Phonon and Plasmon–Phonon Interactions in ZnO Single Crystals and Thin Films

Oleksandr Melnichuk, Lyudmyla Melnichuk and Evgen Venger

INTRODUCTION

Optics of the surface of solids is the field of materials science that attracts considerable attention. It had arisen in the past millennium and it still continues its development. This considerable interest is governed by optical phenomena appeared due to the interaction of electromagnetic waves with lattice vibrations and charge carriers in materials.

The investigation of elementary excitations (such as phonons, plasmons, and polaritons) and their interactions with different materials and structures is one of the interesting problems of modem surface physics. The first theoretical consideration of elementary excitations such as polaritons was performed by K.B. Tolpygo (Tolpygo, 1950) who together with G.I. Pekar called them as “light-excitons.” Later, J. Hopfield proposed the term “polaritons” for such excitations (Hopfield, 1958). Another type of such excitations, known today as “phonon polaritons,” and their dispersion relation was also reported by Tolpygo in 1950 (Tolpygo, 1950) and, independently, by K. Huang in 1951 (Huang, 1951) for ionic crystals.

Surface plasma waves, that is, collective charge carrier motion (also called as surface plasmons (SPs)), are transverse magnetic electromagnetic waves, traveling along interfaces of two different media. The guiding of such surface waves can be considered in similar manner as light guiding in an optical fiber. The highest amplitude of electric field is observed on the solid surface being exponentially decreased from the interface toward media volume.The first study on surface plasmon-polaritons (SPPs), which are the result of the coupling of SPs with light whose wavelength depends on material nature and its geometry, was performed by A.Z. Otto in 1968 (see, for instance, Otto, 1968). SPPs are characterized by shorter wavelength than that of incident light (photons). They can have tighter spatial confinement and higher local field intensity. In the direction perpendicular to the interface, they have subwavelength- scale confinement. SPPs will propagate along the interface until their energy will be lost due to either absorption in the material (metal, semiconductor) or scattering into other directions (such as into free space).

SPPs can be both radiative and nonradiative. Radiative ones, being coupled with plane electromagnetic waves, are involved in emission process or plasma-resonance absorption. Nonradiative SPPs are known as solutions of Maxwell’s equations (their theory concerns the consideration of the complex dielectric function) and result in total reflection. Both types of SPPs have different applications, for instance, in subwavelength optic microscopy or lithography beyond the diffraction limit, in photonic data storage, light generation and biophotonics, in the determination of electrophysical parameters of materials such as carrier concentration, mobility, and conductivity, or even in the measurement of a fundamental property of light itself such as the momentum of a photon in a dielectric medium. The investigation of such excitations and their collective interactions allowed to study not only inorganic solids, but also the organic ones and to develop different sensors. Recently, it was reported that specific light form, which may involve polaritons, could be useful in the development of quantum computers (Liang et ah, 2018).The SPs and SPPs in isotropic solids have been studied in detail (see for instance, Agranovich & Mills, 1985; Dmitruk. Litovchenko, & Stryshevskii. 1989; Tarkhanyan & Uzunoglu, 2006; Venger. Melnichuk, & Pasychnik, 2001; Vinogradov & Dorofeev, 2009). How'- ever, such excitations and their interaction in anisotropic crystals were less addressed due to the difficulties appeared in the theoretical modeling of these excitations (Agranovich & Dubovskii. 1965; Melnichuk, 1998; Melnichuk, Melnichuk. & Pasechnik, 1998; Melnichuk & Pasechnik, 1998; Venger et al., 1995). Indeed, contrary to the consideration of isotropic solids, the modeling of dispersion relations of anisotropic ones requires to involve additional characteristic parameters, that is. the anisotropy of the dispersion laws of surface plasmon-phonon polaritons (SPPPs), effect of free carrier concentration and optical axis orientation with respect to SPPP propagation direction on the number of dispersion branches, anisotropy of damping coefficients of ordinary and extraordinary SPPP, and so on.

