text: π(µ = f ) = π . (4) j j In this setting, the posterior distribution of µ given Y is (cid:16) (cid:17) (cid:80) (cid:80) −(cid:16)(cid:107) − (cid:107) (cid:17) | p(Y |µ = fj)p(µ = fj) exp fj2ωY2 22 πj p(µ = fj Y ) = Mj=1 p(Y |µ = fj)p(µ = fj) = Mj=1 exp −(cid:107)fj2−ωY2 (cid:107)22 πj . In the Bayesian decision theoretical framework considered in this paper, the quantity of interest E is η = Y , and we consider a loss function L(ψ, µ) which we would like to minimize with respect to the posterior distribution. The corresponding Bayes estimator ψˆ minimizes the posterior expected loss from µ as follows: ψˆ = argmin E [L(ψ, µ)|Y ] . (5) ∈R ψ n It is worth pointing out that the above Bayesian framework is only used to obtain decision theoretically motivated model averaging estimators (Bayesian estimators have good theoretical properties such as admissibility, etc). In particular we do not assume that the model itself is correctly speciﬁed. That is, in this paper we allow misspeciﬁed models, where the parameters µ and ω2 are not necessarily equal to the true mean η and the true variance σ2 in (2), and η does { } not necessarily belong to the dictionary f , . . . , f . 1 M · · The Bayesian estimator of (5) depends on the underlying loss function L( , ). For example, (cid:107) − (cid:107) under the standard least squares loss L(ψ, µ) = ψ µ 2, the Bayes estimator is the posterior 2 mean, which leads to the Exponential Weighted Model Aggregation (EWMA) estimator (Rigollet (cid:16) (cid:17) (cid:80) and Tsybakov, 2012) : (cid:16) (cid:17) (cid:80) −(cid:107) − (cid:107) M exp fj Y 22 π f ψ(cid:96)2(ω2) = jM=1 exp −(cid:107)f2jω−2Y (cid:107)22 πj j . (6) j=1 2ω2 j This estimator is optimal in expectation (Dalalyan and Tsybakov, 2007, 2008), but suboptimal in deviation (Dai et al., 2012). In this paper, we introduce the following exponentiated least squares loss motivated from the (cid:18) (cid:19) exponential moment technique for proving large deviation tail bounds for sums of random variables: − 1 ν (cid:107) − (cid:107) L(ψ, µ) = exp ψ µ 2 , (7) 2ω2 2 ∈ where the parameter ν (0, 1). It is easy to verify that the Bayes estimator deﬁned by (5) with the loss function deﬁned in (7) can be written as ψ (ω2, ν) = argmin J(ψ) , (8) X ∈R ψ n where 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: π(µ

text: f

text: π

text: j

text: j

text: In

text: this

text: setting,

text: the

text: posterior

text: distribution

text: of

text: µ

text: given

text: Y

text: is

text: p(Y

text: |µ

text: fj)p(µ

text: fj)

text: exp

text: πj

text: p(µ

text: fj

text: Y

text: p(Y

text: |µ

text: fj)p(µ

text: fj)

text: exp

text: πj

text: In

text: the

text: Bayesian

text: decision

text: theoretical

text: framework

text: considered

text: in

text: this

text: paper,

text: the

text: quantity

text: of

text: interest

text: E

text: is

text: η

text: Y

text: and

text: we

text: consider

text: a

text: loss

text: function

text: L(ψ,

text: µ)

text: which

text: we

text: would

text: like

text: to

text: minimize

text: with

text: respect

text: to

text: the

text: posterior

text: distribution.

text: The

text: corresponding

text: Bayes

text: estimator

text: ψˆ

text: minimizes

text: the

text: posterior

text: expected

text: loss

text: from

text: µ

text: as

text: follows:

text: ψˆ

text: argmin

text: E

text: [L(ψ,

text: µ)|Y

text: ∈R

text: ψ

text: n

text: It

text: is

text: worth

text: pointing

text: out

text: that

text: the

text: above

text: Bayesian

text: framework

text: is

text: only

text: used

text: to

text: obtain

text: decision

text: theoretically

text: motivated

text: model

text: averaging

text: estimators

text: (Bayesian

text: estimators

text: have

text: good

text: theoretical

text: properties

text: such

text: as

text: admissibility,

text: etc).

text: In

text: particular

text: we

text: do

text: not

text: assume

text: that

text: the

text: model

text: itself

text: is

text: correctly

text: speciﬁed.

text: That

text: is,

text: in

text: this

text: paper

text: we

text: allow

text: misspeciﬁed

text: models,

text: where

text: the

text: parameters

text: µ

text: and

text: ω2

text: are

text: not

text: necessarily

text: equal

text: to

text: the

text: true

text: mean

text: η

text: and

text: the

text: true

text: variance

text: σ2

text: in

text: and

text: η

text: does

text: not

text: necessarily

text: belong

text: to

text: the

text: dictionary

text: f

text: f

text: M

text: The

text: Bayesian

text: estimator

text: of

text: depends

text: on

text: the

text: underlying

text: loss

text: function

text: L(

text: For

text: example,

text: under

text: the

text: standard

text: least

text: squares

text: loss

text: L(ψ,

text: µ)

text: ψ

text: µ

text: the

text: Bayes

text: estimator

text: is

text: the

text: posterior

text: mean,

text: which

text: leads

text: to

text: the

text: Exponential

text: Weighted

text: Model

text: Aggregation

text: (EWMA)

text: estimator

text: (Rigollet

text: and

text: Tsybakov,

text: M

text: exp

text: fj

text: Y

text: π

text: f

text: exp

text: πj

text: j

text: j

text: This

text: estimator

text: is

text: optimal

text: in

text: expectation

text: (Dalalyan

text: and

text: Tsybakov,

text: but

text: suboptimal

text: in

text: deviation

text: (Dai

text: et

text: al.,

text: In

text: this

text: paper,

text: we

text: introduce

text: the

text: following

text: exponentiated

text: least

text: squares

text: loss

text: motivated

text: from

text: the

text: exponential

text: moment

text: technique

text: for

text: proving

text: large

text: deviation

text: tail

text: bounds

text: for

text: sums

text: of

text: random

text: variables:

text: ν

text: L(ψ,

text: µ)

text: exp

text: ψ

text: µ

text: where

text: the

text: parameter

text: ν

text: It

text: is

text: easy

text: to

text: verify

text: that

text: the

text: Bayes

text: estimator

text: deﬁned

text: by

text: with

text: the

text: loss

text: function

text: deﬁned

text: in

text: can

text: be

text: written

text: as

text: ψ

text: ν)

text: argmin

text: J(ψ)

text: X

text: ∈R

text: ψ

text: n

text: where

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: R However, unlike the primal objective function J(ψ) which is deﬁned on n, the dual objective R function Q(λ) is deﬁned on M. When M is large or inﬁnity, the optimization of Q(λ) is non-trivial. Although greedy algorithms are proposed in (Dai et al., 2012), they cannot handle the standard KL-divergence; instead they can only work with the linear entropy where ρ(t) = 1; it gives a larger penalty than the standard KL-divergence (and thus worse resulting oracle inequality), and it cannot be generalized to handle continuous dictionaries (because in such case, the linear entropy ∞ with ρ(t) = 1 will always be + ). Therefore the numerical greedy procedures of (Dai et al., 2012) converges to a solution with a worse oracle bound than that of the solution for the primal formulation considered in this paper. The following two corollaries (Corollary 1 and Corollary 2) are listed for illustration convenience, they are directly implied from Theorem 1 and optimality of f in Theorem 3.1 of Dai et al. (2012). λQ ∈ ≥ ∈ Corollary 1. Assume that ν (0, 1) and ω2 σ2 − . For any λ ΛM, the following oracle min(ν,1 ν) inequality holds (cid:88) M (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) − (cid:107) − (cid:107) K ψ (ω2, ν) η 2 ν λ f η 2 + (1 ν) f η 2 + 2ω2 (λ, πδ) , (26) X 2 j j 2 λ 2 j=1 − with probability at least 1 δ. M(cid:88)oreover, M E(cid:107) − (cid:107) ≤ (cid:107) − (cid:107) − (cid:107) − (cid:107) K ψ (ω2, ν) η 2 ν λ f η 2 + (1 ν) f η 2 + 2ω2 (λ, π) . (27) (cid:80) X 2 (cid:80) j j 2 λ 2 j=1 (cid:107) − (cid:107) (cid:107) − (cid:107) (cid:107) − (cid:107) (cid:80)Since ν M λ f η 2 = ν M λ f f 2+ν f η 2, our theorem implies that ψ (ω2, ν) j=1 j j 2 j=1 j j λ 2 λ 2∈ X can compete with an arbitrary f in the convex hull with λ ΛM as long as the variance term (cid:107) − (cid:107) λ ν M λ f f 2 and the divergence term K(λ, π) are small. j=1 j j λ 2 H Although for notation simplicity, the result is stated for ﬁnite dictionary , the analysis of the ∞ paper directly applies to inﬁnite dictionaries where M = as well as continuous dictionaries. For ∈ R × example, given a matrix X n d, we may consider a continuous dictionary indexed by vector w H { ∈ R ∈ R (cid:107) (cid:107) ≤ } (cid:82) as = f n : f = Xw (w d; w 1) . We may consider the uniform prior π on w, w w 2 ∈ R and the corollary is well-deﬁned as long as a distribution λ on w d is concentrated around a (cid:107) − (cid:107) single model (so that the variance term corresponding to λ f f 2dw is small) with ﬁnite KL w w λ 2 divergence with respect to π. For example, λ can be chosen as the uniform distribution on a small { (cid:107) − (cid:107) ≤ } ball w : w w r . In such case, f = f and the variance term is small when r is small. 0 2 λ w0 Corollary 1 can be applied to derive an oracle inequality that competes with any single model f . w0 In the case that M is ﬁnite, we can more directly obtain an oracle inequality that competes with the best single model, which is the situation that λ is at a vertex of the simplex ΛM: (cid:26) (cid:18) (cid:19)(cid:27) Corollary 2. Under the assumptions of Corollary 1, ψ (ω2, ν) satisﬁes X (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) 1 ψ (ω2, ν) η 2 min f η 2 + 2ω2 log , (28) X 2 j∈1,...,M j 2 πjδ − (cid:26) (cid:18) (cid:19)(cid:27) with probability at least 1 δ. Moreover, E(cid:107) − (cid:107) ≤ (cid:107) − (cid:107) 1 ψ (ω2, ν) η 2 min f η 2 + 2ω2 log . (29) X 2 j∈1,...,M j 2 πj 8 ##LTLine## ##LTLine## ##LTLine##

text: R

label: paragraph

text: However,

text: unlike

text: the

text: primal

text: objective

text: function

text: J(ψ)

text: which

text: is

text: deﬁned

text: on

text: n,

text: the

text: dual

text: objective

text: R

text: function

text: Q(λ)

text: is

text: deﬁned

text: on

text: M.

