Low-Frequency Vibrations of Semiconductor Nanoplatelets - The

Subscriber access provided by CLARKSON UNIV

C: Physical Processes in Nanomaterials and Nanostructures

Low-Frequency Vibrations of Semiconductor Nanoplatelets Serguei V. Goupalov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00831 • Publication Date (Web): 12 Apr 2019 Downloaded from http://pubs.acs.org on April 12, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Low-Frequency Vibrations of Semiconductor Nanoplatelets Serguei V. Goupalov∗,†,‡ † Department of Physics, Jackson State University, Jackson, Mississippi 39217, USA ‡ Ioffe Institute, St. Petersburg 194021, Russia E-mail: [email protected] Abstract Colloidal semiconductor nanoplatelets have recently emerged as exciting materials for optoelectronic and biological applications. Their vibrational excitations, or phonon modes, are responsible for energy relaxation of charge carriers, excitons, trions, and other electronic excitations and can participate in their spin relaxation. In the present study, theory of elastically isotropic thin plates is generalized to account for crystalline anisotropy of nanoplatelets. Dispersion relations for acoustic and flexural vibrations of CdSe nanoplatelets with a zinc-blende crystal lattice grown along the [001] crystallographic axis are found. The rest of the low-frequency modes are approached from the model of isotropic elastic medium confined between two planes and assuming the free-standing boundary conditions at the planes. This provides a systematic way to classify low-frequency vibrational modes of a nanoplatelet.

Introduction In recent years great success has been achieved in the synthesis of quasi-2D nanostructures by methods of colloidal chemistry. 1–4 The resulting nanoplatelets are free-standing single-crystal 1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

structures having atomically controlled thickness. Their optical properties 5–10 and electronic structure 5,11,12 are being intensely investigated due to their remarkable potential for optoelectronic and biological applications. 4 As these studies progress, it becomes necessary to understand the vibrational properties of nanoplatelets. Indeed, phonons are responsible for energy relaxation of charge carriers and other electronic excitations and can participate in their spin relaxation. For instance, Shornikova et al. 9 have recently proposed a phenomenological model, where transitions between the bright and dark exciton states, separated by an energy gap of several meV, are assisted by emission or absorption of a phonon. This phonon should have energy matching the gap and carry angular momentum projection for the process to be allowed. A systematic study of nanoplatelet vibrations has begun only recently. Girard et al. 13 performed low-frequency Raman measurements on CdSe and CdS nanoplatelets and reported observation of the fundamental thickness breathing mode. Lebedev 14 calculated phonon spectra of CdSe nanoplatelets from first principles within the density-functional theory and modeled their Raman and infrared absorption spectra. Methods of the theory of elasticity usually very well describe vibrational properties of nanostructures. A traditional theory of thin elastic plates 15 accounts for acoustic and flexural modes of elastically isotropic nanoplatelets and expresses their energy spectra in terms of bulk elastic moduli. However, ab initio calculations 14 suggest that sound velocities for acoustic phonons of CdSe nanoplatelets, having a zinc-blende crystal lattice and grown along the [001] axis, possess considerable anisotropy. In addition, the acoustic and flexural modes do not exhaust all low-frequency modes inherited from the acoustic vibrations of a bulk semiconductor, in particular the fundamental thickness breathing mode observed in the Raman spectra. 13 In the present paper we seek to compensate for these deficiencies. In the first part of the paper, we propose a modification of the theory of thin elastic plates, which aims to account for the elastic anisotropy of semiconductor nanoplatelets. We obtain anisotropic

