# Taylor spectrum for modules over Lie algebras

Boris Bilich  Department of Mathematics, HSE University, Usacheva str. 6,
119048, Moscow, Russian Federation<EMAIL_ADDRESS>
###### Abstract.

In this paper we generalize the notion of the Taylor spectrum to modules over
an arbitrary Lie algebra and study it for finite-dimensional modules. We show
that the spectrum can be described as the set of simple submodules in case of
nilpotent and semisimple Lie algebras. We also show that this result does not
hold for solvable Lie algebras and obtain a precise description of the
spectrum in case of Borel subalgebras of semisimple Lie algebras.

###### Key words and phrases:

Taylor spectrum, Lie algebra cohomology

###### 2020 Mathematics Subject Classification:

Primary 17B56, Secondary 17B30, 47A13

##  1\. Introduction

In 1970, Taylor introduced the notion of the joint spectrum for a finite tuple
of commuting operators $T=(T_{1},...,T_{n})$ on a Banach space $V$ in [16]. It
was defined as the set of points
$\lambda=(\lambda_{1},...,\lambda_{n})\in\mathbb{C}^{n}$ such that a certain
Koszul complex associated to $\lambda$ is not exact. The Taylor spectrum of a
single operator coincides with the classical notion of spectrum. The same
year, Taylor estabilished the existence of the holomorphic functional calculus
in a neighborhood of the spectrum in [17]. In 1972, he proposed a framework
for a noncommutative functional calculus in [18] but the notion of spectrum
for non-commuting tuples of operators was not yet developed.

The first step in this direction was made by Fainshtein in [11]. He
generalized the Taylor spectrum to tuples of operators generating a finite
dimensional nilpotent Lie algebra. In [3], Boasso and Larotonda independently
introduced the Taylor spectrum for representations of solvable Lie algebras.
It is given by

$\sigma^{BL}_{\mathfrak{g}}(V)=\\{\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^{*}\colon\mathrm{Tor}^{U\mathfrak{g}}_{k}(V,\mathbb{C}_{\lambda})\neq
0\text{ for some }k\in\mathbb{Z}_{\geq 0}\\}$ (1)

where $\mathfrak{g}$ is a solvable Lie algebra and $V$ is a right Banach
$\mathfrak{g}$-module.

The spectrum $\sigma^{BL}$ is equivalent to the definition of Fainshtein in
the nilpotent case. Moreover, we can think about commuting tuples
$T=(T_{1},...,T_{n})$ as representations of the $n$-dimensional abelian Lie
algebra and it turns out that the spectrum for abelian Lie algebras is
equivalent to the original Taylor spectrum.

The Taylor spectrum for Lie algebras was actively studied in works of several
authors. A variety of classical theorems were generalized, such as the
projection property and different kinds of the spectral mapping theorem. The
details can be found in the monograph [1] and in a series of works by Dosi
(see for example [7, 8, 9]). In 2010, Dosi estabilished a noncommutative
holomorphic functional calculus in a neighborhood of the spectrum for
supernilpotent Lie algebras of operators [10].

In this paper we introduce the notion of Taylor spectrum for modules over an
arbitrary finite-dimensional Lie algebra and study it for finite-dimensional
modules. Let $\hat{\mathfrak{g}}$ be the set of isomorphism classes of simple
$\mathfrak{g}$-modules. The spectrum of a $\mathfrak{g}$-module is given as a
subset of $\hat{\mathfrak{g}}$ by the non-vanishing property of some homology
groups (Definition 3.1).

The article is organized as follows. In Section 2 we recall neccesary facts
about Lie algebra cohomology. In Section 3 we give the definition of the
spectrum and study its properties. We obtain a complete description for the
spectrum of finite-dimensional modules over a semisimple Lie algebra in
Theorem 3.5. In this case, the spectrum coincides with the set of isomorphism
classes of simple submodules. This result holds even for Banach modules as
shown in Corollary 3.7. We finish the section with Proposition 3.8, which
shows the behaviour of the spectrum for Lie algebra extensions.

Section 4 is devoted to the spectrum for solvable Lie algebras. Although this
case was studied before (see [1, 3]), the authors considered mostly infinite-
dimensional modules. We explore the spectrum for finite-dimensional modules.
It turns out that the spectrum is hard to compute even for the trivial module.
An approximation is given by Theorem 4.1, which happens to be a refolmutaion
in terms of the Taylor spectrum of an earlier result by Millionschikov [15].
We describe completely the spectrum in case of nilpotent Lie algebras (Theorem
4.4) and Borel subalgebras of semisimple Lie algebras (Theorem 4.7).

######  Acknowledgments.

The author is grateful to his scientific advisor Alexei Pirkovskii for his
careful guidance during writing this paper. Thanks should also go to Egor
Alimpiev and Marat Rovinsky for language corrections to an early draft and to
Simon Wadsley and Yves de Cornulier for useful discussions on the Mathoverflow
website (see answers to [2]).

##  2\. Preliminaries

###  2.1. Notation

In this article all algebras, including Lie algebras, are defined over the
field of complex numbers. We use the notation $U\mathfrak{g}$ for the
enveloping algebra of a Lie algebra $\mathfrak{g}$. We denote by
$\operatorname{\mathbf{\mathfrak{g}-mod}}$ and
$\operatorname{\mathbf{mod-\mathfrak{g}}}$ the categories of left and right
$\mathfrak{g}$-modules, respectively. We write $\hat{\mathfrak{g}}$ for the
set of isomorphism classes of simple finite-dimensional $\mathfrak{g}$-modules
and $\mathbb{C}$ for the trivial module. If $S$ is a simple
$\mathfrak{g}$-module we also denote by $S\in\hat{\mathfrak{g}}$ its
isomorphism class. This will not lead to a confusion.

###  2.2. Functors on categories of modules

For the rest of this section we denote by $\mathfrak{g}$ an arbitrary finite-
dimensional Lie algebra. Let $V$ be a $\mathfrak{g}$-module. We define vector
spaces

$V^{\mathfrak{g}}=\\{v\in V\colon g\cdot v=0~{}\forall g\in\mathfrak{g}\\},$

called invariants and

$V_{\mathfrak{g}}=V/\mathfrak{g}V,$

called coinvariants. The assignments $V\mapsto V^{\mathfrak{g}}$ and $V\mapsto
V_{\mathfrak{g}}$ define functors from
$\operatorname{\mathbf{\mathfrak{g}-mod}}$ (or
$\operatorname{\mathbf{mod-\mathfrak{g}}}$) to the category of vector spaces.
These functors are isomorphic to
$\operatorname{Hom}_{U\mathfrak{g}}(\mathbb{C},V)$ and
$\mathbb{C}\otimes_{U\mathfrak{g}}V$ respectively.

We define two functors
$\square^{*}\colon\operatorname{\mathbf{\mathfrak{g}-mod}}^{op}\to\operatorname{\mathbf{\mathfrak{g}-mod}}$
and
$\square^{\circ}\colon\operatorname{\mathbf{\mathfrak{g}-mod}}\to\operatorname{\mathbf{mod-\mathfrak{g}}}$
as follows. The duality functor $\square^{*}$ maps a $\mathfrak{g}$-module V
to its dual vector space with the left $\mathfrak{g}$-action given by

$(g\cdot f)(v)=-f(g\cdot v),~{}\text{ for all }f\in V^{*},~{}v\in
V,~{}g\in\mathfrak{g}.$

The antipode functor $\square^{\circ}$ maps $V$ to itself as a vector space
with the right $\mathfrak{g}$-action given by

$v\cdot g=-g\cdot v,~{}\text{ for all }v\in V,~{}g\in\mathfrak{g}.$

The antipode functor defines an equivalence of the categories
$\operatorname{\mathbf{\mathfrak{g}-mod}}$ and
$\operatorname{\mathbf{mod-\mathfrak{g}}}$. We also denote by $\square^{*}$
and $\square^{\circ}$ the functors on the category of right
$\mathfrak{g}$-modules, defined in the same way. It is easy to see that
$(\square^{\circ})^{\circ}$ is naturally isomorphic to the identity functor.
However, $(\square^{*})^{*}$ is naturally isomorphic to the identity functor
only on the subcategory of finite-dimensional modules.

Another important functor is
$\square\otimes_{\mathbb{C}}\square\colon\operatorname{\mathbf{mod-\mathfrak{g}}}\times\operatorname{\mathbf{\mathfrak{g}-mod}}\to\operatorname{\mathbf{\mathfrak{g}-mod}}$.
If $V\in\operatorname{\mathbf{mod-\mathfrak{g}}}$ and
$W\in\operatorname{\mathbf{\mathfrak{g}-mod}}$, then $V\otimes_{\mathbb{C}}W$
is the tensor product of $V$ and $W$ as vector space with the action of
$\mathfrak{g}$, which is determined by the formula

$g\cdot(v\otimes w)=v\otimes(g\cdot w)-(v\cdot g)\otimes w,\text{ for all
}w\in W,~{}v\in V,~{}g\in\mathfrak{g}.$

For $V,W\in\operatorname{\mathbf{\mathfrak{g}-mod}}$ (resp.
$\operatorname{\mathbf{mod-\mathfrak{g}}}$), we denote by $V\otimes W$ the
left (resp. right) $\mathfrak{g}$-module $V^{\circ}\otimes_{\mathbb{C}}W$
(resp. $V\otimes_{\mathbb{C}}W^{\circ}$).

