Jie Wu
email: jie.wu@math.cnrs.fr

ELLIOTT-HALBERSTAM CONJECTURE AND VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES

Keywords: Mathematics Subject Classification. 11N05, 11N25, 11N36 Shifted prime, Friable integer, Sieve

Denote by P the set of all primes and by P + (n) the largest prime factor of integer n 1 with the convention P + (1) = 1. For each η > 1, let c = c(η) > 1 be some constant depending on η and P a,c,η := {p ∈ P : p = P + (q -a) for some prime q with p η < q c(η)p η }.

In this paper, under the Elliott-Halberstam conjecture we prove, for y → ∞,

according to values of η. These complement for some results of Banks-Shparlinski [1], of Wu [12] and of Chen-Wu [2].

Introduction

Denote by P the set of all prime numbers and by P + (n) the largest prime factor of the positive integer n 1 with the convention P + (1) = 1. Banks & Shparlinski [START_REF] Banks | On values taken by the largest prime factor of shifted primes[END_REF] proposed to estimate the number of primes p that occur as the largest prime factor of a shifted prime q -a when q ∈ P lies in a certain interval determined by p. This question has applications in theoretical computer science and has been considered by Vishnoi [START_REF] Vishnoi | Theoretical aspects of randomization in computation[END_REF].

Let Z * be the set of non-zero integers. For a ∈ Z * , c > 1 and η > 0, we put P a,c,η := {r ∈ P : r = P + (q -a) for some prime q with r η < q cr η } and π a,c,η (y) := |{r y : r ∈ P a,c,η }|, π(y) := |{r y : r ∈ P}|.

Banks & Shparlinski [1, Theorem 1.1] proved that for each η ∈ ( 32 17 , 1 + 3 4 √ 2), there exists a constant c = c(η) > 1 such that the asymptotic formula (1.1) π a,c,η (y) = π(y) + O A,a,c,η y (log y) A (y → ∞)

holds for every fixed non-zero integer a ∈ Z * and any constant A > 1. Moreover for 2 η < 1 + 3 4 √ 2 ≈ 2.0606, this estimate holds for any constant c > 1. Very recently, Wu [START_REF] Wu | On values taken by the largest prime factor of shifted primes[END_REF] extended Banks-Shparlinski's interval ( 32 17 , 1 + 3 4 √ 2) to ( 32 17 , η 0 ), where η 0 ≈ 2.142 is the unique solution of the equation η -1 -4η log(η -1) = 0 in (1, ∞). Banks & Shparlinski [1, page 144] also remarked that the asymptotic formula (1.1) holds for η ∈ (1, 32 17 ] if we assume the Elliott-Halberstam conjecture (see EH prime [ε] below). Subsequently, Chen & Wu [START_REF] Chen | On values taken by the largest prime factor of shifted primes (II)[END_REF] further extended the domain of η at the price proportion positive instead of density 1. More precisely, they proved that

(1.2) π a,c,η (y) log 4 √ 2 η -1 η 1 -4 log(η -1) -δ log(η -1) 8 c -1 π(y),
where δ = δ(c, η) is sufficiently small positive number. Clearly, (1.2) implies π a,c,η (y) π(y)

provided η < 1 + 4 √ e.
This is complement for the results of Banks-Shparlinski and of Wu mentioned above. It seems rather natural to pose the following question.

Question 1. Is the asymptotic formula (1.1) true for all η > 1 ?

In this paper, we shall try to answer this question under the well-known Elliott-Halberstam conjecture. Firstly we state two versions of this conjecture for prime numbers.

Conjecture 1 (Elliott-Halberstam). Let a ∈ Z * and ε ∈ (0, 1) be fixed constants.

(i) For any A > 0, the inequality

q x 1-ε (a,q)=1 p x p≡a(mod q) 1 - π(x) ϕ(q) A,a,ε x (log x) A (EH prime [ε])
holds uniformly for all x 3, where the letter p always denotes prime numbers, ϕ(q) is the Euler function and the implied constant depends on A, a and ε.

(ii) Let κ 1 (m) and κ 2 (m) be the characteristic functions of the odd integers and of even integers, respectively. Then for any A > 0, we have

q x 1-ε (a,q)=1 mp x mp≡a(mod q) κ i (m) - 1 ϕ(q) mp x (mp,q)=1 κ i (m) A,a,ε x (log x) A (EH * prime [ε])
uniformly for all x 3, where the letter p always denotes prime numbers and the implied constant depends on A and a.

Remark 1. According to the classical Bombieri-Vinogradov theorem and Proposition 2.2 of Wu [START_REF] Wu | On values taken by the largest prime factor of shifted primes[END_REF], the Elliott-Halberstam conjectures EH prime [ε] and EH * prime [ε] hold for all ε ∈ ( 1 2 , 1). Secondly we also need a version of this conjecture for friable numbers.

Conjecture 2 (Elliott-Halberstam). Let a ∈ Z * and ε ∈ (0, 1) be fixed constants. For any A > 0, we have

q x 1-ε (a,q)=1 n x n≡a(mod q), P + (n) y 1 - 1 ϕ(q) n x (n,q)=1, P + (n) y 1 A,a,ε x (log x) A (EH friable [ε]) uniformly in x y 2.
Remark 2. According to Wolke's work [START_REF] Wolke | Über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen[END_REF] (see also [START_REF] Fouvry | Entiers sans grand facteur premier en progressions arithmétiques[END_REF]Theorem 6]), EH friable [ε] holds unconditionally for all ε ∈ ( 1 2 , 1).

Our results are as follows.

Theorem 1. Let a ∈ Z * and c > 1 be fixed constants.

(i) Let η ∈ (1, 32 17 ] and assume the Elliott-Halberstam conjecture EH prime [ε] with ε = 1 -1/η > 0. Then for any A > 1 we have

(1.3) π a,c,η (y) = π(y) + O A,a,c,η y (log y) A , as y → ∞.
(ii) Let η 1 ≈ 2.3303 be the unique positive zero of the equation η -1 -2η log(η -1) = 0 in (1, ∞). For each η ∈ [η 0 , η 1 ), there is a sufficiently small positive number ε = ε(η) such that assuming the Elliott-Halberstam conjecture EH prime [ε], for any A > 1 we have

(1.4) π a,c,η (y) = π(y) + O A,a,c,η y (log y) A , as y → ∞.
Theorem 2. Let a ∈ Z * and c > 1 be fixed constants. For every η ∈ [2, ∞), there is a sufficiently small positive number ε = ε(η) such that under the Elliott-Halberstam conjecture EH friable [ε], we have

(1.5) π a,c,η (y) log 4 √ 2 η -1 η π(y) 1 + O a,c,η 1 3 √ log y + ε , as y → ∞.
Remark 3. (a) For comparaison, we have

1 + 3 4 √ 2 ≈ 2.060, η 0 ≈ 2.142, η 1 ≈ 2.330, 1 + 4 √ e ≈ 2.284, 1 + √ e ≈ 2.648.
(b) The first assertion of Theorem 1 is due to Banks & Shparlinski [1, page 144]. Here we give a proof for convenience of the reader. By combining Theorem 1 with results of [START_REF] Wu | On values taken by the largest prime factor of shifted primes[END_REF], we see that the asymptotic formula (1.1) holds for 1 < η < η 1 under the Elliott-Halberstam conjecture.

(c) Theorem 2 improves signaficantively Chen-Wu's result (1.2) in two aspects : result and method. Firstly the proportion in (1.5) increases from log 2 8 to log 2 4 when η runs over [2, ∞), and that in (1.2) tends toward 0 as η → 1 + 4 √ e. Secondly our proof of (1.5) needs information of prime numbers in "arithmetical progressions" {a + mq} m friable with friable indice. For (a, q) = 1 and x y 2, define the counting function (1.6) π(x, y; q, a) := p x p≡a(mod q) P + ((p-a)/q) y 1.

A systematic study on the asymptotical behaviour has been done by Liu, Wu & Xi [START_REF] Liu | Primes in arithmetic progressions with friable indices[END_REF], recently. We need a theorem of Bombieri-Vinogradov type and an inequality of Brun-Titichmarsh for this new counting function (see Lemmas 2.2-2.3 below).

Some preliminary lemmas

In this section, we present three lemmas, which will be useful later.

The Rosser-Iwaniec linear sieve.

The first lemma is due to Iwaniec [START_REF] Iwaniec | Rosser's sieve[END_REF][START_REF] Iwaniec | A new form of the error term in the linear sieve[END_REF]. and

d|P P (z) λ + d w(d) d p z p∈P 1 - w(p) p F (s) + O e √ L-s 3 √ log D (2.2) d|P P (z) λ - d w(d) d p z p∈P 1 - w(p) p f (s) + O e √ L-s 3 √ log D (2.3)
for any z ∈ [2, D], s = (log D)/ log z, set of prime numbers P and multiplicative function w satisfying

0 < w(p) < p (p ∈ P), (2.4) u<p v, p∈P 1 - w(p) p -1 log v log u 1 + L log u (2 u v), (2.5)
where P P (z) := p z, p∈P p and the implied O-constants are absolute. Here F, f are defined by the continuous solutions to the system

         sF (s) = 2e γ (1 s 2) sf (s) = 0 (0 < s 2) (sF (s)) = f (s -1) (s > 2) (sf (s)) = F (s -1) (s > 2)
where γ is the Euler constant.

2.2.

Bombieri-Vinogradov theorem of for π(x, y; q, a).

The second lemma is a theorem of Bombieri-Vinogradov type for the counting function π(x, y; q, a) defined as in (1.6) (see [START_REF] Liu | Primes in arithmetic progressions with friable indices[END_REF]Theorem 2]). Lemma 2.2. Let a ∈ Z * , A > 0 and κ a non-negative arithmetic function. Assuming the Elliott-Halberstam conjecture EH prime [ε], the following estimate q Q (q,a)=1 κ(q) π(x, y; q, a) -π(x) ϕ(q) ρ log(x/q) log y

a,A x (log x) A q x κ(q) 2 q + π(x)εu q Q κ(q) ϕ(q)
holds uniformly in x 2, exp (log x) 2/5+ε y x and Q min{y, √ x}, where ρ(u) is the Dickman function.

2.3.

Brun-Titichmarsh inequality for π(x, y; q, a).

The lemma below is a variant of [8, Theorem 1].

Lemma 2.3. Let a ∈ Z * and c > 1 be fixed constants. For any ε > 0, we have

(2.6)
π(cx, y; q, a) -π(x, y; q, a) 4(c -1)x ϕ(q) log(x/q) ρ log(x/q) log y 1 + O a,c,ε

1 3 √ log x + O A,a,c x q(log x) A
uniformly in exp{log log x) 5/3+ε } y x and 1 q min{y, √ x} with (a, q) = 1.

Proof. Denote by S the quantity on the left-hand side of (2.6). Without loss of generality, we can assume q is even and a is odd. Put P 2a (z) := p z,p 2a p. By the Möbius inversion, we can write

S = x<a+mq cx (mq,a)=(a+mq,P 2a ( √ cx))=1 P + (m) y 1 + O(x ε ) = x<a+mq cx (a,mq)=1, P + (m) y d|(a+mq,P 2a ( √ cx)) µ(d) + O(x ε ),
λ + d + O(x ε ) = d D d|P 2aq ( √ cx) λ + d (x-a)/q<m (cx-a)/q m≡-aq (mod d) (a,m)=1, P + (m) y 1 + O(x ε ),
where {λ + d } d 1 is an upper bound sieve of level D as in Lemma 2.1 and q is the inverse of q modulo d (i.e. qq ≡ 1 (mod d)). To apply the Elliott-Halberstam conjecture EH friable [ε], we would like to remove the restriction (a, m) = 1 by Möbius inversion, so that

S |a µ( ) d D d|P 2aq ( √ cx) λ + d (x-a)/q<m (cx-a)/q m≡-aq (mod d) |m, P + (m) y 1 + O(x ε ) = |a P + ( ) y µ( ) d D d|P 2aq ( √ cx) λ + d (x-a)/ q<m (cx-a)/ q m≡-a q (mod d) P + (m) y 1 + O(x ε ).
We are now in a good position to employ the Elliott-Halberstam conjecture EH friable [ε] with D = (x/q) 1-ε , getting

S S + + O x q(log x) A , (2.7)
where

S + := |a µ( ) d D d|P 2aq ( √ cx) λ + d ϕ(d)
x/ q<m cx/ q (d,m)=1, P + (m) y 1.

and we have used the trivial bound

|a d D d|P 2aq ( √ cx) |λ + d | ϕ(d) cx/ q<m (cx-a)/ q (d,m)=1, P + (m) y 1 a log x.
Here we removed the restriction that P + ( ) y since we henceforth assume y > |a|.

