# On digraphs without onion star immersions††thanks: This work is a part of
the projects CUTACOMBS (ŁB, OD, KO) and BOBR (MP) that have received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreements No 714704 and 948057,
respectively).

Łukasz Bożyk  Institute of Informatics, University of Warsaw, Poland  Oscar
Defrain  Institute of Informatics, University of Warsaw, Poland  LIS, Aix-
Marseille Université, France
Karolina Okrasa  Institute of Informatics, University of Warsaw, Poland
Faculty of Mathematics and Information Science,
Warsaw University of Technology, Poland  Michał Pilipczuk  Institute of
Informatics, University of Warsaw, Poland

(November 29, 2022)

###### Abstract

The $t$-onion star is the digraph obtained from a star with $2t$ leaves by
replacing every edge by a triple of arcs, where in $t$ triples we orient two
arcs away from the center, and in the remaining $t$ triples we orient two arcs
towards the center. Note that the $t$-onion star contains, as an immersion,
every digraph on $t$ vertices where each vertex has outdegree at most $2$ and
indegree at most $1$, or vice versa.

We investigate the structure in digraphs that exclude a fixed onion star as an
immersion. The main discovery is that in such digraphs, for some duality
statements true in the undirected setting we can prove their directed
analogues. More specifically, we show the next two statements.

  * •

There is a function $f\colon\mathbb{N}\to\mathbb{N}$ satisfying the following:
If a digraph $D$ contains a set $X$ of $2t+1$ vertices such that for any
$x,y\in X$ there are $f(t)$ arc-disjoint paths from $x$ to $y$, then $D$
contains the $t$-onion star as an immersion.

  * •

There is a function $g\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}$
satisfying the following: If $x$ and $y$ is a pair of vertices in a digraph
$D$ such that there are at least $g(t,k)$ arc-disjoint paths from $x$ to $y$
and there are at least $g(t,k)$ arc-disjoint paths from $y$ to $x$, then
either $D$ contains the $t$-onion star as an immersion, or there is a family
of $2k$ pairwise arc-disjoint paths with $k$ paths from $x$ to $y$ and $k$
paths from $y$ to $x$.

20(-1.9, 7.38)  20(-2.15, 7.68)

##  1 Introduction

A graph $H$ is an immersion of a graph $G$ if one can injectively map vertices
of $H$ to vertices of $G$ and edges of $H$ to edge-disjoint paths in $G$ so
that the image of every edge connects the images of its two endpoints. Thus,
immersions are an embedding notion for (undirected) graphs that is based on
edge-disjointness and edge cuts, as opposed to the notions of minors and of
topological minors, which are based on vertex-disjointness and (vertex)
separations.

The Graph Minor series of Robertson and Seymour brought fundamental
understanding of the minor order in graphs. The key components of this
understanding are the Grid Minor Theorem [RS86], which connects the existence
of large grid minors with a dual notion of tree-likeness—the treewidth—and the
Structure Theorem [RS03], which describes the structure in graphs that exclude
a fixed graph $H$ as a minor. The analogues of these two tools have been
understood also in the setting of immersions. In [Wol15], Wollan introduced
the parameter tree-cut width and proved the Wall Immersion Theorem, a duality
theorem connecting the tree-cut width to the largest size of a wall that can
be found in a graph as an immersion. In the same work, he also gave a
structure theorem that describes graphs excluding a fixed graph $H$ as an
immersion. Also, as a part of the Graph Minors series, Robertson and Seymour
[RS10] proved that the immersion order is a well quasi-order on graphs. We
invite the reader to [DMMS13, MW14, DW16, GPR+21, GKRT21, Liu21, Liu22,
BDOP22] for other works on structural properties of graphs with forbidden
immersions.

The notion of an immersion can be naturally generalized to directed graphs
(digraphs) by considering mapping every arc to a directed path leading from
the image of the tail to the image of the head. Recently, there has been
significant interest in proving directed analogues of the advances of the
theory of (undirected) minors. In particular, Kawarabayashi and Kreutzer
[KK15] proved the directed variant of the Grid Minor Theorem, while a series
of recent papers [GKKK20, GKKK22] is gradually working towards a directed
analogue of the Structure Theorem. From this point of view, it is natural to
ask whether a similar structure theory can be developed for directed
immersions. It seems that so far, not much is known in this direction. We
remark that a meaningful structure theory for tournaments with forbidden
immersions was developed, see [CS11, COS12, Ray18, FP19, BP22], but this
restricted setting is very different from the setting of general digraphs.
Also, there has been work on finding appropriate degree restrictions that
force the existence of immersions of large complete digraphs or large
transitive tournaments, see [DMMS12, Loc19].

The aim of this paper is to make the first modest steps towards a structure
theory for digraphs excluding a fixed digraph as an immersion. A more specific
motivation is to provide opening moves towards a statement linking directed
wall immersions with suitable width measures for digraphs.

### Our contribution.

Consider the $t$-onion star: the digraph depicted in Figure 1 below. It is
obtained from the star with $2t$ leaves by replacing every edge with a triple
of arcs; for $t$ edges two of the arcs are oriented towards the center and one
away from the center, and for the remaining $t$ edges we use the reverse
orientation. Note the following.

Figure 1: The onion star.

######  Observation 1.1.

Suppose $D$ is a digraph on $t$ vertices where every vertex has outdegree at
most $2$ and indegree at most $1$, or vice versa. Then the $t$-onion star
contains $D$ as an immersion.

Therefore, excluding the $t$-onion star as an immersion, for some fixed
$t\in\mathbb{N}$, is a weaker condition than excluding any fixed digraph $D$
satisfying the premise of 1.1; this in particular applies to any reasonable
notion of a directed wall. The $t$-onion star is a directed analogue of the
graph $S_{3,t}$ that was used by Wollan in [Wol15] as an obstruction commonly
found in various avenues of his proof of the (undirected) Wall Immersion
Theorem.

For two vertices $x,y$ in a digraph $D$, let $\mu(x,y)$ be the maximum number
of arc-disjoint paths from $x$ to $y$ that one can find in $D$. As an opening
step of his proof, Wollan proved the following statement.

######  Theorem 1.2 (Wollan, [Wol15, Lemma 1]).

Suppose a graph $G$ contains a set $X$ consisting of $t+1$ vertices such that
for all distinct $x,y\in X$, we have $\mu(x,y)\geqslant t^{2}$. Then $G$
contains the complete graph $K_{t}$ as an immersion.

The proof is a relatively easy application of flow-cut duality, which
nonetheless crucially relies on the undirectedness of the graph. The main
result of this paper is the following weak directed analogue of theorem 1.2.

######  Theorem 1.3.

There exists a function $f\colon\mathbb{N}\to\mathbb{N}$ such that the
following holds. Suppose a digraph $D$ contains a set $X$ consisting of $2t+1$
vertices such that for all distinct $x,y\in X$, we have $\mu(x,y)>f(t)$. Then
$D$ contains the $t$-onion star as an immersion.

Note that in theorem 1.3 we obtain an obstruction that is weaker than that of
theorem 1.2: the $t$-onion star instead of a complete digraph.

The proof of theorem 1.3 applies the same basic flow-cut strategy as the proof
of theorem 1.2 due to Wollan, but there is a major issue. For any $x,y\in X$,
the premise of theorem 1.3 provides the existence of a large family
$\mathcal{P}_{x\to y}$ of arc-disjoint paths from $x$ to $y$, and of a large
family $\mathcal{P}_{y\to x}$ of arc-disjoint paths from $y$ to $x$. However,
in principle every path of $\mathcal{P}_{x\to y}$ could intersect every path
of $\mathcal{P}_{y\to x}$, while for the abovementioned strategy to work, we
need a family containing many arc-disjoint paths from $x$ to $y$ and many from
$y$ to $x$ that are also arc-disjoint between each other. In general digraphs,
there is no hope for exposing such a family: in Figure 2 we give an example
where $\mu(x,y)$ and $\mu(y,x)$ can be arbitrarily large, but one cannot find
even two arc-disjoint paths: one from $x$ to $y$ and one from $y$ to $x$.
However, we prove that the desired statement holds under the assumption of
excluding an onion star.

Figure 2: An example of a digraph that contains a large family of arc-disjoint
paths from $x$ to $y$ (in blue), a large family of arc-disjoint paths from $y$
to $x$ (in red), but no two arc disjoint paths such that one goes from $x$ to
$y$, and the other one from $y$ to $x$.

######  Theorem 1.4.

There exists a function $g\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ such
that the following holds. Suppose $D$ is a digraph and $x,y$ are two distinct
vertices in $D$ such that $\mu(x,y)>g(t,k)$ and $\mu(y,x)>g(t,k)$, for some
$t,k\in\mathbb{N}$. Then at least one of the following holds:

  * •

$D$ contains the $t$-onion star as an immersion;

  * •

in $D$ there is a family of $2k$ arc-disjoint paths consisting of $k$ paths
from $x$ to $y$ and $k$ paths from $y$ to $x$.

theorem 1.4 is the main technical component in the proof of theorem 1.3, and
its proof spans most of this paper. We believe it is of independent interest,
as in essence the statement shows the following: under the assumption of
excluding a fixed onion star, directed flows can be to some extent uncrossed
so that we obtain basic properties observable in undirected flows.

##  2 Preliminaries

For a positive integer $n$, by $[n]$ we denote the set $\\{1,\ldots,n\\}$, and
by $\mathbb{N}$ the set of positive integers. If $f$ is a function whose
domain and codomain are equal, and $t\geqslant 0$ is an integer, by
$f^{\langle t\rangle\\!}(x)$ we mean

$\underbrace{f(f(\ldots f}_{\text{$t$ times}}(x)\ldots)).$

### Digraphs and cuts.

For a directed graph $D$, by $V(D)$ and $A(D)$ we denote the vertex and the
arc set of $D$, respectively. We allow the existence of parallel arcs
(multiple arcs with the same pair of endpoints), but we do not allow loops
(arcs with both endpoints at the same vertex). For $xy\in A(D)$, we call $x$
and $y$ the _tail_ and the _head_ of $xy$, respectively. For an arc $a\in
A(D)$ we denote by $\operatorname{tail}(a)$ and $\operatorname{head}(a)$ its
tail and head, respectively. For a set $X\subseteq V(D)$, by $D[X]$ we denote
the subgraph induced by $X$. Moreover we define

$\displaystyle\delta^{+}(X)$ $\displaystyle\coloneqq\\{a\in
A(D):\operatorname{tail}(a)\in X\text{ and }\operatorname{head}(a)\notin
X\\},$ $\displaystyle\delta^{-}(X)$ $\displaystyle\coloneqq\\{a\in
A(D):\operatorname{tail}(a)\notin X\text{ and }\operatorname{head}(a)\in
X\\}.$

If $X=\\{x\\}$, we omit internal brackets and write $\delta^{+}(x)$ and
$\delta^{-}(x)$, respectively.

A _path_ in a digraph $D$ is a sequence $(a_{1},a_{2},\ldots,a_{k})$ of
pairwise distinct arcs of $D$ with the property that
$\operatorname{head}(a_{i})=\operatorname{tail}(a_{i+1})$ for every
$i\in[k-1]$. Given a path $P=(a_{1},a_{2},\ldots,a_{k})$ we put
$\operatorname{first}(P)\coloneqq a_{1}$, $\operatorname{last}(P)\coloneqq
a_{k}$, and $A(P)\coloneqq\\{a_{1},a_{2},\ldots,a_{k}\\}$. Path $P$ is
_simple_ if $\operatorname{head}(a_{i})=\operatorname{tail}(a_{j})$ if and
only if $j=i+1$, i.e., if the tail of the first arc and the heads of all arcs
form $k+1$ distinct vertices. Let $<_{P}$ be the natural linear order of
$A(P)$ defined by $a_{i}<_{P}a_{j}$ if and only if $i<j$. We define _trimmed
paths_ :

$\displaystyle P(\to a_{i})$
$\displaystyle\coloneqq(a_{1},a_{2},\ldots,a_{i-1}),$ $\displaystyle P(\to
a_{i}]$ $\displaystyle\coloneqq(a_{1},a_{2},\ldots,a_{i}),$ $\displaystyle
P(a_{i}\to)$ $\displaystyle\coloneqq(a_{i+1},a_{i+2},\ldots,a_{k}),$
$\displaystyle P[a_{i}\to)$
$\displaystyle\coloneqq(a_{i},a_{i+1},\ldots,a_{k}),$ $\displaystyle
P(a_{i},a_{j})$ $\displaystyle\coloneqq(a_{i+1},a_{i+2},\ldots,a_{j-1}).$

If $s=\operatorname{tail}(a_{1})$ and $t=\operatorname{head}(a_{k})$, then $P$
is called an _$(s,t)$ -path_. If $\mathcal{P}$ is a family of paths and $Q$ is
a path, then we put $A(\mathcal{P})\coloneqq\bigcup_{P\in\mathcal{P}}A(P)$ and
$\mathcal{P}(Q)\coloneqq\\{P\in\mathcal{P}\mid A(P)\cap A(Q)\neq\emptyset\\}$.
For two arc-disjoint paths $P=(p_{1},\ldots,p_{n})$ and
$Q=(q_{1},\ldots,q_{m})$ such that
$\operatorname{head}(p_{n})=\operatorname{tail}(q_{1})$, we denote by $PQ$ the
_concatenation_ of $P$ and $Q$, i.e., the path
$(p_{1},\ldots,p_{n},q_{1},\ldots,q_{m})$.