text: When

text: M

text: is

text: large

text: or

text: inﬁnity,

text: the

text: optimization

text: of

text: Q(λ)

text: is

text: non-trivial.

text: Although

text: greedy

text: algorithms

text: are

text: proposed

text: in

text: (Dai

text: et

text: al.,

text: they

text: cannot

text: handle

text: the

text: standard

text: KL-divergence;

text: instead

text: they

text: can

text: only

text: work

text: with

text: the

text: linear

text: entropy

text: where

text: ρ(t)

text: it

text: gives

text: a

text: larger

text: penalty

text: than

text: the

text: standard

text: KL-divergence

text: (and

text: thus

text: worse

text: resulting

text: oracle

text: inequality),

text: and

text: it

text: cannot

text: be

text: generalized

text: to

text: handle

text: continuous

text: dictionaries

text: (because

text: in

text: such

text: case,

text: the

text: linear

text: entropy

text: with

text: ρ(t)

text: will

text: always

text: be

text: Therefore

text: the

text: numerical

text: greedy

text: procedures

text: of

text: (Dai

text: et

text: al.,

text: converges

text: to

text: a

text: solution

text: with

text: a

text: worse

text: oracle

text: bound

text: than

text: that

text: of

text: the

text: solution

text: for

text: the

text: primal

text: formulation

text: considered

text: in

text: this

text: paper.

text: The

text: following

text: two

text: corollaries

text: (Corollary

text: and

text: Corollary

text: are

text: listed

text: for

text: illustration

text: convenience,

text: they

text: are

text: directly

text: implied

text: from

text: Theorem

text: and

text: optimality

text: of

text: f

text: in

text: Theorem

text: of

text: Dai

text: et

text: al.

text: λQ

text: Corollary

text: Assume

text: that

text: ν

text: and

text: ω2

text: σ2

text: For

text: any

text: λ

text: ΛM,

text: the

text: following

text: oracle

text: ν)

text: inequality

text: holds

text: M

text: K

text: ψ

text: ν)

text: η

text: ν

text: λ

text: f

text: η

text: ν)

text: f

text: η

text: (λ,

text: πδ)

text: X

text: j

text: j

text: λ

text: with

text: probability

text: at

text: least

text: δ.

text: M

text: K

text: ψ

text: ν)

text: η

text: ν

text: λ

text: f

text: η

text: ν)

text: f

text: η

text: (λ,

text: π)

text: X

text: j

text: j

text: λ

text: ν

text: M

text: λ

text: f

text: η

text: ν

text: M

text: λ

text: f

text: f

text: f

text: η

text: our

text: theorem

text: implies

text: that

text: ψ

text: ν)

text: j

text: j

text: j

text: j

text: λ

text: λ

text: 2∈

text: X

text: can

text: compete

text: with

text: an

text: arbitrary

text: f

text: in

text: the

text: convex

text: hull

text: with

text: λ

text: ΛM

text: as

text: long

text: as

text: the

text: variance

text: term

text: λ

text: ν

text: M

text: λ

text: f

text: f

text: and

text: the

text: divergence

text: term

text: K(λ,

text: π)

text: are

text: small.

text: j

text: j

text: λ

text: H

text: Although

text: for

text: notation

text: simplicity,

text: the

text: result

text: is

text: stated

text: for

text: ﬁnite

text: dictionary

text: the

text: analysis

text: of

text: the

text: paper

text: directly

text: applies

text: to

text: inﬁnite

text: dictionaries

text: where

text: M

text: as

text: well

text: as

text: continuous

text: dictionaries.

text: For

text: R

text: example,

text: given

text: a

text: matrix

text: X

text: n

text: d,

text: we

text: may

text: consider

text: a

text: continuous

text: dictionary

text: indexed

text: by

text: vector

text: w

text: H

text: R

text: R

text: as

text: f

text: n

text: f

text: Xw

text: (w

text: d;

text: w

text: We

text: may

text: consider

text: the

text: uniform

text: prior

text: π

text: on

text: w,

text: w

text: w

text: R

text: and

text: the

text: corollary

text: is

text: well-deﬁned

text: as

text: long

text: as

text: a

text: distribution

text: λ

text: on

text: w

text: d

text: is

text: concentrated

text: around

text: a

text: single

text: model

text: (so

text: that

text: the

text: variance

text: term

text: corresponding

text: to

text: λ

text: f

text: f

text: is

text: small)

text: with

text: ﬁnite

text: KL

text: w

text: w

text: λ

text: divergence

text: with

text: respect

text: to

text: π.

text: For

text: example,

text: λ

text: can

text: be

text: chosen

text: as

text: the

text: uniform

text: distribution

text: on

text: a

text: small

text: ball

text: w

text: w

text: w

text: r

text: In

text: such

text: case,

text: f

text: f

text: and

text: the

text: variance

text: term

text: is

text: small

text: when

text: r

text: is

text: small.

text: λ

text: Corollary

text: can

text: be

text: applied

text: to

text: derive

text: an

text: oracle

text: inequality

text: that

text: competes

text: with

text: any

text: single

text: model

text: f

text: In

text: the

text: case

text: that

text: M

text: is

text: ﬁnite,

text: we

text: can

text: more

text: directly

text: obtain

text: an

text: oracle

text: inequality

text: that

text: competes

text: with

text: the

text: best

text: single

text: model,

text: which

text: is

text: the

text: situation

text: that

text: λ

text: is

text: at

text: a

text: vertex

text: of

text: the

text: simplex

text: ΛM:

text: Corollary

text: Under

text: the

text: assumptions

text: of

text: Corollary

text: ψ

text: ν)

text: satisﬁes

text: X

text: ψ

text: ν)

text: η

text: min

text: f

text: η

text: log

text: X

text: j

text: πjδ

text: with

text: probability

text: at

text: least

text: δ.

text: Moreover,

text: ψ

text: ν)

text: η

text: min

text: f

text: η

text: log

text: X

text: j

text: πj

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ (Dated: May 26, 2020) We investigate the origin of the fast recombination dynamics of bound and free excitons in GaN nanowire ensembles by temperature-dependent photoluminescence spectroscopy using both continuous-wave and pulsed excitation. The exciton recombination in the present GaN nanowires is dominated by a nonradiative channel between 10 and 300K. Furthermore, bound and free excitons in GaN NWs are strongly coupled even at low temperatures resulting in a common lifetime of these states. By solving the rate equations for a coupled two-level system, we show that one cannot, in practice, distinguish whether the nonradiative decay occurs directly via the bound or indirectly via the free state. The nanowire surface and coalescence-induced dislocations appear to be the most obvious candidates for nonradiative defects, and we thus compare the exciton decay times measured for a variety of GaN nanowire ensembles with different surface-to-volume ratio and coalescence degrees. The data are found to exhibit no correlation with either of these parameters, i.e., the dominating nonradiative channel in the GaN nanowires under investigation is neither related to the nanowire surface, nor to coalescence-induced defects for the present samples. Hence, we conclude that nonradiative point defects are the origin of the fast recombination dynamics of excitons in GaN nanowires. I. INTRODUCTION show that the donor-bound and free exciton states exhibit a common effective lifetime revealing an efﬁcient coupling of both states even at low temperatures. An important con- GaN nanowires (NWs) offer novel opportunities for the sequence of this coupling is the fact that situations where integration of optically active nanostructures based on GaN the nonradiative decay occurs either via the bound or the with electronic components based on silicon.51–54 In con- free states are practically indistinguishable, since they re- trast to planar GaN layers, they can be fabricated with a sult in the same decay time for the coupled system. Next, low density of structural defects directly on Si substrates we compare the effective exciton lifetimes at low temper- by molecular-beam epitaxy (MBE).51,55,56 Because of the atures measured for 19 different GaN NW ensembles with absence of threading dislocations, exciton recombination in different surface-to-volume ratio and coalescence degree. GaN NWs is assumed to be predominantly radiative.57 How- No systematic variation with either of these quantities is ob- ever, this expectation is inconsistent with the experimentally served, suggesting that the dominating nonradiative channel observed short decay times of bound excitons in GaN NWs. in the present GaN NWs is neither related to the surface nor Even at low temperatures, time-resolved photoluminescence to coalescence-induced defects. (TRPL) measurements yield typically58 decay times in the range of a few tens to about 200 ps,59–65 i. e., signiﬁcantly II. EXPERIMENTAL DETAILS shorter than the radiative lifetime of the bound exciton state in bulk GaN of at least 1 ns.66,67 Since NWs inevitably pos- The investigated GaN NW ensembles were grown by sess a large surface-to-volume ratio, it is often suspected plasma-assisted MBE on Si(111) substrates relying on either that nonradiative surface recombination causes these fast their spontaneous formation under suitable conditions56,75 or decay times.60–63,68 Recent experimental results, however, employing selective-area growth (SAG) on a thin AlN buffer show that also comparatively thick NWs exhibit short decay layer deposited on the Si substrate prior to NW growth.76 times which cannot be ascribed to surface recombination The substrate temperatures were in the range of 780 to alone.64,65 830 °C, and the Ga/N ﬂux ratio was changed accordingly Coalescence of NWs is a phenomenon less frequently to guarantee effectively N-rich conditions.77 considered as the possible origin of nonradiative recombi- The morphological properties of the as-grown GaN NW nation in GaN NWs. Given that the NW density in spon- ensembles were studied by ﬁeld-emission scanning electron taneously formed GaN NW ensembles is on the order of microscopy as described in detail in Ref. 78. In particular, − 109 to 1010 cm 2, coalescence of adjacent NWs during their top-view micrographs were used to determine the area and formation is inevitable. Basal-plane stacking faults69,70 and perimeter of several hundreds of individual NWs for each of chains of dislocations70 have been observed as consequences the samples under investigation. of the coalescence of GaN NWs. The former defects capture In order to investigate the optical properties of the GaN free excitons efﬁciently, but manifest themselves by char- NWs, the as-grown samples were mounted onto the cold- acteristic radiative transitions at speciﬁc energies.71–74 The ﬁnger of a microscope cryostat. Continuous-wave pho- latter, in contrast, were found to act as efﬁcient nonradiative toluminescence (cw PL) spectroscopy was performed by centers for excitons.69 exciting the ensembles with the 325 nm (3.814 eV) line of In the present paper, we investigate the radiative and non- a He-Cd laser focused onto the sample with an excitation radiative decay of free and bound excitons in GaN NW density of less than 1 W/cm2, leading to a photogenerated − ensembles grown by molecular-beam epitaxy (MBE) on carrier density below 1014 cm 3. The PL signal was spec- Si(111) substrates. We ﬁrst focus on a representative sample trally dispersed by a 80 cm monochromator providing a to demonstrate that the exciton lifetime is dominated by a spectral resolution of 0.25 meV and detected with a cooled nonradiative decay channel between 10 and 300 K. We also charge-coupled device array. The TRPL measurements were

text: (Dated:

text: May

label: date

text: We

label: abstract

text: investigate

text: the

text: origin

text: of

text: the

text: fast

text: recombination

text: dynamics

text: of

text: bound

text: and

text: free

text: excitons

text: in

text: GaN

text: nanowire

text: ensembles

text: by

text: temperature-dependent

text: photoluminescence

text: spectroscopy

text: using

text: both

text: continuous-wave

text: and

text: pulsed

text: excitation.