Switching summations, it follows that

S + = |a µ( ) x/ q<m cx/ q P + (m) y d D d|P 2amq ( √ cx) λ + d ϕ(d) = x/q<m cx/q (a,m)=1, P + (m) y d D d|P 2amq ( √ cx) λ + d ϕ(d) •
From Lemma 2.1, we deduce (2.8)

S + log √ x log(x/q) 2e γ + O 1 3 √ log x x/q<m cx/q (a,m)=1, P + (m) y p< √ x p 2amq 1 - 1 p -1 log √ x log(x/q) 2e γ + O 1 3 √ log x p< √ x p 2aq 1 - 1 p -1 x/q<m cx/q (a,m)=1, P + (m) y H(m),
where H(m) is the multiplicative function, defined by

H(p ν ) = 1 if p | 2q or p > x 1/2 p-1 p-2
if p 2q and p x 1/2 for all ν 1. According to [8, (3.5)], we have

m∈S(x/q,y) (a,m)=1 H(m) = Ψ x q , y ϕ(a) a p< √ x p 2aq 1 + 1 p(p -2) 1 + O (log log x) 2 log y ,
where S(x, y) := {n x : P + (n) y} and Ψ(x, y) := |S(x, y)|. Combining this with (2.8), we find that (2.9)

S + log x log(x/q) 2e γ + O 1 3 √ log x ϕ(a) a Ψ cx q , y -Ψ x q , y p √ x p aq 1 - 1 p .
By the Mertens formula, it follows that

p x 1/2 p aq 1 - 1 p = aq ϕ(a)ϕ(q) • 2e -γ log x 1 + O 1 log x .
On the other hand, according to [5, Theorem 1], we have Ψ(x, y) = xρ log x log y 1 + O log((log x)/ log y + 1) log y uniformly for x 3 and exp{(log log x) 5/3+ε } y x. Combining these with (2.9) and (2.7), we can get the required inequality (2.6).

Proof of Theorem 2

For each prime r ∈ ( 1 2 y, y], consider (3.1)

Q r (y) :=

x<q cx P + (q-a)=r 1.
Noticing that P + (q -a) = r ⇔ q ≡ a (mod r) and P + (q -a) r, we can write

y<r 2y Q r (y) (log 2)(c -1)ρ(η -1) π(x) log y 1 + O a,c,η 1 log y + ε .
On the other hand, the Brun-Titchmarsh inequality (2.6) give us 1 log

Q r (y) 4(c -1)x ϕ(r) log(x/r) ρ(η -1) 1 + O a,η 1 3 √ log x 4(c -1) ρ(η -1) η -1 • x y log y 1 + O a,η 1 3
4 √ 2 η -1 η π(y) 1 + O a,c,η 1 3 √ log x + ε .
This completes the proof of Theorem 2.

Proof of Theorem 1

As in [START_REF] Wu | On values taken by the largest prime factor of shifted primes[END_REF], the letters p, q, r and are always used to denote prime numbers, and d, m, and n always denote positive integers. In what follows, let a ∈ Z * and η ∈ (1, 32 17 ] ∪ [η 0 , η 1 ). Let δ be a sufficiently small positive constant and let c > 1 be a parameter to be chosen later. Let x 0 (A, a, c, η, δ) be a large constant depending on A, a, c, η, δ at most. For x x 0 (A, a, c, η, δ) and r ∈ ( Let r ∈ R g (y) and let Q r (y) be defined as in (3.1). When η ∈ (1, 32 17 ], we have r > y = x 1/η x 17/32 > (cx) 1/2 . Thus the definition of R g (y) allows us to write

(4.4) Q r (y) = π(cx; r, a) -π(x; r, a) (c -1 -3δ) π(x) ϕ(r) > 0,
where we have used the inequality π(cx) (c -δ)π(x) for x x 0 (a, c, δ). By the definition of P a,c,η and (4.4), it is easy to see that R g (y) ⊂ P a,c,η ∩ [y, 2y]. In view of (4.3), we find that π a,c,η (2y) -π a,c,η (y) = π(2y) -π(y) + O A,a,c,η y (log y) A . This implies the first assertion of Theorem 1, thanks to standard dyadic split. 4.2. The case of η ∈ [η 0 , η 1 ).

In this case, for every prime r ∈ R g (y), we can write It is easy to see that G(η) is decreasing on [2, ∞) and G(2) = 1. Therefore there is a unique real number η 1 ∈ (2, ∞) such that G(η 1 ) = 0 and for η ∈ [2, η 1 ) we have the inequality (4.9) Q r (y) A,a,δ,η π(y) ϕ(r)

for y y 0 (A, a, δ, η). As before, (4.9) allows us to deduce that R g (y) ⊆ P a,c,η ∩ (y, 2y]. Combining this with (4.3) leads to π a,c,η (2y) -π a,c,η (y) = π(2y) -π(y) + O A,a,c,δ,η y (log y) A . This implies the required asymptotic formula (1.4).

Lemma 2 . 1 .

 21 Let D 2 and let µ(n) be the Möbius function. Then there are two sequences {λ ± d } d 1 , vanishing for d > D or µ(d) = 0, satisfying |λ ± d | 1, such that (2.1)

1 =Since η 2 , 3

 123 (mod r), P + (q-a) y y<r 2y π(cx, y; r, a) -π(x, y; r, a) = M + E, , y; r, a) -E(x, y; r, a) , and E(x, y; r, a) := π(x, y; r, a)we have y = x 1/η x 1/2 and Q = min(y, √ x) = y. Using Lemma 2.2 with the characteristic function of prime numbers in (y, 2y] in place of κ(q), we easily derive, under the conjecture of Elliott-Halberstam EH prime [ε], that (3.3) |E| y<r 2y |E(cx, y; r, a)| + |E(x, y; r, a)| a x (log x) 3 + εη π(x) log y a,η ε π(x) log y for all x x 0 (ε), where we have used the following bound (constant depends on a, η at most. According to [9, Corollary III.5.8.3], we have |ρ (u)| ρ(u) log u (u > 1). Thus for all r ∈ (y, 2y], we have ρ log(x/r) log y = ρ(η -1) 1 + O η 5) and (3.3) into (3.2), it follows that (3.6)

√

  log x for all primes r ∈ (y, 2y]. This implies that (3.7) y<r 2y Q r (y) ρ(η -

1 2 y 4 . 1 .R

 241 , y], put x := r η . As usual, for (a, d) = 1 define π(x; d, a) := p x p≡a(mod d) 1. The case of η ∈ (1, 32 17 ]. For η > 1, c > 1, y 3 and x = y η , put R b (y) := y < r 2y : π(x; r, a)b (y) := y < r 2y : π(cx; r, a) -Noticing that y = x 1/η = x 1-(1-1/η) , the Elliott-Halberstam conjecture EH prime [ε] with ε = 1 -1/η allows us to deduce tthat δ π(x) y |R b (y)| y<r 2y π(x; r, a) -π(x) ϕ(r)A,a,δ,ηx (log x) A+1 , which gives immediately (4.1) |R b (y)| A,a,δ,η y (log y) A • Similarly (4.2) |R b (y)| A,a,c,δ,η y (log y) A • Define R g (y) := y < r 2y : π(x; r, a) (1 + δ) π(x) ϕ(r) , R g (y) := y < r 2y : π(cx; r, a) (1 -δ) π(cx) ϕ(r) , and R g (y) := R g (y) ∩ R g (y). Clearly R g (y) ⊂ P ∩ (y, 2y] ⊂ R b (x) ∪ R b (x) ∪ R g (y). Thus the estimations (4.1) and (4.2) imply that (4.3) |R g (y)| = π(2y) -π(y) + O A,a,c,δ,η y (log y) A (y 2).

(4. 5 ) 3 √ 2 √ δ -δ η - 1 η - 1 -

 53211 Q r (y) = π(cx; r, a) -π(x; r, a) -Q r (y)(c -1 -3δ) π(x) ϕ(r) -Q r (y).for x x 0 (a, c, δ), where(4.6) Q r (y) := x<q cx q≡a(mod r), P (q-a)>r 1.Similar to [12, Proposition 2.1] * , we can prove(4.7) Q r (y) (c -1 + 2δ) 2η log(η -1) η -1 • π(y) ϕ(r) 1 + O a,c,δ,η,ε1log r for y 3, r ∈ (y, 2y] and η 2. Inserting (4.7) into (4.4) and taking c = 1 + 2 √ δ, we can find that Q r (y) 2η log(η -1) ) := η -1 -2η log(η -1).

* The proof is identical and the only difference is that we can take z = (y/q) (1-ε)/2 and D = z 2 thanks to the Elliott-Halberstam conjecture EH * prime [ε], instead of z = (y/q) 1/4 /(log y) B and D = z 2

Acknowledgements. This work is supported in part by Scientific Research Innovation Team Project Affiliated to Yangtze Normal University (No. 2016XJTD01).


Kui Liu
email: liukui@qdu.edu.cn

Jie Wu
email: jie.wu@math.cnrs.fr

WEIGHTED ERD ŐS-KAC THEOREM IN SHORT INTERVALS

Keywords: Mathematics Subject Classification. 11N60, 11N36, 11N37 Erdős-Kac theorem, Central limit theorem, number of distinct prime factors, number of divisors

In this paper, we generalize Elliott's weighted Erdős-Kac theorem to the case of short intervals.

Introduction

Recently Elliott [START_REF] Elliott | Central limit theorem for classical cusp forms[END_REF][START_REF] Elliott | Corrigendum: central limit theorem for classical cusp forms[END_REF] established a weighted central limit theorem for the Fourier coefficients of cusp form. For comparison, he also gave a weighted Erdős-Kac theorem on the value distribution of the function ω(n) that counts the number of distinct prime divisors of the positive integer n. As usual, denote by d(n) the classic divisor function. For α ∈ R, define

(1.1) D α (x) := n x d(n) α .
Elliott's weighted Erdős-Kac theorem can be stated as follows (see [START_REF] Elliott | Corrigendum: central limit theorem for classical cusp forms[END_REF]Theorem]): for each λ, we have

(1.2) 1 D α (x) n x ω(n)-2 α log 2 x λ(2 α log 2 x) 1/2 d(n) α → Φ(λ)
as x → ∞, where log k denotes the k-fold iterated logarithm and Φ(λ) is the Gaussian law defined by

(1.3) Φ(λ) := 1 √ 2π λ -∞
e -τ 2 /2 dτ.

The case of α = 0 of (1.2) firstly was established by Erdős & Kac [START_REF] Erdős | Gaussian law of errors in the theory of additive functions[END_REF] in 1939. This problem has a long and rich history. The best actual result is due to Delange [START_REF] Delange | Sur des formules dues à Atle Selberg[END_REF]. The aim of this paper is to generalize Elliott's result (1.2) to the case of short intervals. Similar to (1.1), write

(1.4) D α (x, y) := x<n x+y d(n) α .
Our result is as follows.

Theorem 1.1. (i) Let α ∈ R and ε > 0. Then for each real number λ, we have

(1.5) 1 D α (x, y) x<n x+y ω(n)-2 α log 2 x λ(2 α log 2 x) 1/2 d(n) α = Φ(λ) + O α,ε 1 log 2 x
uniformly for x → ∞ and x 7/12+ε y x, where the implied constant depends on α and ε only. The error term in (1.5) is optimal.

