# Flat bands and multi-state memory devices from chiral domain wall
superlattices in magnetic Weyl semimetals

Vivian Rogers<EMAIL_ADDRESS>Chandra Department of Electrical and
Computer Engineering, The University of Texas at Austin, Austin TX 78712
Swati Chaudhary<EMAIL_ADDRESS>Department of Physics, The
University of Texas at Austin, Austin, Texas 78712, USA  Department of
Physics, Northeastern University, Boston, Massachusetts 02115, USA  Department
of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Richard Nguyen  Chandra Department of Electrical and Computer
Engineering, The University of Texas at Austin, Austin TX 78712  Jean Anne
Incorvia<EMAIL_ADDRESS>Chandra Department of Electrical and
Computer Engineering, The University of Texas at Austin, Austin TX 78712

###### Abstract

In this work, we study the electronic structure and transport in magnetic Weyl
semimetal (MWSM) chiral domain wall (DW) superlattices. We show that the
device conductance is strongly modulated by the number of DWs between
contacts, providing a means to encode multiple resistance states in a single
racetrack device. Additionally, we show that periodic exchange fields can
generate tunable flat bands in the direction of transport as well as a
chirality filtering mechanism for Bloch DWs. This work elucidates both the
physical behavior and nanoelectronic potential for DW-MWSM devices.

##  I Introduction

Topological materials such as topological insulators, Dirac semimetals, and
Weyl semimetals have gained much interest for their litany of novel electron
transport behaviors and potential utility in nanoelectronic devices [1, 2].
Weyl semimetals in particular must break either spatial inversion or time-
reversal symmetry (TRS). In a magnetic Weyl semimetal (MWSM), a Dirac cone is
split into two Weyl cones with opposite chiralities when TRS is broken. This
opens up an additional chirality degree of freedom which can significantly
influence the electronic transport [3, 4].

Figure 1:  Diagram of DW lattice MWSM device. $x_{0}$ represents the start
position of a series of chiral DWs (red and blue domains depicted). In the
bottom device, few MWSM DWs are present between nonmagnetic electrodes. In the
top device, $x_{0}$ is increased, injecting DWs into the active region,
increasing resistance between the grey, nonmagnetic contacts.

In intrinsically magnetic Weyl semimetals, the helicity of Weyl fermions can
be controlled by changing the magnetization of the material, giving rise to
large longitudinal magnetoresistances (MRs) via the helicity mismatch of
carriers in differently-magnetized portions of a sample. Previous works [5, 6]
have looked at this MR associated with transport across MWSM domain walls
(DWs) and a magnetic tunnel junction (MTJ) constructed from MWSMs, both of
which conclude that on/off ratios could be improved significantly
($10^{5}\%\text{ vs }10^{2}\%$) over the MRs utilized in traditional CoFeB/MgO
MTJs. This giant Weyl MR arising from spatially-varying magnetic textures is
expected to be resilient to disorder and large when compared to the DW
resistances attributed to anisotropic MR or spin-mistracking in a
nontopological DW [7, 8]. Experimentally, an anomalous longitudinal MR of
$\approx 7\%$ was observed by nucleating DWs in a sample of Weyl ferromagnet
Co3Sn2S2 [9]. Other works [10, 11, 12] conclude that the chirality-
magnetization locking will cause carriers to localize at discontinuities in
the magnetization profile, which along with the recently observed topological
hall torque [13, 14], opens up novel and orders-of-magnitude more efficient
means for electronic control of magnetic information in MWSMs.

The carrier localization and chirality-magnetization locking in MWSMs
motivates interest in transport and the electronic structure of the periodic
exchange superlattices, with the starting intuition that transport should be
strongly forbidden in the direction of alternating magnetic texture. Previous
works [10, 11] have shown that Néel DWs will localize carriers in 1D coplanar
to the DW, but so far no works have investigated the connection between
periodic magnetic textures and electronic bandstructure in 3D. Experimentally,
such a system could potentially be realized in a MWSM helimagnet with or via
injection of chiral DWs into a magnetic racetrack [15] with mean-free-path
larger than the superlattice length.

In this work, we outline the working principles of a multi-state MWSM DW
memory device and investigate the electronic structure of MWSMs in a helical
spin texture. We show that the conductance is modulated with the number of DWs
between contacts in a MWSM and show the emergence of flat bands in the
periodically-magnetized MWSM superlattice. Not only is this of interest for
the physics of topological magnetic superlattices, but this highly-tunable
readout of magnetic texture could allow for a novel and efficient mechanism to
convert magnetization to charge information—a necessity of spintronic
circuits, with great utility in emerging neuromorphic computing platforms
[16].

##  II  Domain wall superlattices and plane-wave analysis

We consider two models to elucidate the electronic structure and transport in
periodically-magnetized MWSM superlattices: one using the continuum plane-wave
basis model and another using the standard tight-binding approximation for
transport. For both models, we consider the cases of Néel and Bloch magnetic
DW lattices with the exchange field defined as

$\displaystyle\bm{\beta}_{N}(\bm{r})=$
$\displaystyle\,J(\cos\theta_{r}\,\hat{y}+\sin\theta_{r}\,\hat{x}),$ (1)
$\displaystyle\bm{\beta}_{B}(\bm{r})=$
$\displaystyle\,J(\cos\theta_{r}\,\hat{y}+\sin\theta_{r}\,\hat{z})$ (2)

where $\theta_{r}$ is a spatially-dependent magnetization angle. For the
infinite superlattice along $\hat{x}$, $\theta_{r}$ can be defined as
$\theta_{r}=2\pi x/a_{S}=\vec{b}_{x}\cdot\vec{R}$, where $a_{S}=n_{x}\,a_{0}$
is the superlattice length. Later, we define $\theta_{r}$ for a series of
chiral DWs. For the continuum model, we consider the four-band model
Hamiltonian of a TRS-broken MWSM [1]:

$\hat{H}(k)=v_{f}\tau_{x}\otimes(\mathbf{k}\cdot\bm{\sigma})+m\tau_{z}\otimes\sigma_{0}+\bm{\beta}\cdot\bm{\mathbf{\sigma}},$
(3)

defined in the spin ($\sigma$) and pseudospin ($\tau$) spaces. Imposing a 1D
superlattice structure in $\hat{x}$ and adding the Fourier-transformed
exchange field $\widetilde{\bm{\beta}}$ 111See Supplemental Material at [URL
will be inserted by publisher] to the Hamiltonian off-diagonals results in

$\hat{H}(k)=\sum_{G\in\mathbb{Z}\vec{b}_{x}}c_{G}^{+}c_{G}(v_{f}\tau_{x}\otimes\mathbf{(k-G)}\cdot\bm{\sigma}+m\tau_{z}\otimes\sigma_{0})\\\
+\sum_{G^{\prime},G\in\mathbb{Z}\vec{b}_{x}}c_{G^{\prime}}^{+}c_{G}\tau_{0}\otimes\widetilde{\bm{\beta}}_{(\mathbf{G}^{\prime}-\mathbf{G})}\cdot\bm{\mathbf{\sigma}}.$
(4)

We set the Fermi velocity $v_{f}=1.5\times 10^{6}\frac{m}{s}$
222$v_{f}=1.5\times 10^{6}\frac{m}{s}=\frac{a_{0}t}{\hbar}=\frac{1nm\times
1eV}{\hbar}$. This is used for comparison with the tight-binding model. In
real MWSMs, this is roughly an order of magnitude lower [13], but this does
not change the underlying physics. and focus on $m=0$ in this paper. An
exchange splitting magnitude of $J=0.25\;eV$ is taken to split the Weyl nodes
with a small $\Delta k=0.04\frac{2\pi}{a_{0}}$ with $a_{0}=1$ nm, and for
comparison with previous work [5].

Figure 2:  Chirality $\langle\gamma^{5}\rangle$ projected plane-wave band
structures for superlattice period $a_{s}=100\,a_{0},J=0.25$ eV with differing
exchange field textures. $\mathbf{k_{0}}$ denotes the position of the
$|L\rangle$ Weyl point in the uniformly magnetized case. (a) Uniformly
magnetized MWSM shows $|L\rangle$-chiral Weyl node with band folding. (b) Néel
DW superlattices show band flattening and axial Landau levels in
$\hat{x}\text{ and }\hat{y}$, while (c) Bloch DW superlattices only show band
flattening in $\hat{x}$.