A _cut_ in $D$ is a partition $(A,B)$ of $V(D)$. An _$(a,b)$ -cut_ in $D$,
where $a,b\in V(D)$, is a cut $(A,B)$ with $a\in A$ and $b\in B$. The _size_
of the cut $(A,B)$ is $|\delta^{+}(A)|=|\delta^{-}(B)|$.

The classical theorem of Menger describes the relation between the size of
$(a,b)$-cuts and the number of arc-disjoint paths connecting $a$ and $b$.

######  Theorem 2.1 (Menger [Men27]).

Let $D$ be a digraph and let $a,b\in V(D)$. The maximum number of arc-disjoint
$(a,b)$-paths equals the minimum size of an $(a,b)$-cut.

If $\mathcal{P}$ and $\mathcal{Q}$ are two families of paths in a digraph $D$,
then by _intersection graph_ of the pair $(\mathcal{P},\mathcal{Q})$ we mean
the undirected bipartite graph with bipartition classes $\mathcal{P}$,
$\mathcal{Q}$, and an edge between $P\in\mathcal{P}$ and $Q\in\mathcal{Q}$ if
$P$ and $Q$ have at least one common arc.

### Immersions.

We say that a digraph $D$ admits a digraph $H$ as an _immersion_ if there
exists an _immersion model_ of $H$ in $D$, that is, a mapping $\pi$ defined on
the vertices and arcs of $H$ as follows:

  * •

$\pi$ maps vertices of $H$ to pairwise different vertices of $D$;

  * •

$\pi$ maps each arc $xy\in A(H)$ to a path in $D$ with tail $\pi(x)$ and head
$\pi(y)$;

  * •

paths in $\\{\pi(a):a\in A(H)\\}$ are pairwise arc-disjoint.

It is clear that $D$ admits $H$ as an immersion if and only if there exists an
immersion model of $H$ in $D$ in which all arcs are mapped to simple paths.

Let $t\geqslant 1$. An _onion_ 111We note that onions are directed variants of
pumpkins, which were studied e.g. in [JPS+14]. is the digraph
$\smash{\overrightarrow{O}}$ which consists of two vertices $x$ and $y$, two
arcs $yx$ and one arc $xy$. The vertices $x$ and $y$ are called _roots_ of the
onion, with $x$ being also called the _sink_ of the onion, and $y$ being the
_source_ of the onion. If $\pi$ is an immersion model of
$\smash{\overrightarrow{O}}$ in $D$, then the vertices $\pi(x)$, $\pi(y)$ of
$D$ will be called roots (accordingly sink and source) of this immersion. A
_${t}$ -onion-star_ is the digraph $\smash{\overrightarrow{S_{t}}}$ with the
set of vertices $\\{x,y_{1},\ldots,y_{t},z_{1},\ldots,z_{t}\\}$ and set of
arcs $\bigcup_{i=1}^{t}A_{i}$, where each $A_{i}$ consists of single arcs
$y_{i}x$ and $xz_{i}$, and double arcs $xy_{i}$ and $z_{i}x$ (see Figure 3).

Figure 3: An onion (left) with sink $x$ and source $y$ and a ${t}$-onion-star
(right). Blue and red arcs are incoming and outgoing arcs of $x$,
respectively.

Observe that the graph
$\smash{\overrightarrow{S_{t}}[\\{y_{1},\ldots,y_{t},z_{1},\ldots,z_{t}\\}]}$
is arcless and for every $i=1,2,\ldots,t$, the graphs
$\smash{\overrightarrow{S_{t}}[\\{x,y_{i}\\}]}$ and
$\smash{\overrightarrow{S_{t}}[\\{x,z_{i}\\}]}$ are onions. Moreover, $x$ is
the source of $\smash{\overrightarrow{S_{t}}[\\{x,y_{i}\\}]}$ and the sink of
$\overrightarrow{S_{t}}[\\{x,z_{i}\\}]$.

### Graphs.

If $G$ is a simple undirected graph, we denote by $V(G)$ and $E(G)$ its sets
of vertices and edges, respectively. For a vertex $v\in V(G)$, let
$\deg_{G}(v)$ be the number of neighbors of $v$ in $G$. We say that two
disjoint sets $A,B\subseteq V(G)$ are _complete_ to each other if every $a\in
A$ is adjacent to every $b\in B$. Similarly, $A$ and $B$ are _anti-complete_
to each other if every $a\in A$ is nonadjacent to every $b\in B$.

An undirected graph $G$ is _bipartite_ if there exists a partition of $V(G)$
into sets $X$ and $Y$ such that each edge of $G$ has one endpoint in $X$ and
another in $Y$. A bipartite graph is _balanced_ if there exists a choice of
the bipartition classes of $G$ such that the numbers of vertices in the two
classes are equal. By $K_{n,n}$ we denote the complete balanced graph on $2n$
vertices. Let $n$ be a positive integer. By $b(n)$ we denote the minimum
integer such that in every balanced graph on $2b(n)$ vertices, with
bipartition classes $X$ and $Y$, either there exists an induced subgraph
isomorphic to $K_{n,n}$, or there exist two sets $A\subseteq X,B\subseteq Y$,
each of size $n$, that are anti-complete to each other. The existence of
$b(n)$ for each $n$ follows from [Tho82].

######  Theorem 2.2 (Thomason [Tho82]).

For every $n\geqslant 1$ we have $b(n)\leqslant 2^{n}(n-1)+1.$

In other words, if a bipartite graph $G$ has sufficiently many vertices then
we can always find in it two sets of size $n$, each contained in a different
bipartition class, that are either complete or anti-complete to each other. In
the same flavor, the following classical result of Kővári, Sós and Turán
[KST54] gives a lower bound on the number of edges of a (not necessarily
bipartite) graph $G$ so that it contains a large complete balanced subgraph.

######  Theorem 2.3 (Kővári, Sós, Turán [KST54]).

There exists a function $c\colon\mathbb{N}\to(0,\infty)$ such that if an
$n$-vertex graph $G$ has at least $c(k)n^{2-1/k}$ edges for some
$k\in\mathbb{N}$, then it contains $K_{k,k}$ as a subgraph.

##  3 Onion Harvesting Lemma

This section is devoted to the proof of lemma 3.1, the main conceptual piece
of this work.

Let $\mathcal{P}$ be a family of paths in a digraph $D$ and let $X\subseteq
V(D)$. We say that $\mathcal{P}$ _starts at $X$_ if for every
$P\in\mathcal{P}$ it holds that
$\operatorname{tail}(\operatorname{first}(P))\in X$, and that it _ends at $X$_
if for every $P\in\mathcal{P}$ it holds that
$\operatorname{head}(\operatorname{last}(P))\in X$. If $X=\\{x\\}$, we omit
the brackets and say, respectively, that $\mathcal{P}$ starts or ends at $x$.

Consider a pair $(\mathcal{P},\mathcal{Q})$ where $\mathcal{P}$ and
$\mathcal{Q}$ are families of pairwise arc-disjoint simple paths in $D$ such
that there exists $x\in V(D)$ with $\mathcal{P}$ starting at $x$ and
$\mathcal{Q}$ ending at $x$. We say that $(\mathcal{P},\mathcal{Q})$ is a
_well-crossing pair rooted at $x$_ (or simply _well-crossing pair_) if for
every $P\in\mathcal{P}$ and $Q\in\mathcal{Q}$ we have $A(P)\cap
A(Q)\neq\emptyset$, i.e., if the intersection graph of
$(\mathcal{P},\mathcal{Q})$ is complete.

We shall prove that, for every $t\geqslant 1$, if a digraph $D$ contains a
well-crossing pair $(\mathcal{P},\mathcal{Q})$ with both families
$\mathcal{P}$ and $\mathcal{Q}$ sufficiently large (in terms of $t$), then $D$
admits a $t$-onion-star as an immersion. This is formalized by the following
lemma.

######  Lemma 3.1 (Onion Harvesting Lemma).

Let $t\geqslant 1$ be an integer and $D$ be a digraph. There exists a function
$F\colon\mathbb{N}\to\mathbb{N}$ with the following property: if in $D$ there
exists a well-crossing pair $(\mathcal{P},\mathcal{Q})$ with
$|\mathcal{P}|\geqslant F(t)$ and $|\mathcal{Q}|\geqslant F(t)$, then in $D$
there exists an immersion model of a $t$-onion-star.

Let $(\mathcal{P},\mathcal{Q})$ be a well-crossing pair. In the following, we
call _crossing_ of $(\mathcal{P},\mathcal{Q})$ an arc $e$ that belongs to
$A(P)\cap A(Q)$ for some $P\in\mathcal{P}$ and $Q\in\mathcal{Q}$, and specify
_$P$ -crossing_ (resp.  _$Q$ -crossing_, _$(P,Q)$ -crossing_) for a crossing
of $(\mathcal{P},\mathcal{Q})$ contained in $P$ (resp. $Q$, $A(P)\cap A(Q)$).
A crossing $e$ is _$(P,Q)$ -safe_ (or simply _safe_) if there exist at least
$|\mathcal{Q}|/3$ paths in $\mathcal{Q}\setminus\\{Q\\}$ whose $<_{P}$-minimal
crossing with $P$ precedes $e$ in $<_{P}$. Otherwise, it is said to be
$(P,Q)$-_dangerous_ (or simply _dangerous_). We say that a path $P$ is _$Q$
-dangerous_ if it contains a dangerous $(P,Q)$-crossing. If a path
$Q\in\mathcal{Q}$ is trimmed to $\smash{\overline{Q}}$, then we say that $P$
is _$\smash{\overline{Q}}$ -dangerous_ if there exists a dangerous
$(P,Q)$-crossing belonging to $\smash{\overline{Q}}$.

In order to prove Lemma 3.1, we will repeatedly “harvest” immersions of single
onions rooted at the root of $(\mathcal{P},\mathcal{Q})$, keeping
appropriately large well-crossing pairs disjoint with previously found onions
to enable gaining new ones.

######  Lemma 3.2.

Let $n\geqslant 1$ be an integer. There exists a function
$f\colon\mathbb{N}\to\mathbb{N}$ with the following property: If in a digraph
$D$ there exists a well-crossing pair $(\mathcal{P},\mathcal{Q})$ rooted at
$x$ with $|\mathcal{P}|=f(n)$ and $|\mathcal{Q}|=f(n)$, then in $D$ there
exist an immersion model $\pi$ of an onion with source $x$, and a well-
crossing pair $(\mathcal{P}^{*},\mathcal{Q}^{*})$ rooted at $x$ with
$|\mathcal{P}^{*}|=n$, $|\mathcal{Q}^{*}|=n$ such that:

  * •

$A(\mathcal{P}^{*})\subseteq A(\mathcal{P})$, and $A(\mathcal{Q}^{*})\subseteq
A(\mathcal{Q})$;

  * •

all paths in $\mathcal{P}^{*}\cup\mathcal{Q}^{*}$ are arc-disjoint with $\pi$;

  * •

no arc in $A(\mathcal{P}^{*})\cap A(\mathcal{Q}^{*})$ has the sink of $\pi$
for tail.

###### Proof of lemma 3.2.

Let $c$ be the function from Theorem 2.3. Define functions $g$ and $f$ by:

$g(n)=\max\left\\{8,\left\lceil\frac{1}{2}(16c(2n))^{2n}\right\rceil\right\\},\qquad
f(n)=\max\left\\{6n,3\cdot\left\lceil\frac{1}{4}(36c(g(n)))^{g(n)}\right\rceil\right\\}.$

Note that since the values of $f$ are divisible by $3$, so are the sizes of
$\mathcal{P}$ and $\mathcal{Q}$ for $(\mathcal{P},\mathcal{Q})$ a well-
crossing pair satisfying the assumptions of the lemma.

In the following, we consider one such pair $(\mathcal{P},\mathcal{Q})$ rooted
at a vertex $x$ and fix $Q\in\mathcal{Q}$. Let $e$ be the $<_{Q}$-smallest arc
of $Q$ such that $|\mathcal{P}(Q(e\to))|=|\mathcal{P}|/3$. Observe that $e$ is
a crossing satisfying $|\mathcal{P}(Q[e\to))|=|\mathcal{P}(Q(e\to))|+1$. Let
$\smash{\overline{Q}}:=Q[e\to)$. We distinguish two cases depending on the
number of $\smash{\overline{Q}}$-dangerous paths in
$\mathcal{P}(\smash{\overline{Q}})$. These cases are depicted in fig. 4 and
fig. 5, respectively.

Case 1. Suppose that $\mathcal{P}(\smash{\overline{Q}})$ contains at least two
$\smash{\overline{Q}}$-dangerous paths. Let $d$ be the $<_{Q}$-greatest
dangerous crossing in $Q$ and let $P_{2}$ be the path such that $d$ is a
$(P_{2},Q)$-crossing. Then $d$ belongs to $\smash{\overline{Q}}$. Let
$d^{\prime}$ be the $<_{P_{2}}$-smallest $(P_{2},Q)$-crossing belonging to
$\smash{\overline{Q}}(e^{\prime}\to)$. Clearly since $d$ is dangerous and
belongs to $\smash{\overline{Q}}$, $d^{\prime}$ must be dangerous (with
possibly $d=d^{\prime}$). Let $e^{\prime}$ be the $<_{Q}$-greatest dangerous
crossing in $A(Q)\setminus A(P_{2})$ and let
$P_{1}\in\mathcal{P}(\smash{\overline{Q}})\setminus\\{P_{2}\\}$ be the path
such that $e^{\prime}$ is a $(P_{1},Q)$-crossing (with possibly
$e=e^{\prime}$).