2


Page 2 of 21

Page 3 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dispersion relations for acoustic and flexural phonon modes of the [001]-grown nanoplatelets with a zinc-blende crystal lattice in terms of bulk elastic moduli. Although CdSe with zincblende crystal structure does not exist in bulk, the corresponding elastic moduli have been calculated using various ab initio approaches. 14,16,17 In the second part of the paper, we model a nanoplatelet using a slab of isotropic homogeneous elastic medium confined between two planes and assume the free-standing boundary conditions on these planes. This simple model provides a natural way to classify the lowfrequency vibrational modes of a nanoplatelet. While an elastically anisotropic nanoplatelet has different spectra of vibrations for various directions of plane wave propagation, the vibrational spectrum of an elastically isotropic nanoplatelet is degenerate with respect to the direction of propagation at all non-zero wave vectors and represents an average over all directions. This degeneracy is equivalent to a degeneracy over projections of angular momentum for cylindrical waves. The solution in the form of a cylindrical wave is particularly convenient for problems involving phonon-assisted spin-flip transitions. 9 Assuming a solution in this form allows one to apply a well developed mathematical apparatus of the quantum theory of angular momentum providing a straightforward and concise derivation of the equations of motion. The model of a slab of isotropic elastic medium enables one to account for all low-frequency solutions, including the thickness breathing mode, while the acoustic and flexural modes following from the isotropic limit of the theory considered in the first part will serve as special cases. We then show that the dispersion equations obtained for cylindrical waves coincide with the dispersion equations obtained for plane waves in the classical Rayleigh-Lamb theory. 18–21

3



Page 4 of 21

Theory Theory of thin elastic plates for [001]-grown nanoplatelets A bulk semiconductor with a zinc-blende crystal lattice has the Td point symmetry. The Lagrangian density depending on the displacement field u(r) and invariant under symmetry operations of the point group Td takes the form 15

L=

λ1 2 ρ 2 u˙ x + u˙ 2y + u˙ 2z − uxx + u2yy + u2zz − λ2 (uxx uyy + uxx uzz + uyy uzz ) 2 2

(1)

−2 λ3 u2xy + u2xz + u2yz , where ρ is the mass density, uαβ

1 = 2

∂uα ∂uβ + ∂xβ ∂xα

is the strain tensor, and λ1 ≡ λxxxx , λ2 ≡ λxxyy , λ3 ≡ λxyxy are the bulk elastic moduli. 15 Here we assume that the plane of the nanoplatelet is perpendicular to the axis z and that the axes x, y, and z are chosen along [100], [010], and [001], respectively. The key approximation of the theory of thin elastic plates 15 is the assumption that the components of the stress tensor satisfy

σxz = σyz = σzz = 0

(2)

everywhere inside the plate. In our case these conditions yield

2 σxz = −

∂L = 4 λ3 uxz = 0 , ∂uxz

(3)

2 σyz = −

∂L = 4 λ3 uyz = 0 , ∂uyz

(4)

4


Page 5 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and σzz = −

∂L = λ1 uzz + λ2 (uxx + uyy ) = 0 . ∂uzz

(5)

From Eqs. (3), (4) we get uxz = uyz = 0 while from Eq. (5) we obtain

uzz = −

λ2 (uxx + uyy ) . λ1

Substituting these expressions into Eq. (1) we obtain ρ 2 L= u˙ x + u˙ 2y + u˙ 2z − 2

λ1 λ2 − 2 2 2λ1

u2xx

+

u2yy

λ2 − λ2 − 2 λ1

uxx uyy − 2 λ3 u2xy .

(6)

Note that a nanoplatelet grown along the [001] axis has the D2d point symmetry. 22 One can check that all the terms in Eq. (6) are invariant under symmetry operations of the point group D2d (cf. Ref. 15 ). In the isotropic limit λ1 → ρ c2l , λ2 → ρ (c2l − 2 c2t ), λ3 → ρ c2t , where s cl =

E(1 − σ) , ρ (1 + σ) (1 − 2 σ) s

ct =

E 2 ρ (1 + σ)

are the bulk longitudinal and transverse sound velocities, 15 E is Young’s modulus and σ is the Poisson ratio. In the isotropic limit the Lagrangian density (6) becomes ρ L= 2

u˙ 2x

+

u˙ 2y

+

u˙ 2z

−

3c2t

4c4 − 2t cl

2

(uxx + uyy ) −

c2t

2 2 (uxx − uyy ) + 4uxy .

(7)

The coefficient 3c2t −

4c4t E = = vl2 − c2t , 2 cl 2ρ(1 − σ)

5


(8)


Page 6 of 21

where s vl =

E ρ (1 − σ 2 )

is the longitudinal in-plane sound velocity for a thin elastic plate. 15 It is convenient to rewrite Eq. (8) as vl2

=

4c2t

c2t 1− 2 . cl

(9)

The transverse in-plane sound velocity for a thin elastic plate coincides with its bulk counterpart. 15 The Lagrangian density (7) describes the in-plane motion of a thin elastically isotropic plate. 23 Now we will complement Eq. (6) to account for the out-of-plane motion of the nanoplatelet. We will decompose the displacement of the nanoplatelet into the displacement u(0) of the plane crossing the nanoplatelet at half of its thickness and the displacement u(1) relative to this plane:

u(x, y, z) = u(0) (x, y) + u(1) (x, y, z) .