(ii) The same result also holds if the summation condition on ω(n) may be replaced by

log d(n)/ log 2 -2 α log 2 x λ(2 α log 2 x) 1/2 .
Remark 1. The exponent 7 12 in Theorem 1.1 comes from Huxley's zero-density bound for the Riemann ζ-function [START_REF] Huxley | The difference between consecutive primes[END_REF]. This constant can be reduced to 1 2 if we assume the zero-density hypothesis.

In order to prove that the error term in (1.5) is optimal, we need to establish a weighted Laudan prime number theorem in short intervals. For α ∈ R and k ∈ N, define

(1.6) π k,α (x, y) := x<n x+y ω(n)=k d(n) α .
We have the following result.

Theorem 1.2. Let α ∈ R, B > 0 and ε > 0. Then we have 2 .

(1.7) π k,α (x, y) = y log x (2 α log 2 x) k-1 (k -1)! λ α k -1 2 α log 2 x + O log 2 x k log x + k -1 (log 2 x)
uniformly for x 3, x 7/12+ε y x and 1 k B2 α log 2 x, where

(1.8) λ α (z) = 2 α Γ(2 α z + 1) p 1 + ν 1 (ν + 1) α z p ν 1 - 1 p 2 α z
and the implied constant depend on α, B and ε only.

A principal tool for the proof of Theorems 1.1 and 1.2 is a rather general result of Cui, Lü and Wu [START_REF] Cui | The Selberg-Delange method in short intervals with some applications[END_REF] on the Selberg-Delange method in short intervals (see Lemma 2.1 below). A more result can be found in [8].

Some preliminary lemmas

Let f (n) be an arithmetic function and let its Dirichlet series be defined by (2.1)

F(s) := ∞ n=1 f (n)n -s .
Let z ∈ C, w ∈ C, α > 0, δ 0, A 0, B > 0, C > 0, M > 0 be some constants. We say that the Dirichlet series F(s) is of type P(z, w, α, δ, A, B, C, M ) if the following conditions are verified: (a) for any ε > 0 we have

(2.2) |f (n)| ε M n ε (n 1),
where the implied constant depends only on ε;

(b) we have

∞ n=1 |f (n)|n -σ M (σ -1) -α (σ > 1); (c) the Dirichlet series (2.3) G(s; z, w) := F(s)ζ(s) -z ζ(2s) -w
can be analytically continued to a holomorphic function in (some open set containing) σ 1 2 and, in this region, G(s; z, w) satisfies the bound

(2.4) |G(s; z, w)| M (|τ | + 1) max{δ(1-σ),0} log A (|τ | + 1)
uniformly for |z| B and |w| C, where and in the sequel we implicitly define the real numbers σ and τ by the relation s = σ + iτ and choose the principal value of the complex logarithm.

The following result is Corollary 1.2 of [START_REF] Cui | The Selberg-Delange method in short intervals with some applications[END_REF], which constitutes the key tool for the proof of Theorem 1.1.

Lemma 2.1. If the Dirichlet series F(s) is of type P(z, w, α, δ, A, B, C, M ), then for any ε > 0, we have

(2.5) x<n x+y f (n) = y(log x) z-1 λ(z, w) + O M log x uniformly for x 2, x (7+5δ)/(12+5δ)+ε y x, |z| B and |w| C, where λ(z, w) := G(1; z, w)ζ(2) w Γ(z)
and the implied constant in the O-term depends only on A, B, α, δ and ε. Lemma 2.2. Let B > 0 and ε > 0. Then we have

(2.6) x<n x+y d(n) α z ω(n) = y(log x) 2 α z-1 zλ α (z) + O B,ε 1 log x
uniformly for x 2, x 7/12+ε y x and |z| B, where

(2.7) λ α (z) = 2 α Γ(2 α z + 1) p 1 + ν 1 (ν + 1) α z p ν 1 - 1 p 2 α z
.

In particular, we have

(2.8) D α (x, y) = y(log x) 2 α -1 λ α + O ε 1 log x
uniformly for x 2 and x 7/12+ε y x, where (2.9)

λ α := λ α (1) = 1 Γ(2 α ) p 1 + ν 1 (ν + 1) α p ν 1 - 1 p 2 α . Proof. Since the function n → d(n) α z ω(n) is multiplicative, for e s > 1 we can write F α,z (s) := n 1 d(n) α z ω(n) n s = p 1 + ν 1 (ν + 1) α z p νs = ζ(s) zα ζ(2s) wα G(s; z α , w α ),
where

z α := 2 α z, w α := -2 2α-1 z 2 -(2 α-1 -3 α )z and the Euler product (2.10) G(s; z α , w α ) := p 1 + ν 1 (ν + 1) α z p νs 1 - 1 p s zα 1 - 1 p 2s wα .
This Euler product is expandable as a Dirichlet series

G(s; z α , w α ) = n 1 b(n)n -s ,
where n → b(n) is the multiplicative function whose values on prime powers are given by the identity

1 + ν 1 b(p ν )ξ ν = 1 + ν 1 (ν + 1) α zξ ν 1 -ξ zα 1 -ξ 2 wα (|ξ| < 1).
In particular, we have (2.12)

|b(p ν )| M 1 (B)2 ν/6 (ν 3, |z| B)
with

M 1 (B) := max |z| B max |ξ| 2 -1/6 1 + ν 1 (ν + 1) α zξ ν 1 -ξ zα 1 -ξ 2 wα .
With the help of (2.11) and (2.12), for σ > 1 3 and |z| B we easily deduce that

p ν 1 |b(p ν )|p -νσ M 1 (B) p ν 3 2 -1/6 p σ -ν = M 1 (B) p (2 -1/6 p σ ) -3 1 -(2 -1/6 p σ ) -1 2 1/2 M 1 (B) 1 -2 -1/6 p 1 p 3σ •
This shows that the Euler product G(s; z α , w α ) converges absolutely for σ > 1 3 and (2.13)

|G(s; z α , w α )| M (B) (σ 1 2 , |z| B) with M (B) := exp 2 1/2 M 1 (B) 1 -2 -1/6 p 1 p 3/2 .
Consequently, F α,z (s) is a Dirichlet series of type

P(z α , w α , |z α |, 0, 0, 2 α B, 2 2α-1 B 2 + (3 α -2 α-1 )B, M (B)).
Applying Lemma 2.1, we get the required asymptotic formula (2.8).

The third lemma is the Berry-Esseen inequality (see [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF]Theorem II.7.14]).

Lemma 2.3. Let F , G be two distribution functions with respective characteristic functions f and g. Suppose that G is differentiable and that G is bounded on R. Then we have

F -G ∞ 16 G ∞ T + 6 T -T f (τ ) -g(τ )
τ dτ for all T > 0, where H ∞ := sup λ∈R |H(λ)| for any real-valued function H defined on the real numbers.

Proof of Theorem 1.2

Recall π k,α (x, y) defined as in (1.6). Noticing that

x<n x+y d(n) α z ω(n) = k π k,α (x, y)z k ,
we can apply the the Cauchy formula to write, with r := k/(2 α log 2 x),

π k,α (x, y) = 1 2πi |z|=r x<n x+y d(n) α z ω(n) dz z k+1 • By Lemma 2.2, it follows that (3.1) π k,α (x, y) = y log x • I k,α (x; r) + O α,B,ε y (log x) 2 • (2 α log 2 x) k k! ,
uniformly for x 2, x 7/12+ε y x and k 2 α B log 2 x, where

I k,α (x; r) := 1 2πi |z|=r (log x) 2 α z λ α (z) z k dz
and we have used the following estimations

|z|=r (log x) 2 α e z |z| k+1 | dz| 2 α log 2 x k k 2π 0 e k cos θ dθ 2 α log 2 x k k π/2 0 e k cos θ dθ + 1 (t = k(1 -cos θ)) 2 α log 2 x k k e k √ k k 0 e -t t -1/2 dt + 1 (2 α log 2 x) k k! ,
thanks to the Stirling formula. It remains to evaluate I k,α (x; r). We shall discuss two cases: k = 1 or k 2. Since z → λ α (z) is analytic for |z| B, we have

I 1,α (x; r) = 1 2πi |z|=r e z2 α log 2 x λ α (z) z dz = λ α (0) = 2 α .
Inserting it into (3.1), we obtain that

π 1,α (x, y) = 2 α y log x 1 + O α,ε log 2 x log x .
This proves (1.7) for k = 1.

Next we suppose that k 2. Since z → λ α (z) is analytic for |z| B, we have I k,α (x; r) = I k,α (x; r 0 ) with r 0 := (k -1)/(2 α log 2 x). Writing the Taylor expansion of λ α (z) at z = r 0 :

(3.2) λ α (z) = λ α (r 0 ) + λ α (r 0 )(z -r 0 ) + (z -r 0 ) 2 1 0 (1 -t)λ α (r 0 + t(z -r 0 )) dt,
we shall estimate the contributions of three terms on the right-hand side of (3.2) to I k,α (x; r 0 ). Firstly those of the first two terms are, respectively,

(3.3) λ α (r 0 ) 2πi |z|=r e z2 α log 2 x z k dz = (2 α log 2 x) k-1 (k -1)! λ α k -1 2 α log 2 x and (3.4) λ α (r 0 ) 2πi |z|=r e z2 α log 2 x (z -r 0 ) z k dz = λ α (r 0 ) (2 α log 2 x) k-2 (k -2)! -r 0 (2 α log 2 x) k-1 (k -1)! = 0.
For 0 t 1 and |z| = r 0 , we have

|r 0 + t(z -r 0 )| = |r 0 (1 -t) + tz| r 0 (1 -t) + t|z| = r 0 . Since z → λ α (z) is analytic for |z| B, there is a positive constant C α such that |λ α (z)| C α for |z| B.
Thus the contribution of the third term on the right-hand side of (3.2) to I k,α (x; r 0 ) is (3.4) and (3.5) into (3.1), we find that

(3.5) α 2π 0 e (k-1) cos θ r -(k-3) 0 |e iθ -1| 2 dθ α r -(k-3) 0 π/2 0 e (k-1) cos θ (1 -cos θ) dθ + π α r -(k-3) 0 e k-1 (k -1) -3/2 k-1 0 e -t t 1/2 dt + π α (2 α log 2 x) k-1 (k -1)! • k -1 (2 α log 2 x) 2 • Inserting (3.3),
π k,α (x, y) = y log x • (2 α log 2 x) k-1 (k -1)! λ α k -1 2 α log 2 x + O α,B,ε y (log x) 2 • (log 2 x) k k! + y log x • (2 α log 2 x) k-1 (k -1)! • k -1 (log 2 x) 2 ,
which is equivalent to (1.7).