There are several techniques used to investigate phonon and plasmon systems of materials as well as the plasmon-phonon coupling. Among them, the most addressed are high-resolution electron energy loss spectroscopy and inelastic neutron scattering. Coupled w'ith theoretical calculations, these techniques provide relevant information on free-carrier concentrations in the bulk and in the subsurface regime, band bending, phonon frequency dispersions, electron mobility, and plasmon damping for homogeneously doped materials. However, both techniques require specific complex equipment. In this regard, optical methods such as Raman scattering and infrared spectroscopy become more attractive. These methods are nondestructive and sensitive to materials' nature.

Among different approaches of infrared spectroscopy used for the investigation of elementary excitations and their interactions, the most attractive are specular infrared reflection (1RR) and attenuated total reflection (ATR). Both of them are sensitive to the interaction of the plasmons with the phonons because their coupling affects significantly the shape of the 1RR and ATR spectra in the range of residual rays when the plasmon frequency (v_p) is approaching the frequency of longitudinal optical phonon (v_L). Modeling of these spectra allows, for instance, the concentration and mobility of free carriers, as well as material conductivity to be determined in semiconductor materials.

For the modeling of the 1RR and ATR spectra in the range of residual rays of corresponding materials, the dielectric permittivity of the whole system is represented by the sum of the “phonon” and “plasmon” terms taking into account plasmon-phonon interaction (Melnichuk, 1998; Venger, Melnichuk. Melnichuk, & Semikina, 2016). When v_p < v_L, such interaction is insignificant, thus for the modeling of 1RR spectrum only the contribution of phonon subsystem in dielectric permittivity of the material can be used. By contrast, for v_p > v_L, both “phonon” and “plasmon” terms are important. Otherwise, significant discrepancy will appear between the theoretical simulation and experimental spectra in the range of “residual rays” of the investigated materials.

In this chapter, main attention will be paid to theoretical consideration of surface polaritons and SPPPs in anisotropic materials. Besides, the effect of magnetic field on such collective excitations will be considered. As an example of anisotropic materials, zinc oxide (both single crystals and thin films) will be addressed.

ZnO has wurtzite structure with the C%_r(P6jmc) space group and show's strong anisotropy of the properties of the phonon and plasmon-phonon subsystems (Dmitruk et al., 1989; Kuzmina & Nikitenko, 1984; Venger et al., 2001). It attracts much attention among different polar materials because of the combination of unique optical, mechanical, and piezoelectric properties. This wide-bandgap semiconductor (E_g = 3:37 eV) possesses high chemical inertness and durability in ambient conditions (Ellmer, Klein, & Rech, 2008). The growth of ZnO single crystals is a complicated process that requires a specific high- temperature equipment. Due to some technological difficulties in the growth of ZnO crystals and, as a consequence, their high cost, the production of ZnO films becomes more attractive today. One of the advantages of ZnO films is their capability to keep typical properties of single crystals. Another one is the possibility to grow with high structural quality at low temperature (even at room or near-room temperature) (Fortunato et al., 2005); Gieraltowska, Wachinski. Witkowski. Godlewski, & Guziewicz, 2012). These films can be used in different optical devices such as light-emitting devices, optical detectors, as well as in solar-cell windows, ultraviolet detectors, and so on (Ellrner et ah, 2008; Fortunato et ah, 2005).

One of the specific feature of ZnO is high anisotropy of phonon system and a low anisotropy of plasmon one. Consequently, the interaction of long-wavelength optical vibration of the lattice with electron plasma in ZnO materials will be different for the cases E±c and Ec. This will affect IRR spectra obtained for both the orientations and their analysis permits to reveal the anisotropy of optical and electrophysical properties of these materials. Such effect was earlier demonstrated for pure and doped ZnO single crystals (Venger et ah, 2001), as well as for pure and rare-earth-doped ZnO films grown semiconductor and insulating substrates such as 6H-SiC, Si, Si0₂, and A1₂0₂ (Korsunska et ah, 2019; Melnichuk, Melnichuk, Korsunska, Khomenkova, & Venger, 2019; Melnichuk, Melnichuk, Tsykaniuk, et ah, 2019; Venger et ah, 2001. 2016). Using nondestructive and fast enough IRR method, free carrier concentration and their mobility as well as damping coefficients were estimated for such textured films when the results of direct electrical measurements were controversial due to a specific, column-like structure of the films (Korsunska et ah, 2019; Melnichuk, Melnichuk, Tsykaniuk, et ah, 2019).

