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Russian Physics Journal, Vol. 57, No. 4, August, 2014 (Russian Original No. 4, April, 2014)

INFLUENCE OF EFFECTS OF SELF-POLARIZATION AND

EXCITON-PHONON INTERACTIONS ON THE EXCITON ENERGY

IN LEAD IODIDE NANOFILMS

V. M. Kramar and O. V. Pugantseva UDC 538.915; 538.975

In the approximation of effective masses for electronic and phononic – dielectric continuum – systems, the influence of spatial bounding, self-polarization, and exciton-phonon interactions on the exciton state in a flat double nanoheterostructure (a nanofilm) – lead iodide in a polymer matrix –is theoretically investigated for the model of a single infinitely deep quantum well. It is demonstrated that the dominating factor determining the energy of the bottom of the ground exciton band and its binding energy is spatial bounding. The relationship between two other effects depends on the nanofilm thickness, namely, the influence of the self-polarization effect in ultrathin films significantly exceeds that of exciton-phonon interaction.

Keywords: exciton, binding energy, lead iodide, nanofilm, quantum well.

INTRODUCTION

Lead iodide and low dimensional nanostructures on its basis are perspective materials for the development of temperature sensors, photodetectors, biomedical sensors, and x-ray and γ-radiation detectors that can operate in a wide temperature interval (from –200 to +130°C) [1]. Large values of the charge numbers of Pb and I atoms, gap widths, and binding exciton energy in these structures cause exciton effects in their optical absorption, transmission, and luminescence spectra [2].

It is well known that spatial bounding (SB) of quasiparticles in low-dimensional structures cause additional quantization of their state characteristics. The nature, character and degree of changes of the quasiparticle states are determined by the characteristic sizes of the structure. This is manifested through the experimentally observed dependences of the optical properties of the structure, including the exciton region of the spectrum, on the structure dimensions [3]. Theoretical investigations of the exciton states in semiconductor nanoheterostructures with single quantum well (QW) have been carried out by a number of authors using different methods in various approximations (for example, see [4–7]); however, the quantum wells in layered semiconductors were not considered.

In this work, results of theoretical investigation into the influence of spatial bounding, self-polarization (SP), and exciton-phonon interaction (EPI) on the energy of the ground exciton state in a flat lead iodine nanofilm (NF) embedded in an organic dielectric medium – ethylene-methacrylic acid (E-MAA) copolymer – are presented. Such low- dimensional ultrathin crystal structures were synthesized and investigated in [8, 9]. In particular, in [8] the positions of exciton peaks in low-temperature (T = 2 K) absorption spectra of PbI2 platelet microcrystallites embedded in E-MAA were experimentally determined. The dependence of the positions of exciton peaks on the platelet microcrystallite thickness was established, namely, with increasing platelet thickness, the maximum of the exciton band in the spectrum was displaced toward longer wavelengths.

Assuming that the platelet thickness is much less than its lateral diameter, it can be considered as a quasi–two-dimensional structure. In the case of its modeling by a flat 2D heterostructure of type I – a nanofilm – analytical formulas have been obtained that allow the energy of the ground exciton state and its binding energy to be calculated.

Yuriy Fedkovych Chernivtsi National University, Chernivtsi, Ukraine, e-mail: v.kramar@chnu.edu.ua. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 110–116, April, 2014. Original article submitted February 15, 2013.

1064-8887/14/5704-0545 2014 Springer Science+Business Media New York

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The exciton energy calculated for this model of the E-MAA/PbI2/E-MAA nanoheterostructure is characterized by a nonlinearly decreasing dependence on the nanofilm thickness. For any arbitrary thickness, spatial bounding gives the main contribution to the exciton energy. The influence of the SP and EPI effects is much weaker and depends on the nanofilm thickness. The contribution of the SP effect to the exciton energy in ultrathin nanofilms exceeds significantly the contribution of its change due to EPI. With increasing nanofilm thickness, the influence of the SP effect decreases, and of the EPI effect increases.

1. MODEL AND METHODS OF THEORETICAL INVESTIGATION

Let us consider a nanofilm with thickness a comprising layered semiconductor crystal embedded into an organic dielectric medium. The layered crystals have an atomically smooth surface and small number of broken bonds [10], which allows us to consider the heterojunction to be unstrained and the quantum well to be rectangular and infinitely deep in the presence of the dielectric medium.