Proof of Theorem 1.1

Denote by F x,y (λ) the left-hand side of (1.5) and by ϕ x,y (τ ) its characteristic function, i.e., (4.1)

ϕ x,y (τ ) := +∞ -∞ e iτ λ dF x,y (λ) = 1 D α (x, y) x<n x+y d(n) α exp iτ ω(n) -2 α log 2 x 2 α log 2 x = e -iτ T D α (x, y) x<n x+y d(n) α e i(τ /T )ω(n) ,
where T = 2 α log 2 x. By using Lemma 2.3 with (F, G) = (F x,y , Φ), it follows that

F x,y -Φ ∞ 16 √ 2πT + 6 T -T ϕ x,y (τ ) -e -τ 2 /2 τ dτ.
Thus it suffices to show that

(4.2) T -T ϕ x,y (τ ) -e -τ 2 /2 τ dτ 1 T
uniformly for x 2 and x 7/12+ε y x. Applying Lemma 2.2 with z = e it , we have 1) A(e it ) + O ε 1 log x uniformly for t ∈ R, x 2 and x 7/12+ε y x, where λ α (z) and λ α are defined as in (2.7) and (2.9) respectively, and A(z) := zλ α (z)/λ α is an entire function of z such that A(1) = 1. Taking t = τ /T , the preceding asymptotic formula implies that (4.3) ϕ x,y (τ ) = (log x) 2 α (e i(τ /T ) -1) A(e i(τ /T ) )e -iτ T + O ε (log x) -1 uniformly for x 2, x 7/12+ε y x and |τ | T . In view of the inequality cos t -1 -2(t/π) 2 (|t| 1), we have (log x) 2 α (e i(τ /T ) -1) = e (cos(τ /T )-1)T 2 e -2(τ /π) 2 , from which we deduce that ϕ x,y (τ ) ε e -2(τ /π) 2 uniformly for x 2, x 7/12+ε y x and |τ | T . Thus (4.4)

1 D α (x, y) x<n x+y d(n) α e itω(n) = (log x) 2 α (e it -
±T ±T 1/3 ϕ x,y (τ ) -e -τ 2 /2 τ dτ T T 1/3 e -2(τ /π) 2 dτ 1 T •
With the help of the Taylor developments

A(e i(τ /T ) ) = 1 + O(τ /T ), e i(τ /T ) -1 = i(τ /T ) -1 2 (τ /T ) 2 + O((τ /T ) 3 ), we deduce that (log x) 2 α (e i(τ /T ) -1) A(e i(τ /T ) )e -iτ T = e -τ 2 /2+O(τ 3 /T ) 1 + O |τ | T = e -τ 2 /2 1 + O |τ | + |τ | 3 T for |τ | T 1/3 . Inserting into (4.3), it follows that ϕ x,y (τ ) = e -τ 2 /2 1 + O |τ | + |τ | 3 T + O ε 1 log x
uniformly for x 2, x 7/12+ε y x and |τ | T 1/3 . With the help of this evaluation, we can deduce that (4.5)

±T 1/3 ±1/ log x ϕ x,y (τ ) -e -τ 2 /2
τ dτ

T 1/3 1/ log x e -τ 2 /2 1 + τ 2 T + 1 τ log x dτ 1 T + log 2 x log x 1 T • For |τ | (log x) -1 ,

( 2 .

 2 11) b(p) = b(p 2 ) = 0 for all primes p and

Now ( 4 . 2 )√ 2 α

 422 we have trivially|τ (ω(n) -2 α log 2 x)/ 2 α log 2 x| (|τ | log x)/T.Thus we can writeexp iτ ω(n) -2 α log 2 x 2 α log 2 x = 1 + O |τ | log x T .Inserting into (4.1), it follows thatϕ x,y (τ ) = 1 + O |τ | log x T .From this and the relation e -τ 2 /2 = 1 + O(τ 2 ), we derive that (4.6)1/ log x -1/ log x ϕ x,y (τ ) -e -τ 2 follows from (4.4), (4.5) and (4.6) immediately. Finally we prove that the error term in (1.5) is optimal. DefineR λ (x, y) := 1 D α (x, y) x<n x+y ω(n)-2 α log 2 x λ(2 α log 2 x) 1/2 d(n) α -Φ(λ), R(x, y) := sup λ∈R |R λ (x, y)|. Let k := [2 α log 2 x] and θ := k -2 α log 2 x. Then we have (4.7) π k,α (x, y) D α (x, y) = F x,y θ 2 α log 2 x -F x,y log 2 xe -τ 2 /2 dτ + 2R(x, y) 1 2 π2 α+1 log 2 x + 2R(x, y).

Acknowledgements. This work is supported in part by the National Natural Science Foundation of China (Grant Nos. 12071238, 11771121, 11971370 and 12071375) and by the NSF of Chongqing (Grant No. cstc2019jcyj-msxm1651).

On the other hand, Theorem 1.2 and (2.8) imply that

thanks to the Stirling formula. From (4.7) and (4.8), we derive that R(x, y)

uniformly for x 2 and x 7/12+ε y x.

This completes the proof of Theorem 1.1.


Yurong Yang

Charles Paillard

Bin Xu

L Bellaiche

Photostriction and elasto-optic response in multiferroics and ferroelectrics from first principles

The present work reviews a series of recent first-principles studies devoted to the description of the interaction of light and strain in ferroelectric and multiferroic materials. Specifically, the modelling schemes used in these works to describe the so-called photostriction and elasto-optic effects are presented, in addition to the results and analysis provided by these abinitio calculations. In particular, the large importance of the piezoelectric effect in the polar direction in the photostriction of ferroelectric materials is stressed out. Similarly, the occurrence of low-symmetry phases in lead titanate thin films under tensile strain is demonstrated to result in large elasto-optic constants. In addition, first-principle calculations allow to gain microscopic knowledge of subtle effects, for instance in the case of photostriction, where the deformation potential effect in directions perpendicular to the polar axis is shown to be almost as significant as the piezoelectric effect. As a result, the numerical methods presented here could propel the design of efficient opto-mechanical devices.

Introduction 1.Ferroelectric and Multiferroic Materials

Perovskite ferroelectrics possess a spontaneous electrical polarization and form an important class of materials [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF]. Such compounds exhibit properties of technological relevance, since they can, e.g., convert various forms of energies. For instance, some ferroelectric (FE) systems can efficiently convert mechanical to electrical energies and vice-versa, because of their large piezoelectric coefficients, which is at the heart of piezoelectric transducers and actuators. Ferroelectric materials are also of importance for other applications, such as non-volatile memories, dielectrics for microelectronics and wireless communication, and pyroelectric arrays Perovskite multiferroics (MFEs) constitute another intensively studied type of materials because of the inherent coexistence between ferroelectricity and magnetism. The primary initial interest of MFE materials resides in the promise of designing new devices exploiting the magnetoelectric coupling between magnetic and electrical properties [START_REF] Catalan | Physics and Applications of Bismuth Ferrite[END_REF]. As a matter of fact, controlling magnetic properties by an electric field would be of large technological interest.

There are properties and phenomena of FEs and MFEs that have been much less studied than, e.g., piezoelectricity and magnetoelectricity, despite the numerous studies devoted to these two types of materials. Such properties and phenomena are related to the interaction of light and matter, and hold promise for exciting discoveries -such as the use of ferroelectrics as novel and efficient photovoltaic or photoconductive materials [START_REF] Ilya Grinberg | Perovskite oxides for visible-light-absorbing ferroelectric and photovoltaic materials[END_REF][START_REF] Riad Nechache | Bandgap tuning of multiferroic oxide solar cells[END_REF][START_REF] Spanier | Power conversion efficiency exceeding the Shockley-Queisser limit in a ferroelectric insulator[END_REF].

In this Review article, we will concentrate on recent computational, ab-initio efforts devoted to two specific light-matter interactions, that are photostriction and elasto-optic effects.

Photostriction

Most ferroelectric materials are semiconductors or insulators, and thus have a finite bandgap. As such, when light made of photons with energy higher than the bandgap hits a ferroelectric material, it is partially absorbed. Its absorption can convert the energy carried by light into:

(i) Electrical energy, i.e., the photovoltaic effect, whose origin may stem from various contributions [START_REF] Paillard | Photovoltaics with Ferroelectrics: Current Status and Beyond[END_REF][START_REF] Lopez-Varo | Physical aspects of ferroelectric semiconductors for photovoltaic solar energy conversion[END_REF]: the so-called Bulk Photovoltaic Effect (BPVE) generated by the lack of inversion symmetry inherent to all ferroelectric materials [START_REF] Fridkin | Anomalous photovoltaic effect in ferroelectrics[END_REF][START_REF] Fridkin | The experimental investigation of the photovoltaic effect in some crystals without a center of symmetry[END_REF][START_REF] Sturman | The Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials[END_REF][START_REF] Fridkin | Bulk photovoltaic effect in noncentrosymmetric crystals[END_REF], the ferroelectric-metal Schottky interface, the presence of internal electric fields caused by defects, etc.

(ii) Chemical energy, i.e., the photocatalysis effect [START_REF] Cui | Effect of Ferroelectricity on Solar-Light-Driven Photocatalytic Activity of BaTiO 3 -Influence on the Carrier Separation and Stern Layer Formation[END_REF].

(iii) Mechanical energy, i.e., the photostriction effect, which is the main focus of this subsection.

Photostriction in ferroelectric materials was studied as early as in 1966 in SbSI crystals, with a 004 % change in the length of the polar axis under white light illumination [START_REF] Tatsuzaki | Strain along c-axis of sbsi caused by illumination in dc electric field[END_REF]. An order of magnitude larger effect was observed later on by Xiao et al. in Fe-doped lithium niobate crystals [START_REF] Dingquan | Photostriction effect of doped linbo 3 single crystals[END_REF]. Interestingly, the photostrictive properties of ferroelectrics have mostly been interpreted as being driven by their piezoelectric properties (strain linearly induced by a change in polarization or electric field) [START_REF] Kundys | Wavelength dependence of photoinduced deformation in bifeo 3[END_REF]. As a result, investigations of materials with large piezoelectric properties, such as the ones usually found in so-called Morphotropic Phase Boundaries (MPBs), were at the forefront of research in the 90's. In particular, (Pb 1-x La x )(Zr 1-y Ti y )O 3 (PLZT) was comprehensively studied [START_REF] Chu | Impurity doping effect on photostriction in PLZT ceramics[END_REF][START_REF] Uchino | High voltage photovoltaic effect in PbTiO3-based ceramics[END_REF][START_REF] Poosanaas | Photostrictive effect in lanthanummodified lead zirconate titanate ceramics near the morphotropic phase boundary[END_REF], and devices based on the photostriction effect and PLZT were designed to take advantage of its large coupling of light and strain [START_REF] Uchino | New applications of photostrictive ferroics[END_REF]. However, most classical ferroelectric materials, including PLZT, have large bandgaps that often rise above 3 eV [START_REF] Scott | Ferroelectric Memories[END_REF], which lies at best at the extreme side of the visible range, and most often, in the ultraviolet (UV) domain. As a result, research on the conversion of solar energy into mechanical energy (but also into electrical energy) using ferroelectrics has been at a standstill for a decade or so. Recently though, new prospects have appeared in the field, mainly driven by the appearance of low band-gap FE and MFE materials. For instance, Nechache et al. have achieved a low band-gap of 1.5 eV in multiferroic Bi 2 FeCrO 3 , eventually resulting in one of the most efficient perovskite oxide-based solar cell [START_REF] Riad Nechache | Bandgap tuning of multiferroic oxide solar cells[END_REF]. Most recent photostriction experiments have been conducted on multiferroic bismuth ferrite, whose band-gap ranges from 2.2 to 2.7 eV [START_REF] Catalan | Physics and Applications of Bismuth Ferrite[END_REF][START_REF] Bai | Size effect on optical and photocatalytic properties in bifeo 3 nanoparticles[END_REF], and is therefore suitable for visible light absorption. As of right now, photostriction of 0.004% was reported in bismuth ferrite [START_REF] Kundys | Light-induced size changes in BiFeO3 crystals[END_REF][START_REF] Kundys | Wavelength dependence of photoinduced deformation in bifeo 3[END_REF]. The photostriction effect in bismuth ferrite has recently been used to increase the coercivity of a Nickel film deposited on a BiFeO 3 single crystal by 50% [START_REF] Iurchuk | Optical Writing of Magnetic Properties by Remanent Photostriction[END_REF], opening the way to strain-mediated optical control of magnetic properties. Note also that the photostriction effects described so far are taking place at the scale of seconds or even minutes in the case of lithium niobate [START_REF] Dingquan | Photostriction effect of doped linbo 3 single crystals[END_REF]. On the other hand, pump-probe experiments have demonstrated the occurence of photo-generated strain fields at the scale of picoseconds and below [START_REF] Pascal Ruello | Photoexcitation of gigahertz longitudinal and shear acoustic waves in BiFeO3 multiferroic single crystal[END_REF][START_REF] Daranciang | Ultrafast Photovoltaic Response in Ferroelectric Nanolayers[END_REF][START_REF] Wen | Electronic origin of ultrafast photoinduced strain in bifeo 3[END_REF][START_REF] Lejman | Giant ultrafast photoinduced shear strain in ferroelectric BiFeO3[END_REF][START_REF] Schick | Localized excited charge carriers generate ultrafast inhomogeneous strain in the multiferroic bifeo 3[END_REF][START_REF] Wen | Capturing ultrafast photoinduced local structural distortions of bifeo 3[END_REF][START_REF] Li | Giant optical enhancement of strain gradient in ferroelectric bifeo 3 thin films and its physical origin[END_REF][START_REF] Lejman | Ultrafast acousto-optic mode conversion in optically birefringent ferroelectrics[END_REF]. Interestingly, Ref. [START_REF] Lejman | Giant ultrafast photoinduced shear strain in ferroelectric BiFeO3[END_REF] showed the possibility, depending on the orientation of BiFeO 3 grains with respect to the probe, to generate preferentially shear or longitudinal acoustic waves.