For a constant magnetization, the exchange field $\beta=J\hat{M}$ will split
the $|L\rangle\text{ or }|R\rangle$ ($\text{i.e. }\langle\gamma^{5}\rangle$ =
-1,+1) Weyl points in momentum space as
$\mathbf{k}_{\pm}=\mp\mathbf{k}_{0}=\mp\Delta k\hat{M}=\mp\frac{1}{\hbar
v_{f}}\sqrt{J^{2}-m^{2}}\hat{M}$. Then a low-energy model for the Weyl cone
dispersions can be taken as [1, 10, 11]

$\hat{H}_{k,\pm}=\pm\hbar
v_{f}[\mathbf{k\mathrm{+}k}_{\pm}]\cdot\bm{\sigma}=\pm\hbar
v_{f}[\mathbf{k}\mp\frac{e}{\hbar}\mathbf{A}^{5}]\cdot\bm{\sigma},$ (5)

where $\mathbf{A}^{5}$ leads to a chirality-dependent magnetic field
determined by the exchange field texture:

$\mathbf{B}^{5}=\nabla\times\mathbf{A}^{5}=\frac{J}{ev_{f}}\nabla\times\mathbf{M}(\mathbf{r}).$
(6)

This generates $\mathbf{B}^{5}_{N}(\bm{r})=\frac{-2\pi
J}{a_{S}ev_{f}}(\sin\theta_{r}\hat{z})$ and
$\mathbf{B}^{5}_{B}(\bm{r})=\frac{-2\pi
J}{a_{S}ev_{f}}(\cos\theta_{r}\hat{y}+\sin\theta_{r}\hat{z})$ for the Néel and
Bloch wall cases, the inclusion of which significantly modifies the band
structure. In Fig 2, we show the chirality-projected bands for the uniformly
magnetized, Néel, and Bloch helical-exchange superlattices, where the
chirality operator $\gamma^{5}=\tau_{x}\otimes\sigma_{0}$ and
$\langle\gamma^{5}\rangle=\frac{-1}{\pi}Tr(G_{k}^{R}(E)\gamma^{5})$ with the
time-retarded Green’s function $G^{R}_{k}=(E+i\eta-H_{k})^{-1}$[19]. While
straightforward to project the Hamiltonian eigenstates as $\langle
u_{n}(k)|\gamma^{5}|u_{n}(k)\rangle$, it would be misleading in the Néel wall
case as the $|L\rangle$ and $|R\rangle$ states are chirality-polarized but
degenerate in $k_{z}$, thus obscuring one or the other. In the Bloch wall
case, the lowest-energy $|R\rangle\text{ and }|L\rangle$-chiral bands are
lowered and raised in energy with $\Delta
E_{\pm}=\mp(\sqrt{J^{2}+\hbar^{2}v_{f}^{2}\pi^{2}/a_{S}^{2}}-\hbar
v_{f}\pi/a_{S})\approx\mp J$ at $\Gamma$ (see Eq. 9) for a right-handed
magnetization spatial helicity (as opposed to the spin helicity
$\hat{\mathbf{p}}\cdot\hat{\bm{\sigma}}$). It can also be seen that the axial
magnetic fields, $\mathbf{B}^{5}_{N}$ and $\mathbf{B}^{5}_{B}$, will localize
carriers in $\hat{x}$. Notably, the Néel case will also localize carriers in
$\hat{y}$, while the Bloch wall case remains dispersive in $\hat{y}\text{ and
}\hat{z}$, as mentioned in Ref. [10].

Focusing on the Néel wall Hamiltonian $H_{N}$, In Fig. 3, we sample the
spectrum of $H_{N}(\Gamma)$ and, as also discussed in Ref. [11], generate
axial Landau levels (LLs) corresponding to

Figure 3:  (a) 3D surfaces of the chirality-projected but degenerate bands are
shown for a slice of $H_{N}(k)$, with $k_{x}$ fixed to 0. (b) the eigenvalue
spacing is sampled at $\Gamma$, plotted with the observed correspondence to
the analytical Landau level spacing for bands under $E=J$.

$\displaystyle\text{max}(|\textbf{B}^{5}|)=\frac{2\pi J}{a_{S}ev_{f}};$ (7)
$\displaystyle
E_{\nu}\approx\text{sgn}(\nu)\sqrt{2|\nu|\frac{Jhv_{f}}{a_{S}}}.$ (8)

The good fitting of the low-energy spectrum to max($\mathbf{B}^{5}$) $=10.47$
T implies that the wavefunctions are strongly localized to regions of highest
axial magnetic field, i.e. around $x=a_{s}(2\mathbb{Z}+1)/4$, or where the
Néel magnetization is collinear with the direction of the superlattice. While
an analogous axial LL physics should manifest for the Bloch wall case, the
continuously changing direction of $\mathbf{B}^{5}$ makes it somewhat obscure:
$\mathbf{B}^{5}$ rotates with constant magnitude in the $y-z$ plane as a
function of $x$. As a result, the zeroth Landau level disperses in different
directions at different positions which can be seen in Fig. 2 (c) for the
$k_{y}$ and $k_{z}$ directions. Magnetization profile
$\bm{\beta}_{B}(\mathbf{r})$ (Eq. 2), which decides the separation of two Weyl
nodes, leads to a nodal-ring structure in the spectrum in the $k_{y}-k_{z}$
plane centered at $\Gamma$. Interestingly, the dispersion for $k_{||}=\Gamma$
can be obtained analytically 333See Supplemental Material at [URL will be
inserted by publisher] for full derivation and is given by:

$E_{n}=\mp\sqrt{J^{2}+\hbar^{2}v_{f}^{2}\left(k_{x}+\frac{\pi}{a_{S}}(2n+1)\right)^{2}}+\hbar
v_{f}\frac{\eta\pi}{a_{S}}.$ (9)

In both cases, we notice that the electronic bands are significantly flattened
in the direction of superlattice. Knowing that periodic magnetic textures in
MWSMs can modify dispersions in the direction of transport and provide dynamic
control over the electronic structure, along with the intuition that DWs will
reflect carriers via the helicity-mismatch mechanism, we move to a device
picture to assess one application of MWSM DW lattices.

##  III Quantum transport and tight-binding analysis

For transport and spatial resolution of Fermi surfaces, we employ the
canonical four-band tight-binding model of a MWSM [5, 21, 22] to approximate
Eq. 4, again choosing the $\tau_{x}$ and $\tau_{z}$ matrices for the momentum
and mass 444It is necessary to include a k-dependent mass term in order to
create a single pair of Weyl cones in the tight-binding model, though
$m(k)\propto(1-cos(ka))$ is near-zero in the region of the Weyl points terms,
respectively:

$\hat{H}_{\mathrm{site\,i}}=\sum_{j\in\\{x,y,z\\}}[\frac{t}{2}(c_{\text{i}}^{+}c_{\text{i}}\hat{\tau}_{0}-c_{\mathrm{i}+\hat{j}}^{+}c_{\mathrm{i}}\hat{\tau}_{z})\otimes\hat{\sigma}_{0}-\\\
\frac{it}{2}(c_{\mathrm{i}+\hat{j}}^{+}c_{\mathrm{i}}\hat{\tau}_{x}\otimes\hat{\sigma}_{j})+\text{h.c}]+\bm{\beta}(r_{\mathrm{i}})\cdot\bm{\mathbf{\sigma}},$
(10)

with hopping parameter $t=$ 1 eV, lattice parameter $a_{0}=$ 1 nm for
simplicity, $n_{x}=$100 sites in $\hat{x}$, and the Bloch phase prefactors
applied to hoppings in $\hat{y}$ and $\hat{z}$ to construct a supercell
$H(k_{||})$. Here, $\theta_{r}$ of a single $180^{\circ}$ DW is defined in
[24] and convolved with a semi-infinite Dirac comb to generate the periodic
magnetization angle:

$\theta_{r}=(2\text{tan}^{-1}(\pi
e^{-\frac{x+d}{d}}))\star\sum_{\mathrm{i}=0}^{\infty}\delta(\mathrm{i}n_{x}a_{0}-x_{0}-x),$
(11)

with $\star$ being the convolution operator, $x_{0}$ referring to the start
position of the chiral DW lattice, the DW width $d=8$ nm, and the superlattice
periodicity $a_{s}=30$ nm. While these parameters are not necessarily physical
with relevance to the modelling of real magnetic racetrack systems, they are a
minimal model that is computationally tractable and allows us to demonstrate
underlying physics. In a real system, thicker domain walls or a longer
superlattice would decrease max($\mathbf{B}^{5}$) [11] thus decreasing the
axial Landau Level spacing and zone-folded subband spacing.

Figure 4:  Chirality-resolved mixed-space Fermi surface with $\mu=0.1$ eV and
$\eta=10^{-2.5}$ eV are shown with corresponding exchange field textures for
$x_{0}$ fixed to 63 nm. (a) and (b) show the Neel DW lattice, while (c) and
(d) show the helical bloch DW lattice.

