text: (cid:32) (cid:33) − − 1/n(cid:48) − σel pl σel pl εel pl = a + a . (13) a E K N 4. Solve the Coﬃn-Manson-Basquin equation for , sur (cid:48) − σ N (cid:48) N εel pl = f (2 )b + ε (2 )c. (14) a E sur f sur (cid:48) (cid:48) Here σ , ε > 0 and b, c < 0 are material constants. f f Let us now turn to the case, when, in addition to a mechanical load, a temperature change from the baseline temperature T to the temperature ﬁeld T takes place. It is 0 then usually assumed that the durability changes with temperature. Dissemination of displacements through the crystal is facilitated by the thermal excitation of the atomic oscillations in the lattice. Therefore, LCF life to crack initiation usually decreases with temperature. Here, we take a simplistic Arrhenius law [4] for the temperature dependence of LCF life. This is an ’import’ from creep damage modelling and not necessarily the most adequate temperature model. Alternatively one could consider temperature dependent (cid:48) CMB parameters σ (T ), ε (T ), b(T ) and c(T ) modelled by continuous functions. The ﬁeld f f of temperature models in fatigue however is vast and lies beyond the scope of this article. We thus choose · − − N · N ( ) = e Q(T T0) ( ). (15) sur sur Here Q plays the rôle of an activation energy. Using (14) it is easily shown that (cid:48) (cid:48) this corresponds to b(T ) = b and c(T ) = c constant and σf(T) = e(cid:16)−Qb(T(cid:17)−T0) σf and (cid:48) − − (cid:48) E E ε (T ) = e Qc(T T0)ε . Wrapping up these modelling steps, we obtain the following: f f ∇ m Lemma 2.11. Let N = N ( u, T ) be deﬁned as above. Then 1 depends ∇ sur sur Nsur continuously on u and T . ∇ Note that models that include notch support factors [4] also require derivatives 2u. Therefore second order derivatives will enter into the deﬁnition of N . sur F We have thus shown that in realistic models, the functions from (4) can be vol/sur assumed to be continuous, as long as the temperature does not exceed a certain limit T . m 3 The Thermomechanic Formulation of Linear Elasticity Up to here, we have seen that volume and surface forces as well as changes in temper- atures that are imposed to a device have an impact on its durability. In linear thermo- elasticity both aspects are taken into account, see [17]. 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: σel

text: pl

text: σel

text: pl

text: εel

text: pl

text: a

text: a

text: a

text: E

text: K

text: N

text: Solve

text: the

text: Coﬃn-Manson-Basquin

text: equation

text: for

text: sur

text: σ

text: N

text: N

text: εel

text: pl

text: f

text: )b

text: ε

text: )c.

text: a

text: E

text: sur

text: f

text: sur

text: Here

text: σ

text: ε

text: and

text: b,

text: c

text: are

text: material

text: constants.

text: f

text: f

text: Let

text: us

text: now

text: turn

text: to

text: the

text: case,

text: when,

text: in

text: addition

text: to

text: a

text: mechanical

text: load,

text: a

text: temperature

text: change

text: from

text: the

text: baseline

text: temperature

text: T

text: to

text: the

text: temperature

text: ﬁeld

text: T

text: takes

text: place.

text: It

text: is

text: then

text: usually

text: assumed

text: that

text: the

text: durability

text: changes

text: with

text: temperature.

text: Dissemination

text: of

text: displacements

text: through

text: the

text: crystal

text: is

text: facilitated

text: by

text: the

text: thermal

text: excitation

text: of

text: the

text: atomic

text: oscillations

text: in

text: the

text: lattice.

text: Therefore,

text: LCF

text: life

text: to

text: crack

text: initiation

text: usually

text: decreases

text: with

text: temperature.

text: Here,

text: we

text: take

text: a

text: simplistic

text: Arrhenius

text: law

text: for

text: the

text: temperature

text: dependence

text: of

text: LCF

text: life.

text: This

text: is

text: an

text: ’import’

text: from

text: creep

text: damage

text: modelling

text: and

text: not

text: necessarily

text: the

text: most

text: adequate

text: temperature

text: model.

text: Alternatively

text: one

text: could

text: consider

text: temperature

text: dependent

text: CMB

text: parameters

text: σ

text: (T

text: ε

text: (T

text: b(T

text: and

text: c(T

text: modelled

text: by

text: continuous

text: functions.

text: The

text: ﬁeld

text: f

text: f

text: of

text: temperature

text: models

text: in

text: fatigue

text: however

text: is

text: vast

text: and

text: lies

text: beyond

text: the

text: scope

text: of

text: this

text: article.

text: We

text: thus

text: choose

text: N

text: N

text: e

text: Q(T

text: sur

text: sur

text: Here

text: Q

text: plays

text: the

text: rôle

text: of

text: an

text: activation

text: energy.

text: Using

text: it

text: is

text: easily

text: shown

text: that

text: this

text: corresponds

text: to

text: b(T

text: b

text: and

text: c(T

text: c

text: constant

text: and

text: σf(T)

text: σf

text: and

text: E

text: E

text: ε

text: (T

text: e

text: Qc(T

text: Wrapping

text: up

text: these

text: modelling

text: steps,

text: we

text: obtain

text: the

text: following:

text: f

text: f

text: m

text: Lemma

text: Let

text: N

text: N

text: u,

text: T

text: be

text: deﬁned

text: as

text: above.

text: Then

text: depends

text: sur

text: sur

text: Nsur

text: continuously

text: on

text: u

text: and

text: T

text: Note

text: that

text: models

text: that

text: include

text: notch

text: support

text: factors

text: also

text: require

text: derivatives

text: Therefore

text: second

text: order

text: derivatives

text: will

text: enter

text: into

text: the

text: deﬁnition

text: of

text: N

text: sur

text: F

text: We

text: have

text: thus

text: shown

text: that

text: in

text: realistic

text: models,

text: the

text: functions

text: from

text: can

text: be

text: vol/sur

text: assumed

text: to

text: be

text: continuous,

text: as

text: long

text: as

text: the

text: temperature

text: does

text: not

text: exceed

text: a

text: certain

text: limit

text: T

text: m

text: The

text: Thermomechanic

text: Formulation

text: of

text: Linear

text: Elasticity

text: Up

text: to

text: here,

text: we

text: have

text: seen

text: that

text: volume

text: and

text: surface

text: forces

text: as

text: well

text: as

text: changes

text: in

text: temper-

text: atures

text: that

text: are

text: imposed

text: to

text: a

text: device

text: have

text: an

text: impact

text: on

text: its

text: durability.

text: In

text: linear

text: thermo-

text: elasticity

text: both

text: aspects

text: are

text: taken

text: into

text: account,

text: see

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: → ∞ (cid:32) (cid:32) Φ(Ω ) as k , as well as T T and u u for the corresponding solutions T b P ∈nk k→∞ nk k→∞ and u to (Ω), where (T, u) V (Ω). P Proof. In terms of Lemma 5.4 there is a convergent subsequence (Φ ) ∈N tending to some ∈ → ∞ (cid:107) (cid:107) ≤ (cid:48) nl l −O→ Φ C4,α(Ωext) as l concerning . (cid:48) , 0 α < α. Hence, Ω Ω := [Ck,α (Ωext)]3 ∈ n Φ(Ω ), when passing to the limit. According to Theorem 6.11 T C2,φ(Ω ) for every n ∈bN. Moreover, this theorem leads to (cid:107)T (cid:107) ≤ Ct for evnerl y n . Let pnl := p1,2,φ bel the extension operator in Deﬁnition 7.1nlaCnd2,φs(ΩetnlT) ext := p T . lBecauseΩnolf LemΩmnla (cid:107) (cid:107) ≤nl (cid:107) Ωnl(cid:107) nl ≤ A.3 there is a constant Cext such that T ext Cext T CextCt. The constant Cext can again be chosen unifonrl mlCy2,wφ(iΩtehxtr)egard to Onl, Cw2h,φa(tΩnwl)as shown in the (cid:107) (cid:107) previous lemma. This results in a uniform bound for all T ext . By means of → nl C∗2,φ(Ωex→t) ∞ Lemma A.1, there is again a convergent subsequence T ext T ext, as j concerning (cid:107) (cid:107) ∗ ∈ (cid:48) nj . (cid:48) . The limit T ext, C2,φ (Ωext) even is an element of C2,φ(Ω), see the proof C2,φ (Ωext) of Lemma 5.4. Thus, Ω −O→ Ω, T ext −C−2−,φ→(cid:48) T ext,∗ for some T ext,∗ ∈ C2,φ(Ω). nj nj ∗ Because, in particular, all derivatives of T ext up to order two converge to those of T ext, → ∞ ∗ (cid:22) nj as j the function T ext, solves (17). For Theorem 6.11 holds, (17) has a unique Ω ∗ solution T , and T ext, has to be an extension of T to Ωext. ∈ We apply the same arguments to the solutions u [C3,ϕ(Ω )]3 that are allocated nj nj to Φ and T ext. In this way, we ﬁnally obtain a further subsequence (Ω , T ext, uext) ∈N, nj nj nk nk nk k where uext = p3,3,ϕu and nk Ωnk nk Ω −O→ Ω, T ext −C−2−,φ→(cid:48) T ext,∗, uext −C−3−,ϕ→(cid:48) uext,∗ for some uext,∗ ∈ C3,ϕ(Ω). nk nk nk Actually (T ext,∗, uext,∗) (cid:22) solves (19), because T ext,∗ (cid:22) solves (17) and uext −C−→3 uext,∗, Ω Ω u ∗ ∗ l where (T ext, , uext, ) is an extension of the unique solution (T, u) of (19). (cid:80) (cid:80) Lemma 7.4 (Continuity of Local Cost Funktionals). Let F , F ∈ C0(Rd) with vol sur O d = 3 + s 3j + r 3j+1, s = 2, r = 3, and let the set only consist of C0-admissible j=0 ∈ O j=∈0 ∈ shapes. For Ω , T C2(Ω) and u C3(Ω) consider the volume integral J (Ω, T, u) vol and the surface integral J (Ω, T, u) of (38). sur ∈ O −O→ → ∞ ∈ (cid:32) Let (Ω ) ∈N with Ω Ω as n , (u ) ∈N [C3(Ω )]3 be a sequence with u u n n n n n n n ∈ (cid:32) (q = 3, β = 0) and (T ) ∈N C2(Ω ) be a sequence with T T (q = 2, β = 0). Then, n n n n J → → ∞ (i) (Ω , T , u ) J (Ω, T, u) as n . vol n n n vol O J → (ii) If the family consists only of C1-admissible shapes, then (Ω , T , u ) J (Ω, T, u) sur n n n sur → ∞ as n . Proof. (i) Statement (i) can be proofed in the same way as Lemma 6.3 in [15]. 29 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Φ(Ω

text: as

text: k

text: as

text: well

text: as

text: T

text: T

text: and

text: u

text: u

text: for

text: the

text: corresponding

text: solutions

text: T

text: b

text: P

text: ∈nk

text: k→∞

text: nk

text: k→∞

text: and

text: u

text: to

text: (Ω),

text: where

text: (T,

text: u)

text: V

text: (Ω).

text: P

text: Proof.

text: In

text: terms

text: of

text: Lemma

text: there

text: is

text: a

text: convergent

text: subsequence

text: (Φ

text: ∈N

text: tending

text: to

text: some

text: nl

text: l

text: −O→

text: Φ

text: as

text: l

text: concerning

text: α

text: α.

