The well-known notion of domination in a graph abstracts the idea of protecting locations with guards. This paper introduces a new graph invariant, the autonomous domination number, which abstracts the idea of defending a collection of locations with autonomous agents following a simple protocol to coordinate their defense using only local information.

§ INTRODUCTION

§.§ Domination and Variations

A dominating set in a graph is a subset of vertices satisfying the condition that every vertex is either contained in the subset or is adjacent to such a vertex. One thinks of the members of the set as location at which guards are placed. A guard protects its own location as well as all locations that are immediately adjacent. The domination number of a graph is the least size of a dominating set.

One can ask how well a configuration of guards (i.e. a dominating set) can respond to an attack, or series of attacks, by an adversary. In the first case, that of a single attack, one arrives at the notion of secure dominating set. A dominating set is said to be securely dominating if, given an attack, a guard can move to the attack location and the new guard configuration is still dominating. No condition is made, though, on what might happen in a subsequent attack.

One can consider an arbitrarily long sequence of attacks. This leads to eternal dominating sets and foolproof eternal dominating sets. The former case refers to a configuration of guards which can respond to any series of attacks of arbitrary length under the direction of a strategic planner. The latter refers to a guard configuration which responds to attacks but without any coordination whatsoever, either by a master planner or among the guards. Instead, any guard adjacent to an attack moves to defend it.

There is a natural intermediate notion between eternal domination and foolproof eternal domination, called autonomous domination, and it is the purpose of the present work to introduce and study this idea. Autonomous domination amounts to abstracting the idea of general purpose guards which follow simple rules, communicating with their neighbors. In response to an attack, a guard will move to address the attack, provided that other guards can still cover the locations that become exposed by the guard's movement. In the event that multiple guards can move to the attack, a randomization protocol picks one arbitrarily.

§.§ Two Basic Examples

Consider a simple path on four vertices, defended by two guards, as in the following figure. The guards form a dominating set.

Suppose an attack is made at an intermediate vertex, indicated in red below.

Both guards are adjacent to the problem vertex. Only one guard should respond to the attack, though, namely the one to the right. If the guard to the left moves, the defensive position would become unbalanced, exposing the leftmost vertex to subsequent attack. The guards can coordinate their response using only locally available information. The right guard would not lose coverage of any vertex by moving, while the left guard would. Therefore there is a preferred defense that the guards can determine on their own, without higher order coordination.

The preceding example highlights the difference between autonomous domination and what is called foolproof eternal domination. The latter involves guards which respond to attacks without any strategy or coordination at all, so that two guards do not suffice to guard the path, in the foolproof sense.

A second example illustrates the significance of the presence of a master planner. Suppose a graph is defended by three guards, marked in gray in the following figure.

The configuration of guards above is a dominating set. Suppose that a problem arises at the red vertex. Each of the three guards is adjacent to the problem vertex. Moreover, if any guard moves to the problem vertex, the new configuration will still be a dominating set. Local information near the guards does not reveal that one movement is preferable. Suppose, though, that the guards move in the following way.

This is still a dominating set. It is also the case, however, that if a problem arises in one of the bottom two vertices it can not be addressed by the adjacent guard without losing domination.

The issue is that a `bad choice' was made by the guards earlier in the process. The initial problem  should have been addressed by the guard immediately above. Significantly, that information is not available to guards who only know simple local information about possible loss of dominance. Instead, it requires strategic, managerial thinking about the whole graph and the guards within it.

§ DEFINITIONS

The first definition, adjacency of dominating sets, makes the main definition clearer.

Given a graph $G$ and dominating sets $S$ and $S'$, we say that $S$ and $S'$ are adjacent if they differ only by a pair of vertices adjacent in $G$: i.e. there are vertices $v\in S$ and $v' \in S'$ such that $(v,v')$ is an edge of $G$ and $S' = (S \setminus \{v\}) \cup \{v'\}$.

It will be useful at times to use the term `legal guard move' to refer to the pair $(v,v')$ relating two adjacent dominating sets $S$ and $S'$.

Adjacent dominating sets have the same cardinality.

We now proceed to the main definition of the paper. The examples and discussion in the introduction illustrated the notion of domination of a graph subject to ongoing attack in which guards act via a simple protocol to coordinate their movements, but without a master planner. This definition captures the idea of a sufficient collection of autonomous guards.

A collection $\mathcal{F}$ of subsets $S_i \subset V$ is said to be an autonomously dominating family if the following conditions are satisfied.

* Each subset $S_i$ is a dominating set.

* For each subset $S_i$ and each vertex $v \in V \setminus S_i$, there is a subset $S_j \in \mathcal{F}$ which is adjacent to $S_i$ and contains $v$.

* For each subset $S \in \mathcal{F}$, every dominating set which is adjacent to $S$ is also in $\mathcal{F}$.

The first condition simply records that the guard configuration at a given time is capable of responding to an attack at any vertex. The second condition is that there is always some guard that can respond to an attack without losing domination. The first and second conditions imply that each member of the family $\mathcal{F}$ is in fact secure dominating. The third is `closure under all reasonable movements of autonomous guards.' It says that it must be possible for any guard that can move to the attack without losing graph domination to do so. There is no coordinator who, with an eye to the whole, determines which guard moves. What this would mean in a practical situation is that a randomization would pick one guard to move, if multiple guards are free to do so.

Autonomous domination is about defending a graph in perpetuity. As a result the definition of autonomously dominating family does not privilege one of the many configurations as `initial.' Instead all possible arrangements of guards are treated equally. This is the reason for making the definition in terms of a family of subsets, rather than a statement about sequences. A definition using sequences could be made as well but would be more complicated. It is simpler to consider a whole family which is closed under the relation of adjacency for dominating sets.

Having defined autonomous domination for a family, the notion for an individual set follows.

A set of vertices in a graph is said to be an autonomous dominating set if it belongs to an autonomous dominating family.

It is now possible to define a graph invariant, the autonomous domination number.

The autonomous domination number $\adn(G)$ of a graph $G$ is the least size of an autonomous dominating set.

Autonomous domination differs from eternal and eternal foolproof domination in ways made evident by the form of the definition. In Definition <ref>, one obtains eternal domination by dropping the third condition. All that is required is the existence of a possible guard (second condition). The master planner will determine which one moves. Eternal domination was introduced in [1] and has been further studied, including in [2] and [3].

On the other hand, the third condition can be strengthened as follows.

Strengthened Third Condition: For each subset $S_i$, each vertex $v \in V \setminus S_i$, and each vertex $w \in S_i$, if $v$ and $w$ are adjacent, then $(S_i \setminus \{w\}) \cup \{v\}$ is also contained in $\mathcal{F}$.

This gives foolproof eternal domination, in which guards move without considering whether such movement is prudent. The foolproof eternal domination number itself is very simple to compute (see Lemma <ref>, derived from  <cit.>). Variations have been investigated, such as the situation with multiple simultaneous guard movements [5].

Many forms of domination are listed in the appendix of [4], a comprehensive reference on domination in graphs. Eternal domination and variants were introduced subsequent to its publication.

§ PROPERTIES AND COMPUTATIONAL METHODS

§.§ Elementary Bounds

The foolproof eternal domination number is $n-\delta$ by <cit.>, where $n$ is the order of the graph and $\delta$ is the minimal degree. It was noted above that foolproof eternal domination arises by strengthening the definition of autonomous domination, so that the following inequality is immediate.

Let $G$ be a graph with order $n$ and minimal degree $\delta$. Then $\adn(G) \le n - \delta$.

The upper bound can be attained. For example, the complete graphs attain the upper bound.

It is also evident that the autonomous domination number is bounded below by the eternal domination number, since an eternal dominating set is subject to fewer conditions. In eternal domination a `manager' is able to have an eye to the whole of the network being defended, and this is reflected in the following inequality.

The autonomous domination number is bounded below by the eternal domination number.

§.§ Some means of computation

An adversary can attack at a series of independent vertices. Each such attack must be addressed by a separate guard, since the locations are independent. The next lemma records this.

Let $G$ be a graph and $\mathcal{F}$ an autonomous family of dominating sets. Given any independent set $I$ of vertices of $G$, there is a set $S$ in the automonous family such that $I \subset S$.

Let $S'$ be an element of $\mathcal{F}$. Let $S_0 = S' \cap I$, and define  $T = I \setminus S_0$. If $T$ is empty there is nothing to show. Suppose $T$ non-empty, and enumerate its elements $T = \{t_1,t_2,\dots,t_n\}$.

By the definition of autonomous family, there is a dominating set $S_1$ adjacent to $S'$ which also contains $t_1$. By the independence of $I$, the guard moving to $t_1$ does not come from a vertex in $S_0$. Continuing in this manner, we find a sequence of dominating sets $S_i\in \mathcal{F}$ such that $S_i$ is adjacent to $\S_{i-1}$ and $|S_i \cap I| = |S_{i-1} \cap I| + 1$. Ultimately we arrive at $S_n$, which includes all vertices of $I$. This is the set $S$ that was sought.

When considering concrete examples the following corollary is useful. When a graph contains an independent dominating set, it is easy to explore possible guard configurations.

Suppose that $G$ is a connected graph, $\mathcal{F}$ an autonomous dominating family, and $I$ an independent dominating set. Suppose further that the subgraph induced by the complement of $I$ is connected. Then any set $T$ of vertices of $G$  which contains $I$ and has the same cardinality as elements of $\mathcal{F}$ is contained in the autonomous family $\mathcal{F}$.

By Lemma <ref> there is some $S \in \mathcal{F}$ which contains $I$. Let the remaining guards move along paths from their initial locations to the remaining elements of $T$. These constitute a series of legal guard moves.

A usful bound arises in the case that all dominating sets of a given size are secure dominating.

If every dominating set of size $k$ is secure dominating, then $\adn \le k$.

Define an autonomous family consisting of all dominating sets of size $k$.

The converse does not hold. Consider a $K_3$ with two leaves from one vertex. The autonomous domination number of this graph is $3$, but the set consisting of the 3 $K_3$ vertices is dominating but not secure dominating.

§.§ A partition count

The eternal domination number is bounded above by a count of certain subcliques of a graph, given in <cit.>. More specifically, if the vertices of a graph can be divided among $c$ subsets such that the induced graph on each subset is complete, then the eternal domination number is bounded above by $c$.

This partition number $c$, though bounding the eternal domination number, does not provide a bound of $\adn$. Observe that a path $P_n$ can be partitioned (by taking adjacent pairs of vertices, with possibly one left over) into $\lceil \frac{n}{2} \rceil$ such subsets, all of whose induced graphs are complete. On the other hand, $\adn(P_n)$, computed in Proposition <ref>, exceeds this bound by an amount growing arbitrarily large as $n$ increases.

There is, however, the following technical proposition involving a partition of the set of vertices, which determines the autonomous domination number at least in some cases. It will be applied in Section <ref> to compute autonomous domination numbers of products of complete graphs.

Suppose that a graph $G$ is such that the vertices of $G$ may be partitioned into $k$ sets $S_i$ so that the induced subgraph on each $S_i$ is complete. Suppose further that $|S_i| > k$ for each $i$, and that there is at most one edge from a vertex in one partition $S_i$ to some other partition $S_j$. Then $\adn(G) = k$.

Let $S$ be a set of vertices of size $k$ containing one element of each of the sets $S_i$. This is a dominating set by the hypothesis that the induced graphs on the sets $S_i$ are complete.

With guard configuration $S$, consider an attack at an arbitrary unoccupied vertex $v$. The vertex $v$ is contained in some $S_i$, and as such is defensible by the (single) guard also contained in the subset $S_i$. It is possibly adjacent to a guard at a vertex $w$, contained in a distinct partition set $S_j$, $j \ne i$. We claim that there is no dominating set adjacent to $S$ containing $w$.

There are at least $k+1$ vertices in $S_j$ by hypothesis. There are, in addition to the guard at $w$, $k-1$ other guards. If the guard at $w$ moves to $v$, the vertex $w$ is still guarded (by the guard that just moved). The $k$ or more other vertices of $S_j$ must be guarded. Since the $k-1$ guards are contained in partition sets other than $S_j$, those $k-1$ guards each defend at most one vertex of $S_j$, by the connectivity hypothesis. We see that the set $S$ is not adjacent to any dominating set which contains no element of $S_j$. Therefore each guard must remain in its respective partition set.

