## Abstract

Intrinsic radiative lifetime is an essential physical property of low-dimensional excitons that represents their optical transition rate and wavefunction, which directly measures the probability of finding an electron and a hole at the same position in an exciton. However, the conventional method that is used to determine this property via measuring the temperature-dependent photoluminescence (PL) decay time involves uncertainty due to various extrinsic contributions at high temperatures. Here, we propose an alternative method to derive the intrinsic radiative lifetime via temperature-independent measurement of the absorption cross section and transformation using Einstein's A-B-coefficient equations derived for low-dimensional excitons. We experimentally verified our approach for one-dimensional (1D) excitons in high-quality 14 × 6 nm^{2} quantum wires by comparing it to the conventional approach. Both independent evaluations showed good agreement with each other and with theoretical predictions. This approach opens a promising path to studying low-dimensional exciton physics.

## Introduction

Low-dimensional free excitons have optical transition rates that are concentrated within states having a center-of-mass wave vector *K* ≤ *k*_{0} = *nω*_{ex}/*c*, where is the exciton energy, *n* is the refractive index, and *c* is the speed of light^{1,2,3,4,5,6,7,8,9,10,11}. Their radiative lifetime reflects the extent of center-of-mass motion and the tightly bound internal motion (wavefunction). Due to differences in the confinement dimensions, the radiative lifetime of 1D excitons becomes shorter than that of 0D excitons in molecules or dots and longer than that of 2D excitons. The 1D state is unique and essential in low-dimensional exciton physics because it is the lowest dimension that permits the motion of excitons.

The absolute values of the inverse radiative lifetime *τ _{K}*

^{−1}of 1D excitons are scaled and characterized by the inverse intrinsic radiative lifetime

*τ*

_{1D}

^{−1}or the averaged

*τ*

_{K}^{−1}over all

*K*between 0 and

*k*, which is given by

_{0}^{2,3,4,5}as schematically shown in Fig. 1a. Here,

*φ*(0) is the exciton internal wavefunction

*φ*(z) at an electron-hole relative distance

*z*= 0;

*μ*is the amplitude of the interband-transition dipole moment vector (

*μ*,

_{x}*μ*,

_{y}*μ*); and

_{z}*ε*

_{0}is the vacuum dielectric constant. The key equations, derivations, and related quantities of

*τ*

_{1D}

^{−1}shown below are described in detail in the supplementary information. The intrinsic radiative lifetime

*τ*

_{1D}has been studied intensively in various 1D systems

^{2,3,4,5,6,7,8}. In (In)GaAs, the value of

*τ*

_{1D}was theoretically calculated to be 150 ps

^{2}in a 10-nm diameter wire;

*τ*

_{1D}values in 2D wells and 0D dots have been reported to be shorter (by approximately 10 ps)

^{1,12}and longer (by approximately 1 ns)

^{13}, respectively. For carbon nano-tubes, an

*ab initio*calculation predicted a short

*τ*

_{1D}of 8–19 ps

^{6}, which is comparable to many experimental results

^{7,8,9,10}.

However, *τ*_{1D} has been conventionally evaluated^{3,4,5,6,7,8,9,10} using the temperature-dependent PL decay time *τ*(*T*) of thermalized 1D excitons, which is expressed as where *k _{B}* is the Boltzmann constant,

*T*is the temperature, and

*M*is the exciton effective mass. We note here that the PL lifetime at elevated temperatures may be sensitive to extrinsic contributions, such as non-radiative decay, carrier migrations to/from adjacent regions, population migration to dark levels and/or higher levels, and band non-parabolicity. All of these factors induce uncertainty in the evaluation of

*τ*

_{1D}via PL lifetime. Hence, a different approach without relying on temperature dependence has been demanded.