Figure 4: The situation of Case 1 when $e\neq e^{\prime}$ and $d\neq
d^{\prime}$. Paths of $\mathcal{P}$ are depicted in black, $Q$ is in red and
$Q$-crossings are in thick red. An onion immersion model with source $x$ and
sink $y$ is underlined in gray.

Let $y=\operatorname{tail}(d^{\prime})$. We claim that the $(x,y)$-paths
$P_{1}(\to e^{\prime}]Q(e^{\prime},d^{\prime})$, $P_{2}(\to d^{\prime})$ and
the $(y,x)$-path $Q[d^{\prime}\to)$ form an immersion model $\pi$ of an onion
with source $x$ and sink $y$. Observe that $P_{1}(\to e^{\prime}]$ is arc-
disjoint with $Q(e^{\prime}\to)$ as $e^{\prime}$ is the $<_{Q}$-greatest
dangerous $(P_{1},Q)$-crossing, and having a safe $(P_{1},Q)$-crossing in
$A(P_{1}(\to e^{\prime}])\cap A(Q(e^{\prime}\to))$ would imply that
$e^{\prime}$ is also safe, a contradiction. Similarly, the
$<_{P_{2}}$-minimality of $d^{\prime}$ asserts that $P_{2}(\to d^{\prime})$ is
arc-disjoint with $Q[e^{\prime}\to)$. Note that these arguments hold even if
$e=e^{\prime}$ and $d=d^{\prime}$. Finally, $Q(e^{\prime},d^{\prime})$ and
${Q[d^{\prime}\to)}$ are arc-disjoint as $Q$ is simple. Hence, the path
${P_{1}(\to e^{\prime}]Q(e^{\prime},d^{\prime})}$ is a well-defined (although
not necessarily simple) $(x,y)$-path, and paths $P_{1}(\to
e^{\prime}]Q(e^{\prime},d)$, $P_{2}(\to d^{\prime})$, and $Q[d^{\prime}\to)$
are pairwise arc-disjoint, yielding an immersion model $\pi$ of an onion with
source $x$ in $D$.

Let $\mathcal{Q}^{\circ}\subseteq\mathcal{Q}$ be the set of paths which are
arc-disjoint both with $P_{1}(\to e^{\prime}]$ and $P_{2}(\to d^{\prime}]$ (so
in particular $Q\notin\mathcal{Q}^{\circ}$). Since $e^{\prime}$ and
$d^{\prime}$ are both dangerous,
$|\mathcal{Q}^{\circ}|>|\mathcal{Q}|-|\mathcal{Q}|/3-|\mathcal{Q}|/3=|\mathcal{Q}|/3\geqslant
2n$ by the definition of $f$. Let us pick an arbitrary collection
$\mathcal{P}^{*}\subseteq\mathcal{P}\setminus\\{P_{1},P_{2}\\}$ of paths not
intersecting $Q[e^{\prime}\rightarrow)$ with $|\mathcal{P}^{*}|=n$. Such a
collection exists as $|\mathcal{P}(Q(e\to))|=|\mathcal{P}|/3$ yielding
$\smash{\frac{2}{3}}|\mathcal{P}|\geqslant 4n$ candidate paths for
$\mathcal{P}^{*}$. Note that each path in such a collection has at most one
arc whose tail is $y$ (as the paths are simple), so there are at most
$|\mathcal{P}^{*}|=n$ arcs in $A(\mathcal{Q}^{\circ})\cap A(\mathcal{P}^{*})$
whose tail is $y$. Let $\mathcal{Q}^{*}\subseteq\mathcal{Q}^{\circ}$ comprise
of any $n$ paths disjoint with these arcs. Clearly, since
$\mathcal{P}^{*}\subseteq\mathcal{P}$ and
$\mathcal{Q}^{*}\subseteq\mathcal{Q}$, $(\mathcal{P}^{*},\mathcal{Q}^{*})$ is
a well-crossing pair. Hence $\pi$, $\mathcal{P}^{*}$ and $\mathcal{Q}^{*}$
satisfy all the desired properties.

Case 2. Suppose that there is at most one $\smash{\overline{Q}}$-dangerous
path in $\mathcal{P}(\smash{\overline{Q}})$. Let $\mathcal{S}$ be the set of
those paths in $\mathcal{P}(\smash{\overline{Q}})$ whose all crossings with
$\smash{\overline{Q}}$ are safe, and let

$\smash{\overline{\mathcal{S}}}\coloneqq\left\\{P(\to\min{{}_{<_{P}}}\\{A(P)\cap
A(\smash{\overline{Q}})\\})\mid P\in\mathcal{S}\right\\}.$

This is the set of paths from $\mathcal{S}$ trimmed so that they start at $x$,
have no arcs in common with $\smash{\overline{Q}}$ and are maximally long.
Since $|\mathcal{P}(\smash{\overline{Q}})|=|\mathcal{P}|/3+1$ and there is at
most one dangerous path in $\mathcal{P}(\smash{\overline{Q}})$, we derive
$|\smash{\overline{\mathcal{S}}}|=|\mathcal{S}|\geqslant|\mathcal{P}(\smash{\overline{Q}})|-1=|\mathcal{P}|/3$.
We shall first exhibit large well-crossing subfamilies of
$\smash{\overline{\mathcal{S}}}$ and $\mathcal{Q}$.

Figure 5: The situation of Case 2. Paths in $\smash{\overline{\mathcal{S}}}$
are depicted in blue, $Q$ is in red and $Q$-crossings are in thick red. Paths
$P_{1}$ and $P_{2}$ are those of 3.4. An onion immersion model with source $x$
and sink $y$ is underlined in gray.

######  Claim 3.3.

There exist subfamilies
$\smash{\overline{\mathcal{S}}}^{\prime}\subseteq\smash{\overline{\mathcal{S}}}$,
$\mathcal{Q}^{\prime}\subseteq\mathcal{Q}\setminus\\{Q\\}$ such that
$|\smash{\overline{\mathcal{S}}}^{\prime}|=g(n)$,
$|\mathcal{Q}^{\prime}|=g(n)$ and
$(\smash{\overline{\mathcal{S}}}^{\prime},\mathcal{Q}^{\prime})$ is well-
crossing.

###### Proof of claim.

Let $G$ be the intersection graph of
$(\smash{\overline{\mathcal{S}}},\mathcal{Q})$. By definition, each trimmed
path $\smash{\overline{P}}$ in $\smash{\overline{\mathcal{S}}}$ originates
from a path $P$ in $\mathcal{S}$ whose $(P,Q)$-crossings $e$ within
$\smash{\overline{Q}}$ are safe, i.e., there are at least $|\mathcal{Q}|/3$
distinct paths among $\mathcal{Q}\setminus\\{Q\\}$ crossing
$\smash{\overline{P}}$ before $e$ with respect to $<_{\smash{\overline{P}}}$.
Since moreover $|\smash{\overline{\mathcal{S}}}|\geqslant|\mathcal{P}|/3$, we
get

$|E(G)|=\sum_{\smash{\overline{P}}\in\smash{\overline{\mathcal{S}}}}\deg_{G}(\smash{\overline{P}})\geqslant\sum_{\smash{\overline{P}}\in\smash{\overline{\mathcal{S}}}}\frac{|\mathcal{Q}|}{3}\geqslant\frac{|\mathcal{P}|}{3}\cdot\frac{|\mathcal{Q}|}{3}=\left(\frac{f(n)}{3}\right)^{2}.$

Note in addition that
$\frac{4}{3}f(n)=\frac{|\mathcal{P}|}{3}+|\mathcal{Q}|\leqslant|V(G)|\leqslant
2f(n)$. By Theorem 2.3 there exists a constant $c_{1}=c(g(n))$ such that if

$|E(G)|\geqslant c_{1}\cdot|V(G)|^{2-\frac{1}{g(n)}},$

then $G$ contains $K_{g(n),g(n)}$ as a subgraph, corresponding to the desired
well-crossing pair
$(\smash{\overline{\mathcal{S}}}^{\prime},\mathcal{Q}^{\prime})$. For the
above inequality to hold it is enough to guarantee that

$\frac{|E(G)|}{|V(G)|^{2}}\geqslant\frac{c_{1}}{|V(G)|^{\frac{1}{g(n)}}}.$ (1)

By $|V(G)|\leqslant 2f(n)$ and $|E(G)|\geqslant\frac{1}{9}f(n)^{2}$ we first
get

$\frac{|E(G)|}{|V(G)|^{2}}\geqslant\frac{\frac{1}{9}f(n)^{2}}{4f(n)^{2}}=\frac{1}{36}.$
(2)

From the definition of $f$ it follows that
$f(n)\geqslant\frac{3}{4}(36c_{1})^{g(n)}$, so equivalently

$\frac{1}{36}\geqslant\frac{c_{1}}{\left(\frac{4}{3}f(n)\right)^{\frac{1}{g(n)}}}.$
(3)

Moreover, $|V(G)|\geqslant\frac{4}{3}f(n)$ gives

$\frac{c_{1}}{\left(\frac{4}{3}f(n)\right)^{\frac{1}{g(n)}}}\geqslant\frac{c_{1}}{|V(G)|^{\frac{1}{g(n)}}}.$
(4)

Putting together inequalities eq. 2, eq. 3, eq. 4, we obtain eq. 1, which
finishes the proof.  $\lrcorner$

Consider the pair
$(\smash{\overline{\mathcal{S}}}^{\prime},\mathcal{Q}^{\prime})$ given by 3.3
and let $\mathcal{S}^{\prime}$ be the set of paths in $\mathcal{S}$ from which
$\smash{\overline{\mathcal{S}}}^{\prime}$ originates (i.e., the paths from
$\smash{\overline{\mathcal{S}}}^{\prime}$ before trimming). For two distinct
paths $P_{1},P_{2}\in\mathcal{S}^{\prime}$, we denote

$\smash{\overline{\mathcal{Q}}}^{\prime}(P_{1},P_{2})\coloneqq\left\\{R(\max{{}_{<_{R}}}\\{A(\\{P_{1},P_{2}\\})\cap
A(R)\\}\to)\ \mid\ R\in\mathcal{Q}^{\prime}\right\\}.$

This is the set of paths from $\mathcal{Q}^{\prime}$ trimmed so that they end
at $x$, have no arcs in common with $P_{1}$ or $P_{2}$ and are longest
possible.

######  Claim 3.4.

There exist distinct $P_{1},P_{2}\in\mathcal{S}^{\prime}$ and subfamilies
$\mathcal{P}^{\circ}\subseteq\smash{\overline{\mathcal{S}}}^{\prime}$,
$\mathcal{Q}^{\circ}\subseteq\smash{\overline{\mathcal{Q}}}^{\prime}(P_{1},P_{2})$
such that $|\mathcal{P}^{\circ}|=2n$, $|\mathcal{Q}^{\circ}|=2n$ and
$(\mathcal{P}^{\circ},\mathcal{Q}^{\circ})$ is well-crossing.

###### Proof of claim.

Let $k\coloneqq
g(n)=|\mathcal{Q}^{\prime}|=|\smash{\overline{\mathcal{S}}}^{\prime}|=|\mathcal{S}^{\prime}|$.
Choose a pair $\\{P_{1},P_{2}\\}\subseteq\mathcal{S}^{\prime}$ uniformly at
random (from the set of all $\binom{k}{2}$ two-element subsets). Let
$\smash{\overline{\mathcal{Q}}}^{\prime}\coloneqq\smash{\overline{\mathcal{Q}}}^{\prime}(P_{1},P_{2})$
and let $G$ be the intersection graph of
$(\smash{\overline{\mathcal{S}}}^{\prime},\smash{\overline{\mathcal{Q}}}^{\prime})$.
Clearly, $|\smash{\overline{\mathcal{Q}}}^{\prime}|=|\mathcal{Q}^{\prime}|=k$.
For $\smash{\overline{R}}\in\smash{\overline{\mathcal{Q}}}^{\prime}$ let
$N_{\smash{\overline{R}}}$ be the random variable denoting the number of paths
$P\in\smash{\overline{\mathcal{S}}}^{\prime}$ such that
$A(\smash{\overline{R}})\cap A(P)\neq\emptyset$, i.e., the degree of
$\smash{\overline{R}}$ in $G$.

