# On how generalised entropies without parameters impact information
optimisation processes

Jesús Fuentes<EMAIL_ADDRESS>Octavio Obregón  Departamento de
Física, División de Ciencias e Ingenierías Campus León,
Universidad de Guanajuato, A.P. E-143, C.P. 37150, León, Guanajuato, México.

###### Abstract

As an application of generalised statistical mechanics, it is studied a
possible route toward a consistent generalised information theory in terms of
a family of non-extensive, non-parametric entropies $H^{\pm}_{D}(P)$. Unlike
other proposals based on non-extensive entropies with a parameter dependence,
our scheme is asymptotically equivalent to the one formulated by Shannon,
while it differs in regions where the density of states is reasonably small,
which leads to information distributions constrained to their background. Two
basic concepts are discussed to this aim. First, we prove two effective coding
theorems for the entropies $H^{\pm}_{D}(P)$. Then we calculate the channel
capacity of a binary symmetric channel (BSC) and a binary erasure channel
(BEC) in terms of these entropies. We found that processes such as data
compression and channel capacity maximisation can be improved in regions where
there is a low density of states, whereas for high densities our results
coincide with the Shannon’s formulation.

PACS numbers

02.50.-r; 89.70.+c; 05.40.Ca

keywords

data compression, coding theorems, non-extensive entropies

##  I Introduction

The first rigorous study on data compression was published by Claude Shannon
in 1948 1, in a seminal paper that constituted a firm setting in the
foundations of classical information theory. In that work it was considered a
$D$-ary alphabet as well as a collection of uncorrelated random sources of
letters (whose probabilities depend only on the $k$ letters that immediately
precede them) with the aim to examine to what extent could one compress $N$
symbols emitted from a particular source. Shannon found that any process of
data compression results limited by the entropy itself such that a codeword
whose average length attains the entropy can be regarded as optimal. In other
words, this result led to quantify the average amount of information (absence
of redundancy) by simply measuring the entropy.

Nevertheless, the entropy as an information measure may have different
definitions in its functional form and conceptual purport. Although, in
general, we can classify them into two different groups: extensive and non-
extensive entropies. The class of non-extensive entropies is typically related
to statistical systems out of equilibrium and whose components are interacting
with themselves or with external agents. These systems are often referred as
non-equilibrium processes, and play a direct role in modern trends of
information theory 2. In this sense, a non-extensive entropy can be understood
as a generalisation to the (Boltzmann-Gibbs) Shannon’s entropy

$H_{D}^{S}(P)=-\sum_{i}p(x_{i})\ln(p(x_{i})),$

since the latter succeeds as long as a large system can be treated in
equilibrium. Which shall not be disregarded given the equilibrium
configuration is realisable for a large number of situations up to a good
approximation.

Thus the question whether the introduction of non-extensive entropies in those
systems that were hardly approximated in equilibrium can really bring some
significant novelty, has a positive answer and, in fact, is worth attention
even when the description of the problem turns complicated via a non-extensive
measure.

In information theory the story is not different. Below we shall show how the
rate of data compression for distributions associated with small systems (i.e.
setups with a few number of accessible micro states $N$) can attain a more
efficient bound compared with the standard formulation if the process is
accomplished by means of the non-extensive entropy $H_{D}^{-}(P)$ application,
whereas $H_{D}^{-}(P)$ and $H_{D}^{S}(P)$ will tend to coincide as the number
of the accessible states $N$ grows. Actually, the entropies $H_{D}^{\pm}(P)$
and $H_{D}^{S}(P)$ are asymptotically equivalent as seen from Fig. 1, see Ref.
3 for further details. This means that the entropies $H_{D}^{\pm}(P)$ truly
resemble the standard theory whenever the probability distribution $P$
concerns a large system.

A similar discussion will be addressed to the computation of the channel
capacity of a BSC and a BEC. In the first case, we will show how the entropy
$H_{D}^{+}(P)$ improves the outcome obtained from $H_{D}^{S}(P)$, recalling
that the partial optimisation will be valid in the regime of small systems. In
the case of a BEC, there is an interesting switching on the obtained bounds,
going from upper to lower and vice versa, depending on what ratio of errors is
present over the communication channel.

In turn, we shall introduce the functional form of the entropies
$H_{D}^{\pm}(P)$:

$H_{D}^{\pm}(P)=-\sum_{i}^{N}p(x_{i})\log^{\pm}_{D}(p(x_{i})),$ (1)

where $\log^{\pm}$ are generalised logarithms (see Appendix A) and
$\\{p(x_{1}),\ldots,p(x_{N})\\}\in P$ are the probabilities of the codewords
$x_{1},\ldots,x_{N}$ emitted by an uncorrelated random source $X$. The
subindex $D$ refers to the size of the alphabet, e.g. a binary alphabet has
$D=2$ symbols.

The information measures (1) were originally proposed by one of us in 4,
within the superstatistics framework 5 and years after, their quantum
equivalents were obtained by means of a generalised replica trick 6, i.e. a
useful method to compute the partition function in spin crystals.

Indeed, the entropies $H_{D}^{\pm}(P)$ demand another generalised quantities
as we will see below. For instance, we cannot find effective coding theorems
in terms of these measures if the constraint $\sum_{i}^{N}D^{-l_{i}}\leq 1$ is
imposed to minimise the corresponding codeword length, which leads to the
necessity of a generalised constraint, as stated in Prop. 1.

On the basis of the entropies (1), a third one can be observed as well,
expressly $H_{D}^{0}(P)=\frac{1}{2}\left[H_{D}^{+}(P)+H_{D}^{-}(P)\right]$.
Yet along this paper we are not to analyse explicitly this latter measure, but
only the fundamental entropies $H_{D}^{\pm}(P)$, nonetheless, given that
$H_{D}^{0}(P)$ is an average of the other two entropy measures, the following
discussion directly comprises this case.

Note that $H_{D}^{\pm}(P)$ do not depend on any free parameter but only on the
probability distribution. Both non-extensive entropies, furthermore, resemble
the Shannon’s entropy at first order, which is evident from the following
series expansion:

$\begin{split}-p&(x_{i})\log^{\pm}_{D}(p(x_{i}))=-p(x_{i})\log_{D}(p(x_{i}))\\\
&\mp\frac{1}{2}[p(x_{i})\log_{D}(p(x_{i}))]^{2}-\frac{1}{6}[p(x_{i})\log_{D}(p(x_{i}))]^{3}+\cdots\end{split}$
(2)

due to $p(x_{i})\in[0,1]$ the higher order terms become subdominant but their
actual contribution is not entirely negligible in a regime of high
probabilities.

The special case of a uniform distribution $p(x_{i})=\frac{1}{N}$ gives us
further information about the behaviour of $H_{D}^{\pm}(P)$. Substituting this
distribution into (1) yields $H_{D}^{\pm}(P)=\pm N\mp N^{1\mp\frac{1}{N}}$,
and its series expansion:

$H_{D}^{\pm}(P)=\sum_{i}^{\infty}(\mp 1)^{i+1}\frac{\log^{i}N}{i!N^{i-1}},$
(3)

which produces the plots in Fig. 1. Our previous discussion can be summarised
in that figure.

From an intuitive viewpoint, one could ask to what degree could the standard
information theory be modified by implementing the information measures (1),
since Fig. 1 tells in advance the existence of upper and lower bounds on the
Shannon’s entropy in certain regions of $N$.

Figure 1: Behaviour of the entropies (1) with respect to the Shannon’s entropy
in terms of a uniform distribution $p(x_{i})=\frac{1}{N}$ for all $x_{i}\in
X$.

As well, it is worthy of note that the Shannon’s entropy and the entropies
$H_{D}^{\pm}(P)$ are written in the fashion of
$H_{D}(P)=\sum_{i}^{N}G_{D}(p(x_{i}))$, where $G_{D}(x)$ is a strictly
monotonic increasing function known as the entropic form. There are other
entropy measures that adopts such structure, such as the Tsallis’s $q$-entropy
7 (see also 8), which has been widely studied in different disciplines 9.
Regarding information theory, a coding theorem in terms of the $q$-entropy has
been found by means of an algebraic method 10. Yet, it has been argued in 11,
12, that its free parameter $q$ may be an obstacle in the theoretical
explanation of some problems, as well as its divergence from the standard
entropy for a large number of symbols (or micro states) even if $q\approx 1$.
The hassle of free parameters is absent from $H_{D}^{\pm}(P)$, also it has
been proved in 3 that $H_{D}^{\pm}(P)$ are stable measures for any $N$ and any
$P$, thus we think such entropies are good candidates for a generalised
information theory.

The structure of our work is as follows. In Sec. II we obtain coding theorems
for the two entropies defined in (1) and discuss the implications of the
resulted data compression in comparison with the Shannon’s entropy. Sec. III
is devoted to the computation of the channel capacity of a BSC and a BEC, we
compare our results with the ones in the standard theory. Finally, our
conclusions and general outlook are presented in Sec. IV.

