# Anomalous and normal dislocation modes in Floquet topological insulators

Tanay Nag<EMAIL_ADDRESS>SISSA, via Bonomea 265, 34136 Trieste, Italy  Bitan
Roy<EMAIL_ADDRESS>Department of Physics, Lehigh University, Bethlehem,
Pennsylvania, 18015, USA

###### Abstract

Electronic band structure featuring nontrivial bulk topological invariant
manifest through robust gapless modes at the boundaries, e.g., edges and
surfaces. As such this bulk-boundary correspondence is also operative in
driven quantum materials. For example, a suitable periodic drive can convert a
trivial state into a Floquet topological one, accommodating nondissipative
dynamic gapless modes at the interfaces with vacuum. Here we theoretically
demonstrate that dislocations, ubiquitous lattice defects in crystals, can
probe Floquet topological and an unconventional $\pi$-trivial insulators in
the bulk of a driven quantum system by supporting normal and anomalous modes
at its core. Respectively they reside at the Floquet zone center and boundary.
We exemplify these outcomes specifically for two-dimensional Floquet Chern
insulator and $p_{x}+ip_{y}$ superconductor, where these localized modes are
respectively constituted by charged and neutral Majorana fermions. Our
findings should be instrumental in probing Floquet topological phases in the
state-of-the-art experiments in driven quantum crystals and metamaterials
through bulk topological lattice defects.

_Introduction_. Quantum electronic materials can be classified into two broad
categories: topological and trivial. In the former family, electronic
wavefunctions in the bulk of the system feature nontrivial topological winding
number, resulting in gapless metallic states at an interface with vacuum
hasan-kane:RMP ; qi-zhang:RMP . This so-called the bulk-boundary
correspondence beyond the territory of static systems also extends to the
world of driven quantum materials oka-aoki ; berg-demler ; galitski ; moessner
; berg-levin ; dsenakdutta ; seradjeh ; zoller-budich ; cooper ; roy-harper ;
eckardt ; takashi ; nag-juricic-roy . But, due to the nontrivial role of the
time dimension the bulk-boundary correspondence in driven systems is more
subtle. For example, a static featureless electronic wavefunction subject to a
suitable periodic drive can acquire nontrivial winding in the time direction,
and due to the time translational symmetry the emergent dynamic boundary modes
in the resulting Floquet topological phases are nondissipative. Thus far
however identification of dynamic topological matters somewhat exclusively
relied on their boundary modes at the edges and surfaces gedik ; alu ; azameit
; Exp-2 ; Exp-4 ; Exp-5 ; Exp-MM-1 ; Exp-MM-3 ; Exp-MM-4 . Here we
theoretically demonstrate that bulk topological lattice defects, namely
_dislocation_ [Fig. 1], can be instrumental in probing Floquet topological
phases [Fig. 2], as they accommodate robust nondissipative modes, localized
near the dislocation core [Figs. 3-5], detectable in tunneling spectroscopy
measurements.

Dislocation in a two-dimensional (2D) lattice is created by removing a line of
atoms (open cyan circles), ending at a site, known as its center or core
(orange circle), and subsequently joining (represented by the dashed lines)
the sites (blue and red dots) across the missing line of atoms, such that the
translational symmetry is restored everywhere in the system, except near the
core of the dislocation, see Fig. 1. As a result any closed loop around the
dislocation center exhibits a missing translation by the Burgers vector ${\bf
b}$, with ${\bf b}=a{\bf e}_{x}$ in Fig. 1, where $a$ is the lattice spacing.
When an electron with momentum ${\bf K}$ encircles a dislocation it therefore
picks up a hopping phase, given by $\exp[i\Phi_{\rm dis}]$, where $\Phi_{\rm
dis}={\bf K}\cdot{\bf b}$ (modulo $2\pi$). For topological phases of matter
with the band inversion at momentum ${\bf K}_{\rm inv}$, this phase is given
by $\Phi^{\rm top}_{\rm dis}={\bf K}_{\rm inv}.{\bf b}$ (modulo $2\pi$) ran-
zhang-vishwanath . In a 2D topological insulator (electrical or thermal),
introduction of a dislocation via reconnecting the sites across the line of
removed atoms, therefore causes a level repulsion between the modes living on
the edges across it (blue and red lines in Fig. 1). With the band inversion at
a finite momentum of the Brillouin zone (BZ), as in the case of the $M$ phase
with the band inversion at the $M=(1,1)\pi/a$ point of the BZ, such level
repulsion between the edge modes is captured by a domain wall mass. It
originates from the nontrivial $\pi$ hopping phase across the missing line of
atoms (since $\Phi^{\rm top}_{\rm dis}=\pi$). The dislocation then binds
topological zero-energy modes in the close vicinity of its core ran-zhang-
vishwanath ; teo-kane ; juricic-PRL ; nagaosa ; juricic-natphys ; hughes-yao-
qi ; you-cho-hughes ; roy-juricic-dislocation , following the spirit of the
Jackiw-Rebbi mechanism jackiw-rebbi .

Here we demonstrate applicability of this general protocol in periodically
driven Floquet topological insulators (FTIs). In particular, we show that
dislocation defects can unveil 2D _translationally active_ FTIs [Fig. 2],
featuring finite momentum Floquet-Bloch band inversion, by localizing
topologically robust nondissipative modes at its core. Otherwise, topological
dislocation modes can be found either at the Floquet zone center (ZC) with
vanishing quasienergy ($\mu=0$) [Figs. 3 and 4, (c) and (e)] or at the zone
boundary (ZB) with $\mu=\pm\omega/2$ [Figs. 3 and 4, (d) and (f)], where
$\omega$ is the frequency of the external periodic drive. More intriguingly,
dislocations allow us to identify an unconventional phase, which we refer to
as the _$\pi$ -trivial insulator_, that simultaneously supports normal and
anomalous dislocation modes, respectively residing at the Floquet ZC and ZB,
despite possessing net zero topological invariant [Fig. 5]. We exemplify these
outcomes for 2D Floquet Chern insulators and $p_{x}+ip_{y}$ superconductor for
spinless or spin polarized fermions, respectively describing electrical and
thermal insulators. While the former system accommodates charged dislocation
modes, nondissipative localized Majorana fermions appear at the dislocation
core of a translationally active Floquet $p_{x}+ip_{y}$ superconductor.