text: The

text: exciton

text: recombination

text: in

text: the

text: present

text: GaN

text: nanowires

text: is

text: dominated

text: by

text: a

text: nonradiative

text: channel

text: between

text: and

text: Furthermore,

text: bound

text: and

text: free

text: excitons

text: in

text: GaN

text: NWs

text: are

text: strongly

text: coupled

text: even

text: at

text: low

text: temperatures

text: resulting

text: in

text: a

text: common

text: lifetime

text: of

text: these

text: states.

text: By

text: solving

text: the

text: rate

text: equations

text: for

text: a

text: coupled

text: two-level

text: system,

text: we

text: show

text: that

text: one

text: cannot,

text: in

text: practice,

text: distinguish

text: whether

text: the

text: nonradiative

text: decay

text: occurs

text: directly

text: via

text: the

text: bound

text: or

text: indirectly

text: via

text: the

text: free

text: state.

text: The

text: nanowire

text: surface

text: and

text: coalescence-induced

text: dislocations

text: appear

text: to

text: be

text: the

text: most

text: obvious

text: candidates

text: for

text: nonradiative

text: defects,

text: and

text: we

text: thus

text: compare

text: the

text: exciton

text: decay

text: times

text: measured

text: for

text: a

text: variety

text: of

text: GaN

text: nanowire

text: ensembles

text: with

text: different

text: surface-to-volume

text: ratio

text: and

text: coalescence

text: degrees.

text: The

text: data

text: are

text: found

text: to

text: exhibit

text: no

text: correlation

text: with

text: either

text: of

text: these

text: parameters,

text: i.e.,

text: the

text: dominating

text: nonradiative

text: channel

text: in

text: the

text: GaN

text: nanowires

text: under

text: investigation

text: is

text: neither

text: related

text: to

text: the

text: nanowire

text: surface,

text: nor

text: to

text: coalescence-induced

text: defects

text: for

text: the

text: present

text: samples.

text: Hence,

text: we

text: conclude

text: that

text: nonradiative

text: point

text: defects

text: are

text: the

text: origin

text: of

text: the

text: fast

text: recombination

text: dynamics

text: of

text: excitons

text: in

text: GaN

text: nanowires.

text: I.

text: INTRODUCTION

text: show

text: that

text: the

text: donor-bound

text: and

text: free

text: exciton

text: states

text: exhibit

text: a

text: common

text: effective

text: lifetime

text: revealing

text: an

text: efﬁcient

text: coupling

text: of

text: both

text: states

text: even

text: at

text: low

text: temperatures.

text: An

text: important

text: con-

text: GaN

text: nanowires

text: (NWs)

text: offer

text: novel

text: opportunities

text: for

text: the

text: sequence

text: of

text: this

text: coupling

text: is

text: the

text: fact

text: that

text: situations

text: where

text: integration

text: of

text: optically

text: active

text: nanostructures

text: based

text: on

text: GaN

text: the

text: nonradiative

text: decay

text: occurs

text: either

text: via

text: the

text: bound

text: or

text: the

text: with

text: electronic

text: components

text: based

text: on

text: In

text: con-

text: free

text: states

text: are

text: practically

text: indistinguishable,

text: since

text: they

text: re-

text: trast

text: to

text: planar

text: GaN

text: layers,

text: they

text: can

text: be

text: fabricated

text: with

text: a

text: sult

text: in

text: the

text: same

text: decay

text: time

text: for

text: the

text: coupled

text: system.

text: Next,

text: low

text: density

text: of

text: structural

text: defects

text: directly

text: on

text: Si

text: substrates

text: we

text: compare

text: the

text: effective

text: exciton

text: lifetimes

text: at

text: low

text: temper-

text: by

text: molecular-beam

text: epitaxy

text: Because

text: of

text: the

text: atures

text: measured

text: for

text: different

text: GaN

text: NW

text: ensembles

text: with

text: absence

text: of

text: threading

text: dislocations,

text: exciton

text: recombination

text: in

text: different

text: surface-to-volume

text: ratio

text: and

text: coalescence

text: degree.

text: GaN

text: NWs

text: is

text: assumed

text: to

text: be

text: predominantly

text: How-

text: No

text: systematic

text: variation

text: with

text: either

text: of

text: these

text: quantities

text: is

text: ob-

text: ever,

text: this

text: expectation

text: is

text: inconsistent

text: with

text: the

text: experimentally

text: served,

text: suggesting

text: that

text: the

text: dominating

text: nonradiative

text: channel

text: observed

text: short

text: decay

text: times

text: of

text: bound

text: excitons

text: in

text: GaN

text: NWs.

text: in

text: the

text: present

text: GaN

text: NWs

text: is

text: neither

text: related

text: to

text: the

text: surface

text: nor

text: Even

text: at

text: low

text: temperatures,

text: time-resolved

text: photoluminescence

text: to

text: coalescence-induced

text: defects.

text: (TRPL)

text: measurements

text: yield

text: decay

text: times

text: in

text: the

text: range

text: of

text: a

text: few

text: tens

text: to

text: about

text: i.

text: e.,

text: signiﬁcantly

text: II.

text: EXPERIMENTAL

text: DETAILS

text: shorter

text: than

text: the

text: radiative

text: lifetime

text: of

text: the

text: bound

text: exciton

text: state

text: in

text: bulk

text: GaN

text: of

text: at

text: least

text: Since

text: NWs

text: inevitably

text: pos-

text: The

text: investigated

text: GaN

text: NW

text: ensembles

text: were

text: grown

text: by

text: sess

text: a

text: large

text: surface-to-volume

text: ratio,

text: it

text: is

text: often

text: suspected

text: plasma-assisted

text: MBE

text: on

text: substrates

text: relying

text: on

text: either

text: that

text: nonradiative

text: surface

text: recombination

text: causes

text: these

text: fast

text: their

text: spontaneous

text: formation

text: under

text: suitable

text: or

text: decay

text: Recent

text: experimental

text: results,

text: however,

text: employing

text: selective-area

text: growth

text: (SAG)

text: on

text: a

text: thin

text: AlN

text: buffer

text: show

text: that

text: also

text: comparatively

text: thick

text: NWs

text: exhibit

text: short

text: decay

text: layer

text: deposited

text: on

text: the

text: Si

text: substrate

text: prior

text: to

text: NW

text: times

text: which

text: cannot

text: be

text: ascribed

text: to

text: surface

text: recombination

text: The

text: substrate

text: temperatures

text: were

text: in

text: the

text: range

text: of

text: to

text: °C,

text: and

text: the

text: Ga/N

text: ﬂux

text: ratio

text: was

text: changed

text: accordingly

text: Coalescence

text: of

text: NWs

text: is

text: a

text: phenomenon

text: less

text: frequently

text: to

text: guarantee

text: effectively

text: N-rich

text: considered

text: as

text: the

text: possible

text: origin

text: of

text: nonradiative

text: recombi-

text: The

text: morphological

text: properties

text: of

text: the

text: as-grown

text: GaN

text: NW

text: nation

text: in

text: GaN

text: NWs.

text: Given

text: that

text: the

text: NW

text: density

text: in

text: spon-

text: ensembles

text: were

text: studied

text: by

text: ﬁeld-emission

text: scanning

text: electron

text: taneously

text: formed

text: GaN

text: NW

text: ensembles

text: is

text: on

text: the

text: order

text: of

text: microscopy

text: as

text: described

text: in

text: detail

text: in

text: Ref.

text: In

text: particular,

text: to

text: cm

text: coalescence

text: of

text: adjacent

text: NWs

text: during

text: their

text: top-view

text: micrographs

text: were

text: used

text: to

text: determine

text: the

text: area

text: and

text: formation

text: is

text: inevitable.

text: Basal-plane

text: stacking

text: and

text: perimeter

text: of

text: several

text: hundreds

text: of

text: individual

text: NWs

text: for

text: each

text: of

text: chains

text: of

text: have

text: been

text: observed

text: as

text: consequences

text: the

text: samples

text: under

text: investigation.

text: of

text: the

text: coalescence

text: of

text: GaN

text: NWs.

text: The

text: former

text: defects

text: capture

text: In

text: order

text: to

text: investigate

text: the

text: optical

text: properties

text: of

text: the

text: GaN

text: free

text: excitons

text: efﬁciently,

text: but

text: manifest

text: themselves

text: by

text: char-

text: NWs,

text: the

text: as-grown

text: samples

text: were

text: mounted

text: onto

text: the

text: cold-

text: acteristic

text: radiative

text: transitions

text: at

text: speciﬁc

text: The

text: ﬁnger

text: of

text: a

text: microscope

text: cryostat.

text: Continuous-wave

text: pho-

text: latter,

text: in

text: contrast,

text: were

text: found

text: to

text: act

text: as

text: efﬁcient

text: nonradiative

text: toluminescence

text: (cw

text: PL)

text: spectroscopy

text: was

text: performed

text: by

text: centers

text: for

text: exciting

text: the

text: ensembles

text: with

text: the

text: nm

text: eV)

text: line

text: of

text: In

text: the

text: present

text: paper,

text: we

text: investigate

text: the

text: radiative

text: and

text: non-

text: a

text: He-Cd

text: laser

text: focused

text: onto

text: the

text: sample

text: with

text: an

text: excitation

text: radiative

text: decay

text: of

text: free

text: and

text: bound

text: excitons

text: in

text: GaN

text: NW

text: density

text: of

text: less

text: than

text: leading

text: to

text: a

text: photogenerated

text: ensembles

text: grown

text: by

text: molecular-beam

text: epitaxy

text: (MBE)

text: on

text: carrier

text: density

text: below

text: cm

text: The

text: PL

text: signal

text: was

text: spec-

text: substrates.

text: We

text: ﬁrst

text: focus

text: on

text: a

text: representative

text: sample

text: trally

text: dispersed

text: by

text: a

text: cm

text: monochromator

text: providing

text: a

text: to

text: demonstrate

text: that

text: the

text: exciton

text: lifetime

text: is

text: dominated

text: by

text: a

text: spectral

text: resolution

text: of

text: meV

text: and

text: detected

text: with

text: a

text: cooled

text: nonradiative

text: decay

text: channel

text: between

text: and

text: K.