text: Hence,

text: Ω

text: Ω

text: [Ck,α

text: n

text: Φ(Ω

text: when

text: passing

text: to

text: the

text: limit.

text: According

text: to

text: Theorem

text: T

text: for

text: every

text: n

text: ∈bN.

text: Moreover,

text: this

text: theorem

text: leads

text: to

text: Ct

text: for

text: evnerl

text: y

text: n

text: Let

text: pnl

text: bel

text: the

text: extension

text: operator

text: in

text: Deﬁnition

text: ext

text: p

text: T

text: lBecauseΩnolf

text: LemΩmnla

text: ≤nl

text: nl

text: there

text: is

text: a

text: constant

text: Cext

text: such

text: that

text: T

text: ext

text: Cext

text: T

text: CextCt.

text: The

text: constant

text: Cext

text: can

text: again

text: be

text: chosen

text: unifonrl

text: to

text: Onl,

text: shown

text: in

text: the

text: previous

text: lemma.

text: This

text: results

text: in

text: a

text: uniform

text: bound

text: for

text: all

text: T

text: ext

text: By

text: means

text: of

text: nl

text: Lemma

text: there

text: is

text: again

text: a

text: convergent

text: subsequence

text: T

text: ext

text: T

text: ext,

text: as

text: j

text: concerning

text: nj

text: The

text: limit

text: T

text: ext,

text: (Ωext)

text: even

text: is

text: an

text: element

text: of

text: see

text: the

text: proof

text: (Ωext)

text: of

text: Lemma

text: Thus,

text: Ω

text: −O→

text: Ω,

text: T

text: ext

text: T

text: ext,∗

text: for

text: some

text: T

text: ext,∗

text: nj

text: nj

text: Because,

text: in

text: particular,

text: all

text: derivatives

text: of

text: T

text: ext

text: up

text: to

text: order

text: two

text: converge

text: to

text: those

text: of

text: T

text: ext,

text: nj

text: as

text: j

text: the

text: function

text: T

text: ext,

text: solves

text: For

text: Theorem

text: holds,

text: has

text: a

text: unique

text: Ω

text: solution

text: T

text: and

text: T

text: ext,

text: has

text: to

text: be

text: an

text: extension

text: of

text: T

text: to

text: Ωext.

text: We

text: apply

text: the

text: same

text: arguments

text: to

text: the

text: solutions

text: u

text: that

text: are

text: allocated

text: nj

text: nj

text: to

text: Φ

text: and

text: T

text: ext.

text: In

text: this

text: way,

text: we

text: ﬁnally

text: obtain

text: a

text: further

text: subsequence

text: (Ω

text: T

text: ext,

text: uext)

text: ∈N,

text: nj

text: nj

text: nk

text: nk

text: nk

text: k

text: where

text: uext

text: and

text: nk

text: Ωnk

text: nk

text: Ω

text: −O→

text: Ω,

text: T

text: ext

text: T

text: ext,∗,

text: uext

text: uext,∗

text: for

text: some

text: uext,∗

text: nk

text: nk

text: nk

text: Actually

text: (T

text: ext,∗,

text: uext,∗)

text: solves

text: because

text: T

text: ext,∗

text: solves

text: and

text: uext

text: −C−→3

text: uext,∗,

text: Ω

text: Ω

text: u

text: l

text: where

text: (T

text: ext,

text: uext,

text: is

text: an

text: extension

text: of

text: the

text: unique

text: solution

text: (T,

text: u)

text: of

text: Lemma

text: (Continuity

text: of

text: Local

text: Cost

text: Funktionals).

text: Let

text: F

text: F

text: with

text: vol

text: sur

text: O

text: d

text: s

text: r

text: s

text: r

text: and

text: let

text: the

text: set

text: only

text: consist

text: of

text: O

text: shapes.

text: For

text: Ω

text: T

text: and

text: u

text: consider

text: the

text: volume

text: integral

text: J

text: (Ω,

text: T,

text: u)

text: vol

text: and

text: the

text: surface

text: integral

text: J

text: (Ω,

text: T,

text: u)

text: of

text: sur

text: O

text: −O→

text: Let

text: (Ω

text: ∈N

text: with

text: Ω

text: Ω

text: as

text: n

text: (u

text: ∈N

text: be

text: a

text: sequence

text: with

text: u

text: u

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: (q

text: β

text: and

text: (T

text: ∈N

text: be

text: a

text: sequence

text: with

text: T

text: T

text: (q

text: β

text: Then,

text: n

text: n

text: n

text: n

text: J

text: (i)

text: (Ω

text: T

text: u

text: J

text: (Ω,

text: T,

text: u)

text: as

text: n

text: vol

text: n

text: n

text: n

text: vol

text: O

text: J

text: (ii)

text: If

text: the

text: family

text: consists

text: only

text: of

text: shapes,

text: then

text: (Ω

text: T

text: u

text: J

text: (Ω,

text: T,

text: u)

text: sur

text: n

text: n

text: n

text: sur

text: as

text: n

text: Proof.

text: (i)

text: Statement

text: (i)

text: can

text: be

text: proofed

text: in

text: the

text: same

text: way

text: as

text: Lemma

text: in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: with (cid:82) (cid:82) J F ∇ ∇ ∇ ∇ (Ω, u, T ) = (t, x, T, T, u, u, 2u, 3u) dx Jvol,t Ω Fvol ∇ ∇ ∇ ∇ . (4) (Ω, u, T ) = (t, x, T, T, u, u, 2u, 3u) dA sur,t ∂Ω sur Then the optimal reliability problems Deﬁnition 2.5 can be translated to the following shape optimization problems: ∈ O Lemma 2.6. Let the crack initiation process γ = γ(Ω) for some Ω be a PPP with intensity measure (1). ∗ ∈ O ∈ R (i) A shape Ω solves the optimal reliability problem (i) at ﬁxed time t if and + only if J ∗ ≤ J ∀ ∈ O (Ω , u, T ) (Ω, u, T ) Ω . (5) t t ∗ (ii) Furthermore a shape Ω solves the optimal reliability problem 2.5 in the sense of ﬁrst ∗ ∈ R order stochastic dominance 2.4 (ii), if and only if Ω solves (5) for all t . + ∗ (iii) Finally, a shape Ω also solves the optimal reliability problem in the sense of instan- taneous hazard, if an only if J ∗ J d t(Ω , u, T ) ≤ d t(Ω, u, T ) ∀t ∈ R , Ω ∈ O. (6) dt dt + ≤ − −J Proof. This follows from P (τ t) = 1 e vol,t(Ω,u,T), see Lemma 2.3 and equations (1) J ∗ and (3–4). Thus, h (t) = d t(Ω ,u,T). τ dt Remark 2.7: Another perspective to the problem of optimal reliability is given by the notion of acceptability functionals: Let A(τ) be an acceptability functional in the sense of [19]. Common choices include the life expectation A(τ) = E[τ] or risk adjusted versions of it, e.g. A(τ) = E[τ] − δVar[τ] for some δ > 0. Furthermore, for φ : R → R measurable, A (τ) = E[φ(τ)] also deﬁnes an φ acceptability functional, provided the φ(τ) is in L1(Ξ, P ). Then, the design Ω is more or (cid:48) A ≥ A (cid:48) equally reliable than Ω with respect to A, if (τ) (τ ). φ φ ∗ Hence, a shape Ω solves the optimal reliability problem in the sense of acceptability for all A with increasing φ, if and only if it solves (5) ∀t ∈ R , see the equivalent formulations φ + of ﬁrst order stochastic dominance in [19, Theorem 1.13 (i)]. 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

label: equation

text: J

text: F

text: (Ω,

text: u,

text: T

text: (t,

text: x,

text: T,

text: T,

text: u,

text: u,

text: dx

text: Jvol,t

text: Ω

text: Fvol

text: (Ω,

text: u,

text: T

text: (t,

text: x,

text: T,

text: T,

text: u,

text: u,

text: dA

text: sur,t

text: ∂Ω

text: sur

text: Then

text: the

text: optimal

text: reliability

text: problems

text: Deﬁnition

text: can

text: be

text: translated

text: to

text: the

text: following

text: shape

text: optimization

text: problems:

text: O

text: Lemma

text: Let

text: the

text: crack

text: initiation

text: process

text: γ

text: γ(Ω)

text: for

text: some

text: Ω

text: be

text: a

text: PPP

text: with

text: intensity

text: measure

label: list

text: O

text: R

text: (i)

text: A

text: shape

text: Ω

text: solves

text: the

text: optimal

text: reliability

text: problem

text: (i)

text: at

text: ﬁxed

text: time

text: t

text: if

text: and

text: only

text: if

text: J

text: J

text: O

text: (Ω

text: u,

text: T

text: (Ω,

text: u,

text: T

text: Ω

text: t

text: t

text: (ii)

text: Furthermore

text: a

text: shape

text: Ω

text: solves

text: the

text: optimal

text: reliability

text: problem

text: in

text: the

text: sense

text: of

text: ﬁrst

text: R

text: order

text: stochastic

text: dominance

text: (ii),

text: if

text: and

text: only

text: if

text: Ω

text: solves

text: for

text: all

text: t

text: (iii)

text: Finally,

text: a

text: shape

text: Ω

text: also

text: solves

text: the

text: optimal

text: reliability

text: problem

text: in

text: the

text: sense

text: of

text: instan-

text: taneous

text: hazard,

text: if

text: an

text: only

text: if

text: J

text: J

text: d

text: t(Ω

text: u,

text: T

text: d

text: t(Ω,

text: u,

text: T

text: ∀t

text: R

text: Ω

text: O.

text: dt

text: dt

text: −J

text: Proof.

text: This

text: follows

text: from

text: P

text: (τ

text: t)

text: e

text: vol,t(Ω,u,T),

text: see

text: Lemma

text: and

text: equations

text: J

text: and

text: Thus,

text: h

text: (t)

text: d

text: t(Ω

text: ,u,T).

text: τ

text: dt

text: Remark

text: Another

text: perspective

text: to

text: the

text: problem

text: of

text: optimal

text: reliability

text: is

text: given

text: by

text: the

text: notion

text: of

text: acceptability

text: functionals:

text: Let

text: A(τ)

text: be

text: an

text: acceptability

text: functional

text: in

text: the

text: sense

text: of

text: Common

text: choices

text: include

text: the

text: life

text: expectation

text: A(τ)

text: E[τ]

text: or

text: risk

text: adjusted

text: versions

text: of

text: it,

text: e.g.

text: A(τ)

text: E[τ]

text: δVar[τ]

text: for

text: some

text: δ

text: Furthermore,

text: for

text: φ

text: R

text: R

text: measurable,

text: A

text: (τ)

text: E[φ(τ)]

text: also

text: deﬁnes

text: an

text: φ

text: acceptability

text: functional,

text: provided

text: the

text: φ(τ)

text: is

text: in

text: P

text: Then,

text: the

text: design

text: Ω

text: is

text: more

text: or

text: A

text: A

text: equally

text: reliable

text: than

text: Ω

text: with

text: respect

text: to

text: A,

text: if

text: (τ)

text: (τ

text: φ

text: φ

text: Hence,

text: a

text: shape

text: Ω

text: solves

text: the

text: optimal

text: reliability

text: problem

text: in

text: the

text: sense

text: of

text: acceptability

text: for

text: all

text: A

text: with

text: increasing

text: φ,

text: if

text: and

text: only

text: if

text: it

text: solves

text: ∀t

text: R

text: see

text: the

text: equivalent

text: formulations

text: φ

text: of

text: ﬁrst

text: order

text: stochastic

text: dominance

text: in

text: Theorem

text: (i)].