##  II Effective coding theorems

Below we shall present coding theorems in terms of the non-extensive entropies
(1). To this aim, a variational method is applied to find the average length
that a codeword can attain by compressing the data subject to an adequate
constraint. This constraint is not arbitrary. As we will show there is an
intertwining between the entropy measure and the minimisation constraint such
that the functional structure of the entropy is preserved.

For our purposes, in the following discussion we assume that the communication
channel is perfectly noiseless, that is we shall neglect any external
perturbation. As well, we take for granted that the codes are prefix-free,
i.e. there is no a single codeword which is the initial segment of any other
available codeword in the set generated by $X$.

The aim of the method reported below is to find codes that minimise a
monotonic increasing function of the form

$L=\varphi^{-1}\left(\sum_{i}^{N}p(x_{i})\varphi(l(x_{i}))\right),$ (4)

where $l(x_{i})\in\mathbb{Z}_{+}$ is the length of the $i$-th codeword related
to the Nagumo-Kolmogorov function $\varphi$ such that the cost of managing a
sequence of length $l(x_{i})$ is
$\varphi(l(x_{i})):\mathbb{R}_{+}\to\mathbb{R}_{+}$. The quantity defined in
(4) is called the average length for the cost function $\varphi$, in
accordance with Campbell 13, however, for brevity we refer to it simply as the
average length.

A number of cost functions $\varphi$ have been studied in 14, although for
most entropies of the generalised form, we have that $\varphi(z)=z$. For
instance, in the case of the Shannon’s entropy, the average length
$L_{S}=\varphi^{-1}(\sum_{i}^{N}p_{S}(x_{i})\varphi(l_{S}(x_{i})))=\sum_{i}^{N}p_{S}(x_{i})l_{S}(x_{i})$
can be optimised via the functional

$J_{S}=L_{S}+\gamma K_{S},$ (5)

where $\gamma\in\mathbb{R}$ is an undetermined Lagrange multiplier and the
constraint

$K_{S}=\sum_{i}^{N}D^{-l_{S}(x_{i})}\leq 1,$ (6)

is the Kraft (McMillan) inequality 15, accounting for the fundamental control
function in the optimisation of this case.

To solve the optimisation problem in (5), one merely differentiate $J_{S}$
with respect to $l_{S}(x_{i})$ and equates to zero, implying that the rule to
find the extrema of (5) is the same for every $x_{i}\in X$. After the
elementary calculation one finds:

$l_{S}^{*}(x_{i})=-\log_{D}p_{S}(x_{i}),$ (7)

where the superscript * indicates that such average length is regarded as
optimal, namely it solves the problem (5).

Below we are to proceed in an analogous way but in terms of the entropy
measures (1). Nonetheless, a generalisation to (6) shall be devised to
optimise the mean lengths related to $H_{D}^{\pm}(P)$.

This is an interesting aspect of our scheme, for the quantity to be minimised
will grant a different rate of data compression than the estimated by the
classical theory.

Hence, in our case, the problem consists of choosing codes that become minima
when they are subject to the quantities in the following definition.

######  Proposition 1.

The entropies $H_{D}^{\pm}(P)$ define an optimisable functional of the class
(5)

$J_{\pm}=L_{\pm}+\gamma K_{\pm}$ (8)

where $\gamma$ is a Lagrange multiplier and the average lengths $L_{\pm}$ are
univocally determined by the constraints:

$K_{\pm}=\sum_{i}^{N}\sum_{j}^{\infty}a_{\pm}(j)\Gamma[j+1,-\log
D^{-l_{\pm}(x_{i})}]\leq\text{const.},$ (9)

the individual lengths $l_{\pm}(x_{i})$ will eventually be related to the
probability distributions $p_{\pm}(x_{i})$, the real coefficients $a_{\pm}(j)$
are given in Appendix A and

$\Gamma(y,x)=\int_{x}^{\infty}dz\,z^{y-1}e^{-z},$

is the incomplete gamma function.

###### Proof.

Since (1) can be expressed in the generalised form, the corresponding Nagumo-
Kolmogorov function is $\varphi(x)=x$. Consistently, the average lengths in
(8) read as

$L_{\pm}=\sum_{i}^{N}p_{\pm}(x_{i})l_{\pm}(x_{i}).$ (10)

To find the optimal individual lengths $l_{\pm}^{*}(x_{i})$, we now
differentiate (8) with respect to $l_{\pm}(x_{i})$, that is:

$\frac{\partial J_{\pm}}{\partial l_{\pm}(x_{i})}=\frac{\partial
L_{\pm}}{\partial l_{\pm}(x_{i})}+\gamma\frac{\partial K_{\pm}}{\partial
l_{\pm}(x_{i})},$ (11)

yet we are looking for a global minimum, it follows that the equality must
vanish term to term, therefore

$\begin{split}\frac{\partial L_{\pm}}{\partial
l_{\pm}(x_{i})}&=\sum_{i}^{N}p_{\pm}(x_{i})\\\ &=-\gamma\frac{\partial
K_{\pm}}{\partial l_{\pm}(x_{i})}\\\ &=\gamma\log
D\sum_{i}^{N}\exp^{\pm}\left[-l_{\pm}(x_{i})\log D\right]\\\
&=\sum_{i}^{N}\exp^{\pm}\left[-l_{\pm}(x_{i})\log D\right],\end{split}$ (12)

where the Lagrange multiplier has been selected as $\gamma=\frac{1}{\log D}$
and $\exp^{\pm}$ are stretched exponential functions (see Appendix A).

One can observe the equality in (12) is satisfied for all $x_{i}\in X$ iff
$l_{\pm}^{*}(x_{i})=-\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)$. At this point
it should be evident that we have obtained two optimal individual lengths as
given by each probability distribution $p_{\pm}(x_{i})$, for that reason we
have appended the label $\pm$ to $l_{\pm}^{*}(x_{i})$, since the lengths and
the distributions are univocally related. That completes the proof. ∎

It follows that the expected lengths $L_{\pm}$ are lower and upper bounded by
the entropy measures $H_{D}^{\pm}(P)$, which is consistent with the standard
formulation. Thus we are entitled to introduce the coding Theorems 1 and 2.

However, before the formal introduction of such theorems, some comments
deserve attention. First, note that the individual lengths $l_{\pm}$ and the
probabilities $p_{\pm}$ are determined by each other, therefore the
constraints (9) are not arbitrary, but must be constructed such that entropy
measures can be recovered at the optimal points.

We also remark that the standard theory is straightforwardly recoverable from
our scheme. First note how the Kraft inequality (6) is obtained by truncating
$K_{\pm}$ at first order, that is:

$\begin{split}K_{\pm}&=a_{\pm}(0)\sum_{i}^{N}D^{-l_{\pm}(x_{i})}\\\
&\quad+a_{\pm}(1)\sum_{i}^{N}\Gamma[2,-\log[D^{-l_{\pm}(x_{i})}]\\\
&\qquad+a_{\pm}(2)\sum_{i}^{N}\Gamma[3,-\log[D^{-l_{\pm}(x_{i})}]+\cdots,\end{split}$
(13)

with $a_{\pm}(0)=1$, see Appendix A.

Equivalently the limit $N\to\infty$ means that the system will possess pretty
low probabilities due to a high number of accessible states, in that case our
proposal will be asymptotically equivalent to the Shannon’s theory 3,
symbolically expressed as $p_{S}(x)=\lim_{N\to\infty}p_{\pm}(x)$.

######  Theorem 1.

The expected lengths defined by Eq. (10) for a $D$-ary alphabet regarding the
entropies $H_{D}^{\pm}(P)$, satisfy $L_{\pm}\geq H_{D}^{\pm}(P)$, with
equality iff $l_{\pm}^{*}(x_{i})=-\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)$
for every $x_{i}$ in $X$.

###### Proof.

It follows directly by writing the difference between the expected lengths and
the entropies. Then one gets

$\begin{split}L_{\pm}-H_{D}^{\pm}(P)&=\sum_{i}^{N}p_{\pm}(x_{i})l_{\pm}(x_{i})\\\
&\hskip
36.98866pt+\sum_{i}^{N}p_{\pm}(x_{i})\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)\\\
&=\sum_{i}^{N}p_{\pm}(x_{i})\left[l_{\pm}(x_{i})+\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)\right]\\\
&\geq 0,\end{split}$ (14)

necessarily leading to
$l_{\pm}(x_{i})\geq-\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)$, for the reason
that every $l_{\pm}(x_{i})$ is an integer. Then, it follows that the equality
is attained iff the individual lengths $l_{\pm}(x_{i})=l_{\pm}^{*}(x_{i})$ are
optimal. And the theorem is demonstrated. ∎

Likewise, the entropies $H_{D}^{\pm}$ amount to a lower bound on the expected
lengths $L_{\pm}$, yet as we are to show these lengths are within one dit of
the lower bound as well.