There is a variety of mechanisms contributing to photostriction [START_REF] Kundys | Photostrictive materials[END_REF][START_REF] Thomsen | Surface generation and detection of phonons by picosecond light pulses[END_REF][START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]. First, light-matter interaction, through the interaction of photoexcited carriers with the lattice, creates heating of the material, resulting in thermal dilation. Second, the electronic density redistribution generated by the excitation causes lattice to deform through a mechanism known as the deformation potential [START_REF] Bardeen | Deformation Potentials and Mobilities in Non-Polar Crystals[END_REF][START_REF] Herring | Transport and Deformation-Potential Theory for Many-Valley Semiconductors with Anisotropic Scattering[END_REF][START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF] (often termed electronic pressure as well). Third, in materials lacking inversion symmetry, such as ferroelectrics, the converse piezoelectric effect caused by the appearance of a photo-induced electric field results in the appearance of a photo-induced deformation [START_REF] Kundys | Wavelength dependence of photoinduced deformation in bifeo 3[END_REF][START_REF] Kundys | Photostrictive materials[END_REF][START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF]. The latter mechanism is particularly appealing, as it provides polar materials (and ferroelectric materials in particular) with an additional handle compared to centrosymmetric materials. In addition, it is known how to design large piezoelectric and ferroelectric materials, by mixing a ferroelectric material (such as lead titanate) with a relaxor or antiferroelectric material (such as lead magnesium niobate or lead zirconate). Such mixing results in the appearance of low symmetry (often monoclinic, sometimes orthorhombic) phases for a specific compositional range defining the aforementionned MPB [START_REF] Noheda | A monoclinic ferroelectric phase in the pb(zr 1-x ti x )o 3 solid solution[END_REF][START_REF] Bellaiche | Finite-Temperature Properties of Pb(Zr$ {1-x}$Ti$ x$)O$ 3$ Alloys from First Principles[END_REF][START_REF] Noheda | Phase diagram of the ferroelectric relaxor (1x)pbmg 1/3 nb 2/3 o 3 -xpbtio 3[END_REF][START_REF] V A Shuvaeva | The macroscopic symmetry of pb(mg 1/3 nb 2/3 ) 1-x ti x o 3 in the morphotropic phase boundary region[END_REF]. The existence of these lowsymmetry phases in the phase diagram has been demonstrated to host giant piezoelectric properties [START_REF] Zhao | Dielectric and piezoelectric performance of PMN-PT single crystals with compositions around the MPB: influence of composition, poling field and crystal orientation[END_REF].

The origin of photostriction can be qualitatively captured using a simple Landau model [START_REF] Kundys | Wavelength dependence of photoinduced deformation in bifeo 3[END_REF][START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF]. Consider a uniaxial ferroelectric material. Its thermodynamic potential Φ F (T, P , η), in the dark, can be expressed as a Taylor expansion in terms of the order parameters (polarization P and strain η) with respect to the reference paraelectric phase,

Φ F (T, P , η) = Φ P (T ) + α(T ) 2 P 2 + β 4 P 4 + + γ 6 P 6 + C 2 η 2 + Q 2 P 2 η. (1)
In this equation, α(T ) is a second rank tensor. β and γ are fourth and sixth rank tensors, which are considered temperature independent. Q is the fourth-rank electrostrictive tensor, and C is the second-rank stiffness tensor. At equilibrium, the thermodynamic potential must reach a stationary point, and thus the following set of equations must be solved,

∂Φ F ∂P = 0 = α(T )P + βP 3 + γP 5 + QP η, (2)
∂Φ F ∂η = 0 = Cη + 1 2 QP 2 . ( 3
)
The perturbation caused by illumination can be considered as adding the energy of the photo-excited electrons and substracting those of the holes [START_REF] Kundys | Wavelength dependence of photoinduced deformation in bifeo 3[END_REF], in a similar fashion to what has been done to study heavily doped ferroelectric materials [START_REF] Pasynkov | On some problems of the phenomenological theory of ferroelectric-semiconductors[END_REF]. As a result, the thermodynamic potential of the gas of photoexcited electrons and holes must be considered alongside that of Eq. ( 1). The total thermodynamic potential is therefore Φ(T, P , η) = Φ F (T, P , η) + Φ ph (T, P , η),

with

Φ ph (T, P , η) = n,k δn nk ε nk . (5)
In Eq. ( 5), δn nk is the change of occupancy of the Bloch state nk with respect to the system in the dark, and ε nk is the energy of that state. Since photon absorption creates the same amount of holes and electrons, the condition n,k δn nk = 0 must be satisfied. In the simple case of a two-level system, for which holes are created at the top of valence band ε v while electrons are photoexcited at the bottom of the conduction band ε c , the former equation can be simply written as,

Φ ph (T, P , η) = δn e E v→c , (6)
with δn e being the number of photoexcited electrons in the conduction band, and E v→c = ε cε v . The latter is a function of temperature, polarization and strain. Assuming that the perturbation introduced by switching on the light is small, the order parameters can be expanded around their equilibrium in the dark. That is, they can be written P = P 0 + δP and η = η 0 + δη, with the quantities having the subscript "0" referring to the equilibrium values in the dark, which are solutions of Eqs. ( 2) & (3). It is then straightforward to define, as done in Ref. [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF], a photo-induced stress σ photo and photo-induced electric field E photo ,

σ photo = -δn e ∂E v→c ∂η P0,η0 , (7)
E photo = -δn e ∂E v→c ∂P P0,η0 , (8)
allowing to re-write Eq. ( 6) as

Φ ph (T, P , η) = Φ ph (T, P 0 , η 0 ) -σ photo .δη -E photo .δP . (9)
As a result, the equilibrium conditions under illumination, in this simple model, are

∂Φ ∂P = 0 = α(T )P + βP 3 + γP 5 + QηP -E photo , (10)
∂Φ ∂η = 0 = Cη + 1 2 QP 2 -σ photo ., (11)
which, under the assumption that δP and δη are indeed small compared to the equilibrium values in the dark, can be linearized, leading to [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF]

δP = χ ph E photo + d T ph σ photo , (12)
δη = S ph σ photo + d ph E photo . (13)
In those latter equations, χ ph is the dielectric susceptibility under illumination, d ph is the piezoelectric constant under illumination, and S ph the elastic compliance under illumination.

From Eq. ( 13), the contribution of the potential deformation mechanism (through the photo-induced stress σ photo ) and the piezoelectric mechanism (caused by the photo-induced electric field E photo ) appear naturally. To be exhaustive, one should also include the contribution of lattice heating and thermal dilatation. This can be done by adding the energy of the lattice/phonon gas, Φ L = n,q δn ph,nq ω nq , in which δn ph,nq and ω nq represent the change in population and energy of the n th phonon mode at wavevector q [START_REF] Thomsen | Surface generation and detection of phonons by picosecond light pulses[END_REF].

Elasto-optic effect

Elasto-optic effect describes the change of permittivity, ε, due to mechanical strain, η. It can be used for creating or designing acousto-optic modulators, acousto-optic tunable filter, electro-optic deflection, etc [START_REF] Ic Chang | acoustooptic devices and applications[END_REF]. Such effect can be quantified by the elasto-optic tensor, p ij , that is defined via

∆ 1 ε i = j p ij η j , (14)
where i are j are both integers that can vary from 1 to 6 (when adopting the Voigt notation). Introducing the extinction coefficient k and refractive index n tensors via

ε = ε + iε = (n + ik) 2 , (15)
Under low-frequency field (the Pockels effect), ε is about zero, and Eq. ( 14) can be rewritten as

∆ 1 n 2 i = i p ij η j . (16)
Eq. ( 16) was used in Ref. [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF] to compute the elasto-optic coefficient. Note that, as indicated above, both refractive index n and strain η have 6 components when using the Voigt notation, implying that there are 36 components for the elastooptic coefficient p ij . However, common elasto-optic materials (such as SiO 2 and LiNbO 3 ) have relatively high symmetry, and, as a result, only a few of the elasto-optic tensor components are finite.

Methods

Photostriction using the ∆SCF method

When performing Density Functional Theory (DFT) calculations, one is typically minimizing the energy functional with respect to density by self-consistently resolving the ground state density, in accordance with the two Hohenberg-Kohn theorems at the foundation of DFT [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF]. There are several schemes to achieve such calculations, however one simple, popular and practical way is to assume that one can find a system of non-interacting electrons with the same energy and ground state density as the true system of fully interacting electrons. Such approximation is called the Kohn-Sham (KS) ansatz [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF], and provides a physically intuitive one-electron picture, within which one is left to fill monoelectronic orbital states (determined from a monoelectronic Schrödinger equation) by increasing levels of energy. As described in Fig.

(1), a KS calculation would start by guessing an input potential V in , thereby fully determining the one-electron Hamiltonian. This Hamiltonian is then solved, resulting in a set of monoelectronic orbitals ϕ i with eigenvalues ε i . The lowest energy lying states would then be filled in order to determine the ground state, and the new potential V out generated by the distribution of electrons in these occupied orbitals becomes the input for a new cycle, until convergence of the ground state density is reached up to a desired accuracy. Figure 1: (a) Regular Kohn-Sham calculation to find the ground state electronic density: from an input potential V in , the now known Kohn-Sham Hamiltonian can be diagonalized. Its lowest eigenstates ("orbitals") are then filled with electrons, and a new potential is calculated based on the latter electronic distribution. The output potential is then plugged in as input potential until convergence to a reasonable accuracy is reached. (b) ∆SCF calculation: at filling, not all low energy lying eigenstates are filled, but some electrons are imposed to occupy excited orbitals. An output potential is then calculated based on this electronic distribution, and the density is self-consistently solved under the constraint of certain occupation numbers. Reproduced from Ref. [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF]. Copyright 2017 American Physical Society.

The ∆SCF method [START_REF] Martin | Electronic Structure: Basic Theory and Practical Methods[END_REF] is essentially analog to a regular Kohn-Sham calculations: starting from an input potential V in , the monoelectronic Schrödinger equation is solved. The main difference arises when the filling of the orbitals is performed. As sketched in Fig. (1.b), some electrons are constrained to occupy excited states in the conduction band, while leaving holes in the valence band. Therefore, the system is converged under the constraint of having some electrons lying in higher energy states, and holes in lower energy states. This rather simple approach has been proven to be rather successful in describing resonance level in finite systems such as molecules adsorbed on metallic surfaces [START_REF] Gavnholt | δ self-consistent field method to obtain potential energy surfaces of excited molecules on surfaces[END_REF], or the ligand-field splitting in Fe-phtalocyanine [START_REF] Nakamura | Constraint density functional calculations for multiplets in a ligand-field applied to Fe-phthalocyanine[END_REF]. In infinite systems however, the ∆SCF method does not improve much the description of excited states energies such as the bandgap energy when using traditional functionals [START_REF] Perdew | Density functional theory and the band gap problem[END_REF]. One should resort to more accurate models, such as the GW approximation [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF] or Time Dependent DFT (TD-DFT) [START_REF] Ullrich | Time-Dependent Density-Functional Theory, Concepts and Applications[END_REF], which are rather computationally expensive. Performing structural relaxation with the former technique is even practically untractable. Furthermore, we note that many GW calculations are typically used as corrections to eigenenergies and wavefunctions determined with a more traditional functional [START_REF] Deslippe | BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures[END_REF]. As a result, the ∆SCF method is a useful tool to apprehend photostrictive phenomena.

