Tightly bound indirect exciton in single-layer hybrid organic-inorganic perovskite semiconductor

Accepted Manuscript Tightly bound indirect exciton in single-layer hybrid organic-inorganic perovskite semiconductor Jing Li, Tao Liu, Timothy C.H. Liew PII:

S0749-6036(17)31991-2

DOI:

10.1016/j.spmi.2017.08.056

Reference:

YSPMI 5231

To appear in:

Superlattices and Microstructures

Received Date: 22 August 2017 Accepted Date: 24 August 2017

Please cite this article as: J. Li, T. Liu, T.C.H. Liew, Tightly bound indirect exciton in single-layer hybrid organic-inorganic perovskite semiconductor, Superlattices and Microstructures (2017), doi: 10.1016/ j.spmi.2017.08.056. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Tightly bound indirect exciton in single-layer hybrid organicACCEPTED MANUSCRIPT

inorganic perovskite semiconductor Jing Li1,2, Tao Liu1,* and Timothy C. H. Liew1 1

Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,

Nanyang Technological University, 21 Nanyang Link, Singapore 637371 2

School of Physics and Engineering, Qufu Normal University, Qufu, China 273165

Abstract

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* [email protected]

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We theoretically study the direct and indirect excitons (IXs) in a single-layer hybrid organicinorganic perovskite (HOIP) semiconductor. Due to the 2D nature, the single-layer HOIP supports the large binding energy of IXs and direct excitons over a wide range of applied electric fields, which exceed the thermal energy of room temperature. Moreover, the ground-state IX has a lower energy than that of direct exciton, which will extend the coherence and relaxation time of

devices of IXs.

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IXs. This is beneficial to optoelectronic applications and excitonic information processing

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Keywords: Single-layer perovskite, indirect exciton, binding energy

1. Introduction

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An exciton is a quasiparticle consisting of an electron bound to a hole in a semiconductor. Its bosonic nature has motivated a considerable effort and interest in exploring intriguing excitonic phenomena and devices such as possible condensation into a quantum-degenerate state [1], optoelectronic transistors [2,3] and valleytronics [4]. However, the short lifetime of excitons formed by electron and hole in the same quantum layer (i.e. direct exciton) greatly limits the application and realization of exciton optoelectronic devices in spite of its large optical response.

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An indirect exciton (IX) is a bound pair of an electron and a hole in spatially separated quantum layers, which shows unique properties: long coherence and spin-relaxation time, long propagation distance and formation of coherent quantum gases and control by voltage in-situ [5,6]. Thus IXs have the potential to develop conceptually new excitonic information processing devices and

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explore high-performance optoelectronic devices. The extraordinary electronic and optical properties of IXs in conventional semiconductor quantum-well (QW) structures have been widely

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investigated [5–9]. However, the binding energy of IXs in conventional semiconductor QWs is much smaller than the thermal energy, which prevents the achievement of room-temperature excitonic devices. Therefore, searching and studying materials supporting large binding energy of IXs with easy fabrication processes, which exceeds the thermal energy of room temperature is quite necessary [10–12].

The recently discovered atomically thin two-dimensional (2D) semiconductor materials e.g.

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transition metal dichalcogenides (TMDs) and hybrid organic-inorganic perovskite (HOIP) crystals provide us with a chance to explore tightly bound IXs. Their 2D nature and reduced dielectric screening make the many-body interaction much stronger than that of conventional

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semiconductor QWs. However, in spite of large many-body interaction for TMDs [9], the complex fabrication processes of heterobilayers supporting IXs recently makes the applications of TMDs challenging. Recently, hybrid organic-inorganic perovskites have been attracting much

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attention due to distinctive optical and electronic properties, e.g. huge optical non-linearity with ultrafast response, large and tunable spin-orbit coupling and long carrier diffusion length, from its unique crystal structures [13–21]. Most noticeably, the inorganic layers and organic barrier layers of perovskite naturally form multiple QW structures. Although direct excitons with large binding energy have been investigated in lead-iodide-based HOIPs [18,22–24], one important opening question is whether a single-layer HOIP supports tightly bound IXs and the ground state energy of IXs can be smaller than that of direct excitons. By solving the Schrodinger equation including many-body Coulomb interaction, we calculate the ground state of electron-hole interaction in single layer perovskite. A large binding energy of IX is proven in these materials, which paves the way towards studying room-temperature excitonic dynamics and devices. Most noticeably, the ground-exciton state of IX is the lowest eigenstate of the electron-hole many-body

system. This will extend the coherence and relaxation time of IX which is beneficial to its

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optoelectronic applications and excitonic information processing devices.