Thus the family generated by $S$ under sequences of legal guard moves is autonomous dominating.

Similar reasoning shows that no collection of $k-1$ vertices forms a dominating set.

§.§ Chromatic Relation

The eternal domination number of a graph is bounded above by the chromatic number of the graph complement, shown in <cit.> and following discussion. The idea is this. Color vertices of the complementary graph, assign a guard to each color, and give instructions to each guard to address attacks only on vertices of its assigned color.

This chromatic bound does not suffice for autonomous domination. The order six graph in Section <ref> has a three-colorable complement but its autonomous domination number is $4$.

§.§ Connectedness

Ordinarily it will be of interest only to consider connected graphs. The following lemma is straightforward.

The autonomous domination number of a disjoint union of graphs is the sum of the autonomous domination numbers of the summands.

§ ELEMENTARY EXAMPLES

It is not difficult to check that $\adn(P_2) = 1$ and $\adn(P_3) = 2$. For longer paths, the following result gives the autonomous domination number.

The autonomous domination number of the path $P_n$ is $n-2$ for $n \ge 4$.

First, consider $n\ge 5$. Let $\{a_1,a_2,\dots,a_n\}$ be the vertices of $P_n$, with edges $(a_i,a_{i+1})$. The set $I = \{a_2, a_4,\dots,a_{\lfloor \frac{n}{2} \rfloor}\}$ is independent and dominating, so by Lemma <ref> it is a subset of an autonomous dominating set. Let $S$ be a dominating set with $|S|\le n-3$ which contains $I$. Let $j$ be an odd integer such that $a_j \in S$ and $a_{j+2} \notin S$ and $j+2\le n$. Then the legal guard moves $(a_{j+1},a_{j+2})$ followed by $(a_j,a_{j+1})$ show that the guards outside $I$ can accumulate among the vertices with higher indices. After a series of such moves, by the cardinality of $S$ we arrive at a dominating set which contains $\{a_2,a_4\}$ and which does not contain $\{a_1,a_3,a_5\}$.  Then $(a_4,a_5)$ is a legal guard move. The resulting set of vertices is no longer secure dominating, since only the guard at $a_2$ defends $a_3$, but $(a_2,a_3)$ is not a legal guard move since it would leave $a_1$ without defense.

In the case $n=4$, there is no dominating set of size $n-3 = 1$.

Having established the lower bound, now consider any dominating set $S$ of size $n-2$. Either the omitted vertices are adjacent, or they are not. If they are adjacent, they must be interior to the path (by the fact that $S$ is dominating), in which case $S$ is secure dominating. If they are not adjacent it is immediate that $S$ is secure dominating. By Theorem <ref> and the lower bound already established, we see that $\adn(P_n) = n-2$.

The foolproof eternal domination number of $P_n$ is $n-1$, and the eternal domination number is $\lceil \frac{n}{2} \rceil$ as computed in [1], so that the autonomous domination number is intermediate, but closer to the foolproof for large $n$.

A similar relation is present in the case of cycles. For small cycles, it is straightforward to check that $\adn(C_3) = 1$, $\adn(C_4) = 2$, and ${\adn(C_5)=3}$. The general result is the following.

The autonomous domination number of the cycle $C_n$ is $n-3$ for $n \ge 6$.

Let $n$ be at least $8$, and let $\{a_1,\dots,a_n\}$ be the vertices of $C_n$, with edges $(a_i,a_{i+1})$ and $(a_1,a_n)$. Let $I$ be the set of all odd-indexed vertices with index strictly less than $n$. Let $S$ be any dominating set of size $|S| \le n-4$ which contains $I$. Reasoning as in the proof of Proposition <ref>, $S$ is connected through a series of legal guard moves to a dominating set which contains the vertices of $I$ as well as the vertex $a_n$ and all vertices with sufficiently large even index. In particular, by cardinality, the vertices $\{a_2,a_4,a_6, a_8\}$ are unguarded. The series of legal guard moves $(a_3,a_2)$ and $(a_7,a_8)$ yields a dominating set which is not secure dominating, since the guard at $a_5$ must defend both $a_4$ and $a_6$ but cannot do so.

In the case of $n=6$, there is no secure dominating set of size $2$. In the case of $n=7$, the independent set $\{a_1,a_4,a_6\}$ is dominating and so would necessarily be contained in an autonomous family whose elements have size $3$. But $(a_6,a_7)$ is a legal guard move, yielding $\{a_1,a_4,a_7\}$ which is not secure dominating, since only $a_4$ defends $a_3$ and $a_5$.

Every dominating set of size $n-3$ is secure dominating (there is no room to end up with an `isolated guard'), so by Theorem <ref> we conclude that $\adn(C_n) = n-3$ for $n\ge 6$.

The comparable results are that the eternal domination number of $C_n$ is $\lceil \frac{n}{2} \rceil$ and the foolproof eternal domination number is $n-2$, again given in [1].

§ AUTONOMOUS DOMINATION NUMBERS FOR CERTAIN FAMILIES OF GRAPHS

Let $p \le q$ be natural numbers. In <cit.> it is shown that the eternal domination number of $K_p \times K_q$ is $p$, while the foolproof eternal domination number is $pq-(p+q)+2$, so that the two differ widely in general. For this family of graphs the autonomous domination number happens to coincide with the eternal domination number.

Suppose $p$ and $q$ are natural numbers with $p \le q$. Then $\adn(K_p \times K_q) = p$.

First consider the case that $p < q$. Partition the graph $K_p \times K_q$ into the $p$ sets of vertices of the copies of $K_q$, each of which has size $q$. Then apply Proposition <ref> to infer that $\adn(K_p \times K_q) = p$.

It remains to treat the case $p=q$. By <cit.> the eternal domination number is $p$, so the elementary bound of Lemma <ref> implies that $\adn$ is at least $p$. We now show that $p$ guards suffice.

Enumerate the vertices of each $K_p$ with non-negative integers $\{0,\dots,p-1\}$. The vertices of the product $K_p \times K_p$ are then ordered pairs $(i,j)$ of such integers.

Distribute $p$ guards to $p$ vertices with distinct first coordinates; the second coordinate of each location is arbitrary. This is a dominating set by the completeness of the factor $K_p$.

Consider a legal guard move whose result is that there is an integer $i$ such that there is no guard at $(i, j)$ for all $j$. In other words, the collection of vertices $(i, \bullet)$ in $K_p \times K_p$ has been vacated. For this to be the case, there must have been guards at $(j_k, k)$ for all $k$ (the first coordinates need not be distinct) so that every complete subgraph of the form $(i, \bullet)$ remains dominated.

Therefore, at every time, legal guard movements will be such that either every guard has a distinct first coordinate, or every guard has a distinct second coordinate, and all such subsets of vertices are dominating sets.

Since the eternal domination number $p$ is a lower bound for the autonomous domination number, we conclude that $\adn(K_p\times K_p) =p$.

Ladder graphs are also an instance in which eternal and foolproof eternal domination numbers diverge widely. In this case the autonomous domination number differs negligibly from the foolproof one. By <cit.> the eternal domination number of $P_2 \times P_n$ is $n$, and the foolproof eternal domination number is $2n-2$.

The ladder graph $P_2 \times P_n$ has autonomous domination number ${\adn(P_2 \times P_n) = 2n-3}$.

Let the vertices of $P_2\times P_n$ be $\{a_1,\dots,a_n,b_1,\dots,b_n\}$ with edges $(a_i,a_{i+1})$, $(b_i,b_{i+1})$, and $(a_i,b_i)$. Let $S$ be a dominating set of size $n \le |S| \le 2n-4$ which contains the independent dominating set $\{a_1,b_2,a_3,b_4,\dots\}$. Suppose that $S$ contains $a_{2k}$ and does not contain $a_{2k+2}$. The pairs $(a_{2k+1},a_{2k+2})$ and $(a_{2k},a_{2k+1})$ are legal guard moves whether or not $b_{2k+1}$ is occupied. Thus excess guards in the $a$-subgraph can be moved to vertices of higher index. The same can be done with guards among the $b$ subgraph vertices. After many such moves, we obtain a dominating set $S'$ connected to $S$ through a series of legal guard moves does not contain $\{a_2,a_4,b_1,b_3\}$. Then $(b_2,b_3)$ and $(a_3,a_4)$ are legal guard moves, but the resulting dominating set is not secure dominating since the guard at $a_1$ is isolated and cannot defend both $b_1$ and $a_2$.

To show that $2n-3$ autonomous guards suffice, consider any dominating set of size $2n-3$. All vertices but those on the ends of the ladder have degree 3, which means that an adjacent vertex is necessarily occupied simply by cardinality. Since the set is presumed dominating, either at least one of $\{a_1,b_1\}$ is included or both $a_2$ and $b_2$ are included. In either case the set is secure dominating. The same reasoning applies for the other end of the ladder.

§ COUNTEREXAMPLES

When people act with limited information, circumstances that are in principle advantageous can nonetheless yield detrimental results. The following two examples show that autonomous domination reflects this, in failure that arises through an excess either of choices or of strength. The addition of an edge, which offers new ways for guards to defend locations, can lead them away from the most strategically suitable positions. Additional guards, which in theory make a defensive arrangement stronger, can allow for poor positioning due to temporary security arising from the abundance of defenders.

The examples below are evidence of the utility of the concept of autonomous domination. The parameter does not decrease monotonically with edge addition, and it is not a super-hereditary. These suggest that autonomous domination is a good abstraction that can account for difficulties that arise when agents act tactically with restricted information.

§.§ Edge addition

The autonomous domination number can decrease when an edge is added. A simple example is completing a path on three vertices to the complete graph on these three vertices, in which case the autonomous domination number decreases from 2 to 1.

Adding an edge can increase the autonomous domination number, however. A new edge means that guards can be drawn away from one portion of the graph and into another, and this could leave the source portion vulnerable.

The graph on the left in Figure <ref> has autonomous domination number $2$.

The vertices of the graph separate into two induced $K_3$ subgraphs, the lower and the upper levels, and Proposition <ref> applies.

Edge addition can increase the autonomous domination number

The additional diagonal edge added to obtain the graph on the right causes the autonomous domination number to increase.

The autonomous domination number of the graph on the right in Figure <ref> is $3$.

The autonomous domination number must be at least $2$, since the graph has an independent set of size $2$. Consider the independent dominating set consisting of the upper left and lower central vertices. This is adjacent to the dominating set consisting of the upper left and upper central vertices. But this is not secure dominating since an attack on the upper right vertex cannot be safely covered.

Another simple example is adding an edge to connect the disjoint union of $C_5$ and $K_3$. The graph $C_5 \coprod K_3$ has autonomous domination number 5, by the additivity statement Lemma <ref>. When a bridge is built between them, a guard can be `siphoned off' of the $C_5$ subgraph in a way that leaves it exposed to attack. The autonomous domination number of the new graph is 6, rather than 5.

§.§ Failure through excess of guards

In can be that $n$ guards suffice for autonomous domination but $n+1$ guards do not. Thus autonomous domination is not a super-hereditary property.

Define a graph with the vertices
and the following edges.

* Edges such that the set $\{a_i\}_{i=1}^4$ induces a complete subgraph.

* Edges such that the set $\{b_i\}_{i=1}^5$ induces a complete subgraph.

* $(a_i,b_i)$ for $1 \le i \le 4$.

* $(b_1,a_2)$

* There are no other edges.

The graph is depicted in Figure <ref>. The vertices $a_i$ are below and the vertices $b_i$ are above. The topmost vertex is $b_5$, not adjacent to any of the $a_i$.

The graph of Definition <ref>

The eternal domination number of this graph is $2$. This is also the autonomous domination number. Yet there is no autonomous set with three guards. The reason is that an arbitrary dominating set is connected through a series of legal guard moves with $\{b_1,a_3,a_4\}$, which is connected by a pair of legal guard moves with $\{b_1,b_3,b_4\}$. This last set is not secure dominating, since it is not adjacent to a dominating set containing $b_5$. Informally, with three guards it is possible that they all move `upstairs,' and this is a bad decision.