In the tight-binding model, we consider trivial metallic electrodes 555We take
a simple, spin and orbital-degenerate toy metal hamiltonian
$\hat{H}_{\mathrm{site}\;i}=\mathop{\sum_{d\in\\{+,-\\},j\in\\{x,y,z\\}}}{}\
[tc_{i+d\hat{j}}^{+}c_{i}\tau_{0}\otimes\sigma_{0}]+(3t-\varepsilon_{0})[c_{i}^{+}c_{i}\tau_{0}\otimes\sigma_{0}]$
with an arbitrary $t=1$ eV and $\varepsilon_{0}=1eV$ and semi-infinite MWSM
electrodes to elucidate the underlying physics 666The semi-infinite MWSM
electrodes are useful in the large scattering limit, where an electron will
locally see an infinite lattice in all directions. They are also most useful
for resolving the mixed-space Fermi-surfaces, where surface states will
confound the fermi surface braiding. For the case with MWSM electrodes, the
magnetizations $\bm{\beta}(x=-a_{0})\text{ and }\bm{\beta}(x=a_{s}+a_{0})$ are
copied for the left and right contact and extend to infinity for the
generation of the Sancho-Rubio [27] contact self-energies $\Sigma_{L,R}$ to
retain the continuity of the magnetization field texture.

Figure 5:  Device conductance is modulated by injecting DWs between contacts
via sweeping $x_{0}$. (a) Conductance modulation by orders of magnitude is
shown for all combinations of MWSM and simple metallic electrodes with Bloch
and Néel DW lattices. (b) A phenomenological broadening term $\eta$ is swept,
showing discrete stairsteps in conductance, thus multi-weight behavior, as
$\eta$ increases (i.e. carrier lifetime decreases). (c) and (d) show
k-resolved transmission maps for the Néel and Bloch DW cases, respectively,
sweeping $x_{0}$, over center of surface Brillouin zone. Hotspots in
transmission $T(k_{||})$ are visible where the bulk states from the left and
right electrodes overlap, connected by twisted Fermi arcs of character
determined by the magnetic texture. $x_{0}$ position is labeled in white for
each sub-plot.

Of particular interest is the braiding of the Fermi surfaces in connection
with the band structures of Fig. 2, which are useful to understand transport.
Fig. 4 shows the mixed-space Fermi surface of the periodically magnetized
racetrack devices for energies close to the Weyl node. Néel devices show a
$k_{y}$-symmetric braiding of the L and R chiral states, corresponding to the
degenerate states in Fig. 4 (b). This can be understood from the profile of
$\bm{\beta}_{N}$ for the Néel case which decides the position of two Weyl
nodes in $k_{y}-k_{z}$ plane. Additionally, this magnetization pattern
generates $\mathbf{B}_{N}^{5}$ pointing along $z$ which flattens the bands in
$k_{y}$. It should be noted here that these Fermi surfaces can also be
interpreted as zeroth LL [11]. On the other hand, for Bloch DW case,
$\mathbf{B}_{B}^{5}$ rotates in $y-z$ leading to these twisted Fermi-surfaces
which can also be understood as the Fermi contour of the zeroth LL shown in
Fig. 2 (c) where for an infinitesimal non-zero energy, $|L\rangle$ and
$|R\rangle$ states are at a different radius in the $k_{y}-k_{z}$ plane. As a
result, the $|L\rangle$ chiral states wrap around the expected position of the
twisted Weyl cones, while the $|R\rangle$ chiral states wrap closely to
$k_{||}=\Gamma$. This twisted Fermi-surface motivates analysis of the chiral
anomaly and Weyl orbits with periodic real and axial magnetic fields [28].

For device modeling, we neglect the orbital effects of the magnetic field and
consider transport in the Landauer limit of the NEGF quantum transport
formalism 777See [19]; $G^{R}_{k_{||}}=(E+i\eta-
H_{k_{||}}-\Sigma_{R,k_{||}}-\Sigma_{L,k_{||}})^{-1};\Gamma_{i}=i(\Sigma_{i}-\Sigma_{i}^{\dagger});T=tr(\Gamma_{L}G^{R}\Gamma_{R}G^{A});G^{A}=(G^{R})^{\dagger}$.
With a minimal carrier lifetime broadening of $\eta=10^{-4}$ eV, in Fig. 5 (a)
we show that device conductance can be modulated by orders of magnitude via
the injection of DWs into the active region of the device, and that this
effect persists for both magnetization patterns, with both MWSM and simple
metal electrodes. In Fig. 5 (b), we sweep the broadening parameter $\eta$,
corresponding to a phenomenological inelastic momentum-preserving scattering
self-energy [30], and show that it 1) smooths out conductivity vs. $x_{0}$ and
2) forms stairsteps in conductance with to the number of DWs in the active
region of the device–as one might expect with an increasing number of
helicity-mismatch carrier reflections. This form of nonvolatile, texture-
dependent conductivity is desirable for use in emerging analog memory devices
[16].

Curiously, even a high-resolution 401 x 401 k-grid over the zoomed-in portion
of the surface Brillouin zone (SBZ) is unable to converge the value of the
Landauer conductance for small broadening parameters ($\eta=10^{-3},10^{-4}$
eV), leading to unphysical spikes in conduction. To explain this, we consider
the $k_{||}$-resolved transmission in Fig. 5 (c) and Fig. 5 (d) to resolve
conductance behavior from the distorted Weyl cones in the periodic
magnetization region. We see that the distorted Weyl cones form
infinitesimally-thick curtains of conductance in the SBZ which are challenging
to capture numerically: In Fig. 5 (c), the Néel DW lattice conduction is
dominated by the distorted $|L\rangle$ and $|R\rangle$ Weyl cones in the
periodic region of the device, especially where they overlap with the contact
electrodes’ bulk Weyl cones. In contrast, the Bloch DW device in Fig. 5 (d)
has minimal conduction through the distorted Weyl cones which wrap around the
bulk Weyl cones in either contact. Thus, the majority of the conductance
tunnels through the periodic region of the device from the Bulk states in both
contacts, leading to oscillations in Fig. 5 (a). Interestingly, due to the
broadening parameter providing finite-lifetime states from the contacts,
conductance from the contact Weyl cone overlap decreases and the inner
$|R\rangle$-chiral transmission ring will begin to plateau in the total
conductance, giving rise to a texture-dependent chirality filtering mechanism
to generate or reflect chiral currents with potential application in emerging
devices [31].

##  IV  Discussion

We have studied the electronic structure and transport in a MWSM superlattice
device, have shown magnetization-tunable flat bands in the direction of
transport, and have demonstrated that multiple conductance states could
potentially be encoded in such a device. We have also shown that the multi-
state behavior persists through all electrode and magnetic superlattice
configurations we have considered, quite unlike the spin transport in
traditional spin filtering systems. With regards to the superlattice picture,
more analysis needs to be done to fully understand the connection between
exchange textures and topological or electronic structure, the exploration of
which is only nascent in systems such as semiconductor nanowires [32] or
magnetic superlattice graphene [33, 34] under skyrmion exchange superlattices
which show topological flat bands. It should be noted that this effect only
relies on the mean-field exchange splitting of a periodic ferromagnet, acting
on spin space with contributions on the order of 100 to 1000 meV in bulk
ferromagnets, as opposed to a real magnetic field or proximity-induced
exchange, the effects of which would show with much weaker Zeeman
perturbations or contribution to an effective gauge field. Thus, we expect the
electronic structure in periodic MWSM lattices to be more robust closer to
room temperatures, supposing a MWSM with Curie temperature above 300K as in
Mn3Sn[35], Co2MnGa[36], or Fe3Sn2[37]. Nevertheless, other perspectives on
scattering at MWSM DWs imply that skew scattering [38] or heavily-tilted Weyl
cones [39] could significantly decrease the MR.