Figure 1: Construction of a single edge dislocation through Volterra cut-and-
glue procedure on a square lattice by removing a line of atoms (open cyan
circles), ending at the center or core of the dislocation (orange circle) and
subsequently reconnecting (represented by the dashed lines) the sites (red and
blue dots) living on the edges (red and blue lines) across it. The
corresponding Burgers vector is ${\bf b}=a{\bf e}_{x}$.

_Model_. To illustrate the general principle of probing Floquet insulators
through bulk dislocation defects, here we focus on 2D time-reversal symmetry
breaking insulators, described by the static Hamiltonian

$~{}H=\sum_{\bm{k}}\Psi^{\dagger}_{\bm{k}}\;\hat{h}(\bm{k})\;\Psi_{\bm{k}},\>\>\>{\rm
with}\>\>\>\hat{h}(\bm{k})=\sum^{3}_{\alpha=0}\sigma_{\alpha}\;d_{\alpha}(\bm{k}).$
(1)

For a collection of charged fermions, the two-component spinor is defined as
$\Psi^{\top}_{\bm{k}}=\left(c_{A,\bm{k}},c_{B,\bm{k}}\right)$. Here
$c_{X,\bm{k}}$ is the fermion annihilation operator on orbital $X=A,B$ with
momentum $\bm{k}$. The particle-hole asymmetry is captured by
$d_{0}(\bm{k})=[m+t\\{\cos(k_{x}a)+\cos(k_{y}a)\\}]$. Hereafter we set the
lattice spacing $a=1$. The Pauli matrices $\\{\sigma_{\mu}\\}$ then operate on
the orbital indices. By contrast, for a $p_{x}+ip_{y}$ superconductor of
spinless or spin polarized fermions the spinor reads
$\Psi^{\top}_{\bm{k}}=\left(c_{\bm{k}},c^{\star}_{-\bm{k}}\right)$. Then
$\\{\sigma_{\mu}\\}$ operate on the Nambu or particle-hole indices, and
$d_{0}(\bm{k})\equiv 0$ due to the charge conjugation symmetry. As
$d_{0}(\bm{k})$ does not play any role in the topology of the above model,
throughout we set $d_{0}(\bm{k})=0$. The remaining components of the
$\bm{d}$-vector are given by

$\bm{d}(\bm{k})=\bigg{(}t_{1}\sin(k_{x}),t_{1}\sin(k_{y}),m_{0}-t_{0}\sum_{j=x,y}\cos(k_{j})\bigg{)}.$

Figure 2: Global phase diagram of a periodically driven time-reversal symmetry
breaking insulator for $t_{1}=t_{0}=1$ and $m_{0}=3$ [Eq. (1)], such that the
static system is a trivial insulator. Here $m_{1}$ and $\omega(=2\pi/T)$ are
respectively the amplitude and frequency of the external periodic drive [Eq.
(3)]. In the high frequency regime ($\omega\gtrsim 8$) the system only
supports FTIs with $C=\pm 1$ and a trivial ones with $C=0$ [Eq. (2)]. On the
other hand, in the intermediate and low frequency regime ($\omega\lesssim 8$),
the system in addition accommodates FTIs with $C=\pm 2$ and the $\pi$-trivial
insulator. The Floquet insulators supporting normal (anomalous) dislocation
modes at quasienergy $\mu=0$ ($\pm\omega/2$) at the Flqouet zone center
(boundary) are marked by $\otimes$ ($\oplus$). By contrast, the phases
simultaneously supporting both normal and anomalous dislocation modes, such as
the $\pi$-trivial insulator, are identified by $\odot$.

This model features both topological and trivial insulating phases,
respectively for $|m_{0}/t_{0}|<2$ and $|m_{0}/t_{0}|>2$. Furthermore, within
the topological regime there exist two distinct phases for (a)
$0<m_{0}/t_{0}<2$ with the band inversion at the $\Gamma=(0,0)$ point of the
BZ, also known as the $\Gamma$ phase, and (b) $-2<m_{0}/t_{0}<0$ with the band
inversion at the $M=(\pi,\pi)$ point of the BZ, also known as the $M$ phase.
In the $p_{x}+ip_{y}$ paired state, the pairing in these two phases takes
place in the close vicinity of a Fermi surface, realized near the $\Gamma$ and
$M$ points, respectively. The system is then in the weak coupling BCS regime.
By contrast, in the trivial paired state, pairing occurs in the absence of a
Fermi surface, and the system is then in the strong coupling BEC regime.
Irrespective of these details all the phases are characterized by the integer
topological invariant, the first Chern number

$~{}C=\frac{1}{4\pi}\int_{\rm
BZ}\;d^{2}\bm{k}\;\left(\partial_{k_{x}}\hat{\bm{d}}(\bm{k})\times\partial_{k_{y}}\hat{\bm{d}}(\bm{k})\right)\cdot\hat{d}(\bm{k}),$
(2)

where $\hat{\bm{d}}(\bm{k})=\bm{d}(\bm{k})/|\bm{d}(\bm{k})|$. The momentum
integral is performed over the first BZ. While $C=0$ in a normal insulator, in
the $\Gamma$ ($M$) phase possesses $C=-1$ ($+1$). Both of them support
topologically protected one-dimensional chiral edge modes, yielding quantized
charge and thermal Hall conductivities, given by $\sigma_{xy}=Ce^{2}/h$ TKNN ;
haldane  and $\kappa_{xy}=C\pi^{2}k^{2}_{B}T/(3h)$ as $T\to 0$ read-green ,
respectively, in a Chern insulator and $p_{x}+ip_{y}$ paired state.