Because autonomous domination is not super-hereditary, it does not suffice to show that a certain number of guards fail in order to find a lower bound on the autonomous domination number. Instead, every possibility greater than or equal to the eternal domination number must be considered.

§ REALIZABILITY OF PARAMETER VALUES

It is straightforward to observe that the eternal domination number is bounded below by the domination number, and similarly that the autonomous domination number is bounded below by the eternal domination number (Lemma <ref>). A question following from these bounds is whether every possible collection of three natural numbers satisfying the inequalities constitute the domination number, eternal domination number, and autonomous domination number of a graph. This is so (excluding a trivial exception), as will be shown below. The proof will require reference to a number of families of graphs constructed for the purpose. These will be presented first, and their various parameters computed. The theorem then collects these results.

Given a natural number $n\ge 2$ define a graph $\Acal_n$ with vertices

$$\{a_1, a_2, \dots, a_{2n+1},b_1, b_2, \dots, b_{2n}, c\}$$
and  the following edges.

* The vertices $a_i$ induce a complete subgraph.

* The vertices $b_i$ induce a complete subgraph.

* $(a_i,b_i)$ for all $1 \le i \le 2n$

* $(a_i,b_j)$ for all $1 \le i \le n$ and $1 \le j \le 2n - i$ (observe that some of these were included in the previous item)

* $(c,a_i)$ fo all $1 \le i \le 2n+1$

* $(c,b_j)$ fo all $1 \le j \le 2n$

The graph $\Acal_2$ is sketched in Figure <ref>. Some simplification has been made there to avoid clutter. The ovals denote induced complete subgraphs, and the vertex $c$, adjacent to all other vertices, is shown with just a few half edges.

The graph $\Acal_2$

The parameters of $\Acal_n$ are these: ${\gamma(\Acal_n)=1}$, ${\edn(\Acal_n)=2}$, and  ${\adn(\Acal_n)=n+1}$.

The vertex $c$ is adjacent to every other vertex, so $\gamma(\Acal_n)=1$. The vertices $a_{2n+1}$ and $b_{1}$ are not adjacent, so $\edn(\Acal_n)\ge 2$. That two guards suffice follows from observing that $\{c,a_1,a_2,\dots,a_{2n+1}\}$ and $\{b_1, b_2, \dots, b_{2n}\}$ induce complete subgraphs.

Consider any natural number $g$ satisfying $2 \le g \le n$. There is no autonomous dominating set of size $g$. Any dominating set is adjacent to one which includes $c$. Since $c$ is adjacent to every vertex, we see that any two sets of the same size, both of which contain $c$, are adjacent through a series of legal guard moves. An arbitrary dominating set of size $g$ is thus connected to $\{a_{2n--g+2}, a_ {2n-g+3} \dots, a_{2n},c\}$. This set is adjacent to $\{a_{g-1},a_{2n-g+2}, a_ {2n-g+3} \dots, a_{2n}\}$, which is a dominating set. It is not secure dominating, however, since an attack on $a_{2n+1}$ leads to failure.

It remains to show that $\Acal_{n}$ has an autonomous dominating set of size $n+1$. Any dominating set of $n+1$ vertices among the $a_i$ is secure dominating. No set consisting only of $b$ vertices is dominating, since none is adjacent to $a_{2n+1}$. Thus the only remaining case to check is that a dominating set including $c$ is secure dominating. It follows immediately from the existence of induced complete subgraphs that a dominating set including both $a$ and $b$ vertices, as well as $c$, is secure dominating. The set $\{c,b_1,b_2,\dots,b_{2n}\}$ is also secure dominating, since it is dominating, and exchanging any element with an adjacent $a$ vertex leads to a set containing both $a$ and $b$ vertices. Such a set is dominating.

Given a non-negative integers $m$ and $n$, define the graph $\Bcal_{m,n}$ with vertices

$$\{a_1,a_2,\dots a_{2n+3},b_1,b_2,\dots,b_{2n+4},c^1_1,c^2_1,c^1_2,c^2_2,\dots,c^1_m,c^2_m\}$$
and the following edges.

* The vertices $a_i$ induce a complete subgraph.

* The vertices $b_i$ induce a complete subgraph.

* $(a_i,b_i)$ for all $1 \le i \le n$

* $(c^1_i,c^2_i)$ for all $1 \le i \le m$

* $(b_{n+1},c^1_i)$ for all $1 \le i \le m$

* $(a_1,b_2)$ and $(a_2,b_1)$

* $(b_{3+i},a_{3+j})$ for each $0 \le i \le n$ and $0 \le j \le 2n - i$

* There are no other edges.

A sketch of such a graph is in Figure <ref>. There the parameter values are $m=2$ and $n=0$.

The graph $\Bcal_{2,0}$

The domination parameters of $\Bcal_{m,n}$ are:  ${\gamma(\Bcal_{m,n})=\edn(\Bcal_{m,n}) = m+2}$, and ${\adn(\Bcal_{m,n})=m+n + 3}$.

The set $I = \{a_{2n+3},b_{1},c^2_1,c^2_2,\dots,c^2_m\}$ is an independent dominating set of size $m+2$. There is no smaller dominating set, since at least one guard is needed for each of the $m$ branches $\{c^1_i,c^2_i\}$ and none of these will guard $b_i$ for $i \le 2n+3$ or any vertex among the $a_j$.

The eternal domination number is also $m+2$ by the clique bound of <cit.>, since one guard can be allocated to each of the $m$ pairs $\{c^1_i,c^2_i\}$, one can be allocated to the vertices $a_i$, and finally one to the vertices $b_j$.

We now show that the autonomous domination number is not less than $m+n + 2$. Let $k$ be a natural number satisfying $m+2 \le m+1+k < m+n+3$, and let $m+1+k$ be the size of some dominating set $S$ of vertices which contains the independent set $I$. Let $T$ be the (possibly empty) set of $k-1$ vertices $\{b_{2n+3 - k + 2},b_{2n+3-k+3}, \dots, b_{2n+3}\}$. Since the graph induced by the complement of $I$ is connected, Corollary <ref> implies that there is a series of legal guard moves connecting $S$ with the dominating set $S' = \{a_{2n+3}, b_1, c_1^2,c_2^2,\dots,c_m^2\} \cup T$.

Let $S'$ and $S''$ be adjacent by the legal move $(a_{2n+3},b_{2+k})$. Observe that $S''$ is dominating, since each element of $T$ is adjacent to exactly one of $\{a_i\}_{i=2n+3 - k + 2}^{2n+3}$ and $b_{2+k}$ is adjacent to each of $\{a_3, a_4, \dots, a_{2n+3-k+1}\}$.

The set $S''$ is not, however, secure dominating. Given an attack at $b_{2n+1}$, no guard can cover without losing domination. This shows that the autonomous domination number is greater than $m+n+2$.

Now we must show that $m+n+3$ guards suffice for autonomous domination. Consider any dominating set of this size. It necessarily contains at least $m$ vertices from among $c^j_i$. It follows that the set contains at most $n+3$ vertices from $\{a_i\}$ and $\{b_j\}$. For the sake of being a dominating set, we also know that there are at least two such vertices.

There are a few cases to consider.

Case 1: Guards are at $a$ and $b$ vertices.
This case is straightforward. Since $\{a_i\}$ and $\{b_i\}$ are each cliques, the set is secure dominating.

Case 2: No guard is at any $b$ vertex.
In order for the set to be dominating, it necessarily contains the $n$ vertices $\{a_{n+4}, a_{n+5}, \dots, a_{2n+3}\}$ (each of which is the unique vertex among the $\{a_i\}$ adjacent to the similarly indexed $b_i$). The set must in addition contain at least one of $\{a_1,a_2\}$.

Either some guards are at $a$ vertices and some are at $b$ vertices, in which case the set is secure dominating, or all guards are on $a$ vertices. Only the latter case deserves further consideration. An attack at a $b$ vertex can be covered, by domination and the fact that the $a$ and $b$ vertices induce respective complete graphs. An attack at an $a$ vertex is securely coverable, which is seen as follows. Let $a_i$ be the first occupied $a$ vertex (counting up from index $1$). Then all guards in the vertices from $a_{i+1}$ to $a_{2n-i}$ are redundant. That means that at most the vertices from $a_{2n-i+1}$ up to $a_{2n}$ are additional necessary guards. That is $i$ total. By counting, $i$ is at most $n$. (The situation $i=n+1$ is impossible, since then the occupied vertices would be $n+1$ through $2n+1$, and this is not a dominating set.)

Given a natural number $m\ge 2$ and a natural number $n$, define the graph $\Ccal_{m,n}$ with vertices are
$$\{a_1 a_2, b_1, b_2, \dots,b_m, c_1, c_2, \dots, c_n\}$$
and the following edges.

* Each vertex $a_i$ ($i=1,2$) is adjacent to every other vertex.

* The set of vertices $\{c_i\}_{i=1}^n$ induces a complete subgraph.

* There are no other edges.

The graph $\Ccal_{3,3}$ is depicted in Figure <ref>.

The graph $\Ccal_{3,3}$

The domination parameters of $\Ccal_{m,n}$ are these: ${\gamma(\Ccal_{m,n}) = 1}$, ${\edn(\Ccal_{m,n}) = m + 1}$, and ${\adn(\Ccal_{m,n}) = m+n}$.

Since $a_1$ is adjacent to every vertex the domination number is 1.

The set $\{b_1, b_2, \dots. b_m, c_1\}$ is an independent set, so that the eternal domination number is at least $m+1$. In fact $m+1$ suffice. Let one guard be assigned to $\{b_1, a_1, a_2\}$, and let one guard be assigned to each vertex $b_i$ ($i>2$). Finally, let a guard be assigned to $\{c_1, c_2, \dots , c_n\}$.

Let any number $k$ of guards from $m +1$ to $m+n-1$ be given, and consider any dominating set of size $k$. Such a dominating set is adjacent to one containing the vertex $a_1$. Then, via a series of legal guard moves, we arrive at the dominating set $\{a_1, c_1, c_2, \dots, c_n, b_1, b_2 , \dots , b_{k-n-1}\}$. This is not a secure dominating set, since an attack at $b_{k-n}$ can only be covered by the guard at $a_1$, and then no guard is adjacent to $b_{k-n+1}$.

A set with $m+n$ vertices either contains all the peripheral vertices $b_i$ and $c_j$, or it contains one of the central vertices $a_k$. In either case such a set is secure dominating.

Given natural numbers $m$ and $n$, define the graph $\Dcal_{m,n}$  with vertices
$$\{a_1, a_2, b_1^1, b_1^2, b_2^1, b_2^2, \dots , b_m^1, b_m^2, c_1, c_2, c_3, \dots , c_n\}$$
and the following edges.

* The set of vertices $\{a_1, a_2, b_1^1, b_2^1, \dots, b_m^1\}$ induces a complete subgraph.

* $(b_i^1,b_i^2)$ for each $i$, $1 \le i \le m$.

* The set of vertices $\{a,_2,c_1, c_2, \dots , c_n\}$ induces a complete subgraph.

* There are no other edges.

The graph $\Dcal_{3,3}$ is depicted in Figure <ref>. The oval encloses vertices inducing a complete subgraph.

The graph $\Dcal_{m,n}$

The domination parameters of $\Dcal_{m,n}$ are these: $\gamma(\Dcal_{m,n}) = m+1$, $\edn(\Dcal_{m,n}) = m+2$, and $\adn(\Dcal_{m,n}) = m+n+1$.

The vertices $\{b_1^1, b_2^1, \dots, b_m^1, a_2\}$ form a dominating set. No smaller dominating set exists. At least $m$ guards are required for the $m$ leaves $b_i^2$, and another is required for vertices $c_i$.

The set $\{a_1,b_1^2, b_2^2, \dots, b_m^2, c_1\}$ is an independent set of size $m+2$, giving a lower bound for the eternal domination number. In fact $m+2$ guards suffice, since $\{a_1,a_2\}$, $\{\{b_i^1, b_i^2\}_{i=1}^m\}$, and $\{c_i\}_{i=1}^n$ is a partition of the vertices into $m+2$ subsets all inducing complete subgraphs.

