# A real open book not fillable by a real Lefschetz fibration

Feri̇t Öztürk  Boğazi̇çi̇ Üni̇versi̇tesi̇, Department of Mathematics, TR-34342
Bebek, İstanbul, Turkey<EMAIL_ADDRESS>and  Nermi̇n Salepci̇
Institut Camille Jordan, Université Lyon I, 43, Boulevard du 11 Novembre 1918
69622 Villeurbanne Cedex, France<EMAIL_ADDRESS>
###### Abstract.

A real 3- or 4-manifold has by definition an orientation preserving smooth
involution acting on it. We consider Lefschetz fibrations of 4-dimensional
manifolds-with-boundary and open book decompositions on their boundary in the
existence of a real structure. We prove that there is a real open book which
cannot be filled by a real Lefschetz fibration, although it is filled by non-
real Lefschetz fibrations.

##  1\. Introduction and basic definitions

There is a very close relation between open book decompositions and Lefschetz
fibrations over $D^{2}$, with fibers surfaces with boundary. It is known that
any open book decomposition is filled by a Lefschetz fibration if its
monodromy can be factorized as a product of positive Dehn twists (work of
Eliashberg, Giroux, Gompf, Loi-Piergallini, Akbulut-Ozbagci etc. See e.g.
[4]). Moreover, two fibrations filling the same open book are isomorphic if
and only if the two such factorizations are Hurwitz equivalent [3].

In the present work we consider Lefschetz fibrations and open book
decompositions on manifolds with a real structure. A real structure on an
oriented $4$-manifold (respectively $3$-manifold) with boundary (possibly
empty) is defined to be an involution which is orientation preserving and the
fixed point set of which is of dimension $2$ (respectively $1$), if it is not
empty. Hence, if $M$ is the oriented boundary of an oriented $4$-manifold $X$,
a real structure on $X$ restricts to a real structure on $M$. We call a
manifold $X$ together with a real structure a real manifold and denote by
$c_{X}$ the real structure on it.

Let $(L,\pi)$ be an open book decomposition of a real 3-manifold $(M,c_{M})$.
Here $L\subset M$ is the binding and $\pi:M-L\rightarrow S^{1}$ is a fibration
with fibers the page surfaces. We say that $(L,\pi)$ is a real open book
decomposition, if $\rho\circ\pi=\pi\circ c_{M}|_{M-L}$ where $\rho:S^{1}\to
S^{1}$ is a reflection [6].

Let $p:E\to B$ be a genus-$g$ Lefschetz fibration of a 4-manifold $E$. A _real
Lefschetz fibration_ is a Lefschetz fibration together with a pair of real
structures $c_{E}:E\to E$ and $c_{B}:B\to B$ commuting with the fiber
structure [8].

In this article we answer a natural question that relates real open books and
real Lefschetz fibrations. We show that there is a Lefschetz fibration that
does not admit a real structure, while the canonical open book on its boundary
is real. We also show that this real open book cannot be filled by any real
Lefschetz fibration with the same fiber topology and with arbitrary number of
singular fibers (Theorem 6).

Acknowledgements. The second author would like to thank A. Degtyarev for
helpful discussions. The work in this article is supported by the Scientific
and Technological Research Council of Turkey [TUBITAK-ARDEB-109T671]. The
second author is supported by the European Community’s Seventh Framework
Programme ([FP7/2007-2013] [FP7/2007-2011]) under grant agreement no [258204].

##  2\. Construction

Let $E$ be an oriented smooth 4-manifold and $B$ an oriented smooth surface. A
_genus- $g$ Lefschetz fibration_ of $E$ is a proper smooth projection $p:E\to
B$ such that $p$ has only finitely many critical points in int($E$) with
pairwise distinct images around which one can choose complex charts such that
the projection takes the form $(z_{1},z_{2})\to z_{1}^{2}+z_{2}^{2}$.
Moreover, the inverse image of a regular value is a closed oriented smooth
surface of genus $g$. It follows from the definition that if $\partial
B\neq\varnothing$, then $\partial E=p^{-1}(\partial B)$ is a fiber bundle over
$\partial B$. The notion of Lefschetz fibration can be slightly generalized to
cover the case of fibers with boundary. Then $E$ turns into a manifold with
corners and its boundary, $\partial E$, becomes naturally divided into two
parts: $p^{-1}(\partial B)$ and $\partial F\times B$. In this case, if, in
particular, $B=D^{2}$, then $\partial E$ admits a canonical open book
decomposition with binding $p^{-1}(0)$ and projection $p|_{\partial E}$. A
_real Lefschetz fibration_ is a fibration together with a pair of real
structures $c_{E}:E\to E$ and $c_{B}:B\to B$ commuting with the fiber
structure.

It is known that around a singular fiber, Lefschetz fibrations are determined
by the monodromy that is a single positive Dehn twist around a simple closed
curve, the _vanishing cycle_. Algebraically, Lefschetz fibrations over
$B=D^{2}$ can be encoded by the factorization $(t_{a_{1}},\dots,t_{a_{n}})$ of
the monodromy $f=t_{a_{n}}\circ\ldots\circ t_{a_{1}}$ (because of the
composition notation the order is reversed) along $\partial D^{2}$ into a
product of positive Dehn twists considered up to Hurwitz equivalence. Namely,
two such factorizations are called _Hurwitz equivalent_ if one can get from
one factorization to the other by a finite sequence of Hurwitz moves:

$\begin{array}[]{lcr}(\ldots,t_{a_{i}},t_{a_{i+1}},\ldots)\to(\ldots,t_{a_{i}}^{-1}\circ
t_{a_{i+1}}\circ t_{a_{i}},t_{a_{i}},\ldots),\\\
(\ldots,t_{a_{i}},t_{a_{i+1}},\ldots)\to(\ldots,t_{a_{i+1}},t_{a_{i+1}}\circ
t_{a_{i}}\circ t_{a_{i+1}}^{-1},\ldots);\end{array}$

and possibly a global conjugation.