Figure 3: Dislocation as a bulk probe of Floquet topological insulators (FTIs)
in the high frequency regime. Quasienergy ($\mu_{n}$) spectra in a semi-
infinite open system of linear dimension $L=100$ in the $y$ direction for (a)
$\omega=12.9$, $m_{1}=-1.84$, yielding $C=+1$, and (b) $\omega=11.0$,
$m_{1}=-5.3$, yielding $C=-1$. Here blue and red [green] states are
respectively localized on the right (R) and left (L) edges [in the bulk (B)],
confirming that the Floquet-Bloch band inversion takes place at $k_{x}=\pi$ at
the Floquet zone center in (a) and boundary in (b). (c) Normal (red) and (d)
anomalous (blue) dislocation modes at quasienergies $\mu=0$ and $\pm\omega/2$,
respectively, well separated from the bulk states (black), in a system with
periodic boundaries and a pair of edge dislocation-antidislocation with the
Burgers vector ${\bf b}=\pm a{\bf e}_{x}$. Local density of states (LDoS) for
(e) normal and (f) anomalous dislocation modes.

Even though the edge modes are protected by the nontrivial bulk topological
invariant in the $\Gamma$ and $M$ phases, they respond distinctly when the
translational symmetry, on the other hand, is broken in the bulk of the system
by introducing a dislocation. Since $\Phi^{\rm top}_{\rm dis}=\pi$ ($0$) in
the $M$ ($\Gamma$) phase, only the $M$ phase supports topological modes at the
dislocation core, according to the ${\bf K}\cdot{\bf b}$ rule, and is
therefore translationally active juricic-PRL ; juricic-natphys . The
dislocation mode gets pinned at zero energy due to an _antiunitary_ particle-
hole symmetry, generated by $A=\sigma_{1}{\mathcal{K}}$, as
$\\{\hat{h}(\bm{k}),A\\}=0$, where ${\mathcal{K}}$ is the complex conjugation
broy-antiunitary . The band inversion momentum (${\bf K}_{\rm inv}$) can be
recognized from the zero-energy edge modes in a semi-infinite system with
$k_{x}$ as a good quantum number and open boundaries in the $y$ direction, for
example, which in the $\Gamma$ ($M$) phase appear near $k_{x}=0$ ($\pi$), see
Fig. S1 of the Supplemental Information (SI). The topological dislocation
modes this way manifest the bulk-boundary correspondence in a static
translationally active phase. In the rest of the paper, we show that the ${\bf
K}\cdot{\bf b}$ rule is instrumental in identifying Floquet insulators, which
may or may not have any analogues in the static system.

Figure 4: Edge modes of FTIs with (a) $C=2$ for $m_{1}=-5.17$ and (b) $C=-2$
for $m_{1}=-8.32$ in a semi-infinite system at an intermediate frequency
$\omega=5.2$. The Floquet-Bloch band inversion occurs at $k_{x}=\pi$ ($0$) at
(a) Floquet ZC (ZB) and (b) Floquet ZB (ZC). Respectively, these two phases
support (c) normal (red) and (d) anomalous (blue) modes in a periodic system
with a pair of dislocation-antidislocation with ${\bf b}=\pm a{\bf e}_{x}$.
The LDoS for (e) normal and (f) anomalous dislocations modes. Color bars are
the same as in Fig. 3.

_Floquet insulators_. To establish the applicability of the ${\bf K}\cdot{\bf
b}$ rule within the Floquet framework, we focus on a static trivial insulator.
For the rest of the discussion we, therefore, set $m_{0}=3$ and
$t_{1}=t_{0}=1$, and periodically drive this system by the on site staggered
potential

$~{}\hat{V}(t)=m_{1}\sigma_{3}\sum_{r=-\infty}^{\infty}\delta(t-rT),$ (3)

where $r$ is an integer, $m_{1}$ is the amplitude of the periodic drive, and
$T$ denotes its period. The corresponding drive frequency is $\omega=2\pi/T$.
The Floquet operator after a single kick in the momentum space takes the form

$\displaystyle~{}U(\bm{k},T)={\rm
TO}\left(\exp\left[-i\int^{T}_{0}\left[\hat{h}(\bm{k})+\hat{V}(t)\right]dt\right]\right),$
(4)

where ‘TO’ stands for the time ordered product. The Floquet operator also
satisfy the antiunitary particle-hole symmetry as
$A^{-1}U(\bm{k},T)A=U^{\star}(-\bm{k},T)$. The effective Floquet Hamiltonian
associated with the Floquet operator, given by $\hat{h}_{\rm
Flq}(\bm{k})=i\ln(U(\bm{k},T))/T$, takes the form $\hat{h}_{\rm
Flq}(\bm{k})=\bm{d}_{\rm ren}(\bm{k})\cdot{\bm{\sigma}}$. The explicit form of
the $\bm{d}_{\rm ren}(\bm{k})$-vector in terms of the renormalized hopping and
mass parameters is somewhat lengthy, which we show in Sec. S2 of the SI. In
terms of the components of the Floquet Hamiltonian we then compute the first
Chern number $C$ after taking $\bm{d}(\bm{k})\to\bm{d}_{\rm ren}(\bm{k})$ in
Eq. (2) and construct the global phase diagram of the time-reversal symmetry
breaking Floquet insulators, shown in Fig. 2.

