Dispersion model of two-phonon absorption: application to c-Si

Daniel Franta; David Nečas; Lenka Zajíčková; Ivan Ohlídal

doi:10.1364/OME.4.001641

1. Introduction

Infrared absorption in crystalline silicon, as well as in other homopolar materials (diamond or crystalline germanium) is caused by multi-phonon absorption processes only. Due to their low probability, silicon exhibits low absorption in the infrared, especially in the range from 1.5 to 5 μm (2000–7500 cm⁻¹). Therefore, single crystal silicon (c-Si) is one of the popular materials for infrared refractive optics [1, 2].

The polished c-Si wafers are also used as substrates for the deposition of thin films because they are available at reasonable price, considering their smoothness (roughness below 1 nm) and well-controlled properties. They are used frequently for optical characterization of deposited films in ultraviolet and visible (UV/VIS) spectral region for two reasons. The optical constants of c-Si in UV/VIS can be found reliably as tabulated data and they are not influenced by impurities in c-Si. Moreover, the optical characterization in vacuum ultraviolet region can be performed without modeling of the scattering effects at the film–substrate boundary because of very good surface quality.

Besides optical measurements in UV/VIS, the transmittance measurements in the entire range from Si electronic gap in near-infrared to far-infrared or even terahertz region provide excellent sensitivity to changes in optical properties of weakly absorbing films on double-side polished c-Si. Knowing the IR absorption of the optical films is crucial for determining their structure because their structure may vary from amorphous to polycrystalline [3]. This cannot be studied in the UV/VIS spectral range. In the region where the films are transparent, the optical methods are not sufficiently sensitive to obtain the structural information, given only by the refractive index profile, whereas in the region of electronic interband transitions, the information is limited by the penetration depth of the light reflected from the film. Unlike the UV/VIS range, the fitting of IR measurements is performed seldom, although the chemical structure of films on c-Si is often studied qualitatively by identifying the absorption peaks in the middle IR range. The problem is that the absorption peaks in the films overlap in the region below 2000 cm⁻¹ with the c-Si multi-phonon absorption and the absorption on impurities, such as interstitial oxygen. It is usually solved by calculating the relative transmittance, i. e. the transmittance of the whole sample divided by the transmittance of bare c-Si [4–6]. However, it leads to creation of artifacts in the analyzed data [7]. Therefore, the IR analysis should be performed preferentially by fitting the data taking into account optics of thin films.

The fitting of IR optical measurements of thin films on c-Si substrates requires the knowledge of c-Si optical constants that vary significantly with the concentration of impurities and temperature. It is often necessary to know the optical constants of thin films at temperatures different from the room temperature. It is also necessary to take into account slight variations in the concentrations of dopants and oxygen within one ingot. These lead to differences between individual silicon pieces. Although, the c-Si wafers can be prepared in high purity by the float-zone method, they are very expensive compared to c-Si wafers prepared by the Czochralski method [8], commonly used in electronic industry.

The problems discussed above of IR characterization of thin films on c-Si substrates can be solved quite economically by using double-side polished slightly doped Czochralski silicon wafers, the optical constants of which are determined together with the optical constants of the film during the fitting of experimental data. It requires a suitable parametrization of c-Si optical constants and identification of a reasonably small number of parameters that are set free during the fitting. It was shown that variations of the optical constants of phosphorus-doped c-Si can be described by only three parameters, temperature and concentrations of interstitial oxygen and phosphorus [9]. In this work, the most important IR absorption process in c-Si, i. e. two-phonon absorption [10–14], is discussed and a parametrization of c-Si optical constants in IR is suggested, which is suitable for fitting of the spectrophotometric and ellipsometric data.

2. Two-phonon absorption

The transition strength function of a system at temperature T described by orthogonal initial and final states |i〉, |f〉 with corresponding energies E_i, E_f can be expressed as follows [15, 16]

F (E, T) = \frac{π}{ε_{0} V} {(\frac{e \bar{h}}{m_{e}})}^{2} \sum_{i, f}^{i \neq f} exp (\frac{Ω - E_{i}}{k_{B} T}) \frac{{| 〈 f | {\hat{p}}_{x} | i 〉 |}^{2}}{E_{f} - E_{i}} [δ (E_{f} - E_{i} - E) + δ (E_{i} - E_{f} - E)],

where e, h̄, Ω, ε₀, V, m_e, k_B and T are the elementary charge, the reduced Planck constant, the thermodynamic potential, the vacuum permittivity, the sample volume, the electron mass, the Boltzmann constant and the thermodynamic temperature, respectively. The operator p̂_x is formed by combining total momentum operators of electrons and nuclei, i. e. operators p̂_xe and p̂_xSi in the case of crystalline silicon:

{\hat{p}}_{x} = {\hat{p}}_{x e} - \frac{Z_{Si} m_{e}}{m_{Si}} {\hat{p}}_{x Si},

where Z_Si = 14 is the proton number of silicon and m_Si is the mass of its nucleus [10, 15].