In what follows, we consider a genus-$1$ Lefschetz fibration $p:X\to D^{2}$
with exactly two singular fibers. Choose a base point $d\in\partial D^{2}$ and
consider a basis $(\gamma_{1},\gamma_{2})$ of
$\pi_{1}(D^{2}\setminus\\{\mbox{critical values}\\},d)$ obtained by connecting
the base point $d$ to the positively oriented simple loops, each surrounding
the corresponding critical value once. Fix an identification of
$F_{d}=p^{-1}(d)$ with $T^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$. Denote by $a$,
the class on $T^{2}$ of $(1,0)\in\mathbb{R}^{2}$ and by $b$, the class of
$(0,1)$ and consider the curves $u,v$ represented respectively by $3a+5b$ and
$a$ on $T^{2}$. We assume that the monodromy along $\gamma_{1}$ and
$\gamma_{2}$ are given respectively by the positive Dehn twists $t_{u}$ and
$t_{v}$. In other words, the corresponding singular fibers are obtained from
$T^{2}$ by pinching the curves $u$ and $v$. The total monodromy of the
fibration, thus, becomes the composition $f=t_{v}\circ t_{u}$.

Recall that $f\mapsto f_{*}$ defines an isomorphism from the mapping class
group Map($T^{2}$) (the identity component of the space of diffeomorphisms) of
the torus to the group of automorphisms of
$H_{1}(T^{2},\mathbb{Z})\cong\mathbb{Z}a\oplus\mathbb{Z}b$. The latter is
isomorphic to

$\textup{SL(2,\,$\mathbb{Z}$)}=\left\\{[t_{a}]=\small{\left[\begin{array}[]{rr}1&1\\\
0&1\end{array}\right]},[t_{b}]=\small{\left[\begin{array}[]{rr}1&0\\\
-1&1\end{array}\right]}:\begin{array}[]{ll}[t_{a}][t_{b}][t_{a}]=[t_{b}][t_{a}][t_{b}]\\\
([t_{a}][t_{b}])^{6}=\textrm{id}\end{array}\right\\}$

where $[t_{\circ}]$ refers to the matrix representation of the automorphism
$t_{\circ*}$. With respect to the above presentation, $[f]=[t_{v}][t_{u}]$ is
given by the matrix $\small{\left[\begin{array}[]{rr}-39&25\\\
-25&16\end{array}\right]}$.

######  Proposition 1.

$p:X\to D^{2}$ does not admit a real structure.

Proof: Suppose that $p:X\to D^{2}$ admits a real structure. That is to say,
there exist real structures $c_{X}:X\to X$ and $c_{D^{2}}:D^{2}\to D^{2}$ such
that $p\circ c_{X}=c_{D^{2}}\circ p$. Therefore, the critical points as well
as their images are invariant under the action of the real structures $c_{X}$
and $c_{D^{2}}$, respectively, so either they are both real or they are
interchanged by the real structures. By definition of real Lefschetz
fibrations, the decomposition $(t_{u},t_{v})$ associated to
$(\gamma_{1},\gamma_{2})$ is Hurwitz equivalent to a decomposition
$(t_{c_{X}(v)},t_{c_{X}(u)})$ associated to
$(c_{D^{2}}(\gamma_{2}),c_{D^{2}}(\gamma_{1}))$. Although, the positions of
$(\gamma_{1},\gamma_{2})$ can be arbitrary with respect to the real structure
$c_{D^{2}}$, by conjugation by a power of $f$ (such a conjugation preserves
Hurwitz classes) and by moving, if necessary, the base point on the boundary,
we can assume that we have only the two positions shown in Figure 1. In the
former case (the case shown on the left in Figure 1), the fiber $F_{d}$ can be
endowed with a real structure $c$ by pulling a real structure on a real fiber
between the two real singular fibers. Up to isotopy, we can assume that
vanishing cycles $u$, $v$ are invariant under the action of the real structure
$c$. Lemma 2 concerns the intersection number of invariant curves and
prohibits this case, (since the intersection number of $u$ and $v$ is equal to
5).

Figure 1. Two possible positions of $(\gamma_{1},\gamma_{2})$ on base $D^{2}$
regarding the real structure reflection with respect to horizontal.

Now assume that critical points are exchanged by the real structure (this is
the case depicted on the right in Figure 1). Let $c:T^{2}\to T^{2}$ be the
real structure conjugate to the inherited real structure on the base fiber
$F_{d}$. Then, by the assumption we have $c(u)=v$; as a consequence, $u+v$ and
$u-v$ are elements of $\pm$ eigenspaces $H^{c}_{+}$ and, respectively,
$H^{c}_{-}$ of $c_{*}:H_{1}(T^{2},\mathbb{Z})\to H_{1}(T^{2},\mathbb{Z})$.
Thus, the primitive forms of the classes of $u+v$ and $u-v$ form a basis,
respectively of $H^{c}_{+}$ and $H^{c}_{-}$. Moreover, for any real structure
$c$, the intersection number of the bases of $H_{+}^{c}$ and $H_{-}^{c}$ is at
most 2 which is violated by our choice of $u$ and $v$. $\Box$

######  Lemma 2.

The number of intersection points of two invariant curves on a real torus can
only be 0, 1 or 2.