2. Single-particle states We first consider the single-particle states of a single-layer lead-iodide-based HOIP of (C6H13NH3)2(CH3NH3)m−1PbmI3m+1 where inorganic layers and organic barrier form multiple QW structures. The barrier and well width are 1.0 nm and 1.2 nm, respectively. We assume the z

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direction is perpendicular to the layer of HOIP. In the effective mass approximation, the singleparticle Schrodinger equation can be expressed as: H e,h ( z ) ϕe,h ( z ) = Ee,hϕe,h ( z )

h ∂ 1 ∂ + Ve,h ± eFz 2 ∂z me,h ∂z

(2)

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H e,h ( z ) = −

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Where the electron and hole Hamiltonians are

(1)

Here me and mh are the electron and hole effective masses. Ve = 0.9 eV and Vh = 2.9 eV are conduction band and valance confinement potentials, and F is an applied external electric field amplitude.

By applying the shooting method [25], we can numerically solve the above single-particle Schrodinger equation, where the wavefunctions of single-layer perovskites at different electric

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field strengths are shown in Fig. 1. As the strength of the electric field increases, the energy-level splitting between the ground state and the first-excited states of the electron and hole bands rises (see Fig.2 (a) and (b)). Moreover, the energy difference between the electron band e1 and hole band h1 can be much larger than that between the electron band e2 and hole band h1 (see

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Fig.2(c)). This large energy-level splitting at high electric field may induce the ground-exciton state of IXs formed primarily by bands e1 and h1, lower than that of the direct exciton formed

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primarily by bands e2 and h1 (while the obtained exciton states are always mixtures of IXs and direct excitons, we refer to IX and direct exciton states as those mostly composed of electrons and holes in different and the same quantum well, respectively). In this case, the ground-exciton state of IXs as the first-excited state of electron-hole many-body eigenstates can induce steady condensation of IXs and extend the optoelectronic applications.

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Fig. 1 The wavefunction of electron and hole ground and first-excited states of single-layer lead-iodide-based perovskites at different electric fields. The gray lines present the potentials of HOIPs multiple QW structure. The

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band gap is 1.7 eV.

Fig. 2 The energy difference of first-excited and ground states of (a) valence band and (b) conduction band at different electric fields. (c) The energy difference of conduction band and ground valence band at different electric fields.

3. Excitonic states

By including the electron and hole Coulomb interactions, we can calculate the excitonic states.

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The Hamiltonian of exciton states can be expressed as:

H ( ze , zh ) = He ( ze ) + H h ( zh ) + H X ( ze , zh , ρ ) + Eg

(3)

where H X ( ze , zh , ρ ) represents the electron-hole relative motion and Coulomb potential which has the form as h2  ∂ 2 ∂   2+ − ρ∂ρ  ε 2µ  ∂ρ

e2

( ze − z h )

b

(4) 2

+ρ

2

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H X ( ze , zh , ρ ) = −

here µ is the exciton reduced mass, ρ is electron-hole relative coordinate and εb is the background dielectric constant. In order to solve the Schrodinger equation of the exciton i.e.

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H ( ze , zh )ψ = EXψ , we expand the exciton wavefunction into a superposition of the electron and

hole wavefunction products as:

∑

i = e1,e 2 j = h1

aijφij ( ze , zh , ρ )

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ψ=

(5)

where we only consider the ground state of the valence band and the ground-state and firstexcited state of conduction band, which will not cause a problem due to the same energy splitting between first-excited and ground states of both valance and conduction bands. The higher excited

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states are neglected due to large energy separation from fist-excited state. φij has the form, in analogy to the 1s state of a hydrogen atom

2 1

π λij

e

− ρ λij

(6)

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φij ( ze , zh , ρ ) = ϕei ( ze ) ϕhj ( zh )

Here λij are variational parameters. The coefficient aij represents the probability amplitude of

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finding an electron with band index i and a hole with band index j. The unknown parameters λij , aij and the exciton eigenvalue E X satisfy the following equation