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 J. CHAPPELON, L. MART´INEZ-SANDOVAL, L. MONTEJANO, L.P. MONTEJANO, AND J.L. RAM´IREZ ALFONS´IN ∈ (cid:48) (cid:48) position there is x X which is not in H . Let H be the hyperplane generated by H and x. − Consequently, H contains exactly d points of X and as above there are exactly k 3 points of X ∩ − at each side of H. This implies that L is transversal to every triangle of X (H L), otherwise, − these three points together with the k 3 points outside H, but in the same side, give rise to a k-set whose convex hull does not intersect L. But this is a contradiction because the triangle { } (cid:3) with vertices v, w, x does not intersect L. − − It is not diﬃcult to generalize Theorem 2.5 for λ > 3 if we ask d 2(λ 1) points of X to be − contained in a (d λ)-plane transversal L and a collection of simplices (formed with the rest of points) of diﬀerent dimensions (not necessarily intervals) to be all intersected by L. However, we have not been able to prove that also for λ > 3 every simplex can be chosen to be an interval. (cid:0) (cid:1) We may now prove Theorem 2.1. { } Proof of Theorem 2.1. Let us ﬁrst introduce some notation. Let [n] = 1, . . . , n and let [n] be s− the collection of all subsets of size s. We denote by Ω the ﬁnite set of partitions of [d + 2(k λ)] − (cid:116) (cid:116) · · · (cid:116) | | − − | | of the form [d + 2(k λ)] = B B B , where B = d 2(λ 1) and B = 2, for every (cid:54) (cid:54) 1 2 k 1 i 2 i k. Let X be our collection of points that we suppose, without loss of generality, in general − position (if not, we slighly perturb X). Let L be a non-complete (d λ)-plane transversal to the convex hulls of all k-sets of X. Since X is in general position then, by Theorem 2.5, we have | ∩ | − − − − that X L = d 2(λ 1) and the other 2(k 1) points of X are matched in k 1 pairs in such a way that L intersects the corresponding closed segments determined by them. Therefore, this ∈ non-complete Kneser transversal L naturally yields a partition α Ω. If this is so, we shall say that L is a Kneser transversal of type α. We have the following immediate consequences of this deﬁnition. ∈ (1) To every non-complete Kneser transversal it corresponds a unique type α Ω. − (cid:62) (2) By Lemma 2.3, since k λ 2, any two non-complete Kneser transversals of the same type coincide. (3) Since Ω is a ﬁnite set then X admits a ﬁnite number of Kneser transversals. ∈ (4) Given X and α Ω, there exists (cid:15) > 0 such that if X does not admit a Kneser transversal (cid:48) { (cid:48) (cid:48) } R of type α and X = x , . . . , x is a collection of points in d in general position such | − (cid:48)| 1 ∈n (cid:48) that x x < (cid:15), for every i [n], then X does not admit a Kneser transversal of type ∈ i i α Ω. ∈ Our next goal is to show that if X admits a Kneser transversal of type α Ω and (cid:15) > 0 is R (cid:48) { (cid:48) (cid:48) } given, then we can choose a collection of points in d in general position, X = x , . . . , x , | − (cid:48)| ∈ (cid:48) 1 n such that x x < (cid:15), for every i [n] and with the property that X does not admit a Kneser i ∈ transversal of type α Ω. To see this, we may assume without loss of generality that { − − } { − − } { − − } α = 1, . . . , d 2(λ 1) , d 2λ + 3, d 2λ + 4 , . . . , d + 2k + 2λ 1, d + 2k 2λ − { } − and L is a (d λ)-plane that contains x , . . . , x − − and intersects the k 1 closed segments 1 d 2(λ 1) with extreme points { } { } x − , x − , . . . , x − , x − . d 2λ+3 d 2λ+4 d+2k+2λ 1 d+2k 2λ (cid:48) | − (cid:48) | Choose x − such that x − x − < (cid:15) and in such a way that L does not d+2k 2λ {d+2k 2λ d+(cid:48) 2k 2λ } (cid:48) ∪ { (cid:48) } − intersect the line generated by x − , x − . Suppose now X = X x − { } (cid:48) d+2k+2λ 1 d+2k (cid:48)2λ d+2−k 2λ(cid:62) x − and let L be a Kneser transversal of X of type α. By Lemma 2.3, since k 2 λ, d+2k 2λ (cid:48) L must be equal than L , which is imposible because L does not intersect the line generated { (cid:48) } by x − , x − . Furthermore, (cid:15) > 0 is so small that if X does not contain a Kneser d+2k+2λ 1 d+2k∈ 2λ (cid:48) transversal of type β Ω, X either. As a consequence of our last statement and of (1)-(4), it follows that given (cid:15) > 0 suﬃciently R (cid:48) { (cid:48) (cid:48) } small, there is a collection of points in d in general position X = x , . . . , x , such that | − (cid:48)| ∈ 1 n x x < (cid:15), for every i [n] and with the property that the number of non-complete Kneser i i (cid:48) transversals of X is bigger that the number of non-complete Kneser transversals of X . Since the number of non-complete Kneser transversals of X is ﬁnite, this completes the proof of our (cid:3) theorem.

text: J.

text: CHAPPELON,

text: L.

text: MART´INEZ-SANDOVAL,

text: L.

text: MONTEJANO,

text: L.P.

text: MONTEJANO,

text: AND

text: J.L.

text: RAM´IREZ

text: ALFONS´IN

text: position

text: there

text: is

text: x

text: X

text: which

text: is

text: not

text: in

text: H

text: Let

text: H

text: be

text: the

text: hyperplane

text: generated

text: by

text: H

text: and

text: x.

text: Consequently,

text: H

text: contains

text: exactly

text: d

text: points

text: of

text: X

text: and

text: as

text: above

text: there

text: are

text: exactly

text: k

text: points

text: of

text: X

text: at

text: each

text: side

text: of

text: H.

text: This

text: implies

text: that

text: L

text: is

text: transversal

text: to

text: every

text: triangle

text: of

text: X

text: (H

text: L),

text: otherwise,

text: these

text: three

text: points

text: together

text: with

text: the

text: k

text: points

text: outside

text: H,

text: but

text: in

text: the

text: same

text: side,

text: give

text: rise

text: to

text: a

text: k-set

text: whose

text: convex

text: hull

text: does

text: not

text: intersect

text: L.

text: But

text: this

text: is

text: a

text: contradiction

text: because

text: the

text: triangle

text: with

text: vertices

text: v,

text: w,

text: x

text: does

text: not

text: intersect

text: L.

text: It

text: is

text: not

text: diﬃcult

text: to

text: generalize

text: Theorem

text: for

text: λ

text: if

text: we

text: ask

text: d

text: points

text: of

text: X

text: to

text: be

text: contained

text: in

text: a

text: (d

text: λ)-plane

text: transversal

text: L

text: and

text: a

text: collection

text: of

text: simplices

text: (formed

text: with

text: the

text: rest

text: of

text: points)

text: of

text: diﬀerent

text: dimensions

text: (not

text: necessarily

text: intervals)

text: to

text: be

text: all

text: intersected

text: by

text: L.

text: However,

text: we

text: have

text: not

text: been

text: able

text: to

text: prove

text: that

text: also

text: for

text: λ

text: every

text: simplex

text: can

text: be

text: chosen

text: to

text: be

text: an

text: interval.

text: We

text: may

text: now

text: prove

text: Theorem

text: Proof

text: of

text: Theorem

text: Let

text: us

text: ﬁrst

text: introduce

text: some

text: notation.

text: Let

text: [n]

text: n

text: and

text: let

text: [n]

text: be

text: s−

text: the

text: collection

text: of

text: all

text: subsets

text: of

text: size

text: s.

text: We

text: denote

text: by

text: Ω

text: the

text: ﬁnite

text: set

text: of

text: partitions

text: of

text: [d

text: λ)]

text: of

text: the

text: form

text: [d

text: λ)]

text: B

text: B

text: B

text: where

text: B

text: d

text: and

text: B

text: for

text: every

text: k

text: i

text: i

text: k.

text: Let

text: X

text: be

text: our

text: collection

text: of

text: points

text: that

text: we

text: suppose,

text: without

text: loss

text: of

text: generality,

text: in

text: general

text: position

text: (if

text: not,

text: we

text: slighly

text: perturb

text: X).

text: Let

text: L

text: be

text: a

text: non-complete

text: (d

text: λ)-plane

text: transversal

text: to

text: the

text: convex

text: hulls

text: of

text: all

text: k-sets

text: of

text: X.

text: Since

text: X

text: is

text: in

text: general

text: position

text: then,

text: by

text: Theorem

text: we

text: have

text: that

text: X

text: L

text: d

text: and

text: the

text: other

text: points

text: of

text: X

text: are

text: matched

text: in

text: k

text: pairs

text: in

text: such

text: a

text: way

text: that

text: L

text: intersects

text: the

text: corresponding

text: closed

text: segments

text: determined

text: by

text: them.

text: Therefore,

text: this

text: non-complete

text: Kneser

text: transversal

text: L

text: naturally

text: yields

text: a

text: partition

text: α

text: Ω.

text: If

text: this

text: is

text: so,

text: we

text: shall

text: say

text: that

text: L

text: is

text: a

text: Kneser

text: transversal

text: of

text: type

text: α.

text: We

text: have

text: the

text: following

text: immediate

text: consequences

text: of

text: this

text: deﬁnition.

text: To

text: every

text: non-complete

text: Kneser

text: transversal

text: it

text: corresponds

text: a

text: unique

text: type

text: α

text: Ω.

text: By

text: Lemma

text: since

text: k

text: λ

text: any

text: two

text: non-complete

text: Kneser

text: transversals

text: of

text: the

text: same

text: type

text: coincide.

text: Since

text: Ω

text: is

text: a

text: ﬁnite

text: set

text: then

text: X

text: admits

text: a

text: ﬁnite

text: number

text: of

text: Kneser

text: transversals.

text: Given

text: X

text: and

text: α

text: Ω,

text: there

text: exists

text: such

text: that

text: if

text: X

text: does

text: not

text: admit

text: a

text: Kneser

text: transversal

text: R

text: of

text: type

text: α

text: and

text: X

text: x

text: x

text: is

text: a

text: collection

text: of

text: points

text: in

text: d

text: in

text: general

text: position

text: such

text: ∈n

text: that

text: x

text: x

text: for

text: every

text: i

text: [n],

text: then

text: X

text: does

text: not

text: admit

text: a

text: Kneser

text: transversal

text: of

text: type

text: i

text: i

text: α

text: Ω.

text: Our

text: next

text: goal

text: is

text: to

text: show

text: that

text: if

text: X

text: admits

text: a

text: Kneser

text: transversal

text: of

text: type

text: α

text: Ω

text: and

text: is

text: R

text: given,

text: then

text: we

text: can

text: choose

text: a

text: collection

text: of

text: points

text: in

text: d

text: in

text: general

text: position,

text: X

text: x

text: x

text: n

text: such

text: that

text: x

text: x

text: for

text: every

text: i

text: [n]

text: and

text: with

text: the

text: property

text: that

text: X

text: does

text: not

text: admit

text: a

text: Kneser

text: i

text: transversal

text: of

text: type

text: α

text: Ω.