(10)

We further assume that u(1) has only in-plane components. Since in the z-direction the nanoplatelet consists of no more than several monolayers, we will consider u(1) as a small correction. Then the conditions uxz = uyz = 0 allow one to write (1)

(0)

(1)

∂ux ∂uz =− , ∂z ∂x

(0)

∂uy ∂uz =− ∂z ∂y

or (0)

(0)

u(1) x = −z

∂uz , ∂x

u(1) y = −z

∂uz . ∂y

(11)

Substituting Eqs. (10), (11) into Eq. (6) and neglecting u˙ (1) we will get an expression depending only on u(0) . Let us average this expression over the thickness of the nanoplatelet,

6


Page 7 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


h: 1 L¯ = h

Zh/2 L dz .

(12)

−h/2

The resulting Lagrangian density can be obtained from Eq. (6) by the substitution

uαβ uηζ

1 → huαβ uηζ i = h

Zh/2

(0)

uαβ uηζ dz =

(0) uαβ

(0)

∂ 2 uz h2 ∂ 2 uz . + 12 ∂xα ∂xβ ∂xη ∂xζ

(0) uηζ

(13)

−h/2

Omitting the superscripts we obtain ρ 2 u˙ x + u˙ 2y + u˙ 2z − L¯ = 2 λ2 − λ2 − 2 λ1

λ2 λ1 − 2 2 2λ1

"

h2 u2xx + u2yy + 12

∂ 2 uz ∂x2

2

h2 + 12

∂ 2 uz ∂y 2

2 # (14)

" 2 2 # 2 h2 ∂ 2 uz ∂ 2 uz ∂ uz h 2 uxx uyy + . − 2 λ u + 3 xy 12 ∂x2 ∂y 2 12 ∂x∂y

The Lagrangian equations ∂ L¯ ∂ L¯ d ∂ L¯ d ∂ L¯ d2 − − + ∂uη dt ∂ u˙ η dxα ∂ ∂uη dxα dxβ ∂ ∂ 2 uη ∂xα

=0

(15)

∂xα ∂xβ

yield 2 λ22 ∂ uy ∂ 2 ux ∂ 2 ux + λ − + λ + λ , 2 3 3 ∂x2 λ1 ∂x∂y ∂y 2

(16)

2 λ22 ∂ 2 uy λ22 ∂ ux ∂ 2 uy ρ u¨y = λ1 − + λ − + λ + λ , 2 3 3 λ1 ∂y 2 λ1 ∂x∂y ∂x2

(17)

2 λ22 h2 ∂ 4 uz ∂ 4 uz λ22 h ∂ 4 uz ρ u¨z = − λ1 − + − λ − + 2 λ . 2 3 λ1 12 ∂x4 ∂y 4 λ1 6 ∂x2 ∂y 2

(18)

ρ u¨x =

λ2 λ1 − 2 λ1

7



Page 8 of 21

Table 1: Bulk elastic moduli for zinc-blende CdSe and in-plane sound velocities in a [001]grown CdSe nanoplatelet. λ1 , GPa 86.2 88.1 72.6

λ2 , GPa 30.3 53.6 53.1

[100]

λ3 , GPa 36.8 27.4 24.9

16 17 14

[100]

, 105 2.55 2.20 2.10

cm s

k 4 − 4 λ1 + λ2 −

2λ22 λ1

, 105 3.65 3.13 2.44

Ref. vl

cm s

vt

[110]

vl

, 105 3.87 3.41 2.94

cm s

[110]

vt

, 105 2.22 1.74 1.70

Assuming u ∝ ei(kx x+ky y−ωt) we obtain

2 ω1,2 =

λ1 −

λ22 λ1

r + λ3

2ρ

λ1 −

k2 ±

λ22 λ1

− λ3

2

(λ1 − λ2 − 2 λ3 ) kx2 ky2

2ρ (19)

from the first two equations and

ω32

h2 = 12 ρ

λ2 λ1 − 2 λ1

(kx4

+

ky4 )