The Hamiltonian of the exciton-phonon system of the nanofilm is represented in the form of the sum

ex ph intˆ ˆ ˆ ˆH H H H

of the one-particle Hamiltonians of the free exciton ( exH ) and phonon ( phH ), respectively, and of the exciton-phonon

interaction operator ( intH ). We now place the origin of the Cartesian system of coordinates in the nanofilm center and

orient this system of coordinates so that the OZ axis coincided with the C crystallographic axis of the layered crystal, and go on to the center-of-mass system for the electron and hole coupled by the Coulomb interaction. Then the Hamiltonian of the free exciton in the nanofilm assumes the form

ex e-hˆ ˆ ( , )j h

j e,hH H V z ,ze

ρ . (1)

Here

2 2 22

sp|| 2ˆ ( ) ( )

22

jj j j

jj j

H V z V zmm z

(2)

is the one-particle Hamiltonian of the electron (j = e) or hole (j = h) in the corresponding quantum well assigned by the confining potential ( )jV z ,

2 | |

sp (0)|| 0

( )| ( 1) |2

n

j nj jn

eV z

z z na

(3)

are the potentials of their interaction with their electrostatic images in the heterojunction plane (SP potentials) [11], and

2 2 | |

e-h (0)(0) (0) (0) (0) (0)2 2 2 20|| ||

( , , )[ ( ) / ] [ ( 1) ] /

n

e hnne h e h

e eV z z

z z z z na

ρ (4)

is the potential of the Coulomb interaction of the electron and hole (the second term describes the interaction of electrostatic images of each quasiparticle in the heterojunction planes [4]), m||

j (mj) is the effective mass of the

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quasiparticle characteristic for motion along the C axis (in the plane of the layered package) of the layered crystal, ρ = (xe – xh, ye – yh), and

(0) (1) (0) (1)( ) / ( ) ,

where 0 (1) is the superscript that denotes the well (barrier) material. In the center-of-mass system for the exciton, we write Eq. (1) in the form

2 | |

ex sp0 (0) (0) (0)2 20 [||

ˆ ˆ ˆ[ ( )] ( , , ) ( 1) ] /

n||

j SS ej nj e,h n ze h

eH H V z H U z zh

z na

ρ , (5)

where

22

||0 || 2

ˆ ( )2

jjj j

H V zm z

(6)

is the energy operator of almost free motion of the carrier along the OZ axis in the quantum well,

2 2 *2

2 (0)

1ˆ2SS

d d m eH

d d

(7)

is the modified (through the introduction of the effective charge e* = βe, where β is the variational parameter [7]) Shinada–Sugano Hamiltonian [12] describing relative motion of the electron and hole that form a 2D exciton in the XOY plane, 1/μ = 1/m

e + 1/mh, and

*2 2

(0) (0) (0) (0)2 2||

( , )[ ( ) / ]

e h

e h

e eU z ,z

z z

ρ (8)

is the correction considering the difference between the exciton in the nanofilm and the 2D exciton. Considering potential (8) to be a weak perturbation, we take advantage of the fact that consideration of the SP

potential Vsp can lead to an energy change of the almost free carrier without significant transformation of its stationary state functions [13]. Then in the first approximation of perturbation theory, the eigenfunction ψ of operator (5) can be represented as a product of the electron and hole wave functions in the infinitely deep QW:

cos( / ), 1,3,5, ,2

( )sin( / ), 2, 4,6, ,

jn j

j

n z a nz

n z a na

(9)

by the function ( )nm ρ of the 2D exciton described by Hamiltonian (7). The explicit forms of its eigenfunctions and

eigenvalues were presented in [12]. We note that they include the variational parameter β determining the effective charge e*. Its value is determined from the condition of a minimum of the exciton energy in the nanofilm:

ex exˆmin [ | | / | ]E H . (10)

The Hamiltonians of free phonons and electron-phonon interactions in a 2D flat heterojunction were presented

in [14]; their application allowed us to derive intH in the representation of secondary quantization over all variables:

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λint ex-LO ex-I λ

, , ,

ˆ ˆ ˆ ˆˆ ˆ ( )[j n n n nj j j jn n λj e,h j j

H H H F a a B

k q k qk q

q

,

ˆˆ ˆ( ) ]n n n nj j j jF a a B

k q k qq

q

and to find an explicit dependence of the electron-phonon function n nj jF

(q) on the transverse component of the

phonon wave vector q = (q||, q) for the model of rectangular quantum well of finite depth [7]. Here the assumption has

been accepted that exciton is coupled with phonons through the interactions with them of the electron and hole, and the subscript α specifies the phonon branch type. For the branch of the longitudinal optical (LO) phonons with the longitudinal component of the wave vector in the nanofilm of thickness a taking values λπ/a (here λ = 1, 2, …, N and N is the number of layers of the well material), symbol λ is used for the subscript α. For interface (I) phonons, the subscript σ (σ = S for symmetric and σ = A for antisymmetric modes) is used for α. They differ by the forms of the dispersive dependence of the phonon energy (Ω) on q