From all the possible photon absorption and emission processes included in Eq. (1), it is possible to separate only those involving the creation or annihilation of two phonons, obtaining the two-phonon transition strength function:

\begin{array}{l} F_{2 ph} (E, T) = \frac{π}{ε_{0} V | E |} {(\frac{e \bar{h}}{m_{e}})}^{2} \sum_{A, B} \sum_{k \in BZ} \sum_{i} exp (\frac{Ω - E_{i}}{k_{B} T}) \\ \times [\sum_{n_{A, k} = 0}^{\infty} \sum_{n_{B, - k} = 0}^{\infty} {| 〈 i, n_{A, k} + 1, n_{B, k} + 1 | {\hat{p}}_{x} | i, n_{A, k}, n_{B, - k} 〉 |}^{2} c_{A, B, k}^{+} δ [E_{A + B} (k) - | E |] \\ - \sum_{n_{A, k} = 1}^{\infty} \sum_{n_{B, - k} = 1} {| 〈 i, n_{A, k} - 1, n_{B, - k} - 1 | {\hat{p}}_{x} | i, n_{A, k}, n_{B, - k} 〉 |}^{2} c_{A, B, k}^{+} δ [E_{A + B} (k) - | E |] \\ + \sum_{n_{A, k} = 0}^{\infty} \sum_{n_{B, k} = 1} {| 〈 i, n_{A, k} + 1, n_{B, k} - 1 | {\hat{p}}_{x} | i, n_{A, k}, n_{B, k} 〉 |}^{2} c_{A, B, k}^{-} δ [E_{A - B} (k) - | E |] \\ - \sum_{n_{A, k} = 1}^{\infty} \sum_{n_{B, k} = 0} {| 〈 i, n_{A, k} - 1, n_{B, k} + 1 | {\hat{p}}_{x} | i, n_{A, k}, n_{B, k} 〉 |}^{2} c_{A, B, k}^{-} δ [E_{A - B} (k) - | E |]], \end{array}

where

c_{A, B, k}^{\pm} = exp (- \frac{n_{A, k} E_{A} (k) + n_{B, \mp k} E_{B} (\mp k)}{k_{B} T}), E_{A \pm B} (k) = E_{A} (k) \pm E_{B} (\mp k) .

The symbol i now represents the initial state of the system, excluding the two phonons (A, k) and (B, ±k) that are written explicitly using their occupation numbers n_A,k and n_B,±k both in the state vectors and summations. Also the energy E_i excludes the energies of these two phonons, i. e. E_A(k), E_B(±k). The summation over the final states is replaced by the summation over all k inside the Brillouin zone and all phonon branches A ≠ B satisfying E_A(k) > E_B(k). The six branches are denoted TA₁, TA₂, LA, LO, TO₁ and TO₂ (see the phonon dispersion relations E_p(k) in Fig. 1). The partition function is now

exp (- \frac{Ω}{k_{B} T}) = \sum_{i} \sum_{n_{A, k} = 0}^{\infty} \sum_{n_{B, \pm k} = 0}^{\infty} exp (- \frac{E_{i}}{k_{B} T}) c_{A, B, k}^{\pm} .

Fig. 1 Phonon dispersion curves of c-Si calculated from the first principles calculation taken from [17]. The points denote phonon frequencies determined using the dispersion model presented here.

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If the momentum matrix elements in Eq. (3) do not depend on the particular initial state |i, n_A,k, n_B,±k〉 but only on the difference in phonon occupation numbers between the final and initial states, the momentum matrix elements can be factored out of the summations over the initial states. This is a basic assumption in the quasiparticle approach:

{| 〈 i, n_{A, k} + 1, n_{B, - k} + 1 | {\hat{p}}_{x} | i, n_{A, k}, n_{B, - k} 〉 |}^{2} \approx (n_{A, k} + 1) (n_{B, - k} + 1) {| 〈 i, 1, 1 | {\hat{p}}_{x} | i, 0, 0 〉 |}^{2}

and

{| 〈 i, n_{A, k} + 1, n_{B, k} - 1 | {\hat{p}}_{x} | i, n_{A, k}, n_{B, k} 〉 |}^{2} \approx (n_{A, k} + 1) (n_{B, k}) {| 〈 i, 1, 0 | {\hat{p}}_{x} | i, 0, 1 〉 |}^{2} .

The summation can then be performed and F_2ph formally written as a sum of two terms

F_{2 ph} (E, T) = \sum_{A, B} [F_{A + B} (E, T) + F_{A - B} (E, T)],

where F_A+B corresponds to processes in which the both phonons are either created or annihilated, whereas F_A−B corresponds to processes in which one phonon is created and one is annihilated. The total number of branch combinations and thus also terms in Eq. (8) is 30. All the terms have the same form:

F_{A \pm B} (E, T) = \sum_{k \in BZ} f_{A \pm B} (k, T) P_{A \pm B} (k) δ [E_{A \pm B} (k) - | E |] .