Fix $\smash{\overline{R}}\in\smash{\overline{\mathcal{Q}}}^{\prime}$ and let
$R\in\mathcal{Q}^{\prime}$ be the path from which $\smash{\overline{R}}$
originates (i.e., the untrimmed counterpart of $\smash{\overline{R}}$). For
every $P\in\smash{\overline{\mathcal{S}}}^{\prime}$ let
$a(P)\coloneqq\max_{<_{R}}\\{A(P)\cap A(R)\\}$ be the $<_{R}$-greatest common
arc of $P$ and $R$. As
$(\smash{\overline{\mathcal{S}}}^{\prime},\mathcal{Q}^{\prime})$ is well-
crossing, the $k$-element set $\\{a(P)\mid
P\in\smash{\overline{\mathcal{S}}}^{\prime}\\}\subseteq A(R)$ is naturally
linearly ordered by $<_{R}$. Label all paths from
$\smash{\overline{\mathcal{S}}}^{\prime}$ with numbers from $1$ to $k$
accordingly with the order of appearance of arcs $a(P)$ on $R$, and let
$\ell(P)$ be the label of $P$. Note that $N_{\smash{\overline{R}}}\geqslant
k-\max(\ell(P_{1}),\ell(P_{2}))$ and for every $i=0,1,\ldots,k-1$,

$\mathbb{P}(\max(\ell(P_{1}),\ell(P_{2}))=i+1)=\frac{i}{\binom{k}{2}}.$

By a straightforward computation we obtain

$\mathbb{E}N_{\smash{\overline{R}}}=\sum_{i=0}^{k-1}\mathbb{P}(N_{\smash{\overline{R}}}=k-i-1)(k-i-1)=\frac{1}{\binom{k}{2}}\sum_{i=0}^{k-1}i(k-i-1)=\frac{k-2}{3}.$

Note that the distribution of $N_{\smash{\overline{R}}}$ depends only on
labels of $P_{1}$ and $P_{2}$ and since these are chosen randomly, for
different
$\smash{\overline{R}},\smash{\overline{S}}\in\smash{\overline{\mathcal{Q}}}^{\prime}$
the variables $N_{\smash{\overline{R}}}$, $N_{\smash{\overline{S}}}$ are
identically distributed. By the above observations and since $k=g(n)\geqslant
8$, we have

$\mathbb{E}|E(G)|\geqslant\mathbb{E}\sum_{\smash{\overline{S}}\in\smash{\overline{\mathcal{Q}}}^{\prime}}N_{\smash{\overline{S}}}\geqslant
k\cdot\frac{k-2}{3}\geqslant\left(\frac{k}{2}\right)^{2}.$

It follows that there exists a choice of $P_{1}$, $P_{2}$ such that

$|E(G)|\geqslant\left(\frac{g(n)}{2}\right)^{2}.$ (5)

Similarly as in the proof of 3.3, we observe by Theorem 2.3 that there exists
a constant $c_{2}=c(2n)$ such that if

$|E(G)|\geqslant c_{2}\cdot|V(G)|^{2-\frac{1}{2n}},$

then $G$ contains $K_{2n,2n}$ as a subgraph, corresponding to the desired
well-crossing pair $(\mathcal{P}^{\circ},\mathcal{Q}^{\circ})$. For the above
inequality to hold it is enough to guarantee that

$\frac{|E(G)|}{|V(G)|^{2}}\geqslant\frac{c_{2}}{|V(G)|^{\frac{1}{2n}}}.$

This is obtained observing that

$\frac{|E(G)|}{|V(G)|^{2}}=\frac{|E(G)|}{4(g(n))^{2}}\geqslant\frac{1}{16}\geqslant\frac{c_{2}}{(2g(n))^{\frac{1}{2n}}}=\frac{c_{2}}{|V(G)|^{\frac{1}{2n}}},$

where the first inequality follows from eq. 5, and the second inequality
follows from the definition of $g$ ensuring that
$g(n)\geqslant\frac{1}{2}(16c_{2})^{2n}$.  $\lrcorner$

Consider $P_{1}$, $P_{2}$ and $(\mathcal{P}^{\circ},\mathcal{Q}^{\circ})$ as
in the claim above and denote
$e_{i}=\min_{<_{P_{i}}}\\{A(\smash{\overline{Q}})\cap A(P_{i})\\}$ for
$i=1,2$. Assume without loss of generality that $e_{1}<_{Q}e_{2}$, i.e., that
the $<_{P_{1}}$-smallest $(P_{1},\smash{\overline{Q}})$-crossing is further
from $x$ on $\smash{\overline{Q}}$ than the $<_{P_{2}}$-smallest
$(P_{2},\smash{\overline{Q}})$-crossing. Let $y=\operatorname{tail}(e_{2})$.
By the choice of $e_{1}$ and $e_{2}$, the $(x,y)$-paths $P_{2}(\to e_{2})$,
$P_{1}(\to e_{1}]Q(e_{1},e_{2})$ and the $(y,x)$-path $Q[e_{2}\to)$ are arc-
disjoint and hence form an immersion model $\pi$ of an onion with source $x$
and sink $y$. Moreover, this model is disjoint with
$A(\mathcal{P}^{\circ}\cup\mathcal{Q}^{\circ})$ as
$\mathcal{Q}^{\circ}\subseteq\smash{\overline{\mathcal{Q}}}^{\prime}$ with
$\smash{\overline{\mathcal{Q}}}^{\prime}$ arc-disjoint with both $P_{1}$ and
$P_{2}$.

The last step (to ensure that $y$ will not be the tail of a crossing in the
found well-crossing pair) is performed similarly as in the first case: we pick
an arbitrary collection $\mathcal{P}^{\ast}$ of exactly $n$ paths among the
$2n$ paths in $\mathcal{P}^{\circ}$, and select a subfamily
$\mathcal{Q}^{\ast}$ consisting of $n$ paths which do not contain crossings of
tail $y$ among the $2n$ paths in $\mathcal{Q}^{\circ}$.

That concludes the proof of Lemma 3.2. ∎

Now, we observe that by iterating Lemma 3.2 $t$ times, we can obtain an
immersion model of a digraph consisting of $t$ onions with a common source.

######  Corollary 3.5.

Let $n$, $t$ be positive integers. Suppose that in a digraph $D$ there exists
a well-crossing pair $(\mathcal{P},\mathcal{Q})$ rooted at $x$ with
$|\mathcal{P}|=|\mathcal{Q}|=f^{\langle t\rangle\\!}(n)$, where $f$ is the
function from Lemma 3.2. Then in $D$ there exists a family $\mathcal{H}$ of
$t$ arc-disjoint immersion models of onions with common source $x$, whose all
sinks are pairwise different, and a well-crossing pair
$(\mathcal{P}^{*},\mathcal{Q}^{*})$ rooted at $x$ with $|\mathcal{P}^{*}|=n$,
$|\mathcal{Q}^{*}|=n$ such that all paths in
$\mathcal{P}^{*}\cup\mathcal{Q}^{*}$ are arc-disjoint with every element of
$\mathcal{H}$.

###### Proof.

Put $\mathcal{P}_{0}\coloneqq\mathcal{P}$ and
$\mathcal{Q}_{0}\coloneqq\mathcal{Q}$. For $i=0,1,\ldots,t-1$ let us apply
Lemma 3.2 to the pair $(\mathcal{P}_{i},\mathcal{Q}_{i})$ to get an immersion
model of an onion $\pi_{i+1}$ with sink $y_{i+1}$, and a well-crossing pair
$(\mathcal{P}_{i+1},\mathcal{Q}_{i+1})$.

Now it is enough to take $\mathcal{P}^{*}=\mathcal{P}_{t}$,
$\mathcal{Q}^{*}=\mathcal{Q}_{t}$ and $\mathcal{H}=\\{\pi_{i}\\}^{t}_{i=1}$.
Indeed, to see that the sinks $y_{1},\ldots,y_{t}$ are distinct, recall that
for every $i\in\\{0,\ldots,t-1\\}$ no arc in $A(\mathcal{P}_{i+1})\cap
A(\mathcal{Q}_{i+1})$ has $y_{i+1}$ as a tail. Since
$A(\mathcal{P}_{i})\subseteq A(\mathcal{P}_{j})$, and
$A(\mathcal{Q}_{i})\subseteq A(\mathcal{Q}_{j})$ for every $0\leqslant
j\leqslant i$, vertices $y_{1},\ldots,y_{i}$ do not appear as tails of arcs in
$A(\mathcal{P}_{i})\cap A(\mathcal{Q}_{i})$, hence $y_{i+1}$ must be distinct
from $y_{1},\ldots,y_{i}$. The remaining properties of the immersion model are
straightforward to verify. ∎

An analogous lemma can be designed to enable finding onions whose sink (not
source) is $x$. The proof basically follows from reversing all arcs in the
arguments delivered previously.

######  Corollary 3.6.

Let $n$, $t$ be positive integers. Suppose that in a digraph $D$ there exists
a well-crossing pair $(\mathcal{P},\mathcal{Q})$ rooted at $x$ with
$|\mathcal{P}|=|\mathcal{Q}|=f^{\langle t\rangle\\!}(n)$, where $f$ is the
function from Lemma 3.2. Then in $D$ there exists a family $\mathcal{H}$ of
$t$ arc-disjoint immersion models of onions with common sink $x$, whose all
sources are pairwise different, and a well-crossing pair
$(\mathcal{P}^{*},\mathcal{Q}^{*})$ rooted at $x$ with $|\mathcal{P}^{*}|=n$,
$|\mathcal{Q}^{*}|=n$ such that all paths in
$\mathcal{P}^{*}\cup\mathcal{Q}^{*}$ are arc-disjoint with every element of
$\mathcal{H}$.

###### Proof.

It is enough to apply Corollary 3.5 to the digraph $\overleftarrow{D}$, in
which each arc of $D$ is replaced with a reversed arc, and to the (swapped)
well-crossing pair
$\smash{(\overleftarrow{\mathcal{Q}},\overleftarrow{\mathcal{P}})}$ consisting
of families of reversed paths from $\mathcal{Q}$ and $\mathcal{P}$,
respectively. ∎

Finally, we use corollary 3.5 and corollary 3.6 to prove Lemma 3.1.

###### Proof of Lemma 3.1.

Take $F(t)=f^{\langle 4t\rangle\\!}(1)$, where $f$ is the function satisfying
Lemma 3.2. Let $x$ be the root of the pair $(\mathcal{P},\mathcal{Q})$.
Applying Corollary 3.5 to $D$, $(\mathcal{P},\mathcal{Q})$, $n=f^{\langle
2t\rangle\\!}(1)$ and $2t$ (in place of $t$) we find a family
$\mathcal{H}^{+}=\\{\pi^{+}_{i}\\}_{i=1}^{2t}$ of $2t$ arc-disjoint immersion
models of onions with source $x$ and mutually different sinks
$\\{y_{i}\\}_{i=1}^{2t}$, along with a well-crossing pair
$(\mathcal{P}^{*},\mathcal{Q}^{*})$ in which both families have size at least
$f^{\langle 2t\rangle\\!}(1)$ and are arc-disjoint with elements of
$\mathcal{H}^{+}$. Now applying Corollary 3.6 to $D$,
$(\mathcal{P}^{*},\mathcal{Q}^{*})$, $n=1$ and $2t$ (in place of $t$), we find
a family $\mathcal{H}^{-}=\\{\pi^{-}_{j}\\}_{j=1}^{2t}$ of $2t$ arc-disjoint
immersion models of onions with sink $x$ and mutually different sources
$\\{z_{j}\\}_{j=1}^{2t}$. Moreover, $A(\mathcal{H}^{+})\cap
A(\mathcal{H}^{-})=\emptyset$. It now remains to note that it is possible to
find two sets $I,J\subseteq[2t]$ such that $|I|=|J|=t$ and $\\{y_{i}:i\in
I\\}\cap\\{z_{j}:j\in J\\}=\emptyset$. The union of immersions $\pi^{+}_{i}$
where $i\in I$ and $\pi^{-}_{j}$ where $j\in J$ forms an immersion of a
${t}$-onion-star. ∎

##  4 Proof of theorem 1.3

In this section we prove Theorem 1.4 and then Theorem 1.3.

For a digraph $D$, a vertex $y\in V(D)$ and a set $Z\subseteq
V(D)\setminus\\{y\\}$ we denote by $\mu(y,Z)$ ($\mu(Z,y)$, respectively) the
maximum number of arc-disjoint paths from $y$ to $Z$ (from $Z$ to $y$,
respectively) that can be found in $D$.

Since our proof of theorem 1.3 requires an iterative application of Theorem
1.4, to simplify this procedure we actually prove the following, slightly more
general version of the latter one. Note that Theorem 1.4 follows from Theorem
4.1 by taking $Y=\\{y\\}$.

######  Theorem 4.1.

There exists a function $g\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ such
that the following holds. Suppose $D$ is a digraph, $y\in V(D)$, and $Z$ is a
subset of $V(D)\setminus\\{y\\}$. Let $\mathcal{P}$ and $\mathcal{Q}$ be
maximum families of pairwise arc-disjoint simple paths such that elements of
$\mathcal{P}$ start at $y$ and end at $Z$, and elements of $\mathcal{Q}$ start
at $Z$ and end at $y$. Assume that $|\mathcal{P}|>g(t,k)$ and
$|\mathcal{Q}|>g(t,k)$, for some $t,k\in\mathbb{N}$. Then at least one of the
following holds:

  * •

$D$ contains the $t$-onion star as an immersion;

  * •

there is a subset $\mathcal{P}^{\prime}$ of $\mathcal{P}$, and a subset
$\mathcal{Q}^{\prime}$ of $\mathcal{Q}$, each consisting of $k$ paths, such
that the elements of $\mathcal{P}^{\prime}\cup\mathcal{Q}^{\prime}$ are
pairwise arc-disjoint.