For $V\in\operatorname{\mathbf{\mathfrak{g}-mod}}$ and and a simple
$\mathfrak{g}$-module $S\in\hat{\mathfrak{g}}$, we write $V_{S}$ for the
$\mathfrak{g}$-module $S\otimes_{\mathbb{C}}V$. If $S$ is one-dimensional,
then it can be described by a character
$\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^{*}$. In this case we
write $V_{\lambda}$ for $V_{S}$. For example, let $\mathbb{C}$ be the trivial
$\mathfrak{g}$-module. With use of the above notation $\mathbb{C}_{\lambda}$
stands for the one-dimensional module with the action given by $g\cdot
s=\lambda(g)s$ for all $s\in\mathbb{C}_{\lambda}$ and $g\in\mathfrak{g}$. We
also use the notation $V_{-S}$ for the module $S^{*}\otimes_{\mathbb{C}}V$. If
$S$ is one-dimensional of character $\lambda$, then $V_{-S}$ is isomorphic to
$V_{-\lambda}$.

###  2.3. Lie algebra (co)homology

In this subsection we briefly recall necessary definitions and properties of
Lie algebra cohomology. For details, we refer the reader to any related
textbook (see for example [19, Chapter 7]).

######  Definition 2.1.

Let $V$ be a left $\mathfrak{g}$-module. For all $k\in\mathbb{Z}_{\geq 0}$,
the $k$-th homology of $\mathfrak{g}$ with coefficients in $V$ is defined as

$H_{k}(\mathfrak{g},V)=\mathrm{Tor}_{k}^{U\mathfrak{g}}(\mathbb{C},V).$

Dually, the $k$-th cohomology is defined as

$H^{k}(\mathfrak{g},V)=\operatorname{Ext}^{k}_{U\mathfrak{g}}(\mathbb{C},V).$

(Co)homologies are functors from the category
$\operatorname{\mathbf{\mathfrak{g}-mod}}$ to the category of vector spaces.
They can be computed with the Chevalley-Eilenberg free resolution of the
trivial $\mathfrak{g}$ module $\mathbb{C}$ (see [19, Definition 7.7.1]). It
has $F_{k}=U\mathfrak{g}\otimes_{\mathbb{C}}\bigwedge^{k}\mathfrak{g}$ in
degree $k$ with the differential given by

$d(u\otimes g_{1}\wedge\cdots\wedge
g_{p})=\sum_{i=1}^{p}(-1)^{i+1}ug_{i}\otimes
g_{1}\wedge\cdots\wedge\hat{g}_{i}\wedge\cdots\wedge g_{p}+$
$+\sum_{i<j}(-1)^{i+j}u\otimes[g_{i},g_{j}]\wedge\cdots\wedge\hat{g}_{i}\cdots\wedge\hat{g}_{j}\cdots\wedge
g_{p},~{}\text{where }u\in U\mathfrak{g},~{}g_{i}\in\mathfrak{g}.$

We need the following fact about homology.

######  Lemma 2.2.

Let $V\in\operatorname{\mathbf{mod-\mathfrak{g}}}$ and
$W\in\operatorname{\mathbf{\mathfrak{g}-mod}}$. Then

$\mathrm{Tor}_{k}^{U\mathfrak{g}}(V,W)\cong\mathrm{Tor}_{k}^{U\mathfrak{g}}(\mathbb{C},V\otimes_{\mathbb{C}}W)=H_{k}(\mathfrak{g},V\otimes_{\mathbb{C}}W),$

for all $k\in\mathbb{Z}_{\geq 0}$.

###### Proof.

Since the functors
$\mathbb{C}\otimes_{U\mathfrak{g}}(\square\otimes_{\mathbb{C}}W)$ and
$\square\otimes_{U\mathfrak{g}}W$ are naturally isomorphic, it suffices to
show that if $P_{\bullet}\to V$ is a flat resolution of $V$, then
$P_{\bullet}\otimes_{\mathbb{C}}W$ is a flat resolution of
$V\otimes_{\mathbb{C}}W$.

By definition, the flatness of $P_{k}$ means the exactness of the functor
$P_{k}\otimes_{U\mathfrak{g}}\square$. We use the associativity of the tensor
product to obtain a natural isomorphism of functors
$(P_{k}\otimes_{\mathbb{C}}W)\otimes_{U\mathfrak{g}}\square$ and
$P_{k}\otimes_{U\mathfrak{g}}(W\otimes_{\mathbb{C}}\square)$. The last functor
is the composition of exact functors, so it is also exact. If follows that
$P_{k}\otimes_{\mathbb{C}}W$ is flat and we obtain the natural isomorphism
$\mathrm{Tor}_{k}^{U\mathfrak{g}}(V,W)\cong\mathrm{Tor}_{k}^{U\mathfrak{g}}(\mathbb{C},V\otimes_{\mathbb{C}}W)$.
∎

A useful variation of the Poincaré duality holds for finite dimensional Lie
algebras. Let $n$ be the dimension of $\mathfrak{g}$. We endow the one-
dimensional vector space $\bigwedge^{n}\mathfrak{g}$ with the structure of a
left $\mathfrak{g}$-module, which extends the adjoint action by the Leibnitz
rule. That is,

$g\cdot g_{1}\wedge\cdots\wedge
g_{n}=\sum_{i=1}^{n}g_{1}\wedge\cdots\wedge[g,g_{i}]\wedge\cdots\wedge g_{n}$

for $g,g_{i}\in\mathfrak{g}$.

######  Proposition 2.3 (Poincaré duality).

For $0\leq k\leq n$ and a left $\mathfrak{g}$-module $V$, there are vector
space isomorphisms

$H^{k}(\mathfrak{g},V^{*})\cong H_{k}(\mathfrak{g},V)^{*}~{}\text{ and
}~{}H^{k}(\mathfrak{g},V)\cong
H_{n-k}(\mathfrak{g},(\bigwedge^{n}\mathfrak{g})^{*}\otimes_{\mathbb{C}}V),$

natural in $V$. Consequently

$H^{k}(\mathfrak{g},V^{*})\cong
H^{n-k}(\mathfrak{g},\bigwedge^{n}\mathfrak{g}\otimes_{\mathbb{C}}V)^{*}.$

###### Proof.

[14, Theorem 6.10]. ∎

##  3\. Taylor spectrum of $\mathfrak{g}$-modules

###  3.1. Definition and first properties

Let $\mathfrak{g}$ be an arbitrary Lie algebra of dimension $n$ and $V$ be a
left $\mathfrak{g}$-module. Recall that $\hat{\mathfrak{g}}$ is the set of
isomorphism classes of simple finite-dimensional $\mathfrak{g}$-modules.

######  Definition 3.1.

The Taylor spectrum of $V$ is the subset of $\hat{\mathfrak{g}}$ defined as

$\sigma_{\mathfrak{g}}(V)=\\{S\in\hat{\mathfrak{g}}\colon\mathrm{Tor}^{U\mathfrak{g}}_{k}(S^{*},V)\neq
0\text{ for some }k\geq 0\\}.$

We will occasionaly use the notation $\sigma(V)$ instead of
$\sigma_{\mathfrak{g}}(V)$ when it is clear what $\mathfrak{g}$ is meant. We
will also write the spectrum for the Taylor spectrum.

Suppose that $\mathfrak{g}$ is a solvable Lie algebra. By Lie’s theorem, every
simple $\mathfrak{g}$-module is one-dimensional. Thus, $\hat{\mathfrak{g}}$
can be identified with the space
$(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^{*}$ of characters. In this case,
the Taylor spectrum can be expressed in terms of the spectrum (1), defined by
Boasso and Larotonda in [3]:

$\sigma_{\mathfrak{g}}(V)=-\sigma^{BL}_{\mathfrak{g}}(V^{\circ}).$

We may think about $\sigma_{\mathfrak{g}}$ as a generilization of
$\sigma^{BL}_{\mathfrak{g}}$.

The spectrum has other useful characterizations.

######  Proposition 3.2.

For an arbitrary $S\in\mathfrak{g}$, the following are equivalent:

  1. (1)

$S\in\sigma(V)$;

  2. (2)

$H_{k}(\mathfrak{g},V_{-S})\neq 0$ for some $k$;

  3. (3)

$H^{k}(\mathfrak{g},\bigwedge^{n}\mathfrak{g}\otimes_{\mathbb{C}}V_{-S})\neq
0$ for some $k$;

  4. (4)

$S^{*}\otimes_{\mathbb{C}}\bigwedge^{n}\mathfrak{g}\in\sigma(V^{*})$.

###### Proof.

  * $1\Leftrightarrow 2$

This follows immediatley from Lemma 2.2.

  * $2\Leftrightarrow 3$

This is just the Poincaré duality (Proposition 2.3).

  * $3\Leftrightarrow 4$

Again, by the Poincaré duality we have

$H^{k}(\mathfrak{g},\bigwedge^{n}\mathfrak{g}\otimes_{\mathbb{C}}V_{-S})^{*}=H^{n-k}(\mathfrak{g},(V_{-S})^{*}).$

Observe that $(V_{-S})^{*}$ is isomorphic to
$S\otimes_{\mathbb{C}}V^{*}\cong\bigwedge^{n}\mathfrak{g}\otimes_{\mathbb{C}}(\bigwedge^{n}\mathfrak{g})^{*}\otimes_{\mathbb{C}}S\otimes
V^{*}\cong\bigwedge^{n}\mathfrak{g}\otimes_{\mathbb{C}}(V^{*})_{-S^{*}\otimes_{\mathbb{C}}\bigwedge^{n}\mathfrak{g}}$.
Using the equivalence of (1) and (3) we obtain the desired result.

∎

######  Lemma 3.3.

Let $V$, $V^{\prime}$, and $V^{\prime\prime}$ be $\mathfrak{g}$-modules such
that there is a short exact sequence

$0\to V^{\prime}\to V\to V^{\prime\prime}\to 0.$

Then

  1. (1)

$\sigma_{\mathfrak{g}}(V)\subset\sigma_{\mathfrak{g}}(V^{\prime})\cup\sigma_{\mathfrak{g}}(V^{\prime\prime})$.

  2. (2)

If the short exact sequence splits, i.e., $V\cong V^{\prime}\oplus
V^{\prime\prime}$ then
$\sigma_{\mathfrak{g}}(V)=\sigma_{\mathfrak{g}}(V^{\prime})\cup\sigma_{\mathfrak{g}}(V^{\prime\prime})$.