######  Theorem 2.

For a $D$-ary alphabet and a source distribution $X$ let $l_{\pm}(x_{i})$ be
the optimal individual lengths that solve the optimisation problem (8), where
the associated mean lengths are defined by Eq. (10). Then

$H_{D}^{\pm}(P)\leq L_{\pm}<H_{D}^{\pm}(P)+1.$ (15)

###### Proof.

As stated by Theorem 1, the choice of codeword lengths
$l_{\pm}^{*}(x_{i})=-\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)$ results in
$L_{\pm}^{*}=H_{D}^{\pm}$. However, to assure that every $l_{\pm}(x_{i})$ is
an integer, then we shall take
$l_{\pm}(x_{i})=\left\lceil-\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)\right\rceil$.
In that way the individual lengths do really satisfy

$-\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)\leq
l_{\pm}(x_{i})<-\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)+1,$ (16)

multiplying each member by $p_{\pm}(x_{i})$ and summing over all $x_{i}$,
leads to

$\begin{split}-\sum_{i}^{N}p_{\pm}(x_{i})\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)&\leq\sum_{i}^{N}p_{\pm}(x_{i})l_{\pm}(x_{i})\\\
&<-\sum_{i}^{N}p_{\pm}(x_{i})\log^{\pm}_{D}\left(p_{\pm}(x_{i})\right)\\\
&\hskip 42.67912pt+\sum_{i}^{N}p_{\pm}(x_{i}),\end{split}$ (17)

then from Eqs. (1) and (10) together with the normalisation of the
probability, we finally arrive at the expression

$H_{D}^{\pm}(P)\leq L_{\pm}<H_{D}^{\pm}(P)+1,$ (18)

and we have the theorem. ∎

We would like to remark that the expected lengths $L_{\pm}$ satisfy
$H_{D}^{\pm}\leq L_{\pm}<H_{D}^{\pm}+1$. Nonetheless the optimal code in view
of $H_{D}^{\pm}$ can only be better that the prescribed by $L_{\pm}$, thus one
is addressed to the Theorem 2.

Figure 2: Random processes for $N=20$ events. Figure 3: Random processes for
$N=150$ events.

To illustrate our discussion, we have generated two random processes with
uncorrelated sources, that we shall use as datasets to compute the average
lengths $L\pm$ as well as $L_{S}$, the usual length in information theory 16.

Considering the random process in Fig. 2, with a binary alphabet $D=2$, we
have obtained an average length $L_{S}^{*}$=4.6bits, whereas the average
lengths regarding entropies $H^{+}(P)$ and $H^{-}(P)$ are $L_{+}^{*}$=4.1bits
and $L_{-}^{*}$=4.75bits, respectively. That means that a more efficient
transmission process would result from the entropy $H^{+}(P)$ in comparison to
a code compressed via $H^{S}(P)$. However as the number of random events
increases, the lengths $L_{\pm}^{*}$ tend to coincide with $L_{S}^{*}$. For
instance, with respect to the process in Fig. 3, the standard average length
is $L_{S}^{*}$=7.42bits, while $L_{+}^{*}$=7.3bits and $L_{-}^{*}$=7.49bits,
hence diminishing the difference between $L_{S}^{*}$ and $L_{\pm}^{*}$. This
is a natural consequence since the entropies $H_{D}^{\pm}(P)$ converge
asymptotically to $H_{D}^{S}(P)$, see Ref. 3, but maintaining a subtle
difference between $H_{D}^{\pm}(P)$ and $H_{D}^{S}(P)$ whenever the number of
events is reasonably small.

##  III Channel capacity

It would be apparently simple to expand the capacity of a communication system
by merely increasing the number of different signalling events transiting from
$A$ to $B$. Why not sending indiscriminately thousands of different voltages
per symbol across the communication channel? As the number of different
signals grows at $A$, a special difficulty arises at $B$. All these emitted
signals are limited by the environmental noise that cannot be entirely
avoided. Thus the difference between signalling events shall be greater than
the intrinsic noise level to keep ambiguities away from the recovered signals
by the acts of $B$.

To circumvent this problem, the notion of channel capacity becomes fundamental
to know the maximum amount of symbols per unit time, $n$, that can be emitted
and the many differences per symbols, $s$, that can be selected from the
message space $s^{n}$.

This leads to a decision tree on the actual possibilities that spans the
message space, therefore, producing a probability distribution on the
codewords computed at $B$. Yet, a one-to-one relation between the input and
the output is not given a priori, i.e for two different inputs we can obtain
the same outcome, which makes the signals to look ambiguous. A possible choice
of unambiguous inputs is realisable with a BSC, permitting a signal
reconstruction with an error of no importance.

We shall use such scheme in the next few paragraphs to determine the maximum
rate at which the signal can be reconstructed in terms of the entropies (1),
leading to a new set of bounds on the channel capacity due to Shannon.

######  Definition 1.

Let $P$ and $Q$ be two different probability distributions. The amount of
information shared between them is measured via the mutual information
$I(P,Q)=H_{D}(Q)-H_{D}(Q|P)$, where $H_{D}(Q|P)$ is the conditional entropy.
Then the channel capacity is defined as:

$C=\max_{P}I(P,Q).$ (19)

The BSC, see the diagram below, is one the simplest models of communication
channels with errors. Each input is complemented with probability $p$.

$p$$p$$1-p$$1-p$$1$$1$$0$$0$

In case that an error occurs, an input value 1 (0) will be regarded as 0 (1),
hence we are unable to identify those error bits and at the end we get an
untrusted set of messages. For that reason, we shall assume that every bit
sent by $A$ has a negligible probability of error.

To calculate the channel capacity of a BSC, first note the mutual information
$I(P,Q)$ is bounded by 15

$I(P,Q)\leq 1-H(P),$ (20)

(henceforth all operations are binary, $D=2$, thus we preferably drop such
label from our notation), observe also, that the equality is attained when a
uniform distribution feeds the input, it follows from (19) that

$C_{\text{BSC}}=1-H(P),$ (21)

and we have the general expression for the channel capacity of a BSC.

Figure 4: Capacity of a BSC in terms of entropy measures that depend only on
the probability distribution. The entropy $H^{-}(P)$ estimates a slightly
greater capacity than the one obtained from the Shannon’s entropy.

Particularly, the channel capacity (21) of a BSC communication system whose
information weight is the Shannon’s entropy, reads

$C_{\text{BSC}}^{S}=1+p\log p+(1-p)\log(1-p),$ (22)

where the logarithms are base 2.

Likewise, we can determine the channel capacity of a BSC in terms of the
entropies $H_{D}^{\pm}(P)$, which yields

$C_{\text{BSC}}^{+}=\frac{\sqrt{2}-p^{p}-(1-p)^{1-p}}{\sqrt{2}-2},$ (23)

and

$C_{\text{BSC}}^{-}=\frac{2\sqrt{2}p^{-p}-(1-p)^{-(1-p)}}{2\sqrt{2}-2},$ (24)

where we have normalised the capacities $C_{\text{BSC}}^{\pm}$ to be
comparable with the standard case $C_{\text{BSC}}^{S}$, see Fig. 4.

In particular, the channel capacity $C_{\text{BSC}}^{-}$ exhibits a modest
improvement with respect to $C_{\text{BSC}}^{S}$, indicating a tentative upper
limit, above Shannon’s, on how many bits can be transmitted per second over
the channel without errors. Consistently, at $p=\frac{1}{2}$ the three
capacities coincide, since at this point occurs the highest degree of
uncertainty, resembling the initial situation in which one cannot form any
judgment from the received bits at $B$.

Another interesting scenario that we shall discuss is that of a BEC: A
situation in which a fraction $\alpha$ of bits is erased or lost during the
transmission process, but the receiver actually knows those bits. As in the
case of BSC, there are two inputs but now there will be three outputs:

$0$$0$$1$$1$$*$$\alpha$$\alpha$$1-\alpha$$1-\alpha$

This diagram means that there is an input $X$, which emits 0s and 1s with
probabilities $p$ and $1-p$, respectively. Whereas the output casts 0s, 1s or
$*$s with probabilities $(1-\alpha)p$, $(1-\alpha)(1-p)$ or $\alpha$, in that
order. Furthermore, if the source emits a 0 or 1, the probabilities of
receiving these bits without errors are $(1-\alpha)p$ and $(1-\alpha)(1-p)$,
otherwise the probabilities that these bits become interchanged over the
communication channel are $p\alpha$ and $(1-p)\alpha$.

It follows that the channel capacity of a BEC with an erasure probability
$\alpha$, is calculated as:

$C_{\text{BEC}}=\max_{P}\,H(X)+H(Y)-H(X,Y),$ (25)

where $H(X,Y)$ is the joint entropy computed in terms of the joint
distributions, which alternatively can be done via transition matrices, see
for instance 17.

In the case of the Shannon’s entropy, the calculation of the capacity (25)
gives:

$C_{\text{BEC}}^{S}=\max_{P}\,(1-\alpha)H^{S}(P),\quad P=\\{p,1-p\\},$ (26)

since the entropy attains its maximum for a uniform distribution,
$p=\frac{1}{2}$, the channel capacity of BEC reduces to $C^{S}=1-\alpha$, a
well known result in the Shannon’s theory.

Figure 5: Channel capacity of a BEC, for entropy measures that depend only on
the distribution $P$.