text: We

text: also

text: charge-coupled

text: device

text: array.

text: The

text: TRPL

text: measurements

text: were


text: 5 nonradiative decay channel over the whole investigated tem- (a) Case 'A' (b) Case 'B' perature range. Furthermore, the results imply that the free n n F F and bound exciton states are coupled at lower temperatures than expected from the binding energy of the (D0,XA) com- γc γe γc γe plex. This ﬁnding corroborates the results from our previous n n y D D work, where the inevitable presence of electric ﬁelds in GaN g r NbeWtwseewnasalsluegxgceitsotendicasstatthees.6c4auIsneoordfearntoenghaainncaeddeceopueprluinng- Ene γD γ0F.1= ns-1 1 γnDs=-1 γF derstanding of the consequences of this coupling and to aid the interpretation of the effective decay rate of the system as 0 0 a whole, we proceed by investigating a rate equation system 100 describing the dynamics of the two coupled states. (c) D. Rate equation model describing the coupling at low x 4 temperatures s) 10-1 Case ’A’: ergTyhsecchoeumpeleids (dDis0p,lXaAye)d(cid:28)inXFAig+sD. 40(say)satnedm4, f(obr),wishidcehscarnibeend- rb. unit 10-2 Case ’B’: eff = 152 ps bloywtheexcfiotalltoiownindgensseittyo:f linear rate equations in the limit of sity (a 10-3 eff = 152 ps n e nt ddntF = −γcnF +γenD −γFnF (2) PL I 10-4 / (D0,XA) X ddntD = γcnF −γenD −γDnD. (3) 10-50.0 0.2A 0.4 0.6 0.8 1.0 Time (ns) Here, n and n denote the time-dependent densities F D of free and donor-bound excitons with the initial densi- FIG. 4. (color online) (a) and (b) Schematic energy diagrams vi- ties nF(0) = n0F and nD(0) = 0, respectively, while γF and sreusapliezcitnivgeElyq.s.T(h2e)tawnod s(t3a)tefsorintvhoelvtwedoadrieffdeerennottecdasbeys |’nA’(cid:105),aanndd’Bth’e, γD represent the sums of the respe{ctive}radiative (γi,r) and crystal ground state is represented by |0(cid:105). The red soilid arrows nonradiative rates (γi,nr) with i = F, D . The coupling is symbolize the effective decay channel, which dictates the dynam- represented by γ , the capture rate of free to bound excitons, c ics of the coupled system, while the green dotted arrows represent and γe, the rate for the dissociation of the (D0,XA) complex purely radiative transitions. (c) Simulated TRPL transients of the and the subsequent return of the exciton to its free state. (D0,XA) and XA given by γi,rni, respectively, employing the effec- This rate equation system can be solved analytically, tive decay rates γ as summarized in Table I and radiative decay i − which yields the time-dependent densities for both states: rates γ for the free and bound exciton of 0.1 and 1ns 1, respec- (cid:104) (cid:105) i,r tively. nF(t) = 2nβ0F ((cid:104)β −δ)e−12(α−β)t +(β +(cid:105) δ)e−12(α+β)t (4) nF dire(cid:2)ctly after exc(cid:3)itation. At longer times, the system re- veals its coupled nature by exhibiting a common effective nD(t) = γcβn0F e−12(α−β)t −e−12(α+β)t , (5) decay r−ate, w−hich is represented by the terms proportional to exp 1(α β)t in Eqs. (4) and (5). 2 If donor-bound and free excitons do not couple (i. e., γe = with 0), the common decay rate of both states naturally disap- | | − | | pears since β δ = 0, and the effective decay rates of (cid:113) the free and bound exciton states are given by γ +γ and γ , c F D α = γc +γe +γD +γF, (6) respectively, as expected for the uncoupled system. (cid:28) β = α−2 −−4[γeγF +γD (γc +γF)], (7) is i(cid:48)Tnvhae(cid:48)rdiayn(cid:48)ntawmi(cid:48)itchsreosfptehcetctoouaptlreadnssfyosrtmemati(oDn0(,γXcA,)γe,γDX,Aγ+F)D→0 δ = γc γe γD +γF. (8) (γ c,γ e,γ D,γ F), if α, β, and δ remain constant. Such a transformation always exists, and we will now discuss a particular example representing two limiting cases. (cid:2) (cid:3) The time-dependent densities [Eqs. (4)–(5)] are governed Let us ﬁrst consider the situation depicted in Fig. 4(a) by two exponentials with different decay constants. The (hereafter referred to as ’case A’). The free exciton is as- − − terms proportional to exp 12(α +β)t describe the in- sumed to have an effective decay rate of γF = γF,r = 0.1 ns 1, crease of the donor-bound exciton population n and the i. e., equal to its radiative decay rate γ estimated in Ref. 93. D F,r simultaneous fast decrease of the free exciton population To account for the experimentally observed rapid decay, the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: nonradiative

text: decay

text: channel

text: over

text: the

text: whole

text: investigated

text: tem-

text: (a)

font: default

text: Case

text: 'A'

text: (b)

text: Case

text: 'B'

text: perature

text: range.

text: Furthermore,

text: the

text: results

text: imply

text: that

text: the

text: free

text: n

text: n

text: F

text: F

text: and

text: bound

text: exciton

text: states

text: are

text: coupled

text: at

text: lower

text: temperatures

text: than

text: expected

text: from

text: the

text: binding

text: energy

text: of

text: the

text: com-

text: γc

text: γe

text: γc

text: γe

text: plex.

text: This

text: ﬁnding

text: corroborates

text: the

text: results

text: from

text: our

text: previous

text: n

text: n

text: y

text: D

text: D

text: work,

text: where

text: the

text: inevitable

text: presence

text: of

text: electric

text: ﬁelds

text: in

text: GaN

text: g

text: r

text: Ene

text: γD

text: γF

text: derstanding

text: of

text: the

text: consequences

text: of

text: this

text: coupling

text: and

text: to

text: aid

text: the

text: interpretation

text: of

text: the

text: effective

text: decay

text: rate

text: of

text: the

text: system

text: as

text: a

text: whole,

text: we

text: proceed

text: by

text: investigating

text: a

text: rate

text: equation

text: system

text: describing

text: the

text: dynamics

text: of

text: the

text: two

text: coupled

text: states.

text: (c)

text: D.

text: Rate

text: equation

text: model

text: describing

text: the

text: coupling

text: at

text: low

text: x

text: temperatures

text: s)

text: Case

text: ’A’:

text: ergTyhsecchoeumpeleids

text: f(obr),wishidcehscarnibeend-

text: rb.

text: unit

text: Case

text: ’B’:

text: eff

text: ps

text: linear

text: rate

text: equations

text: in

text: the

text: limit

text: of

text: sity

text: (a

text: eff

text: ps

text: n

text: e

text: nt

text: ddntF

text: −γcnF

text: +γenD

text: −γFnF

text: PL

text: I

text: X

text: ddntD

text: γcnF

text: −γenD

text: −γDnD.

text: Time

text: (ns)

text: Here,

text: n

text: and

text: n

text: denote

text: the

text: time-dependent

text: densities

text: F

text: D

text: of

text: free

text: and

text: donor-bound

text: excitons

text: with

text: the

text: initial

text: densi-

text: FIG.

text: (color

text: online)

text: (a)

text: and

text: (b)

text: Schematic

text: energy

text: diagrams

text: vi-

text: ties

text: and

text: respectively,

text: while

text: γF

text: and

text: γD

text: represent

text: the

text: sums

text: of

text: the

text: respe{ctive}radiative

text: (γi,r)

text: and

text: crystal

text: ground

text: state

text: is

text: represented

text: by

text: The

text: red

text: soilid

text: arrows

text: nonradiative

text: rates

text: (γi,nr)

text: with

text: i

text: F,

text: D

text: The

text: coupling

text: is

text: symbolize

text: the

text: effective

text: decay

text: channel,

text: which

text: dictates

text: the

text: dynam-

text: represented

text: by

text: γ

text: the

text: capture

text: rate

text: of

text: free

text: to

text: bound

text: excitons,

text: c

text: ics

text: of

text: the

text: coupled

text: system,

text: while

text: the

text: green

text: dotted

text: arrows

text: represent

text: and

text: γe,

text: the

text: rate

text: for

text: the

text: dissociation

text: of

text: the

text: complex

text: purely

text: radiative

text: transitions.

text: (c)

text: Simulated

text: TRPL

text: transients

text: of

text: the

text: and

text: the

text: subsequent

text: return

text: of

text: the

text: exciton

text: to

text: its

text: free

text: state.

text: and

text: XA

text: given

text: by

text: γi,rni,

text: respectively,

text: employing

text: the

text: effec-

text: This

text: rate

text: equation

text: system

text: can

text: be

text: solved

text: analytically,

text: tive

text: decay

text: rates

text: γ

text: as

text: summarized

text: in

text: Table

text: I

text: and

text: radiative

text: decay

text: i

text: which

text: yields

text: the

text: time-dependent

text: densities

text: for

text: both

text: states:

text: rates

text: γ

text: for

text: the

text: free

text: and

text: bound

text: exciton

text: of

text: and

text: respec-

text: i,r

text: tively.

text: nF(t)

text: +(β

text: nF

text: after

text: At

text: longer

text: times,

text: the

text: system

text: re-

text: veals

text: its

text: coupled

text: nature

text: by

text: exhibiting

text: a

text: common

text: effective

text: nD(t)

text: decay

text: r−ate,

text: w−hich

text: is

text: represented

text: by

text: the

text: terms

text: proportional

text: to

text: exp

text: β)t

text: in

text: Eqs.

text: and

text: If

text: donor-bound

text: and

text: free

text: excitons

text: do

text: not

text: couple

text: (i.

text: e.,

text: γe

text: with

text: the

text: common

text: decay

text: rate

text: of

text: both

text: states

text: naturally

text: disap-

text: pears

text: since

text: β

text: δ

text: and

text: the

text: effective

text: decay

text: rates

text: of

text: the

text: free

text: and

text: bound

text: exciton

text: states

text: are

text: given

text: by

text: γ

text: +γ

text: and

text: γ

text: c

text: F

text: D

text: α

text: γc

text: +γe

text: +γD

text: +γF,

text: respectively,

text: as

text: expected

text: for

text: the

text: uncoupled

text: system.

text: β

text: α−2

text: −−4[γeγF

text: +γD

text: (γc

text: +γF)],

text: is

text: δ

text: γc

text: γe

text: γD

text: +γF.

text: (γ

text: c,γ

text: e,γ

text: D,γ

text: F),

text: if

text: α,

text: β,

text: and

text: δ

text: remain

text: constant.