###### Proof.

The tensor product over $\mathbb{C}$ is exact. So for any simple
$\mathfrak{g}$-module $S$ the sequnce

$0\to V^{\prime}_{-S}\to V_{-S}\to V^{\prime\prime}_{-S}\to 0$

is exact. It induces the homology long exact sequence

$\xrightarrow{\partial}H_{k}(V^{\prime}_{-S})\to H_{k}(V_{-S})\to
H_{k}(V^{\prime\prime}_{-S})\xrightarrow{\partial}H_{k-1}(V^{\prime}_{-S})\to.$

If $H_{k}(V^{\prime}_{-S})=H_{k}(V^{\prime\prime}_{-S})=0$ for all $k$, then
$H_{k}(V_{-S})$ are also zero. This proves (1).

For (2) observe that the connecting homomorphisms $\partial$ are all zero if
the exact sequence splits. So, $H_{k}(V_{-S})=H_{k}(V^{\prime}_{-S})\oplus
H_{k}(V^{\prime\prime}_{-S})$ and
$\sigma_{\mathfrak{g}}(V)=\sigma_{\mathfrak{g}}(V^{\prime})\cup\sigma_{\mathfrak{g}}(V^{\prime\prime})$.
∎

###  3.2. Semisimple Lie algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra of dimension $n$. Note that
$\bigwedge^{n}\mathfrak{g}$ is isomorphic to the trivial
$\mathfrak{g}$-module. It follows from Proposition 3.2 that the Taylor
spectrum of a $\mathfrak{g}$-module $V$ can be described as the set

$\sigma_{\mathfrak{g}}(V)=\\{S\in\hat{\mathfrak{g}}\colon
H^{k}(\mathfrak{g},V_{-S})\neq 0\text{ for some }k\\}.$

The next lemma helps us to compute the spectrum for finite-dimensional
$\mathfrak{g}$-modules.

######  Lemma 3.4.

Let $S$ be a simple $\mathfrak{g}$-module. Assume that $S\not\cong\mathbb{C}$.
Then

$H^{k}(\mathfrak{g},S)=0\text{ for all }k.$

###### Proof.

[19, Theorem 7.8.9] ∎

If the module is trivial, then, obviously,
$H^{0}(\mathfrak{g},\mathbb{C})\cong\mathbb{C}$. The following theorem gives a
complete description of the spectrum for semisimple Lie algebras.

######  Theorem 3.5.

Let $V$ be a finite-dimensional $\mathfrak{g}$-module. The Taylor spectrum of
$V$ coincides with the set of simple components of $V$:

$\sigma_{\mathfrak{g}}(V)=\\{S\in\hat{\mathfrak{g}}\colon S\text{ is
isomorphic to a submodule of }V\\}.$ (2)

###### Proof.

By Lemma 3.3(2), the spectrum of $V$ is the union of spectra of its simple
submodules, so we may assume that $V$ is simple. Let $S$ be a simple
$\mathfrak{g}$-module. Let $V_{-S}=\oplus_{i=1}^{m}S_{i}$ be a simple
decomposition of $V_{-S}$. Then we have
$H^{k}(\mathfrak{g},V_{-S})\cong\oplus_{i=1}^{m}H^{k}(\mathfrak{g},S_{i})$. By
Lemma 3.4, $H^{k}(\mathfrak{g},V_{-S})$ is nonzero for some $k$ if and only if
$S_{i}\cong\mathbb{C}$ for some $i$. On the other hand, the module $V_{-S}$ is
isomorphic to $\operatorname{Hom}_{\mathbb{C}}(S,V)$ and
$(V_{-S})^{\mathfrak{g}}\cong\operatorname{Hom}_{\mathfrak{g}}(S,V)$, so it is
one-dimensional if $S\cong V$ and zero otherwise. It follows that $S$ lies in
the spectrum of $V$ if and only if it is isomorphic to a submodule of $V$. ∎

In fact, formula (2) holds even for Banach $\mathfrak{g}$-modules. We need the
following fact to prove this.

######  Proposition 3.6.

Any Banach module $V$ over a semisimple Lie algebra $\mathfrak{g}$ is the
union of its finite-dimensional submodules.

###### Proof.

[1, Corollary 5, §30] ∎

In other words, $V$ is the colimit of the filtered diagram $\mathfrak{V}$ of
its finite-dimensional submodules. Recall that the homology and the tensor
product commute with filtered colimits. Thus, we can strengthen Theorem 3.5.

######  Corollary 3.7.

The Taylor spectrum of a Banach $\mathfrak{g}$-module $V$ coincides with the
set of isomorphism classes of simple submodules of $V$.

###### Proof.

The filtered diagram $\mathfrak{V}$ of modules induces the filtered diagram
$H_{*}(\mathfrak{g},\mathfrak{V}_{-S})$ of vector spaces. Moreover,
$H_{*}(\mathfrak{g},V_{-S})$ is the colimit of
$H_{*}(\mathfrak{g},\mathfrak{V}_{-S})$. Since all the maps in $\mathfrak{V}$
are inclusions, so are the maps in $H_{*}(\mathfrak{g},\mathfrak{V}_{-S})$.
Thus, $H_{*}(\mathfrak{g},V_{-S})$ is nonzero if and only if $S\in\sigma(W)$
for some $W\subset V$. This means that $S\in\sigma(V)$ if and only if $S$ is
isomorphic to a submodule of $V$. ∎

###  3.3. Extensions of Lie algebras

Let $\mathfrak{h}$ be an arbitrary Lie algebra and $\lambda$ be a character of
$\mathfrak{h}$. By a one-dimensional extension of $\mathfrak{h}$ we mean an
exact sequence of Lie algebras

$0\to\mathbb{C}_{\lambda}\rightarrow\mathfrak{g}\xrightarrow{\pi}\mathfrak{h}\to
0.$

Here $\mathbb{C}_{\lambda}$ is an ideal of $\mathfrak{g}$ and the commutator
in $\mathfrak{g}$ satisfies $[g,c]=\lambda(\pi(g))c$ for all
$g\in\mathfrak{g}$ and $c\in\mathbb{C}_{\lambda}$. The fundamental result of
the Lie algebra cohomology theory states that isomorphism classes of such
extensions are in one-to-one correspondence with the set
$H^{2}(\mathfrak{h},\mathbb{C}_{\lambda})=\operatorname{Ext}^{2}_{U\mathfrak{h}}(\mathbb{C},\mathbb{C}_{\lambda})$.
We recall how to construct the bijection. Let $\xi\in
Hom_{\mathbb{C}}(\bigwedge^{2}\mathfrak{h},\mathbb{C}_{\lambda})$ be a
cocycle. We define a Lie bracket on the vector space
$\mathfrak{g}=\mathfrak{h}\oplus\mathbb{C}_{\lambda}$ by

$[(h_{1},c_{1}),(h_{2},c_{2})]_{\xi}=([h_{1},h_{2}],\lambda(h_{1})c_{2}-\lambda(h_{2})c_{1}+\xi(h_{1}\wedge
h_{2})),$

for all $h_{i}\in\mathfrak{h},~{}c_{i}\in\mathbb{C}_{\lambda}$. It can be
shown that this formula satisfies the axioms of a Lie bracket and that
cohomologous cocycles induce isomorphic extensions (see [19, Theorem 7.6.3]).

We adapt the techniques for computing cohomology of Lie algebra extensions
from [12, Chapter II.§6]. From now on and till the end of the section we write
$\mathfrak{g}$ for a one-dimensional extension of $\mathfrak{h}$, represented
by a cocycle $\xi\in
Hom_{\mathbb{C}}(\bigwedge^{2}\mathfrak{h},\mathbb{C}_{\lambda})$. We also use
the notation $\mathfrak{c}$ for the ideal
$\mathbb{C}_{\lambda}\subset\mathfrak{g}$. Let $V$ be a $\mathfrak{h}$-module.
We denote by $V^{\pi}$ the $\mathfrak{g}$-module which is obtained from $V$
via the homomorphism $\pi$. Note that if $S\in\hat{\mathfrak{h}}$ is an
ireducible $\mathfrak{h}$-module, then $S^{\pi}$ is also irreducible as a
$\mathfrak{g}$-module. We use this to identify $\hat{\mathfrak{h}}$ with the
subset of $\hat{\mathfrak{g}}$.

Assume that we have an $\mathfrak{h}$-module $V$ and we want to compute the
spectrum $\sigma_{\mathfrak{g}}(V^{\pi})$. The following proposition gives us
some clues.

######  Proposition 3.8.

Let $\mathfrak{g}$ and $V$ be as above. Then

  1. (1)

$\sigma_{\mathfrak{g}}(V^{\pi})\subset\hat{\mathfrak{h}}\subset\hat{\mathfrak{g}}$;

  2. (2)

If $S\in\sigma_{\mathfrak{g}}(V^{\pi})$, then either
$S\in\sigma_{\mathfrak{h}}(V)$ or $S_{-\lambda}\in\sigma_{\mathfrak{h}}(V)$;

  3. (3)

If the extension is central, i.e., $\lambda=0$, then
$\sigma_{\mathfrak{g}}(V^{\pi})=\sigma_{\mathfrak{h}}(V)$.

###### Proof.

Let $S\in\hat{\mathfrak{g}}\setminus\hat{\mathfrak{h}}$. It is easy to verify
that $\mathfrak{c}S=\\{c\cdot s\colon s\in S,~{}c\in\mathfrak{c}\\}$ is a
submodule of $S$. Since $S$ is irreducible, it is either a zero submodule or
the whole $S$. If it is zero, then $S$ is actually an $\mathfrak{h}$-module
and it contradicts our assumption. Thus, $\mathfrak{c}S=S$ and
$S_{\mathfrak{c}}=0$. The same argument can be used to show that
$S^{\mathfrak{c}}=0$.