This highly-tunable electronic structure and conductivity via control of
magnetization in MWSMs would be of great use to nanoelectronics, which we will
discuss briefly. While it is hard to make definitive claims from our model
without incoherent momentum relaxation – we see longitudinal Weyl MRs over
$10^{14}\%$ – we estimate that, using the DW resistances observed in [9] an
effective MR
$=\frac{R_{\text{off}}-R_{\text{on}}}{R_{\text{off}}}=\frac{n_{DW}R_{DW}}{R_{0}}=(\frac{L}{d}+1)\frac{R_{DW}}{R_{0}}=32.1\%$
could be achievable for the observed $2R_{DW}\approx 4\,\Omega$ and
$R_{0}\approx 53\,\Omega$, for a domain width $d=80$ nm and device length
$L=600$ nm. This is 14x the 3.93% read margin observed in CoFeB transverse-
read DW memory [8]. In itself, this becomes competitive with other GMR devices
(with regards to on/off ratio) or multi-state TMR devices [16], but without
the need of a magnetic tunnel junction to measure the magnetization-induced
resistance change. Supposing a deliberate engineering of the chemical
potential or DW width could bring about a modest $\text{MR}_{DW}=25\%$ with
$L=1\mu$m, and $d=40$ nm, on/off ratios upwards of 650% become possible.
Another potential advantage of this approach, compared to traditional DW-MTJ
devices, is that one could avoid the need for precise MgO deposition to avoid
pinholes, thus making high MRs accessible to industry and academic labs
without expensive, lengthy fabrication processes. Here, one could construct
memory devices using simple metallic thin films, so long as the chirality-
magnetization locking in the MWSM is preserved and surface fermi arcs do not
dominate the conductance. Predicted orders-of-magnitude-improved readout
performance over existing MTJs [5] could further increase device viability,
supposing one could amplify the Weyl MR or suppress scattering in a real
system.

##  V Acknowledgements

The authors acknowledge funding from the UT CDCM MRSEC supported under NSF
Award Number DMR-1720595, funding from Sandia National Laboratories, and
computational resources from the Texas Advanced Computing Center (TACC) at the
University of Texas at Austin. The authors also thank Gregory A. Fiete for
helpful discussions, as well as Kerem Camsari and Shuvro Chowdhury for their
discussions on the theory and implementation of the NEGF transport formalism.

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  * Note [1] See Supplemental Material at [URL will be inserted by publisher].
  * Note [2] $v_{f}=1.5\times 10^{6}\frac{m}{s}=\frac{a_{0}t}{\hbar}=\frac{1nm\times 1eV}{\hbar}$. This is used for comparison with the tight-binding model. In real MWSMs, this is roughly an order of magnitude lower [13], but this does not change the underlying physics.
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  * Note [5] We take a simple, spin and orbital-degenerate toy metal hamiltonian $\hat{H}_{\mathrm{site}\tmspace+{.2777em}i}=\mathop{\sum@\slimits@_{d\in\\{+,-\\},j\in\\{x,y,z\\}}}{}\ [tc_{i+d\hat{j}}^{+}c_{i}\tau_{0}\otimes\sigma_{0}]+(3t-\varepsilon_{0})[c_{i}^{+}c_{i}\tau_{0}\otimes\sigma_{0}]$ with an arbitrary $t=1$ eV and $\varepsilon_{0}=1eV$.
  * Note [6] The semi-infinite MWSM electrodes are useful in the large scattering limit, where an electron will locally see an infinite lattice in all directions. They are also most useful for resolving the mixed-space Fermi-surfaces, where surface states will confound the fermi surface braiding.
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##  Appendix A Supplementary information

###  A.1 Bloch wall hamiltonian spectrum with $k_{||}$ = 0

First, one can choose the equivalent block-diagonal $\tau_{z}$ matrix for the
momentum term in Eq.4 and set m = 0. This decouples the left (-) and right (+)
helical states $\eta$, and allows one to express the Hamiltonian of the
infinitely-periodic hamiltonian with helical exchange field as follows. Let
$\Omega=\frac{J}{2\hbar v_{f}}$, and let $G=\frac{2\pi}{a_{S}}=\vec{b}_{x}$ be
the superlattice reciprocal vector. We use the definition for $\bm{\beta}(r)$
in section II:

$\displaystyle\bm{\beta}_{N}(r)=J(\cos(\theta_{r})\hat{y}+\sin(\theta_{r})\hat{x})=\frac{J}{2}(e^{iGx}\hat{y}+e^{-iGx}\hat{y}+-ie^{iGx}\hat{x}+ie^{-iGx}\hat{x}),$
(12)
$\displaystyle\bm{\beta}_{B}(r)=J(\cos(\theta_{r})\hat{y}+\sin(\theta_{r})\hat{z})=\frac{J}{2}(e^{iGx}\hat{y}+e^{-iGx}\hat{y}+-ie^{iGx}\hat{z}+ie^{-iGx}\hat{z}).$
(13)

In Fourier space, this can be written as:

$\displaystyle\widetilde{\bm{\beta}}^{N}_{G^{\prime}-G}=\frac{J}{2}[(i\hat{x}+\hat{y})\delta(G^{\prime}-G-\vec{b}_{x})+(-i\hat{x}+\hat{y})\delta(G^{\prime}-G+\vec{b}_{x})];$
(14)
$\displaystyle\widetilde{\bm{\beta}}^{B}_{G^{\prime}-G}=\frac{J}{2}[(i\hat{z}+\hat{y})\delta(G^{\prime}-G-\vec{b}_{x})+(-i\hat{z}+\hat{y})\delta(G^{\prime}-G+\vec{b}_{x})].$
(15)

We will now focus on the Bloch wall case.

$\hat{H}_{B}(k)=\hbar
v_{f}\begin{bmatrix}\ddots&\Omega(\sigma_{y}+i\sigma_{z})&&&\\\
\Omega(\sigma_{y}-i\sigma_{z})&\eta(k_{x}+G)\sigma_{x}&\Omega(\sigma_{y}+i\sigma_{z})&&\\\
&\Omega(\sigma_{y}-i\sigma_{z})&\eta(k_{x})\hat{\sigma}_{x}&\Omega(\sigma_{y}+i\sigma_{z})&\\\
&&\Omega(\sigma_{y}-i\sigma_{z})&\eta(k_{x}-G)\hat{\sigma}_{x}&\Omega(\sigma_{y}+i\sigma_{z})\\\
&&&\Omega(\sigma_{y}-i\sigma_{z})&\ddots\end{bmatrix}$ (16)

Then, the basis transformation corresponding to a $120^{\circ}$ rotation about
$-\hat{x}+\hat{y}+\hat{z}$ can be used to transform the spin matrices:

$\displaystyle\sigma_{y}\rightarrow-\sigma_{x}$
$\displaystyle\sigma_{z}\rightarrow\sigma_{y}$
$\displaystyle\sigma_{x}\rightarrow-\sigma_{z}$

and, knowing that
$\sigma_{+}=\sigma_{x}+i\sigma_{y};\sigma_{-}=\sigma_{x}-i\sigma_{y}$ (we use
the pauli matrices without prefactor 1/2), the Hamiltonian can be rewritten
as:

$\hat{H}_{B}(k)=-\hbar v_{f}\begin{bmatrix}\ddots&\Omega(\sigma_{-})&&&\\\
\Omega(\sigma_{+})&\eta(k_{x}+G)\sigma_{z}&\Omega(\sigma_{-})&&\\\
&\Omega(\sigma_{+})&\eta(k_{x})\hat{\sigma}_{z}&\Omega(\sigma_{-})&\\\
&&\Omega(\sigma_{+})&\eta(k_{x}-G)\hat{\sigma}_{z}&\Omega(\sigma_{-})\\\
&&&\Omega(\sigma_{+})&\ddots\end{bmatrix}$ (17) $=-\hbar
v_{f}\bigoplus_{n}\left(\begin{bmatrix}-\eta(k_{x}+(n+1)G)&2\Omega\\\
2\Omega&\eta(k_{x}+nG)\end{bmatrix}\right)=-\hbar
v_{f}\bigoplus_{n}\left(-\eta k_{x}\sigma_{z}+2\Omega\sigma_{x}-\eta
nG\sigma_{z}-\eta G\frac{1}{2}(\sigma_{z}+\sigma_{0})\right)$ (18)

$=-\hbar v_{f}\bigoplus_{n}(\left(\begin{smallmatrix}2\Omega\\\ 0\\\
-\eta(k_{x}+G(n+\frac{1}{2}))\end{smallmatrix}\right)\cdot\bm{\sigma}-\eta
G\frac{1}{2}\sigma_{0})=\bigoplus_{n}\hat{H}_{n}$ (19)

with the spectrum of $\hat{H}_{n}$ (thus, a diagonal subblock of
$\hat{H}_{B}(k_{||}=\Gamma)$) given as:

$E_{n}=-\hbar
v_{f}\left(\pm\sqrt{4\Omega^{2}+(k_{x}+G(n+\frac{1}{2}))^{2}}-\eta
G/2\right)=\mp\sqrt{J^{2}+\hbar^{2}v_{f}^{2}(k_{x}+\frac{\pi}{a_{S}}(2n+1))^{2}}+\hbar
v_{f}\frac{\eta\pi}{a_{S}}$ (20)


# Tunable Casimir equilibria with phase change materials: from quantum
trapping to its release

Lixin Ge<EMAIL_ADDRESS>School of Physics and Electronic Engineering,
Xinyang Normal University, Xinyang 464000, China  Xi Shi  Department of
physics, Shanghai Normal University, Shanghai, 200234, China  Zijun Xu  School
of Physics and Electronic Engineering, Xinyang Normal University, Xinyang
464000, China  Ke Gong  School of Physics and Electronic Engineering, Xinyang
Normal University, Xinyang 464000, China

###### Abstract

A stable suspension of nanoscale particles due to the Casimir force is of
great interest for many applications such as sensing, non-contract nano-
machines. However, the suspension properties are difficult to change once the
devices are fabricated. Vanadium dioxide (VO2) is a phase change material,
which undergoes a transition from a low-temperature insulating phase to a
high-temperature metallic phase around a temperature of 340 K. In this work,
we study Casimir forces between a nanoplate (gold or Teflon) and a layered
structure containing a VO2 film. It is found that stable Casimir suspensions
of nanoplates can be realized in a liquid environment, and the equilibrium
distances are determined, not only by the layer thicknesses but also by the
matter phases of VO2. Under proper designs, a switch from quantum trapping of
the gold nanoplate (“on” state) to its release (“off” state) as a result of
the metal-to-insulator transition of VO2, is revealed. On the other hand, the
quantum trapping and release of a Teflon nanoplate is found under the
insulator-to-metal transition of VO2. Our findings offer the possibility of
designing switchable devices for applications in micro-and nano-
electromechanical systems.