7.2 SPPPS IN ZINC OXIDE SINGLE CRYSTALS

The study of the influence of strong anisotropy of the crystal lattice and weak anisotropy of the effective mass of electrons on the excitation and propagation of nonradiative SPPPs in uniaxial semiconductors has attracted the attention of the researchers only in last decade. Hereafter, the theory of the propagation of electromagnetic waves in hexagonal ZnO single crystals will be considered. The relationship of surface electromagnetic waves with plasma oscillations of free charge carriers (electrons) and with long-wavelength optical lattice vibrations is reported. It should be noted that the developed theoretical models can be implemented for the determination of the optical and electrophysical parameters of not only undoped and doped ZnO single crystals and thin films, but also of other anisotropic materials.

7.2.1 General Theory of Surface Plasmon-Phonon Polaritons in Anisotropic Crystals

The theory of SPPPs in anisotropic crystals was described by Gurevich and Tarkhanyan (1975) and developed later by Tarkhanyan and Uzunoglu (2006). According to this theory, a uniaxial crystal occupies the region г < 0, and the optical axis of the crystal lies in the xz plane and has an angle в with the z axis. Then, the tensor of dielectric constant c^fv) has the form presented by Eq. (7.1):

where and f. are the components of /;,Д v), which in the region of plasmon phonon interaction (without decay) are given by the expressions (7.2) - (7.4):

where v is the frequency (in cm^-1), £_тсх,ц and e₀x>|| ^are the main values of the high- frequency and static permittivity tensors perpendicular and parallel to the C-axis; v_rj_,|| and v_T, л are the frequencies of transverse and longitudinal optical phonons; /1_х_ц are the main values of the dimensionless tensor of the inverse effective mass.

When the wave vector of surface vibration К is perpendicular to the C-axis (/LlQ and xyC (в = tt/2), the SPPP is characterized by the dispersion relation:

In this case, the dispersion of surface waves is determined by dielectric function e_x(v) only, when £y(v) < — 1.

It will be considered only the solutions of Eq. (7.5) for which> 1 (K>co/c), where со is the angular frequency and c is light velocity), that is, non-radiation SPPPs. In this case, two branches of SPPP appear. The high-frequency branch v⁺ begins at the point v = vn. With the К increasing (K»colc), the frequency v approaches asymptotically the value:

this corresponds to the solution of the dispersion equation in the case of no delay of e_x = -1 (biquadratic equation). The low-frequency branch v“ exists for all К values, and the SPPP frequency v increases with К rising from zero to the solution of Eq. (7.6) taken with sign

For K±C, xy±C (в - 0), the SPPPs are characterized by the dispersion relation: and inequalities:

The condition (7.8) along with (7.7) is correct either for ец >xl or for ец < - (l/|cj_|), that is,

This case differs qualitatively from the isotropic one, that is, new SPPP branches appear. Their number and the range of existence depend on the concentration of the electrons in the conduction band as well as on the values of г_Хх,ц, v> , ц, у_Рх>ц, v_x ||⁺’“, and £2_х.ц⁺’^- that are determined by the relations:

and

For a given К value, the maximal number of branches is 4. The curves v = v,(K) (i - 1-4), that are the solution of Eq. (7.7), increase with К rise and at K»colc that approach asymptotically the frequencies on the surface phonons corresponding to the solutions of the equation

However, only the segments of the dispersion curves that lie in the (v, K) plane and for those the condition (7.8) is satisfied, correspond to the surface waves. Equation (7.12) is a fourth-order one with respect to v”, but only one, two, or three solutions lie in the range where e_x < 0. Consequently, three, two, or one of the dispersion curve(s) end at finite К values, whereas the number of such solutions depends on the concentration of electrons in the conduction band.