The Hochschild-Serre spectral sequence
$E^{2}_{p,q}=H_{p}(\mathfrak{h},H_{q}(\mathfrak{c},V_{-S}))$ converges to
$H_{k}(\mathfrak{g},V_{-S})$ (see [19, 7.5]). It suffices to prove that
$H_{q}(\mathfrak{c},V_{-S})=0$ for all $q\geq 0$. The only possible nonzero
homology groups are
$H_{0}(\mathfrak{c},V_{-S})=(V_{-S})_{\mathfrak{c}}\cong(S^{*})_{\mathfrak{c}}.\otimes_{\mathbb{C}}V=0$
and
$H_{1}(\mathfrak{c},V_{-S})=(V_{-S})^{\mathfrak{c}}\cong(S^{*})^{\mathfrak{c}}\otimes_{\mathbb{C}}V=0$
since $V$ is $\mathfrak{c}$-invariant. Therefore $S$ is not in
$\sigma_{\mathfrak{g}}(V^{\pi})$ by Proposition 3.2. We have proved (1).

Suppose now that $S\in\hat{\mathfrak{h}}$. Then the $\mathfrak{h}$-modules
$H_{0}(\mathfrak{c},V_{-S})$ and $H_{1}(\mathfrak{c},V_{-S})$ are isomorphic
to $V_{-S}$ and $V\otimes_{\mathbb{C}}\mathbb{C}_{\lambda}$ respectively.
Since $H_{k}(\mathfrak{c},V_{-S})=0$ for $k>1$, the Hochschild-Serre spectral
sequence stabilizes on the third page. The differntials in $E^{2}$ are
$d_{k}\colon H_{k}(\mathfrak{h},V_{-S})\to
H_{k-2}(\mathfrak{h},V_{-S}\otimes_{\mathbb{C}}\mathbb{C}_{\lambda})$. The
spectral sequence collapses if and only if all the differentials are
isomorphisms of vector spaces.

If $d_{k}$ is not an isomorphism for some $k$, then either
$H_{k-2}(\mathfrak{h},V_{-S}\otimes_{\mathbb{C}}\mathbb{C}_{\lambda})$ or
$H_{k}(\mathfrak{h},V_{-S})$ are nonzero. This means that $S$ or
$S_{-\lambda}$ are in $\sigma_{\mathfrak{h}}(V)$. This proves (2).

Suppose that $\lambda=0$. Since $S\cong S_{-\lambda}$, we have
$\sigma_{\mathfrak{g}}(V^{\pi})\subset\sigma_{\mathfrak{h}}(V)$ by (2). Let
$S$ be in $\sigma_{\mathfrak{h}}(V)$. There is an integer $k\geq 0$ such that
$H^{k}(\mathfrak{h},V_{-S})\neq 0$ and $H^{i}(\mathfrak{h},V_{-S})=0$ for
$i<k$. The differential $d_{k}\colon H_{k}(\mathfrak{h},V_{-S})\to
H_{k-2}(\mathfrak{h},V_{-S})$ is not an isomorphism, so
$H_{k}(\mathfrak{g},V_{-S})\neq 0$. This proves (3). ∎

##  4\. Case of solvable Lie algebras

###  4.1. Trivial module

In this section $\mathfrak{g}$ denotes an arbitrary solvable Lie algebra of
dimension $n$. By Lie’s theorem, every simple $\mathfrak{g}$-module is one-
dimensional, so we identify $\hat{\mathfrak{g}}$ with the space
$(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^{*}$ of characters.

Let $V$ be a finite-dimensional $\mathfrak{g}$-module. By _weights_
$\omega(V)\subset\hat{\mathfrak{g}}$ we mean the set of diagonal matrix
entries in a triangular basis for $V$. It is independent on the choice of such
basis and can be also described as the set of one-dimensional subquotients of
$V$ (Jordan-Hölder theorem). Consider the adjoint representation
$\operatorname{ad}\mathfrak{g}\in\operatorname{\mathbf{\mathfrak{g}-mod}}$.
The weights $\omega(\operatorname{ad}\mathfrak{g})$ are called _Jordan-Hölder
values_ of $\mathfrak{g}$. We denote by $2\rho$ the sum of all Jordan-Hölder
values with multiplicites. If $(g_{1},...,g_{n})$ is a triangular basis of
$\operatorname{ad}\mathfrak{g}$, then we have

$g\cdot g_{1}\wedge\cdots\wedge g_{n}=2\rho(g)g_{1}\wedge\cdots\wedge g_{n}$

for $g\in\mathfrak{g}$. So, $\bigwedge^{n}\mathfrak{g}$ is a one-dimensional
$\mathfrak{g}$-module with character $2\rho$.

######  Theorem 4.1.

Let $\mathfrak{g}$ be a solvable Lie algebra of dimension $n$. If
$\lambda\in\hat{\mathfrak{g}}$ is in the spectrum
$\sigma_{\mathfrak{g}}(\mathbb{C})$, then it is the sum of at most $n$ Jordan-
Hölder values of $\mathfrak{g}$. Moreover, if
$\lambda\in\sigma_{\mathfrak{g}}(\mathbb{C})$, then $2\rho-\lambda$ is also in
$\sigma_{\mathfrak{g}}(\mathbb{C})$.

###### Proof.

The proof is by induction on $n$. The theorem is trivial for $n=1$. Assume
that the statement holds for all solvable Lie algebras of dimension $n-1$.
Choose any one-dimensional ideal $\mathfrak{c}$ in $\mathfrak{g}$ and denote
the corresponding character by $\mu$. By Proposition 3.8.(2) we know that the
spectrum $\sigma_{\mathfrak{g}}(\mathbb{C})$ is the subset of
$\\{0,\mu\\}+\sigma_{\mathfrak{g}/\mathfrak{c}}(\mathbb{C})$. By induction,
any $\nu\in\sigma_{\mathfrak{g}/\mathfrak{c}}(\mathbb{C})$ is the sum of at
most $n-1$ Jordan-Hölder values of $\mathfrak{g}/\mathfrak{c}$, which are also
Jordan-Hölder values of $\mathfrak{g}$. Since $\mu$ is a Jordan-Hölder value
of $\mathfrak{g}$, we conclude that any element of the spectrum
$\sigma_{\mathfrak{g}}(\mathbb{C})$ is the sum of at most $n$ Jordan-Hölder
values of $\mathfrak{g}$.

For the second assertion we use Proposition 3.2. The trivial module is
isomorphic to its dual, so if $\mathbb{C}_{\lambda}$ is in the spectrum of
$\mathbb{C}$, then
$\mathbb{C}_{\lambda}^{*}\otimes_{\mathbb{C}}\bigwedge^{n}\mathfrak{g}\cong\mathbb{C}_{2\rho-\lambda}$
is also in the spectrum. ∎

######  Remark.

An equivalent statement was first obtained by Millionschikov in [15, Theorem
3.1]. Theorem 4.1 is a slight reformulation of that result in terms of the
Taylor spectrum.

Obviously, $0$ is always in the spectrum and, hence, so is $2\rho$. So,
generally, there are often more than one element in the spectrum of
$\mathbb{C}$.

######  Example 4.2.

Let $\mathfrak{g}$ be the 3-dimensional solvable Lie algebra with basis
$e_{1},e_{2},e_{3}$ and the commutator given by $[e_{1},e_{2}]=e_{2}$ and
$[e_{1},e_{3}]=\lambda e_{3}$ for some $\lambda\in\mathbb{C}$. The space of
characters is one-dimensional, so we identify it with $\mathbb{C}$ by the
evaluation at $e_{1}$. Then
$\sigma_{\mathfrak{g}}(\mathbb{C})=\\{0,1,\lambda,1+\lambda\\}$. Indeed, $0$
and $2\rho=1+\lambda$ are always in the spectrum and for $1$ and $\lambda$ the
first homology groups are non-vanishing.

###  4.2. Nilpotent Lie algebras

The following well-known fact in representation theory for nilpotent Lie
algebras is crucial for the computation of the Taylor spectrum.

######  Lemma 4.3.

Let $V$ be a finite-dimensional indecomposable module over nilpotent Lie
algebra $\mathfrak{g}$. Then the set of weights $\omega(V)$ consists of one
element.

###### Proof.

[4, Chapter VII, Proposition 9] ∎

We call modules with only one weight monoweighted. Lemma 4.3 shows us that any
finite-dimensional module over a nilpotent Lie algebra can be decomposed in
the sum of monoweighted submodules. We are ready to formulate the main result.

######  Theorem 4.4.

Let $\mathfrak{g}$ be a nilpotent Lie algebra and $V$ be a finite-dimensional
$\mathfrak{g}$-module. Then the spectrum $\sigma_{\mathfrak{g}}(V)$ coincides
with the set of isomorphism classes of one-dimensional submodules of $V$.

###### Proof.

The spectrum of $V$ is the union of spectra of its indecomposable submodules
by Lemma 3.3(2), so we may assume that $V$ is indecomposable. Then by Lemma
4.3, $V$ is monoweighted. If $\mu$ is the only weight of $V$, then
$\omega(V_{-\mu})=\\{0\\}$ and
$\sigma_{\mathfrak{g}}(V_{-\mu})=\sigma_{\mathfrak{g}}(V)-\mu$, so we may
additionally assume that $\omega(V)=\\{0\\}$. The proof is by induction on the
dimension of $V$.

By Engel’s theorem, all Jordan-Hölder values of $\mathfrak{g}$ are zero. Using
Theorem 4.1 we conclude that the assertion holds for $V\cong\mathbb{C}$. If
$\lambda$ is a character of $\mathfrak{g}$, then
$\sigma_{\mathfrak{g}}(\mathbb{C}_{\lambda})=\sigma_{\mathfrak{g}}(\mathbb{C})+\lambda$.
This means that the assertion holds for one-dimensional modules.