It shall not escape attention, that the channel capacities (22) and (26) are
completely additive, that is, given two distributions $p_{1}$ and $p_{2}$, we
have $C^{S}(p_{1}p_{2})=C^{S}(p_{1})+C^{S}(p_{2})$, a direct result that
follows from
$C^{S}(p_{1}p_{2})=\text{max}_{p_{X_{1},X_{2}}}I(X_{1},X_{2};Y_{1},Y_{2})$
with $I(X_{1},X_{2};Y_{1},Y_{2})=I(X_{1},Y_{1})+I(X_{2},Y_{2})$. Although this
property will not be satisfied by the non-extensive entropies $H^{\pm}(P)$, in
which case the additivity is asymptotically achieved, rather we are not
interested in such aspects but precisely in the non-additive consequences and
possible applications, if any.

In turn, we shall compute the channel capacity of a BEC but in terms of (1).
As in the example of a BSC, we have to renormalise the corresponding
capacities to compare with the standard result (25). We get:

$C_{\text{BEC}}^{+}=\max_{P}\,\frac{1-\alpha^{\alpha}+H^{+}(P)-H^{+}(P\alpha)}{2-\sqrt{2}},$
(27)

and

$C_{\text{BEC}}^{-}=\max_{P}\,\frac{\alpha^{-\alpha}-1+H^{-}(P)-H^{-}(P\alpha)}{2\sqrt{2}-2},$
(28)

as in the case (26), the maximum will be attained at $p=\frac{1}{2}$.

See Fig. 5, where we have plotted the channel capacities $C_{\text{BEC}}^{S}$
and $C_{\text{BEC}}^{\pm}$. Interestingly $C_{\text{BEC}}^{\pm}$ behave such
that the three measures coincide at the critical points $p=0,p=1$ and
$p=\frac{1}{2}$, but $C_{\text{BEC}}^{\pm}$ flips their character at
$p=\frac{1}{2}$, which shows the flexibility of our approach.

The flipping behaviour of $C_{\text{BEC}}^{\pm}$ is interpreted as follows.
Note that $C_{\text{BEC}}^{-}$ will establish an upper bound on
$C_{\text{BEC}}^{S}$ as long as the fraction of errors $\alpha$ does not
dominate the communication channel, i.e. $<50\%$. Otherwise there is a
tradeoff and $C_{\text{BEC}}^{-}$ will be slightly reduced immediately after
the threshold at $p=\frac{1}{2}$. For the same reason, the lower bound
suggested by $C_{\text{BEC}}^{+}$, will eventually be promoted to an upper
bound, with respect to $C_{\text{BEC}}^{S}$, when the ratio $\alpha$ dominates
the channel ($>50\%$).

##  IV Conclusions and Outlook

In the first part of this paper we have utilised a variational method to
minimise the average length of a given prefix-free code on the basis of the
non-extensive entropy measures $H_{D}^{\pm}(P)$, which depend only on the
distribution. We have shown that such optimisation problem is solved as long
as the adequate constraint is introduced —in our case $K_{\pm}$.

These constraints defined in Prop. 1 are not arbitrary, provided the average
lengths $L_{\pm}$ must equal the entropies $H_{D}^{\pm}(P)$ when they reach
their optimal values.

In consequence we have introduced the Theorems 1 and 2, that account for an
effective data compression at different average rates, either in terms of
$H_{D}^{+}(P)$ or $H_{D}^{-}(P)$. Therefore the corresponding average lengths
$L_{\pm}$ must be bounded from below and above in terms of the information
measures themselves and the associated optimal individual lengths
$l_{\pm}(x_{i})$.

Specifically, we think that $H_{D}^{+}(P)$ could bring some novelty with
respect to the data compression estimated via the Shannon’s entropy
$H_{D}^{S}(P)$, iff the involved system possess only few accessible states. We
suggest that such efficiency, however, could be prolonged for even a larger
system if one pinpoints a reasonable way to divide it into a number of
subsystems in order to outperform successively the data compression.

In the second part, we have analysed the channel capacity of two simple and
generic models, a BSC and a BEC, regarding the entropies $H_{D}^{\pm}(P)$. We
have obtained a channel capacity of a BSC higher than the calculated using the
Shannon’s theory, $C_{\text{BSC}}^{-}\geq C_{\text{BSC}}^{S}$, where the
equality is attained at the critical points $p=0,p=1$ and $p=\frac{1}{2}$.

Yet an interesting behaviour occurs in the capacities of a BEC. We have
obtained a twofold behaviour in each of the capacities $C_{\text{BEC}}^{+}$
and $C_{\text{BEC}}^{-}$. The first one estimates a lower bound on
$C_{\text{BEC}}^{S}$ but only if the population of errors is below the $50\%$
of the received bits, otherwise it will establish a new bound, but now above
$C_{\text{BEC}}^{S}$, whereas the inverse situation is furnished by
$C_{\text{BEC}}^{-}$. We recall that such capacities are not strictly computed
as functions of some distribution $P$, but depend now on a new parameter
$\alpha$ that represents the ratio of errors over the communication channel.

## Acknowledgments

The authors appreciate the early discussions and observations given by H.
García-Compeán. We also are grateful for the general comments provided by J.L.
López. J.F. would like to thank the financial support granted by CONACYT
(Mexico). O.O. thanks the support of CONACYT Project 257919, UG Projects and
PRODEP.

## Data Availability

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

##  Appendix A Generalised logarithms and exponentials

The generalised logarithm functions define as

$\begin{split}\log^{+}(x)&\equiv-\frac{1-x^{x}}{x}\\\
\log^{-}(x)&\equiv-\frac{x^{-x}-1}{x},\end{split}$ (29)

for $x>0$, otherwise the functions become undefined. From such definitions it
becomes evident that the functions $\log^{(\pm)}$ do not fulfil the three laws
of logarithms. In addition, the corresponding inverse functions of the
generalised logarithms (29) do not posses a closed form hence, to subdue this
technicality, a numerical representation has to be taken into account. These
functions have been constructed as

$\exp^{\pm}(x)\equiv\exp(-x)\sum_{j=0}^{\infty}a_{\pm}(j)x^{j},\quad
a_{\pm}(j)\in\mathbb{R},$ (30)

whose first nine coefficients $a_{\pm}(j)$ are given in Table 1.

|  $a_{+}(j)$ |  $a_{-}(j)$
---|---|---
$j=8$ |  -0.000157095 |  0.000105402
$j=7$ |  0.00373467 |  -0.00211934
$j=6$ |  -0.0362676 |  0.0166679
$j=5$ |  0.186358 |  -0.0675544
$j=4$ |  -0.546751 |  0.16867
$j=3$ |  0.905157 |  -0.317048
$j=2$ |  -0.709322 |  0.3725
$j=1$ |  0.0228963 |  0.0147449
$j=0$ |  1 |  1
Table 1: $a^{(\pm)}$ Coefficients.

## References

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# Efficient and improved prediction of the band offsets at semiconductor
heterojunctions from meta-GGA density functionals

Arghya Ghosh  Department of Physics, Indian Institute of Technology,
Hyderabad, India  Subrata Jana  Corresponding author<EMAIL_ADDRESS>Department of Chemistry & Biochemistry, The Ohio State University, Columbus,
OH 43210, USA  Tomáš Rauch  Institut für Festkörpertheorie und -Optik,
Friedrich-Schiller-Universität Jena, 07743 Jena, Germany  European Theoretical
Spectroscopy Facility  Fabien Tran  VASP Software GmbH, Sensengasse 8, A-1090
Vienna, Austria  Miguel A. L. Marques  Institut für Physik, Martin-Luther-
Universität Halle-Wittenberg, 06120 Halle/Saale, Germany  European Theoretical
Spectroscopy Facility  Silvana Botti  Institut für Festkörpertheorie und
-Optik, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany  European
Theoretical Spectroscopy Facility  Lucian A. Constantin  Istituto di
Nanoscienze, Consiglio Nazionale delle Ricerche CNR-NANO, 41125 Modena, Italy
Manish K Niranjan  Department of Physics, Indian Institute of Technology,
Hyderabad, India  Prasanjit Samal  School of Physical Sciences, National
Institute of Science Education and Research, HBNI, Bhubaneswar 752050, India

###### Abstract

Accurate theoretical prediction of the band offsets at interfaces of
semiconductor heterostructures can of- ten be quite challenging. Although
density functional theory has been reasonably successful to carry out such
calculations and efficient and accurate semilocal functionals are desirable to
reduce the computational cost. In general, the semilocal functionals based on
the generalized gradient approximation (GGA) significantly underestimate the
bulk band gaps. This, in turn, results in inaccurate estimates of the band
offsets at the heterointerfaces. In this paper, we investigate the performance
of several advanced meta-GGA functionals in the computational prediction of
band offsets at semiconductor heterojunctions. In particular, we investigate
the performance of r2SCAN (revised strongly-constrained and appropriately-
normed functional), rMGGAC (revised semilocal functional based on cuspless
hydrogen model and Pauli kinetic energy density functional), mTASK (modified
Aschebrock and Kümmel meta-GGA functional), and LMBJ (local modified Becke-
Johnson) exchange-correlation functionals. Our results strongly suggest that
these meta-GGA functionals for supercell calculations perform quite well,
especially, when compared to computationally more demanding GW calculations.
We also present band offsets calculated using ionization potentials and
electron affinities, as well as band alignment via the branch point energies.
Overall, our study shows that the aforementioned meta-GGA functionals can be
used within the DFT framework to estimate the band offsets in semiconductor
heterostructures with predictive accuracy.