Hereafter, the effects mentioned above will be discussed, at first, for different ZnO single crystals grown by hydrothermal method. Their electrophysical parameters such as carrier concentration (n₀), their mobility (/0 and effective masses (m_± * and m_t*), as well as conductivity (0) given for different directions of electric field (E±C and £||Q are collected in Table 7.1. The concentration and mobility of carriers were estimated using Hall measurements. The modeling of the crystal properties will be performed using self-consistent parameters taken from Melni- chuk (1998); Venger et al. (2001) and are collected in the Table 7.2.

7.2.2 The Effect of Anisotropy on the High-Frequency SPPP Dispersion in Zinc Oxide

In this section, the effect of the crystal anisotropy on the SPPP dispersion dependences is discussed for three different orientations of hexagonal ZnO. Figure 7.1a (symbols) shows three experimental dispersion curves of v_s(X), obtained for ZnO crystal with electron concentration «о = 9.3 x 10¹⁶ cm^-3 (sample Z02-3, Table 7.1). They are plotted versus the dimensionless wave vector q = Kc/co_T, where К is the SPPP wave vector, c is the light speed, and co_T|| is the angular frequency of the transverse optical ZnO phonon when the vector of electric field of the infrared radiation E is parallel to the C-axis (E||C).

The high-frequency v⁺ branches of SPPPs were obtained in accordance with Eqs. (7.5) and (7.7), taking into account the anisotropy of electron effective mass in ZnO (Melnichuk, 1998; Venger et al., 2001). Curve 1 corresponds to the case /f||C and xyC (.v and у axes lie on the surface of the sample), whereas curve 2 was obtained at KA.C and xy±C. Both these dependences are the features of extraordinary SPPPs. The ordinary SPPPs are manifested in

TABLE 7.1

Electrophysical parameters of ZnO crystals grown by hydrothermal method

Sample
Z02-3	9.3 x 1016	150	167	22	25	0.21	1.23
ZOl-3	6.6 x 1017	80	96	85	101	0.23	1.13
Z06-B	2.0 x 1018	74	50	237	243	0.22	1.18
ZC1M	4.2 x Ю18	2.2	—	148	—	—	—
Z09	5.0 x Ю19	1.5	—	1250	—	—	—

TABLE 7.2

Self-consistent parameters for ZnO single crystals used for theoretical simulation

	8.1	3.95	412	591
	9.0	4.05	380	570

the orientation КJ.C and xy±C (curve 3). As one can see, the experimental results and their theoretical simulation are in a good agreement (Figure 7.1).

The optical parameters of ZnO (such as £_x(v) and ец(у)) (Table 7.2) were obtained based on the dispersion analysis of 1RR spectra, taking into account the anisotropy of electron effective mass (Table 7.1). The plasma frequencies of free carriers v_px and у_рц are linked by the relation

where ш_х,ц is the transverse and longitudinal effective mass of electrons. Since for ZnO т*_±/тТ, = 1.15, hence у_рц = 1.1 v_px. The frequencies of plasmon-phonon oscillations are y_pf = 548 cm^-1 (curve 2) and 561 cm^-1 (curve 3) when the plasma frequencies v_px = 90 cm^-1 and v_p|| = 100 cm^-1 (I£±C. EC). The anisotropy of SPPPs manifests itself at Kc/coy > 2. Thus, when Kclcor - 2, the difference 5v = 18 cm^-1 is seen between curves 1 and 3. At the same time, when A7->0, curves 1 and 2 tend to achieve the frequency v_pf = 548 cm^-1.

The inset (a) in Figure 7.1 shows in detail the segment of the v_s(/0 dependence in the 380 420 cm^-1 range. It is seen that up to v = 412 cm^-1, the SPPPs of the second type are manifested. Their existence is limited by the conditions e_x(v) < 0 and c_z(v) > K~. The symbols represent the experimental data that are consistent with calculated ones (solid curve). The SPPPs of type 1 start at v = 412 cm^-1 when e_x(v) < 0 and t;_z(v) < 0.