Let $k$ be a number of guards $m+2 \le k \le m+n$. Consider a dominating set of size $k$ containing the independent set of vertices $\{a_1, b_1^2, b_2^2, \dots , b_m^2, c_1\}$. This set has size $m+2$, so that there are $k-m-2$ remaining guards. Then via a series of legal guard moves we obtain the set
$$\{a_1,  b_1^2, b_2^2, \dots , b_m^2, c_1, c_2, \dots, c_{k-m-1}\}$$
which is adjacent to
$$\{a_1,  b_1^1, b_2^2, \dots , b_m^2, c_1, c_2, \dots, c_{k-m-1}\}$$
which is adjacent to
$$\{a_2,  b_1^1, b_2^2, \dots , b_m^2, c_1, c_2, \dots, c_{k-m-1}\}$$
which is adjacent to
$$\{  b_1^1, b_2^2, \dots , b_m^2, c_1, c_2, \dots, c_{k-m}\}$$
which is not secure dominating, since an attack at $b_1^2$ is only covered by the guard at $b_1^1$, leaving $a_1$ unprotected.

Let $m$ and $n$ be natural numbers. Define the graph $\Ecal_{m,n}$ with vertices
$$\{a_1, a_2, a_3, \dots, a_{m+3}, b_1, b_2, c_1, c_2, \dots , c_n\}$$
and the following edges.

* The vertices $a_i$ induce a complete subgraph.

* $(a_i,b_j)$, for each pair $(i,j) \in \{1,2\}\times \{1,2\}$.

* $(a_{m+3}, c_i)$ for each $i \in \{1, 2, \dots, n\}$.

The graph $\Ecal_{3,3}$ is depicted in Figure <ref>. The vertices $a_i$, inducing a complete subgraph, are enclosed by the oval and their edges are omitted.r

The graph $\Ecal_{3,3}$

The domination parameters of $\Ecal_{m,n}$ are these: ${\gamma(\Ecal_{m,n}) = 2}$, ${\edn(\Ecal_{m,n}) = n+3}$, and ${\adn(\Ecal_{m,n}) = m+n+3}$.

There is no single vertex adjacent to all vertices, but each vertex is adjacent to either $a_1$ or $a_{m+3}$. This establishes the domination number.

The set $\{b_1, b_2, a_3, c_1, c_2, \dots, c_n\}$ is an independent set of size $n+3$, establishing a lower bound for the eternal domination number. That $n+3$ guards suffice is seen by noting that stationary guards can be assigned to each $b_i$ and $c_j$, and one final guard is needed for the complete graph induced by $\{a_i\}$.

Let $k$ be a natural number $n+3 \le k \le m + n + 2$ and consider a dominating set of size $k$ which contains the independent set $\{b_1, b_2, a_3, c_1, c_2, \dots, c_n\}$. Through a series of legal guard moves we obtain a dominating set which contains $b_1$ and $b_2$ but neither $a_1$ nor $a_2$ nor $a_3$. This is adjacent to one containing $a_1$ but not $b_1$, and this is adjacent to one containing $a_2$ but not $b_2$. This is adjacent to the dominating set containing $a_3$ but not $a_2$. This set is not secure dominating. An attack at one of the vertices $b_i$ can only be addressed by the guard at $a_1$, at which point the other $b$ vertex is left without defense.

That $m+n+3$ guards suffice follows from counting. With $m+n+3$ guards at least two of $\{a_1,a_2,b_1,b_2\}$ will be occupied, hence neither $b_1$ nor $b_2$ will be unguarded.

Given natural numbers $\ell$, $m$, and $n$, define the graph $\Fcal_{\ell,m,n}$ with vertices
$$\{a_1, a_2, \dots, a_\ell, b_1, b_2, \dots, b_m, c_1, c_2, c_3,\dots, c_{\ell+n}\}$$
and the following edges.

* $(a_i,c_i)$ for each $i \in \{1,2,\dots,\ell\}$.

* $(b_i,c_j)$ for each $i \in \{1,2,\dots,m\}$ and each $j \in \{1,2,\dots,\ell\}$.

* The vertices $c_i$ induce a complete subgraph.

* There are no other edges.

The graph $\Fcal_{4,3,3}$ is depicted in Figure <ref>. The vertices $c_i$, inducing a complete subgraph, are enclosed in the oval and their edges are omitted.

The domination parameters of $\Fcal_{m,n}$ are these: ${\gamma(\Fcal_{\ell,m,n}) = \ell}$, ${\edn(\Fcal_{\ell,m,n}) = \ell+ m + 1}$, and ${\adn(\Fcal_{\ell,m,n}) = \ell + m + n}$.

One vertex of each pair $\{a_i,c_i\}$ must be occupied in order to obtain a dominating set, so that the domination number is at least $\ell$. Provided that at least one guard is stationed at $c_i$, the set is dominating, since each vertex $c_i$ is adjacent to every $b$ and $c$ vertex.

The set $\{a_1,a_2,\dots,a_\ell,b_1,b_2,\dots,b_m,c_{\ell+1}\}$ is an independent set of size $\ell + m +1$, which gives a lower bound for the eternal domination number. That many guards suffice, since stationary guards can be assigned to each $a_i$ and $b_j$ and one guard can defend the remaining vertices $c_i$ which induce a complete subgraph.

Let $k$ be a natural number $\ell +m+1 \le k \le \ell +m +n - 1$. Consider a dominating set of size $k$ which contains the independent set $\{a_1,a_2,\dots,a_\ell,b_1,b_2,\dots,b_m,c_{\ell+1}\}$. This is adjacent to one with $c_1$ replacing $a_1$. Then all the guards on the $b_i$ can move to $c_i$ without losing domination. In the end, since the $\ell$ leaves are occupied, as well as interior vertices $\{c_{\ell+1},\dots,c_{\ell + n}\}$, then because $k \le \ell + m + n - 1$ we see that at most $m-1$ of the vertices ${c_1,\dots,c_\ell}$ are occupied. There is then no defense against the series of attacks $b_1$, $b_2$, …$b_m$.

We finally show that $\ell +m +n$ guards suffice for autonomous domination. Consider any dominating set of that size. Since the set is dominating, at least one member of each pair $\{a_i,c_i\}$ is contained in the set. Moreover, either no $c_i$ is in the set, in which case all of the vertices $b_i$ are, or at least one of the $c_i$ is in the set. In the former case secure domination is straightforward: every vertex $a_i$ is occupied, and every vertex $b_i$ is occupied, and there are $n>0$ guards remaining necessarily contained in the set $\{c_i\}$ which induces a complete subgraph. In the latter case, let $j$ be the number of vertices $c_i$ contained in the dominating set $1 \le j \le \ell$. Then $\ell-j$ guards occupy leaves, since the set is dominating. At most $n$ guards are contained in the subset of vertices $\{c_{\ell+1},\dots,c_{\ell + n}\}$, which means at least $\ell+m+n - (j + (\ell-j) +n) = m$ guards are contained in the vertices $b_i$. But there are only $m$ such vertices. Thus the only empty vertices are either among pairs $\{a_i,c_i\}$ (which contain a guard by hypothesis) or are among the vertices $\{c_{\ell+1},\dots,c_{\ell+n}\}$ (which induce a complete subgraph and contain a guard). Thus the set is secure dominating.

The graph $\Fcal_{4,3,3}$

Suppose that $G$ is a graph and $\edn(G)=1$. Then ${\adn(G)=1}$.

If $\edn(G)=1$ then there is no independent set of vertices of size $2$. Therefore $G$ is necessarily a complete graph.

The previous proposition is the reason for excluding the case $a=b=1$ and $c \ne 1$ in the following theorem. Otherwise there is no limit on how the various domination numbers can be related, beyond their ordering.

Let $a$, $b$, and $c$ be natural numbers such that $a \le b \le c$ and either $c=1$ or $b>1$. Then there is a graph $G$ such that $\gamma(G)=a$, $\edn(G)=b$, and $\adn(G)=c$.

It is necessary only to put together the examples collected previously. We organize the cases in a sort of lexicographic order, thinking of the sizes of $a$, $b$, and $c$. There are some extraordinary situations for small parameter values, leading to a number of cases.

Case $c=1$: Then $K_n$ is such a graph.

Case $a=1$ and $b=2$
If $c > 2$, then $\Acal_{c-1}$ is such a graph.
If $c=2$, then $P_3$ is such a graph.

Case $a=1$ and $b>2$

The graph $\Ccal_{b-1,b-1 + c}$ is such a graph.

Case $a=2$ and $b=2$
If $c=2$ then $P_4$ is such a graph.
If $c>2$ then $\Bcal_{0,c-1}$ is such a graph.

Case $a=2$ and $b=3$
If $c=3$, then $K_{2,3}$ is such a graph.
If $c>3$, then $\Dcal_{1,c-2}$ is such a graph.

Case $a=2$ and $b > 3$
If $b=c$, then $K_{2,b}$ is such a graph.
If $b<c$, then $\Ecal_{c-b,b-3}$ is such a graph.

Case $a \ge 3$ and $b=a$
The graph $\Bcal_{b-2, c-b-1}$ is such a graph.

Case $a \ge 3$ and $b=a+1$
The graph $\Dcal_{a-1,c-a}$ is such a graph.

Case $a \ge 3$ and $b=a\ge 2$
The graph $\Fcal_{a,b-a-1,c-b+1}$ is such a graph.

§ ACKNOWLEDGEMENTS

I thank Professor Kiran Bhutani for helpful remarks, which include her suggestion of the realizability question which is answered in Theorem <ref>.

[1]
Burger, A. P and Cockayne, E. J. and Gründlingh, W. R. and Mynhardt, C. M. and van Vuuren, J. H. and Winterbach, W.
Infinite order domination in graphs,
J. Combin. Math. Combin. Comput., 50 (2004) 179–194.

[2]
Goddard, W. and Hedetniemi, S. and Hedetniemi, S.
Eternal Security in Graphs.
J. Combin. Math. Combin. Comput., 52 (2005) 160–180.

[3]
Goldwasser, Klostermeyer
Tight Bounds for Eternal Dominating Sets in Graphs.
Discrete Mathematics 308 12 (2008) 2589–2593.

[4]
Haynes, Teresa W. and Hedetniemi, Stephen T. and Slater, Peter J.
Fundamentals of domination in graphs
Monographs and Textbooks in Pure and Applied Mathematics, 208. 1998. Marcel Dekker, Inc., New York.

[5]
Klostermeyer, MacGillivray
Foolproof Eternal Domination in the All-Guards Move Model.
Mathematica Slovaca 52 (2012) No. 4 595-610.


# Tunneling in ABC trilayer graphene superlattice

Mouhamadou Hassane Saley  Laboratory of Theoretical Physics, Faculty of
Sciences, Chouaïb Doukkali University, PO Box 20, 24000 El Jadida, Morocco
Jaouad El-hassouny  Laboratory of Theoretical Physics, Faculty of Sciences,
Chouaïb Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Abderrahim
El Mouhafid  Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Ahmed Jellal
<EMAIL_ADDRESS>Laboratory of Theoretical Physics, Faculty of Sciences,
Chouaïb Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Canadian
Quantum Research Center, 204-3002 32 Ave Vernon, BC V1T 2L7, Canada

###### Abstract

We investigate the transport properties of Dirac fermions in ABC trilayer
graphene (ABC-TLG) superlattices. Based on the transfer matrix method and
using the continuity conditions of the system, we calculate the transmission
probabilities and the corresponding conductance. In the context of two-band
tunneling, Klein tunneling is observed, but it decreases with an increase in
the number of cells. An interlayer bias opens a gap when the number of cells
is increased. Furthermore, increasing the barrier/well width and the cell
number results in an increase in the number of gaps and oscillations in both
two-band and six-band cases. Asymmetry is found in the scattered transmission
due to the presence of the interlayer bias. The conductance decreases when the
number of cells increases and a gap region is found. Our results indicate that
adjusting the number of cells, the width of the barrier/well, and the barrier
heights makes it possible to control electron tunneling and the gap number in
ABC-TLG. These findings provide valuable insights for the development of
electronic devices using graphene materials.

###### pacs:

72.80.Vp, 73.21.Ac, 73.23.Ad
Keywords: Trilayer graphene, ABC-Stacking, barrier superlattice, energy
spectrum, transmissions, Klein tunneling, conductance.

