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Infrared absorption spectra of a spatially dispersive polar semiconductor nanowire Afshin Moradi n Department of Engineering Physics, Kermanshah University of Technology, Kermanshah, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 20 January 2015 Received in revised form 5 April 2015 Accepted 7 April 2015 by F. Peeters Available online 13 April 2015

To model the infrared wave absorption of a polar semiconductor nanowire with spatially dispersive permittivity, a new extension of the Mie theory is applied. This is achieved by imposing the usual Maxwell boundary conditions as well as appropriate additional boundary condition of the continuity of the normal component of the displacement ﬁeld at the surface of the system. Examples of calculated infrared absorption spectra are presented, and it is found that by decreasing the wire radius, the main absorption peaks, due to the coupled surface plasmon–LO-phonon modes, are blue-shifted from their classical positions and two groups of subsidiary peaks, due to coupled bulk plasmon–LO-phonon modes, appear in the optical spectra of the system. & 2015 Elsevier Ltd. All rights reserved.

Keywords: A. Polar semiconductor nanowire D. Infrared absorption property

Several years ago, Yokota [1] and Varga [2] presented theoretical discussions of the interaction between bulk plasmons and bulk longitudinal-optical (LO) phonons in polar semiconductors. Generally, in polar semiconductors containing free carriers, the plasmons and the long wave LO phonons are coupled by the macroscopic polarization ﬁeld which is associated with both types of excitations. The coupling is strongest when the free carrier concentration is such that the plasma frequency ωp is close to the LO phonon frequency [1,2]. In the presence of surfaces there also appear surface modes of mixed plasmon–phonon character, and these dominate the infrared absorption properties of small crystallites [3–6]. The dispersion properties of coupled surface plasmon–optical phonons (SPOP) oscillations in polar semiconductors have been extensively studied by several workers in a number of geometries such as semi-inﬁnite plane surfaces [7,8], thin ﬁlms [9], and cylinders [10], within the local theory approximation. Also, the dispersion relations for SPOP coupled waves are derived in non-retardation limit, for thin ﬁlms [11] and spheres [12], using the hydrodynamical (HD) model. In this way and in view of the growing interest in polar semiconductor cylindrical nanostructures [13–17], recently we investigated the hybridization of the surface plasmon oscillations with the surface optical phonon waves in cylindrical polar semiconductor nanowires [18], by employing the linearized HD theory in conjunction with the Poisson equation [19] and applying the appropriate additional boundary condition.

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http://dx.doi.org/10.1016/j.ssc.2015.04.002 0038-1098/& 2015 Elsevier Ltd. All rights reserved.

However, the nonlocal effects on the infrared absorption spectra of polar semiconductor nanowires have not been investigated as yet. This is the aim of the present communication. We note that the nonlocal effects are well known to inﬂuence the optical properties of metallic nanowires [20–30]. In this regard, by using the HD dielectric function of the polar semiconductor materials [10] as well as appropriate additional boundary condition of the continuity of the normal component of the displacement ﬁeld at the surface of the system [31], we extend the classical Mie theory for the absorption of electromagnetic radiation by a spatially dispersive polar semiconductor wire that is irradiated by a normally incident plane wave. Since for the parallel polarization with respect to the nanowire axis, no surface modes are excited [20] only the case of perpendicular polarization will be discussed. Let us consider a cylindrical nanowire of radius a and inﬁnite length that aligned along the z-axis embedded in a medium with dielectric function εd. We use cylindrical coordinate ðρ; ϕ; zÞ for an arbitrary point in space. Here, the system supports both the usual transverse and longitudinal waves. Also, it is assumed that the nonlocal responses of the system are dominated by the nonlocality induced by free electron gas, while the optical phonons only contribute to local responses. Thus in the HD model, the dielectric properties of the present system are characterized by both the usual Drude transverse dielectric function, as

εT ¼ εn ðωÞ

ε1 ω2p ; ωðω þiγ Þ

ð1Þ

A. Moradi / Solid State Communications 212 (2015) 10–13

and the HD longitudinal dielectric function

ε1 ω2p ε L ¼ ε n ð ωÞ ; ωðω þ iγ Þ α2 k2L

ð2Þ

where γ is the free carrier collision frequency and ωp is the plasma frequency, deﬁned by ω2p ¼ 4π n0 e2 =mn ε1 . Here n0 is the free carrier concentration and mn is the effective mass. Also, ω is the freqpﬃﬃﬃﬃﬃﬃﬃﬃ uency, kL is the nonlocal longitudinal wave vector, α ¼ 3=5vF and

ε n ð ωÞ ¼ ε 1 þ

ε0 ε1 : ω 2 ω 1 iΓ ωTO ωTO

ð3Þ

In the above equation, ε0 and ε1 , are the static and the high frequency dielectric constants, Γ is the damping factor and ωTO is the frequency of the long wavelength transverse optical phonons. Now, we assume that the system be exposed by a normally incident beam in which the electric ﬁeld of it is perpendicular to the cylinder axis. The vector cylindrical harmonics functions can be deﬁned according to Ref. [32] as follows: Mm ¼ ∇ ½ez Z m ðkρÞexpðimϕÞ;

