{1}-\psi^{\mathrm{FM}}_{2}&=E\psi^{\mathrm{FM}}_{1}\,.\end{split}$
(54)

Here, $k$ and $-k$ correspond to the same physical state, and we will only
consider $k\geq 0$. Note also that there are only two possible choices for the
oscillatory phase $\varphi$ of the wave function in the leads that give real
energies: $0$ and $\pi$. Here we need to consider the two cases separately: An
oscillatory phase $\varphi=0$ gives energies above the upper band, while an
oscillatory phase $\varphi=\pi$ gives energies in the gap.

Let us first consider the states above the upper band, $\varphi=0$. Inserting
the ansatz, Eqs. (50) and (51), into the boundary equations Eq. (54), we get
the following equation for $k$:

$2\left(\cos
k-W\right)\frac{e^{p(k)}}{1+e^{p(k)}}=\frac{\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[k-k\frac{L+2}{2}\right]}{\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[-k\frac{L+2}{2}\right]}\,,$
(55)

with $p(k)=\operatorname{arccosh}\left[2\cos^{2}k-2W\cos k-1\right]$. Again,
we can quite easily count the number of solutions by considering the behaviour
of the functions on the two sides of the equation.

The left-hand side

$\mathrm{LHS}^{0}(k)=2\left(\cos
k-W\right)\frac{e^{\operatorname{arccosh}\left[2\cos^{2}k-2W\cos
k-1\right]}}{1+e^{\operatorname{arccosh}\left[2\cos^{2}k-2W\cos k-1\right]}}$
(56)

is a function starting at $\mathrm{LHS}^{0}(k)=\cos k-W$ for
$k=\arccos{\left[\left(-W-\sqrt{W^{2}+4}\right)/2\right]}\equiv\beta\pi$,
which decreases monotonically as $k\to\pi$; see Fig. 8. This is similar to
what we had for the gated segment, but now the value of
$\mathrm{LHS}^{0}(\beta\pi)$ varies with $W$. It can be shown that the line
$C(k)$ giving the end point of LHS for a certain $W$ is given by

$C(k)=\frac{1}{\cos k}\,.$ (57)

The right-hand side (of which we have one for the symmetric eigenstates and
another for the antisymmetric eigenstates)

$\mathrm{RHS}(k)=\frac{\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[k-k\frac{L+2}{2}\right]}{\genfrac{}{}{0.0pt}{}{\cos}{\sin}\left[-k\frac{L+2}{2}\right]}$
(58)

is a function with derivative $\mathrm{RHS}^{\prime}(k)\geq 0$ for $k\geq 0$
(and $\mathrm{RHS}^{\prime}(k)=0$ only for $k=0,\pi$).

The functions $\mathrm{LHS}^{0}(k)$ and $\mathrm{RHS}(k)$ are plotted in Fig.
8, where we show $\mathrm{RHS}(k)$, which is independent of $W$, for $L=12$,
and $\mathrm{LHS}^{0}(k)$ for a range of different values of $W$.

Figure 8: The pink lines illustrate $\mathrm{LHS}^{0}(k)$ for different values
of $W$. The blue lines represent $\mathrm{RHS}(k)$ for $L=12$. The black
dashed line shows $C(k)$. The blue lines (RHS) cross $C(k)$ for $q_{n}\pi/Ln$.
The pink lines (LHS) touch $C(k)$ for $k=\beta\pi$.

To find an analytical expression for the number of solutions $k_{i}$ of Eq.
(55) for an arbitrary length $L$ of the FM segment, we need to find the
solutions of

$C(q)=\mathrm{RHS}(q)$ (59)

and count the number of solutions $q$ which are larger than $\beta\pi$. It is
straight forward to show that $C(q)=\mathrm{RHS}(q)$ for $q_{n}=\pi n/L$ with
$n\in\\{\mathrm{ceil}(L/2),...,L\\}$.

Counting the number of solutions now reduces to counting how many $q_{n}=\pi
n/L>\beta\pi$. Note again that we only allow solutions corresponding to real
$p(k)>0$, and $k=\beta\pi$ is thus not an allowed solution. We thus end up
with

$N^{0}_{\mathrm{loc}}=L-\mathrm{floor}\left(\beta L\right)$ (60)

states localised on the FM segment with energies above the upper AFM band.

Figure 9: The pink lines illustrate $\mathrm{LHS}^{\pi}(k)$ for different
values of $W$. The blue lines represent $\mathrm{RHS}(k)$ for $L=12$. The
black dashed lines shows $C(k)=0$ and $k=\pi/2$. The blue lines (RHS) cross
$C(k)$ for $q_{n}=\pi/Ln$. The pink lines (LHS) touch $C(k)$ for $\gamma\pi$
and diverge to $-\infty$ for $k\to\pi/2$.

To count the number of localised states with energies in the gap we must set
$\varphi=\pi$. $\mathrm{LHS}(k)$ then becomes

$\mathrm{LHS}^{\pi}(k)=-2\left(\cos
k-W\right)\frac{e^{\operatorname{arccosh}\left[-2\cos^{2}k+2W\cos
k+1\right]}}{1-e^{\operatorname{arccosh}\left[-2\cos^{2}k+2W\cos k+1\right]}}$
(61)

while $\mathrm{RHS}(k)$ stays the same as in Eq. (58). $\mathrm{LHS}^{\pi}(k)$
starts at $k=\gamma\pi\equiv\arccos W$ with $\mathrm{LHS}^{\pi}(\gamma\pi)=0$
($k=0$ for $W>1$ with
$\mathrm{LHS}^{\pi}(0)=-2(1-W)e^{\operatorname{arccosh}(2W-1)}/(1-e^{\operatorname{arccosh}(2W-1)})$)
and decreases continuously as $k\to\pi/2$, where it diverges to $-\infty$.