##  I Introduction

Micro- and nano-electromechanical systems (MEMS and NEMS), which integrate
electrical and mechanical functionality on the micro- and nano-scales, have
attracted enormous attention Lyshevski (2018); Craighead (2000). Thanks to
small sizes, the MEMS and NEMS exhibit low mass, high mechanical resonance
frequencies and quantum effects, leading to a broad range of applications such
as biological/chemical detections Eom et al. (2011), accelerometers Xu et al.
(2011) and micro/nanomachines Wang (2013). One major problem in MEMS and NEMS
is the $stiction$ which makes the systems collapse and permanent adhesion
caused by the attractive Casimir forces Buks and Roukes (2001); Chan et al.
(2001). The Casimir force is a macroscopic quantum effect which arises from
quantum fluctuations of the electromagnetic field Casimir (1948). In most
cases, two neutral, parallel plates consisted of the same materials are
attractive to each other, and the magnitudes of the attraction depend on
several parameters such as separations, geometric thicknesses, finite
conductivities and temperatures (see, e.g., the review Klimchitskaya et al.
(2009) and Refs.Yampol’skii et al. (2008, 2010). Therefore, repulsive Casimir
forces are highly required for non-contact and low-friction MEMS and NEMS. The
repulsive Casimir forces have been intensively studied in many systems Woods
et al. (2016) including liquid-separated environments Munday et al. (2009);
van Zwol and Palasantzas (2010); Phan and Viet (2011); Dou et al. (2014),
meta-materials Rosa et al. (2008); Zhao et al. (2009, 2011); Song et al.
(2018), topological insulators Grushin and Cortijo (2011); Chen and Wan
(2012); Nie et al. (2013) and specific geometrics Tang et al. (2017); Levin et
al. (2010). In addition, the concept of Casimir equilibria was also
investigated, using the enclosed geometries Rodriguez et al. (2008); Rahi and
Zaheer (2010) and dispersive materials Rodriguez et al. (2010a). Lately,
stable Casimir equilibria of nanoplates above a Teflon-coated gold substrate
were reported by Zhao et al Zhao et al. (2019). However, the Casimir
equilibria of previous studies were mainly in passive systems. Once the
devices are fabricated, the trapping properties are difficult to change. Thus,
the tunable trapping or even the switching from the trapping to its release by
external stimuli (e.g., heating, electric fields or optical waves) is highly
desired in MEMS and NEMS.

In order to active modulate the Casimir effect, one straight way is to change
the dielectric properties of materials under external means Torricelli et al.
(2012); Sedighi et al. (2013); Torricelli et al. (2010). Vanadium dioxide
(VO2) Shao et al. (2018); Zylbersztejn and Mott (1975) is a phase change
material(PCM), which undergoes a transition from a low-temperature insulating
phase to a high-temperature metallic phase at critical temperature 340 K. The
phase transition of VO2 is accompanied by a structural transformation from the
monoclinic phase to the tetragonal one. Meanwhile, the dielectric function of
VO2 changes dramatically during the phase transition, leading to many
interesting applications Wu et al. (2017); Liu et al. (2017); Kats et al.
(2012); van Zwol et al. (2012). In general, the phase transition of VO2 can be
induced by changing the temperature of systems. Alternatively, the phase
transition can be driven by optical lasers Cavalleri et al. (2001); Rini et
al. (2008) or electrical gratings Qazilbash et al. (2008); Nakano et al.
(2012) on a sub-picosecond timescale. Recently, VO2 has been employed to study
the tunable Casimir effect in the vacuum Galkina et al. (2009); Pirozhenko and
Lambrecht (2008); Castillo-Garza et al. (2007). For a large separation (e.g.,
$>$1 $\mu$m), the contrast of Casimir forces due to the phase-transition is
quite large (e.g., over 2 times for two semi-infinite plates of VO2, this
value could be even larger for the case of finite thickness Galkina et al.
(2009); Pirozhenko and Lambrecht (2008)). As the separation is small (e.g.,
$\sim$100 nm), however, the modulation of Casimir forces owning to the phase
transition and finite-thickness decreases greatly Pirozhenko and Lambrecht
(2008); Castillo-Garza et al. (2007). Nonetheless, the Casimir forces are
always attractive and only magnitude modulations have been reported in a
vacuum-separated configuration. The influences of phase transition of VO2 on
the sign modulation of Casimir forces (e.g., from attraction to repulsion) are
yet less explored. In a liquid environment, the function of sign modulation
and the related phenomena such as tunable Casimir equilibria are expected
based on the phase transition of VO2.

Here, the Casimir forces between a nanoplate and a layered structure separated
by a liquid are investigated. The layered structure consists of two kinds of
materials, i.e., Vanadium dioxide (VO2) and Teflon. It is found that stable
Casimir equilibria of gold nanoplates can be realized when a VO2 film is
buried under a semi-infinite Teflon. The properties of Casimir equilibria are
determined, not only by the layer thicknesses but also by the matter phases of
VO2. For thick-film VO2, the Casimir equilibria and quantum traps can be
achieved for both the metallic and insulating phases. On the other hand, a
switch from quantum trapping of the gold nanoplate(“on” state) to its release
(“off” state) can be triggered by the metal-to-insulator phase transition when
the thickness of VO2 is thin (e.g., 20 nm). Finally, stable suspensions of
Teflon nanoplates are also proposed with a complementary design, where the
Teflon substrate is coated by a VO2 film. Unlike the case of gold nanoplates,
the quantum trapping of Teflon nanoplates and its release correspond to the
insulating and metallic phases of VO2. Moreover, the switching phenomena can
be realized only with a several-nanometers thickness of VO2.

Figure 1: (color online) (a) Schematic view of a gold nanoplate suspended in a
liquid environment. (b) The permittivity of different materials (gold, VO2,
bromobenzene and Teflon) as a function of imaginary frequency.

##  II Theoretical models

The system in this work is schematically shown in Fig. 1(a), where a gold
nanoplate with thickness $L_{g}$ is suspended in a liquid of bromobenzene. The
separation between the nanoplate and the substrate is $d$. The substrate is
composed of a VO2 film buried under a semi-infinite plate of Teflon. The
thicknesses of the top-layer Teflon and VO2 are denoted as $L_{T}$ and
$L_{\mathrm{V}}$, respectively. The in-plane dimension of the gold nanoplate
is much larger than $L_{g}$ and $d$, and it is considered as a slab during our
calculations. The Casimir force is calculated by $F_{c}=-\partial
E_{c}(d)/\partial d$, where $E_{c}(d)$ is the Casimir energy between the gold
nanoplate and the substrate, having the form Nie et al. (2013); Zhao et al.
(2019)

$E_{c}(d)=A\hbar\int_{0}^{\infty}\frac{d\xi}{2\pi}\int\frac{d^{2}\mathbf{k_{\|}}}{(2\pi)^{2}}\log\det\left[1-\mathbf{R_{1}}\cdot\mathbf{R_{2}}e^{-2k_{3}d}\right],$
(1)

where $\hbar$ is the reduced Planck constant, $A$ is the in-plane area,
$\mathbf{k_{\parallel}}$ is the parallel wavevector,
$k_{3}=\sqrt{k_{\parallel}^{2}+\varepsilon_{liq}(i\xi)\xi^{2}/c^{2}}$ is the
vertical wavevector, $c$ is the speed of light in vacuum,
$\varepsilon_{liq}(i\xi)$ is the permittivity of the intervening liquid
evaluated with imaginary frequency $\omega=i\xi$, $\mathbf{R}_{1,2}$ is the
$2\times 2$ reflection matrix for layered structures, having the form