ð4Þ

Nm ¼ ð1=kÞ∇ Mm ;

ð5Þ

Lm ¼ ∇½Z m ðkρÞexpðimϕÞ:

ð6Þ pﬃﬃﬃﬃﬃ Herek is given by kT ¼ ε T ω=c for the transverse modes and 2 kL ¼ ω2 þ iγω ε1 ω2p =εn ðωÞ =α2 , for the longitudinal modes pﬃﬃﬃﬃﬃ inside the non-magnetic nanowire and by kd ¼ εd ω=c outside it, where c is the light speed in vacuum. Also eρ , eϕ , and ez are unit vectors in ρ, ϕ, and z directions, respectively, and Z m ðkρÞ represents a cylindrical Bessel or Hankel function, and is chosen as follows. Inside the cylinder J m ðkT ρÞ and J m ðkL ρÞ are used for the transverse and the longitudinal modes, respectively. Outside the cylinder J m ðkd ρÞ and H m ðkd ρÞ are used for the incident and scattered waves, respectively. We note that the Hankel function is chosen to indicate that the scattered ﬁeld is a wave traveling in the outward radial direction. The expansion of the incident electric ﬁeld is Ei ¼ i

þ1 X m ¼ 1

Em Mm ðkd ρÞ;

ð7Þ

where Em ¼ E0 ð iÞm =kd . The expansion of the transmitted and scattered electric ﬁelds can be represented as Et ¼

þ1 X m ¼ 1

Es ¼

þ1 X m ¼ 1

g m Em Mm ðkT ρÞ;

ð8Þ

iam Em Mm ðkd ρÞ:

ð9Þ

Also, in the polar semiconductor wire, at the same frequency ω, there is a longitudinal wave (bulk plasmons) that can be described by the following electric ﬁeld: EL ¼

þ1 X m ¼ 1

hm Em Lm ðkL ρÞ:

ð10Þ

The unknown expansion coefﬁcients am, gm, and hm, can be determined by the boundary conditions at the surface of the cylinder. The usual two classical boundary conditions at the plasma–dielectric interface require the continuity of the tangential components of the electric and magnetic ﬁelds across the interface. We note that in the nanowire both the transverse and longitudinal (usually neglected) waves give a contribution to the value of electric ﬁeld, we have

E i ϕ þ E s ϕ j ρ ¼ a ¼ E t ϕ þ E Lϕ j ρ ¼ a ;

ðH iz þ H sz Þj ρ ¼ a ¼ H tz j ρ ¼ a :

11

ð11Þ ð12Þ

Also, a additional boundary condition must be introduced at the cylindrical interface. According to the results obtained by Yan et al. [31], the third well-known boundary condition at the plasma– dielectric interface requires the continuity of the normal component of the displacement ﬁeld. The third boundary condition gives εd Eiρ þ Esρ j ρ ¼ a ¼ εn ðωÞ Etρ þELρ j ρ ¼ a : ð13Þ Solving the system of Eqs. (11)–(13), we ﬁnd that the coefﬁcients of the scattered wave are given by

cm þ kd J 0m ðkT aÞ J m ðkd aÞ kT J m ðkT aÞJ 0m ðkd aÞ

; ð14Þ am ¼ cm þkd J 0m ðkT aÞ H m ðkd aÞ kT J m ðkT aÞH 0m ðkd aÞ where cm ¼

m2 J m ðkL aÞ εd k T k d J : ð k a Þ T ðkL aÞJ 0m ðkL aÞ m εn ðωÞ kd a kT a

ð15Þ

Here the primes denote differentiation with respect to the argument of the radial functions. When cm ¼ 0, the coefﬁcients am reduce to the standard cylinder scattering coefﬁcients [32]. The absorption width (absorption cross section per unit length) of the cylinder is given by (in units of the geometric width) Ca ¼