##  I Introduction

Over the past three decades, semiconductor heterostructure engineering has
greatly influenced the development of nanoelectronic, optoelectronic, and
other multifunctional devices Facchetti and Marks (2010); Franciosi and Van de
Walle (1996); Vurgaftman  _et al._ (2001); Mönch (2017); Robertson (2013). In
recent years, the interest in heterostructures with atomically sharp
interfaces has increased manifold due to their huge potential in advanced
nanoscale technologies as well as the possibility of fascinating emergent
interfacial phenomena in them Alferov (2001); del Alamo (2011); Harrison  _et
al._ (1978); Nakagawa  _et al._ (2006). For instance, integer and fractional
quantum hall effects have been discovered in high mobility two-dimensional
electron gas formed at the interface of GaAs/AlGaAs heterostructures Ikebe
_et al._ (2010); Stier  _et al._ (2015); Dziom  _et al._ (2019); Anversa  _et
al._ (2014). The 2D topological insulator phase also exhibits quantum spin
hall effect was first observed in quantum well heterostructures such as
HgTe/CdTe Bernevig  _et al._ (2006); König  _et al._ (2007).

One of the most important parameters in heterostructures is the band offset or
band alignment at the interface. The band offset critically controls the out-
of-plane electrical transport and capacitance behavior in heterostructure
based devices. Naturally, the control and tunability of band offsets are
highly desirable since it renders the control of the overall device
performance and functionality Facchetti and Marks (2010); Franciosi and Van de
Walle (1996); Van de Walle (1989); Robertson (2013). In general, the band
offsets in heterostructures can critically depend on the interfacial atomic
structure Franciosi and Van de Walle (1996); Frensley and Kroemer (1977);
Tejedor and Flores (1977); Pickett  _et al._ (1978); Van de Walle (1989); Van
de Walle and Martin (1987). Therefore, one way to control band offsets is via
atomic level manipulation of the heterointerface Van de Walle (1989); Van de
Walle and Martin (1987). In this context, the accurate theoretical estimates
and prediction of band offsets can be quite advantageous as these can guide
the development and fabrication of heterostructures based devices. Over the
years, ab-initio calculations based on the density functional theory (DFT)
Hohenberg and Kohn (1964); Kohn and Sham (1965); Engel and Dreizler (2013);
Burke (2012); Jones (2015) have been widely used to estimate the band offsets
Franciosi and Van de Walle (1996); Van de Walle (1989); Pickett  _et al._
(1978); Van de Walle and Martin (1987). Further, these past studies have been
mostly performed within the local density approximation (LDA) Perdew and Wang
(1992) and/or semi-local generalized gradient approximation (GGA) Perdew  _et
al._ (1996, 2008) functionals Weston  _et al._ (2018). However, the DFT
calculations with GGA (LDA) functionals suffer from the drawback to severely
undersetimate the band gaps Perdew  _et al._ (2017); Tran  _et al._ (2007);
Borlido  _et al._ (2020); Patra  _et al._ (2019a); Tran  _et al._ (2018a);
Jana  _et al._ (2018a, b); Tran and Blaha (2017); Tran  _et al._ (2019) and
thereby the band offsets Weston  _et al._ (2018); Hinuma  _et al._ (2014);
Grüneis  _et al._ (2014). Several earlier studies have suggested that band gap
accuracy of semilocal GGAs is quite inadequate as compared to that of beyond-
semilocal methods, like (system or dielectric-dependent) hybrid DFT Heyd  _et
al._ (2003); Krukau  _et al._ (2006); Heyd and Scuseria (2004); Jana and Samal
(2019); Jana  _et al._ (2020a, b); Zheng  _et al._ (2019); Brawand  _et al._
(2016, 2017); Chen  _et al._ (2018); Cui  _et al._ (2018); Jana  _et al._
(2022) or Green function based many-body calculations ($\rm GW$ method) Hedin
(1965); Hybertsen and Louie (1986). Though hybrids and $\rm GW$ calculations
allow to calculate band gaps and band offsets closer to the experimental
magnitudes Zhang  _et al._ (2020); Weston  _et al._ (2018); Bischoff  _et al._
(2020); Borlido  _et al._ (2018), those methods are not computationally
feasible for large supercell. Hence, as an alternative, the computationally
efficient meta-generalized gradient approximations (meta-GGAs) functionals
Perdew  _et al._ (2009); Sun  _et al._ (2015); Mejia-Rodriguez and Trickey
(2018); Patra  _et al._ (2019b, a); Furness  _et al._ (2020); Jana  _et al._
(2021); Neupane  _et al._ (2021); Aschebrock and Kümmel (2019); Della Sala
_et al._ (2016) could be a good choice for estimating heterostructure band
offsets, especially, as recent studies have shown that those can be applied
reliably and in an improved way for obtaining the band gaps of bulk and
layered solids Borlido  _et al._ (2020); Patra  _et al._ (2020, 2021a); Tran
_et al._ (2021); Aschebrock and Kümmel (2019); Neupane  _et al._ (2021); Tran
_et al._ (2019). Recently, a methodology based on hybrid (HSE06) and GGA
functional has also been proposed for estimating band offsets Weston  _et al._
(2018). In this method, bulk band structures are calculated using a hybrid
functional(HSE). However, atomic relaxations and the potential alignment in
the superlattice are calculated using the GGA functional which is highly less
expensive computationally as compared to HSE. The present study is different
from that presented in Ref. Weston  _et al._ (2018) in that entire supercell
calculations are done using meta-GGA functionals.

In fact, motivated by those improved reliabilities for band gaps, in this
paper we use meta-GGA functionals to study the band offsets at $14$
semiconductor heterojunctions composed of Zincblende and diamond structures
with (110) interfaces. We assess the performance of the functionals by
considering both homovalent (e.g. GaAs/AlAs and GaP/AlP) and heterovalent
heterojunctions (e.g., Ge/GaAs, Si/GaP, Ge/AlAs, ZnSe/GaAs, and Ge/ZnSe). We
calculate the band offsets using (i) the ionization potential (IP) Hinuma  _et
al._ (2014), (ii) the branch point energy (BPE), or charge neutrality level
(CNL) Tersoff (1984, 1985); Schleife  _et al._ (2009); Robertson (2013); Mönch
(1996); Guo and Robertson (2017); Guo  _et al._ (2019), and (iii) the average
potential in the supercell containing heterostructures Facchetti and Marks
(2010); Weston  _et al._ (2018); Hinuma  _et al._ (2013); Höffling  _et al._
(2012); Grüneis  _et al._ (2014).

At the meta-GGA level of approximation, we mainly focus on the r2SCAN Furness
_et al._ (2020) (revised strongly-constrained and appropriately-normed (SCAN)
Sun  _et al._ (2015)), rMGGAC Jana  _et al._ (2021) (revised semilocal
functional based on cuspless hydrogen model and Pauli kinetic energy density
(MGGAC) Patra  _et al._ (2019b)), mTASK Neupane  _et al._ (2021) (modified
Thilo Aschebrock and Stephan Kümmel (TASK) Aschebrock and Kümmel (2019)
functional), and LMBJ Rauch  _et al._ (2020a) (local modified Becke-Johnson
(MBJ) Tran and Blaha (2009)) semilocal functionals. We have chosen the
aforementioned recently proposed meta-GGAs instead of earlier proposed meta-
GGAs (like Tao-Perdew-Staroverov-Scuseria Tao  _et al._ (2003)) since these
new functionals show considerable improvement for the band gaps of bulk and 2D
layered materials Borlido  _et al._ (2020); Patra  _et al._ (2020, 2021a);
Tran  _et al._ (2021); Aschebrock and Kümmel (2019); Neupane  _et al._ (2021)
and are therefore better suited for estimating the band offsets.

This paper is organized as follows. In sec. II, we briefly present the meta-
GGA functionals considered in this study. In section III, we discuss all
results. The conclusions are presented in Sec. IV.

##  II Brief overview of methods

Here, we briefly discuss the exchange-correlation (XC) functionals considered
this study, starting from the standard GGA-PBE functional Perdew  _et al._
(1996). In general, the band gap estimates are not accurate enough when PBE is
used. Nevertheless, since it is a widely used functional and its performance
will serve as a reference for more accurate functionals. r2SCAN Sun  _et al._
(2015); Furness  _et al._ (2020) and rMGGAC Patra  _et al._ (2019b); Jana  _et
al._ (2021) are general purpose meta-GGA functionals. While r2SCAN is
constructed by satisfying as many constraints as possible, rMGGAC satisfies
fewer constraints. Also, the construction of r2SCAN is non-empirical, while
some of the parameters of rMGGAC are fitted by using test sets. In fact, both
functionals show improved performance to evaluate the band gaps of solids with
respect to PBE Furness  _et al._ (2020); Jana  _et al._ (2021); Patra  _et
al._ (2021b); Jana  _et al._ (2018b, a). Unlike r2SCAN and rMGGAC, mTASK
Aschebrock and Kümmel (2019); Neupane  _et al._ (2021) should not really be
considered as a general purpose functional, but more specialized for the band
gap of solids Borlido  _et al._ (2020); Patra  _et al._ (2020, 2021a); Tran
_et al._ (2021); Aschebrock and Kümmel (2019); Neupane  _et al._ (2021).