text: To

text: see

text: this,

text: we

text: may

text: assume

text: without

text: loss

text: of

text: generality

text: that

text: α

text: d

text: d

text: 2λ

text: d

text: 2λ

text: d

text: 2λ

text: d

text: 2λ

text: and

text: L

text: is

text: a

text: (d

text: λ)-plane

text: that

text: contains

text: x

text: x

text: and

text: intersects

text: the

text: k

text: closed

text: segments

text: d

text: with

text: extreme

text: points

text: x

text: x

text: x

text: x

text: d

text: d

text: 2λ

text: Choose

text: x

text: such

text: that

text: x

text: x

text: and

text: in

text: such

text: a

text: way

text: that

text: L

text: does

text: not

text: 2λ

text: 2λ

text: 2λ

text: intersect

text: the

text: line

text: generated

text: by

text: x

text: x

text: Suppose

text: now

text: X

text: X

text: x

text: x

text: and

text: let

text: L

text: be

text: a

text: Kneser

text: transversal

text: of

text: X

text: of

text: type

text: α.

text: By

text: Lemma

text: since

text: k

text: λ,

text: 2λ

text: L

text: must

text: be

text: equal

text: than

text: L

text: which

text: is

text: imposible

text: because

text: L

text: does

text: not

text: intersect

text: the

text: line

text: generated

text: by

text: x

text: x

text: Furthermore,

text: is

text: so

text: small

text: that

text: if

text: X

text: does

text: not

text: contain

text: a

text: Kneser

text: 2λ

text: transversal

text: of

text: type

text: β

text: Ω,

text: X

text: either.

text: As

text: a

text: consequence

text: of

text: our

text: last

text: statement

text: and

text: of

text: it

text: follows

text: that

text: given

text: suﬃciently

text: R

text: small,

text: there

text: is

text: a

text: collection

text: of

text: points

text: in

text: d

text: in

text: general

text: position

text: X

text: x

text: x

text: such

text: that

text: n

text: x

text: x

text: for

text: every

text: i

text: [n]

text: and

text: with

text: the

text: property

text: that

text: the

text: number

text: of

text: non-complete

text: Kneser

text: i

text: i

text: transversals

text: of

text: X

text: is

text: bigger

text: that

text: the

text: number

text: of

text: non-complete

text: Kneser

text: transversals

text: of

text: X

text: Since

text: the

text: number

text: of

text: non-complete

text: Kneser

text: transversals

text: of

text: X

text: is

text: ﬁnite,

text: this

text: completes

text: the

text: proof

text: of

text: our

text: theorem.


text: candidates for sensing and (quantum) information applications. Gd N@C [2] 3 80 and N@C [3] are further experimentally feasible magnetic endofullerenes with 60 interesting applications[2, 3]. The present study relies on the idea of transferring light orbital angular momentum to bound degenerate states in fullerenes leading to the emergence of a magnetic orbital moment. We evaluate then the magnetic ﬁeld inside the fullerene cage due to the respective surface charge currents. To this end two established facts are important: i) Recent experiments [4] and ﬁrst principle calculations [5, 6, 7, 8, 9, 10, 11, 12] unveiled the existence of so-called super atomic molecular orbitals (SAMOs) in C which are diﬀuse, unoccupied orbitals bound by the (molecular cage) central 60 potential and have thus well-deﬁned orbital angular momenta (s, p, and d); they extend way beyond the site-deﬁned bound π orbitals and hence they are less aﬀected by doping or modiﬁcations of the geometry. Indeed, ab initio calcula- tions suggest the existence of SAMOs for larger or doped fullerenes. Even, for various bowl-shaped fragments of π-conjugated molecules SAMOs are present [13]. Here we exploit these properties of SAMOs for photo-generating surface charge current loops, and hence internal, nanoscale magnetic ﬁeld pulses which act on encapsulated magnetic moments, e.g. N@C [3]. As our ab-initio calcu- 60 lations explicitly demonstrate N@C [3] possess indeed SAMOs that we may 60 employ for magnetic pulse generation (cf. supplementary materials). ii) Special type of structured light pulses will be used, namely optical vortices with a topological charge that sets the amount of orbital angular momentum (OAM) transferrable when interacting with matter. Such OAM beams have found an impressively wide range of applications [14, 15, 16, 17, 18, 19, 20, 21, 22]. Of particular relevance here are the recently generated fs OAM pulses [23] and OAM sources in the VUV and EUV regimes [24, 25, 26]. Due to phase sin- gularity, the intensity of the OAM beam vanishes on the optical axis and rises to form a ring shape with a radius determined by the beam’s waist (cf. Fig.1). The advantage of pumping current-carrying SAMOs with linear polarized OAM beam is that one is not restricted to the conventional optical propensity rules. So the current (and therefore the magnetic moment) can be enhanced by increas- ing the topological charge for ﬁxed moderate pulse intensity, as demonstrated below. Furthermore, for larger systems, e.g. ring structures [27, 28], one may use the OAM spatial structure to couple locally to the system (i.e., only to the sphere region but not to the endohedral objects). As schematically shown in Fig.1, in the case of C the extension of the diﬀraction-limited beam is much 60 larger than that of the cage. Hence, it is relevant to inspect the charge-current dependence on the distance ρ between C center and the optical axis. Perhaps 0 60 unexpected, we ﬁnd a smooth dependence of the magnetic moment on ρ . The 0 increase of the ﬁeld intensity with ρ counter-acts the amount of OAM (with 0 respect to the center of the cage) acquired by SAMOs, which means, in a di- lute gas-phase collection of C distributed over the OAM beam spot roughly 60 the same magnetic ﬁeld is generated in the cage. As demonstrated here, this magnetic ﬁeld can be tuned in sign and amplitude by varying the properties of the OAM beam, such as the topological charge, the intensity, the waist, or the 2

text: candidates

text: for

text: sensing

text: and

text: (quantum)

text: information

text: applications.

text: Gd

text: N@C

text: and

text: N@C

text: are

text: further

text: experimentally

text: feasible

text: magnetic

text: endofullerenes

text: with

text: interesting

text: The

text: present

text: study

text: relies

text: on

text: the

text: idea

text: of

text: transferring

text: light

text: orbital

text: angular

text: momentum

text: to

text: bound

text: degenerate

text: states

text: in

text: fullerenes

text: leading

text: to

text: the

text: emergence

text: of

text: a

text: magnetic

text: orbital

text: moment.

text: We

text: evaluate

text: then

text: the

text: magnetic

text: ﬁeld

text: inside

text: the

text: fullerene

text: cage

text: due

text: to

text: the

text: respective

text: surface

text: charge

text: currents.

text: To

text: this

text: end

text: two

text: established

text: facts

text: are

text: important:

text: i)

text: Recent

text: experiments

text: and

text: ﬁrst

text: principle

text: calculations

text: unveiled

text: the

text: existence

text: of

text: so-called

text: super

text: atomic

text: molecular

text: orbitals

text: (SAMOs)

text: in

text: C

text: which

text: are

text: diﬀuse,

text: unoccupied

text: orbitals

text: bound

text: by

text: the

text: (molecular

text: cage)

text: central

text: potential

text: and

text: have

text: thus

text: well-deﬁned

text: orbital

text: angular

text: momenta

text: (s,

text: p,

text: and

text: d);

text: they

text: extend

text: way

text: beyond

text: the

text: site-deﬁned

text: bound

text: π

text: orbitals

text: and

text: hence

text: they

text: are

text: less

text: aﬀected

text: by

text: doping

text: or

text: modiﬁcations

text: of

text: the

text: geometry.

text: Indeed,

text: ab

text: initio

text: calcula-

text: tions

text: suggest

text: the

text: existence

text: of

text: SAMOs

text: for

text: larger

text: or

text: doped

text: fullerenes.

text: Even,

text: for

text: various

text: bowl-shaped

text: fragments

text: of

text: π-conjugated

text: molecules

text: SAMOs

text: are

text: present

text: Here

text: we

text: exploit

text: these

text: properties

text: of

text: SAMOs

text: for

text: photo-generating

text: surface

text: charge

text: current

text: loops,

text: and

text: hence

text: internal,

text: nanoscale

text: magnetic

text: ﬁeld

text: pulses

text: which

text: act

text: on

text: encapsulated

text: magnetic

text: moments,

text: e.g.

text: N@C

text: As

text: our

text: ab-initio

text: calcu-

text: lations

text: explicitly

text: demonstrate

text: N@C

text: possess

text: indeed

text: SAMOs

text: that

text: we

text: may

text: employ

text: for

text: magnetic

text: pulse

text: generation

text: (cf.

text: supplementary

text: materials).

text: ii)

text: Special

text: type

text: of

text: structured

text: light

text: pulses

text: will

text: be

text: used,

text: namely

text: optical

text: vortices

text: with

text: a

text: topological

text: charge

text: that

text: sets

text: the

text: amount

text: of

text: orbital

text: angular

text: momentum

text: (OAM)

text: transferrable

text: when

text: interacting

text: with

text: matter.

text: Such

text: OAM

text: beams

text: have

text: found

text: an

text: impressively

text: wide

text: range

text: of

text: applications

text: Of

text: particular

text: relevance

text: here

text: are

text: the

text: recently

text: generated

text: fs

text: OAM

text: pulses

text: and

text: OAM

text: sources

text: in

text: the

text: VUV

text: and

text: EUV

text: regimes

text: Due

text: to

text: phase

text: sin-

text: gularity,

text: the

text: intensity

text: of

text: the

text: OAM

text: beam

text: vanishes

text: on

text: the

text: optical

text: axis

text: and

text: rises

text: to

text: form

text: a

text: ring

text: shape

text: with

text: a

text: radius

text: determined

text: by

text: the

text: beam’s

text: waist

text: (cf.

text: The

text: advantage

text: of

text: pumping

text: current-carrying

text: SAMOs

text: with

text: linear

text: polarized

text: OAM

text: beam

text: is

text: that

text: one

text: is

text: not

text: restricted

text: to

text: the

text: conventional

text: optical

text: propensity

text: rules.

text: So

text: the

text: current

text: (and

text: therefore

text: the

text: magnetic

text: moment)

text: can

text: be

text: enhanced

text: by

text: increas-

text: ing

text: the

text: topological

text: charge

text: for

text: ﬁxed

text: moderate

text: pulse

text: intensity,

text: as

text: demonstrated

text: below.

text: Furthermore,

text: for

text: larger

text: systems,

text: e.g.

text: ring

text: structures

text: one

text: may

text: use

text: the

text: OAM

text: spatial

text: structure

text: to

text: couple

text: locally

text: to

text: the

text: system

text: (i.e.,

text: only

text: to

text: the

text: sphere

text: region

text: but

text: not

text: to

text: the

text: endohedral

text: objects).

text: As

text: schematically

text: shown

text: in

text: in

text: the

text: case

text: of

text: C

text: the

text: extension

text: of

text: the

text: diﬀraction-limited

text: beam

text: is

text: much

text: larger

text: than

text: that

text: of

text: the

text: cage.

text: Hence,

text: it

text: is

text: relevant

text: to

text: inspect

text: the

text: charge-current

text: dependence

text: on

text: the

text: distance

text: ρ

text: between

text: C

text: center

text: and

text: the

text: optical

text: axis.