þ1

2 X j am j 2 þ Reðam Þ : kd a m ¼ 1

ð16Þ

In order to interpret the optical spectra in terms of the normal modes of the system, we will calculate the frequencies of the eigenmodes. In the retarded case, the frequencies of the coupled SPOP and coupled bulk plasmon–optical phonons (BPOP) modes can be deduced from Eq. (14) by equating to zero the denominator of this expression. In the non-retarded case, applicable for the nanowires, we have ε1 ω2 εd J m ðkL aÞ 1 m : ð17Þ ω2 ¼ n p 1 ðkL aÞJ 0m ðkL aÞ ε ð ωÞ ε d þ ε n ð ωÞ The roots of the above equation provide the dispersion relation of the modes, including both coupled SPOP modes (for imaginary values of kL) and coupled BPOP modes (for real kL). In the following, with the above equations, we present the simulation results and discussion of the problem. As an example, let us discuss the behavior of the absorption spectra of the InAs cylindrical wires in vacuum [5], using the parameters given by Palik et al. [33]. For the our sample ωp ¼ 236 cm 1 , and the damping frequency of the free carriers γ ¼ 30 cm 1 . The other parameters are ωTO ¼ 219 cm 1 , ωLO ¼ 243 cm 1 , ε1 ¼ 11:8 and the phonon damping frequency Γ ¼ 2 cm 1 . In Fig. 1 [panels (a1)–(a4)] the absorption spectra of four nanowires with different radii are shown. The results of standard local absorption theory are also shown by the dashed red curves. We observe that the local model displays three peaks for large values of a [see the inset of panel (a4)]. The middle one being due to the transverse optical phonons, i.e., ωTO . The other two peaks are due to the SPOP modes. In the latter case, for the present nanowires the absorption frequencies are close to the non-retarded values obtained from εT ¼ 1 for γ ¼ Γ ¼ 0. One can see that the results of nonlocal absorption theory exhibit the main SPOP peaks (ω þ and ω ) that shift towards the high frequencies side. Furthermore, the subsidiary peaks are observed in the frequency regions Ω o ω o ωLO and ω 4 Ω þ (Ω and Ω þ are the frequencies of the coupled BPOP modes of an inﬁnite medium in the presence of the nonlocal effects, i. e., the roots of equation εL ¼ 0) where these peaks originate from the excitation of coupled BPOP modes and ωLO is the LO phonons frequency. Comparing the inset of panel (a1) with the inset of

A. Moradi / Solid State Communications 212 (2015) 10–13

4 Log Ca

(a1) =20nm

Log Ca

6

3

4 5 6 7 8 9 10

4

(a2) =40nm

5 0.5

1.0

1.5

2.0

8

Log Ca

12

BPOP Mode

6 7

10

8

12

10

BPOP Modes

9

1.5

2.0

2.5

3.0

0.5

2

(a3) =80nm

4

3

5

4

Log Ca

Log Ca

3

1.0

6

1.0

1.5

(a4) =160nm Log Ca

0.5

5

7

6

8

7

2.0

2.5

3.0

2 3 4 5 6 7 8 0.5

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8

9 0.5

1.0

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2.0

2.5

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1.0

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3.0

Fig. 1. (Color online) Nonlocal infrared absorption spectra (in units of the geometric width) of four InAs nanowires in vacuum with different values of a (full blue curve). The dashed red curves show the result of the classical local model according to Ref. [5].

3.0

3.0

2.5

2.5

BPOP Modes

2.0

2.0

1.5

1.5 LO

BPOP Modes

1.0

1.0

=40nm =20nm

0.5

=160nm

=80nm

0.5

(a1) m=1

(a2) m=2

0.0

0.0 0

2

4

6

8

10

12

0

3.0

3.0

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(a4) m=4

(a3) m=3 0.0

0.0 0

2

4

6

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10

12

0

2

4

6

8

10

12

Fig. 2. Dispersion curves ω=ωTO for the surface and bulk modes (a1) m¼ 1, (a2) m¼2, (a3) m¼ 3, and (a4) m¼ 4, in a cylindrical InAs nanowires in vacuum, versus the parameter η ¼ aωTO =α.

A. Moradi / Solid State Communications 212 (2015) 10–13

panel (a4), it can be seen that by decreasing the radius of the system, the absorption at transverse optical phonons frequency will be suppressed. Also, we ﬁnd by increasing the radius of a polar semiconductor nanowire, the number of subsidiary peaks, due to the coupled BPOP modes, increases, but their relative amplitude decreases in the considered energy interval, while the shift of the SPOP peaks from their classical local positions decreases. This means that the blue shift of the SPOP resonances from their standard nonlocal positions could be enhanced by decreasing the nanowire radius. Finally, the assignment of the absorption peaks can easily be performed using the calculated frequencies of the surface and bulk modes, as obtained from Eq. (17). In this way, we illustrate in Fig. 2 [panels (a1)–(a4)] the dependence of the dimensionless frequency ω=ωTO on the dimensionless variable η ¼ aωTO =α for different values of m. Physically, for a thin nanowire, only the uniform surface mode m¼1 is observable in the optical spectra (Fig. 1). When the radius increases the contribution of the other surface modes are also noticeable. As can be seen from panel (a1) of Fig. 2, for the case of a nanowire of radius 20 nm, there appears two SPOP modes' peaks with m¼1 (ω þ and ω ) and one group of bulk modes in the frequency region Ω o ω o ωLO in the considered energy interval. One can see that in the considered energy interval, by increasing the radius of the nanowire, the number of bulk modes in the frequency region Ω o ω o ωLO , increases and a new group of bulk modes appears in the frequency region ω 4 Ω þ . To summarize, we have studied the infrared absorption properties of spatially dispersive polar semiconductor nanowires. We have obtained the absorption spectra of the system, and investigated their dependence on the nonlocal effects. Numerical results of the absorption spectra show the two main absorption peaks, due to the coupled SPOP modes, are blue shifted from their classical local positions and subsidiary peaks, due to coupled BPOP excitations, appear in the frequency regions Ω o ω o ωLO and ω4Ωþ . References [1] I. Yokota, J. Phys. Soc. Jpn. 16 (1961) 2075.

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