Proof: The real structure restricted to an invariant curve defines an action
on the curve. This action can be the identity, a reflection or an antipodal
involution; let us call those curves a real curve, a reflection curve and an
antipodal curve respectively. Recall that up to equivariant diffeomorphisms,
the real structure on a torus are distinguished by the number of real
components that can be 0,1 or 2. for each real structure we have

  * •

if $c$ has no real components, then there exist two $c$-equivariant isotopy
classes of antipodal curves and no classes of other types;

  * •

if $c$ has one real component, then $T^{2}$ contains a unique $c$-equivariant
isotopy class of non-contractible reflection curves, a unique class of
antipodal curves, and a unique real curve;

  * •

if $c$ has two real components, then $T^{2}$ contains no $c$-equivariant
isotopy class of antipodal curves, a unique class of reflection curves, and
two classes of real curves.

The result follows from the following observations: a reflection curve has two
real points, so the intersection with a real curve must be at 0, 1 or 2
points. Moreover, a reflection curve must intersect an antipodal curve at 0 or
2 points, and real curves and antipodal curves are disjoint. $\Box$

######  Proposition 3.

Let $M=\partial X$; then $M$ as a surface bundle is real.

Proof: By [7, Proposition 4], it is enough to check whether its monodromy is
real, i.e. it admits a decomposition into a product of two real structures.
Real structures on $T^{2}$ correspond to involutive elements of determinant
$-1$ in GL(2, $\mathbb{Z}$). Here we have
$[f]=\small{\left[\begin{array}[]{rr}-39&25\\\
-25&16\end{array}\right]}=\small{\left[\begin{array}[]{rr}5&-3\\\
8&-5\end{array}\right]\left[\begin{array}[]{rr}-120&77\\\
-187&120\end{array}\right]}$, a product of two involutive elements of
determinant $-1$. $\Box$

######  Remark 1.

In [7], an explicit classification of real elements of SL(2, $\mathbb{Z}$) as
well as a particular decomposition of a product of two real structures for
each conjugacy class is given. According to [7, Proposition 4], a hyperbolic
element, i.e. an element with absolute value of trace greater than $2$, of
SL(2, $\mathbb{Z}$) is real if and only if its _cutting period cycle_
$[a_{1},\ldots,a_{2k}]$ is odd-bipalindromic. For hyperbolic matrices, the
cutting period cycle together with the sign of the trace is a complete
invariant of conjugacy classes and it can be recovered from the trace and the
periodic tail of the continued fraction expansion of the slope of the
eigenvectors. In our specific example the cutting period cycle of $f$ is found
to be $[1313]$ which is odd-bipalindromic, i.e. up to cyclic ordering, it has
two palindromic pieces $1$, $313$ of odd length.

So far we have constructed a $T^{2}$-fibered Lefschetz fibration cannot be
real while the $T^{2}$ bundle at its boundary is real. Now we show that a
similar construction can be cooked up to obtain a real open book at the
boundary.

Denote by $\breve{p}:\breve{X}\to D^{2}$ the Lefschetz fibration with boundary
obtained from $p:X\to D^{2}$ above by taking out a neighborhood of a section.
Note that such a section always exists for genus 1 Lefschetz fibrations [5].
The regular fibers of $\breve{p}$ are tori with one boundary component. The
fibration $\breve{p}$ has no real structure, since otherwise $p$ would have
one. However, we have:

######  Proposition 4.

The canonical open book of $\breve{M}=\partial\breve{X}$ admits a real
structure.

Proof: As a consequence of Lemma 1 in [6] it is enough to show that the
monodromy of $\breve{p}|_{\breve{M}}$ is real. Note that the monodromy of
$\breve{p}|_{\breve{M}}$ is an element of
$\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)$, the group of
relative isotopy classes of orientation preserving diffeomorphisms which are
identity on the boundary.

The crucial observation is that being real in Map($T^{2}$) is equivalent to
being real in $\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)$.
Namely, there is a short exact sequence

$0\to\langle
t_{\partial}\rangle\to\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)\to\textup{Map($T^{2}$)}\to
0$

given by the central extension of Map($T^{2}$), where $t_{\partial}$ is the
Dehn twist along the boundary component. Obviously, images of real elements of
$\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)$ are real. For
the converse, suppose im$[f]=[c][c^{\prime}]$ for some
$[f]\in\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)$ and real
structures $[c],[c^{\prime}]\in\textup{GL(2,\,$\mathbb{Z}$)}$. If necessary,
by replacing $c$ by $c\circ g$ and $c^{\prime}$ by $g^{-1}\circ c^{\prime}$
for some diffeomorphism $g$, we can assume that $c$ and hence $c^{\prime}$
leaves $\partial T^{2}$ invariant. We have
$t_{\partial}=c^{\prime}t_{\partial}^{-1}c^{\prime}$. Thus
$t_{\partial}^{-1}c$ is a real structure which we denote by
$c^{\prime\prime}$. As $[f]=[c][c^{\prime}][t_{\partial}]^{k}$, then for some
integer $k$, $[f]=[c][c^{\prime}][c^{\prime}\circ
c^{\prime\prime}]^{k}=[c][c^{\prime\prime}\circ(c^{\prime}\circ
c^{\prime\prime})^{k-1}]$. Being conjugate to a real structure,
$c^{\prime\prime}\circ(c^{\prime}\circ c^{\prime\prime})^{k-1}$ is a real
structure. Thus, $[f]$ is real. $\Box$

With similar hands-on approach as the one above, we prove the following
further result.

######  Proposition 5.

Up to isomorphism preserving the identification $T^{2}\to F_{d}$, there are
exactly two Lefschetz fibrations with exactly two singular fibers filling the
canonical open book on $\breve{M}$; moreover, neither of the fibrations admit
a real structure.