In addition to the XC approximations listed above, which are based on an
energy functional, we will also consider the potential-only LMBJ Rauch  _et
al._ (2020a) method. The MBJ XC potential is an improvement of the Becke-
Johnson potential Becke and Johnson (2006) that in addition uses an average of
the reduced density gradient $\frac{|\nabla\rho|}{\rho}$. While in MBJ the
average of the reduced density gradient is calculated in the whole unit cell,
in LMBJ it is local and calculated at every point of space, allowing for the
application of the potential also to systems with vacuum and inhomogeneous
materials Rauch  _et al._ (2020a). The LMBJ potential contains four
parameters, and we chose the values $\alpha=0.488$, $\beta=0.5$ Bohr,
$\sigma=2$ Å, and $r_{s}^{\mbox{th}}=7$ Bohr, based on previous optimizations
Koller  _et al._ (2012); Rauch  _et al._ (2020b). MBJ is currently the most
accurate semilocal functional for the band gap of bulk solids Borlido  _et
al._ (2020); Patra  _et al._ (2020), while LMBJ belongs to the group of
accurate functionals for two-dimensional materials Patra  _et al._ (2021a);
Tran  _et al._ (2021).

##  III Results and Discussions

In this paper, we calculate the band offsets using three different schemes,
namely: (i) Ionization potential (IP) based, (ii) Branch-point energy (BPE)
based, and (iii) Average potential (APM) based. The IP and BPE based methods
involve calculation of bulk and surface components only. On the other hand,
the APM requires calculation for entire superlattice. Therefore, the APM or
average potential based calculations are typically the most expensive,
followed by the IP-based and BPE-based ones. One may also note that the
interface contribution to band offsets may only be estimated using average
potential method as discussed later. In the following we briefly discuss these
methods.

###  III.1 Ionization potential and electron affinity

We first define the ionization potential (IP), electron affinity (EA) and
their connection with the band alignments of the heterostructure. For solids,
the IPs are defined as the difference between the vacuum level of the
electrostatic potential and the valance band maximum (VBM),

$\epsilon_{\mathrm{IP}}=[\epsilon_{\mathrm{Vac,s}}-\epsilon_{\mathrm{Ref,s}}]-[\epsilon_{\mathrm{VBM,b}}-\epsilon_{\mathrm{Ref,b}}]~{},$
(1)

where $\epsilon_{\mathrm{Vac,s}}-\epsilon_{\mathrm{Ref,s}}$ is calculated for
the surface supercell constructed in the (110) direction (for the zincblende
structure) taking into account the macroscopic average of the local
electrostatic potential in the vacuum region ($\epsilon_{\mathrm{Vac,s}}$) and
in the bulk region ($\epsilon_{\mathrm{Ref,s}}$) of the supercell.
$\epsilon_{\mathrm{VBM,b}}-\epsilon_{\mathrm{Ref,b}}$ is determined from the
bulk calculation, with $\epsilon_{\mathrm{VBM,b}}$ being the position of VBM
and $\epsilon_{\mathrm{Ref,b}}$ the reference level for the bulk calculation,
i.e., the average of the electrostatic potential in the unit cell. Then, the
EAs are obtained using the formula
$\epsilon_{\mathrm{EA}}=\epsilon_{\mathrm{IP}}-\epsilon_{\mathrm{g}}$, where
$\epsilon_{\mathrm{g}}$ is the band gap of the bulk structure. The details of
the calculation procedure of IPs and EAs are given in Section III.4.

From IPs and EAs obtained for different semiconductors as described above, one
can in principle extract the band offsets between two materials by comparing
the differences between their IPs and EAs. To obtain the valance band offset
(VBO) ($\Delta\epsilon_{v}$) between materials A and B, we evaluate

$\Delta\epsilon_{v}=\epsilon_{\mathrm{IP}}^{\mathrm{B}}-\epsilon_{\mathrm{IP}}^{\mathrm{A}}.$
(2)

It should be noted that this method neglects the effects occurring during the
actual formation of the interface, such as charge transfer, interface dipole,
or atomic relaxations. Still, it can be applied to interfaces of
semiconductors with similar lattice constants where such effects are expected
to be weak and can thus be neglected.

Finally, we did not calculate the IPs and EAs using the LMBJ potential, since
some of us recently demonstrated in Ref. Rauch  _et al._ , 2021 that the band
edges of semiconductors are predicted to be located too high with respect to
the vacuum level by the LMBJ potential.

###  III.2 Band offsets from branch point energy

In the context of band alignment, another important quantity that serves as an
energy reference level is the branch point energy (BPE) Tersoff (1984, 1985);
Schleife  _et al._ (2009); Robertson (2013); Mönch (1996); Guo and Robertson
(2017); Guo  _et al._ (2019). The BPE can be approximated as the following
averaging rule of high-lying valence bands (VB) and low-lying conduction bands
(CB),

$\epsilon_{\mathrm{BPE}}=\frac{1}{2\mathcal{N}_{k}}\sum_{k}\Big{[}\frac{1}{\mathcal{N}_{\mathrm{VB}}}\sum_{i}^{\mathcal{N}_{\mathrm{VB}}}\epsilon_{i}^{v}({\bf{k}})+\frac{1}{\mathcal{N}_{\mathrm{CB}}}\sum_{j}^{\mathcal{N}_{\mathrm{CB}}}\epsilon_{j}^{c}({\bf{k}})\Big{]}~{},$
(3)

where $\mathcal{N}_{k}$ is the number of ${\bf{k}}$ points in the Brillouin
zone, $\epsilon_{i}^{v}({\bf{k}})$ and $\epsilon_{j}^{c}({\bf{k}})$ denote the
$i^{th}$ highest VB and $j^{th}$ lowest CB at wave vector ${\bf{k}}$,
respectively. The sum over $i$ and $j$ runs over $\mathcal{N}_{\mathrm{CB}}$
lowest CBs and $\mathcal{N}_{\mathrm{VB}}$ highest of VBs. In Ref. Hinuma  _et
al._ , 2014, for the zincblende and diamond phases
$\mathcal{N}_{\mathrm{VB}}=2$ and $\mathcal{N}_{\mathrm{CB}}=1$ was
considered. Here, we include spin-orbit coupling in bulk calculations and we
consider a (110) oriented bulk unit cell with four atoms. Therefore, we chose
$\mathcal{N}_{\mathrm{VB}}=8$ and $\mathcal{N}_{\mathrm{CB}}=4$.

Finally, given the IP and BPE of two materials A and B, the VBO of the A/B
heterojunction is calculated using the linear model Cowley and Sze (1965); Wei
and Zunger (1987); Guo  _et al._ (2019),

$\Delta\epsilon_{v}=(\epsilon_{\mathrm{IP}}^{\mathrm{B}}-\epsilon_{\mathrm{BPE}}^{\mathrm{B}})-(\epsilon_{\mathrm{IP}}^{\mathrm{A}}-\epsilon_{\mathrm{BPE}}^{\mathrm{A}})+{\mathcal{S}}(\epsilon_{\mathrm{BPE}}^{\mathrm{B}}-\epsilon_{\mathrm{BPE}}^{\mathrm{A}})~{},$
(4)

where $\mathcal{S}$ is the dimensionless pinning factor Guo  _et al._ (2019).
Following Ref. Guo and Robertson, 2017, the VBOs from the IP rule (see Eq.
(2)) and BPE-matching rule corresponds to $S=1$ and $S=0$, respectively. In
the latter, we substitute the IP with the VBM. Thus,
$\epsilon_{\mathrm{BPE}}-\epsilon_{\mathrm{VBM}}$, which enters Eq. (4), can
be obtained from a bulk calculation.

###  III.3  Band offsets from average potential method

To estimate the band offsets, we also consider the average potential method
(APM). The band offset in this case is determined using the lineup of the
local electrostatic potential in the supercell including the interface and the
position of the VBM in bulk calculations of the materials A and B forming the
interface. We calculate the VBO as Weston  _et al._ (2018); Hinuma  _et al._
(2013)

$\Delta\epsilon_{v}=(\epsilon_{\mathrm{VBM,b}}^{\mathrm{B}}-\epsilon_{\mathrm{Ref,b}}^{\mathrm{B}})-(\epsilon_{\mathrm{VBM,b}}^{\mathrm{A}}-\epsilon_{\mathrm{Ref,b}}^{\mathrm{A}})+\Delta\mathrm{V}~{},$
(5)

where $\epsilon_{\mathrm{VBM,b}}^{\mathrm{A/B}}$ is the VBM position and
$\epsilon_{\mathrm{Ref,b}}^{\mathrm{A/B}}$ the reference potential (unit cell
average) of bulk material A/B. $\Delta\mathrm{V}$ is the difference of the
macroscopic averages of the electrostatic potential in the bulk-like regions
of the A and B parts of the supercell. To obtain the average potential, we
first calculate the in-plane average of the local potential and then obtain
the macroscopic average from its Gaussian average. The macroscopic average is
almost constant in the two bulk-like regions of the supercell. This method is
different from the other methods discussed above in that it takes into account
the atomic and electronic structure of the interface which may critically
influence the band offset. It should be further noted that to evaluate band
offsets from Eq. (5) we consider strained bulk structures, as described in
Sec. III.4. Thus, we calculate band offset for what is referred to here as
strained interface Hinuma  _et al._ (2013).