text: Such

text: a

text: transformation

text: always

text: exists,

text: and

text: we

text: will

text: now

text: discuss

text: a

text: particular

text: example

text: representing

text: two

text: limiting

text: cases.

text: The

text: time-dependent

text: densities

text: [Eqs.

text: are

text: governed

text: Let

text: us

text: ﬁrst

text: consider

text: the

text: situation

text: depicted

text: in

text: Fig.

text: by

text: two

text: exponentials

text: with

text: different

text: decay

text: constants.

text: The

text: (hereafter

text: referred

text: to

text: as

text: ’case

text: A’).

text: The

text: free

text: exciton

text: is

text: as-

text: terms

text: proportional

text: to

text: exp

text: +β)t

text: describe

text: the

text: in-

text: sumed

text: to

text: have

text: an

text: effective

text: decay

text: rate

text: of

text: γF

text: γF,r

text: ns

text: crease

text: of

text: the

text: donor-bound

text: exciton

text: population

text: n

text: and

text: the

text: i.

text: e.,

text: equal

text: to

text: its

text: radiative

text: decay

text: rate

text: γ

text: estimated

text: in

text: Ref.

text: D

text: F,r

text: simultaneous

text: fast

text: decrease

text: of

text: the

text: free

text: exciton

text: population

text: To

text: account

text: for

text: the

text: experimentally

text: observed

text: rapid

text: decay,

text: the

text: ##LTFigure##

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text: 7 0.24 0.24 (a) 10 K (b) 10 K 0.21 1 J/cm2 1 J/cm2 0.21 0.18 Lifetimes taken from: 0.18 Ref. (10) ns) 0.15 Ref. (13) 0.15 ns) (eff 0.12 / Ref. (14) 0.12 (eff Investigated samples: / new samples 0.09 0.09 sample III from Ref. (15) sample 4 from Ref. (49) 0.06 0.06 sample 1 from Ref. (50) 50 100 150 200 250 0.4 0.6 0.8 1.0 Æd*æ (nm) C FIG. 5. (color online) Overview of the effective (D0,XA) lifetime obtained by ﬁtting the short component of the respective TRPL transient recorded at low temperatures with a single-exponential decay for 19 different GaN NW ensembles. Solid symbols represent samples grown using the self-induced growth mode, while the open symbols are samples grown by SAG.76 (a) Dependence of the effective lifetime τ on the parameter (cid:104)d∗(cid:105). The solid line depicts the expected lifetimes for an effective surface recombination velocity of S˜ = 6.3×103 cmef/fs.63 ∗ The horizontal bars indicate the typical standard deviation of the d -distribution in the NW ensemble for the self-induced growth (upper bar) and SAG (lower bar). (b) Dependence of τ on the coalescence degree σ of the sample. eff C the respective distribution to characterize the ensemble. For trend towards shorter values for higher coalescence degrees. ∗ sample R, the distribution of d (not shown) is symmetric IV. CONCLUSION and is described well by a normal distribution with a mean (cid:104) ∗(cid:105) of d = 72 nm. This value is 20% smaller than the mean (cid:104) (cid:105) Our results show conclusively that the exciton lifetime in equivalent disk diameter ddisk = 99 nm [cf. Fig. 1(b)], un- the investigated GaN NW ensembles is limited by nonradia- derlining the importance of considering the actual shape of tive recombination. The lifetime is independent of the en- the NWs for the quantitative analysis of surface-related phe- sembles mean surface-to-volume ratio and coalescence de- nomena. gree, implying that the nonradiative process is neither caused We are now in the position to quantitatively compare the by surface recombination nor by dislocations formed due to (cid:28) effective lifetime of the coupled (D0,XA) XA system f(cid:104)or∗a(cid:105) NW coalescence. The remaining possibility for the origin of number of GaN NW samples with different values of d this nonradiative channel are point defects. In fact, given that grown in several different MBE machines using different the substrate temperatures used in the MBE growth of GaN plasma sources. The results based on samples from Refs. 65, NWs are low compared to those used in both metal-organic 99, and 100 are displayed in Fig. 5(a) together with data and hydride vapor phase epitaxy, the point defect density in reported in Refs. 60 and 64. The effective lifetimes mea- these NWs is necessarily higher than in state-of-the-art GaN sured for these samples remain short compared with the ra- layers, which are grown closer to equilibrium. However, the (cid:104) ∗(cid:105) diative lifetime regardless of the value of d . If the effec- point defect density does not need to be excessively high in tive lifetime τ were fundamentally limited by surface re- order to dominate the exciton decay. We have demonstrated eff combination, the resulting values would be expected to lin- that bound and free excitons in the investigated GaN NWs (cid:104) ∗(cid:105) early increase with d . The solid line shows the expected are strongly coupled even at low temperatures, and the non- × dependence with S˜ = 6.32 103 cm/s as reported by Gor- radiative decay may thus take place via the free state. Given gis et al.63 for bound excitons. Note that we have not yet that the diffusion length of free excitons in GaN is larger − observed decay times longer than expected from this value, than 50 nm, a density as low as 1015 cm 3 may sufﬁce to but only shorter ones. For thin NWs, surface recombination introduce an effective nonradiative decay channel. may therefore be the limiting process after all, but for thicker Attempts to reduce the density of this detrimental de- ones, it is clear that a process not related to the surface de- fects would probably require a higher substrate temperature. termines the exciton decay rate. Figure 5(b) plots the effec- However, due to an exponential increase of both the des- tive lifetime τ from 15 different GaN NW samples versus orption of Ga adatoms and the dissociation of GaN in the eff their coalescence degree σ . With the exception of the GaN high vacuum environment of MBE, it seems unlikely that C NW arrays fabricated by SAG (whose coalescence degree sufﬁciently high substrate temperatures can be reached. A was set to σC = 1 since multiple NW nucleations occur in- perhaps more promising approach for reducing the impact of evitably for each hole76), the values of the effective exciton these defects at least at low temperatures is the manipulation lifetime scatter around 150 ps and do not show any obvious of the strength of the coupling between free and bound states ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: K

text: (b)

text: K

text: Lifetimes

text: taken

text: from:

text: Ref.

text: ns)

text: Ref.

text: ns)

text: (eff

text: Ref.

text: (eff

text: Investigated

text: samples:

text: new

text: samples

text: sample

text: III

text: from

text: Ref.

text: sample

text: from

text: Ref.

text: sample

text: from

text: Ref.

text: Æd*æ

text: (nm)

text: C

text: FIG.

text: (color

text: online)

text: Overview

text: of

text: the

text: effective

text: lifetime

text: obtained

text: by

text: ﬁtting

text: the

text: short

text: component

text: of

text: the

text: respective

text: TRPL

text: transient

text: recorded

text: at

text: low

text: temperatures

text: with

text: a

text: single-exponential

text: decay

text: for

text: different

text: GaN

text: NW

text: ensembles.

text: Solid

text: symbols

text: represent

text: samples

text: grown

text: using

text: the

text: self-induced

text: growth

text: mode,

text: while

text: the

text: open

text: symbols

text: are

text: samples

text: grown

text: by

text: (a)

text: Dependence

text: of

text: the

text: effective

text: lifetime

text: τ

text: on

text: the

text: parameter

text: The

text: solid

text: line

text: depicts

text: the

text: expected

text: lifetimes

text: for

text: an

text: effective

text: surface

text: recombination

text: velocity

text: of

text: S˜

text: The

text: horizontal

text: bars

text: indicate

text: the

text: typical

text: standard

text: deviation

text: of

text: the

text: d

text: -distribution

text: in

text: the

text: NW

text: ensemble

text: for

text: the

text: self-induced

text: growth

text: (upper

text: bar)

text: and

text: SAG

text: (lower

text: bar).

text: (b)

text: Dependence

text: of

text: τ

text: on

text: the

text: coalescence

text: degree

text: σ

text: of

text: the

text: sample.

text: eff

text: C

text: the

text: respective

text: distribution

text: to

text: characterize

text: the

text: ensemble.

text: For

text: trend

text: towards

text: shorter

text: values

text: for

text: higher

text: coalescence

text: degrees.

text: sample

text: R,

text: the

text: distribution

text: of

text: d

text: (not

text: shown)

text: is

text: symmetric

text: IV.

text: CONCLUSION

text: and

text: is

text: described

text: well

text: by

text: a

text: normal

text: distribution

text: with

text: a

text: mean

text: of

text: d

text: nm.

text: This

text: value

text: is

text: smaller

text: than

text: the

text: mean

text: Our

text: results

text: show

text: conclusively

text: that

text: the

text: exciton

text: lifetime

text: in

text: equivalent

text: disk

text: diameter

text: ddisk

text: nm

text: [cf.

text: Fig.

text: un-

text: the

text: investigated

text: GaN

text: NW

text: ensembles

text: is

text: limited

text: by

text: nonradia-

text: derlining

text: the

text: importance

text: of

text: considering

text: the

text: actual

text: shape

text: of

text: tive

text: recombination.

text: The

text: lifetime

text: is

text: independent

text: of

text: the

text: en-

text: the

text: NWs

text: for

text: the

text: quantitative

text: analysis

text: of

text: surface-related

text: phe-

text: sembles

text: mean

text: surface-to-volume

text: ratio

text: and

text: coalescence

text: de-

text: nomena.

text: gree,

text: implying

text: that

text: the

text: nonradiative

text: process

text: is

text: neither

text: caused

text: We

text: are

text: now

text: in

text: the

text: position

text: to

text: quantitatively

text: compare

text: the

text: by

text: surface

text: recombination

text: nor

text: by

text: dislocations

text: formed

text: due

text: to

text: effective

text: lifetime

text: of

text: the

text: coupled

text: XA

text: system

text: NW

text: coalescence.

text: The

text: remaining

text: possibility

text: for

text: the

text: origin

text: of

text: number

text: of

text: GaN

text: NW

text: samples

text: with

text: different

text: values

text: of

text: d

text: this

text: nonradiative

text: channel

text: are

text: point

text: defects.

text: In

text: fact,

text: given

text: that

text: grown

text: in

text: several

text: different

text: MBE

text: machines

text: using

text: different

text: the

text: substrate

text: temperatures

text: used

text: in

text: the

text: MBE

text: growth

text: of

text: GaN

text: plasma

text: sources.

text: The

text: results

text: based

text: on

text: samples

text: from

text: Refs.

text: NWs

text: are

text: low

text: compared

text: to

text: those

text: used

text: in

text: both

text: metal-organic

text: and

text: are

text: displayed

text: in

text: Fig.

text: together

text: with

text: data

text: and

text: hydride

text: vapor

text: phase

text: epitaxy,

text: the

text: point

text: defect

text: density

text: in

text: reported

text: in

text: Refs.