[14]. The subscript η

j distinguishes the contribution to the EPI

energy of the electron (ηe = 1) and hole (ηh = –1); a and a denote quantum operators of the electron state, and B

denotes the quantum operator of the phonon state. At low temperatures (formally, at T = 0 K) only virtual phonons can exist in the system; therefore, the EPI can

be manifested exclusively through processes associated with phonon emission that cause the long-wavelength shift of the energy of stationary states. The shift of the quasiparticle energy level in crystal systems with arbitrary force of coupling with phonons at T = 0 K can be found by the methods of the Green’s functions. In particular, the energy ħω of an electron from the nth miniband in the quantum well is determined by the position of the pole of the Fourier transform of the Green’s function:

1( , ) [ ( ) ( , )]n n nG E M k k k ,

where En(k) is its value disregarding the interaction with phonons and Mn(k, ω) is the mass operator that determines

the shift of the bottom of the nth electron (hole) miniband in the quantum well:

( 0, )n n nM E k .

Here En is a solution of the Schrödinger equation with Hamiltonian ||sp0

ˆ ( )H V z defined on functions (9). Then the

energy of the exciton, formed by the electron and hole being in the ground states in the quantum well (ne = 1 and nh = 1), renormalized due to the interaction with phonons, is given by the expression

exE E , (11)

where

( ) ( )ex

e hg b

E E E E E , (12)

Δ = Δ(e)+Δ(h),

Eg is the gap width of the nanofilm material, Eb is the binding energy of the exciton, and E(j) and Δ(j) are, respectively, the energy of the ground states of the carrier in the quantum well with allowance for the SP effect and its shift due to interaction with phonons.

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2. RESULTS AND DISCUSSION

2. 1. Influence of the Self-Polarization Effect

The energy of the ground (n = 1) state of the carrier in its quantum well is

2

( )sp|| 28

j

j

hE E

m a , (13)

where

2 2 2 2 22 2

2 1sp (0)

0 0 0||

cos cos

( 1 / 2) ( 1 / 2) 2 2

nn

n

e x dx x dxE

x n x n na

(14)

is the correction to the SP defined on functions (9). Using the wave function of the ground state (n = 0 and m = 0) of the 2D exciton [12]

2

exex

4 2 2( , , ) exp cos cose h

e hz z

z za a aaa

ρ , (15)

the analytical expression was derived for calculation of the exciton binding energy:

(0)0||4

ex 0 0 0 0(0)ex 10

644 1 ( , ) ( , )n

b nn

aE R I a I a

a

. (16)

Here aex and Rex are the radius and the Rydberg constant of the exciton in the nanofilm of thickness a,

ex2 22

22 2 22

0 2

ex

ln1 4

( , ) cos sin2 4 4

n

a x

x x aI a nx dx

x x ax

a

(17)

is the function considering at n = 0 the correction caused by perturbation U(ρ, ze, zh) and at n = 1, 2, … the influence of the effect of the nanofilm surface polarization by the exciton, and β0 is the variational parameter minimizing the Eex value.

Results of our calculations performed for the parameters of the 2H–PbI2 crystal [2] and E-MAA polymer [15] demonstrated that the consideration of the polarization potential increased the binding energy of the ground exciton state in the nanofilm (Fig. 1). The value of the correction increases with decreasing nanofilm thickness, exceeding 50 meV for ultrathin (N ≤ 2) films. In this case, the exciton energy also increases (Fig. 2), which is explained by the prevailing increase of the energy of carriers in the quantum well due to spatial bounding and self-polarization effect.

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2. 2. Influence of the Interaction with Phonons

To estimate the shift of the ground electron (hole) state due to the exciton-phonon interaction, the single-phonon approach and the explicit form of the electron-phonon coupling function derived in [7] were used transformed to the case of an infinitely deep quantum well. In this case, the mass operator of the electron-phonon system in the nanofilm is reduced to the form

(LO) (S) (LO) ( )1 11 11 1 1

2( ) ( ) ( ) ( ) ( )][ n n

nM M M M M

, (18)

where the terms

2

2 2(LO) 01 2 (0) 2 2 2 2 2

22LO LO0 LO

LO

14

( ) ln

1 12 2 1

nn

n

a

aeM

Ea

Em a m a

0 2 4 6 8 10 12 14 16

50

100

150

200

250

300 , meV

, nm

2

1

a

E b

0 2 4 6 8 10

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

, eVE ex

N

Fig. 1 Fig. 2

Fig. 1. Dependence of the exciton binding energy on the nanofilm thickness without (curve 1) and with allowance for the self-polarization effect (curve 2).