λ2 + 2 λ2 − 2 + 2 λ3 λ1

kx2

ky2

(20)

from the third one. Thus, along with the anisotropy in the dispersion of acoustic phonons, Eq. (19), we have also obtained an anisotropy in the dispersion of flexural phonons, Eq. (20). In the isotropic limit we get

2 ω1,2

vl2 + c2t 2 vl2 − c2t 2 = k ± k , 2 2

(21)

h2 vl2 4 k 12

(22)

ω32 = in agreement with. 15

In Table 1 are given the bulk elastic moduli of zinc-blende CdSe calculated in Refs. 14,16,17 and the longitudinal, vl , and transverse, vt , in-plane sound velocities for acoustic waves propagating along [100] and [110] directions in a nanoplatelet grown along the [001] axis, calculated using Eq. (19). We used mass density of zinc-blende CdSe from Ref.: 13 ρ = 5.655 g/cm3 . In Table 2 are given the parameters γ of the dispersion relation ω = γk 2 for 8


cm s

Page 9 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2: Parameters γ of the dispersion relation ω = γk 2 for flexural vibrational modes propagating along high-symmetry directions in the plane of a [001]-grown CdSe nanoplatelet of thickness h. structure h, h, MLs nm 3 0.9 4 1.2 5 1.5

Parameters of Ref. 16 Parameters of Ref. 17 γ [100] , γ [110] , γ [100] , γ [110] , 10−2 cm2 /s 10−2 cm2 /s 10−2 cm2 /s 10−2 cm2 /s 0.95 1.00 0.81 0.88 1.27 1.34 1.09 1.18 1.58 1.67 1.36 1.47

Parameters of Ref. 14 γ [100] , γ [110] , 10−2 cm2 /s 10−2 cm2 /s 0.63 0.76 0.85 1.02 1.06 1.27

flexural waves propagating along [100] and [110] directions in 3, 4, and 5 monolayers thick nanoplatelets grown along the [001] axis, calculated using Eq. (20).

Vibrations of an isotropic slab Equations of motion describing the dispacement field in an isotropic elastic medium can be cast in the form 15,24 2 2 ˆ 2 2 cl I ∆ + (ct − cl ) Jˆ ∇ u = −ω 2 u .

(23)

Here u is the vector of displacement understood as a three-component column, Iˆ is the unit matrix, ∆ is the scalar Laplace operator, Jˆα (α = x, y, z) are the matrices of the angular momentum J = 1, and ω is the vibration frequency. The solutions of these equations describe displacements in the longitudinal and transverse sound waves propagating in an infinite isotropic elastic medium. We are interested in the solutions having cylindrical symmetry and regular on the z axis. They can be easily constructed if we use the cyclic vector components rather than the cartesian ones and notice that they should satisfy one of the conditions curl u = 0 (for longitudinal sound) or div u = 0 (for transverse sound). The longitudinal solution is 24

 ei (qz z+M ϕ) (l) uM,qz (r, ϕ, z) = p k 2 + qz2

9

 e JM −1 (kr)  −i qz JM (kr)    √k eiϕ JM +1 (kr) 2

√k  2

   

−iϕ


(24)


Page 10 of 21

while the two transverse solutions can be chosen as follows 24  ei (Qz z+M ϕ) (v1) uM,Qz (r, ϕ, z) = p k 2 + Q2z

Qz √  2

   

JM −1 (kr)  i k JM (kr)  ,  Qz iϕ √ e JM +1 (kr) 2

 i (Qz z+M ϕ)

(v2)

uM,Qz (r, ϕ, z) =

e

√

2

e



−iϕ

(25)



−iϕ

 e JM −1 (kr)     . 0     −eiϕ JM +1 (kr)

(26)

Here M is the projection of the total angular momentum onto the z axis, JM (x) is the Bessel function of order M , and the wave numbers qz and Qz satisfy the dispersion relations ω 2 = c2l (qz2 + k 2 ) = c2t (Q2z + k 2 ), or qz2 =

ω2 − k2 , c2l

(27)

Q2z =

ω2 − k2 . c2t

(28)