To illustrate the response of Floquet insulators to the bulk dislocation
defects, first we diagonalize the Floquet operator from Eq. (4) in a semi-
infinite system with $k_{x}$ as a conserved quantity and open boundaries in
the $y$ direction [Figs. 3(a) and (b), 4(a) and (b), 5(a)]. It provides
information regarding ${\bf K}_{\rm inv}$ of the Floquet-Bloch bands from the
momentum of the corresponding Floquet edge modes. Subsequently, we diagonalize
the Floquet operator in a system with a pair of dislocation-antidislocation
and periodic boundaries. It reveals the existence of normal and/or anomalous
dislocation modes [Figs. 3(c) and (d), 4(c) and (d), 5(b)], well separated
from the bulk states, in translationally active Floquet insulators, for which
$\Phi^{\rm top}_{\rm dis}=\pm\pi$. We further anchor this outcome from the
spatial profile of the local density states (LDoS) of these modes, displaying
that they are highly localized around the dislocation core [Figs. 3(e) and
(f), 4(e) and (f), 5(c) and (d)]. To this end we numerically solve for the
quasimodes $|\mu_{n}\rangle$ with quasienergies $\mu_{n}$, satisfying
$U(\bm{k},T)|\mu_{n}\rangle=\exp\left(i\mu_{n}T\right)|\mu_{n}\rangle$ in the
aforementioned geometries. See Secs. S1 and S2 of the SI.

In the high frequency regime ($\omega\gtrsim 8$) the system supports FTIs with
$C=\pm 1$ and the trivial ones with $C=0$ [Fig. 2]. Only the Floquet
insulators with $C=+1$ and $0$ can be qualitatively similar to the ones
previously discussed for the static system. For example, the FTI with $C=+1$
featuring band inversion at the finite momentum $M$ point at the Floquet ZC
supports _normal_ dislocation modes with $\mu\approx 0$ (due to the finite
size effects). See Fig. 3(a),(c),(e) and compare with Fig. S1 of the SI. The
trivial ones with $C=0$ are devoid of any dislocation mode, as in the static
system. On the other hand, $C=-1$ FTIs can support edge modes at $k_{x}=\pi$,
but at the Floquet ZB. Therefore, in such phases $C_{\rm ZC}=0$, but $C_{\rm
ZB}=1$, ultimately yielding $C=C_{\rm ZC}-C_{\rm ZB}=-1$. Here $C_{\rm ZC}$
($C_{\rm ZB}$) corresponds to the Chern number arising from the Floquet ZC
(ZB) berg-levin . As a result, such $C=-1$ FTIs support _anomalous_
dislocation modes with quasienergies $\mu=\pm\omega/2$, according to the ${\bf
K}\cdot{\bf b}$ rule, but without any analogue in the static system, see Fig.
3(b),(d),(f). Even though these normal and anomalous modes are predominantly
localized at the dislocation cores, as the periodic drive generically
introduces longer range hopping, the corresponding LDoS _weakly_ spreads over
few lattice spacing away from the defect core, in comparison to its
counterpart in a static system [Fig. S1 of SI]. Still the overlap between the
dislocation modes is quite negligible as long as the defect cores are
sufficiently far apart. These features also persist in the medium and low
frequency regimes, which we discuss next.

Figure 5: (a) The edge modes associated with a $\pi$-trivial insulator with
$C=0$ in a semi-infinite system at a low frequency $\omega=1.94$ and for
$m_{1}=-4.5$, supporting band inversions at $k_{x}=\pi$ and $0$ at both
Floquet ZC and ZB. (b) Normal (red) and anomalous (blue) modes bound to a pair
of dislocation-antidislocation. The corresponding LDoS are shown in (c) and
(d), respectively. The colorbars are the same as in Fig. 3.

In the medium and low frequency regime, when the drive frequency is comparable
to the bandwidth of the static system, the adjacent Floquet zones get even
more strongly coupled, which in turn gives rise to Floquet insulators, with no
analogues in the static system. For example, in this regime we find FTIs with
$C=\pm 2$ [Fig. 2]. The appearances of these two phases and the corresponding
Chern number can be appreciated from their edge modes. The FTI with $C=2$
features edge modes at $k_{x}=\pi$ and $0$ at the Floquet ZC and ZB [Fig.
4(a)], respectively, yielding $C_{\rm ZC}=+1$ and $C_{\rm ZB}=-1$, and
therefore $C=C_{\rm ZC}-C_{\rm ZB}=2$. Concomitantly, this phase only supports
normal dislocation modes with $\mu\approx 0$ [Fig. 4(c) and (e)], as the
finite momentum Floquet-Bloch band inversion occurs only at the Floquet ZC. On
the other hand, FTI with $C=-2$ features edge modes at $k_{x}=0$ and $\pi$ at
the Floquet ZC and ZB, respectively, and thereby yielding $C_{\rm ZC}=-1$ and
$C_{\rm ZB}=+1$ [see Fig. 4(b)]. As a result of the finite momentum band
inversion only at the Floquet ZB, this phase supports anomalous dislocation
modes at quasienergies $\mu=\pm\omega/2$ [Fig. 4(d),(f)], also in agreement
with the ${\bf K}\cdot{\bf b}$ rule. Therefore, the FTIs with the Chern number
$C=+2$ and $-2$ become translationally active at the Floquet ZC and ZB,
respectively.

More intriguingly, at low frequency we also find an insulating phase with
$C=0$ [Fig. 2], which we name the $\pi$-trivial insulator, that features edge
modes at $k_{x}=0$ and $\pi$ at both Floquet ZC and ZB [Fig. 5(a)]. Following
the above rule, we find that in this phase $C=0$ as $C_{\rm ZC}=C_{\rm ZB}=0$.
Due to the existence of the edge modes at $k_{x}=\pi$ at both Floquet ZC and
ZB, this phase simultaneously supports both normal and anomalous dislocation
modes [Fig. 5(b)], according to the ${\bf K}\cdot{\bf b}$ rule. These modes
are also strongly localized at the dislocation cores [Figs. 5(c),(d)].
Therefore, the $\pi$-trivial insulator is translationally active at both
Floquet ZC and ZB. In addition, the FTIs with $C=\pm 1$ in the low frequency
regime also support normal and anomalous dislocation modes, in stark contrast
to the situation in the high frequency regime [Fig. 3]. We discuss these
scenarios in Sec. S3 of the SI and the results are displayed in Fig. S2.