##  I Introduction

Nearly two decades have passed since the discovery of graphene by A. Geim and
K. Novoselov [1], sparking significant interest among physicists. This has led
to a multitude of theoretical and experimental studies being published on this
remarkable material. This is due to its attractive properties, which make it a
promising candidate for a large number of electronic, optical, and mechanical
applications. Indeed, besides its two-dimensional structure, graphene boasts
high electron mobility [2, 3, 4], impressive thermal conductivity [5, 6, 7],
transparency [8, 9, 10], flexibility [11, 12] and high mechanical strength
[13, 14]. In addition, within graphene, charge carriers exhibit
characteristics similar to those of relativistic massless particles. Hence,
they are described by the Dirac equation and demonstrate a linear energy
dispersion [15, 16, 17, 18, 19]. Furthermore, graphene is known for its
distinctive properties, including an unconventional quantum Hall effect [15,
20], a minimum conductivity [17, 21], a zero band gap energy [22, 23] and
Klein tunneling phenomena [17, 24, 25, 26, 27, 28]. However, the last two
properties are obstacles to the development of graphene-based electronic
devices. Several techniques have been developed to create a gap in graphene.
These methods include graphene nanoribbons [29, 30, 31], epitaxial substrates
[32, 33], hydrogenation [34, 35], strain engineering [36, 37] and biased AB
bilayer graphene (BLG). [38, 39, 40, 41].

Recent works have placed significant emphasis on trilayer graphene (TLG),
especially on the rhombohedral (ABC) [42, 43, 44, 45, 46, 47, 48] and the
bernal (ABA) [48, 49, 50, 51] stacking arrangements. The ABC-TLG displays a
cubic energy dispersion characterized by two bands touching at low energy. In
contrast, the ABA-TLG stacking configuration features both linear bands akin
to monolayer graphene (MLG) and parabolic bands similar to AB-BLG [52, 53].
Furthermore, it is worth noting that in the presence of a potential barrier,
electrons exhibit perfect transmission, i.e., Klein tunneling [54, 55, 56] in
ABC-TLG, as in MLG. However, as in AB-BLG, the application of an interlayer
bias breaks the inversion symmetry in ABC-TLG, leading to the emergence of a
band gap[52, 55, 56]. In contrast, the ABA stacking maintains its metallic
characteristics under the same constraint. However, the presence of an
interlayer bias breaks the mirror symmetry, resulting in the hybridization of
the linear and parabolic bands [55, 56].

In a recent work [55], it was shown that a gap is achieved for ABC-TLG for a
single barrier. However, in ref. [56] dealing with double barriers, it has
been observed that transmission occurs in the gap region. Motivated by refs.
[55, 56], we want to investigate the transport properties of Dirac fermions in
ABC-TLG superlattices. When the energy is less than the interlayer coupling
$\gamma_{1}\approx 0.4$ eV, only one propagation mode is available, which
results in one channel of transmission. We have shown that increasing the
number of cells results in reduced transmission, and oscillations occur at
non-normal incidence. In the presence of an interlayer bias, a pseudo-gap is
found for a low number of cells ($N=1$) and turns into a gap when the number
of cells is increased. It has also been shown that increasing the barrier
well/width increases the number of oscillations as well as the number of gaps.
For energy greater than $\gamma_{1}$, three modes are available, resulting in
nine transmission channels. In the non-scattered channels, the transmission
decreases with the number of cells, the barrier height, and the barrier/well
width. In addition, the number of gaps can be increased by increasing the
barrier/well width. In the scattered channels, asymmetry with respect to
normal incidence ($k_{y}=0$) is found due to the presence of the interlayer
bias. In channels $T^{(1)}_{2}(k_{y})=T^{(2)}_{1}(-k_{y})$ and
$T^{(2)}_{3}(k_{y})=T^{(3)}_{2}(-k_{y})$ the transmission becomes more
pronounced when the number of cells increases. Moreover, in channels
$T^{(1)}_{3}$ and $T^{(3)}_{1}$, the transmission is very sensitive to the
barrier height. Furthermore, the conductance decreases when the number of
cells increases and this results in a gap region. When the barrier height is
increased, it has found a displacement in the conductance as well as the gap
region.

The present paper is outlined as follows: In Sec. II, we present a theoretical
model and proceed to compute the corresponding eigenvectors and eigenvalues
for both two-band and six-band cases. Sec. III is devoted to determining the
transmission probability and conductance based on continuity conditions at the
interfaces of the system and the transfer matrix method. In Sec. IV, we
numerically present and discuss our results, comparing them with the existing
literature. Finally, in Sec. V, we provide a summary of our findings and draw
conclusions from our study.

Figure 1: (Color online) Schematic diagram of the periodic potential barrier
applied to ABC-TLG.

##  II Theory and methods

We consider an ABC-TLG subjected to a periodic potential barrier along the
$x$-direction as illustrated in Fig. 1. Our system’s unit cell comprises two
distinct regions: the first is identified as a potential barrier region
denoted by $j=B$, and the second is a region with zero potential labeled as
$j=W$, referred to as the ”well” to distinguish it from the barrier region.
The periodic potential is expressed as follows:

$V_{j}(x)=\left\\{\begin{array}[]{ll}U_{j}\mathbb{I}_{s}+\Delta_{j},&\
\text{for barrier}\\\ 0,&\ \text{for well}\end{array}\right.$ (1)

where $U_{j}$ is the barrier height, $\mathbb{I}_{s}$ is the $s\times s$ unit
matrix (with s=2 and 6 for the two band and six band cases, respectively),
$\Delta_{j}$ stands for the interlayer bias with
$\Delta_{j}=\text{diag}\left(\delta_{j},-\delta_{j}\right)$ for the two-band
and
$\Delta_{j}=\text{diag}\left(\delta_{j},\delta_{j},0,0,-\delta_{j},-\delta_{j}\right)$
for the six-band, $d_{B}$ and $d_{W}$ are, respectively, the barrier and well
widths, with the unit cell represented by $d=d_{B}+d_{W}$.

Initially, we will establish a comprehensive solution for the spinor within
the barrier regions. Subsequently, to derive the solution in the well regions,
we simply need to enforce $U_{j}=\delta_{j}=0$. To achieve this, we employ the
eigenvalue equation $H_{j}\Psi_{j}=E_{j}\Psi_{j}$ in conjunction with the
Hamiltonians of two-band and six-band cases and the corresponding spinors
$\Psi_{j}^{\prime}=\left(\psi_{A_{1}j},\psi_{B_{1}j}\right)^{\dagger}$ and
$\Psi_{j}=\left(\psi_{A_{1}j},\psi_{B_{1}j},\psi_{A_{2}j},\psi_{B_{2}j},\psi_{A_{3}j},\psi_{B_{3}j}\right)^{\dagger}$,
respectively, with the symbol $\dagger$ denotes the transpose row vector.
Because of translational invariance in both cases, momentum is conserved in
the $y$-direction ($[H_{j},p_{y}]=0$), and then we can write

$\displaystyle\Psi^{\prime}_{j}(x,y)=e^{ik_{y}y}\left(\psi_{A_{1}j},\psi_{B_{1}j}\right)^{\dagger}$
(2)
$\displaystyle\Psi_{j}(x,y)=e^{ik_{y}y}\left(\psi_{A_{1}j},\psi_{B_{1}j},\psi_{A_{2}j},\psi_{B_{2}j},\psi_{A_{3}j},\psi_{B_{3}j}\right)^{\dagger}.$
(3)

To simplify, we will adopt the length scale $a=\frac{\hbar
v_{F}}{\gamma_{1}}=1.64$ nm along with dimensionless quantities:
$E_{j}\rightarrow\frac{E_{j}}{\gamma_{1}}$,
$U_{j}\rightarrow\frac{U_{j}}{\gamma_{1}}$,
$\delta_{j}\rightarrow\frac{\delta_{j}}{\gamma_{1}}$,
$x\rightarrow\frac{x}{a}$, and $k_{y}\rightarrow ak_{y}.$

###  II.1 Two-band model

In the regime of very low energy, it becomes feasible to introduce an
effective two-band Hamiltonian to describe the ABC-TLG [55, 47, 54, 48]. This
is

$\displaystyle H^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}U_{j}+\delta_{j}&\upsilon\pi^{{\dagger}3}\\\
\upsilon\pi^{3}&U_{j}-\delta_{j}\end{array}\right],$ (6)

where $\upsilon=\frac{\left(\hbar v_{F}\right)^{3}}{\gamma_{1}^{2}}$,
$\pi=q_{x}+ik_{y}$ represents the in-plane momentum, and $v_{F}=10^{6}$ m/s
denotes the Fermi velocity. It is important to note that this approximation
holds true only when the Fermi energy is significantly smaller than
$\gamma_{1}$. The associated eigenstates solution in each region $j=B$ or
$j=W$ can be written in matrix form with plane waves as

$\Psi^{\prime}_{j}(x,y)=\mathcal{L_{\textit{j}}}\mathcal{M_{\textit{j}}}(x)\Lambda_{j}e^{ik_{y}y},$
(7)

and $\mathcal{L_{\textit{j}}}$ is a $6\times 2$ matrix given by

$\mathcal{L_{\textit{j}}}=\left[\begin{array}[]{cccccccc}1&1&1&1&1&1\\\
h_{1j}^{+}&h_{1j}^{-}&h_{2j}^{+}&h_{2j}^{-}&h_{3j}^{+}&h_{3j}^{-}\end{array}\right],$
(8)

where $h_{\tau j}^{\pm}=\frac{\left(\pm q_{\tau
j}+ik_{y}\right)^{3}}{\varepsilon_{j}+\delta_{j}}$, $\tau=1,2,3$ refers to the
propagating modes, $\varepsilon_{j}=E_{j}-U_{j}$. The remaining matrices are

$\displaystyle\mathcal{M_{\textit{j}}}(x)=\text{diag}\left[e^{iq_{1j}x},e^{-iq_{1j}x},e^{iq_{2j}x},\cdots,e^{-iq_{3j}x}\right],$
(9)
$\displaystyle\Lambda_{j}=\left[\alpha_{1j},\beta_{1j},\alpha_{2j},\beta_{2j},\alpha_{3j},\beta_{3j}\right]^{\dagger},$
(10)

and the wave vectors $q_{\tau j}$ of the plane waves are the solutions of the
equation

$\left(q_{\tau
j}^{2}+k_{y}^{2}\right)^{3}+\delta_{j}^{2}=\varepsilon_{j}^{2},$ (11)

giving rise to the dispersion relation

$\varepsilon_{j\pm}=\pm\sqrt{\left(q_{\tau
j}^{2}+k_{y}^{2}\right)^{3}+\delta_{j}^{2}}.$ (12)

###  II.2 Six-band model

The Hamiltonian near the Dirac point for the six-band can be written in the
tight-binding model as [55, 56]

$\displaystyle H_{j}=\left[\begin{array}[]{ccccc}\mathcal{H}&\Gamma&0\\\
\Gamma^{{\dagger}}&\mathcal{H}&\Gamma\\\
0&\Gamma^{{\dagger}}&\mathcal{H}\end{array}\right]+V_{j}(x)\mathbb{I}_{6},$
(16)

where the interlayer coupling $\Gamma$ is given by

$\displaystyle\Gamma=\left[\begin{array}[]{cc}0&0\\\
\gamma_{1}&0\end{array}\right],$ (19)

and $\mathcal{H}=v_{F}\vec{\sigma}\cdot\vec{p}$ is the Hamiltonian of MLG,
with $\vec{\sigma}=\\{\sigma_{x},\sigma_{y}\\}$ being the Pauli matrices,
$\vec{p}=\left(p_{x},p_{y}\right)$ being the in-plane momentum, and
$\mathbb{I}_{6}$ is an $6\times 6$ unit matrix. When Eq. (16) is inserted into
the eigenvalue equation alongside the six spinors Eq. (3), it results in six
interrelated differential equations