The dispersion relations for vibrations of a slab sandwiched between the planes z = ±h/2 are obtained by imposing upon the linear combination (l)

(l)

(v1)

(v1)

(v2)

(v2)

a+ uM,qz + a− uM,−qz + b+ uM,Qz + b− uM,−Qz + c+ uM,Qz + c− uM,−Qz

(29)

of these solutions the boundary condition requiring that the traction forces vanish on the planes. In cylindrical coordinates, the components of the traction force corresponding to the terms with the “+” index in Eq. (29) are as follows " FM,+,r =

i ρ c2t eiM ϕ

2qz dJM (kr) iqz z k 2 − Q2z dJM (kr) iQz z p p −a+ e + b+ e dr dr k 2 + qz2 k k 2 + Q2z

10


Page 11 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


JM (kr) iQz z , −c+ M Qz e kr

(30)

"

2 M qz JM (kr) iqz z M (Q2 − k 2 ) JM (kr) iQz z FM,+,ϕ = ρ c2t eiM ϕ a+ p e + b+ p z e r r k 2 + qz2 k k 2 + Q2z Qz dJM (kr) iQz z +c+ e , k dr

" iM ϕ

FM,+,z = ρ e

JM (kr) a+

c2l

p 2 c2 k 2 k 2 + qz2 − p t k 2 + qz2

(31)

! e

iqz z

# 2 c2t Qz k iQz z − b+ p e . (32) k 2 + Q2z

The components of FM,− can be obtained by changing qz to −qz , Qz to −Qz , and the index “+” to “−”. The boundary condition of FM,+ + FM,− vanishing at z = ±h/2 is imposed. (v2)

One can see that this boundary condition leaves the solutions uM,±Qz , describing the shear waves, decoupled from all other solutions. The dispersion equation for the shear waves reads sin Qz h = 0 .

(33)

The solutions include the acoustic shear wave Qz = 0, or ω = ct k corresponding to the lower sign in Eq. (21) and solutions Qz = πn/h with n being a natural number corresponding to r ω = ct

π 2 n2 + k2 . h2

(34)

The rest of the equations following from the boundary conditions lead to the dispersion equation for the mixed (longitudinal and transverse) waves: c6t 2 2 2 2 4 2 2 2 2 2 4 (ω − 2 ct k ) + 16 2 (ω − cl k ) (ω − ct k ) k sin qz h sin Qz h cl

11



+16 c4t

2

2

k qz Qz (ω −

2 c2t

2 2

k )

qz h 2 Qz h 2 Qz h 2 qz h sin cos + sin cos = 0. 2 2 2 2 2

Page 12 of 21

(35)

Here it is understood that, when qz2 of Eq. (27) becomes negative, qz should be replaced by i κz , sin qz h by i sinh κz h, etc. The case of Q2z < 0 should be addressed similarly. Substituting ω = v k into Eq. (35) and expanding the trigonometric functions up to linear terms one obtains 2

v =

4c2t

c2t 1 − 2 ≡ vl2 cl

in agreement with Eqs. (9), (21). Similarly, substituting ω = γ k 2 into Eq. (35) and expanding the trigonometric functions up to cubic terms one obtains

2

γ =

c2t

c2t 1− 2 cl

h2 h2 vl2 ≡ 3 12

in agreement with Eq. (22). At k = 0, Eq. (35) becomes sin

ωh ωh sin =0 cl ct

(36)

which yields ω = cl πn/h, ω = ct πn/h. The first of these equations yields the frequency of the fundamental thickness breathing mode at n = 1. The fact that Eq. (36) factorizes means that, at k = 0, the modes are no longer mixed and have either purely longitudinal (dilatational, or “quasi-Lamb” in terms of Ref. 14 ) or purely transverse character. As the Bessel functions JM (0) = 0 for all integer M except for M = 0, it follows from Eqs. (24) – (26) that, at k = 0, all vibrational modes are either purely longitudinal (dilatational) and have M = 0, or purely transverse and have |M | = 1. Moreover, in this limit the solutions (25) and (26) are equivalent and represent shear waves. At k 6= 0, all vibrational modes are degenerate with respect to M . The dispersion relations following from Eq. (35) are shown in Fig. 1 for a three monolayers-thick CdSe nanoplatelet.