###### Proof of Theorem 4.1.

We define

$g(t,k)=2^{N}(N-1)+1$

where $N=\max\\{k,F(t)\\}$ and $F$ is the function given by Lemma 3.1. By
Theorem 2.2 applied to the intersection graph of $(\mathcal{P},\mathcal{Q})$,
there exist families $\mathcal{P}^{\prime}\subseteq\mathcal{P}$ and
$\mathcal{Q}^{\prime}\subseteq\mathcal{Q}$ of paths such that
$|\mathcal{P}^{\prime}|,|\mathcal{Q}^{\prime}|\geqslant N$ and either
$(\mathcal{P}^{\prime},\mathcal{Q}^{\prime})$ is a well-crossing pair, or
$A(\mathcal{P}^{\prime})\cap A(\mathcal{Q}^{\prime})=\emptyset$. In the first
case, we conclude by Lemma 3.1 the existence of an immersion model of a
$t$-onion-star in $D$. As of the second outcome, it is precisely as desired. ∎

We are now ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

Define function $f$ as

$f(t)=2t\cdot g^{\langle 4t^{2}\rangle\\!}_{t}(2),$

where $g_{t}(k)\coloneqq g(t,k)$ is the function from Theorem 4.1.

We arbitrarily enumerate the vertices of $X$ as $y,x_{1},\ldots,x_{2t}$.
Consider an auxiliary digraph $D^{\prime}$ obtained from $D$ by adding a
single vertex $v$, $\frac{f(t)}{2t}$ arcs $vx_{i}$ and $\frac{f(t)}{2t}$ arcs
$x_{i}v$ for every $i\in[2t]$. First, observe that in $D^{\prime}$ there
exists a family of $f(t)$ arc-disjoint $(v,y)$-paths. Indeed, let
$\mathcal{P}^{\prime}$ be the maximum family of arc-disjoint $(v,y)$-paths,
and let $k$ be the size of $\mathcal{P}^{\prime}$. By Theorem 2.1, there
exists a $(v,y)$-cut $(A,B)$ of size $k$. Since $|\delta^{+}(v)|=f(t)$, we
have $k\leqslant f(t)$. Observe that if $k<f(t)$, then
$A\setminus\\{v\\}\neq\emptyset$, which implies that there exists $i\in[2t]$
such that $x_{i}\in A$. However, in such a case, $(A\setminus\\{v\\},B)$ is an
$(x_{i},y)$-cut in $D$ of size $k<f(t)$ contradicting the assumption of the
theorem. Hence, $|\mathcal{P}^{\prime}|=f(t)$. By a symmetric argument we
obtain the existence of a family $\mathcal{Q}^{\prime}$ of $f(t)$ arc-disjoint
$(y,v)$-paths in $D^{\prime}$. Clearly, we can assume that
$\mathcal{P}^{\prime}$ and $\mathcal{Q}^{\prime}$ consist of simple paths.

Let $\mathcal{P}$ and $\mathcal{Q}$ respectively denote the families of paths
$\mathcal{P}^{\prime}$ and $\mathcal{Q}^{\prime}$ once every path has been
restricted to the arcs of $D$. Note that each $P\in\mathcal{P}$ (resp.
$Q\in\mathcal{Q}$) is an $(x,y)$-path (resp. $(y,x)$-path) for some $x\in
X\setminus\\{y\\}$. Moreover, for every $i\in[2t]$, the number of
$(x_{i},y)$-paths in $\mathcal{P}$ (resp. $(y,x_{i})$-paths in $\mathcal{Q}$)
is precisely $\frac{f(t)}{2t}$. Hence, if we denote by $\mathcal{P}_{i}$ the
subset of $\mathcal{P}$ that consists of $(x_{i},y)$-paths, for every
$i\in[2t]$, we have that $|\mathcal{P}_{i}|=\frac{f(t)}{2t}=g^{\langle
4t^{2}\rangle\\!}_{t}(2)$. Similarly, by $\mathcal{Q}_{i}$ we denote the
subset of $\mathcal{Q}$ that consists of $(y,x_{i})$-paths, and for every
$i\in[2t]$ we have that $|\mathcal{Q}_{i}|=g^{\langle
4t^{2}\rangle\\!}_{t}(2)$.

Let $p\in[4t^{2}]$. We fix an arbitrary order on the elements of
$[2t]\times[2t]$, and for the $p$-th element $(i,j)$ we proceed as follows. We
apply Theorem 4.1 to families $\mathcal{P}_{i}$ and $\mathcal{Q}_{j}$, setting
$Z=\\{x_{i},x_{j}\\}$, $y$, and $k=g^{\langle 4t^{2}-p\rangle\\!}_{t}(2)$.
Either we conclude that there exists an immersion model of a $t$-onion-star in
$D$, or replace $\mathcal{P}_{i}$ and $\mathcal{Q}_{j}$ by their subsets
$\mathcal{P}^{\prime}_{i}$ and $\mathcal{Q}^{\prime}_{j}$ of size $g^{\langle
4t^{2}-p\rangle\\!}_{t}(2)$. Clearly the theorem holds if the first case
occurs. In the other case, for every $i^{\prime}\neq i,j^{\prime}\neq j$, we
reduce sets $\mathcal{P}_{i^{\prime}}$ and $\mathcal{Q}_{j^{\prime}}$
arbitrarily so that each of them contain exactly $g^{\langle
4t^{2}-p\rangle\\!}_{t}(2)$ paths, and proceed to the next iteration.

After $4t^{2}$ iterations, if no onion-star immersion model is found, for
every $i$ the families $\mathcal{P}_{i}$ and $\mathcal{Q}_{i}$ have precisely
$g^{\langle 0\rangle\\!}(2)=2$ elements each. Let
$\mathcal{S}_{i}=\mathcal{P}_{i}\cup\mathcal{Q}_{i}$ and
$\mathcal{S}=\bigcup_{i\in[2t]}\mathcal{S}_{i}$. We observe that paths from
$\mathcal{S}$ are now pairwise arc-disjoint, and each $\mathcal{S}_{i}$
contains two paths $x_{i}\to y$ and two paths $y\to x_{i}$ for every
$i\in[2t]$. Hence, there exists an $t$-onion-star immersion model in $D$, with
center being $y$. This concludes the proof. ∎

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# Giant thermopower and power factor in magic angle twisted bilayer graphene
at low temperature

S. S. Kubakaddi<EMAIL_ADDRESS>Department of Physics, K. L. E.
Technological University, Hubballi-580031, Karnataka, India

###### Abstract

The in-plane phonon-drag thermopower $S^{g}$, diffusion thermopower $S^{d}$
and the power factor $PF$ are theoretically investigated in twisted bilayer
graphene (tBLG) as a function of twist angle $\theta$, temperature $T$ and
electron density $n_{s}$ in the region of low $T$ (1-20 K). As $\theta$
approaches magic angle $\theta_{m}$, the $S^{g}$ and $S^{d}$ are found to be
strongly enhanced, which is manifestation of great suppression of Fermi
velocity ${\nu_{F}}^{*}$ of electrons in moire flat band near $\theta_{m}$.
This enhancement decreases with increasing $\theta$ and $T$. In the Bloch-
Gruneisen (BG) regime, it is found that $S^{g}\sim{\nu_{F}}^{*-2}$, $T^{3}$
and ${n_{s}}^{-1/2}$. As $T$ increases, the exponent $\delta$ in $S^{g}\sim
T^{\delta}$, changes from 3 to nearly zero and a maximum $S^{g}$ value of
$\sim$ 10 mV/K at $\sim$ 20 K is estimated. $S^{g}$ is larger (smaller) for
smaller $n_{s}$ in low (high) temperature region. On the other hand, $S^{d}$,
taken to be governed by Mott formula, $\sim{\nu_{F}}^{*-1}$, $T$ and
${n_{s}}^{-1/2}$ and $S^{d}<<S^{g}$ for $T>\sim$ 2 K. The power factor $PF$ is
also found to be strongly $\theta$ dependent and very much enhanced.
Consequently, possibility of a giant figure of merit is discussed.In tBLG,
$\theta$ acts as a strong tuning parameter of both $S^{g}$ and $S^{d}$ and
$PF$ in addition to $T$ and $n_{s}$. Our results are qualitatively compared
with the measured out-of-plane thermopower in tBLG.

††preprint: APS

##  I  INTRODUCTION

The recent remarkable and exciting experimental discoveries in twisted bilayer
graphene (tBLG), particularly the existence of correlated insulating phases
and superconductivity at low temperatures, and a highly resistive linear-in-
temperature resistivity $\rho$ at high temperature, have created tremendous
interest in the study of electronic properties of this material Cao  _et al._
(2018a, b); Wu  _et al._ (2018); Yankowitz  _et al._ (2019); Lu  _et al._
(2019); Sharpe  _et al._ (2019); Tomarken  _et al._ (2019); Roy and Juričić
(2019); Polshyn  _et al._ (2019); Wu  _et al._ (2019); Cao  _et al._ (2020);
Serlin  _et al._ (2020); Das Sarma and Wu (2020); Hwang and Das Sarma (2020);
Kubakaddi (2021). A small twist angle $\theta$ between the two layers modifies
the electronic structure in to moire flat band with $\theta$ dependent
suppressed Fermi velocity ${\nu_{F}}^{*}\equiv{\nu_{F}}^{*}$($\theta$), which
is very much smaller than the bare Fermi velocity ${\nu_{F}}$ in monolayer
graphene, and becomes vanishingly small for $\theta$ in the neighborhood of
magic angle $\theta_{m}$ leading to large density of states Wu  _et al._
(2019); Das Sarma and Wu (2020); Hwang and Das Sarma (2020); Bistritzer and
MacDonald (2011). The $\theta$ near $\theta_{m}$, acts as one of the tunable
parameters in limiting the electronic properties of tBLG, apart from the
carrier density $n_{s}$ and temperature $T$ Cao  _et al._ (2018a, b);
Yankowitz  _et al._ (2019); Polshyn  _et al._ (2019); Wu  _et al._ (2019); Cao
_et al._ (2020); Das Sarma and Wu (2020).

The measurements of in-plane electrical resistivity $\rho$ in tBLG, for
$\theta$ closer to $\theta_{m}$, show that $\rho$ scales linear-in-$T$,
strongly increasing with decreasing twist angle and enhanced by about more
than three orders of magnitude over that observed in monolayer graphene (MLG)
Lu  _et al._ (2019); Polshyn  _et al._ (2019); Cao  _et al._ (2020). This
linearity is observed, in some samples, unbelievably down to 0.5 K Lu  _et
al._ (2019); Cao  _et al._ (2020), just before the samples become
superconducting, and in some other samples it persists up to $\sim$ 4 - 6 K Lu
_et al._ (2019); Polshyn  _et al._ (2019); Cao  _et al._ (2020). The linear-
in-$T$ behavior of $\rho$ is generic of electron scattering by acoustic
phonons at high T. The acoustic phonon limited electrical resistivity
$\rho_{p}$ in tBLG is theoretically investigated assuming the linear
dispersion for the Dirac fermions Polshyn  _et al._ (2019); Wu  _et al._
(2019); Das Sarma and Wu (2020). It is found to be strongly $\theta$
dependent, largely enhanced in magnitude and linear-in-$T$ for the
temperatures above $\sim$ 5 K and explaining the experimental data very well.
The strong $\theta$ dependence and enhancement of $\rho_{p}$ is attributed to
the increase in the electron-acoustic phonon (el-ap) scattering due to the
suppression of ${\nu_{F}}^{*}$ induced by the moiré flat band.

Very recently the hot electron power loss $P$, another interesting property,
which involves only el-ap interaction, has been theoretically investigated in
tBLG by the present author Kubakaddi (2021). The profound effect of $\theta$,
near $\theta_{m}$, on $P$ is found to enhance it by $\sim$ 2-3 orders of
magnitude, attributing to the reduced ${\nu_{F}}^{*}$, complimenting the
measurements of $\rho$.

The thermoelectric (TE) power, of any realized new material, is an important
and interesting transport property for the study because of its sensitivity to
electronic structure and scattering mechanisms, besides applications in TE
devices (power generators and refrigerators). It is also known to provide
information complementary to that of electrical resistivity $\rho$ (or
conductivity $\sigma$) and has been a useful tool for probing carrier
transport. The potential of a material for TE applications is determined by
its figure of merit, $Z=S^{2}\sigma/\kappa$, where $S$ is the thermopower i.e.
Seebeck coefficient, $\sigma$ is the electrical conductivity, and $\kappa$ is
the thermal conductivity of the material. Materials, therefore, with an
enhanced power factor $PF=S^{2}\sigma$ and reduced $\kappa$, are suitable
candidates for efficient TE devices Tritt ; Rowe . Thermopower $S$ defined by
$E=S\nabla T$, where $E$ is an electric field set up due to the temperature
gradient $\nabla T$, comprises of two components $S=S^{d}+S^{g}$: the
diffusion thermopower $S^{d}$ arising due to the diffusion of electrons and
the phonon-drag thermopower $S^{g}$ due to the momentum transfer from the
phonon wind to electrons because of the electron-phonon coupling. Since both
$S^{d}$ and $S^{g}$ depend upon Fermi energy and hence the Fermi velocity and
electron density, apart from temperature, we expect them to be strongly
influenced by the twist angle closer to $\theta_{m}$ in tBLG. More
importantly, as we have seen the strong enhancement in the el-ap coupling
dependent $\rho$ Polshyn  _et al._ (2019); Wu  _et al._ (2019); Das Sarma and
Wu (2020) and $P$ Kubakaddi (2021), we expect the strong influence of $\theta$
on $S^{g}$, which depends only upon el-ap interaction. Moreover, the power
factor $PF$ is also expected to be largely governed by $\theta$ in tBLG as
both $\rho$ and $S$ are influenced. Consequently, the twist angle may act as
an additional important tunable parameter for $Z$ in tBLG.