$\mathbf{R_{j}}=\left(\begin{array}[]{cc}r_{j}^{s}&0\\\
0&r_{j}^{p}\end{array}\right),$ (2)

where $r_{j}$ with $j$=1 and $j$=2 are the reflection coefficients for the
upper and lower layered structures, and the superscripts $s$ and $p$
correspond to the polarizations of transverse electric ($\mathbf{TE}$) and
transverse magnetic ($\mathbf{TM}$) modes, respectively. Note that the
temperature $T$ for Eq. (1) equals 0 K and it is an effective approximation as
the separation $d$ is smaller than 1 $\mu m$ for finite temperatures Milton
(2004). For a nanoplate suspended in a liquid, the reflection coefficients can
be given analytically as follows Zhao et al. (2011)

$r^{\alpha}=\frac{r_{0,j}^{\alpha}+r_{j,0}^{\alpha}e^{-2K_{j}L_{j}}}{1+r_{0,j}^{\alpha}r_{j,0}^{\alpha}e^{-2K_{j}L_{j}}},$
(3)

where $\alpha=s$ and $p$, $L_{j}$ is the thickness of the nanoplate,
$K_{j}=\sqrt{k_{\parallel}^{2}+\varepsilon_{j}(i\xi)\xi^{2}/c^{2}}$ is the
vertical wavevector, $\varepsilon_{j}(i\xi)$ is the permittivity of the
nanoplate. The subscripts of $r_{m,n}^{\alpha}$ represent the light is
incident from the medium $m$ to $n$ (0 means the liquid).

Alternatively, the reflection coefficients for layered structures can be
calculated by a transfer matrix method. The general form is given as
$r=M_{21}/M_{11}$, where $M_{21}$ and $M_{11}$ are the elements of the $M$
matrixZhan et al. (2013). The $M$ matrix is the multiplications of
transmission matrices across different interfaces and propagation matrices in
different layers. Considering an arbitrary $N$-layer system, the $M$-matrix is
given as :

$M=D_{0,1}P(L_{1})D_{1,2}P(L_{2})...D_{N-1,N}P(L_{N})D_{N,N+1},$ (4)

where the transmission matrix $D_{j,j+1}$ is given as:

$D_{j,j+1}=\frac{1}{2}\left[\begin{array}[]{cc}1+\eta&1-\eta\\\
1-\eta&1+\eta\end{array}\right],$ (5)

where $\eta=\varepsilon_{j}(i\xi)K_{j+1}/(\varepsilon_{j+1}(i\xi)K_{j})$ for
p-polarization and $\eta=K_{j+1}/K_{j}$ for s-polarization. The propagation
matric in the $j$-th layer (for both $s$ and $p$ polarizations) is written as:

$P(L_{j})=\left[\begin{array}[]{cc}e^{K_{j}L_{j}}&0\\\
0&e^{-K_{j}L_{j}}\end{array}\right].$ (6)

For example, we have $N=2$ for the multilayered substrate in Fig. 1. The $M$
matrix is given by $M=D_{0,1}P(L_{1})D_{1,2}P(L_{2})D_{2,3}$, where the
subscripts 0, 1, 2 and 3 represent the media of liquid, Teflon, VO2 and Teflon
(from top to down); the thicknesses $L_{1}=L_{T}$, $L_{2}=L_{V}$.

##  III Results and discussions

Figure 1(b) shows the permittivity for different materials, where the used
models and parameters are given in the Appendixes. The dielectric function of
VO2 changes dramatically under different temperatures. For temperature
$T>T_{c}$, VO2 is in the metallic phase and it acts as a poor metal. For
$T<T_{c}$, it is in the insulating phase (or called semiconducting phase), and
the corresponding dielectric function nearly matches that of intrinsic silicon
at low frequency Pirozhenko and Lambrecht (2008). To create repulsive Casimir
forces between two dissimilar plates separated by a liquid, the permittivity
should satisfy
$\varepsilon_{1}(i\xi)>\varepsilon_{liq}(i\xi)>\varepsilon_{2}(i\xi)$ for a
vast range of frequency Munday et al. (2009). Clearly, the dielectric
functions of gold and VO2 (either metallic or insulating phase) are larger
than that of bromobenzene over a wide range of frequency. Therefore, the
Casimir force is always attractive for the layered structure of
gold/bromobenzene/VO2. While the Casimir force for the structure of
gold/bromobenzene/Teflon is repulsive instead. Nonetheless, the Casimir
equilibria can not be found for above two layered structures.

Figure 2: (color online) Casimir pressure via different thicknesses of VO2,
where the thickness $L_{T}$=45 nm and $L_{g}$=40 nm are fixed. (a) Thick
films. The solid and dashed lines represent the pressure for the metallic and
insulating phases of VO2, respectively. (b) Thin films. The positive
(negative) sign of the pressure corresponds to the repulsive (attractive)
force.

###  III.1 Tunable Casimir equilibria for gold nanoplates

Now we consider the Casimir forces as the substrate is composed of a VO2 film
and Teflon (see Fig. 1(a)). The Casimir pressure ($P_{c}=F_{c}/A$) for the
thick film of VO2 is given in Fig. 2(a). The results show that the curves are
almost identical for $L_{\mathrm{V}}$=200, 500 and 1000 nm, indicating the
weak impact of the thickness for thick-film configurations. The pressure is
repulsive at small separation (e.g., $d<60$ nm), making the nanoplate stay
away from the substrate. As the separation increases further, the Casimir
equilibria (zero pressure) occur and quantum traps can be realized for both
metallic (solid lines) and insulating phases (dashed lines). In addition, the
equilibrium distance $d_{c}$ is shifted under the phase transition of VO2. On
the other hand, the thin-film thickness and the phase transition of VO2 can
play an important role in Casimir pressure as shown in Fig. 2(b). For the
thickness $L_{\mathrm{V}}$ =10 and 20 nm, quantum traps can be realized for
the metallic phase, whereas no trap is found for the insulating phase. Under
such configurations, a switch from quantum trapping of the nanoplate(“on”
state) to its release (“off” state) can be triggered by the metal-insulator
transition of VO2. However, the quantum trapping occurs for both metallic and
insulating phases as the thickness $L_{\mathrm{V}}$ increases to 30 nm, and
the “off” state disappears. Compared with the vacuum-separated configuration
Castillo-Garza et al. (2007), not only the magnitude of Casimir forces can be
modified in a liquid environment, but also the sign could be switched (e.g.,
from attraction to repulsion for $d$=100 nm, $L_{V}$=30 nm), due to the phase-
transition of VO2.

Figure 3: (color online) Casimir pressure contributed from different
frequencies and different parallel wavevectors. (a) and (b) $d$=30 nm; (c) and
(d) $d$=85 nm (close to critical separation); (e) and (f) $d$=150 nm. (a),
(c)and (e) VO2 in the metallic phase ($T>T_{c}$); (b), (d) and (f) VO2 in the
insulating phase ($T<T_{c}$). The layer thicknesses are set as
$L_{\mathrm{V}}$=20 nm and $L_{T}$=45 nm.

To understand the switch transition from the “on” to the “off” state, the
contour plots of Casimir pressure are shown in Fig. 3 under different
separations. The sign of pressure is determined by the competition of VO2 film
(attraction) and low-refractive-index Teflon (repulsion). For small separation
$d$=30 nm, the pressure is dominant by the repulsive component as shown in
Figs. 3(a) and 3(b). For the metallic phase, the attractive component
increases and it compensates the repulsive one as the separation becomes 85 nm
($d\approx d_{c}$), resulting in Casimir equilibrium (see Fig. 3(c)). While
the repulsion is still dominant for the insulating phase as shown in Fig.
3(d). As $d$ increases further to 150 nm, the Casimir pressure turns out to be
dominantly attractive in Fig. 3(e) for the metallic phase, resulting in a
restoring force for stable trapping. By contrast, the pressure is still
dominant by repulsion for the insulating phase as shown in Fig. 3(f). The
pressure maps between the metallic and insulating phases are almost identical
for large energy (e.g., $>$2 eV), whereas the discrepancy manifests at low
energy. The results indicate that the attractive component appears only at low
frequency and small $k$ vector for metallic VO2, where the field cannot
penetrate the metalZhao et al. (2019). Conversely, the field can penetrate the
thin-film of insulating VO2 easily, leading to repulsive Casimir forces.