During ∆SCF calculations, a k-point A belonging to the valence band is emptied, meanwhile a k-point B belonging to the conduction band is filled with electrons, mimicking what we call a A -→ B transition. In the following works described below, only cases for which the photoexcited holes (electrons resp.) are sitting at the top (bottom) of the valence (conduction) band are considered. This amounts to assume that electron-phonon scattering, responsible for the thermalization of the photo-excited carriers, is extremely fast compared to other processes such as optical recombination of electrons with holes, the latter typically ranging in the nano-to microsecond according to Ref. [START_REF] Thomsen | Surface generation and detection of phonons by picosecond light pulses[END_REF], while electron-phonon thermalization processes typically occur on a picosecond time scale. In other words, the influence of the non-thermalized electrons, which are known to be responsible for ballistic photovoltaic current in ferroelectrics [START_REF] Fridkin | Anomalous photovoltaic effect in ferroelectrics[END_REF][START_REF] Sturman | The Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials[END_REF][START_REF] Fridkin | Bulk photovoltaic effect in noncentrosymmetric crystals[END_REF], is neglected.

Elasto-optic coefficients

First-principle techniques were used in Ref. [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF] to calculate the frequencydependent dielectric tensor once the electronic ground state was determined. More precisely, the imaginary part of dielectric function was computed via the following equation [START_REF] Gajdoš | Linear optical properties in the projector-augmented wave methodology[END_REF]

ε (ω) = 4π 2 e 2 Ω lim q→0 1 q 2 c,v,k 2ω k δ( ck -vk -ω) × < u ck+eαq |u vk >< u ck+e β q |u vk > * , (17)
where c and v refer to band indices of the conduction and valence bands, respectively. u ck is the cell periodic part of the Bloch orbitals at the reciprocal space point k; e α and e β are unit vectors along the α and β Cartesian direction, respectively; and the asterisk symbol denotes complex conjugate. The real part of the dielectric tensor ε αβ is then extracted through the Kramers-Kronig transformation

ε (ω) = 1 + 2 π P ∞ 0 ε αβ (ω )ω ω 2 -ω 2 dω , (18)
where P denotes the so-called principal value [START_REF] Gajdoš | Linear optical properties in the projector-augmented wave methodology[END_REF].

In the low frequency range, complex refractive index, with real part (n) and imaginary part (k), and complex dielectric function, with real part (ε ) and imaginary (ε ), follow the relation

ε = ε + iε = ñ2 = (n + ik) 2 (19)
Reference [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF] concentrated on energies ω smaller than the electronic band gap, which implies that, as consistent with the delta function appearing in Eq.( 17), ε (ω) vanishes for the selected energies. The refractive index is thus simply equal to the square root of ε (ω). Our first-principles computations of the refractive index within its band gap are found to be rather accurate. This is demonstrated by the prediction in Ref. [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF] of an average refractive index, (n 11 + n 22 + n 33 )/3 of 2.8 for PbTiO 3 for an energy about 40% smaller than the band gap, which compares well with the measured value of 2.7 at a wavelength of 633 nm.

Reference [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF] then computed the elasto-optic coefficient, p ij , via the aforementionned Eq. ( 16). Only the '1' and '2' components of the strain tensor (with the '1', '2' and '3' axes being along the pseudo-cubic [100], [010] and [001] directions, respectively) were considered in this formula, with η 1 =η 2 , to reflect epitaxial growth of the film. The resulting effective elasto-optic coefficients are thus obtained from the following relations:

1/(n 2 1 (η 1 )) -1/(n 2 1 (0)) = (p 11 + p 12 )η 1 , 1/(n 2 2 (η 1 )) -1/(n 2 2 (0)) = (p 21 + p 22 )η 1 , and 1/(n 2 3 (η 1 )) -1/(n 2 3 (0)) = (p 31 + p 32 )η 1 .
Note that for the tetragonal, orthorhombic, rhombohedral and the monoclinic Cm phases of epitaxial PbTiO 3 films (see Section 4 for a description of these phases), n 1 = n 2 = n o , which is known as the ordinary refractive index, and n 3 = n e , which is termed the extraordinary refractive index. Note that first-principles methods can predict elasto-optic coefficients accurately. For example, for bulk LiNbO 3 (using the hexagonal setting), one gets p 31 = 0.17 for a photon energy being roughly half-way within its band gap [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF], which is in very good agreement with the corresponding experimental value of 0.18 [START_REF] Fu | Polarization rotation mechanism for ultrahigh electromechanical response in single-crystal piezoelectrics[END_REF].

3 Modelling photostriction in multiferroic and ferroelectric materials

Photostriction in classical ferroelectric materials

Among the vast family of ferroelectric materials, barium and lead titanate emerge as prototypical examples. Both are perovskite oxides, with barium or lead ions occupying the A-sites of the perovskite structure while titanium ions sit on the B-sites. Barium titanate undergoes three successive phase transitions, from the high-symmetry P m 3m paraelectric phase at high temperature to a ferroelectric tetragonal P 4mm phase at 390 K, before transforming to an orthorhombic ferroelectric phase around 280 K, and eventually goes into a ferroelectric rhombohedral R3m phase below 180 K [START_REF] Kwei | Structures of the ferroelectric phases of barium titanate[END_REF]. Lead titanate is even simpler, as it experiences only one transition from the high-symmetry cubic P m 3m phase to a ferroelectric tetragonal P 4mm phase at 766 K [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF].

Both compounds have been successfully described in first-principles calculations using the Local Density Approximation (LDA) functional, and this was the starting point of Ref. [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF], in which other technical details are provided. We describe here only the photostriction induced in the ground state of those compounds, that is the R3m and P 4mm phases of barium and lead titanate respectively. The band structure of both compounds are presented in Fig. [START_REF] Catalan | Physics and Applications of Bismuth Ferrite[END_REF]. The considered "transitions" are Z → Z, Z → Γ, X → Γ and X → Z for PbTiO 3 (PTO) and Z → Γ and Γ → Γ for rhombohedral BaTiO 3 (BTO). Again, the assumption is made that the thermalization of the phonons is sufficiently fast as compared to the photoexcitation and recombination processes to consider that the hole and electrons are located at the top and bottom of valleys in the valence and conduction bands. According to Ref. [START_REF] Thomsen | Surface generation and detection of phonons by picosecond light pulses[END_REF], the relaxation time of the electron to the bottom of the conduction band is typically of the order of 1 ps, while recombination has a much larger (1 ns-1 µs) relaxation time, justifying this approximation. Note also that the first-principle description of ultra-fast phenomena such as those occuring in Refs. [START_REF] Daranciang | Ultrafast Photovoltaic Response in Ferroelectric Nanolayers[END_REF][START_REF] Wen | Electronic origin of ultrafast photoinduced strain in bifeo 3[END_REF][START_REF] Schick | Localized excited charge carriers generate ultrafast inhomogeneous strain in the multiferroic bifeo 3[END_REF][START_REF] Wen | Capturing ultrafast photoinduced local structural distortions of bifeo 3[END_REF] is out of the scope of this review because of the above mentioned approximation.

The results obtained for the aforementioned transitions in PbTiO 3 are presented as filled symbols in Fig. [START_REF] Ilya Grinberg | Perovskite oxides for visible-light-absorbing ferroelectric and photovoltaic materials[END_REF]. Because of the same d x 2 -y 2 character of the conduction states at Γ and X [see Fig. [START_REF] Riad Nechache | Bandgap tuning of multiferroic oxide solar cells[END_REF] in Ref. [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF]], the results are insensitive to the location of the electrons in Γ-Z branch of the conduction band. This can be seen as transitions involving electrons at Γ or Z show the same behavior, provided the hole is located at the same point in the valence band. When the hole sits at X in the valence band, the change of lattice constant in the polar direction is small compared to that induced when the hole sits at Z. In the direction normal to the polar axis (left panel of Fig. (3)), the hole sitting at Z generates a significant expansion, while when located at X, only a minor reduction is found. Note that at X (which is the top of the valence band), the hole has a combined Pb 6s and O 2p x character, while at Z it has Pb 6s and O 2p z character [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF]. As described in Section 1.2, photostriction in polar materials can be understood as originating from three main contributions: potential deformation, converse piezoelectric effect following a photo-induced electric field, and thermal dilatation following photo-induced heating. During ∆SCF calculations, the temperature is 0 K. As a result, thermal dilatation is not taken into account in those calculations. We now address the question of which of the two former mechanisms, between potential deformation and piezoelectric effect, is the dominant mechanism.

δa (×10 -2 %) Z → Z Z → Γ X → Z X → Γ
First, it is noted that the ionic relaxation can be translated into a change of polarization through the Born approximation,

δP i = e V 0 α,j Z α ij δu α j . (20)
In Eq. ( 20), e is the elementary electric charge, α denotes the atom number in the unit cell, δu α j its displacement in the j-th direction with respect to a reference configuration (here the ground-state structure), and Z α ij is the Born effective charge tensor of atom α. Of course, a major approximation is made here: it is assumed that the Born effective charge does not change between the ground state and the excited state, which is, in principle, questionable. However, as discussed in Ref. [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF], because the amount of excited charges is quite small, this approximation seems reasonable. The change of polarization in lead titanate, using Born charges calculated with Density Functional Perturbation Theory (DFPT) [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF], is depicted in the left panel of Fig. [START_REF] Riad Nechache | Bandgap tuning of multiferroic oxide solar cells[END_REF]. For all considered transitions, the polarization decreases. The strongest decrease rate happens for the Z -→ Z and Z -→ Γ transitions, and also correspond to the strongest photostriction effect calculated in Fig. (3).

This change in polarization can be directly related to a change of strain state by the third rank converse piezoelectric tensor g, δη = g.δP .

(

The converse piezoelectric tensor g is trivially related to more well-known piezoelectric tensor d (which relates the change of strain to an electric field) by g = d.χ -1 , with χ being the dielectric susceptibility. After calculating the piezoelectric tensor from DFPT calculations [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF], the piezoelectric-induced change of lattice constant, as estimated from Eq. ( 21 3). It can be observed there that along the polar axis, the piezoelectric model accounts for most of the calculated effect, for all transitions. In the direction perpendicular axis, the situation is not as clear. On one hand, the calculated results agree well with the piezoelectric model for the Z -→ Z and Z -→ Γ transitions, which are the ones with the strongest change of polarization. On the other hand, due to the negative value of the g 13 component, a weak expansion of the lattice is predicted in the plane orthogonal to the polar axis from the converse piezoelectric model for the two transitions departing from the X point (for which the induced change in polarization is weaker) while the ∆SCF data rather shows a small contraction. This latter difference might result from the action of a different mechanism, for instance the potential deformation (also sometimes coined electronic pressure).