text: and

text: The

text: effective

text: lifetimes

text: mea-

text: these

text: NWs

text: is

text: necessarily

text: higher

text: than

text: in

text: state-of-the-art

text: GaN

text: sured

text: for

text: these

text: samples

text: remain

text: short

text: compared

text: with

text: the

text: ra-

text: layers,

text: which

text: are

text: grown

text: closer

text: to

text: equilibrium.

text: However,

text: the

text: diative

text: lifetime

text: regardless

text: of

text: the

text: value

text: of

text: d

text: If

text: the

text: effec-

text: point

text: defect

text: density

text: does

text: not

text: need

text: to

text: be

text: excessively

text: high

text: in

text: tive

text: lifetime

text: τ

text: were

text: fundamentally

text: limited

text: by

text: surface

text: re-

text: order

text: to

text: dominate

text: the

text: exciton

text: decay.

text: We

text: have

text: demonstrated

text: eff

text: combination,

text: the

text: resulting

text: values

text: would

text: be

text: expected

text: to

text: lin-

text: that

text: bound

text: and

text: free

text: excitons

text: in

text: the

text: investigated

text: GaN

text: NWs

text: early

text: increase

text: with

text: d

text: The

text: solid

text: line

text: shows

text: the

text: expected

text: are

text: strongly

text: coupled

text: even

text: at

text: low

text: temperatures,

text: and

text: the

text: non-

text: dependence

text: with

text: S˜

text: cm/s

text: as

text: reported

text: by

text: Gor-

text: radiative

text: decay

text: may

text: thus

text: take

text: place

text: via

text: the

text: free

text: state.

text: Given

text: gis

text: et

text: for

text: bound

text: excitons.

text: Note

text: that

text: we

text: have

text: not

text: yet

text: that

text: the

text: diffusion

text: length

text: of

text: free

text: excitons

text: in

text: GaN

text: is

text: larger

text: observed

text: decay

text: times

text: longer

text: than

text: expected

text: from

text: this

text: value,

text: than

text: nm,

text: a

text: density

text: as

text: low

text: as

text: cm

text: may

text: sufﬁce

text: to

text: but

text: only

text: shorter

text: ones.

text: For

text: thin

text: NWs,

text: surface

text: recombination

text: introduce

text: an

text: effective

text: nonradiative

text: decay

text: channel.

text: may

text: therefore

text: be

text: the

text: limiting

text: process

text: after

text: all,

text: but

text: for

text: thicker

text: Attempts

text: to

text: reduce

text: the

text: density

text: of

text: this

text: detrimental

text: de-

text: ones,

text: it

text: is

text: clear

text: that

text: a

text: process

text: not

text: related

text: to

text: the

text: surface

text: de-

text: fects

text: would

text: probably

text: require

text: a

text: higher

text: substrate

text: temperature.

text: termines

text: the

text: exciton

text: decay

text: rate.

text: Figure

text: plots

text: the

text: effec-

text: However,

text: due

text: to

text: an

text: exponential

text: increase

text: of

text: both

text: the

text: des-

text: tive

text: lifetime

text: τ

text: from

text: different

text: GaN

text: NW

text: samples

text: versus

text: orption

text: of

text: Ga

text: adatoms

text: and

text: the

text: dissociation

text: of

text: GaN

text: in

text: the

text: eff

text: their

text: coalescence

text: degree

text: σ

text: With

text: the

text: exception

text: of

text: the

text: GaN

text: high

text: vacuum

text: environment

text: of

text: MBE,

text: it

text: seems

text: unlikely

text: that

text: C

text: NW

text: arrays

text: fabricated

text: by

text: SAG

text: (whose

text: coalescence

text: degree

text: sufﬁciently

text: high

text: substrate

text: temperatures

text: can

text: be

text: reached.

text: A

text: was

text: set

text: to

text: σC

text: since

text: multiple

text: NW

text: nucleations

text: occur

text: in-

text: perhaps

text: more

text: promising

text: approach

text: for

text: reducing

text: the

text: impact

text: of

text: evitably

text: for

text: each

text: the

text: values

text: of

text: the

text: effective

text: exciton

text: these

text: defects

text: at

text: least

text: at

text: low

text: temperatures

text: is

text: the

text: manipulation

text: lifetime

text: scatter

text: around

text: ps

text: and

text: do

text: not

text: show

text: any

text: obvious

text: of

text: the

text: strength

text: of

text: the

text: coupling

text: between

text: free

text: and

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text: states

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text: 8 in the NWs. As discussed in detail in our previous work,64 we believe that the enhanced coupling in GaN NWs is non- thermal in nature and related to the presence of electric ﬁelds ∂n ∈ A within the NWs. Since these electric ﬁelds arise from the D0 ∂ν (ξ)+εSn(ξ) = 0, ξ ∂ . (A5) pinning of the Fermi level at the NW sidewalls, a passivation of the corresponding M-plane surfaces may drastically di- minish the coupling and result in a higher internal quantum Let us expand n(x,t) in a power series over ε, n(x,t) = efﬁciency at low temperatures. n0 +εn1 +.... Substituting this expansion in Eqs. (A4) and ACKNOWLEDGMENT (A5), we obtain the ﬁrst two terms over the powers of ε as genThfeorauptrhoovrisdwinogualdddliiktieotnoalthsaanmkpPl.eDs ofogranouanrdstMud.yK, nAel.-aKn-. D0∆n0(x,t) = 0, x ∈ A, D0 ∂∂nν0 (ξ) = 0, ξ ∈ ∂A, (A6) Bluhm for the scanning electron micrographs used in Fig. 5, L. Schrottke for a critical reading of the manuscript, and and H. Riechert for continuous support. We gratefully acknowl- edge partial ﬁnancial support of this work by the Deutsche FkionrdsclyhuancgksngoewmleeidngsecshatfhtethsruopupgohrtSFoBf t9h5e1.RKus.sKia.nSaSbceilefneclde ∂n0∂(tx,t) = D0∆n1(x,t)−γn0(x,t)+Gδ(t), x ∈ A, (A7) Foundation under RScF Grant 14-11-00083. Appendix A: Surface recombination for arbitrary NW shapes Let us consider generation, diffusion, and recombination D0 ∂∂nν1 (ξ)+Sn0(ξ) = 0, ξ ∈ ∂A. (A8) of excitons described by the equation It follows from Eq. (A6) that n does not depend on x so ∂n(x,t) − 0 ∂t = D∆n(x,t) γn(x,t)+Gδ(t) (A1) that n0 = n0(t). Now we integrate Eq. (A7) over the domain area A: (cid:90) A within a domain , with the initial condition n(x,0) = 0 and the boundary condition at the border of the domain A∂n∂0t(t) = D0 ∆n1dx−Aγn0(t)+AGδ(t). (A9) A ∂n ∈ A D (ξ)+Sn(ξ) = 0, ξ ∂ . (A2) (cid:90) (cid:90) ∂ν Since ∂n theHdeirfefuns(ixo,nt)coisetfhﬁecieexncti,toγnthdeenrsaidtyia,tGivethreegceonmebraintiaotniornatrea,tDe, A ∆n1dx = ∂A ∂ν1 dξ, S the surface recombination velocity, and ν is a unit normal we arrive using the b(cid:18)oundary (cid:19)condition (A8) at the equation directed outwards of the domain. Our aim is to calculate the PL intensity (normalized to the generation rate G and the dom(cid:90)ain area A) ∂n∂0t(t) = − γ + SAP n0(t)+Gδ(t). (A10) ≡ 1 I(t) n(x,t)dx (A3) In the limit of an inﬁnitely large diffusion coefﬁcient, GA A (cid:112) n0(t) no longer explicitly depends on D and Eq. (A10) gives the desired result in the limit of large diffusivity, i.e. , we thus require that the diffusion length D/γ is large compared to the domain (cid:29) sdiizmeesnositohnaatlDity/γof thAe.syTshteeman, aalnydsiws edoreefsenrobtedloewpetnodthoen dthoe- I(t) = exp[−(γ +γS)t], (A11) main area A and its perimeter P in application to the two- dimensional problem considered in this manuscript. The re- where sults presented below are applicable to a d-dimensional vol- − ume A and its (d 1)-dimensional hypersurface P. SP D0S/iεn,cwe htehree dεififsuasisomnaclloqeufﬁacniteitnyt. TDheisn,lathrgeed,ifwfuesipounteDqu=a- γS = A , (A12) tion (A1) and the boundary condition (A2) read is the surface recombination rate. An account of the sub- sequent terms in the expansion over ε can give the PL inten- ∂n(x,t) − ε ∂t = D0∆n(x,t) εγn(x,t)+εGδ(t), (A4) sity in the case of a large but ﬁnite diffusivity. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: in

text: the

text: NWs.

text: As

text: discussed

text: in

text: detail

text: in

text: our

text: previous

text: we

text: believe

text: that

text: the

text: enhanced

text: coupling

text: in

text: GaN

text: NWs

text: is

text: non-

text: thermal

text: in

text: nature

text: and

text: related

text: to

text: the

text: presence

text: of

text: electric

text: ﬁelds

text: ∂n

label: equation

text: A

text: within

text: the

text: NWs.

text: Since

text: these

text: electric

text: ﬁelds

text: arise

text: from

text: the

text: ∂ν

text: (ξ)+εSn(ξ)

text: ξ

text: pinning

text: of

text: the

text: Fermi

text: level

text: at

text: the

text: NW

text: sidewalls,

text: a

text: passivation

text: of

text: the

text: corresponding

text: M-plane

text: surfaces

text: may

text: drastically

text: di-

text: minish

text: the

text: coupling

text: and

text: result

text: in

text: a

text: higher

text: internal

text: quantum

text: Let

text: us

text: expand

text: n(x,t)

text: in

text: a

text: power

text: series

text: over

text: ε,

text: n(x,t)

text: efﬁciency

text: at

text: low

text: temperatures.

text: Substituting

text: this

text: expansion

text: in

text: Eqs.

text: and

text: ACKNOWLEDGMENT

text: we

text: obtain

text: the

text: ﬁrst

text: two

text: terms

text: over

text: the

text: powers

text: of

text: ε

text: as

text: ofogranouanrdstMud.yK,

text: nAel.-aKn-.

text: x

text: A,

text: ∂∂nν0

text: (ξ)

text: ξ

text: ∂A,

text: Bluhm

text: for

text: the

text: scanning

text: electron

text: micrographs

text: used

text: in

text: Fig.

text: L.

text: Schrottke

text: for

text: a

text: critical

text: reading

text: of

text: the

text: manuscript,

text: and

text: and

text: H.

text: Riechert

text: for

text: continuous

text: support.

text: We

text: gratefully

text: acknowl-

text: edge

text: partial

text: ﬁnancial

text: support

text: of

text: this

text: work

text: by

text: the

text: Deutsche

text: x

text: A,

text: Foundation

text: under

text: RScF

text: Grant

text: Appendix

text: A:

text: Surface

text: recombination

text: for

text: arbitrary

text: NW

text: shapes

text: Let

text: us

text: consider

text: generation,

text: diffusion,

text: and

text: recombination

text: ∂∂nν1

text: ξ

text: ∂A.