For alternative evaluation without measuring temperature dependence, we propose a new approach to derive the intrinsic lifetime *τ*_{1D} via the absorption cross section. Although the absorption probability of 1D excitons is extremely small due to the small volume of a quantum wire and is very difficult to measure, various techniques have so far been developed, such as external- and internal-light transmission measurements for waveguide-structured wires^{14,15,16}, combination of arbitrary-scale sensitive measurements and some absorption standards^{17}, and direct transmission measurements for nanotube/nanowire solutions with controlled and characterized size and density distributions^{18,19}. If the integrated absorption cross section of 1D excitons per unit length is measured by some means, it is then related to the inverse intrinsic lifetime *τ*_{1D}^{−1} of a 1D exciton as follows: where the subscript *i* (*i* = *x*, *y*, or *z*) denotes the direction of incident light polarization. This equation is equivalent to the Einstein's *A–B*-coefficient relationship for 1D excitons. Indeed, assuming the random orientation of the wire with respect to the light field and using , we obtain the angle-averaged expression or, written in a more intuitively acceptable form, (*τ*_{1D}^{−1} = *A*, ). Here, the factor of 2, , and are derived from spin degeneracy, the 2D photon density of states, and the photon energy at the exciton resonance, respectively. Note that the relationship for 1D excitons is different from the original, familiar Einstein *A–B* relationship for atomic dipole systems with a 3D photon density of states due to the low dimensionality (Fig. 1a).

Because such an approach has never been tested nor compared with the conventional approach, it is important to test both using an identical sample. To this end, we employed 100 identical non-doped 14 × 6 nm^{2} GaAs T-shaped quantum wires (T-wires)^{20,21} fabricated by cleaved-edged overgrowth and growth-interrupt annealing using molecular beam epitaxy. Cross-sectional transmission electron microscopy images (Fig. 1c–e) show that all T-shaped intersections of the 14-nm-thick stem wells on the (001) plane and the 6-nm-thick quantum well (arm well) on the (110) plane uniformly form 100 identical 14 × 6 nm^{2} T-wires with equal periods of 56 nm.

To evaluate the absorption cross section of 1D excitons or the absolute absorption probability at the 1D-exciton absorption peak with small uncertainty or errors, we use a reference or a standard of the 2D-continuum absorption plateau in a GaAs quantum well. Similarly to the quantum Hall resistance standard^{22} of a 2D electron gas providing the universal constants of or the fine-structure constant , the optical absorption probabilities of some 2D materials have recently been shown to be equal to universal constants^{17,23,24,25}. Indeed, the optical absorption probability of a single sheet of graphene is shown equal to *πα _{f}* ( = 2.3%)

^{23,24}and that of the 2D-continuum absorption plateau in a GaAs quantum well is to

^{17,26}, where

*ζ*is a correction factor (close to 1) that is only weakly dependent on structure and material

^{17,25}.

## Results

Figure 2 shows the micro-PL and micro-PL excitation (PLE) spectra of the samples at 5 K. The strong PL peak of the T-wires was detected at 1.581 eV (Fig. 2a). The Stokes shift (0.7 meV) of the ground state excitons in the T-wires is smaller than the PLE peak width of 2.4 meV, demonstrating the high quality of the T-wires (Fig. 2b). Even though the excitation light at 1.696 eV initially creates excitons in the arm well and stem wells, the PL from the T-wires is dominant in the PL spectrum. This result indicates that most (greater than 90%) of the carriers created in the arm well flow rapidly to adjacent T-wires (Fig. 1b), which supports our assumption that the PLE spectrum in the spectral region of the T-wires and the arm well is proportional to their absorption probability.

In the PLE spectrum, the sharp peaks at 1.603 and 1.636 eV are assigned to 2D exciton absorption in the arm and stem wells, respectively. The peak at 1.608 eV is due to the absorption of monolayer (ML) islands in the arm well. A flat plateau of the 2D continuum absorption of the arm well is clearly found in the region of 1.610–1.630 eV. The other resonances as the stem wells and the quantum wires are far from there. Thus, we may safely attribute the absorption strength at 1.620 eV solely to the arm-well continuum absorption. We utilized the 2D-continuum absorption as a standard of absorption strength equal to . After evaluating the correction factor *ζ* for the present case^{17}, we obtained a value of 0.7% for the absolute absorption probability of the plateau, thus calibrating the PLE spectrum to the absorption spectrum as indicated in the vertical scale of Fig. 2b. Hence, the peak absorption rate of the 1D exciton peak at 1.582 eV is read as 1.2%. Because the T-wires are arrayed precisely every 56 nm, the peak absorption cross section per length *σ*/*L* becomes 0.67 nm, resulting in the energy-integrated absorption cross section per unit length = 2.43 meV nm. Finally, by using equation (2) and including the estimated uncertainty of ±20%, we obtain *τ*_{1D} = 110 ± 20 ps for the 1D excitons in the T-wires (see supplementary information).