Figure 4: (color online) (a)The equilibrium distances via the thicknesses of
VO2 under three different configurations (see the inset on the right). The
thickness $L_{T}$ is set as 45 nm. The solid (dashed) curves for type III
represent stable (unstable) equilibria. Contour plots of Casimir pressure via
the thicknesses of coating Teflon for (b) metallic VO2 and (c) insulating VO2,
where the thickness $L_{\mathrm{V}}$=20 nm is fixed. In (b) and (c), the gray
zones represent a strong repulsive pressure larger than 1 Pa. The colors of
the curves denote the same meaning as those in (a).

Practically, the influences of gravitation and buoyancy on the force balances
should be taken into account. The condition for the force equilibrium is
written as $\vec{n}\cdot(\mathbf{F}_{c}+\mathbf{F}_{\mathrm{GB}})$=0, where
$\vec{n}$ is the unit vector normal to the surface,
$F_{\mathrm{GB}}=(\rho_{g}-\rho_{liq})gL_{g}A$ is the sum of gravity and
buoyancy, $g$ is the gravitational acceleration, $\rho_{g}\approx$19.3 g/cm3
and $\rho_{liq}\approx$1.50 g/cm3 is the density of gold and liquid
bromobenzene, respectively. The magnitude of $F_{\mathrm{GB}}/A$ is about 7.0
mPa as the thickness $L_{g}$=40 nm. Three types of configurations are depicted
in the inset of Fig. 4(a) for the cross-section views. The type I
configuration corresponds to a zero-projection (or weightlessness in
aerospace), where the switching from quantum trapping (metallic state) to its
release (insulating state) can be obtained as $L_{\mathrm{V}}$ in a proper
range, from about 2 to 22 nm. For type II configuration, the attractive
$F_{\mathrm{GB}}$ can compensate the long-range repulsive Casimir force at
large $d$, leading to stable suspensions for both $T>T_{c}$ and $T<T_{c}$.
However, the equilibrium distances are different, and it can be inferred that
the stiffness of trapping for metallic phase is stronger than that of the
insulating phase. For type III configuration (a flipped down system), the
switching between trapping and its release can also be realized.
Interestingly, there are two equilibrium distances for this configuration. It
is not difficult to know that the smaller equilibrium distance (solid lines)
is stable, whereas the other one (dashed lines) with larger distance is
unstable to small perturbations in position. For both type II and III
configurations, the deviations from Type I become strong as $d_{c}$ is large.

In addition to the thickness of VO2 film, the top-layer Teflon can also play a
significant role in the Casimir effect. The plots of Casimir pressure via the
thicknesses of the coating Teflon $L_{T}$ are shown in Figs. 4(b) and 4(c),
where $L_{\mathrm{V}}$=20 nm is fixed. The results show that the switching
between quantum trapping and it release occurs only when $L_{T}$ is larger
than about 42 nm (no gravity). The larger the $L_{T}$, the larger of the
position for the Casimir equilibrium. As $L_{T}$ is smaller than 42 nm, the
equilibrium distance is also small, and quantum trappings can be realized for
both metallic and insulating phases. For comparison, the gravitation and
buoyancy are taken into account. Again, strong discrepancies among three
configurations occur as the equilibrium positions larger than about 150 nm,
resulting from the comparable magnitude of $F_{GB}$ and the Casimir force. The
impact of $F_{GB}$ can be further reduced by decreasing the thickness $L_{g}$
near the skin depth (about 22 nm) Lisanti et al. (2005).

Figure 5: (color online) Casimir pressure for a complementary design. A thin
film of VO2 with thickness $L_{V}$ is deposited on a Teflon substrate. (a)The
metallic VO2. (b)The insulating VO2. The thickness of the suspended nanoplate
is set as 100 nm.

Figure 6: (color online) Casimir pressure calculated for finite temperatures
and 0 K approximation from Eq. (1). (a)The trapping and release of a gold
nanoplate. The parameters for the substrate are $L_{T}$=45 nm and $L_{V}$=20
nm. (b)The trapping and release of a Teflon nanoplate. The thickness $L_{V}$
is set as 2 nm.

###  III.2 Tunable Casimir equilibria for Teflon nanoplates

The active control of the low-refractive-index nanoplates can also be
significant in many applications. Inspiring by the work Zhao et al. (2019), a
complementary design is schematically shown in the inset of Fig. 5(a). A
Teflon nanoplate is suspended in a liquid of bromobenzene, and the substrate
is a semi-infinite plate of Teflon coated by a VO2 film (high refractive
index). Under such design, the Casimir force is repulsive at very short
separation, due to the dominant interaction between Teflon/bromobenzene/VO2.
As the separation increases, the attractive interaction from
Teflon/bromobenzene/Teflon can be dominant instead, resulting in a stable
Casimir trapping. To verify the design, the Casimir pressure is given
quantitatively in Figs. 5(a) and 5(b) as a function of separation.
Interestingly, the Casimir pressure shows a long-range repulsive behavior for
the metallic VO2, which corresponds to the “off” state. The repulsion pressure
becomes stronger as the thickness $L_{\mathrm{V}}$ enlarges from 2 to 6 nm.
For $L_{\mathrm{V}}$= 2 nm, a Casimir equilibria and strong restoring forces
can be found when VO2 is in the insulating phase. Therefore, the quantum
trapping and release of a Teflon nanoplate can be achieved under the
insulator-to-metal transition of VO2. As the thickness is 4 nm, the restoring
force decreases and the trapping stiffness drops considerably. The calculation
results indicate that the Casimir pressure is quite sensitive to the thickness
of VO2. Due to the low density of Teflon (2.1 g/cm3), the pressure $F_{GB}/A$
for the Teflon nanoplate is about 0.6 mPa, which is reduced significantly
compared with those of gold nanoplates.

###  III.3 Finite temperatures effect

To achieve the phase transition of VO2, the temperatures of the devices need
to be changed. We assume that the dielectric functions of the gold and Teflon
are temperature-independent. For organic liquids, the change of refractive
index due to the temperature Li et al. (1994) is an order of $10^{-4}/$ K, and
the permittivity of bromobenzene is also treated as temperature-independent.
Nonetheless, it is interesting to check the finite temperature effect on
Casimir forces. The integral over frequency $\xi$ in Eq. (1) now is replaced
by a discrete summation Rahi et al. (2009):

$\frac{\hbar}{2\pi}\int_{0}^{\infty}d\xi\leftrightarrow
k_{b}T\overset{\infty}{\underset{n=0}{\sum}}^{\prime},$ (7)

where $\xi$ is replaced by discrete Matsubara frequencies
$\xi_{n}=2\pi\frac{k_{b}T}{\hbar}n(n=0,1,2,3\ldots),$ $k_{B}$ is the
Boltzmann’s constant and the prime denotes a prefactor 1/2 for the term $n$=0.
The Casimir pressures under different temperatures are shown in Figs. 6(a) and
6(b), where two different designs are demonstrated. It is found that the
curves for temperature 320 K (insulating phase) overlap with those calculated
from Eq. (1). For the temperature of 360 K, there is only a small deviation
between 0 K and 360 K. Overall, the calculation results from 320 and 360 K
confirm the accuracy of the 0 K approximation. Recently, the switching between
repulsive and attractive Casimir forces based on PCM has also been reported
Boström et al. (2018), where the equilibrium distances for switching occur
only at several nanometers. The equilibrium distances in our work are more
accessible to experiments, and it can be tuned by designing the geometric
thickness of VO2 and the Teflon.

Figure 7: (color online) The total energy of a suspended gold nanoplate (a)
and a Teflon nanoplate (b) under different types of gravity projection. The
solid and dashed lines represent the cases for the metallic VO2 ($T$=360 K)
and insulating VO2 ($T$=320 K), respectively. The in-plane area $A$ is set as
10 $\mu m\times$10 $\mu m$. Other parameters are kept the same as those in
Fig. 6.