_Discussions_. Considering the simplest realization of 2D Floquet insulators,
here we show that dislocation lattice defects can be instrumental in unveiling
a rich landscape of translationally active dynamic topological phases as they
support normal and anomalous dislocation modes around the defect core. This
mechanism is also applicable for light-induced insulators oka-aoki ; moessner
; gedik , dynamic metamaterials azameit ; Exp-MM-1 ; Exp-MM-3 ; Exp-MM-4 ; alu
; Exp-2  as well as in cold atomic setups Exp-4 ; Exp-5 . Furthermore, the
${\bf K}\cdot{\bf b}$ rule should be applicable to three-dimensional Floquet
phases, capturing their responses to edge and screw dislocations. As we show
that the requisite finite momentum inversion of the Floquet-Bloch band can in
principle be engineered by tuning the amplitude and/or the frequency of the
external periodic drive. This general protocol therefore opens an
experimentally viable route to probe Floquet phases through bulk topological
lattice defects in topological crystals, featuring band inversion at the
$\Gamma$ point or even in trivial systems with no band inversion at all.

With the recent progress in probing static topological phases via lattice
defects edagawa ; beidenkopf , and the existing experimental advancements to
Floquet engineer topological phases gedik ; alu ; azameit ; Exp-2 ; Exp-4 ;
Exp-5 ; Exp-MM-1 ; Exp-MM-3 ; Exp-MM-4 , we expect that the proposed dynamic
dislocation modes can be observed experimentally in the near future. In
addition, our proposal should be germane in driven metamaterials, e.g.,
phonotic azameit ; Exp-MM-1 ; Exp-MM-3 ; Exp-MM-4  and acoustic alu ; Exp-2
lattices, where FTIs has been observed and the lattice defects can be
engineered and manipulated, as it has been already accomplished to probe
translationally active static topological phases photonic-dislocation-1 ;
photonic-dislocation-2 ; photonic-disclination ; mechanical-dislocation .

_Acknowledgments_. T.N. thanks MPIPKS, Dresden for the computation facilities.
B.R. was supported by the Startup grant from Lehigh University. We thank
Vladimir Juričić and Andras L. Szabó for discussions.

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# Nano-assembled open quantum dot nanotube devices

Tim Althuon  Physikalisches Institut, Karlsruhe Institute of Technology, 76131
Karlsruhe, Germany  These authors contributed equally  Tino Cubaynes  Institut
für QuantenMaterialien und Technologien, Karlsruhe Institute of Technology,
76344 Eggenstein-Leopoldshafen, Germany  These authors contributed equally
Corresponding author<EMAIL_ADDRESS>Aljoscha Auer  Physikalisches
Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
Christoph Sürgers  Physikalisches Institut, Karlsruhe Institute of Technology,
76131 Karlsruhe, Germany  Wolfgang Wernsdorfer  Physikalisches Institut,
Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany  Institut für
QuantenMaterialien und Technologien, Karlsruhe Institute of Technology, 76344
Eggenstein-Leopoldshafen, Germany

###### Abstract

A pristine suspended carbon nanotube is a near ideal environment to host long-
lived quantum states. For this reason, they have been used as the core element
of qubits and to explore numerous condensed matter physics phenomena. One of
the most advanced technique to realize complex carbon nanotube based quantum
circuits relies on a mechanical integration of the nanotube into the circuit.
Despite the high-quality and complexity of the fabricated circuits, the range
of possible experiments was limited to the closed quantum dot regime. Here, by
engineering a transparent metal-nanotube interface, we developed a technique
that overcomes this limitation. We reliably reach the open quantum dot regime
as demonstrated by measurements of Fabry-Perot interferences and Kondo physics
in multiple devices. A circuit-nanotube alignment precision of $\pm 200\,$nm
is demonstrated. Our technique allows to envision experiments requiring the
combination of complex circuits and strongly coupled carbon nanotubes such as
the realization of carbon nanotube superconducting qubits or flux-mediated
optomechanics experiments.

## Introduction

Nanocircuits based on carbon nanotubes (CNT) have shown to be a very versatile
platform to explore a broad range of quantum phenomena. As a nearly perfect 1D
system and a pristine material, suspended carbon nanotubes allowed to realize
Wigner 1D crystals [1, 2, 3], spin-qubits [4, 5] or nanomechanics experiments
[6, 7, 8, 9, 10], among others. Because the nano-fabrication technique used to
build such circuits ultimately limits their complexity, the development of new
techniques is a crucial point to envision more advanced experiments. The most
delicate step is the integration of the CNT into the circuits while
maintaining its cleanliness. The most recently developed approach, referred to
as ”stamping” technique relies on a mechanical transfer of the CNT into the
circuit at the last fabrication step [11, 12, 13, 14, 15, 16, 17]. In addition
to producing ultraclean CNTs [11], this technique has the strong advantage to
be a deterministic process in which a single CNT can be pre-characterized via
electrical transport [11] or imaging (this work) and only then integrated at a
precise location within a potentially complex circuit. For instance, it allows
to produce CNT-based circuits with a record high number of gates (up to 16
gates) [18], containing multiple CNTs [18]. In parallel, the stamping
technique establishes compatibility between the CNT technology and more
complex systems, making it a particularly powerful technique. This was
demonstrated with the integration of a CNT inside a microwave resonator [19,
17, 20].

Figure 1: Nano-assembly and contact improvement strategy. a. Simplified
representation of the silicon chip on which suspended carbon nanotubes are
grown between cantilevers. b. Typical circuit (here a double quantum dot) onto
which a CNT is transferred. An argon-milling is performed on the circuit prior
to the loading into the SEM. c. Nano-assembly step in which the CNT is
transferred on the circuit, and disconnected from the cantilevers by driving a
high current in the two outermost sections (see methods). d. Current-induced
annealing step. Most of the resistance of the circuit is localized at the CNT-
metal interface focussing the heating power directly on the interface as
represented with a red halo. e. Radiative thermal annealing of the circuit
using a halogen lamp. The gray global background indicates the steps performed
inside our dedicated SEM.