$\displaystyle-i(\partial_{x}+k_{y})\psi_{B_{1}j}$
$\displaystyle=(\varepsilon_{j}-\delta_{j})\psi_{A_{1}j},$ (20a)
$\displaystyle-i(\partial_{x}-k_{y})\psi_{A_{1}j}$
$\displaystyle=(\varepsilon_{j}-\delta_{j})\psi_{B_{1}j}-\psi_{A_{2}j},$ (20b)
$\displaystyle-i(\partial_{x}+k_{y})\psi_{B_{2}j}$
$\displaystyle=\varepsilon_{j}\psi_{A_{2}j}-\psi_{B_{1}j},$ (20c)
$\displaystyle-i(\partial_{x}-k_{y})\psi_{A_{2}j}$
$\displaystyle=\varepsilon_{j}\psi_{B_{2}j}-\psi_{A_{3}j},$ (20d)
$\displaystyle-i(\partial_{x}+k_{y})\psi_{B_{3}j}$
$\displaystyle=(\varepsilon_{j}+\delta_{j})\psi_{A_{3}j}-\psi_{B_{2}j},$ (20e)
$\displaystyle-i(\partial_{x}-k_{y})\psi_{A_{3}j}$
$\displaystyle=(\varepsilon_{j}+\delta_{j})\psi_{B_{3}j}.$ (20f)

Eliminating one by one the unknowns in Eqs. (20a-20f), we obtain a six-order
differential equation for $\psi_{A_{3}j}$

$\left[\partial_{x}^{6}-\left(\mu_{j}+3k_{y}^{2}\right)\partial_{x}^{4}+\left(\nu_{j}+2\mu_{j}k_{y}^{2}+3k_{y}^{4}\right)\partial_{x}^{2}-k_{y}^{6}-\mu_{j}k_{y}^{4}-\nu_{j}k_{y}^{2}-\rho_{j}\right]\psi_{A_{3}j}=0,$
(21)

where we have set the parameters
$\mu_{j}=-3\varepsilon_{j}^{2}-2\delta_{j}^{2},\nu_{j}=3\varepsilon_{j}^{4}+\delta_{j}^{4}-2\varepsilon_{j}^{2}$,
and
$\rho_{j}=-\varepsilon_{j}^{6}-\varepsilon_{j}^{2}(1+\delta_{j}^{2})^{2}+2\varepsilon_{j}^{4}(\delta_{j}^{2}+1)+\delta_{j}^{2}$.
The solution of Eq. (21) is a linear combination of plane waves

$\psi_{A_{3}j}=\sum_{\tau=\text{1}}^{\text{3}}\left(\alpha_{\tau j}e^{ik_{\tau
j}x}+\beta_{\tau j}e^{-ik_{\tau j}x}\right),$ (22)

where $k_{\tau j}$ is the waves vector in the x-direction. By using Eqs.
(20a-20f) and (22), we can derive the remaining components of the spinors and
then express the general solution in matrix form

$\Psi_{j}(x,y)=L_{\textit{j}}M_{j}(x)\Lambda_{j}e^{ik_{y}y},$ (23)

such as $L_{j}$ is given by

$L_{j}=\begin{pmatrix}\frac{f_{1j}^{-}(\lambda_{1j}\eta_{1j}-\varepsilon_{j}b_{j})}{c_{j}b_{j}f_{1j}^{+}}&\frac{f_{1j}^{+}(\lambda_{1j}\eta_{1j}-\varepsilon_{j}b_{j})}{c_{j}b_{j}f_{1j}^{-}}&\frac{f_{2}^{-}(\lambda_{2j}\eta_{2j}-\varepsilon_{j}b_{j})}{c_{j}b_{j}f_{2j}^{+}}&\frac{f_{2j}^{+}(\lambda_{2j}\eta_{2j}-\varepsilon_{j}b_{j})}{c_{j}b_{j}f_{2j}^{-}}&\frac{f_{3j}^{-}(\lambda_{3j}\eta_{3j}-\varepsilon_{j}b_{j})}{c_{j}b_{j}f_{3j}^{+}}&\frac{f_{3j}^{+}(\lambda_{3j}\eta_{3j}-\varepsilon_{j}b_{j})}{c_{j}b_{j}f_{3j}^{-}}\\\
\frac{(\lambda_{1j}\eta_{1,j}-\varepsilon_{j}b_{j})}{b_{j}f_{1}^{+}}&-\frac{(\lambda_{1j}\eta_{1j}-\varepsilon_{j}b_{j})}{b_{j}f_{1j}^{-}}&\frac{(\lambda_{2j}\eta_{2j}-\varepsilon_{j}b_{j})}{b_{j}f_{2j}^{+}}&-\frac{(\lambda_{2j}\eta_{2j}-\varepsilon_{j}b_{j})}{b_{j}f_{2j}^{-}}&\frac{(\lambda_{3j}\eta_{3j}-\varepsilon_{j}b_{j})}{b_{j}f_{3j}^{+}}&-\frac{(\lambda_{3j}\eta_{3j}-\varepsilon_{j}b_{j})}{b_{j}f_{3j}^{-}}\\\
\frac{(\varepsilon_{j}\eta_{1,j}-b_{j})}{b_{j}f_{1j}^{+}}&-\frac{(\varepsilon_{j}\eta_{1j}-b_{j})}{b_{j}f_{1j}^{-}}&\frac{(\varepsilon_{j}\eta_{2j}-b_{j})}{b_{j}f_{2j}^{+}}&-\frac{(\varepsilon_{j}\eta_{2j}-b_{j})}{b_{j}f_{2j}^{-}}&\frac{(\varepsilon_{j}\eta_{3j}-b_{j})}{b_{j}f_{3j}^{+}}&-\frac{(\varepsilon_{j}\eta_{j}-b_{j})}{b_{j}f_{3j}^{-}}\\\
\frac{\eta_{1j}}{b_{j}}&\frac{\eta_{1j}}{b_{j}}&\frac{\eta_{2j}}{b_{j}}&\frac{\eta_{2j}}{b_{j}}&\frac{\eta_{3j}}{b_{j}}&\frac{\eta_{3j}}{b_{j}}\\\
1&1&1&1&1&1\\\
\frac{f_{1j}^{+}}{b_{j}}&-\frac{f_{1j}^{-}}{b_{j}}&\frac{f_{2j}^{+}}{b_{j}}&-\frac{f_{1j}^{-}}{b_{j}}&\frac{f_{3j}^{+}}{b_{j}}&-\frac{f_{1j}^{-}}{b_{j}}&\end{pmatrix},$
(24)

and $M_{j}(x)$ reads as

$M_{j}(x)=\text{diag}\left[e^{ik_{1j}x},e^{-ik_{1j}x},e^{ik_{2j}x},\cdots,e^{-ik_{3j}x}\right].$
(25)

The matrix $\Lambda_{j}$ is defined in Eq. (10) and will be specified for the
incident and transmitted regions in Sec. III. Here we have set
$b_{j}=\varepsilon_{j}+\delta_{j}$ and $c_{j}=\varepsilon_{j}-\delta_{j}$ as
well as the following parameters

$\displaystyle f_{\tau,j}^{\pm}=k_{\tau j}\pm ik_{y},$ (26a)
$\displaystyle\eta_{\tau j}=b_{j}^{2}-k_{\tau j}^{2}-k_{y}^{2},$ (26b)
$\displaystyle\lambda_{\tau j}=\varepsilon_{j}^{2}-k_{\tau j}^{2}-k_{y}^{2}.$
(26c)

Now, to obtain the eigenvalues, we can simply calculate the determinant
$\det(H_{j}-\mathbb{I}_{6}E_{j})=0$. This process yields a polynomial of
degree six in terms of $\varepsilon_{j}$

$\varepsilon_{j}^{6}+\varepsilon_{j}^{4}\kappa_{1j}+\varepsilon_{j}^{2}\kappa_{2j}+\kappa_{3j}=0,$
(27)

where we have defined the quantities

$\displaystyle\kappa_{1j}=-3k_{j}^{2}-2\delta_{j}^{2}-2,$ (28a)
$\displaystyle\kappa_{2j}=3k_{j}^{4}+2k_{j}^{2}+(1+\delta_{j}^{2})^{2},$ (28b)
$\displaystyle\kappa_{3j}=-\delta_{j}^{2}-(k_{j}^{3}-k_{j}\delta_{j}^{2})^{2},$
(28c)

and $k_{j}=\sqrt{k_{\tau j}^{2}+k_{y}^{2}}$ is the wave vector along
$x$-direction. To solve Eq. (27), we apply Cardano’s method [57] to end up
with the six-band solutions

$\displaystyle\varepsilon_{1j}^{\pm}=\pm\sqrt{2\sqrt{\Xi_{j}}\cos{\left[\frac{\vartheta_{j}+2\pi}{3}\right]}-\frac{\kappa_{1j}}{3}},$
(29a)
$\displaystyle\varepsilon_{2j}^{\pm}=\pm\sqrt{2\sqrt{\Xi_{j}}\cos\left[\frac{\vartheta_{j}+4\pi}{3}\right]-\frac{\kappa_{1j}}{3}},$
(29b)
$\displaystyle\varepsilon_{3j}^{\pm}=\pm\sqrt{2\sqrt{\Xi_{j}}\cos\left[\frac{\vartheta_{j}}{3}\right]-\frac{\kappa_{1j}}{3}},$
(29c)

with the set of the following parameters
$\Xi_{j}=\frac{\kappa_{1j}^{2}-3\kappa_{2j}}{9}$,
$\vartheta_{j}=\arccos\left(\Theta_{j}\sqrt{\frac{1}{\Xi_{j}^{3}}}\right)$ and
$\Theta_{j}=\frac{9\kappa_{1j}\kappa_{2j}-27\kappa_{3j}-2\kappa_{1j}^{3}}{54}$.

By using the six bands Eq. (29), we can derive six longitudinal waves vectors
$k_{\tau j}$ $(\tau=1,2,3)$ for propagating modes. They are given by

$\displaystyle
k_{1j}^{\pm}=\pm\sqrt{2\sqrt{\omega_{j}}\cos\left[\frac{\varpi_{j}+2\pi}{3}\right]-\frac{\mu_{j}}{3}-k_{y}^{2}},$
(30a) $\displaystyle
k_{2j}^{\pm}=\pm\sqrt{2\sqrt{\omega_{j}}\cos\left[\frac{\varpi_{j}+4\pi}{3}\right]-\frac{\mu_{j}}{3}-k_{y}^{2}},$
(30b) $\displaystyle
k_{3j}^{\pm}=\pm\sqrt{2\sqrt{\omega_{j}}\cos\left[\frac{\varpi_{j}}{3}\right]-\frac{\mu_{j}}{3}-k_{y}^{2}},$
(30c)

where we have defined the quantities
$\omega_{j}=\frac{\mu_{j}^{2}-3\nu_{j}}{9}$,
$\varpi_{j}=\arccos\left(\varrho_{j}\sqrt{\frac{1}{\omega_{j}^{3}}}\right)$
and $\varrho_{j}=\frac{9\mu_{j}\nu_{j}-27\rho_{j}-2\mu_{j}^{3}}{54}$.

Figure 2: (Color online) Energy spectrum of ABC-TLG as a function of the
transverse wave vector $k_{y}$. The dashed lines correspond to the spectrum of
the two-band Hamiltonian, while the solid lines correspond to the six-band
Hamiltonian. We choose (a): $\delta=0$, and (b): $\delta=0.2\gamma_{1}$.

###  II.3 Discussions

In Fig. 2 we show the dispersion relation obtained in the two band model
(dashed curves ) and the dispersion relation obtained from the six band model
(solid curves). We clearly see that for $E\ll\gamma_{1}$ and $\delta$, we have
two bands that touch each other at $E=0$, as shown in Fig. 2(a). For energy
exceeding $\gamma_{1}$, we have six bands, two of which touch at $E=0$. The
four others are separated by a gap $\gamma_{1}$ from those that touch at $E=0$
[58]. In the presence of an interlayer bias $\delta\neq 0$ in Fig. 2(b), the
two touched bands at $E=0$ are separated, and a gap is opened between them in
both the two-band and six-band cases. In addition, the bands that cross at
high energy in the six-band case are also separated.