text: Perhaps

text: unexpected,

text: we

text: ﬁnd

text: a

text: smooth

text: dependence

text: of

text: the

text: magnetic

text: moment

text: on

text: ρ

text: The

text: increase

text: of

text: the

text: ﬁeld

text: intensity

text: with

text: ρ

text: counter-acts

text: the

text: amount

text: of

text: OAM

text: (with

text: respect

text: to

text: the

text: center

text: of

text: the

text: cage)

text: acquired

text: by

text: SAMOs,

text: which

text: means,

text: in

text: a

text: di-

text: lute

text: gas-phase

text: collection

text: of

text: C

text: distributed

text: over

text: the

text: OAM

text: beam

text: spot

text: roughly

text: the

text: same

text: magnetic

text: ﬁeld

text: is

text: generated

text: in

text: the

text: cage.

text: As

text: demonstrated

text: here,

text: this

text: magnetic

text: ﬁeld

text: can

text: be

text: tuned

text: in

text: sign

text: and

text: amplitude

text: by

text: varying

text: the

text: properties

text: of

text: the

text: OAM

text: beam,

text: such

text: as

text: the

text: topological

text: charge,

text: the

text: intensity,

text: the

text: waist,

text: or

text: the


text: 4 and the electric ﬁeld along the growth axis (deﬁned by the applied gate voltage). Fig. 4 (a) shows the longest spin lifetime extracted from the ﬁtting procedure described above. The values are given for diﬀerent in-plane mag- netic ﬁelds up to 5 T, for each ﬁeld the gate voltage dependence is shown. Two regimes can be distinguished. They are indicated by diﬀerent background colors in Fig. 4. At low voltage, in the regime where IX energy is higher than that of DX (direct regime), spin relaxation time increases with the magnetic ﬁeld increase. The corre- sponding g-factor slightly increases with bias but remains above g = 0.1, Fig. 4 (b). Using photo-induced reﬂectiv- ity technique described in Ref. [13], we could not detect any measurable modiﬁcation of the exciton lifetime with magnetic ﬁeld in this regime, it remains slightly below ≈ 10 ns up to 10 T. At high voltage, i. e. above 0.3 V, IX becomes the lowest energy exciton state (indirect regime) [31]. In this regime the g-factor decreases substantially and spin lifetime time also changes its behaviour. It de- creases when magnetic ﬁeld increases. It was shown, that FIG. 3. (a) Kerr rotation measured at Vg = 0.8 V as a in this regime, magnetic ﬁeld also leads to strong increase function of the pump-probe delay at B = 0 and B = 1 T, of the IX lifetime, due to the shift of the IX dispersion in E = 1.568 eV, E = 1.569 eV. (b) Three characteristic pp pr k-space [6]. This feature is also reproduced in our photo- decay times extracted from Kerr rotation measurements at induced reﬂectivity experiments. At V = 0.8 V and 7 T V = 0.8 V and at diﬀerent in-plane magnetic ﬁelds. These g g decay times are ascribed to DX, IX, and 2DEG spin relax- exciton lifetime approaches the time delay between the ation. Solid line are ﬁt the spin dephasing model, assum- laser pulses (48 ns), leading to the accumulation of ex- ing g-factor distribution ∆g = 0.016 for IX and ∆g = 0.006 citons in the structure. The 1/B behaviour of the spin for 2DEG. (c) Fourier spectra of Kerr rotation measured at lifetime in the indirect regime can be understood in terms B = 1 T. Zero bias spectrum is compared to V = 0.8 V. g of the inhomogeneous broadening of the long-living and (d) Two precession frequencies extracted from Kerr rotation strongly localized IX as shown in Fig. 3 (b). Strong measurements at V = 0.8 V and at diﬀerent in-plane mag- g gate voltages pushes electron and hole towards QW in- netic ﬁelds. These frequencies are ascribed to IX, and 2DEG spin rprecession terfaces, which contributes to the increasing role of the disorder potential. Additional localisation also leads to the decreasing g-factor [29]. By contrast, the increase of corresponds to diﬀerent g-factors, which may arise from the spin lifetime in the presence of the in-plane magnetic the diﬀerent mass and density and therefore diﬀerent lo- ﬁeld observed in the direct regime does not have analogs calization of IXs and electrons [13]. Indeed, the degree in other electronic or excitonic systems. We will show in of localization is a crucial parameter, that controls the g- the theoretical part of the paper, that this eﬀect is due to factor values in GaAs/AlAs-based heterostructures [29]. the magnetic ﬁeld induced mixing of DX and IX states, Note, that in undoped CQWs identical to the one stud- characteristic of CQWs. ied here, previous studies have found the same value of THEORY the g-factor g = 0.12 [30]. The decay times obtained at Vg = 0.8 V are shown in Fig. 3 (b). While the shortest This section presents the theoretical model of exciton (DX) spin lifetime remains constant, both 2DEG and IX spin relaxation in CQWs in the absence of electric bias spin lifetimes decrease with the increase of the magnetic but in the presence of in-plane magnetic ﬁelds. Using ﬁeld. Solid lines show the ﬁt to 1/B behaviour, consis- two approaches: semi-phenomenological Lindblad equa- tent with the inhomogeneous broadening expected from tion approach and the microscopic approach based on the distribution of g-factors, which gives ∆g = 0.006 for a Shr¨odinger equation, we will show, that magnetic ﬁeld 2DEG and a higher value, ∆g = 0.016, for more localised dependence of the exciton spin lifetime can by explained IXs. by the mixing of DX and IX states. Let us consider Let us summarise the dependence of the spin lifetime CQWs with a pair DX states at energy E and and DX in a system of CQWs on both the in-plane magnetic ﬁeld a pair of IX states at E , Fig. 1 (b, d). At zero bias IX E < E due to reduced binding energy of the latter. DX IX The DX state is characterized by a strong electron-hole exchange interaction, while it is vanishingly small for IX smaller intensity of the signal state. The short range component of this interaction ∆ 0 ##LTFigure## ##LTFigure## ##LTLine##

text: and

text: the

text: electric

text: ﬁeld

text: along

text: the

text: growth

text: axis

text: (deﬁned

text: by

text: the

text: applied

text: gate

text: voltage).

text: Fig.

text: (a)

text: shows

text: the

text: longest

text: spin

text: lifetime

text: extracted

text: from

text: the

text: ﬁtting

text: procedure

text: described

text: above.

text: The

text: values

text: are

text: given

text: for

text: diﬀerent

text: in-plane

text: mag-

text: netic

text: ﬁelds

text: up

text: to

text: T,

text: for

text: each

text: ﬁeld

text: the

text: gate

text: voltage

text: dependence

text: is

text: shown.

text: Two

text: regimes

text: can

text: be

text: distinguished.

text: They

text: are

text: indicated

text: by

text: diﬀerent

text: background

text: colors

text: in

text: Fig.

text: At

text: low

text: voltage,

text: in

text: the

text: regime

text: where

text: IX

text: energy

text: is

text: higher

text: than

text: that

text: of

text: DX

text: (direct

text: regime),

text: spin

text: relaxation

text: time

text: increases

text: with

text: the

text: magnetic

text: ﬁeld

text: increase.

text: The

text: corre-

text: sponding

text: g-factor

text: slightly

text: increases

text: with

text: bias

text: but

text: remains

text: above

text: g

text: Fig.

text: (b).

text: Using

text: photo-induced

text: reﬂectiv-

text: ity

text: technique

text: described

text: in

text: Ref.

text: we

text: could

text: not

text: detect

text: any

text: measurable

text: modiﬁcation

text: of

text: the

text: exciton

text: lifetime

text: with

text: magnetic

text: ﬁeld

text: in

text: this

text: regime,

text: it

text: remains

text: slightly

text: below

text: ns

text: up

text: to

text: T.

text: At

text: high

text: voltage,

text: i.

text: e.

text: above

text: V,

text: IX

text: becomes

text: the

text: lowest

text: energy

text: exciton

text: state

text: (indirect

text: regime)

text: In

text: this

text: regime

text: the

text: g-factor

text: decreases

text: substantially

text: and

text: spin

text: lifetime

text: time

text: also

text: changes

text: its

text: behaviour.

text: It

text: de-

text: creases

text: when

text: magnetic

text: ﬁeld

text: increases.

text: It

text: was

text: shown,

text: that

text: FIG.

text: (a)

text: Kerr

text: rotation

text: measured

text: at

text: Vg

text: V

text: as

text: a

text: in

text: this

text: regime,

text: magnetic

text: ﬁeld

text: also

text: leads

text: to

text: strong

text: increase

text: function

text: of

text: the

text: pump-probe

text: delay

text: at

text: B

text: and

text: B

text: T,

text: of

text: the

text: IX

text: lifetime,

text: due

text: to

text: the

text: shift

text: of

text: the

text: IX

text: dispersion

text: in

text: E

text: eV,

text: E

text: eV.

text: (b)

text: Three

text: characteristic

text: pp

text: pr

text: k-space

text: This

text: feature

text: is

text: also

text: reproduced

text: in

text: our

text: photo-

text: decay

text: times

text: extracted

text: from

text: Kerr

text: rotation

text: measurements

text: at

text: induced

text: reﬂectivity

text: experiments.

text: At

text: V

text: V

text: and

text: T

text: V

text: V

text: and

text: at

text: diﬀerent

text: in-plane

text: magnetic

text: ﬁelds.

text: These

text: g

text: g

text: decay

text: times

text: are

text: ascribed

text: to

text: DX,

text: IX,

text: and

text: spin

text: relax-

text: exciton

text: lifetime

text: approaches

text: the

text: time

text: delay

text: between

text: the

text: ation.

text: Solid

text: line

text: are

text: ﬁt

text: the

text: spin

text: dephasing

text: model,

text: assum-

text: laser

text: pulses

text: ns),

text: leading

text: to

text: the

text: accumulation

text: of

text: ex-

text: ing

text: g-factor

text: distribution

text: ∆g

text: for

text: IX

text: and

text: ∆g

text: citons

text: in

text: the

text: structure.

text: The

text: behaviour

text: of

text: the

text: spin

text: for

text: (c)

text: Fourier

text: spectra

text: of

text: Kerr

text: rotation

text: measured

text: at

text: lifetime

text: in

text: the

text: indirect

text: regime

text: can

text: be

text: understood

text: in

text: terms

text: B

text: T.

text: Zero

text: bias

text: spectrum

text: is

text: compared

text: to

text: V

text: V.

text: g

text: of

text: the

text: inhomogeneous

text: broadening

text: of

text: the

text: long-living

text: and

text: (d)

text: Two

text: precession

text: frequencies

text: extracted

text: from

text: Kerr

text: rotation

text: strongly

text: localized

text: IX

text: as

text: shown

text: in

text: Fig.

text: (b).

text: Strong

text: measurements

text: at

text: V

text: V

text: and

text: at

text: diﬀerent

text: in-plane

text: mag-

text: g

text: gate

text: voltages

text: pushes

text: electron

text: and

text: hole

text: towards

text: QW

text: in-

text: netic

text: ﬁelds.

text: These

text: frequencies

text: are

text: ascribed

text: to

text: IX,

text: and

text: spin

text: rprecession

text: terfaces,

text: which

text: contributes

text: to

text: the

text: increasing

text: role

text: of

text: the

text: disorder

text: potential.