Since the experimental dependence v_s(AT) represented by curve 1 in Figure 7.1 is a continuous line, for its modeling, it was considered the extremum value at у = 412 cm^-1 as “stop point.”

Figure 7.1b represents the experimental ATR curves obtained for the same ZnO crystal (curves 1-5) at different air gaps between ATR crystal and ZnO surface and the angles of IR light incidence. The minima on these spectra corresponds to the frequencies v_m;_n = 408, 450, 496, 518, and 527 cm^-1, whereas their width are T_p = 32, 27, 22, 17, and 15 cm^-1, respectively. The modeling of the ATR spectra was performed for the orientation /f||C and xyC in the extremum vicinity, that is near v = 412 cm^-1. Consequently, for the SPPPs of types 1 and 2, it was performed for v_spi = 413 cm^-1 and v_p2 = 411 cm^-1 taking into account the damping

FIGURE 7.1 (a) Dispersion curves v_s(K) for the SPPPs of ZnO crystal with electron concentration of

«о = 9.3 x 10^l6cm^³. The orientation of wave vector and C-axis is (I) A"||C and .yy||C; (2) KLC and xy±C; (3) K±C and xyC . Inset (a) shows curve I in detail in the range v = 380 — 420cm ¹; /f||C and ,vy||C. (b) ATR spectra of the same crystal recorded with an air gap 26 pm (1,2) and 3 mm (3-5), and the angle of incident light of a = 25.3° (1), 28° (2) 34° (3), 42° (4), and 52° (5). Curves 6 and 7 were calculated with d =12 pm, a = 25.2° (6) and d = 49 pm, a = 25.7° (7) and demonstrate the presence of SPPPs of type 1 (7) and type 2 (6). Curve 8 is the modeling performed for the ZnO crystal with electron concentration n₀ =

4.2 x 10^ls cm ³ that demonstrates the polaritons of type 3.

Reproduced with permission from Melnichuk ct al. (1998). Copyright 1998 Springer Nature coefficient of transverse optical phonons y_f — 11 cm^-1 (Melnichuk, Melnichuk, & Pasechnik. 1994). Here, according to Eq. (7.5), the values of /f_spl and K_sp2 were 1.11054 and 1.10415, respectively. Both theoretical spectra were found to have the same intensity in the minimum and the same width Г_р = 12 cm^-1. The spectra practically overlap indicating that surface polaritons of both types can be generated simultaneously in the presence of damping at an emission frequency of 412 cm^-1. The experimental spectrum with a minimum at v_mj„ = 408 cm^-1 and its width of T_p = 32 cm^-1 (Figure 7.1b, curve 1) corresponds to the damping coefficient of SPPPs T_sp = 4 cm^-1. The cutoff frequencies are determined by Eq. (7.11).

Polaritons of type 2 can be observed only when < 0 and ец > K_x² (Gurevich & Tar- khanyan, 1975; Melnichuk et al., 1998). For ZnO crystals, the existence region of polaritons of type 2 is in the range 380^412 cm^-1 (Figure 7.1a). They are exhibited when the orientation of the sample corresponds to KC and xyC as it was demonstrated experimentally by Melnichuk et al. (1998) for the first time.

One more interesting excitations is the polaritons of type 3 that can be present only in anisotropic crystals with the carrier concentration above a certain value (higher than) no = 2 x 10^l8cirT³ (Melnichuk et al., 1998). For instance, in ZnO crystals with

no = 4.2 x 10¹⁸ cm ³and it is by ATR spectrum (curve 8 in Figure 7.1b). The existence region lies in the low-frequency range and corresponding dispersion curves v_s(K) for such polaritons are bounded by the straight lines ец = 1, e_x = 0 and со = Kc (as it will be discussed later).