We separately measured the temperature-dependent PL decay time of 1D excitons in the same sample and characterized the intrinsic radiative lifetime via the conventional method^{3,4,5}. Figure 3 is a plot of the measured values of the integrated PL intensities and exciton PL lifetimes *τ*(*T*) at various temperatures *T*. Whereas the PL lifetime becomes longer with increasing temperature below 60 K, it decreases rapidly at temperatures greater than 60 K. The integrated PL intensity also decreases significantly at temperatures greater than 60 K, indicating that the decreased lifetime is most likely due to enhanced non-radiative decay or carrier escape from the T-wires. Thus, we compared the observed decay times at temperatures less than 60 K with the theoretically predicted -temperature dependence for the radiative lifetime of thermalized 1D excitons and obtained the best fit to *τ*(*T*) = 0.11 ns. Using the measured value of *τ*(*T*) = 0.11 ns and the estimated exciton mass of *M* = 0.2 m_{0} (Δ = 0.133 meV) for the motion along the T-wires near *K* = 0, we obtained *τ*_{1D} = 150 ps. Due to possible complexities at high temperatures, this approach gives a large uncertainty of 0.16–2, or *τ*_{1D} = 25–300 ps (see supplementary information). Despite the high uncertainty, the value *τ*_{1D} = 150 ps (25–300 ps), which was estimated via the temperature-dependent PL decay time, shows good agreement with the value *τ*_{1D} = 110 ± 20 ps obtained via the absorption cross section. The reason for the low uncertainty in the new method is that it relies mostly on the basic physics of Einstein's relationship and the 2D absorption standard.

## Discussion

The intrinsic radiative lifetime *τ*_{1D} = 110 ps provides a 1D-exciton-wavefunction value of |*φ*(0)| ^{−2} = 20 nm via equation (1). Note that |*φ*(0)|^{−2} represents the 1D exciton size and indicates the probability of finding an electron and a hole at the same position in a 1D exciton. The obtained results, together with the exciton binding energy E_{B } = 14 meV measured for the present T-wires, can be compared with model calculations. Indeed, the value |*φ*(0)|^{−2} = 15–16 nm (*τ*_{1D} = 80 ps) calculated based on simple 1D models^{27,28,29,30} is close to the result obtained in this study. It should also be noted that the obtained 1D exciton size of 20 nm along the wire is not very different from the 2D exciton diameters of 10–20 nm in GaAs wells. This establishes that the longer (by one order of magnitude) intrinsic radiative lifetime of 1D excitons compared with that of 2D excitons cannot be attributed to the internal motion of 1D excitons but alternatively to their center-of-mass motion with lateral confinement, as mentioned in the introduction.

The evaluation of the absorption cross section was performed with such an approximation that radiative coupling effects caused by the dipole-dipole interactions between the arrayed T-wires are ignorable small. This approximation is justified because of a high background dielectric constant of GaAs of around 3.6, a large bulk exciton Bohr radius of about 15 nm, and a small bulk longitudinal-transverse splitting of about 0.1 meV. Indeed, a theoretical calculation^{31} has shown that the effect of dipole-dipole coupling for the case of GaAs QWs separated by 20 nm is very tiny and not measurable in absorption spectra or in time-resolved PL. Our present T-wires are separated by 42 nm and the wire volume is far smaller than QWs. Therefore, the effect of dipole-dipole coupling should be even smaller.

Our proposed method of 1D-exciton intrinsic radiative-lifetime measurement via absorbance and the relevant universal relationships between the Einstein's A–B coefficients of 1D exciton are applicable to nano-tubes^{6,7,8,9,10}, nano-wires^{32,33,34} and nano-columns/needles^{35,36} comprised of various materials, and are useful in establishing intriguing 1D exciton physics.