We used the averaged sound velocities of cl =

12


Page 13 of 21

22

20

18

16

(meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


14

12

10

8

6

4

2

0

0.0

0.5

1.0

ka/

Figure 1: Dispersion relations for the low-frequency mixed modes of an isotropic three monolayers-thick CdSe nanoplatelet (red solid lines). Shown as black dashed lines are the √ 2 dependencies ω = h vl k / 12, ω = vl k, ω = ct π/h, and ω = cl π/h. 3.7 · 105 cm/s, ct = 1.54 · 105 cm/s employed in the isotropic models tailored to wurtzite CdSe 25 (yielding vl = 2.67 · 105 cm/s) and the lattice constant of a = 6.0175 ˚ A. 14 Shown √ in Fig. 1 by dashed lines are the dependencies ω = h vl k 2 / 12, ω = vl k, ω = ct π/h, and ω = cl π/h. One can see that the longitudinal acoustic mode with linear dispersion at low k and the flexural mode with quadratic dispersion at low k are rapidly departing from the patterns predicted by the theory of thin elastic plates, as k increases. At k = 0, the modes having frequencies as multiples of ct π/h (for transverse modes) and cl π/h (for dilatational modes) can be clearly distinguished in Fig. 1. The dispersion relations for the low-frequency shear modes of an isotropic three monolayers-thick CdSe nanoplatelet, following from Eq. (34), are shown in Fig. 2 for the same parameters that were used for Fig. 1. The dispersion relations for the mixed modes are also shown in Fig. 2 by red dashed lines for comparison.

13



22

20

18

16

14

(meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 21

12

10

8

6

4

2

0

0.0

0.5

1.0

ka/

Figure 2: Dispersion relations for the low-frequency shear modes of an isotropic three monolayers-thick CdSe nanoplatelet (blue solid lines). Shown as red dashed lines are the dispersion relations for the low-frequency mixed modes.

Relation to the classical Rayleigh-Lamb theory Let us show that Eq. (35) is equivalent to the classical Rayleigh-Lamb equations. 18–21 Dividing Eq. (35) by cos2

qz h Qz h sin2 2 2

and using Eqs. (27), (28), we obtain

(k 2 − Q2z )4 + 16 qz2 Q2z k 4 x + 4 k 2 qz Qz (k 2 − Q2z )2 1 + x2 = 0 ,

(37)

where x=

tan Q2z h tan qz2h

.

Resolving Eq. (37) over x we obtain the two following solutions: tan Q2z h tan qz2h

=−

4 k 2 qz Qz (k 2 − Q2z )2

14


(38)

Page 15 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and tan Q2z h tan qz2h

=−

(k 2 − Q2z )2 . 4 k 2 qz Qz

(39)

Eqs. (38) and (39) are known as the Rayleigh-Lamb frequency equations for the propagation of symmetric and antisymmetric plane waves in a plate, respectively. 21 We note, however, that Eq. (35) is more convenient for numerical solutions.

Conclusions We have considered two analytical approaches to vibrations of nanoplatelets. The first one represents an approximate model based on the theory of thin elastic plates. It describes only the acoustic and flexural modes but can account for the crystalline anisotropy. This anisotropy reflects the point symmetry of a nanoplatelet and manifests itself in the dispersion relations for the acoustic and flexural modes. These dispersion relations are, respectively, linear and quadratic for the acoustic and flexural plane waves and depend on the direction of propagation. The second approach represents an isotropic model whose solution leads to an analytic frequency equation and is, in this sense, exact. It accounts for low-frequency vibrations not limited by the modes with quadratic and linear dispersion. This model is valid for longwavelength vibrations and helps one to gain an insight into the plethora of vibrational modes resulting from more realistic calculations. The role of the isotropic model is analogous to the role of Lamb’s model of an isotropic homogeneous elastic sphere 26 widely used to describe low-frequency vibrational modes of spherical semiconductor nanocrystals and metal nanoparticles. 27 It was shown 27 that, using a numerical procedure, it is possible to trace the evolution of the vibrational modes of an elastic sphere made of an anisotropic crystalline material to the modes of an elastically isotropic sphere by gradually reducing a parameter responsible for the anisotropy. The anisotropy leads to shifting, splitting, and partial admixture of different modes, but the 15