Proof: Let $u^{\prime},v^{\prime}$ be two simple curves on $T^{2}$ such that
$t_{v^{\prime}}\circ t_{u^{\prime}}=f$. We have
$[f]=[t_{v}][t_{u}]=\small{\left[\begin{array}[]{rr}-39&25\\\
-25&16\end{array}\right]}$. Suppose $v^{\prime}$ is the curve defined by
$\alpha a+\beta b$, then a simple calculation gives
$[t_{v^{\prime}}]=\small{\left[\begin{array}[]{rr}1-\alpha\beta&\alpha^{2}\\\
-\beta^{2}&1+\alpha\beta\end{array}\right]}$. All Dehn twists are represented
by matrices whose traces have absolute value equal to $2$. Therefore, from
$[t_{u^{\prime}}]=[t_{v^{\prime}}^{-1}f]$, we get the identity
$25\alpha^{2}+25\beta^{2}-55\alpha\beta=25$. This quadratic Diophantine
equation has solutions $(\pm 1,0)$, $(0,\pm 1)$ and (the transpose of) these
vectors multiplied from left by the powers of the matrix
$S=\small{\left[\begin{array}[]{rr}-3&5\\\ -5&8\end{array}\right]}$ (see e.g
the step-by-step computation of the online application [2]). Note that $u$ and
$v$ are in the solution set. Note also that $u^{\prime},v^{\prime}$, the
curves with class $b$ and $5a+8b$ respectively, are in the solution set too.
The pairs $(t_{u},t_{v})$ and $(t_{u^{\prime}},t_{v^{\prime}})$ are not
Hurwitz equivalent. Indeed in that case there would be an invertible matrix
$K$ with determinant 1, such that either

$K^{-1}[t^{-1}_{u}][t_{v}][t_{u}]K=[t_{u^{\prime}}]\mbox{ and
}K^{-1}[t_{u}]K=[t_{v^{\prime}}]$

or

$K^{-1}[t^{-1}_{v}][t_{u}][t_{v}]K=[t_{u^{\prime}}]\mbox{ and
}K^{-1}[t_{v}]K=[t_{v^{\prime}}].$

However, by straightforward calculation one can show that there exists no such
$K$.

Note also that the matrices $S$ and $[f]$ commute; hence for any pair $(x,y)$
such that $f=t_{y}\circ t_{x}$, the factorization $(S^{k}x,S^{k}y)$ is Hurwitz
equivalent (with a fixed identification) to the factorization $(x,y)$ for
every $k\in\mathbb{Z}$.

As a consequence, we have two Hurwitz equivalence classes given by the pairs
$(t_{u},t_{v})$ and $(t_{u^{\prime}},t_{v^{\prime}})$. To finish, note that
the number of intersection points of $u^{\prime}$ and $v^{\prime}$ is 5, so
the proof of Proposition 1 applies to $(u^{\prime},v^{\prime})$ to show that
the fibrations defined by the decomposition $(t_{u^{\prime}},t_{v^{\prime}})$
is not real. $\Box$

######  Theorem 6.

The canonical open book on $\breve{M}$ cannot be filled by any real Lefschetz
fibration with the same fiber topology and with arbitrary number of singular
fibers.

Proof: We will show that $\breve{M}$ cannot be filled by a (real or non-real)
Lefschetz fibration with the number of singular fibers different from 2.
Recall that

$\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)=\left\\{[t_{a}],[t_{b}]:[t_{a}][t_{b}][t_{a}]=[t_{b}][t_{a}][t_{b}]\right\\}.$

Therefore, the homomorphism

$\begin{array}[]{lrcl}deg:&\textup{Map}(T^{2}\smallsetminus\textup{int}{D^{2}},\partial)&\to&\mathbb{Z}\\\
&t_{a},t_{b}&\mapsto&1\end{array}$

is well-defined; hence the number of Dehn twists that may constitute a given
element $h$ equals $deg(h)$. In our case, $deg(f)=2$, so it cannot admit a
factorization as a product of an arbitrary number of positive Dehn twists
other than 2. Moreover, all possible factorizations into a product of two Dehn
twists are shown above to be non-real. $\Box$

######  Remark 2.

Elements admitting a factorization into a product of two Dehn twists will be
studied in [1]. There the classification of such factorizations up to Hurwitz
equivalence are presented and their relation to real structures are also
elaborated.

## References

  * [1] Degtyarev A., Salepci N. Products of pairs of Dehn twists and maximal real Lefschetz fibrations. Nagoya Math. J. (to appear)
  * [2] Dario Alpern’s two-integer-variable equation solver, http://www.alpertron.com/QUAD:HTM
  * [3] Kas, A. On the handlebody decomposition associated to a Lefschetz pencil. Pacific J. Math. 89 (1980), 89-104.
  * [4] Loi, A.; Piergallini, R. Compact Stein surfaces with boundary as branched covers of $B^{4}$. Invent. Math. 143 (2001), no. 2, 325–348.
  * [5] Moishezon, B. Complex surfaces and connected sums of complex projective planes. Lecture notes in Math. 603 Springer Verlag(1977).
  * [6] Öztürk, F., Salepci, N. Real open books and real contact structures, preprint.
  * [7] Salepci, N. Real elements in the mapping class group of $T^{2}$, Topology and its Applications 157 (2010) 2580–2590.
  * [8] Salepci, N. Real Lefschetz fibrations. Thèse, Université Louis Pasteur, Strasbourg, 2007. Université Louis Pasteur. Institut de Recherche Mathématique Avancée (IRMA), Strasbourg, 2007