Suppose now that $\dim V=m$ and that the theorem holds for modules of
dimension less than $m$. Choose a one-dimensional trivial submodule of $V$. We
have a short exact sequence of modules

$0\to\mathbb{C}\to V\to V/\mathbb{C}\to 0.$

By Lemma 3.3(1), $\sigma(V)\subset\sigma(\mathbb{C})\cup\sigma(V/\mathbb{C})$.
But $\sigma(\mathbb{C})=\sigma(V/\mathbb{C})=\\{0\\}$ by induction. On the
other hand, $0$ is in the spectrum of $V$ since
$H_{0}(\mathfrak{g},V)=V_{\mathfrak{g}}\neq 0$. This completes the proof. ∎

###  4.3. Borel subalgebras of semisimple Lie algebras

Here we use the terminology from the theory of semisimple Lie algebras. We
refer the reader to [13, Chapter 2] for details.

For the rest of the section $\mathfrak{s}$ is a semisimple Lie algebra and
$\mathfrak{g}$ is a Borel subalgebra of $\mathfrak{s}$. It is known that
$\mathfrak{g}$ is isomorphic to a semidirect product
$\mathfrak{h}\ltimes\mathfrak{n}$, where $\mathfrak{h}$ is a Cartan subalgebra
of $\mathfrak{s}$ and $\mathfrak{n}=[\mathfrak{g},\mathfrak{g}]$. We denote
the root system of $\mathfrak{s}$ relative to $\mathfrak{h}$ by
$\Delta\subset\mathfrak{h}^{*}=\hat{\mathfrak{g}}$. We write
$\Delta^{+}\subset\Delta$ for the subset of positive roots. Elements of
$\Delta^{+}$ are simply nonzero Jordan-Hölder values of $\mathfrak{g}$. We use
the notation $W(\Delta)$ for the Weyl group. For any element $w\in W$ its
length is denoted by $l(w)$.

The aim of the subsection is to describe the spectrum of irreducible
$\mathfrak{s}$-modules considered as $\mathfrak{g}$-modules. First we describe
the cohomology of such modules over $\mathfrak{n}$ by Kostant’s theorem. Then
we use the formula for the cohomology of a semidirect product to compute the
cohomology over $\mathfrak{g}$.

Let $V$ be a $\mathfrak{s}$-module of highest weight $\lambda$. The cohomology
groups $H^{k}(\mathfrak{n},V)$ are naturally $\mathfrak{h}$-modules after a
canonical identification $\mathfrak{h}=\mathfrak{g}/\mathfrak{n}$. Kostant’s
theorem gives an explicit formula for these modules.

######  Lemma 4.5 (Kostant’s theorem).

As an $\mathfrak{h}$-module, $H^{k}(\mathfrak{n},V)$ is the sum of one-
dimensional modules with weights

$w(\lambda+\rho)-\rho,~{}w\in W(\Delta),~{}l(w)=k,$

where $\rho$ is the half-sum of positive roots (or, equivalently, Jordan-
Hölder values).

###### Proof.

[14, Theorem 6.12] ∎

We say that an abelian Lie algebra $\mathfrak{h}$ acts torally on an
$\mathfrak{h}$-module $W$ if $W$ is a direct sum of one-dimensional
submodules. If $\mathfrak{h}$ is a Cartan subalgebra of a semisiple Lie
algebra $\mathfrak{s}$, then $\mathfrak{h}$ acts torally on any finite
dimensional $\mathfrak{s}$-module. If
$\mathfrak{g}=\mathfrak{h}\ltimes\mathfrak{n}$ is the Borel subalgebra of
$\mathfrak{s}$, then $\mathfrak{h}$ also acts torally on $\mathfrak{n}$.

######  Lemma 4.6.

For a finite-dimensional $\mathfrak{s}$-module $V$ we have

$H^{k}(\mathfrak{g},V)=\bigoplus_{p+q=k}\bigwedge^{p}\mathfrak{h}^{*}\otimes_{\mathbb{C}}H^{q}(\mathfrak{n},V)^{\mathfrak{h}}$

###### Proof.

It is a special case of [6, Theorem 4]. ∎

The following theorem gives a complete description of the Taylor spectrum for
Borel subalgebras in the special case of modules restricted from a semisimple
algebra.

######  Theorem 4.7.

Let $\mathfrak{s}$ be a semisimple Lie algebra and $V$ be a simple
$\mathfrak{s}$-module of highest weight $\lambda$. Let $\mathfrak{g}$ be a
Borel subalgebra of $\mathfrak{s}$. Then the Taylor spectrum of $V$ considered
as a $\mathfrak{g}$-module is given by

$\sigma_{\mathfrak{g}}(V)=\\{\rho+w(\lambda+\rho)\colon w\in W(\Delta)\\},$
(3)

where $\rho$ is the half-sum of positive roots and $W(\Delta)$ is the Weyl
group.

###### Proof.

By Proposition 3.2, $\nu\in\sigma_{\mathfrak{g}}(V)$ if and only if
$H^{k}(\mathfrak{g},\bigwedge^{n}\mathfrak{g}\otimes_{\mathbb{C}}V_{-\nu})$ is
nonzero for some $k$. The module $\bigwedge^{n}\mathfrak{g}$ is isomorphic to
$\mathbb{C}_{2\rho}$, so
$\bigwedge^{n}\mathfrak{g}\otimes_{\mathbb{C}}V_{-\nu}\cong V_{2\rho-\nu}$. By
Lemma 4.6, we have

$H^{k}(\mathfrak{g},V_{2\rho-\nu})=\bigoplus_{p+q=k}\bigwedge^{p}\mathfrak{h}^{*}\otimes_{\mathbb{C}}H^{q}(\mathfrak{n},V_{2\rho-\nu})^{\mathfrak{h}}=\bigoplus_{p+q=k}\bigwedge^{p}\mathfrak{h}^{*}\otimes_{\mathbb{C}}(H^{q}(\mathfrak{n},V)_{2\rho-\nu})^{\mathfrak{h}}.$

It follows that $\nu\in\sigma_{\mathfrak{g}}(V)$ if and only if
$(H^{q}(\mathfrak{n},V)_{2\rho-\nu})^{\mathfrak{h}}$ is nonzero for some $q$.
By Lemma 4.5, $H^{q}(\mathfrak{n},V)_{2\rho-\nu}$ is the sum of one-
dimensional modules with weights $w(\lambda+\rho)+\rho-\nu$ with $w\in W$ of
length $l(w)=q$. We conclude that $\nu\in\sigma_{\mathfrak{g}}(V)$ if and only
if $\nu=w(\lambda+\rho)+\rho$ for some $w\in W(\Delta)$. ∎

######  Remark.

A formula equivalent to (3) was obtained for the trivial module in [5, Section
4.4]. Theorem 4.7 generalizes this to arbitrary $\mathfrak{s}$-modules and
gives the description in terms of the Taylor spectrum.

## References

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# The annual energy yield of mono- and bifacial silicon heterojunction solar
modules with high-index dielectric nanodisk arrays as anti-reflective and
light trapping structures

Evgeniia Slivina  Institute of Theoretical Solid State Physics, Karlsruhe
Institute of Technology, Wolfgang-Gaede-Str. 1, 76131, Karlsruhe, Germany
<EMAIL_ADDRESS>Derk Bätzner  Meyer Burger Research AG Rouges-Terres
61, 2068 Hauterive, Switzerland  Raphael Schmager  Institute of Microstructure
Technology, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1,
76344 Eggenstein-Leopoldshafen, Germany  Malte Langenhorst  Institute of
Microstructure Technology, Karlsruhe Institute of Technology, Hermann-von-
Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany  Jonathan Lehr
Light Technology Institute, Karlsruhe Institute of Technology, Engesserstrasse
13, 76131 Karlsruhe, Germany  Ulrich W. Paetzold  Institute of Microstructure
Technology, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1,
76344 Eggenstein-Leopoldshafen, Germany  Uli Lemmer  Light Technology
Institute, Karlsruhe Institute of Technology, Engesserstrasse 13, 76131
Karlsruhe, Germany  Carsten Rockstuhl  Institute of Theoretical Solid State
Physics, Karlsruhe Institute of Technology, Wolfgang-Gaede-Str. 1, 76131,
Karlsruhe, Germany

###### Abstract

While various nanophotonic structures applicable to relatively thin
crystalline sili-con-based solar cells were proposed to ensure effective light
in-coupling and light trapping in the absorber, it is of great importance to
evaluate their performance on the solar module level under realistic
irradiation conditions. Here, we analyze the annual energy yield of relatively
thin heterojunction (HJT) solar module architectures when optimized anti-
reflective and light trapping titanium dioxide (TiO2) nanodisk square arrays
are applied on the front and rear cell interfaces. Our numerical study shows
that upon reducing crystalline silicon (c-Si) wafer thickness, the relative
increase of the annual energy yield can go up to 11.0 %${}_{\text{rel}}$ and
43.0 %${}_{\text{rel}}$ for mono- and bifacial solar modules, respectively,
when compared to the reference modules with flat optimized anti-reflective
coatings of HJT solar cells.

###### keywords:

American Chemical Society, LaTeX

Light Technology Institute, Karlsruhe Institute of Technology, Engesserstrasse
13, 76131 Karlsruhe, Germany Institute of Nanotechnology, Karlsruhe Institute
of Technology, 76344 Eggenstein-Leopoldshafen, Germany IR,NMR,UV

##  1 Introduction

Reducing optical losses is of paramount importance for further developing
photovoltaic (PV) devices. This holds especially true for the market-
dominating single-junction c-Si solar cells, for which optical losses
constitute one of the main limitations to reach their efficiency limit 1, 2.
While a common approach involves employing various uniform anti-reflective
(AR) coatings combined with a chemical texturing of the c-Si wafer, resulting
in the formation of micron-sized pyramidal features 3, in some cases,
alternative approaches to suppress the optical losses are needed. For example,
in industrial solar cells, the micron-sized textures are realized on
relatively thick c-Si wafers, with a current standard of 160 $\mu$m. However,
a transition to the wafer thickness below the standard value and switching to
the foil-like thinner c-Si can allow for lower material consumption. This can
reduce the manufacturing cost and facilitate the acceleration of the expansion
of PV manufacturing 4 to keep up with estimates for global installed PV
capacity 5. However, for such thin devices, the use of micron-sized textures
becomes very challenging due to c-Si wafer handling issues, and novel
approaches have to be identified to reduce the optical losses.