To check whether the potential was of any influence in the obtained results, ∆SCF calculations were performed at fixed atomic positions. In other words, only the volume of the cell is allowed to relax, and thus the polarization is kept approximately constant (up to the change of volume and the minor electronic dipole induced [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF]). The results, plotted as open blue circles in Fig. [START_REF] Spanier | Power conversion efficiency exceeding the Shockley-Queisser limit in a ferroelectric insulator[END_REF] in the case of a Z -→ Z transition in PTO, show that along the polar axis the potential deformation mechanism is a negligible contribution compared to the piezoelectric effect. The picture is however different in the directions perpendicular to the polarization, for which the associated piezoelectric constant is smaller. It is found then that the potential deformation contribution could account for as much as a third of the calculated effect. More importantly, in the case of the X -→ Z transition, although the electronic pressure mechanism brings again a small contribution to the photostriction of the polar axis, it actually now competes and overcomes the (small) expansion caused by the piezoelectric effect, and results in an overall contraction of the a axis [see Fig. [START_REF] Paillard | Photovoltaics with Ferroelectrics: Current Status and Beyond[END_REF]]. Because the electronic pressure competes with the converse piezoelectric effect in the case of the X -→ Z transition, the c axis contraction is also reduced compared to that found in the Z -→ Z transition [see Fig. the ∆SCF data are found to lead to a larger contraction than predicted using the piezoelectric model, despite the competition of the electronic pressure mechanism. It must however be noted that both the piezoelectric model and frozen ion ∆SCF calculations can only be taken as qualitative estimates of the Figure 5: Photo-induced change of lattice constant in tetragonal PbTiO 3 for a Z -→ Z transition, as calculated from a fully relaxed ∆SCF calculation (black squares), as estimated from the piezoelectric model (open black diamonds) and as calculated using frozen atomic position calculations (open blue circles), for the polar axis direction (right panel) and the direction perpendicular to it (left panel). Adapted from Ref. [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF].

importance of the piezoelectric and potential deformation contributions to photostriction, as they rely on quite strong assumptions. Typically, we compute the change of polarization necessary to evaluate the converse piezoelectric effect by using ground state Born charges and finite displacement. As a result, it is most likely that only orders of magnitude and qualitative trends may be safely discussed when decomposing the photostriction effect in terms of those basic mechanisms.

In the case of barium titanate, depicted in Fig. [START_REF] Lopez-Varo | Physical aspects of ferroelectric semiconductors for photovoltaic solar energy conversion[END_REF], the pseudo-cubic lattice constant is found to decrease with increasing concentration of photoexcited for both the Γ -→ Γ and the Z -→ Γ. Meanwhile, the pseudocubic angle slightly increases. A similar analysis in terms of piezoelectric and electronic pressure source terms for the calculated photostriction was conducted in Ref. [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF]. It was shown there that the piezoelectric effect was the dominant mechanism in the Figure 6: Photo-induced change of lattice constant in tetragonal PbTiO 3 for a X -→ Z transition, as calculated from a fully relaxed ∆SCF calculation (black squares), as estimated from the piezoelectric model (open black diamonds) and as calculated using frozen atomic position calculations (open blue circles), for the polar axis direction (right panel) and the direction perpendicular to it (left panel).Adapted from Ref. [START_REF] Paillard | Ab initio approach to photostriction in classical ferroelectric materials[END_REF]. case of the Γ -→ Γ transition, but that potential deformation actually competes with the photo-induced piezoelectric effect for the Z -→ Γ transition.

Photostriction in bismuth ferrite

Bismuth ferrite has a rhombohedral structure with space group R3c below 1103 K [START_REF] Catalan | Physics and Applications of Bismuth Ferrite[END_REF], whose unit cell is sketched in green in Fig. [START_REF] Fridkin | Anomalous photovoltaic effect in ferroelectrics[END_REF]. It is made of two pseudocubic 5-atom perovskite cells (depicted as black cubes in Fig. ( 8)) along the [111] pc pseudocubic direction because of the a -a -a -tilting pattern (in Glazer notation [START_REF] Glazer | The classification of tilted octahedra in perovskites[END_REF]) of the oxygen octahedra [START_REF] Michel | The atomic structure of BiFeO3[END_REF][START_REF] Moreau | Ferroelectric BiFeO3 X-ray and neutron diffraction study[END_REF][START_REF] Kubel | Structure of a ferroelectric and ferroelastic monodomain crystal of the perovskite BiFeO3[END_REF] . The [111] pc direction is also associated with a large polarization of about 100 µC.cm -2 [START_REF] Lebeugle | Room-temperature coexistence of large electric polarization and magnetic order in bifeo 3 single crystals[END_REF][START_REF] Lebeugle | Very large spontaneous electric polarization in bifeo 3 single crystals at room temperature and its evolution under cycling fields[END_REF]. BiFeO 3 (BFO) also exhibits a G-type antiferromagnetic order convoluted with cycloidal arrangement of the antiferromagnetic vector, propagating in the [1 10] pc whilst the antiferromagnetic vector rotates in the plane spanned by [1 10] pc and [111] pc with a period of 62-64 nm [START_REF] Blaauw | Magnetic and structural properties of BiFeO3[END_REF][START_REF] Sosnowska | Spiral magnetic ordering in bismuth ferrite[END_REF][START_REF] Lebeugle | Electric-Field-Induced Spin Flop in BiFeO {3} Single Crystals at Room Temperature[END_REF][START_REF] Catalan | Physics and Applications of Bismuth Ferrite[END_REF]. BiFeO 3 , among ferroelectrics and multiferroics, has a relatively low bandgap, estimated to be of the order of 2.7 eV [START_REF] Rovillain | Polar phonons and spin excitations coupling in multiferroic bifeo 3 crystals[END_REF]. However, possible defect bands at 2.2 eV, as observed in photoconductivity spectroscopy experiments [START_REF] Yang | Electronic Origin and Tailoring of Photovoltaic Effect in BiFeO 3 Single Crystals[END_REF], have for a long time shed confusion on the true value of the bandgap. This relatively "small" bandgap has made BFO one of the prominent multiferroics for the study and application of the coupling between light and ferroic order parameters, such as photovoltaics [START_REF] Choi | Switchable ferroelectric diode and photovoltaic effect in bifeo 3[END_REF][START_REF] Yang | Photovoltaic effects in bifeo 3[END_REF][START_REF] Alexe | Tip-enhanced photovoltaic effects in bismuth ferrite[END_REF], photocatalysis [START_REF] He | Characterization and visible light photocatalytic mechanism of size-controlled bifeo 3 nanoparticles[END_REF][START_REF] Gao | Synthesis of bifeo 3 nanoparticles for the visible-light induced photocatalytic property[END_REF][START_REF] Bai | Size effect on optical and photocatalytic properties in bifeo 3 nanoparticles[END_REF] and photostriction [START_REF] Kundys | Light-induced size changes in BiFeO3 crystals[END_REF][START_REF] Kundys | Wavelength dependence of photoinduced deformation in bifeo 3[END_REF][START_REF] Wen | Electronic origin of ultrafast photoinduced strain in bifeo 3[END_REF]. Steady photostriction in BiFeO 3 was studied in Ref. [START_REF] Paillard | Photostriction in Ferroelectrics from Density Functional Theory[END_REF] using the ∆SCF method in a similar fashion to lead and barium titanates, except that the LSDA+U functional was used, with U = 3.87 eV. The band structure of bismuth ferrite is plotted in Fig. [START_REF] Fridkin | The experimental investigation of the photovoltaic effect in some crystals without a center of symmetry[END_REF]. The bottom of the conduction band (CBM) is located at the Z point, while the top of the valence band (VBM) is located in-between B and Z points. Three transitions, depicted by different arrows in Fig. [START_REF] Fridkin | The experimental investigation of the photovoltaic effect in some crystals without a center of symmetry[END_REF], are considered: Z -→ Z (denoted direct CBM, in green), V BM -→ Z (in red, denoted indirect VBM-CBM) and a vertical transition from the VBM (in blue, denoted direct VBM).

The results, shown in Fig. [START_REF] Sturman | The Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials[END_REF], show that the pseudocubic directions are shrinking upon photoexcitation. Generally speaking, pseudocubic directions "close" to the polar axis (such as [100], [110] and [111]) tend to shorten. On the other hand, due to the opening of the pseudocubic angle with the increase of photo-excited carriers, the directions "away" from the polar axis (such as [1 10] or [ 1 11]) lengthen [START_REF] Paillard | Photostriction in Ferroelectrics from Density Functional Theory[END_REF].

Interestingly, experiments on single crystals of BiFeO 3 under UV light report an elongation of +2.7 × 10 -3 % along the [101] pseudocubic direction [START_REF] Kundys | Wavelength dependence of photoinduced deformation in bifeo 3[END_REF]. According to our calculations, an expansion is only possible if this < 101 > direction is actually a [10 1] direction (i.e., perpendicular to the spontaneous polarization), in which case such expansion would occur for a concentration of photoexcited electrons of roughly 1.6 × 10 19 cm -3 . Note that, according to literature reports, concentration of photoexcited carriers ranging from 5 × 10 17 to 5 × 10 19 cm -3 were measured or estimated [START_REF] Lejman | Giant ultrafast photoinduced shear strain in ferroelectric BiFeO3[END_REF][START_REF] Prashanthi | Enhanced photo-collection in single bifeo 3 nanowire due to carrier separation from radial surface field[END_REF].

To answer the question about whether the origin of the calculated photostric- Taken from Ref. [START_REF] Paillard | Photostriction in Ferroelectrics from Density Functional Theory[END_REF]. Copyright 2016 American Physical Society.

Γ L B 1 B Z Γ X Q F P 1 Z L P -1 0 1 2 3 Energy (eV) ε F
tive effect was driven by the piezoelectric effect or not, we conducted a similar analysis as in lead and barium titanates. The polarization was shown to diminish with the number of photoexcited carriers n e . Meanwhile, the piezoelectric tensor d and the dielectric susceptibility χ, taken from molecular dynamics simulations [START_REF] Graf | Dielectric and piezoelectric properties of bifeo 3 from molecular dynamics simulations[END_REF], allowed the estimation of the piezoelectric constant g and therefore of the piezoelectric photo-induced strain (see open symbols in Fig. [START_REF] Fridkin | Bulk photovoltaic effect in noncentrosymmetric crystals[END_REF]). It is found that for the direct CBM and indirect VBM-CBM transitions, the piezoelectric model is rather accurate. Performing frozen ion ∆SCF calculations indeed proves that the electronic pressure was a minor (cooperating) contribution in those cases [START_REF] Paillard | Photostriction in Ferroelectrics from Density Functional Theory[END_REF]. On the other hand, the photo-induced change of lattice constant in the direct VBM transition case is predicted to be stronger in the piezoelectric model than the calculated values, which was found later to be the result of the competition of the piezoelectric mechanism (which in this case favors shrinking) with the potential deformation (which for this transition favors expansion). Overall, the piezoelectric effect still prevails, but its influence is reduced.

Conclusions about photostriction in ferroelectrics

Overall, the use of first-principles tends to confirm the scenario of a converse piezoelectric effect-driven photostriction in ferroelectric materials. This seems to be particularly true along the polar axis, for which the piezoelectric constant is larger. However, in directions perpendicular to the polar axis, the 12), it appears that lead titanate, at least in the case of the Z -→ Z transition, has the largest photostriction potential, which results mostly from (i) strong photo-induced change of polarization and (ii) large piezoelectric constant. Note also that two dimensional ferroelectric compounds, such as in the monochalcogenide family (SnS, SnSe), have also been predicted to be good photostrictive materials using the same methods [START_REF] Haleoot | Photostrictive Two-Dimensional Materials in the Monochalcogenide Family[END_REF].