From the accuracy point of view, the average potential method (APM) can be
considered the “state-of-art” of band offset calculations, since it includes
the effects of the interface atomic structure, such as the interface dipole.
These are neglected in the BPE- and IP-based methods. Nevertheless, for
lattice-matched heterostructures with no significant interface
reconstructions, the latter two methods can yield reasonable results. For
lattice mismatched systems, the APM method should be chosen.

###  III.4 Calculation details

For benchmark calculations of band offsets, we consider in total $14$
heterostructure interfaces between zincblende ($zb$) (AlAs, AlSb, CdTe, InSb,
GaAs, GaP, ZnSe, and ZnTe) and diamond (Si and Ge) structures. We consider
nonpolar (001) surfaces and interfaces, including the Si and Ge diamond
structures. This set is chosen so that there is minimal strain at the
interfaces Hinuma  _et al._ (2014).

We perform DFT calculations using the plane-wave code Vienna Ab-initio
Simulation Package (VASP) Kresse and Hafner (1993); Kresse and Furthmüller
(1996); Kresse and Joubert (1999); Kresse and Furthmüller (1996). For Ga, Ge,
and In, we used pseudopotentials with relatively deep Ga $3d$, Ge $3d$, and In
$4d$ states treated as valence orbitals. Surface calculations are performed on
slabs consisting of $14$ atomic layers ($18-39$Å) followed by $14$ additional
vacuum layers. For the heterostructures, we consider $11$ atomic layers of
each of the two materials. A kinetic energy cutoff of $520$ eV is used for all
(bulk, surface, and interface) calculations. The Brillouin Zones (BZ) are
sampled using $\Gamma-$centered Monkhorst-Pack (MP) grid with
15$\times$15$\times$15 ${\bf{k}}$-points for bulk and 15$\times$15$\times$1
${\bf{k}}$-points for surface and interface calculations. The atomic
relaxations are performed until the Hellmann-Feynman forces on atoms are
reduced to less than $0.001$ eV/Å. We do not consider dipole corrections as
all the considered surfaces are nonpolar.

For all the considered methods, bulk, surface, and interface calculations are
carried out using the r2SCAN-optimized geometries and ionic positions. Note
that the r2SCAN geometries are very accurate for bulk crystals, being much
closer to experimental values than PBE ones Furness  _et al._ (2020).

For the construction of the heterostructure supercell, we follow the method of
Refs. Hinuma  _et al._ , 2013, 2014. For the in-plane lattice parameter we
take the average value of the bulk lattice parameters (r2SCAN-optimized) of
the two materials forming the (110) interface. After that, the out-of-plane
lattice parameter and internal coordinates of the heterostructure are allowed
to relax at the fixed in-plane parameters. This means that the bulk-like
regions in the A and B parts of the supercell do not agree with the cubic bulk
structures of materials A and B. Therefore, to evaluate Eq. (5) for bulk
calculations we consider tetragonally strained unit cell that agrees with
crystal structures in the bulk-like regions of the supercell. In this case we
will refer strained band offset.

Spin-orbit coupling is considered in the bulk calculations only since its
inclusion in surface and interface calculations has almost no influence on the
electrostatic potential.

###  III.5 Analysis of results

####  III.5.1 IP, EA, and Band gaps

Figure 1: CBMs relative to the vacuum level (negative of the EAs) and VBMs
relative to the vacuum level (negative of the IPs) as obtained from different
levels of theory. The IPs were calculated with Eq. (1) and EAs with
$\epsilon_{\mathrm{EA}}=\epsilon_{\mathrm{IP}}-\epsilon_{\mathrm{g}}$, where
$\epsilon_{\mathrm{g}}$ is the band gap. For comparison, we also show the
experimental and ${\rm{GW}}\Gamma^{1}$@HSE results from Ref. Hinuma  _et al._
, 2014. Numerical values are given in Table S1 of the Supplementary Material
sup . Figure 2: Bulk band gaps as obtained from different levels of theory
with spin-orbit coupling included. For comparison, we also show the
experimental and ${\rm{GW}}\Gamma^{1}$@HSE results from Ref. Hinuma  _et al._
, 2014. All numerical values are given in Table S1 of the Supplementary
Material sup .

In Figs. 1 and 2 we compare the position of the VBM and CB minima (CBM) with
respect to the vacuum level calculated for the (110) surfaces and the bulk
band gaps, respectively, using the PBE GGA functional and different meta-GGA
functionals (r2SCAN, rMGGAC, mTASK, and LMBJ). For comparison, we also present
the experimental results as well as those from ${\rm{GW}}\Gamma^{1}$@HSE
calculations from Ref. Hinuma  _et al._ , 2014. We note that the
${\rm{GW}}\Gamma^{1}$@HSE results reported in Ref. Hinuma  _et al._ , 2014 may
be considered the best available theoretical prediction for the electronic
properties of the semiconductors. Therefore, we will use these throughout this
paper as a reference for comparison with other functionals we studied. All the
calculated values are summarized in Table S1 of the Supplementary Material sup
.

Analyzing the results obtained from different semilocal XC methods, we observe
that the standard GGA PBE functional shows the largest deviation from the
experimental and ${\rm{GW}}\Gamma^{1}$@HSE values for the VBMs and band gaps
of group II-VI semiconductors. On the other hand, CBMs are better reproduced
by this method. Therefore, one can expect that the underestimation of the band
gaps (shown in Fig. 2) obtained using the PBE functional mainly comes from the
deviations from the VBMs. In general, we can conclude that PBE places the VBMs
too high and CBMs slightly low in energy, leading to an overall underestimated
band gap. This observation is consistent with what reported in Ref. Hinuma
_et al._ , 2014. Nevertheless, a better representation of VBMs from PBE may be
obtained using the PBE-$\frac{1}{2}$ Ferreira  _et al._ (2011); Doumont  _et
al._ (2019), or the GLLB-sc Tran  _et al._ (2018b) techniques, where the
orbital information and/or derivative discontinuities are taken into account
explicitly Doumont  _et al._ (2019); Tran  _et al._ (2018b).

Now, we come to the performance of the meta-GGA functionals r2SCAN, rMGGAC,
and mTASK. For r2SCAN and mTASK, the behavior is similarly to that of PBE.
Both functionals underestimate the IPs, but by a lesser margin. Finally,
rMGGAC yields the smallest average error with respect to the experimental and
${\rm{GW}}\Gamma^{1}$@HSE values. No clear trend can be deduced, as the IPs
are in this case overestimated for some materials and underestimated for
others. For the CBMs, we observe a different behavior. r2SCAN places the CBMs
slightly low, similar to PBE. mTASK, on the other hand, predicts too high a
position of the CBMs, due to its overall overestimation of the bulk band gaps.
For rMGGAC, again, no clear trend can be identified, as some CBMs are placed
too high in energy and others low. Finally, looking at the band gaps (Fig. 2),
we see that overall LMBJ yields the most accurate band gaps, which is
inherited from the original MBJ potential. This is in agreement with previous
studies Borlido  _et al._ (2019, 2020). mTASK and rMGGAC predict slightly too
wide band gaps, whereas r2SCAN yields too narrow ones, similar to PBE.

Figure 3: Band alignment based on the IPs (negatives of the VBM to the vacuum
level) and EAs (negatives of CBM to the vacuum level) as obtained using r2SCAN
(red) and rMGGAC (yellow), compared with reference ${\rm{GW}}\Gamma^{1}$@HSE
values (blue) from Ref. Hinuma  _et al._ , 2014. All the surface orientations
are along (110) directions.

In addition, the improved VBM positions from rMGGAC are also very much evident
from Fig. 3, where the ${\rm{GW}}\Gamma^{1}$@HSE reference values Hinuma  _et
al._ (2014) are added for comparison. However, the CBMs are slightly
underestimated for most of the systems considering the
${\rm{GW}}\Gamma^{1}$@HSE benchmark reference Hinuma  _et al._ (2014). For
r2SCAN, on the other hand, we see the systematic overestimation of the VBM
position, whereas the CBM position agrees much better with the
${\rm{GW}}\Gamma^{1}$@HSE reference.