Thermopower has been studied in detail experimentally and theoretically in
conventional two-dimensional electron gas (2DEG) (see Reviews Gallagher and
Butcher ; R. Fletcher and Zeitler ; Tsaousidou ) and in monolayer and
conventional bilayer graphene (BLG) (see Review Sankeshwar  _et al._ ). In
MLG, theories of phonon-drag thermopower Kubakaddi (2009) and diffusion
thermopower Hwang  _et al._ (2009) have been developed and experimentally
investigated by different groups Zuev  _et al._ (2009); Checkelsky and Ong
(2009); Wei  _et al._ (2009); Wu  _et al._ (2011). In conventional BLG,
phonon-drag thermopower and diffusion thermopower are studied theoretically
Kubakaddi and Bhargavi (2010); Bhargavi and Kubakaddi (2013) and compared with
the experimental results Nam  _et al._ (2010).

Recently, thermoelectric transport has also been investigated, for specific
values of $\theta$, in tBLG (across the plane) Mahapatra  _et al._ (2017,
2020), twisted armchair graphene nanoribbons (tAGNR) Liu  _et al._ (2015) and
twisted bilayer graphene nanoribbons junction (tBGNRJ) Deng  _et al._ (2019).
In their very recent work in tBLG, Mahapatra et al Mahapatra  _et al._ (2020)
have reported out-of-plane thermoelectric measurements across the van der
Waals gap and shown that at large $\theta$ ($\sim$ 12.5∘) the thermopower, for
$T=$20-300 K, is entirely driven by a novel phonon-drag effect at subnanometer
scale, while at low twist angles ($\theta<6^{\circ}$) the diffusion component
of the thermopower is recovered that can be described by the Mott relation.
These authors have used the phonon-drag theory of semiconductor quantum wells
Wu  _et al._ (1996), with the layer breathing mode (LBM) of phonons of
quadratic dispersion, to interpret their observations. In tBGNRJ from the
first principle calculations, the enhanced thermoelectric performance with
$ZT=$ (2-6.1) at 300 K, for optimized large $\theta$ ($\sim$ 21.8 ∘ ), is
interpreted as the combination of the reduced $\kappa$, enhanced $S$ and
$\sigma$ Deng  _et al._ (2019). We emphasize that the in-plane thermoelectric
transport in tBLG is yet to be explored.

In the present work, we theoretically investigate the in-plane thermoelectric
transport properties namely thermopower and power factor of electrons in moiré
flat band in tBLG at low temperature for small twist angles closer to magic
angle. The effect of small twist angle on the in-plane phonon-drag and
diffusion thermopower and the power factor in tBLG is explored. In particular,
study of $S^{g}$ is expected to compliment the enhancement of el-ap
interaction found in the in-plane resistivity measurements. $S^{g}$, $S^{d}$
and $PF$ are studied as a function of twist angle, temperature and electron
density. We show that the twist angle $\theta$ acts as one of the strong
tunable parameters of thermopower and power factor.

##  II Theoretical model

As described in Ref Kubakaddi (2021), we assume the electron energy spectrum
in moiré flat band to be given by the Dirac linear dispersion with the density
of states $D(E_{k})=(gE_{k})/[2\pi(\hbar{\nu_{F}}^{*})^{2}]$, where
$E_{k}=\hbar{\nu_{F}}^{*}|k|$ and the degeneracy $g=g_{s}g_{v}g_{l}$, with
$g_{s}$, $g_{v}$ and $g_{l}$ being, respectively, spin, valley and layer
degeneracies each with the value of 2. The assumption of linear energy
dispersion, in this early stage of the development, makes our theory limited
to the carrier density $n_{s}\leq 10^{12}cm^{-2}$ with Fermi energy little
away from the Dirac point Wu  _et al._ (2019); Das Sarma and Wu (2020). Das
Sarma et al Das Sarma and Wu (2020) have shown that the twist angle dependence
of ${\nu_{F}}^{*}(\theta$) can be very well approximated by
${\nu_{F}}^{*}(\theta)\approx 0.5\mid\theta-\theta_{m}\mid{\nu_{F}}$, which
clearly indicates that effect of $\theta$ is very large near $\theta_{m}$.

###  II.1 Phonon-drag thermopower

We calculate the phonon-drag thermopower in tBLG following the well
established Q-approach theory of Cantrell-Butcher (CB) given for conventional
2DEG Cantrell and Butcher (1987), which is also applied in MLG by the present
author Kubakaddi (2009). In this approach, for an in-plane electric field $E$
and temperature gradient $\nabla T$, the coupled Boltzmann equations for
electrons and phonons are solved for electron distribution $f_{\textbf{k}}$ to
obtain phonon-drag current density $J^{g}$ set by transfer of phonon momentum
to electrons due to the electron-phonon coupling. In an open circuit
condition, an electric field $E=S^{g}\nabla T$ is developed to stop this
current density.

Considering the interaction of acoustic phonons of energy $\hbar w_{q}$ and
wave vector $q$ with the Dirac electrons in the moiré miniband tBLG Wu  _et
al._ (2019); Kubakaddi (2021), and following Refs. Kubakaddi (2009); Cantrell
and Butcher (1987), we obtain

$\displaystyle S^{g}=-\frac{geA}{4\pi^{2}\sigma
k_{B}T^{2}\hbar^{5}\nu_{s}{\nu}_{F}^{*}}\int_{0}^{\infty}d(\hbar\omega_{q})\int_{\gamma}^{\infty}dE_{k}(\hbar\omega_{q})^{2}\frac{{\mid
M(q)\mid}^{2}\tau_{p}\tau(E_{k})}{{[1-(\gamma/E_{k})^{2}]}^{1/2}}\times$ (1)
$\displaystyle[\\{1+(\hbar\omega_{q}/E_{k})\\}\\{1+(\hbar\omega_{q}/2E_{k})\\}]f(E_{k})[1-f(E_{k}+\hbar\omega_{q})]N_{q},$

where $e$ is the electron charge, $A$ is the area of the tBLG,
$\gamma=(E_{q}/2),E_{q}=\hbar{\nu_{F}}^{*}q$, ${\mid M(q)\mid}^{2}$ is the el-
ap matrix element,
$f(E_{\textbf{k}})={[exp{(E_{\textbf{k}}-\mu)/k_{B}T}+1]}^{-1}$ is the Fermi-
Dirac distribution with chemical potential $\mu$, $\tau(E_{\textbf{k}})$ and
$\tau_{p}$ are electron and phonon momentum relaxation times, respectively,
$N_{\textbf{q}}={[exp(\hbar\omega_{\textbf{q}}/k_{B}T)-1]}^{-1}$ is the phonon
distribution function, $\omega_{\textbf{q}}=\nu_{s}q$ and $\nu_{s}$ is the
acoustic phonon velocity.

The experimental observations of electrical conductivity Efetov and Kim (2010)
and power loss Betz, A. C. and Vialla, F. and Brunel, D. and Voisin, C. and
Picher, M. and Cavanna, A. and Madouri, A. and Fève, G. and Berroir, J.-M. and
Plaçais, B. and Pallecchi, E. (2012); Baker  _et al._ (2012) in MLG and
linear-in-$T$ resistivity in tBLG Wu  _et al._ (2019); Das Sarma and Wu (2020)
are very well explained by the Dirac electron interaction with only LA
phonons, without screening. Hence, we use the corresponding el-ap interaction
matrix element, given in Refs. Wu  _et al._ (2019); Das Sarma and Wu (2020);
Kubakaddi (2021) for tBLG, in Eq.(1). Taking
$\sigma=e^{2}{\nu_{F}}^{*2}D({E_{F}}^{*})\tau({E_{F}}^{*})/2$
$[=2e^{2}{E_{F}}^{*}\tau({E_{F}}^{*})/\pi\hbar^{2}]$ Wu  _et al._ (2019), at
Fermi energy, as approximated in Refs. Kubakaddi (2009); Cantrell and Butcher
(1987), and the phonon relaxation time in the boundary scattering regime
$\tau_{p}=\Lambda/\nu_{s}$ Kubakaddi (2009); Cantrell and Butcher (1987); Nika
_et al._ (2009); Ghosh  _et al._ (2009) with the phonon mean free path
$\Lambda$ being smallest dimension of the sample, an expression for $S^{g}$ is
found to be

$\displaystyle S^{g}$ $\displaystyle=-\frac{gD^{2}F(\theta)\Lambda}{16\pi
e\rho_{m}k_{B}T^{2}\hbar^{3}{\nu_{s}}^{4}{\nu_{F}}^{*}E_{F}^{*}}\int_{0}^{\infty}d(\hbar\omega_{q})\int_{\gamma}^{\infty}dE_{k}(\hbar\omega_{q})^{3}\times$
$\displaystyle{[1-(\gamma/E_{k})^{2}]}^{1/2}\times[\\{1+(\hbar\omega_{q}/E_{k})\\}\\{1+(\hbar\omega_{q}/2E_{k})\\}]f(E_{k})[1-f(E_{k}+\hbar\omega_{q})]N_{q},$
(2)

where $D$ is the first-order acoustic deformation potential coupling constant
and $\rho_{m}$ is the areal mass
density,$E_{F}^{*}=\hbar{\nu_{F}}^{*}k_{F}^{*}$, $k_{F}^{*}=(\pi
n_{s}/2)^{1/2}$, and $F(\theta)$ is the form factor due to moiré wave
function, modifying the el-ap interaction matrix element in tBLG Wu  _et al._
(2019) as compared with the MLG.

At very low temperature, $q<<2k_{F}$ and $\hbar\omega_{\textbf{q}}\approx
k_{B}T$ and the Bloch-Gruneisen (BG) regime is characterized by the
temperature $T_{BG}=2\hbar\nu_{s}k_{F}^{*}/k_{B}=38\sqrt{N}K$, with $\nu_{s}$
= 2.0×106 cm/s and $n_{s}$ = $N\times$ 1012 cm-2. Additionally, we believe
that, besides $q<<2k_{F}^{*}$, the condition $\hbar\omega_{q}<<E_{F}^{*}$
needs to satisfied because of the very much suppressed ${\nu_{F}}^{*}$ which
is $\theta$ dependent. Consequently, we anticipate the influence of $\theta$
on the temperature range of validity of BG regime power law obtained below. In
tBLG, the $T_{BG}$ is smaller by a factor of $\sqrt{2}$ compared to MLG. For
$T<<T_{BG}$, $q\rightarrow 0$, the $S^{g}$ is given by the power law

$S^{g}=-S^{g0}T^{\delta},\hskip 28.45274pt\textrm{with}\hskip
28.45274pt\delta=3,$ (3)

where

$S^{g0}=\frac{gD^{2}F(\theta)\Lambda
k_{B}^{4}4!\zeta(4)}{2^{7/2}\pi^{3/2}e\rho_{m}\hbar^{4}\nu_{s}^{4}{\nu_{F}}^{*2}n_{s}^{1/2}}.$
(4)

Hence, in the BG regime $S_{g}$ $\sim$ ${\nu_{F}}^{*-2}$ , $T^{3}$, and
$n_{s}^{1/2}$. Moreover, $S_{g}$ $\sim$ $\nu_{s}^{-4}$, indicating that even a
small change in $\nu_{s}$ will result in to a large change in $S^{g}$. The BG
regime relation can be used to determine the el-ap coupling constant $D$ by
experimentally studying $S^{g}$ as a function of temperature.

###  II.2 Diffusion thermopower

The formula for diffusion thermopower $S^{d}$ in tBLG can be obtained from
Boltzmann transport equation in the relaxation time approximation, following
Nag . It is given by

$S^{d}=-\frac{1}{eT}[\\{\frac{\langle\tau(E_{k})E_{k}\rangle}{\langle\tau(E_{k})\rangle}-E_{F}^{*}\\}],$
(5)

where

$\langle x\rangle=\frac{[\int x(-\partial f/\partial
E_{k})D(E_{k})dE_{k}]}{[\int(-\partial f/\partial E_{k})D(E_{k})dE_{k}]}.$

In the low $T$ region and for large $n_{s}$, $S^{d}$ is given by the Mott
formula Kubakaddi (2009), taking the energy dependence of
$\tau(E_{k})=\tau_{0}E_{k}^{s}$

$S^{d}=-[\pi^{2}k_{B}^{2}T(s+1)/3eE_{F}^{*}],$ (6)

where $s$ $\sim|1|$ is the exponent of energy.

Since we need to use the acoustic phonon limited resistivity in tBLG, in the
calculations of power factor, we give here its equation from the Refs. Wu  _et
al._ (2019); Das Sarma and Wu (2020)

$\rho=\frac{32D^{2}F(\theta)k_{F}^{*}}{ge^{2}\rho_{m}\nu_{F}^{*2}\nu_{s}}I(T/T_{BG}),$
(7)

where

$I(z)=\frac{1}{z}\int_{0}^{1}dxx^{4}(1-x^{2})^{1/2}\frac{e^{x/z}}{(e^{x/z}-1)^{2}}.$
(8)

This is the equation which is obtained by appropriate modification of the
equation (2) of Min  _et al._ (2011) for MLG.