ϕ p K ( ρ ) ϕn an + ∑ ϕ pVnm ( ρ ) ϕm am = ( EX − En0 ) ϕ pϕn an 2

(7)

m =1

Where the indices n, p and m represent the electron-hole pair i.e. n, p, m = ( i, j ) where i = e1, e2 and j = h1 , and En0 = Eei + Ehj + Eg . K ( ρ ) , Vnm ( ρ ) and ϕ n have the forms K (ρ) = −

h2  ∂ 2 ∂   2+  2µ  ∂ρ ρ∂ρ 

Vnm ( ρ ) = − ∫∫ d ze dz h

(8)

e 2 Φ n ( ze , z h ) Φ m ( ze , z h )

εb

( ze − z h )

2

+ ρ2

(9)

2 1

ϕACCEPTED e− ρ λ MANUSCRIPT n = π λn

(10)

n

here Φ n ( ze , zh ) = Φ ij ( ze , zh ) = ϕei ( ze ) ϕhj ( zh ) . According to Eq. (1.7), we have the following matrix equation

∑ M (E nm

X

, λn ) am = 0

(11)

m

det M nm ( E X , λn ) = 0

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The above equations have nonzero solutions only when the matrix M nm satisfies (12)

Thus the eigenvalues of the exciton energy E X and variational parameters λn can be determined by

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solving Eq. (1.12). The minimum values of E X corresponding to IXs with primarily n = ( e1, h1) and direct exciton with n = ( e2, h1) , representing two eigenvalues of Hamiltonian Eq. (1.3).

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Figure 3 shows the ground-state exciton energy and corresponding binding energy of the IX and direct exciton at different electric fields. Due to the 2D nature, the single-layer HOIP supports large binding energy of IXs and direct exciton over a wide range of the applied electric fields which exceeds the thermal energy of room temperature. In addition, as shown in Fig. 3(a), as the electric field increases, the binding energy of IX decreases due to the reduced wavefunction overlap at higher electric fields. In contrast, as the electric field rises, the binding

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energy of direct exciton increases (see Fig. 3(b)). This is because the electron and hole bands are well localized in QWs, and thus the probability of finding an electron and hole in the same QW raises with applied electric field due to the quantum confined Stark effect. Most noticeably, the

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ground-state IX has a lower energy than that of the direct exciton (see Fig. 3(c)). Thus the ground-exciton state of IXs is the first-excited state of electron-hole many-body eigenstates. This will extend the coherence and relaxation times of IXs, which is beneficial to its optoelectronic

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applications and excitonic information processing devices.

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Fig. 3 The ground-state exciton energy and binding energy of (a) an IX and (b) direct exciton at different electric fields. (c) The ground-state exciton energy of the IX and direct exciton vs. electric fields are shown in the same diagram.

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4. Summary

The ground-state eigenenergy, binding energy and their dependences on electric fields of the

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direct exciton and IX have been investigated for single-layer HOIP semiconductor showing a multiple QW structure. The single-particle Schrodinger equation shows that the energy difference between an electron band and a hole band for different QWs can be much smaller than that for the same QW at high electric fields. By solving the electron-hole many-body system, we find that the single-layer HOIP supports large binding energy of IXs and direct exciton over a wide range of the applied electric field strength and the binding energies exceed the thermal energy of room temperature. Most noticeably, the ground-exciton state of IXs is the first-excited state of electron-hole many-body eigenstates. In principle, this will extend the coherence and relaxation time of IXs, which is beneficial to their optoelectronic applications and excitonic information processing devices.

Acknowledgement

This work was financially supported by the Ministry of Education Academic Research Fund Tier

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1 grant 2016-T1-1-084. JL thanks the supports from National Science Foundation of China (No. 11604179) and Shangdong Natural Science Foundation (No. ZR2016AQ18).

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Physics of Semiconductor Nanostructures (John Wiley & Sons, 2011).

ACCEPTED MANUSCRIPT 1. Direct and indirect excitons in a single-layer perovskite are investigated.

2. Single-layer perovskite supports the large binding energies of indirect

exceed the thermal energy of room temperature.

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and direct excitons over a wide range of applied electric fields which

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3. The ground-exciton state of IXs is the first-excited state of electron-hole

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many-body eigenstates which is beneficial to their optoelectronic

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applications and excitonic information processing devices.