# Molecular beam epitaxy of three-dimensionally thick Dirac semimetal Cd3As2
films

Y. Nakazawa  Department of Applied Physics and Quantum-Phase Electronics
Center (QPEC), the University of Tokyo, Tokyo 113-8656, Japan  M. Uchida
<EMAIL_ADDRESS>Department of Applied Physics and Quantum-Phase
Electronics Center (QPEC), the University of Tokyo, Tokyo 113-8656, Japan
PRESTO, Japan Science and Technology Agency (JST), Tokyo 102-0076, Japan  S.
Nishihaya  Department of Applied Physics and Quantum-Phase Electronics Center
(QPEC), the University of Tokyo, Tokyo 113-8656, Japan  S. Sato  Department of
Applied Physics and Quantum-Phase Electronics Center (QPEC), the University of
Tokyo, Tokyo 113-8656, Japan  A. Nakao  RIKEN Bioengineering Laboratory, Wako
351-0198, Japan  J. Matsuno  Department of Physics, Osaka University, Osaka
560-0043, Japan  M. Kawasaki  Department of Applied Physics and Quantum-Phase
Electronics Center (QPEC), the University of Tokyo, Tokyo 113-8656, Japan
RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan

###### Abstract

Rapid progress of quantum transport study in topological Dirac semimetal,
including observations of quantum Hall effect in two-dimensional (2D) Cd3As2
samples, has uncovered even more interesting quantum transport properties in
high-quality and three-dimensional (3D) samples. However, such 3D Cd3As2 films
with low carrier density and high electron mobility have been hardly obtained.
Here we report the growth and characterization of 3D thick Cd3As2 films
adopting molecular beam epitaxy. The highest electron mobility ($\mu$ = 3
$\times$ 104 cm2/Vs) among the reported film samples has been achieved at a
low carrier density (n = 5 $\times$ 1016 cm-3). In the magnetotransport
measurement, Hall plateau-like structures are commonly observed in spite of
the 3D thick films (t = 120 nm). On the other hand, field angle dependence of
the plateau-like structures and corresponding Shubunikov-de Haas oscillations
rather shows a 3D feature, suggesting the appearance of unconventional
magnetic orbit, also distinct from the one described by the semiclassical
Weyl-orbit equation.

Topological Dirac semimetal (DSM) is characterized by surface Fermi arcs and
bulk Dirac dispersions along any directions in three-dimensional (3D) momentum
space YoungPRL2012 ; WangPRB2012 ; WangPRB2013 ; YangNatCommun2014 ;
ArmitageRevModPhys2018 . As DSM is driven into other exotic topological phases
including Weyl semimetal, topological insulator, and trivial insulator such as
by symmetry breaking or dimensionality control WangPRB2012 ; WangPRB2013 ;
LiuNatMater2014 ; UchidaNatComm2017 ; CollinsNature2018 , it has attracted
both theoretical and experimental research efforts as an ideal parent
material. DSM has also gathered significant interest because of its
characteristic quantum transport originating from the non-trivial electronic
structure; for examples, surface conduction based on Fermi-arcs WanPRB2011
and negative magnetoresistance induced by chiral anomaly SonPRB2013 ;
BurkovPRL2014 . In paticular, the existence of novel orbital trajectories
under a magnetic field, so called Weyl-orbit, which consists of both the
surface Fermi-arcs and bulk chiral modes, has been predicted PotterNatComm2014
; ZhangSciRep2016 .

Among the candidates of DSM, Cd3As2 is one of the most studied materials
hosting simple band structure aroud the Dirac points NeupaneNatComm2014 ;
LiuNatMater2014 ; JeonNatMater2014 ; BorisenkoPRL2014 ; NishihayaPRB2018 . So
far, a number of studies have been reported on the quantum transport in Cd3As2
HePRL2014 ; LiangNatMater2015 ; WangNatComm2016 ; MollNature2016 , including
observations of quantum Hall effect (QHE) UchidaNatComm2017 ;
NishihayaSciAdv2018 ; ZhangNatComm2017_2 ; ZhangNature2018 ; StemmerPRL2018 ;
LinPRL2019 . Particularly, QHE observed even in three-dimensional Cd3As2 nano-
plates (t = 70 $\sim$ 80 nm) ZhangNatComm2017_2 ; ZhangNature2018 ; LinPRL2019
has indicated the quantization of two-dimensional (2D) surface states to
demand subsequent studies for investigating its origin. However, most of bulk
Cd3As2 samples, synthesized by such as melt growth NeupaneNatComm2014 ;
LiuNatMater2014 ; JeonNatMater2014 ; ZhaoPRX2015 ; NarayananPRL2015  and
chemical vapor deposition ZhangNatComm2017_2 ; ZhangNature2018 ; LinPRL2019 ,
has relatively high carrier density ($\geq$ 1018 cm-3). Therefore, fabrication
of Cd3As2 films with lower carrier density and higher electron mobility has
been required. In order to realize it, molecular beam epitaxy (MBE) is an
useful technique, since the films can be grown by individually controlling the
flux of each element. While there have been some reports on the MBE growth of
Cd3As2 films StemmerAPLMat2016 ; StemmerPRB2017 ; StemmerPRB2018 ; XiuNPG2015
; XiuSciRep2016 ; ChengNJP2016 ; XiuPRB2018 , the films showing quantum
transport have not been thick enough to reflect the 3D nature of topological
Dirac semimetal. In spite of its necessity, high-quality 3D Cd3As2 films have
been hardly obtained mainly because of the high volatility of elements that
require a low growth temperature.

Here we report MBE growth and characterization of 3D thick Cd3As2 films. The
highest electron mobility has been achieved among the reported film samples.
Even in the thickness regime of t $>$ 100 nm, Hall plateau-like structures are
observed together with Shubnikov-de Haas oscillations. Though the details of
mechanisms are still unclear, the unconventional behaviors suggest the
importance of coexistence of surface and bulk quantum states.