overall structure of the vibrational spectrum remains similar to that of an elastically isotropic sphere. Basing on the isotropic model, the low-frequency vibrational modes of nanoplatelets can be divided into shear waves and mixed waves. The shear waves consist of the transverse acoustic mode with linear dispersion characterized by the bulk transverse sound velocity, ct , and modes whose frequencies at k = 0 are multiples of ct π/h, where h is the thickness of the nanoplatelet. The mixed modes consist of the flexural mode with quadratic dispersion at low k, the longitudinal acoustic mode with linear dispersion at low k, characterized by p the sound velocity vl = 2 ct 1 − c2t /c2l , where cl is the bulk longitudinal sound velocity, and modes whose frequencies at k = 0 are multiples of ct π/h and cl π/h. At k = 0, otherwise mixed modes are either purely longitudinal (dilatational) and can be characterized by the projection of angular momentum M = 0 or purely transverse (shear) and have |M | = 1. The latter modes can participate in the phonon-assisted transitions between the exciton bright and dark states. 9 Interaction of low-frequency vibrational modes of nanoplatelets with charge carriers can be determined using expressions for electron-phonon interaction potentials in crystals with Td symmetry. 28 Our results will help obtain reliable estimates for the rates of relaxation processes in nanoplatelets and better understand the phonon spectra obtained by ab initio calculations. These calculations usually involve a systematic error due to the finite character of the meshes in the space of wave vectors used in the calculations. 14 This error may become critical for vibrations at very low frequencies, where our approach based on the theory of thin elastic plates works very well.

16


Page 16 of 21

Page 17 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Acknowledgement The author would like to thank A.I. Lebedev and V.F. Sapega for useful discussions. This work was supported by the National Science Foundation (NSF-CREST Grant HRD1547754).

References 1. Ithurria, S.; Dubertret, B. Quasi 2D Colloidal CdSe Platelets with Thicknesses Controlled at the Atomic Level. J. Am. Chem. Soc. 2008, 130, 16504 – 16505. 2. Li, Zh.; Peng, X. Size/Shape-Controlled Synthesis of Colloidal CdSe Quantum Disks: Ligand and Temperature Effects. J. Am. Chem. Soc. 2011, 133, 6578 – 6586. 3. Ithurria, S.; Bousquet, G.; Dubertret, B. Continuous Transition from 3D to 1D Confinement Observed during the Formation of CdSe Nanoplatelets. J. Am. Chem. Soc. 2011, 133, 3070 – 3077. 4. Nasilowski, M.; Mahler, B.; Lhuillier, E.; Ithurria, S.; Dubertret, B. Two-Dimensional Colloidal Nanocrystals. Chem. Rev. 2016, 116, 10934 – 10982. 5. Ithurria, S.; Tessier, M.D.; Mahler, B.; Lobo, R.P.S.M.; Dubertret, B.; Efros, Al.L. Colloidal nanoplatelets with two-dimensional electronic structure. Nat. Mater. 2011, 10, 936 – 941. 6. Yeltik, A.; Delikanli, S.; Olutas, M.; Kelestemur, Y.; Guzelturk, B.; Demir, H.V. Experimental Determination of the Absorption Cross-Section and Molar Extinction Coefficient of Colloidal CdSe Nanoplatelets. J. Phys. Chem. C 2015, 119, 26768 –26775. 7. Achtstein, A.W.; Antanovich, A.; Prudnikau, A.; Scott, R.; Woggon, U.; Artemyev, M. Linear Absorption in CdSe Nanoplates: Thickness and Lateral Size Dependency of the Intrinsic Absorption. J. Phys. Chem. C 2015, 119, 20156 – 20161. 17