In response to that need, various nanophotonic concepts 6 applicable to c-Si-
based solar cell stacks were proposed to enhance light harvesting through
improved light in-coupling on the front surface. Examples of possible
solutions include plasmonic structures 7, 8, 9, periodically arranged silicon
10, 11 and dielectric 12, 13, 14 nanoscatterers, and biomimetic structures 15.
Moreover, double-sided AR and light trapping (LT) nanostructure gratings
introduced at the front and rear sides of the solar cells were also suggested.
For example, such a concept was investigated for thin-film c-Si 16 and thin-
film hydrogenated nanocrystalline silicon (nc-Si:H) 17 solar cells.

Nevertheless, while a plethora of nanophotonic structures was proposed and
investigated in recent years, it is of paramount importance to analyze their
performance in a full solar module architecture under realistic irradiation
conditions. For example, such analysis was performed for different solar
module architectures where sufficient optical properties were achieved using
strategies involving textured interfaces and/or flat ARCs 18, 19, 20, 21, 22,
23. Since the solar cell is not always illuminated with light at normal
incidence in the realistic scenario, one has to go beyond the analysis of the
ability of the structure to enhance the short-circuit current density under
the standard test conditions and ensure that the proposed AR and/or LT
nanostructure designs are robust concerning irradiation impinging on the solar
module at increasing angles of incidence. Additionally, the absorption of
photons by the solar cell absorber depends on the sun’s position. It is
influenced by the cloud coverage effect on irradiation received by the module
and the module orientation. For solar cells with double-sided photonic
nanostructures, it is also appealing to consider and assess the possible power
output of the solar modules with nanostructured solar cells in the case of a
bifacial module architecture. This module configuration allows for harvesting
the photons that can be absorbed when the sunlight either hits the module from
its back or is reflected from the ground and can be absorbed in the solar
cell.

Here, we study the energy yield of relatively thin wafer-based c-Si solar
modules, for which the solar cell stack is coated with double-sided
nanostructure gratings. Suitably designed square arrays of dielectric high-
index TiO2 nanodisks are used as AR and LT photonic nanostructures. Their
geometrical parameters are subject to optimization. Results are compared to
modules containing more traditional planar thin-film anti-reflective layers.
We consider mono- and bifacial modules and introduce the nanodisk array on the
front surface of the c-Si-based heterojunction (HJT) solar cell for the former
module configuration and both the front and rear side for the latter. The
nanodisk arrays are initially optimized at normal incident light by full-wave
optical simulations concerning the short-circuit current associated with the
reflected portion of the light (details in Sec. Calculation of reflectance,
transmittance, and absorptance 2.4). The optimal design of these nanodisk
arrays depends on whether they are employed on the front side, where they
serve the purpose of suppressing reflectance, or on the rear side, where they
facilitate the light trapping. Therefore, when optimizing the nanodisk arrays,
different designs are found depending on the mechanisms through which they
contribute to enhancing the absorptance in the c-Si wafer. To estimate the
annual energy yield (EY) of the solar module architectures with nanodisk
arrays, the optical response from the optimized nanodisk arrays when placed on
the front and rear solar cell contact layers is simulated using full-wave
optical tools depending on the angle of incidence. This is the primary
information fueled afterward into the EY modeling framework. The annual EY is
assessed for monofacial and bifacial module architectures with and without
TiO2 nanodisk arrays at locations with different climate conditions. The
influence of albedo radiation is also considered, which is especially relevant
to consider in bifacial module configuration 24. Our key contribution is to
show that for the wafer-based c-Si cells with thicknesses for which standard
chemical texturing becomes impractical, and alternative AR and LT structures
are of interest, the nanodisk arrays that we suggest outperform to a
considerable extent the traditional design that relies only on the planar
anti-reflective coatings. In passing, we note that the design of the nanodisk
arrays for front and rear solar cell contacts proposed here is exemplary, and
it is neither restricted to a particular solar cell stack nor the materials
used.

##  2 Module architectures and numerical methods

###  2.1 Investigated module architectures and material properties

The four different architectures studied in this contribution are
schematically depicted in Fig. 1. The annual EY calculations were performed
for two reference monofacial and bifacial module configurations and two
configurations with AR and LT nanodisk arrays introduced on top of the front
and rear ITO contact layers, respectively. The c-Si absorber thickness of all
architectures was varied between 5 and 80 $\mu$m, which is a typical thickness
range between thin PV and conventional wafer-based c-Si PV.

Figure 1: Schematic representation of the four solar module architectures
discussed in this paper: (a) Monofacial standard reference module with
optimized transparent conductive ITO layer on the front, serving as both anti-
reflective coating and front contact, and silver back contact. Front ITO layer
is preceded with window glass and encapsulation (EVA) layers covered with an
anti-reflective coating. For all considered solar module configurations, the
window layers are identical. (b) Monofacial module with optimized anti-
reflective nanodisk array on top of the front ITO contact, and silver back
contact. (c) Bifacial standard reference module with front and rear ITO
contacts with symmetric window layers on both sides of the solar module. (d)
Bifacial module with optimized anti-reflective and light trapping nanodisk
arrays on top of the front and rear ITO contacts with symmetric window layers
on both sides of the solar module, respectively.

In the case of flat reference c-Si HJT solar cell stacks, the front
hydrogenated amorphous silicon (a-Si:H) (passivation intrinsic and n+ doped,
the thickness can be found in 25) and conducting ITO (75 nm) layers were
considered. For the rear side of the HJT solar cell stack, the a-Si:H
(passivation intrinsic and p+ doped) layer was slightly thicker than the
a-Si:H layer on the front, while the ITO layer was thinner than its
counterpart on the front side.

The configurations containing optimized AR nanodisk array had a reduced front
ITO thickness of 10 nm with TiO2 nanodisks arranged in a square lattice of 320
nm pitch, with individual nanodisk having a radius of 125 nm and a height of
90 nm. The bifacial module configuration with added optimized LT TiO2 nanodisk
square array (individual nanodisk with a radius of 215 nm and height of 395
nm, 565 nm pitch) had the same rear ITO thickness as the reference
architectures. These values for the geometrical parameters are the results of
an optimization of the AR and LT nanodisk arrays discussed in Sec. Calculation
of reflectance, transmittance, and absorptance 2.4.

In the case of the monofacial modules, HJT c-Si solar cell had 300 nm thick
silver back contact while the front side contacting metallic grid and its
effect on the optical performance of the module is neglected. The window
module layers comprise encapsulating EVA (400 $\mu$m) and glass (4 mm) with
thin-film anti-reflective MgF2 coating (130 nm). In the case of the bifacial
architecture, the rear side of the solar cell typically has the same
contacting scheme as the front side (no conformal metallic layer). The modules
in the bifacial configuration were considered to have the same window layers
as the ones introduced on the front side. In such a configuration, the module
can absorb the light that is incident on its front and from its rear side.
Additionally, one can harvest albedo radiation since the transmitted portion
of the light is reflected from the ground, and thus, can be reabsorbed in the
silicon, significantly boosting the annual EY. When a bifacial solar module is
tilted, albedo radiation can also impact the annual EY due to the light
reflected from the ground and incident of the front of the module. We note
that when a monofacial solar module is tilted, albedo radiation that is
incident on the front of the module can also be considered.

Refractive indices of c-Si, TiO2, ITO, and Ag used in the calculations were
taken from literature 26, 27, 28, 29. Refractive index data for the front and
rear composite (passivation intrinsic and doped) a-Si:H layers were obtained
using ellipsometry, and corresponding $n$ and $k$ values provided by Meyer
Burger Research AG are plotted in Fig. 2. For the window layers, a non-
absorbing optically thick glass layer was considered to have a nondispersive
refractive index of $n=1.5$, and EVA and MgF2 refractive index data was taken
from 30 and 31, respectively.

Figure 2: Refractive index $n$ and extinction coefficient $k$ of a-Si:H
composite layers. The measured data was fitted to a Tauc-Lorentz model.

###  2.2 Simulation framework

The numerical simulations of the EVA-cell interface for both AR and LT TiO2
nanodisk arrays were performed using the finite element method (FEM) with
commercial software JCMsuite 32. The annual EY was calculated using a
comprehensive modeling framework enabling the quick simulation of various and
sophisticated PV architectures under realistic irradiation conditions
discussed in detail in 33, 34.

###  2.3 Electrical parameters

The electrical parameters corresponding to a typical c-Si HJT solar cell used
in the annual EY calculation are summarized in Table 1. The shadowing by
electrical connections for all considered architectures is disregarded.

Table 1: Electrical parameters of the solar cell Shunt resistance, $R_{\text{sh}}$ [$\Omega\cdot\text{cm}^{2}$] | 5000
---|---
Series resistance, $R_{\text{s}}$ [$\Omega\cdot\text{cm}^{2}$] | 0.7
Reverse-blocking current, $J_{\text{0}}$ [$\text{A}/\text{cm}^{2}$] | 2$\cdot 10^{-13}$
Ideality factor, $n$ | 1.1
Temperature coefficient of $J_{\text{SC}}$, $t_{J_{\text{SC}}}$ [$\%/\text{K}$] | 0.05
Temperature coefficient of $V_{\text{OC}}$, $t_{V_{\text{OC}}}$ [$\%/\text{K}$] | -0.25

A device characterized by these properties would have a $J_{\text{SC}}$ around
38.3 mA/cm2. This yields $V_{\text{OC}}=0.734$ V at temperature
$\text{T}=25^{\circ}$C from the following equation:

$V_{\text{OC}}=nV_{\text{th}}\text{ln}\left(\frac{J_{\text{SC}}}{J_{\text{0}}}+1\right),$
(1)

where the thermal voltage $V_{\text{th}}=kT/q=0.0257$ V, and the values of
ideality factor and reverse-blocking current can be found in Table 1.