## Methods

The sample was fabricated by cleaved-edged over-growth and interrupted annealing using molecular beam epitaxy. The fabrication method is very similar to that used to produce other T-wire structures reported previously and is illustrated in detail in the supplementary information. The sample was grown as follows: During the first MBE growth step on the GaAs (001) substrate, a GaAs buffer layer, a 1-μm Al_{0.5}Ga_{0.5}As bottom carrier-blocking layer, 100 periods of Al_{0.08}Ga_{0.92}As/Al_{0.34}Ga_{0.66}As stem multiple quantum wells with well and barrier thicknesses of 14.15 and 41.88 nm, respectively, a 1-μm Al_{0.5}Ga_{0.5}As top carrier-blocking layer, and a 3-μm GaAs cap layer were grown at a growth temperature of 620°C. After in situ cleavage of the sample, a second MBE growth step was performed at a substrate temperature of 490°C on the fresh (110) surface exposed by the cleavage. In the second growth step, a 6-nm arm quantum well, a 10-nm Al_{0.5}Ga_{0.5}As barrier layer, and 175-nm Al_{0.12}Ga_{0.88}As and 960-nm Al_{0.5}Ga_{0.5}As cladding layers were grown. To improve the interface quality of the arm quantum well, 10-minute growth-interrupt annealing was performed at an elevated substrate temperature of 650°C after the growth of the arm well.

For optical measurements, the samples were set in a custom-built cryostat cooled by flowing liquid helium. A heater was set in the cryostat such that the sample temperature could be varied from 4 K to room temperature. The quality and uniformity of the wires were verified by conducting a micro-PL scan, and the best sample was chosen for study. The micro-PL and micro-PLE scans were performed using a continuous-wave titanium-sapphire (TiS) laser (the wavelength was continuously tuned from 700 to 850 nm in ~0.1 nm steps) and a spectrometer equipped with a liquid-nitrogen-cooled silicon charge-coupled device (CCD). The excitation light from the TiS laser was incident on the central T-wire region with a spot size of approximately 1 μm perpendicularly to the arm well and a polarization parallel to the T-wires (z direction), and PL polarized perpendicularly to the T-wires was detected in the backward scattering geometry. The PLE spectrum was obtained by measuring the integrated intensity of the entire T-wire PL while scanning the excitation photon energy. The temperature-dependent time-resolved PL of 1D excitons was measured in the T-wires between 5 and 100 K via the time-correlated single-photon counting method using a mode-locked TiS laser with a pulse duration of 2 ps and a repetition rate of 80 MHz. All measurements were performed under low-density excitation conditions to prevent the formation of biexcitons or electron-hole plasmas.

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## Acknowledgements

This work was partially supported by KAKENHI (nos. 20104004 and 23104715) for the DYCE research area from the MEXT, Japan. The work at Princeton was partially funded by the Gordon and Betty Moore Foundation and the National Science Foundation MRSEC program through the Princeton Center for Complex Materials (DMR-0819860). A. I. is grateful for the support of the Program to Disseminate Tenure Tracking System of the MEXT, Japan.

## Author information

## Affiliations

### Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

- Shaoqiang Chen
- , Masahiro Yoshita
- , Toshimitsu Mochizuki
- , Shun Maruyama
- & Hidefumi Akiyama

### Interdisciplinary Graduate School of Medical Science and Engineering, University of Yamanashi, Kofu, Yamanashi 400-8511, Japan

- Akira Ishikawa

### Department of Organic and Polymeric Materials, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8500, Japan

- Yuhei Hayamizu

### School of Engineering and Applied Physics, Princeton University, Princeton, New Jersey 08544, USA

- Loren N. Pfeiffer
- & Ken W. West

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### Contributions

H.A., M.Y. and S.C. conceived the project. L.N.P. and K.W.W. fabricated the sample. S.C., M.Y. and S.M. performed the temperature-dependent time-resolved PL measurements. S.C., S.M. and T.M. performed the micro-PL measurements and extracted the PLE spectra. S.C., M.Y. and H.A. calibrated the absorption rate in the PLE spectra. A.I., M.Y., H.A. and S.C. performed the theoretical derivation. S.C., M.Y., H.A. and H.Y. analyzed the results and wrote the paper.

### Competing interests

The authors declare no competing financial interests.

## Corresponding author

Correspondence to Shaoqiang Chen.

## Supplementary information

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### Supplementary Information

Intrinsic radiative lifetime derived via absorption cross section of one-dimensional excitons

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