Cd3As2 films were grown in an EpiQuest RC1100 MBE system on single crystalline
(111) CdTe substrates (see also supplemental material for details). CdTe has a
zincblende structure, and the lattice mismatch between Cd3As2 and CdTe is 2.3
%. The molecular beams were provided from a conventional Knudsen cell
containing Cd (6N, Osaka Asahi Co.) and an MBE-Komponenten valved cracker
source containing As (7N5, Furukawa Co.), respectively. The reservoir
temperature of the cracker source was set to 600 ∘C to sublimate arsenic as
tetramers (As4). The CdTe substrate was etched using 0.01 % Br2-methanol to
remove native oxides just before loading it into the MBE chamber
StemmerAPLMat2018 . Prior to the growth, the substrate was heated up to 500 ∘C
with supplying As flux. After confirming the change of the in situ reflection
high-energy electron diffraction (RHEED) pattern, from a three-dimensional
transmission pattern to a streak pattern, the substrate was cooled down to the
growth temperature of 200 ∘C. The beam equivalent pressures were measured by
an ionization gauge, and were set to 1.4 $\times$ 10-4 Pa for Cd and 1.2
$\times$ 10-4 Pa for As4 during the co-deposition growth. This As-rich growth
condition serves to reduce the carrier density of the Cd3As2 films, because As
deficiency is a major origin of the electron carriers in Cd3As2. The film
thickness was set at 120 nm, and the growth rate was about 0.7
$\mathrm{\AA}$/s.

Figure 1 summarizes fundamental characterization results of a Cd3As2 thick
film (sample A). As shown in the x-ray diffraction (XRD) $\theta$-2$\theta$
scan (Fig. 1(a)), the reflections from the {1 1 2} lattice planes of Cd3As2
are observed without any impurity phases. The rocking curve of the (2 2 4)
Cd3As2 film peak is shown in Fig. 1(b). The full width at half maximum (FWHM)
of the film peak is 0.59∘, which is still broader than the bulk values
HePRL2014 . An XRD reciprocal space map around the (10 2 12) Cd3As2 film and
the (5 1 3) CdTe substrate peaks is shown in Fig. 1(c). The in-plane lattice
constant along [1 $\overline{1}$ 0] and the out-of-plane lattice spacing along
[1 1 2] of the Cd3As2 film are calculated to be 17.81 $\mathrm{\AA}$ and 7.321
$\mathrm{\AA}$, respectively, indicating that the Cd3As2 film is free from
strain by the CdTe substrate.

A RHEED pattern taken along [1 1 $\overline{1}$] azimuth of the Cd3As2 film is
shown in Fig. 1(d). It shows a typical streak pattern, indicative of two-
dimensionally flat film surface. The corresponding atomic force microscopy
(AFM) image and the thickness profile are shown in Fig. 1(e) and (f). The
Cd3As2 film has step and terrace structure, and the step height corresponds to
the lattice spacing of the Cd3As2 (112) planes ($\sim$ 7 $\mathrm{\AA}$).
Figure 1(g) shows temperature dependence of the resistance $R_{\mathrm{xx}}$.
The semiconducting temperature dependence down to 2 K also indicates the
suppression of the As deficiency or electron carriers.

Figures 2(a)–(c) show out-of-plane transverse magnetoresistance
$R_{\mathrm{xx}}$ and Hall resistance $R_{\mathrm{yx}}$ of typical Cd3As2
thick films (samples A, B, and C) measured at 2 K. The Cd3As2 samples were cut
into 5 mm $\times$ 1 mm for transport measurement using the conventional four-
terminal method. For the side electrodes flowing electric current, silver
paste was attached to cover both of the entire sides and to avoid the current-
jetting effect induced by inhomogeneous current flow ReisNJP2016 . Reflecting
the low carrier densities as deduced from the Hall slope, Shubunikov-de Haas
oscillations are few but clearly observed for all the three samples. If an
isotropic 3D Fermi surface is assumed for given carrier densities, the
magnetic field where the system reaches the quantum limit is estimated at
about 4, 5, and 7 T for samples A, B, and C, respectively. On the other hand,
the quantum oscillations corresponding to higher filling factors of $\nu$ = 4
or 8 are observed around these fields. They are not the lowest filling
factors, since the double ($\nu$ = 4, 6, $\ldots$) or four-fold ($\nu$ = 8,
12, $\ldots$) degeneracy is observed within the measurement field range.

The low-temperature electron mobility $\mu$ and carrier density n of our thick
films (red closed circles) are summarized in Fig. 2(d) together with other
reported data. The mobility reaches a maximum of $\mu$ = 3 $\times$ 104 cm2/Vs
for sample A, while the carrier density is reduced down to n = 5 $\times$ 1016
cm-3, which is one-and-a-half order of magnitude lower than the previously
reported PLD-grown films (blue diamonds) UchidaNatComm2017 ;
NakazawaSciRep2018 . The reduction of the carrier density is owing to the
suppression of As deficiency by the As-rich growth condition.

To investigate the origins of the quantum oscillations in $R_{\mathrm{xx}}$
and the plateau-like structure in $R_{\mathrm{yx}}$, the field angle
dependence of $R_{\mathrm{xx}}$ and $R_{\mathrm{yx}}$ was measured. Figures 3
(a) and (b) show their dependence taken for sample A, where the magnetic field
is tilted from the out-of-plane direction ($\theta$ = 0∘) to the current
direction ($\theta$ = 90∘). The quantum oscillations in $R_{\mathrm{xx}}$ and
the plateau-like structures in $R_{\mathrm{yx}}$ remain to be resolved up to
$\theta$ = 75∘. For clarity, $R_{\mathrm{xx}}$,
$\frac{-d^{2}R_{\mathrm{xx}}}{dB^{2}}$, and $\frac{-dR_{\mathrm{yx}}}{dB}$ are
compared for each field angle in Figs. 3(c)–(f). Even under rotation of
$\theta$, the peak and valley positions confirmed in the derivatives do not
largely shift. This field-angle-independent behavior of $R_{\mathrm{xx}}$
indicates that the 3D Fermi surface is realized in the thick Cd3As2 film, also
consistent with thickness dependence previously studied using Cd3As2 films
UchidaNatComm2017 . On the other hand, plateau-like structures in
$R_{\mathrm{yx}}$ are indicative of a 2D electronic state (Figs. 2(a)–(c) and
Fig. 3(b)). Namely, the correspondence of the oscillation phases between the
derivative of the Hall resistance ($\frac{-dR_{\mathrm{yx}}}{dB}$) and the
oscillation component of the magnetoresistance
($\frac{-d^{2}R_{\mathrm{xx}}}{dB^{2}}$) is a sign of the quantum Hall state.
This coexistence of the 2D feature and its 3D angle dependence can not be
explained either by a simple 2D nor 3D model.