##  III Transmission and conductance

We will apply the spinor continuity conditions at each interface of the system
and employ the transfer matrix method to calculate the transmission
probability. Initially, we will define the wave function $\Psi_{j}$ in the
incident and transmitted regions, ensuring equivalence through the relation
$L_{\text{in}}M_{\text{in}}(x)=L_{\text{tr}}M_{\text{tr}}(x)$ and
$k_{\tau,\text{in}}=k_{\text{tr}}$ due to zero potential and interlayer bias.
Consequently, we can eliminate the subscripts from matrices $M(x)$ and $L$ to
express the wave functions as

$\Psi_{\text{in}/\text{tr}}=LM(x)\Lambda_{\text{in}/\text{tr}},$ (31)

where $\Lambda_{\text{in}}$ in the incidence and $\Lambda_{\text{tr}}$ in the
transmission are provided for the six band case by

$\displaystyle\Lambda_{\text{in}}=\left(\delta_{\tau
1},r_{1}^{\tau},\delta_{\tau 2},r_{2}^{\tau},\delta_{\tau
3},r_{3}^{\tau}\right)^{\dagger},$ (32)
$\displaystyle\Lambda_{\text{tr}}=\left(t_{1}^{\tau},0,t_{2}^{\tau},0,t_{3}^{\tau},0\right)^{\dagger},$
(33)

where $\delta_{\tau,i}$ (where $i=1,2,3$) represents the Kronecker symbol,
signifying the right-propagating or evanescent waves. $r_{i}^{\tau}$
corresponds to the left-propagating reflected waves, while $t_{i}^{\tau}$
pertains to the right-propagating transmitted waves. By matching the wave
functions $\Psi_{j}$ at the interface of each region, the coefficients in the
incidence region $\Lambda_{\text{in}}$ and those in the transmission region
$\Lambda_{\text{tr}}$ can be related through the transfer matrix $\chi$ in the
following manner

$\Lambda_{\text{in}}=\chi\Lambda_{\text{tr}},$ (34)

and $\chi$ is given by

$\chi=M^{-1}(0)L^{-1}\Omega^{n}LM(nd),$ (35)

where $n$ is the number of cells and $\Omega$ is

$\Omega=L_{B}M_{B}(0)M_{B}^{-1}(d_{B})L_{B}^{-1}L_{W}M_{W}(d_{B})M_{W}^{-1}(d)L_{W}^{-1}.$
(36)

Then, after some algebras, we can write Eq. (34) in the form

$\begin{pmatrix}t_{1}^{\tau}\\\ r_{1}^{\tau}\\\ t_{2}^{\tau}\\\
r_{2}^{\tau}\\\ t_{3}^{\tau}\\\
r_{3}^{\tau}\end{pmatrix}=\begin{pmatrix}\chi_{11}&0&\chi_{13}&0&\chi_{15}&0\\\
\chi_{21}&-1&\chi_{23}&0&\chi_{25}&0\\\ \chi_{31}&0&\chi_{33}&0&\chi_{35}&0\\\
\chi_{41}&0&\chi_{43}&-1&\chi_{45}&0\\\ \chi_{51}&0&\chi_{53}&0&\chi_{55}&0\\\
\chi_{61}&0&\chi_{63}&0&\chi_{65}&-1\\\
\end{pmatrix}^{-1}\begin{pmatrix}\delta_{\tau,1}\\\ 0\\\ \delta_{\tau,2}\\\
0\\\ \delta_{\tau,3}\\\ 0\end{pmatrix},$ (37)

where $\chi_{uv}$ are elements of matrix $\chi$. Thereafter, we can derive the
expression of the transmission coefficients through Eq. (37) to get the
followings

$\displaystyle
t_{1}^{\tau}=\frac{\left(\chi_{35}\chi_{53}-\chi_{33}\chi_{55}\right)\delta_{\tau,1}+\left(\chi_{13}\chi_{55}-\chi_{15}\chi_{53}\right)\delta_{\tau,2}+\left(\chi_{15}\chi_{33}-\chi_{13}\chi_{35}\right)\delta_{\tau,3}}{\chi_{15}\chi_{33}\chi_{51}-\chi_{13}\chi_{35}\chi_{51}-\chi_{15}\chi_{31}\chi_{53}+\chi_{11}\chi_{35}\chi_{53}+\chi_{13}\chi_{31}\chi_{55}-\chi_{11}\chi_{33}\chi_{55}},$
(38) $\displaystyle
t_{2}^{\tau}=\frac{\left(\chi_{31}\chi_{55}-\chi_{35}\chi_{51}\right)\delta_{\tau,1}+\left(\chi_{15}\chi_{51}-\chi_{11}\chi_{55}\right)\delta_{\tau,2}+\left(\chi_{11}\chi_{35}-\chi_{15}\chi_{31}\right)\delta_{\tau,3}}{\chi_{15}\chi_{33}\chi_{51}-\chi_{13}\chi_{35}\chi_{51}-\chi_{15}\chi_{31}\chi_{53}+\chi_{11}\chi_{35}\chi_{53}+\chi_{13}\chi_{31}\chi_{55}-\chi_{11}\chi_{33}\chi_{55}},$
(39) $\displaystyle
t_{3}^{\tau}=\frac{\left(\chi_{33}\chi_{51}-\chi_{31}\chi_{53}\right)\delta_{\tau,1}+\left(\chi_{11}\chi_{53}-\chi_{13}\chi_{53}\right)\delta_{\tau,2}+\left(\chi_{13}\chi_{31}-\chi_{11}\chi_{33}\right)\delta_{\tau,3}}{\chi_{15}\chi_{33}\chi_{51}-\chi_{13}\chi_{35}\chi_{51}-\chi_{15}\chi_{31}\chi_{53}+\chi_{11}\chi_{35}\chi_{53}+\chi_{13}\chi_{31}\chi_{55}-\chi_{11}\chi_{33}\chi_{55}}.$
(40)

The process is identical for the two-band case, albeit with the additional
consideration of the continuity of the first and second derivatives of the
wave functions at the interfaces [38, 54]. At this point, we need to utilize
the current density to calculate the transmission probabilities. It is given
by

$J=v_{F}\Psi^{\dagger}\alpha\Psi,$ (41)

where $\alpha$ is a $6\times 6$ matrix with Pauli matrix $\sigma_{x}$ on its
diagonal. We are particularly interested in the current density within the
incident and transmitted regions. Thus we inject Eq. (31) into Eq. (41) to get

$J_{\text{in/tr}}=v_{F}\Lambda_{\text{in/tr}}^{\dagger}M(x)^{\dagger}AM(x)\Lambda_{\text{in/tr}}$
(42)

where $A=L^{\dagger}\alpha L$ is a diagonal matrix consisting of traceless
$2\times 2$ blocks, each one corresponds to a propagation mode. It’s important
to highlight that since we are not calculating reflection probabilities,
$J_{\text{in}}$ exclusively considers the right-propagating waves
$\delta_{\tau,i}$ for $\Lambda_{\text{in}}$. Consequently, our transmission
probabilities are expressed as follows:

$T^{\tau}_{i}=\frac{\left|{j}_{\text{tr}}\right|}{\left|{j}_{\text{in}}\right|}=\frac{A_{ii}}{A_{\tau\tau}}\left|t_{i}^{\tau}\right|^{2},$
(43)

where $A_{ii}$ and $A_{\tau\tau}$ are elements of matrix $A$. It is worth
noting that in the two-band case, there exists a single transmission channel
from state $\tau$ to state $i$ with indices $\tau=i=1$, and the corresponding
transmission probability is denoted as $T\neq 0$ [55, 54, 56]. Now, by using
Landauer Butikker’s formula [59] we can calculate the conductance in terms of
transmission. Thus, we will see how it will be affected by the number of
cells. It is given by

$G(E)=G_{0}\frac{L_{y}}{2\pi}\int_{-\infty}^{+\infty}dk_{y}\sum^{3}_{\tau,i=1}T^{\tau}_{i}(E,k_{y}),$
(44)

where $G_{0}=4e^{2}/h$ (the factor 4 emerges from graphene’s valley and spin
degeneracy), and $L_{y}$ refers to the length of the sample in the
y-direction.

##  IV Numerical results

We will examine the numerical results obtained. Two cases will be presented,
depending on the energy value considered. When energy is less than
$\gamma_{1}$, only one propagation mode is available, corresponding to the
two-band case. When energy exceeds $\gamma_{1}$, three propagation modes are
available, leading to nine transmission channels. Given that the potential and
interlayer bias are zero in the well regions ($U_{j}=\delta_{j}=0$ when
$j=W$), we will simplify notation by omitting the subscript $j$. Instead, we
will use the symbols $U$ and $\delta$ to denote the potential and interlayer
bias, respectively, in the barrier regions throughout the discussion.

###  IV.1 Two band tunneling

In Fig. 3, we plot the transmission as a function of energy $E$ and the
transverse wave vector $k_{y}$ for barrier height $U=0.5\gamma_{1}$ and
barrier/well width of $d_{B}=d_{W}=2$nm. Fig. 3(a) illustrates the
transmission for $N=1$ in the absence of an interlayer bias ($\delta=0$). As
found in refs. [54, 55, 56], Klein tunneling occurs at normal incidence due to
the conservation of pseudospin in ABC-TLG. In addition, for energy
$E<0.4\gamma_{1}$, Klein tunneling is observed at non-normal incidence, which
is in contrast with the findings in refs. [55, 56] where the transmission is
zero. This results from the low barrier width that we have considered in
contrast to refs. [55, 56], so in our case, there is a non-zero probability
for the evanescent wave to get through the barrier, as indicated in ref. [60].
In Fig. 3(b), we plot the transmission for $N=8$ and show that as the number
of cells increases, Klein tunneling narrows at normal incidence and disappears
completely when moving away from normal incidence, as the result in refs. [55,
56]. However, at certain energies, narrowed resonances are observed at nearly
normal incidence, as stated in the single barrier case [55] and the double
symmetric barrier case [56]. On the other hand, the transmission shows
oscillations greater than those observed in single and double barrier cases
[55, 56]. Indeed, as the number of cells increases, the number of oscillations
also increases, as demonstrated in Fig. 3(c) for $N=16$. Similar observations
have been reported in refs. [60, 61] dealing on MLG superlattices. However, it
should be noted that these oscillations do not occur at normal incidence, so
Klein tunneling remains intact, as shown in refs. [60, 61, 62]. To investigate
the effect of an interlayer bias, we plot the transmission in Fig. 3(d) with
the same conditions as in Fig. 3(a), but now we have set
$\delta=0.15\gamma_{1}$. As it can be seen, the presence of $\delta$ results
in a reduction of Klein tunneling, both at normal and non-normal incidence,
such that a pseudo-gap is formed at energy $E=0.1\gamma_{1}$. However, a true
gap isn’t achieved due to the narrow barrier width and the lowest number of
cells ($N=1$) [60]. When the number of cells is increased, a gap is opened in
Fig. 3(e) and Fig. 3(f) for $N=8$ and $N=16$, respectively, which is
consistent with the results in ref. [60]. In addition, due to the presence of
the interlayer bias $\delta$, oscillations appear even at normal incidence,
reducing Klein tunneling.

Figure 3: (Color online) Transmission $T$ of ABC-TLG as a function of energy
$E$ and the transverse wave vector $k_{y}$ for $U=0.5\gamma_{1}$ and
$d_{B}=d_{W}=2$ nm. In (a), (b), and (c), we have set the interlayer bias at
$\delta=0$, while for (c), (d), and (e), it is set at $\delta=0.15\gamma_{1}$.
Figure 4: (Color online) The same parameters as in Figs. 3(b) and (e),
respectively, in (a) and (b), but now for $d_{B}=d_{W}=4$nm.

In Figs. 4(a) and (b), we plot the transmission by using the same parameters
as in Figs. 3(b) and (e), respectively, but for increasing barrier/well width,
$d_{B}=d_{W}=4$nm. Fig. 4(a) shows the transmission for zero interlayer bias,
and we notice that increasing the barrier/well width has a significant
influence on the transmission behavior. Indeed, the transmission decreases
strongly at non-normal incidence, thus increasing anti-Klein tunneling (zero
transmission). In addition, a large number of oscillations are found at non-
normal incidence. These oscillations are greater than those obtained in Fig.
3(b) for $d_{B}=d_{W}=2$nm. However, as observed in Figs. 3(b,c) as well as in
refs. [60, 61, 62, 63], these oscillations do not occur at normal incidence.
In Fig. 4(b), when an interlayer bias of $\delta=0.15\gamma_{1}$ is applied,
two gaps are observed in contrast with the results presented in Figs. 3(e,f)
for small barrier/well width. We notice that one of the gaps is wider than the
other, and the transmission is shaped like a ”Mexican hat” between them.
Furthermore, it is noteworthy that Klein tunneling is diminished at normal
incidence and oscillations occur, as in Figs. 3(e,f).