Nevertheless, the stamping approach was until now unable to produce a
transparent CNT-metal interface, which leads to device resistances of stamped
circuits restricted to $\mathrm{R}\geq 200\,\mathrm{k}\Omega$. As a result,
quantum transport measurements have been limited to the closed quantum dot
regime epitomized by Coulomb blockade, which constitutes the main restriction
of the stamping technique. In the carbon nanotube transistors research
community, different methods have been developed to improve the quality of the
CNT-metal contact relying on thermal annealing either via high current [21,
22, 23] or temperature [24, 25], but none has been shown to be compatible with
the stamping approach.

Besides the stamping technique, another approach which has been very efficient
in producing high quality CNT circuits with a transparent CNT-metal interface
relies on the growth of the CNT directly on top of the circuit [26, 7, 27,
28]. Nevertheless, the extreme CVD conditions required to grow CNTs [29] and
the probabilistic nature of this fabrication technique limit it to the
fabrication of simple structures (with the exception of a recent work [28])
preventing to envision more advanced experiments.

In this work, we present an approach combining a custom stamping technique
with a thorough contact improvement procedure allowing to overcome this
limitation. It relies on three key elements: (1) pre-cleaning of the circuit
via argon-milling before the transfer of the CNT, (2) maintaining the nanotube
under vacuum throughout the full process, and (3) annealing the CNT-metal
contact at relatively low temperature (below 300 °C) with a two-step annealing
approach combining current-induced and radiative thermal annealing (RTA). We
could identify the individual effects of each of them by monitoring the device
resistance after each annealing step.

Using this procedure we obtained device resistances down to
$25\,\mathrm{k}\Omega$ at room temperature, hence proving the high
transparency of the CNT-metal interface. This low value of resistance allows
to reach the open-quantum dot regime, as demonstrated by the measurements of
Kondo ridges and Fabry-Pérot interferences. In addition, our technique allows
to position the CNT with an accuracy of $\pm\,$200 nm. The newly accessed open
quantum dot regime combined with the flexibility and the accuracy of the
stamping nano-fabrication technique, paves the way to more advanced
experiments with CNT-based hybrid systems, such as novel superconducting
qubits or flux-mediated optomechanics experiments [30, 31, 32].

Figure 2: Examples of nano-assembled devices. False color scanning electron
micrographs of two different circuits realized with the stamping technique.
Electrodes contacting the CNT are highlighted in yellow. a. A double quantum
dot circuit with 5 gates. b. A circuit with a SQUID geometry (Superconducting
Quantum Interference Device). Two dotted lines define a 800 nm-wide region
where the CNT must be deposited. The positioning of the CNT in the center of
this region allows for a conservative estimation of at most $\pm\,$200 nm
deposition accuracy. Scale bars: 500 nm. Figure 3: Influence of argon-milling
pre-treatment. Device resistance distribution of devices with and without
argon-milling (comprising only devices with R$<$1 M$\Omega$, in total 14
devices without and 29 devices with argon-milling).

## Results and discussion

### Pre-cleanings and transfer of the CNT

Following the principle of the stamping technique, suspended CNTs are grown
between individual cantilevers arranged in an array (see Figure 1.a). In
parallel, the circuit chip on which a CNT will be transferred is fabricated
using standard lithography techniques (see methods). The full device is then
assembled by mechanically transferring a single CNT on top of the circuit
(details in methods). In Figure 2, two examples of fabricated circuits with
different geometries are presented. One specificity of our approach is to
perform the deposition of the CNT inside a dedicated Scanning Electron
Microscope (SEM), which presents multiple advantages. First, it allows to
localize CNTs on the cantilever chip and transfer a single one onto the
circuit while maintaining the CNT and circuit chip under vacuum. Secondly,
e-beam imaging combined with a custom piezo-motor stage allowed us to position
the CNT on the circuit with a precision of $\pm\,200\,$nm (see Figure 2.b).
This second point is crucial to realize complex circuits such as SQUIDs
(Superconducting QUantum Interference Devices) [33, 34], or charge detectors
[18], which require accurate positioning of the CNT. The metal used to contact
the CNT is essential to obtain a low resistance CNT-metal interface. Palladium
has been shown to be an excellent candidate [35] mainly because of its low
workfunction mismatch with CNTs. We thus use palladium as a top layer.

Figure 4: Two steps annealing technique. a. Trace-retrace cycle of an $I$-$V$
curve during the first current-induced annealing step with 1 M$\Omega$ in
series. The inset shows the time evolution of the current between the trace
and the retrace. b. $I$-$V$ curves measured before and after the second
current-induced annealing step. The current measured during this annealing
step is shown in the inset. c. Current measured in an ancillary CNT junction
during the two steps of RTA at a bias of $0.1\,$V. d. Current as a function of
gate voltage measured on device $\\#$10 at different stages of the process.
The ”0 W” label corresponds to the moment just after the current-induced
annealing was finished. Curves labeled ”5.6 W” and ”11 W” are measured after
the RTA step at the corresponding power. e. Evolution of device resistances of
11 transferred CNTs during the full annealing process (measured at a bias of
$0.1\,$V).

In order to obtain a high quality CNT-metal interface, we paid great attention
to the cleanliness of the circuit as well as the CNT throughout the full
assembly process. First of all, we remove adsorbates from the surface of the
circuit before the transfer of the CNT using an argon-milling process
calibrated to etch 5 nm of palladium (see Figure 1.b). Indeed, the first 3-4
nm of palladium can adsorb oxygen [24] which is known to deteriorate the
quality of the CNT-metal interface [36, 24, 37]. The resistance improvement
due to this pre-cleaning step is quantified in Figure 3. In addition, to
maintain a clean surface of the CNT, we developed a transfer technique
allowing to maintain the CNT in vacuum from their synthesis to their
measurement in a cryostat ($P<5\times 10^{-2}\,$mbar during the transfer
steps). We believe this strongly reduces the amount of adsorbates at its
surface. Finally, because the exposure to the e-beam can induce the deposition
of hydrocarbons (see Supplementary Note 2), the region of the circuit where
the CNT is transferred onto, is never exposed to the e-beam. For the same
reason, the section of the CNT which will be connected is only minimally
exposed during localization of the CNT (see Methods).