Novel orbital trajectory, so called Weyl-orbit, has been theoretically
predicted for topological semimetals under a magnetic field PotterNatComm2014
; ZhangSciRep2016 , where the two Fermi arcs on opposite surfaces of the
sample are connected by the bulk chiral mode ($N=0$ Landau level). Quantum
oscillations from the Weyl-orbit occur at the following magnetic fields,

$\frac{1}{B_{n}}=\frac{e}{S_{k}}\left[2\pi(n+\gamma)\mathrm{cos}\theta-t(k_{\mathrm{W}\parallel}(\theta)+2k_{\mathrm{F}\parallel}(\theta))\right]$
(1)

where $n$ is the index of the Landau level, $\gamma$ is the constant phase
offset, $S_{k}$ is the $k$-space area enclosed by the two Fermi arcs combined,
$\theta$ is the tilting angle of the magnetic field from the surface normal.
$k_{\mathrm{W}\parallel}(\theta)$ and $k_{\mathrm{F}\parallel}(\theta)$ are
the field parallel components of the wave vector from +1 to -1 chirality Weyl
nodes and that of the Fermi wave vector, respectively. However, the observed
3D angle dependence can not be explained even by the semiclassical Weyl-orbit
picture. If the observed quantum oscillations can be described by Eq.(1), the
angle-independent behavior is understood in a way that the total change in the
right side of Eq.(1) is nearly independent of $\theta$. But, the second term
$-t(k_{\mathrm{W}\parallel}(\theta)+2k_{\mathrm{F}\parallel}(\theta))$ does
not change such that it compensates the change in the first term
$2\pi(n+\gamma)\mathrm{cos}\theta$. Moreover, the right side of the Eq. (1)
can even take negative values for a part of the measured range of $\theta$,
for such low-carrier-density samples. To understand the characteristic quantum
transport emerging in the 3D thick films, systematic control of the Fermi
level such as by chemical substitution or electrostatic gating will be
necessary as a future work.

In summary, 3D thick Cd3As2 films with low carrier density and high electron
mobility have been obtained by using molecular beam epitaxy. The electron
mobility of $\mu$ = 3 $\times$ 104 cm2/Vs, the highest value among the
reported film samples, has been achieved. The coexistence of Hall plateau-like
structure and 3D field angle dependence can not be interpreted by either
simple 2D or 3D model. The MBE-grown Cd3As2 films will be an ideal platform
for further investigation of the quantum Hall state based on unconventional
magnetic orbits which are also distinct from the one described by the
semiclassical Weyl-orbit equation.

We acknowledge H. Ishizuka, H. Sakai, Y. Araki, K. Muraki, N. Nagaosa, and Y.
Tokura for fruitful discussions. We also thank M. Tanaka and S. Ohya for
technical advice about the handling of arsenides. This work was supported by
JST PRESTO Grant No. JPMJPR18L2 and CREST Grant No. JPMJCR16F1, Japan, and by
Grant-in-Aids for Scientific Research (B) No. JP18H01866 from MEXT, Japan.

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FIG. 1. (a) XRD $\theta$–2$\theta$ scan of a Cd3As2 film (sample A) grown on a
(111) CdTe substrate. (b) Rocking curve of the (224) Cd3As2 film peak. (c) XRD
reciprocal space map around the (10 2 12) Cd3As2 film and the (5 1 3) CdTe
substrate peaks. (d) RHEED image of the Cd3As2 film viewed along the
[11$\overline{1}$] azimuth direction. (e) AFM image and (f) thickness profile
of the film. The thickness profile is measured along the broken line in the
AFM image. (g) Temperature dependence of the longitudinal resistance
$R_{\mathrm{xx}}$ and resistivity $\rho_{\mathrm{xx}}$ measured for the same
film.

FIG. 2. (a)–(c) Out-of-plane transverse magnetoresistances $R_{\mathrm{xx}}$
and Hall resistances $R_{\mathrm{yx}}$ measured for Cd3As2 thick films. The
horizontal bars represent the quantized Hall resistances
$R_{\mathrm{yx}}=-\frac{h}{\nu e^{2}}$ with the filling factors $\nu$. (d)
Electron mobility $\mu$ vs. carrier density $n$ plotted for our present Cd3As2
films (red closed circles). For comparison, previously reported values are
also plotted, including MBE-grown films at the University of California, Santa
Barbara (UCSB, green triangles) StemmerAPLMat2016 ; StemmerPRB2017 ;
StemmerPRL2018 ; StemmerAPLMat2018 ; StemmerPRB2018 ; StemmerPRMat2018 , MBE-
grown films at Fudan University (yellow squares) XiuNPG2015 ; XiuSciRep2016 ;
ChengNJP2016 ; XiuPRB2018 , our PLD-grown films UchidaNatComm2017 ;
NakazawaSciRep2018  (blue diamonds), and bulk samples ($+$ HePRL2014 ,
$\lozenge$ MollNature2016 , $\triangle$ ZhangNatComm2017_2 ,
$\bigtriangledown$ ZhangNature2018 , $\lhd$ NarayananPRL2015 , $\square$
LiangNatMat2015 ,$\rhd$ ZhangNatComm2017 , $\bowtie$ Rosenberg1959 , $\times$
CaoNatComm2015 , $\odot$ Rogers1971 , and $\otimes$ WeberAPL2015 ). The sample
discussed in this paper are denoted as samples A, B, and C.