Elasto-optic effects in ferroelectrics

Reference [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF] investigated elasto-optic effects in PTO films under epitaxial strain. For that, it first computed the total energy of different phases, as shown in Figure (13a). Note that these calculations also predicted that bulk PTO adopts a tetragonal (T) phase with a space group of P 4mm, and having a lattice constant a equal to 3.87 Å as well as an axial ratio of 1.04. Such computational results are rather consistent with experimental results of 1.02 -1.06 [START_REF] Shirane | X-ray study of the phase transition in lead titanate[END_REF]), and the fact that PTO bulk adopts a P 4mm state below 760 K [START_REF] Ronald E Cohen | Origin of ferroelectricity in perovskite oxides[END_REF]. As shown in In the strain window from 0.75% to 1.75%, a monoclinic Cm phase first has a very low energy close to that of the T phase and then becomes the ground state [see Fig. (13a)]. Another monoclinic Pm phase exhibits a very similar energy than the Cm phase, indicating that both Cm and Pm can likely be stabilized in PTO films by, e.g., applying epitaxial strain and/or electric field. Interestingly, both Pm and Cm phases have been observed for PTO films grown on a DyScO 3 substrate (corresponding to a biaxial strain of 1.4%) [START_REF] Catalan | Polar domains in lead titanate films under tensile strain[END_REF]. As shown Fig. (13b), the Cm phase decreases its axial ratio from 1.022 to 0.988 when the epitaxial strain ranges from 0.75% to 1.75%. Unlike the T phase that has only an out-ofplane component of polarization, the Cm phase has both out-of-plane and inplane components of polarization. When this strain varies from 0.75% to 1.75%, the in-plane polarization (P x ) of Cm increases from about zero to 0.66 C/cm 2 while the out-of-plane polarization P z concomitantly decreases from 0.75 C/cm 2 to almost zero. In other words and as shown in Fig. (14a), the polarization of the Cm state continuously rotates from the out-of-plane [001] direction towards the in-plane [110] direction when increasing strain in this intermediate strain range. Regarding the Pm monoclinic phase, the axial ratio decreases from values larger than 1 to less than 1 when the strain is enhanced between 0.75% to 1.75%, and is qualitatively (and even quantitatively) very similar to the axial ratio of the Cm phase. As shown in Figs. (14c) and (14d), the P z out-of-plane polarization of Pm decreases and the P x in-plane polarization increases in this tensile strain range. Note that P y is zero for the Pm phase, which differs from the case of the Cm phase for which P y =P x is non-null. As shown in Fig. (14b), the polarization of the Pm phase rotates from the pseudo-cubic [001] to [100] directions when increasing the tensile strain. These different in-plane polarization can induce different optical properties and elasto-optic effects between the Cm and Pm phases, as we will see below.

At the large strain >1.75%, an orthorhombic (O) phase has the lowest energy and thus becomes the ground state. This O phase has an axial ratio smaller than 1, a large in-plane polarization and a null out-of-plane polarization.

Optical properties and elasto-optic effects were considered in Reference [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF] for these different phases of PTO under tensile strain. The predicted refractive index n for an ω energy being 0.3 eV below the computed band gap is shown in Fig. [START_REF] Kundys | Wavelength dependence of photoinduced deformation in bifeo 3[END_REF]. n o and n e represent the refractive index for ordinary ray and extraordinary ray, respectively. As shown in Fig. (15a), n o is larger then n e in the T phase, while the reverse situation holds for the O phase. The hierarchy between n o and n e in the Cm and P m phases depends on strain. At strain less than about 1.25%, n o is higher than n e , while n o is less than n e for strain larger than 1.25%. To further investigate the hierarchy of n o and n e , Fig. (15b) shows the refractive index as a function of the axial ratio c/a. In first approximation, n o is equal to n e when c/a is 1, and n o is larger (smaller) than n e when c/a is larger (smaller) than 1. Therefore, the hierarchy between n o and n e is dependent on the c/a axial ratio. Such facts may be useful for discovering materials with large elasto-optic effects.

Furthermore, both n o and n e increase with strain in the T phase and decrease In contrast, n e at strain 0.75% of the T phase is much smaller than n e at the strain 1.5% in the O phase. n e in the Cm and Pm phases thus have to sharply increase in the intermediate strain to bridge the n e 's of the T and O phases. This sharply changing n e with strain can induce large elasto-optic effects, according to Eq. ( 16). As a matter of fact and as revealed in Table 1, the effective elasto-optic coefficient (p 31 + p 32 )/2 for the monoclinic Cm and Pm phases indeed has very large values. It is at least two times that of the T phase and three times that of the O state. The large elasto-optic coefficient is related to n e in Cm and n 3 in Pm, which significantly change with strain. These value appear to be the largest elasto-optic coefficient predicted or reported. For instance, they are about twice as large as the coefficient of lithium niobate, for which p 31 = 0.18, and 30-60% Table 1: Elasto-optic coefficients for the four studied phases of epitaxial (001) PTO films under a tensile strain of 1.2%, for an ω energy being 0.3 eV smaller than the computed band gap. The number in parentheses corresponds to (p 21 + p 22 )/2 of the P m phase, which, due to symmetry and unlike in the T , O and Cm phases, differs from (p 11 + p 12 )/2. Note that p 31 = p 32 in the T , O and Cm states, therefore making (p 31 + p 32 )/2 = p 31 in these three phases. Taken from Ref. [START_REF] Chen | Large elasto-optic effect in epitaxial pbtio 3 films[END_REF]. In this part of the review, we described the four phases of PbTiO 3 under tensile strain, and their refractive indices behavior under strain. The fast increase of refractive index n e of the monoclinic Cm and Pm phases with epitaxial strain induces very large elasto-optic effects. Further analysis reveals that such rapid change of refractive index is also related to a sharp change of the axial ratio with strain [see Fig. (15b)]. These results may provide a new route to discover large elasto-optic effect materials by searching for low-symmetry phases possessing a rapid evolution of the axial ratio and polarization with strain. Note that this rapid evolution can occur in epitaxial ferroelectrics films [START_REF] Catalan | Flexoelectric rotation of polarization in ferroelectric thin films[END_REF][START_REF] Rj Zeches | A strain-driven morphotropic phase boundary in bifeo3[END_REF] but also in the morphotropic phase boundaries of some solid solutions [START_REF] Jin | Conformal miniaturization of domains with low domain-wall energy: Monoclinic ferroelectric states near the morphotropic phase boundaries[END_REF][START_REF] Tom Ta Lummen | Thermotropic phase boundaries in classic ferroelectrics[END_REF][START_REF] Ahart | Origin of morphotropic phase boundaries in ferroelectrics[END_REF][START_REF] Rouquette | Pressure-induced rotation of spontaneous polarization in monoclinic and triclinic pbzr 0.52 ti 0.48 o 3[END_REF][START_REF] Wu | Pressure-induced anomalous phase transitions and colossal enhancement of piezoelectricity in pbtio 3[END_REF] when varying composition or hydrostatic pressure. Note also that a recent work demonstrated that large elasto-optic coefficients can occur in epitaxial thin films made of the multiferroic BiFeO 3 [START_REF] Sando | Large elasto-optic effect and reversible electrochromism in multiferroic bifeo3[END_REF].

Conclusion

In summary, this review provided details about (i) the methods used to recently predict photostriction and elasto-optic effects in ferroelectrics and multiferroics from first principles; and (ii) their concomitant results and analyses aimed at microscopically understanding such effects. We hope that these methods and analyses can be used as starting points to design novel FE and MFE materials exhibiting giant couplings between light and physical properties. Such interesting general topic is at its infancy stage, and therefore further developments (such as tackling these couplings using the more sophisticated and complex GW [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF] and time-dependent DFT [START_REF] Ullrich | Time-Dependent Density-Functional Theory: Concepts and Applications[END_REF] schemes as well as exciting related discoveries are likely to occur in a near future.

Figure 2 :

 2 Figure 2: Band structures of (a) tetragonal lead titanate, and (b) rhombohedral barium titanate. The reference energy is placed at the middle of the Kohn-Sham gap. The arrows represent the different transitions that are considered. Adapted from Ref. [42]. Copyright 2017 of the American Physical Society.

Figure 3 :

 3 Figure 3: Change of lattice constant orthogonal (left panel) and parallel (right panel) to the tetragonal axis in the P 4mm phase of lead titanate, as obtained from ∆SCF calculations (filled symbols) and as estimated from the converse piezoelectric model (open symbols). Reproduced from Ref. [42]. Copyright 2017 from the American Physical Society.

  ) and Fig. (4), was plotted as open symbols in Fig. (

Figure 4 :

 4 Figure 4: Change of polarization against the concentration of excited carriers, for different transitions in (left) tetragonal lead titanate and (right) rhombohedral barium titanate. Reproduced from Ref. [42]. Copyright 2017 American Physical Society.

  (6.b)]. Actually, observing Fig. (6.b),

Figure 7 :

 7 Figure 7: Change of pseudo-cubic lattice constant (left panel) and pseudocubic angle (right panel) versus concentration of excited electrons for different transitions in rhombohedral barium titanate. Reproduced from Ref. [42]. Copyright 2017 American Physical Society.

Figure 8 :

 8 Figure 8: Bismuth ferrite rhombohedral unit cell (green) and pseudocubic 5atom unit cell.

Figure 9 :

 9 Figure9: Band of R3c bismuth ferrite, and the different transitions considered in Ref.[START_REF] Paillard | Photostriction in Ferroelectrics from Density Functional Theory[END_REF]. Copyright 2016 American Physical Society.

Figure 10 :

 10 Figure 10: Change of (a) pseudocubic lattice constant, (b) pseudocubic angle and (c) length along the pseudocubic [1 10] direction with the number of photoexcited carriers in rhombohedral bismuth ferrite. Taken from Ref. [74]. Copyright 2016 American Physical Society.

Figure 11 :

 11 Figure 11: Calculated photo-induced change in pseudocubic lattice constant (filled symbols) and estimated change in lattice constant caused by reduction of the polarization and converse piezoelectric effect (open symbols) in R3c BiFeO 3 .Taken from Ref.[START_REF] Paillard | Photostriction in Ferroelectrics from Density Functional Theory[END_REF]. Copyright 2016 American Physical Society.

Figure 12 :

 12 Figure 12: (left panel) Photo-induced change of length in tetragonal PTO, rhombohedral BTO and rhombohedral BFO along several directions, for different transitions; (right panel) Change in pseudocubic angle in rhombohedral BTO versus rhombohedral BFO. Taken from Ref. [42]. Copyright 2017 from the American Physical Society.

  Fig. (13a), this tetragonal T phase is also predicted to be the ground state (i.e., it has the lowest energy) for strains ranging from 0% to 1%. Its out-of-plane polarization (P z ) decreases from 0.87 C/m 2 to 0.80 C/m 2 [see Fig. (13d)], and its axial ratio reduces from 1.04 to 1.02 as the strain increases from 0 to 0.75% [see Fig. (13b)].

Figure 13 :

 13 Figure 13: Total energy (a), axial ratio (b), x-component of the polarization, P x (c), and out-of-plane component of the polarization, P z (d), of PTO films as a function of tensile strain. Note that P y =P x for the O and Cm phases, P y =0 for the T and Pm phases. Taken from Ref. [45]. Copyright 2016 from the American Physical Society.

Figure 14 :

 14 Figure 14: Schematic representations of polarization directions of the Cm, P m, O, and T phases. Polarization of the T phase is along the [001] direction, of the O phase is along the [110] direction. Polarization of Cm is along a [uuv] direction (with u being different with v), which rotates from the [100] direction to the [110] direction under strain, and polarization of P m is along a [u0v] direction (with u being different with v), which rotates from the [001] direction to the [100] direction under strain.

Figure 15 :

 15 Figure 15: Panel (a): Refractive index n of ordinary (open symbols) and extraordinary rays (filled symbols) for PTO in its T (squares), O (dots) and Cm (up triangles) phases as a function of strain. Note that for the monoclinic P m phase, the x-and y-directions are not equivalent by symmetry (due to the polarization direction), and therefore n 1 and n 2 differ. In-plane and out-of-plane refractive indices of the P m phase (down triangles) are also shown. Panels (b) and (c): Similar to Panel (a) but as a function of the axial ratio and magnitude of the polarization, respectively. Taken from Ref. [45]. Copyright 2016 American Physical Society.

  This work is supported by the ONR Grant N00014-17-1-2818 (Y.Y), the DARPA Grant No. HR0011-15-2-0038 under the MATRIX program (C.P.) and the U.S. AFOSR Grant No. FA9550-16-1-0065 (B.X. and L.B.).

  Copyright 2016 American Physical Society. 11 + p 12 )/2 -0.10 0.20 -0.09 (-0.13) -0.04 (p 31 + p 32 )/2 -0.22 0.10 -0.43 -0.35 bigger than that of quartz (p 31 = 0.29) [81].

	Phase	T	O	P m	Cm
	(p