text: of

text: excitons

text: described

text: by

text: the

text: equation

text: It

text: follows

text: from

text: Eq.

text: that

text: n

text: does

text: not

text: depend

text: on

text: x

text: so

text: ∂n(x,t)

text: ∂t

text: D∆n(x,t)

text: γn(x,t)+Gδ(t)

text: that

text: Now

text: we

text: integrate

text: Eq.

text: over

text: the

text: domain

text: area

text: A:

text: A

text: within

text: a

text: domain

text: with

text: the

text: initial

text: condition

text: and

text: the

text: boundary

text: condition

text: at

text: the

text: border

text: of

text: the

text: domain

text: A

text: ∂n

text: A

text: D

text: (ξ)+Sn(ξ)

text: ξ

text: ∂ν

text: Since

text: ∂n

text: A

text: ∂A

text: ∂ν1

text: dξ,

text: S

text: the

text: surface

text: recombination

text: velocity,

text: and

text: ν

text: is

text: a

text: unit

text: normal

text: we

text: arrive

text: using

text: the

text: at

text: the

text: equation

text: directed

text: outwards

text: of

text: the

text: domain.

text: Our

text: aim

text: is

text: to

text: calculate

text: the

text: PL

text: intensity

text: (normalized

text: to

text: the

text: generation

text: rate

text: G

text: and

text: the

text: area

text: A)

text: γ

text: SAP

text: I(t)

text: n(x,t)dx

text: In

text: the

text: limit

text: of

text: an

text: inﬁnitely

text: large

text: diffusion

text: coefﬁcient,

text: GA

text: A

text: no

text: longer

text: explicitly

text: depends

text: on

text: D

text: and

text: Eq.

text: gives

text: the

text: desired

text: result

text: in

text: the

text: limit

text: of

text: large

text: diffusivity,

text: i.e.

text: we

text: thus

text: require

text: that

text: the

text: diffusion

text: length

text: D/γ

text: is

text: large

text: compared

text: to

text: the

text: domain

text: sdiizmeesnositohnaatlDity/γof

text: thAe.syTshteeman,

text: aalnydsiws

text: edoreefsenrobtedloewpetnodthoen

text: dthoe-

text: I(t)

text: exp[−(γ

text: +γS)t],

text: main

text: area

text: A

text: and

text: its

text: perimeter

text: P

text: in

text: application

text: to

text: the

text: two-

text: dimensional

text: problem

text: considered

text: in

text: this

text: manuscript.

text: The

text: re-

text: where

text: sults

text: presented

text: below

text: are

text: applicable

text: to

text: a

text: d-dimensional

text: vol-

text: ume

text: A

text: and

text: its

text: (d

text: hypersurface

text: P.

text: SP

text: htehree

text: dεififsuasisomnaclloqeufﬁacniteitnyt.

text: TDheisn,lathrgeed,ifwfuesipounteDqu=a-

text: γS

text: A

text: tion

text: and

text: the

text: boundary

text: condition

text: read

text: is

text: the

text: surface

text: recombination

text: rate.

text: An

text: account

text: of

text: the

text: sub-

text: sequent

text: terms

text: in

text: the

text: expansion

text: over

text: ε

text: can

text: give

text: the

text: PL

text: inten-

text: ∂n(x,t)

text: ε

text: ∂t

text: εγn(x,t)+εGδ(t),

text: sity

text: in

text: the

text: case

text: of

text: a

text: large

text: but

text: ﬁnite

text: diffusivity.

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text: 11 In what follows, we consider the solution with real ’s, i.e. λ 1 | | (38) kφ? < 2. We assume a power series expansion for the unknown function ( ) f ρ X 0 f (ρ) = n c 0ρn, n, n0 ∈ 2N, (39) n,n n=0 and the eigenvalue p(cid:16)ro(cid:16)bqlem (36) provides (cid:17)th(cid:17)e followi(cid:16)ng diﬀeprence(cid:16)qequation (cid:17)(cid:17) − − − ( + 2) 1 4 2 + + 2 + Λ 1 4 2 + 2 + 2 = 0 (40) cn+2,n0 n kφ? n cn,n0 E? ? kφ? n , with the restriction p (cid:16)q (cid:17) = − Λ 1 − 4 2 + 2 0 + 2 (41) E k n . ? ? φ? The expression of the coeﬃcients can be written in terms of (the analytically continued) Euler c 0 n,n Γ-function as follows [35] (cid:16) q (cid:17) csno,nl 0 = ((Γ−(cid:0)1n2)n++11(cid:1))ΓΓ(cid:16)n21++ 1212q11−−44kkφ2φ2?? +Λ1n?(cid:17)/4 (−1)n2 (cid:0)n20 n−20 !n2 (cid:1)!. (42) and the solution of Eq. (32) thus reads p X √ 0 Ψ 0 (ρ, φ) = C eikφ(cid:126)φ e− Λ2?ρ2 ρ1+ 12−4kφ2? n csol 0ρn. (43) n ,kφ n,n n=0 where is a normalizing factor and with energy eigenvalue C (cid:16)q (cid:17) √ 3Λ = −(cid:126) 1 − 4 2 + 2 0 + 2 (44) En0,kφ 4 kφ? n . Indeed, the procedure we used to solve Eq. (34) is just the generalization of those adopted for a − three-dimensional harmonic oscillator, in which case 2 = ( + 1), being the anPgular momentum. In k l l l φ? fact, the eigenfunction (43) and the eigenvalue (44) correspond to the three-dimensional generalization 0 owfitthheLhaagrumerorneicpoolsycnilolamtoiarlseiggeennefuranlciztieodnsfoarncdoneitginenuvoaulsues. vInalupeasr.ticWuliatrh,inthtehsisummodennl=, 0tchsneolvρanluceoionfcides k k φ φ? determines if there is a physical singularity [? ] in = 0: it is absent for 1 and present for 1 ρ kφ? < 2 kφ? > 2 [36, 37]. Therefore, by assuming the condition (38), we get a nonsingular scenario for the early Universe and the big-bang singularity is removed. Semiclassical states can now be constructed as follows Z X Ψ[k˜φ,n˜] (ρ, φ, t) = A +1/2dkφ? e−(kφ2−αk˜2φ)2 ∞ e−(n2−σn˜2)2 e−iEn,(cid:126)kφt Ψn,kφ(ρ, φ). (45) −1/2 n=1 being a normalizing factor. The behavior of expectation values and distribution variances is presented A in Figs. 7 and 8 for = 0 1. σ . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: what

text: follows,

text: we

text: consider

text: the

text: solution

text: with

text: real

text: ’s,

text: i.e.

text: λ

text: kφ?

text: We

text: assume

text: a

text: power

text: series

text: expansion

text: for

text: the

text: unknown

text: function

text: f

text: ρ

text: X

text: f

text: (ρ)

text: n

text: c

text: n,

text: n,n

text: and

text: the

text: eigenvalue

text: provides

text: Λ

text: n

text: kφ?

text: n

text: E?

text: kφ?

text: n

text: with

text: the

text: restriction

text: p

text: Λ

text: E

text: k

text: n

text: φ?

text: The

text: expression

text: of

text: the

text: coeﬃcients

text: can

text: be

text: written

text: in

text: terms

text: of

text: (the

text: analytically

text: continued)

text: Euler

text: c

text: n,n

text: Γ-function

text: as

text: follows

text: q

text: csno,nl

text: and

text: the

text: solution

text: of

text: Eq.

text: thus

text: reads

text: p

text: X

text: Ψ

text: (ρ,

text: φ)

text: C

text: e−

text: n

text: csol

text: n

text: ,kφ

text: n,n

text: where

text: is

text: a

text: normalizing

text: factor

text: and

text: with

text: energy

text: eigenvalue

text: C

text: 3Λ

text: kφ?

text: n

text: Indeed,

text: the

text: procedure

text: we

text: used

text: to

text: solve

text: Eq.

text: is

text: just

text: the

text: generalization

text: of

text: those

text: adopted

text: for

text: a

text: three-dimensional

text: harmonic

text: oscillator,

text: in

text: which

text: case

text: being

text: the

text: anPgular

text: momentum.

text: In

text: k

text: l

text: l

text: l

text: φ?

text: fact,

text: the

text: eigenfunction

text: and

text: the

text: eigenvalue

text: correspond

text: to

text: the

text: three-dimensional

text: generalization

text: k

text: k

text: φ

text: φ?

text: determines

text: if

text: there

text: is

text: a

text: physical

text: singularity

text: in

text: it

text: is

text: absent

text: for

text: and

text: present

text: for

text: ρ

text: kφ?

text: kφ?

text: Therefore,

text: by

text: assuming

text: the

text: condition

text: we

text: get

text: a

text: nonsingular

text: scenario

text: for

text: the

text: early

text: Universe

text: and

text: the

text: big-bang

text: singularity

text: is

text: removed.

text: Semiclassical

text: states

text: can

text: now

text: be

text: constructed

text: as

text: follows

text: Z

text: X

text: Ψ[k˜φ,n˜]

text: (ρ,

text: φ,

text: t)

text: A

text: Ψn,kφ(ρ,

text: φ).

text: being

text: a

text: normalizing

text: factor.

text: The

text: behavior

text: of

text: expectation

text: values

text: and

text: distribution

text: variances

text: is

text: presented

text: A

text: in

text: Figs.