8. Cassette, E.; Pensack, R.D.; Mahler, B.; Scholes, G.D. Room-temperature exciton coherence and dephasing in two-dimensional nanostructures. Nat. Commun. 2015, 6, 6086. 9. Shornikova, E.V.; Biadala, L.; Yakovlev, D.R.; Sapega, V.F.; Kusrayev, Y.G.; Mitioglu, A.A.; Ballottin, M.V.; Christianen, P.C.M.; Belykh, V.V.; Kochiev, M.V. et al. Addressing the exciton fine structure in colloidal nanocrystals: the case of CdSe nanoplatelets. Nanoscale 2018, 10, 646 – 656. 10. Hu, Z.; Singh, A.; Goupalov, S.V.; Hollingsworth, J.; Htoon, H. Influence of morphology on the blinking mechanisms and the excitonic fine structure of single colloidal nanoplatelets. Nanoscale 2018, 10, 22861 – 22870. 11. Benchamekh, R.; Gippius, N.A.; Even, J.; Nestoklon, M.O.; Jancu, J.-M.; Ithurria, S.; Dubertret, B.; Efros, Al.L.; Voisin, P. Tight-binding calculations of image-charge effects in colloidal nanoscale platelets of CdSe. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 035307. 12. Rajadell, F.; Climente, J.I.; Planelles, J. Excitons in core-only, core-shell and corecrown CdSe nanoplatelets: Interplay between in-plane electron-hole correlation, spatial confinement, and dielectric confinement. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 96, 035307. 13. Girard, A.; Saviot, L.; Pedetti, S.; Tessier, M.D.; Margueritat, J.; Gehan, H.; Mahler, B.; Dubertret, B.; Mermet, A. The mass load effect on the resonant acoustic frequencies of colloidal semiconductor nanoplatelets. Nanoscale 2016, 8, 13251 – 13256. 14. Lebedev, A.I. Lattice dynamics of quasi-two-dimensional CdSe nanoplatelets and their Raman and infrared spectra. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 96, 184306. 15. Landau, L.D.; Lifshitz, E.M. Theory of Elasticity; Pergamon: New York 1970.

18


Page 18 of 21

Page 19 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


16. Rabani, E. An interatomic pair potential for cadmium selenide. J. Chem. Phys. 2002, 116, 258 – 262. 17. Deligoz, E.; Colakoglu, K.; Ciftci, Y. Elastic, electronic, and lattice dynamical properties of CdS, CdSe, and CdTe. Physica B 2006, 373, 124 – 130. 18. Rayleigh, J.W.S. On the Free Vibrations of an Infinite Plate of Homogeneous Isotropic Elastic Matter. Proc. Lond. Math. Soc. 1888, s1-20, 225 – 237. 19. Lamb, H. On the Flexure of an Elastic Plate. Proc. Lond. Math. Soc. 1889, s1-21, 70 – 91. 20. Lamb, H. On Waves in an Elastic Plate. Proc. Roy. Soc. Lond. A 1917, 93, 114 – 128. 21. Graff, K.F. Wave Motion in Elastic Solids; Dover: New York, 1991. 22. Ivchenko, E.L. Optical Spectroscopy of Semiconductor Nanostructures; Alpha Science: Harrow, U.K., 2005. 23. Goupalov, S.V. Continuum model for long-wavelength phonons in two-dimensional graphite and carbon nanotubes. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 71, 085420. 24. Goupalov, S.V. Crystal Structure Anisotropy Explains Anomalous Elastic Properties of Nanorods. Nano Lett. 2014, 14, 1590 – 1595. 25. Saviot, L.; Champagnon, B.; Duval, E.; Kudriavtsev, I.A.; Ekimov, A.I. Size dependence of acoustic and optical vibrational modes of CdSe nanocrystals in glasses. J. Non-Cryst. Sol. 1996, 197, 238 – 246. 26. Lamb, H. On the Vibrations of an Elastic Sphere. Proc. Lond. Math. Soc. 1882, s1-13, 189 – 212.

19



27. Saviot, L.; Mermet, A.; Duval, E. In Handbook of Nanophysics. Nanoparticles and Quantum Dots; Sattler, K. D. Ed.; CRC Press: Boca Raton, FL, 2011; pp 11-1 – 11-16. 28. Pikus, G.E.; Titkov, A.N. In Optical Orientation; Meier, F.; Zakharchenya, B. P. Eds.; North-Holland: Amsterdam, 1984; pp 73 – 131.

20


Page 20 of 21

Page 21 of 21

Graphical TOC Entry "Exact" isotropic model

Vibrational modes

20

of 3 ML CdSe NPL

flexural 15

(meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


acoustic shear mixed (Lamb)

10

5

also subject of approximate anisotropic model

0 0.0

0.5

1.0

ka/

21


Low-Frequency Vibrations of Semiconductor Nanoplatelets - The

Recommend Documents