FIG. 3. (a) Transverse magnetoresistance $R_{\mathrm{xx}}$ and (b) Hall
resistance $R_{\mathrm{yx}}$, measured for the sample A at different field
angles. (c)–(f) Field angle dependence of transverse magnetoresistance, second
derivative of $R_{\mathrm{xx}}$, and first derivative of $R_{\mathrm{yx}}$.

Supplementary materials for
Molecular beam epitaxy of three-dimensionally thick
Dirac semimetal Cd3As2 films
Y. Nakazawa1, M. Uchida1,2,∗, S. Nishihaya1, S. Sato1, A. Nakao3,

J. Matsuno4, and M. Kawasaki1,5

1Department of Applied Physics and Quantum-Phase Electronics Center (QPEC),
the University of Tokyo, Tokyo 113-8656, Japan
2PRESTO, Japan Science and Technology Agency (JST), Tokyo 102-0076, Japan
3Institute for Innovation in International Engineering Education (IIIEE),
the University of Tokyo, Tokyo 113-8656, Japan
4Department of Physics, Osaka University, Osaka 560-0043, Japan
5RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
∗Correspondence to: E-mail<EMAIL_ADDRESS>
Br2-methanol etching of CdTe substrate

The CdTe substrate was etched using 0.01 % Br2-methanol just before loading it
into the MBE chamber. For chemical analysis of the substrate surface, x-ray
photoelectron spectroscopy (XPS) was performed on the CdTe substrate before
and after the etching. Figure S1(a) shows Te 3d spectra of the CdTe substrate
before (red) and after (blue) the Br2-methanol etching. Before the etching,
four Te 3d peaks are observed. While the two peaks at the binding energy of
573 and 584 eV are mainly assigned to Te2- in CdTe, the other two peaks at 577
and 587 eV are assigned to Te4+ in TeO2 [S1, S2], which is the native oxide
formed on the substrate surface [S3]. These TeO2 peaks are more pronounced
when the incident x-ray is tilted by 30∘ from the plane (black), for providing
higher sensitivity to the surface. After the Br2-methanol etching, the peaks
assigned to TeO2 disappear, indicating the oxide layer is successfully removed
from the CdTe substrate surface.

FIG. S1. (a) XPS spectra of the CdTe substrate before (red) and after (blue)
the etching. The spectrum taken with the incident x-ray angle of 30∘ is also
shown (black) to show surface sensitive result. (b) Configuration of the XPS
measurement.

Substrate annealing with supplying As4 flux

Before the growth of Cd3As2 films, the CdTe substrate etched by the
Br2-methanol was annealed at 500 ∘C for 5 minutes inside the MBE chamber with
supplying As4 flux. For clarifying the effect of the annealing process, XPS
was performed for the CdTe substrates annealed with and without supplying As4
flux. The two annealing sequences are shown in Figs. S2 (a) and (b). Figure S2
(c) compares As 2p spectra taken for the CdTe substrates annealed with (blue)
and without (red) supplying the As4 flux. The As 2p peak is observed only for
the substrate annealed with the As4 flux, indicating the surface modification
occurs accompanied with bonding to As.

FIG. S2. (a) and (b) Two annealing processes for the CdTe substrate. (c) XPS
spectra taken for the CdTe substrates annealed with (blue) and without (red)
supplying the As4 flux.

Surface morphology of CdTe substrates and Cd3As2 films

Figure S3 shows the atomic force microscopy (AFM) images with thickness
profiles and reflection high-energy electron diffraction (RHEED) patterns
taken at representative stages during the sequence of the Cd3As2 film growth.
Before the Br2-methanol etching, particle-like structures are found on the
CdTe substrate and RHEED shows a typical three-dimensional spotty pattern with
low intensity (Figs. S3 (b) and (d)). After the Br2-methanol etching, the
particle-like structures are removed and the spotty RHEED pattern becomes
clearer (Figs. S3 (e) and (g)). The RHEED pattern changes to the two-
dimensional streak pattern during the annealing at 500 ∘C with supplying the
As4 flux, indicating the surface modification occurs (Figs. S3(h) and (j)).
After the film growth, RHEED pattern becomes sharp streaks (Fig. S3 (m)) and
the AFM image (Fig. S3 (k)) is free from particles and shows step and terrace
structure, as shown in Fig. 1(e) in the main text.

FIG. S3. (a) Sequence of the Cd3As2 film growth. (b)-(m) AFM images, thickness
profiles along the broken line, and RHEED patterns taken at representative
stages indicated in (a). ((b)-(d) before the Br2-methanol etching (e)-(g)
after the Br2-methanol etching (h)-(j) after annealing at 500 ∘C with
supplying As4 flux (k)-(m) after the film growth).

Supplementary references

[S1] J. F. Moulder, W. F. Stickle, P. E. Sobol, and K. D. Bomben, Handbook of
X-Ray Photoelectron Spectroscopy (Physical Electronics Division, Perkin-Elmer
Corporation, Waltham, 1992).

[S2] I. M. Kotina, L. M. Tukhkonen, G. V. Patsekina, A. V. Shchukarev,and G.
M. Gusinskii, Semicond. Sci. Technol. 13, 890 (1998).

[S3] R. Triboulet and P. Siffert, CdTe and Related Compounds; Physics,
Defects, Hetero- and Nano-structures, Crystal Growth, Surfaces and
Applications (Elsevier, Amsterdam, 2010).