Fig. 2. Energy of the ground exciton state in the nanofilm comprising N layers of 2H–PbI2 calculated disregarding self-polarization effect and exciton-phonon interaction (the dashed curve), taking the self-polarization effect into account (the dashed-dotted curve), and taking the effects of self-polarization and exciton-phonon interaction into account (the solid curve). Dots here indicate positions of peaks of the exciton lines determined according to [8].

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2

2 2 2 2

2

2 2 2 2

1 1cos cos

2 2 ( 1, 3, ..., odd ),(1 ) (1 )

1 1sin sin

2 2 ( 2, 4, ..., even )(1 ) (1 )

,

n n

nn n

n n

nn n

(19)

0

22 /

1( )1 3 2(0)0 2

2

( )| ( )|2

( )

( ) ( ) ( )2

a an

n

n

y f ye

M dya

y y E y ym a

, (20)

consider the change of the energy of the ground state of the carrier caused by emission of virtual phonon from state En. Here

22 2 2 2

( ) ( )cos cos

2 2( ) 1 exp( ) tanh2 [( ) ] [ ] [( ) ] [ ]

nn S

n n n nq a

f a q a qn n q a n n q a

q (21)

if the parities of n and n are identical or

22 2 2 2

( ) ( )sin sin

2 2( ) 1 exp( ) coth2 [( ) ] [ ] [( ) ] [ ]

nn A

n n n nq a

f a q a qn n q a n n q a

q , (22)

if they are different (for n = 1, 3, 5, …, the plus sign is taken, and for n = 2, 4, 6, …, the minus sign is taken),

(0) (0) (0)

0

1 1 1

, (0) (0)( ) [1 exp( )]y y ,

2(0) 2 2 2LO TO

(0) 2 22TOTO0

( )( )

( )

qq

q

,

δσσ’ is the Kronecker delta symbol, and y = aq.

Results of calculations for an E-MAA/2H-PbI2/E-MAA nanofilm demonstrated that the shift of the ground level for the electron (hole) caused by the interaction with the LO phonons, nonlinearly increased with the nanofilm thickness (curves 1 in Fig. 3). In this case, the contribution of higher phonon states (λ ≥ 2) is small in comparison with the ground state (λ = 1). The contribution of higher electronic states (n ≥ 2) to the shift of the ground level increased with the nanofilm thickness. This is explained by the decreased spacing of the levels with increase in the nanofilm thickness that causes the increase of mutual influence of electronic levels in the course of interaction with phonons.

The character of the influence of I phonons is different, namely, with increase in the nanofilm thickness, the long-wavelength shift decreases (curves 2 in Fig. 3). The role of higher (n ≥ 2) levels also increased with increase in the nanofilm thickness due to the same reason as in the case of interaction with the LO phonons.

Results of calculation of the dependence of the total shift of the ground miniband bottom for the carrier in the quanum well caused by its interaction with phonons of both types on the nanofilm thickness (N) are shown by curves 3 in Fig. 3. It can be seen that in ultrathin nanofilms, the long-wavelength shift increases with their thickness, remaining

552

much smaller than the shift in the opposite direction caused by the spatial bounding and self-polarization effects. The contribution of the last effects for N ≤ 3 exceeded the contribution of the exciton-phonon interaction (see Fig. 2). The subsequent increase of the nanofilm thickness is accompanied by the decreasing influence of the self-polarization effect and increasing role of the exciton-phonon interaction (due to the interaction with LO phonons), thereby reducing the exciton energy in comparison with that calculated with allowance for spatial bounding alone.

CONCLUSIONS

The comparison of the calculated results with the available experimental data [8] allows us to conclude that distinctions observed in the positions of the exciton peaks do caused by the difference in the lead iodide platelet thicknesses.

The consideration of the self-polarization and exciton-phonon interaction effects in ultrathin nanofilms approaches values of the exciton energy calculated for the model of infinitely deep quantum well to experimentally measured values. The quantitative discrepancies of the exciton energies in nanoheterostructures calculated by us and

0 2 4 6 8 10

2

4

6

8

10

12

meV ,

N

2

1

3

1 – L0

2 – I

3 – L0+

a

I

5

10

15

20

2

3

1

b

0 2 4 6 8 10

meV ,

N

1 – L0

2 – I

3 – L0+

I

Fig. 3. Shifts of the bottom of the ground electron (a) and hole (b) minibands in the nanofilm comprising N layers of 2H-PbI2 caused by the interaction with LO (curves 1) and I phonons (curves 2). Here curves 3 show the total shift.

553

measured in [8] can be due to the application for calculation of the parameters characteristic for the 2H–PbI2 modification, whereas in [8] the lead iodide polytype was not indicated. It is also obvious that not for all relationships between the microcrystallite lateral size and thickness studied in [8] they can be modeled by a flat nanofilm.

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