###  IV.2 Six-band tunneling

As mentioned above, once the energy exceeds $\gamma_{1}$, three propagation
modes ($k_{1}$, $k_{2}$, and $k_{3}$) become accessible. This, in turn,
results in a total of nine transmission channels. Among these channels, three
of them, $T^{(1)}_{1}$, $T^{(2)}_{2}$, and $T^{(3)}_{3}$, represent the non-
scattered transmission channels, corresponding to propagation via $k_{1}$,
$k_{2}$, and $k_{3}$, respectively. The remaining channels, denoted as
$T^{\tau}_{i}$ with $\tau\neq i$, represent the scattered transmission
channels where electrons incoming through one mode are transmitted via another
one. We have already seen in the two-band case that, in the absence of an
interlayer bias $\delta$, Klein tunneling persists at normal incidence. So in
the six-band case, we plot the transmission by varying the other parameters
and setting the interlayer bias at $\delta=0.2\gamma_{1}$.

In Fig. 5, we present a density plot of the nine transmission channels as a
function of energy $E$ and the transverse wave vector $k_{y}$. The parameters
include a barrier height of $U=2\gamma_{1}$, a barrier/well width of
$d_{B}=d_{W}=0.2$nm and a number of cells, $N=1$. In the non-scattered
channels $T_{1}^{(1)}$, $T_{2}^{(2)}$ and $T_{3}^{(3)}$ we have full
transmission at both normal and non-normal incidence. At normal incidence,
this is the manifestation of Klein tunneling and does not depend on the
barrier width [55]. At non-normal incidence, the full transmission is related
to the barrier width. As discussed in ref. [60], the evanescent waves still
have a certain probability of being transmitted through the barrier when its
width is small. Recall that the condition for having propagation modes $k_{2}$
and $k_{3}$ is that the energy exceeds $\gamma_{1}$, as a result, the
transmission occurs only for $E>\gamma_{1}$ in $T_{2}^{(2)}$ and $T_{3}^{(3)}$
channels. On the other hand, there is no scattering between the three modes
$k_{1}$, $k_{2}$, and $k_{3}$, which results in zero transmission in all the
scattered channels.

Figure 5: (Color online) Transmission of ABC-TLG as a function of incident
energy $E$ and the transverse wave vector $k_{y}$ for barrier height
$U=2\gamma_{1}$, $d_{B}=d_{W}=0.2$nm, and $\delta=0.2\gamma_{1}$. The number
of cells is set at $N=1$.

Fig. 6 shows the transmission with the same parameters as in Fig. 5 except
that we increase the number of cells at $N=5$. Within the channel
$T_{1}^{(1)}$, we notice that the transmission decreases and anti-Klein
tunneling occurs at non-normal incidence, in contrast to the result in Fig. 5.
In addition, due to the presence of the interlayer bias
$\delta=0.2\gamma_{1}$, we observe a cut-off in Klein tunneling at normal
incidence (transmission decreases from unit to half of unit), which contrasts
with the result in ref. [55], where $\delta=0$. However, a gap is not obtained
due to the low number of cells and small barrier width, similar to the result
in ref. [60]. In both channels $T^{(2)}_{2}$ and $T_{3}^{(3)}$, the
transmission decreases considerably when we compare with the results in Fig. 5
for $N=1$. As a result, at normal incidence, Klein tunneling starts to appear
from $E=U$, where, as in Fig. 5, it occurs from $E=U-\gamma_{1}$. In contrast
to the results in Fig. 5, there is a scattering between the modes $k_{1}$ and
$k_{2}$ ($k_{2}$ and $k_{3}$), which results in a weak transmission in the
channels $T^{(1)}_{2}$ and $T^{(2)}_{1}$ ( $T^{(2)}_{3}$ and $T^{(3)}_{2}$).
However, there is no scattering between $k_{1}$ and $k_{3}$, and the
transmission is still zero in the channels $T^{(1)}_{3}$ and $T^{(3)}_{1}$. In
addition, the presence of the interlayer bias results in an asymmetry with
respect to the incidence normal $k_{y}=0$ in the scattered channels, as found
in refs. [56, 38, 40]. Moreover, the scattered channels are equal under the
change of the sign of $k_{y}$, such that we have
$T^{(1)}_{2}(k_{y})=T^{(2)}_{1}(-k_{y})$ and
$T^{(2)}_{3}(k_{y})=T^{(3)}_{2}(-k_{y})$. This is a consequence of the time
reversal symmetry of the system [55]. For example, an electron that is
scattered from $k_{1}$ to $k_{2}$, near the Dirac point $K$, is equivalent to
another that is scattered from $k_{2}$ to $k_{1}$, near the Dirac point
$K^{\prime}$ [55, 38].

Figure 6: (Color online) The same parameters as in Fig. 5 but now for $N=5$.
Figure 7: (Color online) The same parameters as in Fig. 5 but now for $N=20$.

In Figs. 7 and 8, we show the transmission using the same parameters as in
Fig. 5, but for numbers of cells $N=20$ and $N=60$, respectively. In Fig. 7,
where $N=20$, we observe a substantial decrease in Klein tunneling within the
$T^{(1)}_{1}$ channel, at both normal and non-normal incidence. In addition,
we observe the formation of a pseudo-gap. When the number of cells is
increased at $N=60$ in Fig. 8, the transmission decreases even more and a gap
is obtained, as in ref. [60]. On the other hand, in both $T^{(2)}_{2}$ and
$T^{(3)}_{3}$ channels, the transmission becomes zero in the energy region
beneath $U$ when the number of cells increases, as shown in Figs. 7 and 8.
This is contrary to the results in Figs. 5 and 6 as well as the findings in
refs. [55, 56]. Note also that in all the non-scattered channels, when we
compare the results of Fig. 7 with those of Fig. 8, we notice that the
transmission begins to oscillate slightly when the number of cells increases.
In the scattered channels, such as $T^{(1)}_{2}(k_{y})=T^{(2)}_{1}(-k_{y})$
and $T^{(2)}_{3}(k_{y})=T^{(3)}_{2}(-k_{y})$, the transmission becomes more
pronounced as the number of cells increases, see Fig. 8. However, it is still
lower than the results in refs. [55, 56]. The other scattered channels,
$T^{(1)}_{3}$ and $T^{(3)}_{1}$ remain zero and seem to not change when the
number of cells changes.

Figure 8: (Color online) The same parameters as in Fig. 5 but now for $N=60$.

With the same parameters as in Fig. 8, we plot the transmission in Fig. 9 for
barrier height $U=2.5\gamma_{1}$. As a result, increasing the barrier height
decreases the transmission in the non-scattered channels. Indeed, when we
compare the channel $T^{(1)}_{1}$ with the result in Fig. 8, we observe an
upward shift of the gap and an increase in anti-Klein tunneling, similar to
the result in ref. [40]. In channels $T^{(2)}_{2}$ and $T^{(3)}_{3}$,
transmission occurs from $E=2\gamma_{1}$ in Fig. 8, where in this case it
occurs from $E=2\gamma_{1}+\delta$, which results in an increase in anti-Klein
tunneling. In addition, Klein tunneling is narrowed towards the center in all
the non-scattered channels. On the other hand, increasing the barrier height
slightly increases the transmission probability in the scattered channels.
Particularly, there is a scattering between the modes $k_{1}$ and $k_{3}$,
which results in a weak transmission in channels
$T^{(1)}_{3}(k_{y})=T^{(3)}_{1}(-k_{y})$, as in ref. [56].

In Fig. 10, we present the non-scattered channels using the same parameters as
those shown in Fig. 6 (where $N=5$), but with a barrier/well width of
$d_{B}=d_{W}=3$nm. Similar to the two-band tunneling case, an increase in the
barrier/well width results in reduced transmission in all channels. As a
consequence, we have two gaps and pseudo-gaps that occur in channel
$T^{(1)}_{1}$. On the other hand, channel $T^{(2)}_{2}$ shows three gaps of
different sizes, along with one pseudo-gap. In channel $T^{(3)}_{3}$ , two
gaps are observed. However, as in Fig. 6, due to the non-large number of
cells, non-zero transmission is found for $\gamma_{1}<E<U$ in both channels
$T^{(2)}_{2}$ and $T^{(3)}_{3}$ in contrast to the results in Figs. 7 and 8.

Figure 9: (Color online) The same parameters as in Fig. 8 but now for
$U=2.5\gamma_{1}$. Figure 10: (Color online) The same parameters as in Fig. 6
but now for $d_{B}=d_{W}=3$nm. Figure 11: (Color online) (solid lines) Total
conductance for $N=1$, $5$, $20$ and $60$ corresponding to the transmission in
Figs. 5, 6, 7 and 8, respectively. (dashed line) Total conductance for $N=60$
corresponding to the transmission in Fig. 9.

Fig. 11 displays the total conductance for $N=1$ (blue line), $N=5$ (purple
line) , $N=20$ (brown line) and $N=60$ (magenta line) corresponding
respectively to the transmission in Fig. 5, 6, 7 and 8. The black dashed line
shows the conductance associated to transmission in Fig. 9 for $N=60$ and
barrier height $U=2.5\gamma_{1}$. For $N=1$, we have high conductance in the
region $0<E<\gamma_{1}$, which results from the full transmission observed in
Fig. 5 in channel $T^{(1)}_{1}$ due to the small barrier width[60]. When
$E>\gamma_{1}$, there are contribution from $T^{(2)}_{2}$ and $T^{(3)}_{3}$
due to presence of propagation modes $k_{2}$ and $k_{3}$ in this region. This
increases significantly the conductance. As a result, even though there is no
contribution from the scattered channels $T^{\tau}_{i}$ with $\tau\neq i$
(zero transmission), the conductance is higher than the results obtained in
refs. [55, 56]. This is a consequence of the small barrier width that we
considered as discussed in Fig. 5. When the number of cells is increased at
$N=5$, the conductance decreases and moves to right side. From number of cells
$N=20$, the behaviour of the conductance changes and we observe some minima as
in ref [55, 56, 39, 40, 41]. Especially, for $N=60$ a zero conductance is
found, which correspond to a gap region (magenta line), in contrast to the
result in [56]. On the other hand, when we increases the barrier height at
$U=2.5\gamma_{1}$ (black dashed line), we note a displacement of the
conductance to the right side as well as the gap region.

##  V Conclusion

In this study, we have investigated the transport properties of Dirac fermions
through ABC-TLG superlattice. Using the Hamiltonian of the system, we have
computed the corresponding eigenvectors and eigenvalues. Subsequently,
employing the transfer matrix method and considering the continuity conditions
of the spinor at the different interfaces, we have calculated the transmission
probability and the corresponding conductance. In the context of two-band
tunneling, we observe Klein tunneling at both normal and non-normal incidence
for $N=1$. However, as the number of cells increases, Klein tunneling
decreases. In addition, oscillations are found at non-normal incidence, and
their numbers increase with the number of cells. In the presence of an
interlayer bias, a pseudo-gap is found for $N=1$ and turns into a gap when the
number of cells is increased. Moreover, the number of oscillations and gaps
increases with the barrier/well width.

In the case of six-band tunneling and for non-scattered channels, increasing
the number of cells leads to a decrease in the transmission, and oscillations
occur. A pseudo-gap is found for a low number of cells, whereas a gap is
obtained when the number of cells is increased. It has also been shown that
increasing the barrier height at $U=2.5\gamma_{1}$ decreases the transmission
and shifts the gap. On the other hand, in the scattered channels, the
transmission is sensitive to the number of cells and the barrier height.
Especially in channels $T^{(1)}_{3}$ and $T^{(3)}_{1}$, the transmission is
zero for $U=2\gamma_{1}$, whereas a weak transmission occurs for
$U=2.5\gamma_{1}$. Finally, it was shown that, increasing the number of cells
reduces the conductance such that a gap is found for $N=60$. Our results
demonstrate that electron tunneling and gaps can be controlled by adjusting
the barrier/well width, the barrier height, and the number of cells. We hope
that these findings can serve as a valuable reference for the development of
electronic devices using graphene materials.

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