7.2.3 Low-Frequency Dispersion Dependences of SPPPs of Hexagonal ZnO

In the isotropic case, an ordinary polariton has one low-frequency v" branch. It begins at v = 0 and increases up to v“ when К —> oo. The dispersion curves of the surface polaritons have two branches: v⁺ ~( K). Figure 7.2 shows such curves of ZnO when K±C and xyC. With this orientation, ordinary polaritons appear. The calculation of these dependences was performed for the samples with different carrier concentrations: Z02-3 («о = 9.3 x 10¹⁶ cm^-3) and ZOl - 3 (no = 6.6 x 10¹⁷ cm^-3) (Table 7.1). For sample Z02-3 (curves 1 and 2), the parameters were v_px - 90 cm^-1, v,,j ⁺ = 561 cm^-1, and v_pf = 59 cm^-1 (Кс/®тц —* oo), whereas for sample ZOl-3 (curves 3 and 4): v_px = 240 cm^-1, v_pr⁺ = 578 cm^-1, v_pf^_ = 152 cm~^x (Kc/co_T|| —> oo). Index “pf” is used for SPPP boundary conditions corresponding to the values v⁺~ obtained from Eq.(7.6). These calculations showed that SPPP frequency is higher when the concentration of free carriers is higher (electrons for ZnO). All curves in Figure 7.2 correspond to the SPPP of type 1 that exist at any values of x > 1 (Bryksin, Mirlin, & Firsov, 1974).

Let us consider the case 0 = 0, when K±C and xyj.C. For ZnO. the interrelation of the frequencies is v_T|| < vtj. < v_LM < v_Lx. Dispersion curves v_s(K) begin at the frequencies v = 0; v = v_Tx (e_L = oo) and v = £2⁺, v = Пу (ец = 1), which are on the straight line (light line) w - Kc. As

FIGURE 7.2 Dispersion curves v_s(K) for the SPPP ofZnO crystals with orientation K±C and xylC; 1,2 - v_pl = 90 cm ¹ (sample Z02-3, n₀ = 9.3 x 10¹⁶ cm ³); 3, 4 - v_pi = 240 cm ¹ (sample ZOl-3, n_u = 6.6 x Ю¹⁷ cm ³); 1 - v_pf⁺ = 561 cnT¹; 2 - v_pf" = 59 cm 3 - v_pf⁺ = 578 cm^-1; and 4 - v_pf ^_ = 152 cm^-1

mentioned above, under certain conditions, up to four dispersion branches v_s(AT) are possible for an uniaxial semiconductor. At К » cole, their number varies being depending on carrier concentration.

However, if for ZnO crystals the condition

is failure then existence of only three dispersion branches v_s(AT) is possible.

Figures 7.3-7.6 show the dispersion dependences of ZnO with different carrier concentration for the orientation of K±C and xy±C. It is seen from Figure 7.3 that for negligible carrier concentration («о = 4.0 x 10^l5cm^-3, v_p = 1 cm^-1), only the high-frequency v_s(AT) is observed being started at v = vr± = 412 cm^-1 and asymptotically approaches the value of v_pf⁺ = 545.1 cm^-1 at К_с/со_Хц —> oo. This dispersion curve corresponds to extraordinary SPPPs of type 1 since they exist when i:_±_ < 0 and сц <0, and the existence of v_pf^+- for К._с/со_Тц —> oo.

Figure 7.4 shows the dispersion dependence v_s(K) of Z02-3 sample with electron concentration n₀ = 9.3 x 10¹⁶ cm^-3 when K±C and xy±C, v_px = 90 cm^- and v_p^ = 100 cm^-1. In

FIGURE 7.3 Dispersion curve >_S(K) for SPPPs of ZnO (n₀ = 4.0 x 10¹⁵ cm ³) for orientation KlC and xy±C; Vp_X = 1 cm , = 1.1 cm 1. v_pf⁺ = 545.1 cm ¹

FIGURE 7.4 Dispersion curves v_s(K) of the SPPPs in ZnO (sample Z02-3, n₀ = 9.6 x 10¹⁶ cm ³) with orientation KlC and xy±C; v_px = 90 cm^-1; у_рц = 100 cm^-1: 1 - v_pf⁺ = 548 cm^-1; 2 - v_pf“ = 60 cm^-1

this case, two branches v_s(AT) of the SPPPs of type 1 appear approaching the values of v_pf⁺ = 548 cm^-1 and v_pf" = 60 cm^-1 at Кс/сощ —» oo. Curve 2 has the features that are characteristic for plasma polaritons.