##  III Results and discussion

In this section we have numerically evaluated and discussed the results for
phonon-drag thermopower $S^{g}$, diffusion thermopower $S^{d}$ and power
factor $PF$ as a function of twist angle $\theta$ ( 1.1∘, 1.2∘ and 1.3∘),
temperature $T$ (1-20 K), and electron density $n_{s}$(0.5- 1 $n_{0}$) with
$n_{0}=1\times 10^{12}$ cm-2. The material parameters known for graphene are
used Wu  _et al._ (2019); Das Sarma and Wu (2020): $\rho_{m}=7.6\times
10^{-8}$ g/cm2, $\nu_{F}=1\times 10^{8}$ cm/s, $\nu_{s}=2\times 10^{6}$ cm/s,
$D$ = 20 eV, $\theta_{m}=1.02^{\circ}$ and $\Lambda=10\mu$m Wu  _et al._
(2019); Nika  _et al._ (2009). Our choice of $D$ here is the one used by many
research groups to explain their transport data in MLG Efetov and Kim (2010);
Betz, A. C. and Vialla, F. and Brunel, D. and Voisin, C. and Picher, M. and
Cavanna, A. and Madouri, A. and Fève, G. and Berroir, J.-M. and Plaçais, B.
and Pallecchi, E. (2012); Baker  _et al._ (2012); Bistritzer and MacDonald
(2009); DaSilva  _et al._ (2010). In order to see the large influence of the
twist angles the chosen 1.1∘, 1.2∘ and 1.3∘ are closer to the magic angle. The
corresponding effective Fermi velocities ${\nu_{F}}^{*}$ = 4, 9 and 14$\times
10^{6}$ cm/s ($>$ 1.5 $v_{s}$ Das Sarma and Wu (2020)) are much smaller than
the bare Fermi velocity $v_{F}$. The values of the function $F(\theta$) for
these angles are taken from figure 3 of Das Sarma et al Das Sarma and Wu
(2020). The range of $T$ and $n_{s}$ chosen are such that the system remains
degenerate (as required in the present theory), slightly away from the Dirac
point, and within the linear region of moire flat band. For $T<\sim$ 1 K, the
tBLG is likely to be in superconducting state. For the temperature region
considered, in the $S^{g}$ calculation, the phonon relaxation time is assumed
to be due to the boundary scattering Nika  _et al._ (2009). We have set the
specular parameter, which is a measure of the roughness at the graphene edge,
$p=0$ with a reasonable choice of $\Lambda$ Kubakaddi (2009); Nika  _et al._
(2009). It is to be noted that $\Lambda$ and $p(0\leq p\leq 1$) are sample
dependent.

Figure 1: Phonon-drag thermopower $S^{g}$ as a function of temperature $T$ for
different twist angles $\theta$ in tBLG and for MLG with $n_{s}=n_{0}$ .

.

To investigate the effect of twist angle on phonon-drag thermopower in tBLG,
we have shown in figure 1 $S^{g}$ as a function of $T$ for $\theta$ = 1.1∘,
1.2∘ and 1.3∘ for electron density $n_{s}$= $n_{0}$ along with the
corresponding curve for MLG obtained from Ref. Kubakaddi (2009). The behavior
of $S^{g}$ is generic with rapid increase in the low temperature region and
slowly increasing in the higher temperature region and finally flattening with
a knee like structure. At very low temperature all the curves are expected to
show $S^{g}$ $\sim$ $T^{3}$ power law behavior, as expected in the BG regime
($T<<T_{BG}$), and it is manifestation of 2D phonons. In our calculations it
is obeyed for for $T<$ 1 K for $\theta$ = 1.1∘, 1.2∘ and up to $\sim$1 K for
$\theta$ = 1.3∘ and hence not observed in figure 1 for the curves
corresponding to tBLG. It is apparent that, in tBLG, the range of $T$ for this
power law is $\theta$ dependent, and it is smaller for smaller $\theta$, as
found in hot electron power loss Kubakaddi (2021). On the other hand, in MLG
this power law is satisfied up to $\sim$ 2.5 K. Expressing $S^{g}$ $\sim$
$T^{\delta}$, the exponent $\delta$, which depends upon the electron- phonon
interaction, decreases from 3 to nearly zero as $T$ increases. In the case of
the curve for $\theta$ =1.1∘, we notice a kind of upward trend of $S^{g}$ for
about $T>\sim$ 15 K. This may be attributed to the less validity of our model,
which is for the degenerate system and with the increasing $T$ the temperature
dependence of the chemical potential $\mu$ moves too close to the Dirac point
for this $\theta$.

Our very important observation is that $S^{g}$ in tBLG is strongly twist angle
dependent. It is found that, compared to MLG, $S^{g}$ is highly enhanced in
tBLG, attributing to the strongly suppressed effective Fermi velocity
${\nu_{F}}^{*}$ because of the twist angle. Since, in BG regime,
$S^{g}\sim{\nu_{F}}^{*-2}$, $S^{g}$ (tBLG ) /
$S^{g}(MLG)\sim(\nu_{F}/{\nu_{F}}^{*})^{2}$. For $\theta$ = 1.1∘, 1.2∘ and
1.3∘, the ratio ($\nu_{F}/{\nu_{F}}^{*})^{2}=625$, 123.5, and 51.0,
respectively. Consequently, in BG regime, the dramatic effect is that, the
ratio ($v_{F}/{\nu_{F}}^{*})^{2}$ expected to produce the enhancement of
$S^{g}$ in tBLG by these factors, respectively, for $\theta$ = 1.1∘, 1.2∘ and
1.3∘. The enhancement is highest for the $\theta$ closest to $\theta_{m}$ and
as $\theta$ increases $S^{g}$ is decreasing and finally approaching the value
closer to that of MLG for $\theta=3.0^{\circ}$. Moreover, there are other
factors such as $g_{l}$, $F(\theta$) and $k_{F}^{*}$ which also make the
difference (by a factor of $\sim$ 0.7) in the magnitudes of $S^{g}$ (tBLG )
with $S^{g}$ (MLG). With increasing temperature, it is observed from our
calculations that, for a given $\theta$, the magnitude of enhancement of
$S^{g}$ in tBLG decreases. For example, for $\theta$ = 1.1∘, at $T$ = 1, 5 and
10 K, the ratio [$S^{g}$ (tBLG) / $S^{g}$ (MLG)] = 552.63, 456.96 and 272.78,
respectively. These ratios, respectively, for 1.2∘ (1.3∘) are 94.74 (37.67),
80.38 (31.65), 47.633 (19.35). We would like to emphasize that in tBLG the
twist angle acts as an additional strong tunable parameter of $S^{g}$, similar
to the observation made with respect to linear-in-$T$ resistivity in tBLG Wu
_et al._ (2019); Das Sarma and Wu (2020) and hot electron power loss Kubakaddi
(2021). In the present calculations, the predicted largest $S^{g}$ is about 15
mV/K at 20 K for $\theta$ = 1.1∘.

In the BG regime, we find the following important and useful relations in
tBLG. Since the underlying mechanism for phonon limited mobility $\mu_{p}$,
phonon-drag thermopower $S^{g}$ and the hot electron power loss $F_{e}(T)$ is
the same, relations between them are expected. They are established in
conventional 2DEG and 3DEG Gallagher and Butcher ; R. Fletcher and Zeitler ;
Herring (1954); Fletcher  _et al._ (1997); Tsaousidou  _et al._ (2001) and 2D
and 3D Dirac fermions Kubakaddi (2009, 2015, 2017). The relation between
$\mu_{p}$ and $S^{g}$ is a well known Herring’s law
$S^{g}\mu_{p}=-(\Lambda\nu_{s}/T)$ Herring (1954). In the following we shall
find these relations in tBLG. In BG regime, by using the mobility equation
$\mu_{p}(T)=[(15ge\hbar^{4}\rho_{m}\nu_{s}^{5}{\nu_{F}}^{*2}n_{s}^{1/2})T^{-4}]/[16\sqrt{2}\pi^{5/2}D^{2}k_{B}^{4}F(\theta)]$
Kubakaddi (2021), derived from the resistivity equation of Ref. Wu  _et al._
(2019), with our Eq. (3) for $S^{g}$, we find $S^{g}\mu_{p}=-(\Lambda
v_{s}/T)$ and hence Herring’s law is satisfied in tBLG. This relation is the
same as found in conventional 2DEG Gallagher and Butcher ; R. Fletcher and
Zeitler ; Fletcher  _et al._ (1997) and in MLG Kubakaddi (2009). $S^{g}$ is
also related to $F_{e}(T)$, obtained in Ref Kubakaddi (2021), by a simple
equation $S^{g}=(2\Lambda/ev_{s}T)F_{e}(T)$, which is again the same as found
for 2DEG in MOSFET Fletcher  _et al._ (1997) and Dirac fermions in MLG
Kubakaddi (2009). These relations are significant due to the fact that, if one
of them is measured, the other one can be calculated. For example, phonon-
limited mobility of electrons and composite fermions was determined from the
$S^{g}$ measurements in GaAs heterojucntions Tieke  _et al._ (1998).

In the following we will find the temperature at which the maximum momentum
transfer takes place, in the phonon-drag, from phonons to the electron system.
From the simple balance approach, $S^{g}\alpha-f(n_{s},T)C_{v}(T)/(n_{s}e)$
Gallagher and Butcher , where $f(n_{s},T)$ is the fraction of the momentum
transferred from phonons to electrons and $C_{v}(T)$ is the lattice specific
heat which is $\sim T^{2}$ at low temperature in 2D systems. The low - $T$
lattice thermal conductivity also varies as $\sim T^{2}$ in the boundary
scattering regime. By plotting $S^{g}/T^{2}$ as a function of $T$, the
temperature $T_{KA}$ corresponding to a maximum of $S^{g}/T^{2}$ gives us the
temperature at which phonons transfer maximum momentum to the electrons. This
‘Kohn anomaly temperature $T_{KA}$’ is defined by
$2\hbar\nu_{s}k_{F}^{*}=Ck_{B}T_{KA}$ for which the dominant phonon wave
vector $q_{D}=Ck_{B}T/\hbar\nu_{s}$ Gallagher and Butcher ; R. Fletcher and
Zeitler  for $q_{D}=2k_{F}^{*}$. Here $C$ is a dimensionless constant which we
found to be 9.0 in tBLG. In figure 2, we have depicted $S^{g}/T^{2}$ as a
function of $T$ for $\theta$ = 1.1∘ ,1.2∘ and 1.3∘ with $n_{s}=n_{0}$. For
each $\theta$, curve shows a maximum at $T_{KA}$ which is greater for larger
$\theta$ and we find a relation that ($T_{KA}$ / $\theta$) = constant $\sim$
4\. The $T_{KA}$ dependence on $\theta$ is surprising and a new result in
tBLG, although by definition $T_{KA}$ depends upon only $k_{F}^{*}$ and hence
$n_{s}$. This also supplements the observation made in figure 1 regarding the
range of validity of $T^{3}$ power law for different $\theta$. The rapid rise
at low $T$ may be attributed to the increasing number of phonons as their
dominant wave vector $q_{D}\sim k_{B}T$ Gallagher and Butcher ; R. Fletcher
and Zeitler . For $T>T_{KA}$, electrons couple to fewer phonons, as phonons
with $q=2k_{F}^{*}$ are less excited, causing the decrease of $S^{g}/T^{2}$.

In order to investigate the electron density dependence, $S^{g}/T^{2}$ vs $T$
is plotted in figure 3 for $n_{s}$= 0.5, 0.8 and 1.0 $n_{0}$ taking $\theta$
=1.1∘. The curve corresponding to MLG is for $n_{s}=n_{0}$ and it is shown
with the magnification by a factor of 2$\times 10^{2}$. The $T$ dependence of
each of the curves is same as in figure 2, with a maximum occurring at
$T=T_{KA}$. In this case, $T_{KA}$ is shifting to higher value for larger
$n_{s}$, resulting in to a relation ($T_{KA}$ /$\sqrt{n_{s}}$) = 4.25$\times
10^{-6}$ K-cm, which is understandable by definition of $T_{KA}$.
Consequently, $T_{KA}=$ 0.11 $T_{BG}$. It also indicates that $T^{3}$ power
law is satisfied for larger range of $T$ for greater $n_{s}$. At low
temperature $S^{g}$ is greater for smaller $n_{s}$, say upto $T\sim T_{KA}$,
where as it is higher for larger $n_{s}$ for $T>\sim T_{KA}$. In BG regime,
$S^{g}\sim n_{s}^{-1/2}$ dependence is attributed to the 2D nature of Dirac
fermions, as found in MLG Kubakaddi (2009), in contrast to the $n_{s}^{-3/2}$
behavior in conventional 2DEG Fletcher  _et al._ (1997).

At this point we would like to address, for comparison, the dominant phonon
wave vector $q_{D}=Ck_{B}T/\hbar v_{s}$ in the electron-phonon scattering
rate, in different systems. For conventional 2DEG in Si-MOSFET and GaAs
heterojucntions, the value of $C$ is shown to be $\sim$ 5 Gallagher and
Butcher ; R. Fletcher and Zeitler . By revisiting the $S^{g}$ / $T^{2}$ vs $T$
curves in MLG Kubakaddi (2009), monolayer of molybdenum disulphide (MoS2)
Bhargavi and Kubakaddi (2014) and three-dimensional Dirac semimetals (3DDS)
Kubakaddi (2015), from the position of maximum in each of these systems, we
determine $C$ = 7.0, 5.6 and 8.6, respectively.

Figure 2: $S^{g}/T^{2}$ as a function of temperature $T$ for different twist
angles $\theta$ in tBLG with $n_{s}=n_{0}$

.