Figure 5: Kondo regime and Fabry-Pérot interferences. Maps of the differential
conductance measured at 30 mK as a function of the source-drain bias
$V_{\mathrm{sd}}$ and gate voltage $V_{\mathrm{g}}$. a. device $\\#$4, b.
device $\\#$9.

### Current-induced and radiative thermal annealing steps

Nevertheless, having a clean CNT-metal interface is not sufficient to obtain a
highly transparent contact, hence we developed a two-step low temperature
(below 300 °C) annealing process in which the contact is first improved by
driving a large current through the CNT (see Figure 1.a and 1.b) and then
further enhanced via RTA (see Figure 1.d and 1.e).

The first current-induced annealing technique is a very local process since
most of the heating power is dissipated at the contact between the CNT and the
metal (see Figure 1.d). $I$-$V$ curves during current-induced annealing are
presented in figures 4.a, and 4.b. In order to prevent an accidental cut of
the CNT, a first annealing is performed with a $1\,\mathrm{M}\Omega$ resistor
in series. On the voltage ramp-up, a current plateau is visible at low bias
voltage which is reminiscent of a barrier forming at the CNT-metal interface.
It is followed by sudden current jumps suggesting discrete events in the
vicinity of the CNT-metal interface, such as atomic rearrangements, or
desorption of atoms, which are consistent with the local heating scenario.
After this first trace-retrace cycle, the initial current plateau has
disappeared, which confirms the decrease of the barrier due to the local
events. The CNT junction still displays a weak non-ohmic behavior, which is
eliminated by driving the maximum current that the CNT can sustain [38].

The second annealing step consists in a RTA of the entire surface of the
device. For this purpose, we use the thermal radiation of a 70W halogen lamp
which has been installed inside the SEM vacuum chamber. During the RTA step
the device is positioned about 5 mm below the lamp. A tantalum shield
partially surrounds the halogen lamp to prevent excessive heating of the SEM
column. The RTA is subdivided in two steps, at 5.6 W and 11 W, both ranging
from 30 to 60 minutes for most of the devices (see Supplementary Note 6).
These power values gave the best result for the different metals used for the
bulk of the contact electrodes (Ti, Mo, Nb, see Supplementary Note 6). In
contrast to the current-induced annealing, the RTA displays a more continuous
evolution of the device resistance (see Figure 4.c and Supplementary Note 4).
During the cooldown periods, the resistance increases again which we associate
to some thermoelectric effect occurring at metal junctions in the circuit.

The effect of the different annealing steps on the resistance of 11 devices is
presented in Figure 4.e. Importantly, one can notice that all devices have a
resistance above 50 k$\Omega$ after the current-induced annealing, while 7 out
of the 11 devices are below this value after the full annealing procedure,
showing the substantial effect of RTA. Device $\\#$10 could reach a minimal
resistance of 25 k$\Omega$. Besides, a resistance of 300 k$\Omega$ after the
current-induced annealing step strongly helps to reach a final resistance
inferior to 50 k$\Omega$. It shows that an initially clean interface is
required to reach final low resistance values, underlining the need for an
argon-milling pre-treatment step. Moreover, the number of devices for which
this behavior has been observed further emphasizes the robustness of our
technique.

Another effect of the RTA is to switch the CNT transport behavior from
ambipolar to unipolar n-type (see Figure 4.d). This change is accompanied by a
shift of the center of the semiconducting gap towards negative gate values. It
can be explained either by the modification of the workfunction mismatch
between the palladium and the CNT, or by the doping of the CNT [39, 40]. While
the exact mechanism taking place in our situation is not clear, both can
result from the removal of adsorbates either at the CNT-metal interface or on
the CNT itself [41]. In particular, oxygen desorption has been shown to have a
dominant effect in similar circuits [39, 40]. Interestingly, the n-type
behavior resulting from our approach is in clear opposition to the more common
p-type behavior of most CNT-based open quantum circuits realized so far.

It is important to note, that this annealing procedure only requires
relatively low temperatures estimated to not exceed 300 °C (see Supplementary
Note 3). This temperature has to be compared to the range of 800-900 °C
involved in the CNT top-growth fabrication technique [26], as well as the 800
°C needed to form CNT-metal covalent bonds (MoC [25], or TiC formations [42,
43]). We believe RTA is particularly efficient since it directly heats the
surface of the circuit which is where the contact between the CNT and the
metal is located. This relatively low temperature annealing technique is thus
compatible with other quantum technologies such as superconducting circuits
(based on Al, Nb, NbTi, …).

### Open quantum dot regime

The devices with a resistance below 50 k$\Omega$ were further studied with
cryogenic transport measurements (see Figure 5 and Supplementary Note 5). In
both charge stability diagrams presented in Figure 5, the presence of Kondo-
ridges demonstrates that cotunneling processes play a dominant role in the
transport spectrum of the device, proving the presence of large tunnel
coupling rate between the quantum dot and the Fermi reservoirs [44], but still
lower than the charging energy of the quantum dot ($k_{B}T<\Gamma<E_{C}$).
Even higher coupling could be reached as shown by the Fabry-Pérot interference
pattern in Figure 5. In this regime, the tunnel coupling rates to the Fermi
reservoirs are larger than the charging energy of the quantum dot
($k_{B}T<E_{C}<\Gamma$), therefore the electron number in the quantum dot is
not a good quantum number anymore, proving that it cannot be considered as a
closed quantum system. Moreover, the observation of the four-fold degeneracy
of the CNT energy levels confirms the very low amount of disorder in the
circuit.