Figure 7.5 shows the v_s(AT) curves for ZnO crystal with higher electron concentration (sample ZC1M, n₀ - 4.2 x 10¹⁸ cm^-3) for the same orientation and v_px = 605 cm^-1 and Vp,, = 650 cm^-1. The calculation shows the presence of three dispersion dependences (curves 1-3) with the next parameters: v_pf⁺ = 719 cm^-1 (curve 1) and v_pf” = 305 cm^-1 (curve 2). Curve 3 is shown in detail in Figure 7.5b. It is seen that it begins at a frequency of Q|l = 309.9 cm^-1 at Кс/а>т = 0.815 (ец = 0) and ends at a frequency of v_x^_ = 318.4 cm^-1 at Kc/to_T = 1.632 (e_± = 0).

Further increase of carrier concentration results in the transformation of the dispersion curves of the SPPPs of type 3. Figure 7.6a shows such dependences for ZnO crystal with 77₀ = 8.0 x Ю¹⁸ cm^-3, where v_px = 1300 cm^-1 and v,,,, = 1430 cm"¹. In this case, v_pf⁺ = 1273.7 cm^-1, and v_pr” = 363.58 cm^-1 at АГс/штц —* °o. The curve v_s(AT) starts at a frequency of v₃ = 363.8 cm^-1 and tends to the value v₃ = 390 cm^-1 at Kc/cojц —» 0 (Figure 7.6a, curve 3). The calculation of v_s(AT) for ZnO with higher electron concentration (n_Q = 2.0 x Ю¹⁹ cm^-3, v_px = 1500 cm^-1 and v_p,, = 1650 cm^-1 ) is presented in Figure 7.6b. Here, v_pf⁺ = 1450 cm" , v_pf” = 367.7 cm^-1. The cutoff frequency of the dispersion dependence is at v = 395.5 cm"¹ (curve 3).

Thus, the behavior of the third branch of the dependence v_s(K) for ZnO crystals differs from that observed previously in anisotropic crystals. Using the data obtained for the ZC1M sample (Figure 7.5), the ATR spectrum of the SPPPs of type 3 was calculated (Figure 7.7). This spectrum was obtained at = y^ = 6 cm^-1 and y_PL = у = 1 cm"'and an angle of light incidence in the ATR element a - 28°, and the air gap width between the element and the sample cl = 26 pm. The minimum intensity of the spectrum was at frequency v = 312.5 cm”¹ at Kc/co_T = 0.92. An increase in the damping coefficients of phonons and plasmons in ZnO leads to a decrease of the spectrum amplitude.

FIGURE 7.5 (a) Dispersion curves v_s(/Q for SPPPs of ZnO when KiC and xy±C; v_pj_ = 605 cm v_p|| = 650 r/H^-l(sample ZC1M, n₀ = 4.2 * 10^IX cm"³); (b) the detailed presentation of curve 3. =

318.4 cm"'; Kc/coj = 1.632 (e_x =0); Пц = 309.9cm"¹; Кс/со-щ =0.815 (ец = 1); vp^- =306.5 cm"¹; Kc/coj^ = 0,017(ец = 0)

FIGURE 7.6 Dispersion curves v_s(K) for SPPPs of type 3 in ZnO with n₀ = 8.0 x 10^1!i cm ³ (a) and n₀ = 2.0 x 10¹⁹ cm ³ (b) when K±C andxylG

FIGURE 7.7 ATR spectrum of ZnO crystal (sample ZC1M, n₀ = 4.2 x Ю¹⁸ cm ³). Calculation parameters are: lf_L = //у = 6 cm^-1, y_pl = y_pу = 1 cm^-1, a = 28°, d = 26 pm.

7.2.4 The Comparison of Experimental and Theoretical Data

The analysis of these results showed that the SPPPs of type 2 can occur only when e_x < 0 and £|| >/д. (Eq.(7.8)). In the case of ZnO. the range of the existence SPPPs of type 2 is limited by frequencies of 380^412 cm^-1 (see Figure 7.1a). They appear when the orientation of the sample is /f]|C and xyC. The dependence v_s(/Q practically represents a straight line passing towards the v_s(K) of the SPPPs of type 1.