####  III.5.2 Band offsets

Table 1: Strained band offsets (in eV) as calculated using the average potential method. The natural band offsets calculated with ${\rm{GW}}\Gamma^{1}$@HSE are from Ref. Hinuma  _et al._ , 2014. Interfaces | Mismatch | PBE | r2SCAN | rMGGAC | MTASK | LMBJ |  ${\rm{GW}}\Gamma^{1}$@HSE
---|---|---|---|---|---|---|---
ZnTe/AlSb | 1.09 | -0.36 | -0.41 | -0.41 | -0.36 | -0.44 | -0.59
Si/GaP | 0.26 | 0.30 | 0.35 | 0.45 | 0.36 | 0.35 | 0.51
InSb/CdTe | 0.11 | 0.48 | 0.57 | 0.67 | 0.75 | 0.68 | 0.86
InP/InAs | 3.35 | -0.34 | -0.40 | -0.40 | -0.49 | -0.47 | -0.35
Ge/ZnSe | 0.18 | 1.14 | 1.32 | 1.18 | 1.36 | 1.46 | 1.85
Ge/AlAs | 0.07 | 1.02 | 1.16 | 1.05 | 1.35 | 1.12 | 1.19
GaSb/AlSb | 0.74 | 0.34 | 0.39 | 0.36 | 0.52 | 0.38 | 0.48
GaP/AlP | 0.34 | 0.49 | 0.53 | 0.46 | 0.66 | 0.44 | 0.58
GaAs/ZnSe | 0.03 | 0.70 | 0.80 | 0.74 | 0.81 | 0.89 | 1.25
GaAs/InP | 3.99 | 0.25 | 0.26 | 0.25 | 0.22 | 0.32 | 0.43
GaAs/Ge | 0.21 | -0.42 | -0.53 | -0.45 | -0.59 | -0.59 | -0.56
GaAs/AlAs | 0.14 | 0.47 | 0.52 | 0.49 | 0.65 | 0.45 | 0.55
CdSe/ZnTe | 0.44 | -0.66 | -0.71 | -0.81 | -0.95 | -0.81 | -0.92
GaSb/ZnTe | 0.36 | 0.57 | 0.68 | 0.63 | 0.72 | 0.73 | 0.96
Figure 4: PBE and meta-GGA vs ${\rm{GW}}\Gamma^{1}$@HSE VBOs for the set of 14
heterostructure interfaces calculated with the IP-, BPE-, and average
potential based methods. The ${\rm{GW}}\Gamma^{1}$@HSE results are from Ref.
Hinuma  _et al._ , 2014. In the legends of each panel, the values of the mean
absolute percentage error are given.

Here, we finally discuss the valence band offsets (VBO) obtained by the three
methods: IP-based, BPE-based, average potential based. The VBOs obtained from
the IPs and BPEs using different meta-GGA semilocal XC functionals for chosen
$14$ pairs of semiconductors with small lattice mismatch are reported in Table
S3 of the Supplementary Material sup  and those obtained by the APM are listed
in Table 1. For convenience, the BPEs from different XC functionals are also
reported in Table S2 of the Supplementary Material sup . For comparison, we
show all the individual results compared with the ${\rm{GW}}\Gamma^{1}$@HSE
results in Fig. 4. It should be noted that the reference results were obtained
for slightly different supercell models in Ref. Hinuma  _et al._ , 2014. In
that work, the natural band offsets were calculated, for which the bulk VBM
position with respect to the bulk reference level was evaluated at the
equilibrium bulk structure, and the VBOs were then further corrected by
calculating the shift of the VBM with strain. The latter was evaluated from
additional surface calculations. Here, we consider only interfaces of
materials with a small lattice mismatch. Since we found a quite unsatisfying
performance of the IP-based approach obtained by surface calculations, as
shown below, we chose to evaluate the strained VBOs instead. For perfectly
lattice matched materials both results would be identical, and we expect the
discrepancy to remain small for our subset of interfaces. We, therefore,
consider such comparison more meaningful than with experimental results, for
which details of the atomic structure of the interface are often unknown.

Figure 5: Radar plots showing the statistical quantities MAPE, RMSE (in eV),
MAE (in eV), standard deviation ($\sigma$, in eV), and the parameters $a$ and
$b$ (in eV) from the fit ($y=ax+b$) to the theoretical
${\rm{GW}}\Gamma^{1}$@HSE values in Fig. 4 for all XC functionals.

The first and most remarkable finding is the fact that the BPE method yields
overall the best results, despite being the numerically cheapest one. This is
partially attributed to the choice of semiconductors that have quite similar
structures and only a small lattice mismatch. In such situations, the quality
of the description of the bulk electronic structure in the vicinity of the VBM
and CBM becomes important. It is thus not surprising that LMBJ yields the best
result here (mean absolute percentage error (MAPE) of 11%), since the original
MBJ was shown to predict the band gaps with high accuracy Borlido  _et al._
(2019, 2020), closely followed by r2SCAN (MAPE of 14%). rMGGAC and mTASK
perform similarly to PBE when the BPE-based method is used (MAPE of
$\sim$21%).

We also obtain fairly good results from the APM for all the tested meta-GGAs
r2SCAN, rMGGAC, mTASK, and LMBJ, with MAPE slightly above 20%, performing
better than PBE-APM (MAPE of 30%). It should be noted that in presence of
larger lattice mismatch, interface reconstructions, and other inhomogeneities,
the APM is expected to be the most accurate since it is the only one that
takes into account the details of the interface.

Finally, the worst results overall are obtained using the IP-based method.
Only for mTASK, the MAPE is moderate (26 %). r2SCAN yields VBOs with the same
accuracy as PBE (MAPE of $\sim$35%). Interestingly, we obtained the worst
result for rMGGAC with the IP-based method (MAPE of 64 %), even though the
individual IPs were calculated with the best accuracy with this functional.
The systematic underestimation of the IPs in the case of PBE and r2SCAN works
favorably here when compared with rMGGAC.

To further quantify the quality of the chosen meta-GGA functionals for the
calculation of VBOs, we plot the statistical quantities MAPE, root mean square
error (RMSE), mean absolute error (MAE), standard deviation ($\sigma$) in Fig.
5, and the parameters $a$ and $b$ from the fit to the theoretical
${\rm{GW}}\Gamma^{1}$@HSE values in Fig. 4.

Roughly from the radar plots, it can be understood that the smaller the area,
the better the performance of the method. This analysis further confirms our
overall results. The IP-based method is the worst in all cases. On the other
hand, the BPE-based method and the APM performs fairly similarly. Moreover,
r2SCAN, rMGGAC, mTASK, and LMBJ all outperform PBE. However, it should be kept
in mind that we only studied rather ideal interfaces of similar semiconductors
with VBM and CBM with mainly $s$\- and $p$-character. Larger deviations of
these two methods for heterostructures involving highly localized
$d$-electrons close to the Fermi energy and interfaces with larger lattice
mismatch are possible. In particular, it is known that MBJ and LMBJ are
sometimes not very accurate for the band gap of non-magnetic systems with
$d$-electrons (e.g., Cu2OBorlido  _et al._ (2019); Rauch  _et al._ (2020c))
and we expect that this can also influence the VBO.

##  IV Conclusions

Since accurate $\rm GW$ calculations for band offsets using the APM and
sometimes even with the IP- and BPE-based methods might be too expensive, as
an alternative, we show that modern meta-GGA functionals may be the choice for
the calculation of band offsets in heterostructure systems. In summary, we
have shown that the performance of the meta-GGA functionals is quite promising
and reliable in calculating the band offsets. Overall analysis shows that PBE,
r2SCAN, and mTASK provide too high VBM positions (underestimates IPs), while
mostly agreeing well for the CBMs (or EAs), with the exception of mTASK, which
puts the CBMs too high in energy. rMGGAC results agree quite well for VBMs and
CBMs, but no clear trend was observed, as some of the band edges are
overestimated and some are underestimated. For the chosen test set of
semiconductor heterostructures with small lattice mismatch, we observed
improved performance of the meta-GGA functionals in comparison with PBE. For
all meta-GGAs, the BPE-based method yields the smallest error with respect to
theoretical ${\rm{GW}}\Gamma^{1}$@HSE values from Ref. Hinuma  _et al._ ,
2014. The error from the APM is found to slightly higher than that from BPE
method. Finally, the IP-based method proved successful only in the case of
mTASK.

For most of the interfaces, we obtain a good agreement of the VBOs from
r2SCAN, rMGGAC, mTASK, and LMBJ. Furthermore, our comparison of the VBOs
obtained from BPE and APM shows that in most cases all these methods give
quantitatively reasonable results and a possible alternative to the expensive
${\rm{GW}}$ calculation. We hope that the present study will stimulate further
benchmark comparative studies of heterogeneous systems forming a common
interface, as well as for interfaces of 2D systems, a growing topic of present
day electronic structure theory.

## Supplementary Material

See the supplementary material for the complete results reported in this
paper.

###### Acknowledgements.

A.G. would like to thank INSPIRE fellowship, DST, India for financial support.
T.T. and S.B. acknowledge funding from the Volkswagen Stiftung (Momentum)
through the “Dandelion” project. M.K.N acknowledges support from DST-
FIST(SR/FST/PSI-215/2016).

## Data Availability

The data that supports the findings of this study are available within the
article and its supplementary material.

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