## Conclusion

To conclude, we demonstrated that the stamping nano-fabrication technique is
able to produce CNT-based quantum circuits deep into the open quantum dot
regime. For this, we developed a technique relying on the combination of an
argon-milling pre-treatment, vacuum transferred CNTs and a two-step annealing
procedure, enabling us to reach device resistance values below 50 k$\Omega$.
Such low resistance values are sufficient to reach the open quantum dot regime
in which cotunneling processes dominate, leading to Kondo physics as well as
Fabry-Pérot interferences. Moreover, our technique allows to integrate a CNT
into a circuit with a precision of $\pm\,200\,$nm. A more meticulous approach
combined with dedicated markers should in principle allow to reach a precision
down to $\pm\,50\,$nm.

The accuracy of this technique, the relatively low required temperature (below
300 °C) and its capability to reach the open quantum regime make it
particularly attractive to realize hybrid systems and merge different mature
quantum technologies. For instance, one can use a suspended CNT connected with
superconducting electrodes to realize a Josephson junction which could be used
as the basic element of novel superconducting qubits. The low Josephson energy
obtained in nanotube [45] could serve to explore new areas of the parameter
space of superconducting qubit [46].

The developed technique is also perfectly suited to realize optomechanics
experiments based on CNTs. One promising approach relies on the recently
studied flux-mediated phonon-photon coupling [31, 32], for which the large
zero-point fluctuations of the CNT are a powerful advantage [30]. Using such a
coupling scheme, a single-phonon coupling rate g$=1\,$MHz is within reach (see
Supplementary Note 7). This value is sufficient to reach the single photon
strong coupling regime allowing to create mechanical quantum state in the CNT
such as Schrödinger cat states [47].

## Methods

#### CNT growth

The CNTs are grown using a chemical vapor deposition (CVD) process on a custom
cantilever comb chip (see Supplementary Note 1) which allows us to transfer
CNTs suspended between cantilevers onto the circuit chips. The growth is
performed at 880$\,{}^{\circ}$C for 15 min with precursor gas flows of 75 sccm
of CH4 and 50 sccm of H2. We use the same catalyst as in ref [48].

#### Circuit chip fabrication

The fabrication of the circuit chip comprises four layers of lithography
processes on a p-doped SiO2 (300 nm) / Si (500 µm) substrate. First, markers
and large structures are patterned with optical lithography and electron-beam
evaporation of Ti(3 nm) / Au(40 nm). Gate (Ti/Al/AlOx) and contact electrodes
(Ti/electrode material/Pd) are then patterned with electron-beam lithography
and electron-beam evaporation with a lift-off process.

In order to transfer a CNT from the cantilever onto the circuit chip, 12 µm
deep trenches are patterned with optical lithography and reactive-ion etching
using a combination of CHF3/Ar/O2 (SiO2 etching) and SF6/C4F8 (Si etching).

#### Integration of the CNT

We transfer the cantilever chip from the CVD oven into an SEM with a custom
holder using a vacuum transfer arm. The top 60 µm of the cantilever chip are
then imaged to preselect the suspended CNT to be stamped (panorama image
presented in Supplementary Figure S1). Prior to stamping, the surface of the
circuit chip is cleaned with an argon-milling process using a Kaufman source
(for 90 s, at a discharge voltage of 40 V, with a flow of 9 sccm Ar and a beam
acceleration voltage of 200 V). This removes about 5 nm of Pd including the
oxygen adsorbed at the surface [24].

The circuit chip is then micro-bonded and transferred into the SEM usually
within 15-20 min. Because oxygen diffusion at the Pd surface takes place over
several hours [24], we believe the short exposure time to air of 15-20 min
after argon-milling is not detrimental to the quality of the surface. The
cantilever pair with the preselected suspended CNT is laterally aligned with
the stamping position on the sample chip using piezomotors (minimum step size:
10 nm). The cantilevers are then lowered into the trenches of the circuit chip
while tracking the distance between the CNT and the surface of the circuit
with the SEM. Electron-beam exposure of the whole tube is limited to the quick
overview panorama image (see Supplementary Note 1). During the approach only
few microns of the tube edge are exposed to the e-beam while the central part
remains unexposed. We apply a bias of 4 V between the two outermost electrodes
and the next inner electrodes and once the CNT is in mechanical contact a
current through it is measured showing that it has been contacted
electrically. To retract the cantilever chip without affecting the CNT segment
of interest, the circuit chip contains dedicated cutting sections where the
CNT junction can be cut by driving a high current ($\gtrsim\,$10 µA) through
the CNT. After annealing the individual junctions by driving a high current
through the CNT, the chip is radiatively heated with the halogen lamp. Circuit
chips with CNT junctions of resistance below 50 k$\Omega$ at a bias of 0.1 V
are then vacuum-transferred to a dilution cryostat with 30 mK base temperature
where the low-temperature measurements are performed. During the whole
process, the CNTs are thus never exposed to air.

## Acknowledgement

We thank the KIT Nanostructure Service Laboratory (NSL) for the technical
support in the nanofabrication of the devices. T.A. acknowledges financial
support from the Hector Fellow Academy. All authors acknowledge support from
the Leibniz award WE 4458-5, and the DFG grant WE 4458/2-1.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Author contributions

T.A, T.C, and A.A performed the measurements. T.A and T.C fabricated the
devices with the help of A.A. T.A performed the data analysis with the
assistance of T.C and A.A. T.A and T.C developed the nano-assembly method with
the help of A.A, C.S, and W.W. T.C developed the experimental setup with the
help of T.A, A.A, C.S, and W.W. T.C wrote the manuscript, and T.A wrote the
supplementary materials. A.A and W.W proofread the manuscript and the
supplementary materials.

## Competing interests

The authors declare no competing interests.

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