###  III.4 The effect of Brownian motion

In a real configuration, the position of a nanoplate has a fluctuation around
the equilibrium distances due to the Brownian motion. To evaluate the effect
of Brownian motion, the total energy of the suspended nanoplate should be
known, which are written as $U(d)=E_{c}+\Lambda\times(E_{g}+E_{b})$, where
$E_{c}$ is the Casimir energy given by Eq. [1], $E_{g}=\rho_{p}gL_{p}Ad$ and
$E_{b}=-\rho_{liq}gL_{p}Ad$ are respectively the energies caused by the
gravity and buoyancy Phan et al. (2012), $\rho_{p}$ and $L_{p}$ represent the
density and thickness of the suspended nanoplate. The coefficient $\Lambda$ is
the parameter depending on the gravity projection. For type I configuration
(see the inset of Fig. 4), $\Lambda$=0. While we have $\Lambda$=1 and -1 for
type II and type III configurations. The total energy of a gold and Teflon
nanoplate are shown in Figs. 7(a) and 7(b), respectively. The minimum of
$U(d)/k_{B}T$ corresponds to the equilibrium distance $d_{c}$. Clearly, stable
quantum trapping can be realized for a gold (Teflon) nanoplate when VO2 is in
the metallic (insulating) phase. Due to the balance of repulsive Casimir force
and gravity, stable trapping can also be realized for type II configuration.
Theoretically, the transition rate from the equilibrium distance to another
position due to the Brownian motion is proportional to $\exp(-\triangle
U/k_{B}T)$ Phan et al. (2012); Rodriguez et al. (2010b), where $\triangle U$
represents the energy barrier between these two positions. The calculated
results indicate that the transition rates from Casimir equilibria to stiction
are negligible since the energy barriers $\triangle U//k_{B}T$ are quite large
(e.g., over $10^{4}$) for the gold and Teflon nanoplates. For a flipped-down
system (type III), quantum trapping can be realized for gold (Teflon)
nanoplate when VO2 is in the metallic (insulating) phase. However, there is a
nonzero possibility that the nanoplates can escape from the equilibrium
distances to the free-liquid regime ($d\rightarrow\infty$). Fortunately, the
energy barrier $\triangle U/k_{B}T$ for such a transition is the order of
$10^{2}$ as shown in Figs. 7(a) and 7(b), and the transition rate of the
escape is also negligible.

##  IV Conclusions

In summary, the Casimir forces between a nanoplate and a layered structure
containing VO2 films are investigated. In a liquid-separated environment, not
only the magnitude of Casimir forces can be modified, but also the sign could
be switched (e.g., from attraction to repulsion), due to the phase-transition
of VO2. Moreover, a stable Casimir suspension of nanoplates and its tunability
are revealed. For a gold nanoplate, a switch from the quantum trapping to its
release is obtained under the metal-to-insulator transition of VO2. In
addition, the quantum trapping and release of a Teflon nanoplate are
demonstrated with a complementary design. The switching performances due to
the layer thicknesses, gravitation and temperatures are discussed as well.
Theoretically, the bromobenzene can be substituted by other high-refractive-
index liquids (e.g., glycerol and styrene van Zwol and Palasantzas (2010)) as
long as the boiling points are larger than $T_{c}$. The Teflon can also be
replaced by other low-refractive-index materials (e.g., mesoporous silica Dou
et al. (2014)). This work offers the possibility of designing switchable
devices in MEMS/NEMS, resulting from the quantum fluctuations of the
electromagnetic field.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant No. 11804288, No. 11704254, No. 61571386 and No. 61974127), and the
Innovation Scientists and Technicians Troop Construction Projects of Henan
Province. The research of L.X. Ge is further supported by Nanhu Scholars
Program for Young Scholars of XYNU.

##  Appendix A The permittivity of gold

Here, a generalized Drude-Lorentz model is applied for the permittivity of
gold Sehmi et al. (2017):

$\varepsilon(i\xi)=\varepsilon_{D}(i\xi)+\varepsilon_{L}(i\xi),$ (8)

where the Drude term is given by:

$\varepsilon_{D}(i\xi)=\varepsilon_{\infty}+\frac{\gamma\sigma}{\xi(\xi+\gamma)},$
(9)

where $\varepsilon_{\infty}=$0.83409, $\sigma=$3134.5 eV, and $\gamma=$0.02334
eV. The Lorentz term is described by four pairs of poles:

$\varepsilon_{L}(i\xi)=\overset{4}{\underset{j=1}{\sum}}\left(\frac{i\sigma_{j}}{i\xi-\Omega_{j}}+\frac{i\sigma_{j}^{\ast}}{i\xi+\Omega_{j}^{\ast}}\right)$
(10)

where $\sigma_{j}$ and $\Omega_{j}$ are the generalized conductivity and
resonant frequency of the $j$-th Lorentz pole. The star superscripts represent
the operation of complex conjugation. The generalized Drude-Lorentz model
respects causality, and it can represent the exact physical resonances in the
material. The parameters for the model are listed in the Table I.

Table 1: The fitted parameters for Lorentz poles of gold Sehmi et al. (2017). $j$-th |  $\sigma_{j}(\mathrm{eV})$ |  $\Omega_{j}(\mathrm{eV})$
---|---|---
1 |  -0.01743+0.3059*I |  2.6905-0.16645*I
2 |  1.0349+1.2919*I |  2.8772-0.44473*I
3 |  1.2274+2.5605*I |  3.7911-0.81981*I
4 |  9.85+37.614*I |  4.8532-13.891*I

##  Appendix B The permittivity of VO2

For temperature $T>T_{c}$, VO2 is in the metallic phase, and the permittivity
is given by Pirozhenko and Lambrecht (2008); Castillo-Garza et al. (2007)

$\displaystyle\varepsilon(i\xi)$ $\displaystyle=$ $\displaystyle
1+\frac{\omega_{p}^{2}}{\xi(\xi+\gamma)}+\frac{\varepsilon_{\infty}-1}{1+\xi^{2}/\omega_{\infty}^{2}}$
(11)
$\displaystyle+\underset{j=1}{\overset{4}{\sum}}\frac{s_{j}}{1+(\xi/\omega_{j})^{2}+\Gamma_{j}\xi/\omega_{j}},$

where $\varepsilon_{\infty}=3.95,\omega_{p}=3.33$ eV, and $\gamma=0.66$ eV.
The parameters $s_{j}$ and $\Gamma_{j}$ represent respectively the strength
and linewidth of the $j$-th oscillator (resonant frequency $\omega_{j}$).

For temperature $T<T_{c}$, VO2 is in the insulating phase, and the
permittivity is described as

$\varepsilon(i\xi)=1+\frac{\varepsilon_{\infty}-1}{1+\xi^{2}/\omega_{\infty}^{2}}+\underset{j=1}{\overset{7}{\sum}}\frac{s_{j}}{1+(\xi/\omega_{j})^{2}+\Gamma_{j}\xi/\omega_{j}},$
(12)

where $\varepsilon_{\infty}=4.26$ and $\omega_{\infty}=15$ eV. The above
equations for metallic and insulating VO2 are valid for a wide range of
frequency (up to about 10 eV)Castillo-Garza et al. (2007), which are modified
versions of Ref. Verleur et al. (1968). The parameters are listed in Table II.

Table 2: The parameters for the metallic and insulating VO2 Castillo-Garza et al. (2007). $j$-th ($T>T_{c}$) |  $S_{j}$ |  $\omega_{j}(\mathrm{eV})$ |  $\Gamma_{j}$
---|---|---|---
1 |  1.816 |  0.86 |  0.95
2 |  0.972 |  2.8 |  0.23
3 |  1.04 |  3.48 |  0.28
4 |  1.05 |  4.6 |  0.34
$j$-th ($T<T_{c}$) |  $S_{j}$ |  $\omega_{j}(\mathrm{eV})$ |  $\Gamma_{j}$
1 |  0.79 |  1.02 |  0.55
2 |  0.474 |  1.30 |  0.55
3 |  0.483 |  1.50 |  0.50
4 |  0.536 |  2.75 |  0.22
5 |  1.316 |  3.49 |  0.47
6 |  1.060 |  3.76 |  0.38
7 |  0.99 |  5.1 |  0.385
Table 3: The parameters for Teflon(left) and bromobenzene (right)van Zwol and Palasantzas (2010). $j$-th |  $C_{j}$ |  $\omega_{j}(\mathrm{eV})$ |  $C_{j}$ |  $\omega_{j}(\mathrm{eV})$
---|---|---|---|---
1 |  0.0093 |  0.0003 |  0.0544 |  0.00502
2 |  0.0183 |  0.0076 |  0.0184 |  0.0309
3 |  0.139 |  0.0557 |  0.0475 |  0.111
4 |  0.112 |  0.126 |  0.532 |  6.75
5 |  0.195 |  6.71 |  0.645 |  13.3
6 |  0.438 |  18.6 |  0.240 |  24.0
7 |  0.106 |  42.1 |  0.00927 |  99.9
8 |  0.0386 |  77.6 |  |

##  Appendix C The permittivity of Teflon and bromobenzene

The permittivity for the Teflon and bromobenzene are given by the oscillator
model van Zwol and Palasantzas (2010):

$\varepsilon(i\xi)=1+\underset{j}{\overset{}{\sum}}\frac{C_{j}}{1+(\xi/\omega_{j})^{2}},$
(13)

where $C_{j}$ corresponds to the oscillator strength for the $j$-th resonance,
and $\omega_{j}$ is the corresponding resonant frequency. The values of
$C_{j}$ and $\omega_{j}$ listed in Table III are fitted from the experimental
data in a wide range of frequency.

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