text: Additional

text: localisation

text: also

text: leads

text: to

text: the

text: decreasing

text: g-factor

text: By

text: contrast,

text: the

text: increase

text: of

text: corresponds

text: to

text: diﬀerent

text: g-factors,

text: which

text: may

text: arise

text: from

text: the

text: spin

text: lifetime

text: in

text: the

text: presence

text: of

text: the

text: in-plane

text: magnetic

text: the

text: diﬀerent

text: mass

text: and

text: density

text: and

text: therefore

text: diﬀerent

text: lo-

text: ﬁeld

text: observed

text: in

text: the

text: direct

text: regime

text: does

text: not

text: have

text: analogs

text: calization

text: of

text: IXs

text: and

text: electrons

text: Indeed,

text: the

text: degree

text: in

text: other

text: electronic

text: or

text: excitonic

text: systems.

text: We

text: will

text: show

text: in

text: of

text: localization

text: is

text: a

text: crucial

text: parameter,

text: that

text: controls

text: the

text: g-

text: the

text: theoretical

text: part

text: of

text: the

text: paper,

text: that

text: this

text: eﬀect

text: is

text: due

text: to

text: factor

text: values

text: in

text: GaAs/AlAs-based

text: heterostructures

text: the

text: magnetic

text: ﬁeld

text: induced

text: mixing

text: of

text: DX

text: and

text: IX

text: states,

text: Note,

text: that

text: in

text: undoped

text: CQWs

text: identical

text: to

text: the

text: one

text: stud-

text: characteristic

text: of

text: CQWs.

text: ied

text: here,

text: previous

text: studies

text: have

text: found

text: the

text: same

text: value

text: of

text: THEORY

text: the

text: g-factor

text: g

text: The

text: decay

text: times

text: obtained

text: at

text: Vg

text: V

text: are

text: shown

text: in

text: Fig.

text: (b).

text: While

text: the

text: shortest

text: This

text: section

text: presents

text: the

text: theoretical

text: model

text: of

text: exciton

text: (DX)

text: spin

text: lifetime

text: remains

text: constant,

text: both

text: and

text: IX

text: spin

text: relaxation

text: in

text: CQWs

text: in

text: the

text: absence

text: of

text: electric

text: bias

text: spin

text: lifetimes

text: decrease

text: with

text: the

text: increase

text: of

text: the

text: magnetic

text: but

text: in

text: the

text: presence

text: of

text: in-plane

text: magnetic

text: ﬁelds.

text: Using

text: ﬁeld.

text: Solid

text: lines

text: show

text: the

text: ﬁt

text: to

text: behaviour,

text: consis-

text: two

text: approaches:

text: semi-phenomenological

text: Lindblad

text: equa-

text: tent

text: with

text: the

text: inhomogeneous

text: broadening

text: expected

text: from

text: tion

text: approach

text: and

text: the

text: microscopic

text: approach

text: based

text: on

text: the

text: distribution

text: of

text: g-factors,

text: which

text: gives

text: ∆g

text: for

text: a

text: Shr¨odinger

text: equation,

text: we

text: will

text: show,

text: that

text: magnetic

text: ﬁeld

text: and

text: a

text: higher

text: value,

text: ∆g

text: for

text: more

text: localised

text: dependence

text: of

text: the

text: exciton

text: spin

text: lifetime

text: can

text: by

text: explained

text: IXs.

text: by

text: the

text: mixing

text: of

text: DX

text: and

text: IX

text: states.

text: Let

text: us

text: consider

text: Let

text: us

text: summarise

text: the

text: dependence

text: of

text: the

text: spin

text: lifetime

text: CQWs

text: with

text: a

text: pair

text: DX

text: states

text: at

text: energy

text: E

text: and

text: and

text: DX

text: in

text: a

text: system

text: of

text: CQWs

text: on

text: both

text: the

text: in-plane

text: magnetic

text: ﬁeld

text: a

text: pair

text: of

text: IX

text: states

text: at

text: E

text: Fig.

text: (b,

text: d).

text: At

text: zero

text: bias

text: IX

text: E

text: E

text: due

text: to

text: reduced

text: binding

text: energy

text: of

text: the

text: latter.

text: DX

text: IX

text: The

text: DX

text: state

text: is

text: characterized

text: by

text: a

text: strong

text: electron-hole

text: exchange

text: interaction,

text: while

text: it

text: is

text: vanishingly

text: small

text: for

text: IX

text: smaller

text: intensity

text: of

text: the

text: signal

text: state.

text: The

text: short

text: range

text: component

text: of

text: this

text: interaction

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 6 ∝ the long range electron-hole exhange ﬁeld ∆(q) q. We Landau gauge vector potential A = Bde z, correspond- x explore two diﬀerent approaches that allow accounting ing to the external magnetic ﬁeld Be . Here d is the y for this eﬀect. In the ﬁrst one we neglect long-range ex- mean electron-hole separation within the IX, which may change interaction in the DX Hamiltonian, but introduce be approximated by the distance between the CQWs cen- the ﬂuctuating ﬁeld via the phenomenological Lindblad ters. Note that for IXs we neglect both long and short superoperator in the Liouville equation for the density range parts of the electron-hole exchange. matrix. In the second one we directly introduce the ﬂuc- Lindblad equation analysis tuating ﬁeld in the coherent Hamiltonian part and solve the Shroedinger equation on the timescales shorter than In this subsection we use the semi-phenomenological the characteristic momentum scattering time. approach based on quantum Liouville equation with the In the most general way the DX block in the basis of Lindblad superopeoperator in the right-hand part, also − − spin states ( 2, 1, +1, +2) reads: called Lindblad equation. We ﬁrst assume the linear in   k dependence of the long range exhange ∆(k) [12]. It al- lows us to isolate the slowly varying part of the exchange   −  ﬁeld, corresponding to the wave vector K from the fast  ∆ ∆ 0 0  0 B ﬂuctuating part, linear in q : ∆(k) = ∆(K) + ∆(q). We ∆ 0 ∆(k) 0 H = E (k) I + B ∗ . (3) remove the latter from the coherent Hamiltonian and ac- DX DX 4x4 0 ∆(k) 0 ∆ − B count for ﬂuctuating ﬁeld via the Lindblad superoperator 0 0 ∆ ∆ B 0 term. The Lindblad equation for the exciton density ma- trix ρ reads: (cid:126) Here the DX energy E (k) = 2k2/2m +E , ∆ is DX X DX 0 the short range part of the electron-hole exchange inter- dρ − i = (cid:126) [H, ρ] + L(ρ). (5) action, ∆(k) is its long range part, and ∆B = gµBB/2. dt is the electron Zeeman splitting. Note that we neglect the hole spin splitting due to the small heavy hole g-factor. Here the Lindblad superoperator L accounts for the We also neglect the magnetic Stark eﬀect for DXs as it is DX spin relaxation and decay. We neglect relaxation and linear in electron-hole separation distance. On the other decay processes corresponding to the IX. The Lindblad hand, we keep it in the IX Hamiltonian block: term L(ρ) also(cid:18)may be represented in a block mat(cid:19)rix form:     −   0 ρ − γ /2  0 ∆B 0 0  L(ρ) = −ρ − γ /2 L DX(ρIX −DX ) . (6) ∆ 0 0 0 IX DX DX DX DX DX H = E (k) I + B , (4) IX IX 4x4 0 0 0 ∆ B 0 0 ∆ 0 Here ρ − and ρ − are the top-right and bottom- (cid:0)B (cid:1) DX IX IX DX left 4x4 blocks of the total density matrix, which decay at twice slower rate than the DX density γ /2. The where the IX energy EIX(k) = 2m(cid:126)2X k − (cid:126)e Bdex 2 + LDX block, which describes the fast relaxaDtiXon of the E . The correction to the IX energy appears due to the direct part, reads: IX   (cid:16) (cid:17)     0 0 (cid:16) 0 (cid:17) 0   LDX(ρDX−DX) = −ρDX−DXγDX + 00 ρ+DX1,−+−D1ρX+−1,−−ρ1−DX1,γ−−D1X γex ρ−1,−−−1ρ−D−X1,−+ρD1−X1,γ−−e1x γ 00 , (7) DX DX ex DX DX DX DX ex 0 0 0 0 − where γ = (2τ ) 1 is the rate of spin relaxation due a σ+ polarized optical pulse, with a fully relaxed heavy ex ex to the Maialle-Andrada e Silva-Sham mechanism. We hole spin. assume that the hole spin relaxation is by far the fastest The solution of equation (5) yields the dynamics of the process in the system and take as initial condition for the Kerr rotation angle, measured in the experiment. The Lindblad evolution equation the density matrix with only eﬀect itself is produced by spin-dependent exciton res- − − two non-zero elements ρ+1,−+1 = ρ 2,−2 = n/2, which onance shifts, stemming from exciton-exciton exchange DX DX DX DX describes the excitonic system of density n, pumped with interactions. Both direct and indirect components con- ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: long

text: range

text: electron-hole

text: exhange

text: ﬁeld

text: ∆(q)

text: q.

text: We

text: Landau

text: gauge

text: vector

text: potential

text: A

text: Bde

text: z,

text: correspond-

text: x

text: explore

text: two

text: diﬀerent

text: approaches

text: that

text: allow

text: accounting

text: ing

text: to

text: the

text: external

text: magnetic

text: ﬁeld

text: Be

text: Here

text: d

text: is

text: the

text: y

text: for

text: this

text: eﬀect.

text: In

text: the

text: ﬁrst

text: one

text: we

text: neglect

text: long-range

text: ex-

text: mean

text: electron-hole

text: separation

text: within

text: the

text: IX,

text: which

text: may

text: change

text: interaction

text: in

text: the

text: DX

text: Hamiltonian,

text: but

text: introduce

text: be

text: approximated

text: by

text: the

text: distance

text: between

text: the

text: CQWs

text: cen-

text: the

text: ﬂuctuating

text: ﬁeld

text: via

text: the

text: phenomenological

text: Lindblad

text: ters.

text: Note

text: that

text: for

text: IXs

text: we

text: neglect

text: both

text: long

text: and

text: short

text: superoperator

text: in

text: the

text: Liouville

text: equation

text: for

text: the

text: density

text: range

text: parts

text: of

text: the

text: electron-hole

text: exchange.

text: matrix.

text: In

text: the

text: second

text: one

text: we

text: directly

text: introduce

text: the

text: ﬂuc-

text: Lindblad

text: equation

text: analysis

text: tuating

text: ﬁeld

text: in

text: the

text: coherent

text: Hamiltonian

text: part

text: and

text: solve

text: the

text: Shroedinger

text: equation

text: on

text: the

text: timescales

text: shorter

text: than

text: In

text: this

text: subsection

text: we

text: use

text: the

text: semi-phenomenological

text: the

text: characteristic

text: momentum

text: scattering

text: time.

text: approach

text: based

text: on

text: quantum

text: Liouville

text: equation

text: with

text: the

text: In

text: the

text: most

text: general

text: way

text: the

text: DX

text: block

text: in

text: the

text: basis

text: of

text: Lindblad

text: superopeoperator

text: in

text: the

text: right-hand

text: part,

text: also

text: spin

text: states

text: reads:

text: called

text: Lindblad

text: equation.