With orientations K±C, ЛЦС, and .vy||C, the extraordinary SPPPs are excited, which is manifested in the two-component spatial structure of their fields. In Figures 7.3-7.5, in the v,Kc/cotу coordinates, the regions, where condition (7.8) is fulfilled and the dispersion dependences of the SPPPs are observed at в - 0 and v-гц < vn < сц v_s(K) begin at the frequencies of v = 0, v = >_Гх (c_x = °°) and v = Q||⁺, v = 0||^_(ец = 1), coincide with the straight line со - Kc.

The particular interest should be paid to the dispersion dependences v_s(AT), related to the SPPPs of type 3, which manifest themselves only in anisotropic crystals at concentrations of free carriers higher than a certain boundary value. In the case of ZnO, this threshold value is n_Q = 2 x Ю¹⁸ cm^-3 (Melnichuk et al.. 1998; Venger et ah, 2001). The spectral range of the existence of these SPPPs is framed by the straight lines £ц = 1, со = Kc and c_x = 0. Figure 7.5 corresponds to the case vo- <угц). If у_х->у_Гц, then v_s(K) exists for all K>Q^/c (Figure 7.6).

According to Gurevich and Tarkhanyan (1975); Tarkhanyan and Uzunoglu (2006),

where w_x is reduced electron mass. Since for ZnO crystals, т_л * = 0.26m_e (Melnichuk et ah, 1994; Venger, Melnichuk, Melnichuk, & Pasechnik, 1995), then /»_x - (1 l/ij_)m_e and, consequently, //_x = 3.85. Thus, from Eq. (7.15), v₀ = 1.0129v_px and V| = 1091.17 cm^-1 are obtained. If v₀ = vi, then v_px = 1077.27 cm^-1. Since £2ц > 301.1 cm^-1, the SPPPs of type 3 begin to appear at v_px > 550 cm^-1.

Figure 7.5b shows the dependence v_s(AT) for SPPPs of type 3 for ZnO crystal with n₀ - 4.2 x Ю¹⁸ cm^-3 (sample ZC1M, Table 7.1) where v_px = 605 cm^-1. In this case, vjf = 306.5 cm-¹, Kclco = 0.017 (cy = 0); Q.f = 309.9 cm-', Kclco = 0.815 (ey = 1), v_x“ = 318.4 cm^-1, and Kclco = 1.632 (c_x = 0).

Thus, the SPPPs of type 3 exist in the frequency range of v = 309.9-318.4 cm^-1 for this sample. Corresponding dispersion dependences v_s(K) (Figure 7.7) were obtained for K±C, xy±C and vo > vi (v_i_— >V741) with cutoff frequencies V3 = 390 cm^-1 and 395.5 cm^-1. When (vj.^- >vt-||), the SPPPs exist at Кс/соц —* oo, whereas the conditions for the SPPP existence at £2 v < V/ are similar to those for the SPPPs of type 2. However, in the range from V741 < v < v ", the SPPPs of ty|>e 1 appear. It should be noted that the low-frequency v_s~(K) for SPPPs begin at v > 0, when %Г > 1. The increase of free carrier concentration results in the shift of the onset of v_s(AT) from 0.54 to 9.5 cm^-1 (Melnichuk, 1998; Melnichuk et al., 1994, 1998; Venger et al., 2001).

Thus, in this section, the SPPPs of types 1 and 2 in anisotropic ZnO crystals with different carrier concentration were shown for various orientations of the wave vector toward the crystal surface and optical C-axis. The experimental observation of the SPPPs of type 2 is shown. Besides, new dispersion curves for polaritons of type 3 were presented. These excitations appear in the crystals with electron concentration higher than n_Q - 2 x l()^ls cm'³ and the orientation K±C and xy±C. The conditions for the existence of the SPPPs of type 3 are shown along with corresponding calculated dispersion curves and ATR spectrum.