###  2.4 Calculation of reflectance, transmittance, and absorptance

At the EVA-cell interfaces with introduced AR and LT nanodisk arrays,
reflectance and transmittance into all scattering directions at each
wavelength and incidence polar and azimuth angles is calculated as the ratio
of the scattered reflected or transmitted power to the power of the incident
field. At the considered interfaces, both c-Si and EVA are assumed to be semi-
infinite. For a given azimuth angle $\phi_{\text{in}}$, reflectance and
transimittance values form matrices of the size ($N_{\theta_{\text{in}}}$,
$N_{\theta_{\text{r,t}}}$, $N_{\lambda}$), where the entries correspond to all
polar angles of incidence, scattering angles, and wavelengths, respectively.
The polar angle $\theta_{\text{in}}$ is varied from $None$ to $None$ with
$None$ step, and the results are then interpolated at intervals of $None$. In
case of azimuth angle $\phi_{\text{in}}$, the symmetry of the nanodisk coating
is exploited, and only calculations for angles between $None$ and $None$ with
$None$ step are performed. The calculated matrices for different
$\phi_{\text{in}}$ values are subsequently averaged. Total reflectance and
transmittance for a certain wavelength and incident polar and azimuth angles
are calculated according to:

$R=\frac{\sum\limits_{\bm{k}_{\text{r}}}|\tilde{\bm{E}}(k_{\text{r,x}},k_{\text{r,y}})|^{2}\cdot\text{cos}(\theta_{\text{r}})}{|\bm{E}_{0}|^{2}\text{cos}(\theta_{\text{in}})},$
(2)
$T=\frac{\sum\limits_{\bm{k}_{\text{t}}}n_{\text{out}}|\tilde{\bm{E}}(k_{\text{t,x}},k_{\text{t,y}})|^{2}\cdot\text{cos}(\theta_{\text{t}})}{n_{\text{in}}|\bm{E}_{0}|^{2}\text{cos}(\theta_{\text{in}})},$
(3)

where $\bm{k_{\text{r,t}}}$ are the wave vectors of reflected and transmitted
fields, $\theta_{\text{r,t}}=\Re{(\pm k_{\text{z}}/k_{\text{r,t}})}$ are the
scattering angles, $\bm{E}_{0}$ is the amplitude of an incident plane wave
with a mixed TE-TM polarization, and $n_{\text{in}}$ and $n_{\text{out}}$ are
the refractive indices of the media where the incident and scattered waves
propagate, respectively. We use the angular spectrum representation of the
fields, and $\tilde{\bm{E}}(k_{\text{\\{r,t\\},x}},k_{\text{\\{r,t\\},y}})$ in
equations 2 and 3 is calculated by means of the Fourier transform of the
electric fields in real space obtained from full-wave simulations. Absorptance
in each of the thin film layers and nanodisk array was calculated by
integrating the divergence of the Poynting vector across the absorber volume,
thus yielding absorbed power which is normalized to the power of the incident
plane wave. Similarly to reflectance and transmittance, for a given azimuth
angle, absorptance values form a matrix of the size ($N_{m}$,
$N_{\theta_{\text{in}}}$, $N_{\lambda}$), where index $m$ runs over all
absorbing layers in the front or rear solar cell stack. The angular dependent
simulations were performed for optimized AR and LT nanodisk arrays. The
optical performance of the nanodisk arrays was optimized with respect to the
short-circuit current associated with reflectance at a normal light incidence,
which is calculated using the following equation:

$J_{\text{SC,R}}=\int_{\lambda_{1}}^{\lambda_{2}}e\frac{SI_{\text{AM1.5}}(\lambda)R(\lambda)}{E_{\text{ph}}}d\lambda,$
(4)

where $e$ is the electron charge, $E_{\text{ph}}=hc/\lambda$ is the energy of
a photon, and $SI_{\text{AM1.5}}(\lambda)$ is the spectral irradiance. For
this calculation, air mass 1.5 global (AM1.5G) tilted irradiance raw data was
taken from 35, and the total reflectance $R(\lambda)$ was interpolated
accordingly. The short-circuit current due to reflectance $J_{\text{SC,R}}$
was minimized for front nanodisk array and maximized for the rear LT nanodisk
array.

Within the EY modeling framework, where the optical response of the entire
architecture is computed, the light propagation in multi-layer thin-film
stacks is treated coherently, for which the transfer matrix method is
employed. When AR and LT nanodisk arrays are considered instead of those thin-
film layers, the corresponding output matrices for reflectance, transmittance,
and absorptance are integrated into the modeling framework. For thicker
layers, such as the c-Si substrate of the cell and window layers of the
module, the assumption of coherence breaks down. The Beer-Lambert law can
describe the absorption of the light in those layers:

$I(z,\lambda)=I_{\text{0}}\cdot\text{e}^{-\alpha(\lambda)z},$ (5)

where $I_{\text{0}}$ is the initial intensity, $\alpha$ is the absorption
coefficient of the considered medium, and $z$ is the distance traveled in it.

##  3 Results and discussion

###  3.1 Energy yield of solar modules

We analyzed the annual EY of the four solar module architectures introduced in
Fig. 1 for three cities in the United States of America located in different
climate zones 36. Two of the chosen cities, Anchorage, AK and Honolulu, HI,
have highly contrasting irradiation conditions. The former is a cold and
cloudy region (Boreal climate) and the latter a hot and sunny one (Tropical
climate). The additionally chosen Kansas City, MO, has a temperate climate
that receives an annual solar irradiance between Anchorage and Honolulu. By
covering different climate zones, we aimed to highlight the robustness of the
nanodisk arrays performance and their ability to improve the annual EY for all
types of irradiation conditions, albeit with small differences that most
likely originate from the spectral features of the nanodisk arrays. The solar
modules were considered to face south, and the tilt angles $\theta_{\text{m}}$
were optimized for each location. This resulted in $\theta_{\text{m}}$ values
to be $None$ for Anchorage, $None$ for Kansas City, and $None$ for Honolulu,
respectively.

Figure 3 demonstrates the relative improvement of the annual EY when the
nanodisk arrays are used for light management instead of the optimized planar
layers. The increase of the annual EY is shown as a function of the c-Si
absorber thickness. We considered the case when the AR array is introduced on
the front surface of the c-Si HJT cell in the case of the monofacial module
architecture, and both AR and LT nanodisk arrays are applied to the front and
rear surfaces of the cell in the case of the bifacial module architecture. For
this calculation, no albedo was considered. The relative increase of the
annual EY reached up to 11.0 %${}_{\text{rel}}$ and 43.0 %${}_{\text{rel}}$ at
the minimal wafer thickness of 5 $\mu$m for monofacial and bifacial
architectures with nanodisks, respectively.

Figure 3: Relative increase of the annual EY for three locations in case of
(a) the monofacial solar module (comparing (b) to (a) from Fig. 1) and (b) the
bifacial solar module (comparing (d) to (c) from Fig. 1) with varying
thickness of the c-Si absorber.

As expected, for the monofacial case, the module with an AR nanodisk array
(Fig. 1(b)) outperforms the standard flat architecture with an optimized ITO
coating (Fig. 1(a)), with this effect becoming even more apparent when
reducing the c-Si absorber thickness. However, the front side AR array with a
relatively small size of the individual disks does not have strong LT
properties. Its contribution to light harvesting stems mainly from its
broadband AR performance. When instead of a full metallic contact, the window
encapsulation and glass layers are introduced, to take advantage of the solar
irradiation which can hit the module on its back, one has to take care of an
appropriate light trapping structure, which would also act as a decent AR
coating. For this purpose, an LT nanodisk array was designed (Fig. 1(d)). With
the individual disk parameters being significantly larger when compared to the
nanodisk array used as the AR structure at the front interface, the light
which is not absorbed in the silicon and reaches the rear interface of the
cell is effectively scattered in multiple directions, thus improving the LT
properties of the cell. From Fig. 3 (b) it can be seen that it translates into
an even more significant increase of the annual EY than for monofacial module
architecture when comparing to a reference bifacial module architecture (Fig.
1(c)).

If additionally, one considers albedo radiation, the advantage of the bifacial
module architecture with AR and LT nanodisk arrays becomes even more apparent.
The influence of albedo is shown in Fig. 4. Here, the annual EY of the four
module architectures with a selected median c-Si absorber thickness of 40
$\mu$m is shown for the sandstone and grass ground surface compared to EY of
the modules without albedo.

Figure 4: Energy yield of the four module architectures for different
locations when albedo irradiation is taken into account. For all solar module
architectures, the c-Si absorber thickness was 40 $\mu$m.

Since the interfaces of the standard reference modules are flat, without
albedo, the monofacial standard reference module outperforms the bifacial
reference. While for the former, the silver layer reflects the light reaching
it back into the cell, for the latter, a lot of light is lost due to
transmittance when no LT structure is introduced. This difference is bigger
with a smaller module tilt angle since less irradiation can hit the module
from the back. However, as soon as albedo radiation is considered, the
bifacial standard reference outperforms its monofacial counterpart. While for
monofacial architecture albedo radiation does not make a significant
difference, one can see a robust improvement in the annual EY for the bifacial
case. The increase of the annual EY for the monofacial architecture varies
depending on the module tilt and is stronger for locations with a greater
$\theta_{\text{m}}$ (Anchorage). The relative improvement reaches up to around
0.8 %${}_{\text{rel}}$ with sandstone and 3.0 %${}_{\text{rel}}$ with grass as
a ground surface for both the standard reference monofacial module and
monofacial module with AR nanodisk array with insignificant difference between
them. In the case of the bifacial module architecture, a stronger increase of
the annual EY is expected for the sunnier locations (Honolulu). It reaches up
to 8.2 %${}_{\text{rel}}$ with sandstone, and 28.9 %${}_{\text{rel}}$ with
grass ground surface in case of a standard reference bifacial module. For a
bifacial module with AR and LT structures, the relative increase is 8.3
%${}_{\text{rel}}$ and 30.1 %${}_{\text{rel}}$ with sandstone and grass ground
surface, respectively.