text: We

text: ﬁrst

text: assume

text: the

text: linear

text: in

text: k

text: dependence

text: of

text: the

text: long

text: range

text: exhange

text: ∆(k)

text: It

text: al-

text: lows

text: us

text: to

text: isolate

text: the

text: slowly

text: varying

text: part

text: of

text: the

text: exchange

text: ﬁeld,

text: corresponding

text: to

text: the

text: wave

text: vector

text: K

text: from

text: the

text: fast

text: B

text: ﬂuctuating

text: part,

text: linear

text: in

text: q

text: ∆(k)

text: ∆(K)

text: ∆(q).

text: We

text: ∆(k)

text: H

text: E

text: (k)

text: I

text: B

text: remove

text: the

text: latter

text: from

text: the

text: coherent

text: Hamiltonian

text: and

text: ac-

text: DX

text: DX

text: ∆(k)

text: B

text: count

text: for

text: ﬂuctuating

text: ﬁeld

text: via

text: the

text: Lindblad

text: superoperator

text: B

text: term.

text: The

text: Lindblad

text: equation

text: for

text: the

text: exciton

text: density

text: ma-

text: trix

text: ρ

text: reads:

text: Here

text: the

text: DX

text: energy

text: E

text: (k)

text: +E

text: is

text: DX

text: X

text: DX

text: the

text: short

text: range

text: part

text: of

text: the

text: electron-hole

text: exchange

text: inter-

text: dρ

text: i

text: [H,

text: ρ]

text: L(ρ).

text: action,

text: ∆(k)

text: is

text: its

text: long

text: range

text: part,

text: and

text: ∆B

text: dt

text: is

text: the

text: electron

text: Zeeman

text: splitting.

text: Note

text: that

text: we

text: neglect

text: the

text: hole

text: spin

text: splitting

text: due

text: to

text: the

text: small

text: heavy

text: hole

text: g-factor.

text: Here

text: the

text: Lindblad

text: superoperator

text: L

text: accounts

text: for

text: the

text: We

text: also

text: neglect

text: the

text: magnetic

text: Stark

text: eﬀect

text: for

text: DXs

text: as

text: it

text: is

text: DX

text: spin

text: relaxation

text: and

text: decay.

text: We

text: neglect

text: relaxation

text: and

text: linear

text: in

text: electron-hole

text: separation

text: distance.

text: On

text: the

text: other

text: decay

text: processes

text: corresponding

text: to

text: the

text: IX.

text: The

text: Lindblad

text: hand,

text: we

text: keep

text: it

text: in

text: the

text: IX

text: Hamiltonian

text: block:

text: term

text: L(ρ)

text: be

text: represented

text: in

text: a

text: block

text: form:

text: ρ

text: γ

text: ∆B

text: L(ρ)

text: −ρ

text: γ

text: L

text: DX(ρIX

text: −DX

text: IX

text: DX

text: DX

text: DX

text: DX

text: DX

text: H

text: E

text: (k)

text: I

text: B

text: IX

text: IX

text: B

text: Here

text: ρ

text: and

text: ρ

text: are

text: the

text: top-right

text: and

text: bottom-

text: DX

text: IX

text: IX

text: DX

text: left

text: blocks

text: of

text: the

text: total

text: density

text: matrix,

text: which

text: decay

text: at

text: twice

text: slower

text: rate

text: than

text: the

text: DX

text: density

text: γ

text: The

text: where

text: the

text: IX

text: energy

text: EIX(k)

text: k

text: Bdex

text: LDX

text: block,

text: which

text: describes

text: the

text: fast

text: relaxaDtiXon

text: of

text: the

text: E

text: The

text: correction

text: to

text: the

text: IX

text: energy

text: appears

text: due

text: to

text: the

text: direct

text: part,

text: reads:

text: IX

text: 0

text: 0

text: LDX(ρDX−DX)

text: −ρDX−DXγDX

text: γex

text: γ

text: DX

text: DX

text: ex

text: DX

text: DX

text: DX

text: DX

text: ex

text: where

text: γ

text: is

text: the

text: rate

text: of

text: spin

text: relaxation

text: due

text: a

text: σ+

text: polarized

text: optical

text: pulse,

text: with

text: a

text: fully

text: relaxed

text: heavy

text: ex

text: ex

text: to

text: the

text: Maialle-Andrada

text: e

text: Silva-Sham

text: mechanism.

text: We

text: hole

text: spin.

text: assume

text: that

text: the

text: hole

text: spin

text: relaxation

text: is

text: by

text: far

text: the

text: fastest

text: The

text: solution

text: of

text: equation

text: yields

text: the

text: dynamics

text: of

text: the

text: process

text: in

text: the

text: system

text: and

text: take

text: as

text: initial

text: condition

text: for

text: the

text: Kerr

text: rotation

text: angle,

text: measured

text: in

text: the

text: experiment.

text: The

text: Lindblad

text: evolution

text: equation

text: the

text: density

text: matrix

text: with

text: only

text: eﬀect

text: itself

text: is

text: produced

text: by

text: spin-dependent

text: exciton

text: res-

text: two

text: non-zero

text: elements

text: ρ

text: which

text: onance

text: shifts,

text: stemming

text: from

text: exciton-exciton

text: exchange

text: DX

text: DX

text: DX

text: DX

text: describes

text: the

text: excitonic

text: system

text: of

text: density

text: n,

text: pumped

text: with

text: interactions.

text: Both

text: direct

text: and

text: indirect

text: components

text: con-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 tribute to the value of Kerr rotation angle. The value model, but allows to improve the quantitative agreement of the Kerr rotation angle δθ is a sum of two contri- between the theory and the experiment. butions, linear in bright DX and bright IX spin polariza- Microscopic analysis − − − − − − tions ρ+1,−+1 ρ 1,−1 and ρ+1−,+1 ρ 1−, 1 respectively DX DX DX DX IX IX IX IX [39]. Furthermore, the coeﬃcients before the two polar- Assuming a generic spin relaxation mechanism stem- izations, given by the Coulomb carrier exchange, weakly ming from the spin precession in a stochastically ﬂuctu- depend on the electron-hole separation distance and may ating ﬁeld, the spin relaxation rate γ = 1/τ scales with s s be assumed equal for IX and DX contributions, allowing both characteristic value of the ﬂuctuating ﬁeld Ω and us to write: the ﬂuctuation time scale, given by the momentum re- laxation rate τ via Dyakonov-Perel formula γ = Ω(B)2τ s ∼ − − − − − − [12, 40]. In principle, both parameters depend on the de- δθ ρ+1,−+1 ρ 1,−1 + ρ+1−,+1 ρ 1−, 1. (8) DX DX DX DX IX IX IX IX gree of DX-IX coupling: Ω scales with the electron-hole overlap, while τ in drift-diﬀusion regime depends on the Kerr rotation angle decay, as well as the relative spin value of the exciton dipole moment. Here we focus on polarisation, may be extracted from the solution of Eqs. the variation of Ω with magnetic ﬁeld B, which allows (5, 8) as functions of the magnetic ﬁeld B. We ﬁt the us to explain the experimental measurements. Instead experimental data assuming the exciton eﬀective mass of the Lindblad equation we solve the Shr¨odinger equa- m = 0.22m [7] and the interwell distance d = 12 nm. tion, taking the full Hamiltonian (2) with the ﬁeld, stem- X 0 The electron Lande factor g = 0.1 is obtained from exper- ming from the long-range electron-hole exchange ∆(k). iments, and the bright-dark splitting is ∆ = 70 µeV [22]. This approach is valid on the timescales shorter than the 0 In the simulation we neglected the coherent electron-hole momentum relaxation time, as the eﬀective ﬁeld is mo- exchange ﬁeld ∆(K) compared with other ﬁelds in the mentum dependent and its absolute value is in the ﬁrst system for both DX and IX. The decay time obtained approximation linear in the momentum value [41]. − with the ﬁt parameters ∆E = E E = 1.5 meV, Taking a wavefunction, describing a DX with the fully − DX IX − J = 0.15 meV, K = 90 µm 1, 1/γDX = 2 ns, 1/γex = relaxed hole spin and the electron spin se = 1/2, as 400 ps, is shown in Fig. 2 (c) by solid line. It has a initial condition, one may trace the corresponding Kerr maximum in the vicinity of B = 5 T, which corresponds rotation angle δθ(t). Treating it as bright exciton sub- to the maximum DX-IX mixing. One can see, however, space pseudospin projection, we numerically extract the that the relation between the decay times at the peak characteristic frequencies of its rotation Ω(B) from the (cid:28) and at B = 0 is limited by 2. Indeed, in the ideal case Fourier transform of δθ(t). Assuming Ωτ 1, we esti- of negligible DX-IX coupling at B = 0 where the exciton mate the exciton spin decay rate as γs = Ω(B)2τ. The spin relaxation is given by that of DXs, the longest pos- parameters of the numerical calculation are the same as sible decay time is achieved whilst the DX and IX modes for the Lindblad equation. are resonant. Excitons in this case are half-indirect and This approach yields up to 4 times increase of the mea- thus lose the spin polarization at the twice reduced rate, sured spin relaxation time τs = 1/γs at the resonance of according to the solution of the Lindblad equation. In DX and IX energies, as the stochastic rotation frequency (cid:54) the realistic case, where the coupling J = 0, the exci- can be twice lower for a coupled DX-IX state in compar- ton spin dynamics is aﬀected by the IX admixture even ison with a pure DX state. The ﬁt of the experimental at B = 0, therefore this approach yields 2 as the upper data using this microscopic approach (dashed line in Fig. limit for the relation τ (B = 5)/τ (B = 0), whereas the 2 (c)) is therefore more accurate in comparison with the s s measured value is close to 3. semi-phenomenological approach based on the Lindblad The main reason for the discrepancy between this re- equation (solid line), even though it does not take into sult and the experimental data lies in the phenomeno- account the dependence of the transport time τ on the logical nature of γ , introduced as the relaxation rate magnetic ﬁeld. ex of the direct exciton part. Note that this parameter dif- CONCLUSIONS fers from the exciton spin depolarization time in a single QW of the same width as those composing CQWs. The We have studied spin dynamics of excitons in CQWs in classical Dyakonov-Perel picture gives an insight to this the presence of crossed magnetic and electric ﬁelds using diﬀerence. As long as the characteristic exciton trans- time-resolved Kerr rotation spectroscopy. Two qualita- port time τ is longer than the electron tunneling time tively diﬀerent regimes of spin decoherence are identi- (cid:126) − (J/ ) 1, the exciton loses its spin as a whole, rotating ﬁed, depending on the strength of the electric ﬁeld, ap- in a stochastic eﬀective magnetic ﬁeld between scatter- plied along the growth axis. In the presence of the gate ing events, rather than losing it via its DX and IX com- voltage, such that IX becomes the lowest energy exciton ponents independently. Our further microscopic analy- state of the system, the inhomogeneous spin coherence sis of the spin relaxation gives similar qualitative result time is found to be inversely proportional to the mag- as the semi-phenomenological Lindblad equation-based nitude of the in-plane magnetic ﬁeld. This behaviour is

text: tribute

text: to

text: the

text: value

text: of

text: Kerr

text: rotation

text: angle.

text: The

text: value

text: model,

text: but

text: allows

text: to

text: improve

text: the

text: quantitative

text: agreement

text: of

text: the

text: Kerr

text: rotation

text: angle

text: δθ

text: is

text: a

text: sum

text: of

text: two

text: contri-

text: between

text: the

text: theory

text: and

text: the

text: experiment.