The functions $\mathrm{LHS}^{\pi}(k)$ and $\mathrm{RHS}(k)$ are plotted in
Fig. 9 for a range of different values of $W$.

In this case, counting the number of solutions reduces to counting the number
of solutions to $\mathrm{RHS}(q)=0$ with $q>\gamma\pi$ which belongs to an RHS
line that diverges to $-\infty$ for some $k<\pi/2$. $\mathrm{RHS}(q)=0$ for
$q_{n}=\pi n/L$ with $n\in\\{1,\dots,L-1\\}$ and
$\mathrm{RHS}(q^{\infty})\to-\infty$ for $q^{\infty}_{n}=\pi n/(L+2)$ with
$n\in\\{1,\dots,L+1\\}$. To find the number of solutions, we subtract the
number of $q_{n}\leq\gamma\pi$ (for $W>1$, we subtract the number of
$q_{n}<0$, which is trivially zero) from the number of $q^{\infty}_{n}<\pi/2$.
Note that $k=\gamma\pi$ and $k=\pi/2$ are not allowed solutions, as these have
$p=0$.

The total number of $q^{\infty}_{n}<\pi/2$ is $\mathrm{ceil}(L/2)$ and the
total number of $q_{n}\leq\gamma\pi$ is $\mathrm{floor}\left(\gamma L\right)$.
We therefore have

$N^{\pi}_{\mathrm{loc}}=\begin{cases}\mathrm{ceil}\left(L/2\right)-\mathrm{floor}\left(\gamma
L\right),&W\leq 1\\\ \mathrm{ceil}\left(L/2\right),&W>1\end{cases}$ (62)

states which localise in the gap.

The total number of states localised on the FM segment is thus

$\displaystyle N^{\mathrm{FM}}_{\mathrm{loc}}=$ $\displaystyle
N^{0}_{\mathrm{loc}}+N^{\pi}_{\mathrm{loc}}$ $\displaystyle\\!\\!\\!\\!=$
$\displaystyle\begin{cases}L-\mathrm{floor}\left(\beta
L\right)+\mathrm{ceil}\left(L/2\right)-\mathrm{floor}\left(\gamma
L\right),&W\leq 1\\\ L-\mathrm{floor}\left(\beta
L\right)+\mathrm{ceil}\left(L/2\right),&W>1\,.\end{cases}$ (63)

##  Appendix F  Wave function profiles

We present here some illustrative examples of the eigenstate wave functions
localised on the gated / ferromagnetic segment discussed in our work.

Figure 10 shows examples of wave functions with energies $E>2$ for the gated
chain model in Sec. II. The localised behaviour is clearly evident for $E>2$,
when compared to a delocalised state for $E<2$ shown in the top panel for
reference (light grey curve).

Figure 10: Examples of eigenstate wave functions localised on the gated
segment for $V_{0}=1.45$ and $L=50$. Top: $E=2.06$. Middle: $E=2.95$. Bottom:
$E=3.45$. The top panel also shows a delocalised state with $E=0.566$ in light
grey for comparison.

Figure 11 shows similar examples of eigenstate wave functions for the
ferromagnetic segment in an antiferromagnetic chain model in Sec. III.

Figure 11: Examples of eigenstate wave functions localised on the
ferromagnetic segment for $W=0.1$ and $L=50$. Top: $E=0.0265$ (in the gap).
Middle: $E=2.05$. Bottom: $E=2.10$. The top panel also shows a delocalised
state with $E=0.543$ in light grey for comparison.

##  Appendix G  Special cases

While the quasiperiodic behaviour discussed in the main text is generic, there
are fine-tuned values of the system parameters when it disappears. For
completeness, we review them here briefly for the two models considered in our
work.

####  G.0.1 Gated segment

In the case of the gated segment, $N_{\mathrm{loc}}$ shows periodic
fluctuations about its linear behaviour whenever $\alpha$ is rational,
implying that $\alpha\pi=\arccos\left(V_{0}/2-1\right)$ is a rational number
times $\pi$. This occurs for an infinite set of fine-tuned values of $V_{0}$
(e.g., $V_{0}=\\{2-\sqrt{2},1,2\\}$); however, it is a set of measure zero on
the real line.

####  G.0.2 Ferromagnetic segment in an antiferromagnetic chain

Let us now consider the example of a ferromagnetic segment in an
antiferromagnetic chain. For $W\geq 1$, $N^{\mathrm{FM}}_{\mathrm{loc}}$ only
depends on $\beta$ and the situation is very similar to that of the gated
segment. If $\beta$ is rational, which happens for an infinite but sparse
number of values $W$, $N^{\mathrm{FM}}_{\mathrm{loc}}$ shows periodic
fluctuations. In the case of $W<1$, $N^{\mathrm{FM}}_{\mathrm{loc}}$ depends
on both $\beta$ and $\gamma$ and it only loses its quasiperiodic behaviour if
both $\beta$ and $\gamma$ are rational. We only find one value of $W$ where
this happens, namely $W=\sqrt{2}/2$.