###  3.2 Optical performance of solar modules

To better understand the mechanism behind the annual EY improvement when the
nanodisk arrays are introduced on top of the front and back contacts of a
solar cell, one can look at the optical properties of the cell interfaces with
the optimized nanodisk arrays. Figure 5 shows reflectance at normal incidence
of the EVA-cell interfaces for the front and rear HJT solar cell contacts with
AR and LT nanodisk arrays. For both nanodisk arrays and illumination
directions, reflectances of the corresponding reference flat interfaces are
plotted for comparison. The graphs (a) and (b) in Fig. 5 correspond to the
optical response of the front EVA-cell interface. In this case, the objective
was to minimize short-circuit current density corresponding to reflectance
$J_{\text{SC,R}}$ introduced in Eq. 4, which resulted in an AR nanodisk array
with reflectance shown in Fig. 5(a).

Figure 5: Reflectance at normal incidence for the front and rear EVA-cell
interfaces in case of both AR and LT nanodisk arrays. Inset sketches depict
the corresponding interface and illumination direction. Interfaces are the
same for the pairs of graphs (a)-(b) and (c)-(d), respectively.

This design was based on the previous work on helicity preserving TiO2
nanodisk array for the front interface of a c-Si HJT solar cell, where
efficient and broadband backscattering suppression was achieved due to the
ability of the system to suppress cross-talk between opposite handednesses of
the electromagnetic field upon light-matter interaction 25. The requirement
for a system to be helicity preserving is to possess a high enough degree of
rotational symmetry ($n\geq 3$) along the illumination direction. For normal
light incidence, for which the optimization of AR nanodisk array was
performed, the illumination direction is along the symmetry axis of an
individual nanodisk, which essentially means that $n\rightarrow\infty$ in this
case. The resulting reflectance of the AR nanodisk array is lower than the one
of the reference optimized flat AR coating and exceeds it slightly only in the
wavelength region around $\lambda=600$ nm, for which standard AR coating of
the solar cell is typically optimized. However, as shown in Fig. 5(b), the LT
properties of this nanodisk array are not as good as its AR properties. When
the light is impinging from the c-Si absorber, only the long-wavelength
response is relevant since the short-wavelength photons are absorbed before
reaching this interface. This nanodisk coating transmits the light reflected
from the rear of the stack particularly strongly at longer wavelengths and is
not possessing better LT properties than the standard flat reference. This
optical response confirms that the front nanodisk array contributes to the
light harvesting of the considered solar module architectures mainly by its AR
properties.

The graphs (c) and (d) in Fig. 5 demonstrate the optical response of the
optimized LT nanodisk array. Here, the structure parameters strongly differ
from those of the helicity preserving AR nanodisk array. The larger and more
sparsely spaced LT nanodisks allow for improved harvesting of the long-
wavelength photons reaching the rear of the solar cell. The impinging light is
effectively scattered into multiple directions since many diffraction orders
are allowed. Taking into account the absorption depth of c-Si, for the minimal
considered absorber thickness of 5 $\mu$m, the photons that can reach the rear
contact of the solar cell have wavelengths $\lambda\geq 690$ nm. The optimized
reflectance of the LT nanodisk array for the spectral region of interest is
shown in Fig. 5(c). Its LT performance exceeds that of the flat reference
solar cell stack at all wavelengths. Nevertheless, as can be seen from Fig.
5(d), though this nanodisk array outperforms its flat reference counterpart in
terms of AR properties in the longer wavelength range, overall, its main
contribution to the improved absorption in c-Si, and, consequently, the annual
EY of the solar module, is due to its superior LT properties. This way, in the
bifacial module case, when nanodisk arrays with decoupled AR and LT properties
are present on both sides of the solar cell stack, one can achieve a solid
overall solar module performance boost.

Another critical aspect of the solar module’s optical performance is parasitic
absorption. Here, all discussed results are for a selected median c-Si
absorber thickness of 40 $\mu$m and at normal light incidence. Figures 6 and 7
show absorptance in all layers of the monofacial and bifacial module stacks,
respectively, except for the glass layer, which was assumed to be non-
absorbing. Additionally, the absorptance of the rear a-Si:H layer in
monofacial module configuration and of rear and front a-Si:H layers in the
bifacial module configuration for forward and backward illumination direction,
respectively, was negligible, and, thus, it is not shown. Moreover, the LT
TiO2 nanodisk array, which is considered for the bifacial module
configuration, does not introduce any parasitic absorption since the light
absorbed in these nanodisks has short wavelengths and does not reach the rear
cell interface while it is absorbed in c-Si. The short-circuit current
densities indicated on the top of all graphs in Fig. 6 and Fig. 7 were
calculated assuming AM1.5G spectrum using the modified Eq. 4 with absorptance
of c-Si instead of reflectance.

Figure 6: Absorptance in the different layers of (a) the monofacial standrad
reference module and (b) the monofacial module with AR nanodisk array on top
of the front ITO contact at normal light incidence. For both solar module
architectures, the c-Si absorber thickness was 40 $\mu$m. Figure 7:
Absorptance in the different layers of the bifacial solar modules at normal
light incidence: (a) in case of forward illumination direction for the
bifacial standard reference module, (b) in case of forward illumination
direction for the bifacial module with AR and LT nanodisk coatings, (c) in
case of backward illumination direction for the bifacial standard reference
module, (d) in case of backward illumination direction for the bifacial module
with AR and LT nanodisk coatings. For both solar module architectures, the
c-Si absorber thickness was 40 $\mu$m.

In the case of the monofacial standard reference module (Fig. 6(a)), the front
transparent conductive ITO layer serves as an AR coating but also introduces
some parasitic absorption. However, when the AR nanodisk array is introduced
(Fig. 6(b)), ITO thickness for the optimized front solar cell contact is
reduced considerably (from 75 to 10 nm), and parasitic absorption in this
layer is significantly reduced. The nanodisks themselves have a tiny
contribution to the parasitic absorption, considering that EVA absorbs all the
light impinging on the solar module at the wavelengths between 300 and 360 nm
and TiO2 absorbs only at wavelengths shorter than $\lambda=380$ nm. On the
other hand, while the AR nanodisk array increases the absorptance in c-Si
thanks to a better in-coupling of the incident light, it also increases
parasitic absorption in the front a-Si:H layer. However, it should be noted
that in 37 it was shown that the carriers which are absorbed in an intrinsic
a-Si:H layer can still contribute to the short-circuit current, and thus the
parasitic absorption loss in this layer represents an upper bound and can have
a less of an impact in reality. Thus, introducing the TiO2 AR nanodisk array
leads to a broadband enhancement of absorptance in the silicon absorber layer.

In the case of the bifacial standard reference module and forward illumination
direction (Fig. 7(a)), the parasitic absorption for the wavelengths below
$\lambda=700$ nm is similar to the one of the monofacial module reference.
However, the parasitic absorption is slightly lower in the long-wavelength
range since more light is transmitted through the glass and encapsulation
window layers. When AR and LT nanodisk arrays are introduced (Fig. 7(b)), the
front nanodisks improve absorptance in silicon the same way as in the case of
the monofacial module. The rear nanodisk array additionally enhances
aborptance at longer wavelengths. However, due to the strong scattering
enabled by LT nanodisks, parasitic absorption is also increased for longer
wavelengths. When the light is impinging on the rear side of the bifacial
solar module (Fig. 7(d)), the absorptance in c-Si is also improved in the case
when AR and LT arrays are introduced in comparison to the reference flat
module (Fig. 7(c)), even though LT nanodisks are not optimal in terms of their
AR properties and introduce dips in silicon absorptance due to the sharp
spectral features which can be seen in Fig. 5(d).

##  4 Conclusions

We have numerically studied the annual energy yield (EY) under realistic
irradiation conditions for monofacial and bifacial crystalline silicon (c-Si)
heterojunction (HJT) solar module architectures with AR and light trapping
(LT) titanium dioxide (TiO2) nanodisk square arrays introduced on top of the
front and rear ITO layers and compared their power outputs with the ones of
the corresponding standard reference flat solar modules. We have shown that
while reducing the silicon absorber thickness, the relative increase of the
annual EY is reaching up to 11.0 %${}_{\text{rel}}$ and 43.0
%${}_{\text{rel}}$ for monofacial and bifacial modules with nanodisk coatings,
respectively. This improvement is comparable for the locations with different
climate conditions. Moreover, in the case of bifacial module architecture,
taking into account the albedo radiation produces an additional boost of the
module performance.

The designed dielectric nanodisk arrays for the front and rear contacts of
c-Si HJT solar cell have both a significant impact on the light absorption in
the c-Si wafer. At the same time, their AR and LT properties are decoupled.
The front AR nanodisk array has a relatively small individual disk size and
lattice constant, and its broadband backscattering suppression is related to
the helicity preservation condition. In contrast, the rear LT nanodisk array
has larger features and array pitch and allows for efficient scattering into
multiple scattering directions. Furthermore, these AR and LT nanodisk square
array designs are not restricted to a specific material or a particular
photovoltaic solar cell stack and, thus, can be investigated for different
solar module configurations.

The authors acknowledge support by the state of Baden-Württemberg through
bwHPC and the German Research Foundation (DFG) through grant no INST 40/575-1
FUGG (JUSTUS 2 cluster). The authors are also grateful to JCMwave for their
free provision of the FEM Maxwell solver JCMsuite.

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