text: butions,

text: linear

text: in

text: bright

text: DX

text: and

text: bright

text: IX

text: spin

text: polariza-

text: Microscopic

label: section

text: analysis

text: tions

text: ρ

text: and

text: ρ

text: respectively

text: DX

text: DX

text: DX

text: DX

text: IX

text: IX

text: IX

text: IX

text: Furthermore,

text: the

text: coeﬃcients

text: before

text: the

text: two

text: polar-

text: Assuming

text: a

text: generic

text: spin

text: relaxation

text: mechanism

text: stem-

text: izations,

text: given

text: by

text: the

text: Coulomb

text: carrier

text: exchange,

text: weakly

text: ming

text: from

text: the

text: spin

text: precession

text: in

text: a

text: stochastically

text: ﬂuctu-

text: depend

text: on

text: the

text: electron-hole

text: separation

text: distance

text: and

text: may

text: ating

text: ﬁeld,

text: the

text: spin

text: relaxation

text: rate

text: γ

text: scales

text: with

text: s

text: s

text: be

text: assumed

text: equal

text: for

text: IX

text: and

text: DX

text: contributions,

text: allowing

text: both

text: characteristic

text: value

text: of

text: the

text: ﬂuctuating

text: ﬁeld

text: Ω

text: and

text: us

text: to

text: write:

text: the

text: ﬂuctuation

text: time

text: scale,

text: given

text: by

text: the

text: momentum

text: re-

text: laxation

text: rate

text: τ

text: via

text: Dyakonov-Perel

text: formula

text: γ

text: s

text: In

text: principle,

text: both

text: parameters

text: depend

text: on

text: the

text: de-

text: δθ

text: ρ

text: ρ

text: DX

text: DX

text: DX

text: DX

text: IX

text: IX

text: IX

text: IX

text: gree

text: of

text: DX-IX

text: coupling:

text: Ω

text: scales

text: with

text: the

text: electron-hole

text: overlap,

text: while

text: τ

text: in

text: drift-diﬀusion

text: regime

text: depends

text: on

text: the

text: Kerr

text: rotation

text: angle

text: decay,

text: as

text: well

text: as

text: the

text: relative

text: spin

text: value

text: of

text: the

text: exciton

text: dipole

text: moment.

text: Here

text: we

text: focus

text: on

text: polarisation,

text: may

text: be

text: extracted

text: from

text: the

text: solution

text: of

text: Eqs.

text: the

text: variation

text: of

text: Ω

text: with

text: magnetic

text: ﬁeld

text: B,

text: which

text: allows

text: as

text: functions

text: of

text: the

text: magnetic

text: ﬁeld

text: B.

text: We

text: ﬁt

text: the

text: us

text: to

text: explain

text: the

text: experimental

text: measurements.

text: Instead

text: experimental

text: data

text: assuming

text: the

text: exciton

text: eﬀective

text: mass

text: of

text: the

text: Lindblad

text: equation

text: we

text: solve

text: the

text: Shr¨odinger

text: equa-

text: m

text: and

text: the

text: interwell

text: distance

text: d

text: nm.

text: tion,

text: taking

text: the

text: full

text: Hamiltonian

text: with

text: the

text: ﬁeld,

text: stem-

text: X

text: The

text: electron

text: Lande

text: factor

text: g

text: is

text: obtained

text: from

text: exper-

text: ming

text: from

text: the

text: long-range

text: electron-hole

text: exchange

text: ∆(k).

text: iments,

text: and

text: the

text: bright-dark

text: splitting

text: is

text: µeV

text: This

text: approach

text: is

text: valid

text: on

text: the

text: timescales

text: shorter

text: than

text: the

text: In

text: the

text: simulation

text: we

text: neglected

text: the

text: coherent

text: electron-hole

text: momentum

text: relaxation

text: time,

text: as

text: the

text: eﬀective

text: ﬁeld

text: is

text: mo-

text: exchange

text: ﬁeld

text: ∆(K)

text: compared

text: with

text: other

text: ﬁelds

text: in

text: the

text: mentum

text: dependent

text: and

text: its

text: absolute

text: value

text: is

text: in

text: the

text: ﬁrst

text: system

text: for

text: both

text: DX

text: and

text: IX.

text: The

text: decay

text: time

text: obtained

text: approximation

text: linear

text: in

text: the

text: momentum

text: value

text: with

text: the

text: ﬁt

text: parameters

text: ∆E

text: E

text: E

text: meV,

text: Taking

text: a

text: wavefunction,

text: describing

text: a

text: DX

text: with

text: the

text: fully

text: DX

text: IX

text: J

text: meV,

text: K

text: µm

text: ns,

text: relaxed

text: hole

text: spin

text: and

text: the

text: electron

text: spin

text: se

text: as

text: ps,

text: is

text: shown

text: in

text: Fig.

text: (c)

text: by

text: solid

text: line.

text: It

text: has

text: a

text: initial

text: condition,

text: one

text: may

text: trace

text: the

text: corresponding

text: Kerr

text: maximum

text: in

text: the

text: vicinity

text: of

text: B

text: T,

text: which

text: corresponds

text: rotation

text: angle

text: δθ(t).

text: Treating

text: it

text: as

text: bright

text: exciton

text: sub-

text: to

text: the

text: maximum

text: DX-IX

text: mixing.

text: One

text: can

text: see,

text: however,

text: space

text: pseudospin

text: projection,

text: we

text: numerically

text: extract

text: the

text: that

text: the

text: relation

text: between

text: the

text: decay

text: times

text: at

text: the

text: peak

text: characteristic

text: frequencies

text: of

text: its

text: rotation

text: Ω(B)

text: from

text: the

text: and

text: at

text: B

text: is

text: limited

text: by

text: Indeed,

text: in

text: the

text: ideal

text: case

text: Fourier

text: transform

text: of

text: δθ(t).

text: Assuming

text: Ωτ

text: we

text: esti-

text: of

text: negligible

text: DX-IX

text: coupling

text: at

text: B

text: where

text: the

text: exciton

text: mate

text: the

text: exciton

text: spin

text: decay

text: rate

text: as

text: γs

text: The

text: spin

text: relaxation

text: is

text: given

text: by

text: that

text: of

text: DXs,

text: the

text: longest

text: pos-

text: parameters

text: of

text: the

text: numerical

text: calculation

text: are

text: the

text: same

text: as

text: sible

text: decay

text: time

text: is

text: achieved

text: whilst

text: the

text: DX

text: and

text: IX

text: modes

text: for

text: the

text: Lindblad

text: equation.

text: are

text: resonant.

text: Excitons

text: in

text: this

text: case

text: are

text: half-indirect

text: and

text: This

text: approach

text: yields

text: up

text: to

text: times

text: increase

text: of

text: the

text: mea-

text: thus

text: lose

text: the

text: spin

text: polarization

text: at

text: the

text: twice

text: reduced

text: rate,

text: sured

text: spin

text: relaxation

text: time

text: τs

text: at

text: the

text: resonance

text: of

text: according

text: to

text: the

text: solution

text: of

text: the

text: Lindblad

text: equation.

text: In

text: DX

text: and

text: IX

text: energies,

text: as

text: the

text: stochastic

text: rotation

text: frequency

text: the

text: realistic

text: case,

text: where

text: the

text: coupling

text: J

text: the

text: exci-

text: can

text: be

text: twice

text: lower

text: for

text: a

text: coupled

text: DX-IX

text: state

text: in

text: compar-

text: ton

text: spin

text: dynamics

text: is

text: aﬀected

text: by

text: the

text: IX

text: admixture

text: even

text: ison

text: with

text: a

text: pure

text: DX

text: state.

text: The

text: ﬁt

text: of

text: the

text: experimental

text: at

text: B

text: therefore

text: this

text: approach

text: yields

text: as

text: the

text: upper

text: data

text: using

text: this

text: microscopic

text: approach

text: (dashed

text: line

text: in

text: Fig.

text: limit

text: for

text: the

text: relation

text: τ

text: (B

text: (B

text: whereas

text: the

text: (c))

text: is

text: therefore

text: more

text: accurate

text: in

text: comparison

text: with

text: the

text: s

text: s

text: measured

text: value

text: is

text: close

text: to

text: semi-phenomenological

text: approach

text: based

text: on

text: the

text: Lindblad

text: The

text: main

text: reason

text: for

text: the

text: discrepancy

text: between

text: this

text: re-

text: equation

text: (solid

text: line),

text: even

text: though

text: it

text: does

text: not

text: take

text: into

text: sult

text: and

text: the

text: experimental

text: data

text: lies

text: in

text: the

text: phenomeno-

text: account

text: the

text: dependence

text: of

text: the

text: transport

text: time

text: τ

text: on

text: the

text: logical

text: nature

text: of

text: γ

text: introduced

text: as

text: the

text: relaxation

text: rate

text: magnetic

text: ﬁeld.

text: ex

text: of

text: the

text: direct

text: exciton

text: part.

text: Note

text: that

text: this

text: parameter

text: dif-

text: CONCLUSIONS

text: fers

text: from

text: the

text: exciton

text: spin

text: depolarization

text: time

text: in

text: a

text: single

text: QW

text: of

text: the

text: same

text: width

text: as

text: those

text: composing

text: CQWs.

text: The

text: We

text: have

text: studied

text: spin

text: dynamics

text: of

text: excitons

text: in

text: CQWs

text: in

text: classical

text: Dyakonov-Perel

text: picture

text: gives

text: an

text: insight

text: to

text: this

text: the

text: presence

text: of

text: crossed

text: magnetic

text: and

text: electric

text: ﬁelds

text: using

text: diﬀerence.

text: As

text: long

text: as

text: the

text: characteristic

text: exciton

text: trans-

text: time-resolved

text: Kerr

text: rotation

text: spectroscopy.

text: Two

text: qualita-

text: port

text: time

text: τ

text: is

text: longer

text: than

text: the

text: electron

text: tunneling

text: time

text: tively

text: diﬀerent

text: regimes

text: of

text: spin

text: decoherence

text: are

text: identi-

text: (J/

text: the

text: exciton

text: loses

text: its

text: spin

text: as

text: a

text: whole,

text: rotating

text: ﬁed,

text: depending

text: on

text: the

text: strength

text: of

text: the

text: electric

text: ﬁeld,

text: ap-

text: in

text: a

text: stochastic

text: eﬀective

text: magnetic

text: ﬁeld

text: between

text: scatter-

text: plied

text: along

text: the

text: growth

text: axis.

text: In

text: the

text: presence

text: of

text: the

text: gate

text: ing

text: events,

text: rather

text: than

text: losing

text: it

text: via

text: its

text: DX

text: and

text: IX

text: com-

text: voltage,

text: such

text: that

text: IX

text: becomes

text: the

text: lowest

text: energy

text: exciton

text: ponents

text: independently.

text: Our

text: further

text: microscopic

text: analy-

text: state

text: of

text: the

text: system,

text: the

text: inhomogeneous

text: spin

text: coherence

text: sis

text: of

text: the

text: spin

text: relaxation

text: gives

text: similar

text: qualitative

text: result

text: time

text: is

text: found

text: to

text: be

text: inversely

text: proportional

text: to

text: the

text: mag-

text: as

text: the

text: semi-phenomenological

text: Lindblad

text: equation-based

text: nitude

text: of

text: the

text: in-plane

text: magnetic

text: ﬁeld.

text: This

text: behaviour

text: is