The diffusion thermopower, which is sensitive measure of the energy dependence
of the electron momentum relaxation time, for the degenerate electron gas at
low temperature is given by the Mott formula (Eq. (6)). It is shown that for
acoustic phonon scattering, which is the only known resistive source in tBLG
samples Polshyn  _et al._ (2019); Wu  _et al._ (2019); Cao  _et al._ (2020);
Das Sarma and Wu (2020), $s=-1$ for which $S^{d}=0$. The nature of the
impurity scattering in tBLG is not precisely known, although there exist early
calculations of impurity induced resistivity Hwang and Das Sarma (2020). In
view of this, at this initial stage, we have presented a representative
calculation of $S^{d}$ from the Mott formula, for simplicity, taking $s=1$
corresponding to the unscreened impurity scattering in MLG Hwang  _et al._
(2009). Consideration of the detailed energy dependence of $\tau(E_{k})$, in
MLG, has reduced $S^{d}$ by a factor of $\sim$ 2 with parabolic temperature
dependence Hwang  _et al._ (2009). We also point out that, in the temperature
region where acoustic phonon scattering is the dominant resistive mechanism,
applying the Matthiessen’s rule, there may not be contribution of $S^{d}$ from
the resultant relaxation time. It may be noted that in Si-inversion layer,
$n_{s}$ was tuned to make $s=-1$ so that $S^{d}=0$ and the measured
thermopower was only due to $S^{g}$ Fletcher  _et al._ (1997).

Figure 3: $S^{g}/T^{2}$ as a function of temperature $T$ for different
electron densities $n_{s}$ in tBLG for $\theta$ = 1.1∘. The same is plotted
for MLG, with magnification by 200, for $n_{s}=n_{0}$ .

.

In order to see the effect of twist angle, $S^{d}$ is plotted as a function of
$T$ for different $\theta$ in tBLG along with the $S^{d}$ in MLG in figure 4
taking $n_{s}=n_{0}$. It is found that there is a big enhancement of $S^{d}$
compared to that in MLG. For example, at $T=1$, 5 and 10 K, respectively, the
enhancement is by 35, 15 and 10 times. The enhancement is smaller for larger
$\theta$. This is, as expected, due to suppressed Fermi energy, which is
$\theta$ dependent, in tBLG. Hence, $\theta$ is another important tunable
parameter of $S^{d}$ in tBLG. However, enhancement of the $S^{d}$ is much
smaller than that of $S^{g}$.

Figure 4: Diffusion thermopower $S^{d}$ as a function of temperature $T$ for
different twist angles $\theta$ in tBLG and for MLG with $n_{s}=n_{0}$

.

With an objective of finding the temperature region in which either $S^{g}$ or
$S^{d}$ is dominant, we have shown both of them as a function of $T$ for
$\theta$ =1.2∘ and 1.3∘ with the respective curves of MLG in figure 5. We
observe that there is a crossover of $S^{g}$ and $S^{d}$ at a temperature
$T_{c}$, above which $S^{g}$ is dominating, and this $T_{c}$ is $\theta$
dependent. For $\theta$ =1.2∘ (1.3∘) the cross over takes place at $\sim$ 1.3
(1.8) K in tBLG and $T_{c}$ = $\sim$ 3.5 K in MLG. $S^{d}$ is found to be
dominant in a small region of low $T$.

Figure 5: $S^{g}$ and $S^{d}$ as a function of temperature $T$ for different
twist angles $\theta$ in tBLG and for MLG with $n_{s}=n_{0}$.

We would like to make a qualitative comparison of $S$ of our calculations with
the out-of-plane $S$ measured in tBLG Mahapatra  _et al._ (2020) and
theoretically estimated in tBGNR Deng  _et al._ (2019). In tBLG, the
measurements of $S$ are over the range of temperature $T$ = 20-300 K and for
specific $\theta$’s Mahapatra  _et al._ (2020). For large $\theta$ ($\sim$
12.5∘), the authors analyze the measured $S$ by the out-of-plane phonon-drag
$S^{g}$, due to the LBM phonon modes of quadratic dispersion, over the entire
temperature range, with a maximum of $\sim$ 35 $\mu$V/K for $T>\sim$ 100 K,
which shifts to higher $T$ for larger $n_{s}$. This measured $S$ is found to
be smaller for larger $n_{s}$ (i.e. $S=S^{g}$ $\sim{n_{s}}^{-1}$ in Eq.(1) of
Ref.Mahapatra  _et al._ (2017)). In contrast, for small $\theta$
($<6^{\circ}$), measured $S$ is shown to exhibit Mott thermopower ($S^{d}\sim
k_{B}T/E_{F}$) in the experimental temperature range $\sim$ 30-300 K, with a
value of $\sim$ 30 $\mu$V/K at 300 K. However, in their out-of-plane
theoretical analysis of either phonon-drag or diffusion thermopower, we don’t
notice any twist angle dependence. On the other hand, our in-plane calculated
$S^{g}$, for $\theta$ (=1.1∘ \- 1.3∘) close to $\theta_{m}$, at about $\sim$
10 K, is about $\sim$ 103 \- 104 times larger than these measured ones
extrapolated to 10 K. Moreover, for larger $n_{s}$, $S^{g}$ is smaller
(larger) at $T<T_{KA}$ ($T>T_{KA}$). Besides, our calculated $S^{g}>>S^{d}$
for $T>\sim$ 2 K, and both $S^{g}$ and $S^{d}$ are very much $\theta$
dependent through ${\nu_{F}}^{*}$. We point out that our in-plane theory, for
$\theta$’s close to $\theta_{m}$, cannot be applied to interpret the across
plane experimental data of Ref. Mahapatra  _et al._ (2020). In tBGNR, by the
first principles calculation, with Landauer-Buttiker and Boltzmann theories, a
peak value of $S\sim$ 0.75 mV/K is predicted for twist angles 21.8∘ and 38.2∘
at 300 K Deng  _et al._ (2019). This value is closer to our predicted $S=\sim$
1-10 mV/K at $T\sim$ 10 K and for $\theta$’s closer to magic angle.

In figure 6, the power factor is shown as a function of temperature for
$\theta$ =1.2∘ and 1.3∘. In the $PF=(S^{2}/\rho t)$ calculation, with the
thickness $t$ = 0.35 nm Ghosh  _et al._ (2009), we have taken only $S^{g}$
contribution, because $S^{d}<<S^{g}$ in the larger range of $T$ considered,
besides it is likely to be nearly zero in the region where acoustic phonon
scattering is limiting the transport. The required resistivity is calculated
using Eq (7) taken from the Ref. Wu  _et al._ (2019). The $PF$ calculations of
MLG are obtained using equation of $S^{g}$ from the Ref. Kubakaddi (2009) and
$\rho$ of Ref. Min  _et al._ (2011). $PF$ is found to increase with increase
of $T$ in low $T$ region, then slowly saturates and marginally declines in the
high $T$ region. The temperature of near saturation / decline shifts to the
higher temperature as $\theta$ increases. Compared to MLG, we observe a giant
enhancement of $PF$ and this enhancement is again $\theta$ and $T$ dependent,
being largest for smallest $\theta$ and $T$. For example for $\theta$ = 1.2∘,
the enhancement of $PF$ is by a factor $\sim$ 103 at 1 K and for $\theta$ =
1.3∘ it is about 16 times at 20 K. Again, twist angle is found to be a very
significant tuning parameter of the power factor, besides temperature and
electron density. In the present work, for $\theta$ = 1.2∘ the highest power
factor is found to be $\sim$ 75 W/m-K2 at about 8 K. We point out that, it is
an unique situation in tBLG in which the large enhancement of $S\sim S^{g}$ is
nearly equal to the suppression of $\sigma$, due to the strongly increased el-
ap coupling, and hence the giant $PF$.

We roughly estimate the figure of merit $ZT$ in the case of tBLG at $\sim$ 20
K for $\theta$ = 1.2∘. Since, in tBLG it is assumed that acoustic phonon
spectrum is not affected by the twist angle Wu  _et al._ (2019), we can take
the phonon contribution $\kappa_{p}$ to the total thermal conductivity
$\kappa$ = $\kappa_{p}$ \+ $\kappa_{e}$ to be the same as in MLG. Moreover, in
tBLG electronic contribution $\kappa_{e}$ is expected to be very much
suppressed due to the strongly reduced ${\nu_{F}}^{*}$ because by
Wiedemann–Franz law $\kappa_{e}$ $\alpha$ $\sigma$ and hence we take
$\kappa\simeq\kappa_{p}$ . In order to estimate $ZT$, we chose the measured
value $\kappa$ = $\sim$10 W/m-K at $\sim$ 20 K Xu  _et al._ (2010). Then, with
$PF=\sim$70 W/m-K2, using $ZT=(PF\times T)/\kappa$, we get $ZT$ =140, an
extremely large value.

We would like to make the following remarks. It is to be noted that there is
some uncertainty in the value of el-ph coupling constant $D$ (=10- 40 eV) in
the literature Wu  _et al._ (2019); Kubakaddi (2009); Viljas and Heikkilä
(2010) and any change in its value will change $S^{g}$ significantly as
$S^{g}$ $\alpha$ $D^{2}$. Moreover, the strong dependence of $S^{g}$ on the
phonon velocity $v_{s}$ (approximately $S^{g}\sim v_{s}^{-4}$) demands an
accurate knowledge of $v_{s}$. Secondly, an ideal expression for the phonon
relaxation time is given by $\tau_{p}=(\Lambda/v_{s})[(1+p)/(1-p)]$, where $p$
is the specularity parameter. Nika et al Nika  _et al._ (2009) have used
$p=0.9$ in order to explain thermal conductivity data in MLG, and this choice
will increase the effective mean free path and hence $S^{g}$ by nearly 20
times.

Figure 6: Power factor $PF$ as a function of temperature $T$ for different
twist angles $\theta$ in tBLG and for MLG with $n_{s}=n_{0}$.

##  IV Summary

The in-plane thermopower and power factor are calculated theoretically in tBLG
showing the profound effect of suppression of the effective Fermi velocity
${\nu_{F}}^{*}$, due to the twist angle $\theta$, in moiré flat band. Our
minimal naïve theory is analytical and is of more relevant in the early stage
of the development of the subject, to bring out essential features of these
transport properties. Our simple model is for $n_{s}\leq 10^{12}$ cm-2 and
with the chemical potential little away from the Dirac point.

The $\theta$ dependent ${\nu_{F}}^{*}$ is found to strongly enhance the
phonon-drag thermopower $S^{g}$ and diffusion thermopower $S^{d}$, for small
$\theta$ closer to $\theta_{m}$. This effect is more on $S^{g}$ due to the
highly increased electron- acoustic phonon interaction, which is also the
source for the observed large linear-in-$T$ resistivity Wu  _et al._ (2019).
The enhancement of $S^{g}$ up to $\sim$ 500 times that in MLG is predicted
with a large and measurable maximum value of $\sim$ 10 mV/K at about 20 K; and
is found to significantly dominate over $S^{d}$ for large range of
$T$($>\sim$2 K). In tBLG, the twist angle has emerged as a strong tunable
parameter of thermopower, besides $T$ and $n_{s}$. The effect of
${\nu_{F}}^{*}$ on $S^{g}$ and $S^{d}$ is found to reduce with the increasing
$\theta$ and $T$. In the BG regime, power laws $S^{g}\sim{\nu_{F}}^{*-2}$,
$T^{3}$ and $n_{s}^{-1/2}$ are obtained. The temperature range of validity of
$T^{3}$ law is also found to be $\theta$ dependent. In the temperature range
considered, the $T$ dependence of $S^{g}$ is generic with rapid increase at
low $T$ and nearly constant at higher $T$. On the contrary, $S^{d}$ is taken
to be governed by the simple Mott formula with $S^{d}$ $\sim$
${\nu_{F}}^{*-1}$, $T$ and $n_{s}^{-1/2}$. By plotting the $S^{g}/T^{2}$ vs
$T$, for different $\theta$ and $n_{s}$, we have found simple relations of
’Kohn anomaly temperature $T_{KA}$’ with $\theta$, $n_{s}$ and $T_{BG}$,
namely $T_{KA}$/ $\theta$ = constant ($\sim$ 4) and
$T_{KA}$/$\sqrt{n_{s}}=$constant ($\sim$ 4.25 $\times$ 10 -6 K-cm) and
$T_{KA}$/$T_{BG}$= 0.11. In the BG regime, simple relations of $S^{g}$ with
phonon limited mobility $\mu_{p}$ i.e $S^{g}\mu_{p}=-\Lambda\nu_{s}/T$
(Herring’s law) and the hot electron power loss $F_{e}(T)$ i.e
$S^{g}=(2\Lambda/e\nu_{s}T)F_{e}(T)$ are obtained, and these are useful in
determining the one by measuring the other. Our calculations of $S$ are
qualitatively compared with the measured values across the plane of tBLG
Mahapatra  _et al._ (2020). Power factor $PF$ is also investigated as a
function of $\theta$ and $T$. The $PF$ is found to be highly enhanced for
$\theta$ closer to $\theta_{m}$, thereby $\theta$ acting as its strong tunable
parameter. $PF$ as a function of $T$ exhibits a broad maximum and a maximum
value of $PF\sim$ 75 W/m-K2 is predicted for $\theta$ =1.2∘. As a result, it
is discussed that an extremely large figure of merit is possible. We believe,
our findings with this simple analytical model will initiate some experimental
work on the in-plane thermopower in tBLG and be useful to explain the
experimental data.

Acknowledgement: Author wishes to acknowledge Vidyashree Hebbar and Ravi
Kashikar for their help in preparing this manuscript in Latex.

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