text: and

text: for

text: σ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: 13 to introduce a lattice in the conﬁguration space and to approximate the momenta by the action of the translation operator on the minimum lattice distance (polymer scale). In this sense, polymer quanti- zation naturally accounts for the presence of a fundamental discrete structure, as expected in quantum gravity approaches. As a consequence, the quantum dynamics is signiﬁcantly aﬀected at scales which are comparable with the polymer scale, while in the continuum limit there is a substantial overlap with the results obtained in the Schrödinger representation. Let us address polymer quantization for . The associated wave function in the frozen case can be ρ written as Ψpol (ρ, φ) = ζpol (ρ) eikφ(cid:126)φ , (46) where the superscript denotes the polymer part of the wave function. The function belongs to pol ζpol the Hilbert space of square-integrable functions over the Bohr compactiﬁcation of the real line. It can be expanded on the eigenvectors of the polymer operator , whose spectrum is deﬁned on the lattice ρ L = { = ∈ Z}, i.e. ρ nλ , n λ? ? | i | i ˆ = (47) ρ nλ nλ nλ , ? ? ? being the polymer scale. Finite translations act as λ ? | 0i = | 0 + i (48) T ρ ρ ρ , ρ and the momentum operator can be deﬁned in terms of them as follows (cid:126) − pˆpρol = 2iλ (Tλ? T−λ?), (49) ? which formally coincides with the replacement (cid:18) (cid:19) 1 ˆ pˆρ → λ sin λ?(cid:126)pρ . (50) ? The eigenvalue problem (34) b ecomes ! c b 1 + 2 ( ) = − ( ) (51) HPHO k ζpol ρ Eζpol ρ , b φ ρ2 where is the Hamiltonian operator for the harmonic oscillator in the polymer representation, HPHO which in the momentum polarization reads [38] (cid:20) (cid:18) (cid:19)(cid:21) b (cid:126)2 2 (cid:126)2 2˜ ( ) HPHOζ˜pol(pρ) = 8Mλ2 2 − 2 cos λ(cid:126)?pρ ζ˜pol(pρ) − 2 MΩ2 d ζdpopl2pρ , (52) ? ρ where and Ω denote the mass and the frequency of the harmonic oscillator and they take the M following expression in terms of the parameters of the model √ 1 3Λ = Ω = (53) M 3 , 2 . πG Let us assume that the second term on the left-hand side of (51) is a small perturbation and let us solve ﬁrst the eigenvalue problem for the polymer harmonic oscillator: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: to

text: introduce

text: a

text: lattice

text: in

text: the

text: conﬁguration

text: space

text: and

text: to

text: approximate

text: the

text: momenta

text: by

text: the

text: action

text: of

text: the

text: translation

text: operator

text: on

text: the

text: minimum

text: lattice

text: distance

text: (polymer

text: scale).

text: In

text: this

text: sense,

text: polymer

text: quanti-

text: zation

text: naturally

text: accounts

text: for

text: the

text: presence

text: of

text: a

text: fundamental

text: discrete

text: structure,

text: as

text: expected

text: in

text: quantum

text: gravity

text: approaches.

text: As

text: a

text: consequence,

text: the

text: quantum

text: dynamics

text: is

text: signiﬁcantly

text: aﬀected

text: at

text: scales

text: which

text: are

text: comparable

text: with

text: the

text: polymer

text: scale,

text: while

text: in

text: the

text: continuum

text: limit

text: there

text: is

text: a

text: substantial

text: overlap

text: with

text: the

text: results

text: obtained

text: in

text: the

text: Schrödinger

text: representation.

text: Let

text: us

text: address

text: polymer

text: quantization

text: for

text: The

text: associated

text: wave

text: function

text: in

text: the

text: frozen

text: case

text: can

text: be

text: ρ

text: written

text: as

text: Ψpol

text: (ρ,

text: φ)

text: ζpol

text: (ρ)

text: where

text: the

text: superscript

text: denotes

text: the

text: polymer

text: part

text: of

text: the

text: wave

text: function.

text: The

text: function

text: belongs

text: to

text: pol

text: ζpol

text: the

text: Hilbert

text: space

text: of

text: square-integrable

text: functions

text: over

text: the

text: Bohr

text: compactiﬁcation

text: of

text: the

text: real

text: line.

text: It

text: can

text: be

text: expanded

text: on

text: the

text: eigenvectors

text: of

text: the

text: polymer

text: operator

text: whose

text: spectrum

text: is

text: deﬁned

text: on

text: the

text: lattice

text: ρ

text: L

text: Z},

text: i.e.

text: ρ

text: nλ

text: n

text: λ?

text: i

text: i

text: ˆ

text: ρ

text: nλ

text: nλ

text: nλ

text: being

text: the

text: polymer

text: scale.

text: Finite

text: translations

text: act

text: as

text: λ

text: i

text: T

text: ρ

text: ρ

text: ρ

text: ρ

text: and

text: the

text: momentum

text: operator

text: can

text: be

text: deﬁned

text: in

text: terms

text: of

text: them

text: as

text: follows

text: pˆpρol

text: (Tλ?

text: T−λ?),

text: which

text: formally

text: coincides

text: with

text: the

text: replacement

text: ˆ

text: pˆρ

text: λ

text: sin

text: The

text: eigenvalue

text: problem

text: b

text: ecomes

text: c

text: b

text: HPHO

text: k

text: ζpol

text: ρ

text: Eζpol

text: ρ

text: b

text: φ

text: ρ2

text: where

text: is

text: the

text: Hamiltonian

text: operator

text: for

text: the

text: harmonic

text: oscillator

text: in

text: the

text: polymer

text: representation,

text: HPHO

text: which

text: in

text: the

text: momentum

text: polarization

text: reads

text: b

text: 2˜

text: HPHOζ˜pol(pρ)

text: cos

text: ζ˜pol(pρ)

text: d

text: ρ

text: where

text: and

text: Ω

text: denote

text: the

text: mass

text: and

text: the

text: frequency

text: of

text: the

text: harmonic

text: oscillator

text: and

text: they

text: take

text: the

text: M

text: following

text: expression

text: in

text: terms

text: of

text: the

text: parameters

text: of

text: the

text: model

text: 3Λ

text: Ω

text: M

text: πG

text: Let

text: us

text: assume

text: that

text: the

text: second

text: term

text: on

text: the

text: left-hand

text: side

text: of

text: is

text: a

text: small

text: perturbation

text: and

text: let

text: us

text: solve

text: ﬁrst

text: the

text: eigenvalue

text: problem

text: for

text: the

text: polymer

text: harmonic

text: oscillator:

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: 15 Therefore, the dynamical system is now endowed with a ground state, corresponding to the minimum energy eigenvalue, and one can assume the Universe to dynamically relax into such a state. Furthermore, the minimum eigenvalue is negative and the dual dust ﬁeld energy density is positive; thus it behaves as an ordinary dust ﬁeld contribution. Let us now estimate the additional term in (51). By performing the Fourier transform of Mathieu functions it is possible to describe the eigenfunctions (57) and (58) in the coordinate representation. We consider only the Mathieu sine periodic functions, since the same estimate can be repeated for the cosine, and with odd index, since nothing changes for an even one. Hence, se can be expanded as follows X 2n+1 ∞ se2n+1 (u, g) = B22mn++11 (g) sin[(2m + 1)u], , (65) m=0 where B22mn++11 (g) are Fourier coeﬃcients fXor which ∞ |B22mn++11|2 = 1. (66) m=0 The normalized energy eigenstates | i, for which h | i = ˜ , can thus be expanded in terms of |ρi = eiρpρ/(cid:126)|pρi as follows: E2n+1 pρ E2n+1 ζ2pnol+1 X 1 +∞ |E2n+1i = √2 m=−∞(−)m B|22nm++11| |(2m + 1)λ?i. (67) Inverse powers of can be regularized in polymer representation using the expression [39] ρ (cid:16)p p (cid:17) p (cid:92) sgn ( ) 1 ρ → | + | − | − | (68) | | ρ λ ρ λ , ρ λ? ? ? which provides for 1 2 /ρ c (cid:16)p p (cid:17) 1 → 1 | | − | − | 4 + (69) ρ λ ρ λ . ρ2 λ4 ? ? ? H*ence, th(cid:12)e ex(cid:12)pectati+on value of the additional term in (51) on energy eigenstates reads (cid:12)c(cid:12) X (cid:16)p p (cid:17) X (cid:16)p p (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) kφ2 E2n+1 (cid:12)ρ12 (cid:12) E2n+1 = kλφ22 ∞ |B22mn++11|2 |2m + 2| − |2m| 4 ≤ kλφ22 ∞ |2m + 2| − |2m| 4 , ? m=0 ? m=0 (70) where we used the condition (66). Let us note that (cid:0) (cid:1) √ √ 2 + 1 − − 1 4 ∀ 1 (71) n n < n > , 2 n and thus X (cid:16)p p (cid:17) X X ∞ ∞ ∞ |2 + 2| − |2 | 4 4 + 2 = 2 + 2 = 2 + π2 (72) m m < (2 + 1)2 (2 + 1)2 4 , m m m=0 m=1 m=0 such that for the expression (70) one has ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Therefore,

text: the

text: dynamical

text: system

text: is

text: now

text: endowed

text: with

text: a

text: ground

text: state,

text: corresponding

text: to

text: the

text: minimum

text: energy

text: eigenvalue,

text: and

text: one

text: can

text: assume

text: the

text: Universe

text: to

text: dynamically

text: relax

text: into

text: such

text: a

text: state.

text: Furthermore,

text: the

text: minimum

text: eigenvalue

text: is

text: negative

text: and

text: the

text: dual

text: dust

text: ﬁeld

text: energy

text: density

text: is

text: positive;

text: thus

text: it

text: behaves

text: as

text: an

text: ordinary

text: dust

text: ﬁeld

text: contribution.

text: Let

text: us

text: now

text: estimate

text: the

text: additional

text: term

text: in

text: By

text: performing

text: the

text: Fourier

text: transform

text: of

text: Mathieu

text: functions

text: it

text: is

text: possible

text: to

text: describe

text: the

text: eigenfunctions

text: and

text: in

text: the

text: coordinate

text: representation.

text: We

text: consider

text: only

text: the

text: Mathieu

text: sine

text: periodic

text: functions,

text: since

text: the

text: same

text: estimate

text: can

text: be

text: repeated

text: for

text: the

text: cosine,

text: and

text: with

text: odd

text: index,

text: since

text: nothing

text: changes

text: for

text: an

text: even

text: one.

text: Hence,

text: se

text: can

text: be

text: expanded

text: as

text: follows

text: X

text: (u,

text: g)

text: (g)

text: where

text: (g)

text: are

text: Fourier

text: coeﬃcients

text: fXor

text: which

text: The

text: normalized

text: energy

text: eigenstates

text: i,

text: for

text: which

text: h

text: i

text: can

text: thus

text: be

text: expanded

text: in

text: terms

text: of

text: |ρi

text: as

text: follows:

text: pρ

text: X

text: √2

text: m=−∞(−)m

text: Inverse

text: powers

text: of

text: can

text: be

text: regularized

text: in

text: polymer

text: representation

text: using

text: the

text: expression

text: ρ

text: p

text: p

text: sgn

text: ρ

text: ρ

text: λ

text: ρ

text: λ

text: ρ

text: λ?

text: which

text: provides

text: for

text: /ρ

text: c

text: p

text: ρ

text: λ

text: ρ

text: λ

text: ρ2

text: λ4

text: H*ence,

text: value

text: of

text: the

text: additional

text: term

text: in

text: on

text: energy

text: eigenstates

text: reads

text: X

text: p

text: X

text: p

text: where

text: we

text: used

text: the

text: condition

text: Let

text: us

text: note

text: that

text: n

text: n

text: n

text: n

text: and

text: thus

text: X

text: p

text: X

text: X

text: π2

text: m

text: m

text: m

text: m

text: such

text: that

text: for

text: the

text: expression

text: one

text: has

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##