The factors f_A±B are strongly temperature-dependent because they represent the dependence of the transition probability on the mean phonon occupation number [11, 15]:

f_{A + B} (k, T) = n_{A} (k, T) + n_{B} (- k, T) + 1, f_{A - B} (k, T) = n_{B} (k, T) - n_{A} (k, T),

where the mean occupation numbers n_p are given by the Bose–Einstein statistics:

n_{p} (k, T) = f^{BE} [E_{p} (k)] = \frac{1}{exp [E_{p} (k) / k_{B} T] - 1} .

The functions P_A±B correspond to the intrinsic transition probabilities of the processes and can be expressed as follows:

P_{A + B} (k) = \frac{π}{ε_{0} V | E |} {(\frac{e \bar{h}}{m_{e}})}^{2} {| 〈 i, 1, 1 | {\hat{p}}_{x} | i, 0, 0 〉 |}^{2},

P_{A - B} (k) = \frac{π}{ε_{0} V | E |} {(\frac{e \bar{h}}{m_{e}})}^{2} {| 〈 i, 1, 0 | {\hat{p}}_{x} | i, 0, 1 〉 |}^{2} .

It should be noted that although the fundamental temperature dependence of the two-phonon absorption is given by the factors f_A±B, the entire band structure, i. e. states |i, n_A,k, n_B,±k〉, also changes slightly with temperature, mainly due to thermal expansion. Of course, if a phase transition occurs, the band structure can change dramatically.

3. Construction of the dispersion model

The formulas in the preceding section would be useful for ab initio calculations, however, for the construction of practical dispersion models, it is necessary to make further simplifications. First, the three-dimensional summation of delta functions over the Brillouin zone in Eq. (9) is replaced with one-dimensional functions of photon energy E as follows

F_{A \pm B} (E, T) = f_{A \pm B} (E, T) P_{A \pm B} (E),

where the function f_A±B is a corresponding thermal factor and the function P_A±B represents the corresponding intrinsic part of transition strength function exhibiting Van Hove singularities in critical points [18]. The critical points are denoted M_0,A±B, M_1,A±B, M_2,A±B and M_3,A±B, as usual [13]. If there are multiple critical points of the same type they are indexed as follows:

M_{t}^{(j)}

where t = 0, 1, 2, 3 is the type and j distinguishes individual points.

The second major simplification is, that only 15 of the 30 terms in Eq. (8) are considered: TO+TO, TO+LO, TO+LA, LO+LA, TO+TA⁽¹⁾, TO+TA⁽²⁾, TO+TA⁽³⁾, LO+TA, LA+TA⁽¹⁾, LA+TA⁽²⁾, TA+TA, TO−TA, LO−TA, LA−TA and TO−LA. The TA₁ and TA₂ (TO₁ and TO₂) branches are not distinguished because they are degenerate in all points of high symmetry, i. e. Γ, X and L and also in point W (see Fig. 1). Due to multiple critical points of the same type, it is necessary to use 3 terms for TO+TA branches and 2 terms for LA+TA. The terms are distinguished by the superscripts.

The third simplification lies in the assumption that all critical points correspond to the points of symmetry, i. e. Γ, X, L, W and K. While the points of high symmetry (Γ, X and L) must correspond to critical points, this does not generally hold for low symmetry points (W and K). However, the critical points lie close to W and K, and therefore, this simplification is reasonable. The critical point energies E_t,A±B will be expressed using combinations of phonon frequencies in the corresponding points of symmetry:

E_{t, A \pm B} = h [ν_{A (M_{t})} \pm ν_{B (M_{t})}] [1 + ν_{T} (T - 300)] ν_{iso},

where ν_{A(M_t)} and ν_{B(M_t)} are phonon frequencies in branches A and B. The dependence of phonon frequencies on isotopic composition is implemented in the model using a factor ν_iso which would be equal to unity for pure ²⁸Si isotope. In the case of natural silicon isotopic composition, ν_iso = 0.9982. The identification of critical points M_t for individual terms A ± B, which is crucial for the presented model, will be shown in section 5. Instead of energies, phonon frequencies ν = E/h and wavenumbers ν̃ = ν/c will be be used in the following paragraphs, as is usual in the IR region. The symbols h and c denote the Planck constant and speed of light, respectively. The change of phonon frequencies with varying temperature is described by a linear dependency with the coefficient ν_T that is sufficient for a relatively limited temperature range. A more complex dependence could be necessary for a wider range. Even though there are 30 combinations of branches and points of symmetry, many of them correspond to the same phonon frequencies due to degeneration. Therefore, the following simplified notation will be used:

\begin{array}{l} ν_{LA (Γ)} = ν_{{TA}_{1} (Γ)} = ν_{{TA}_{2} (Γ)} \equiv ν_{A (Γ)} = 0, ν_{LO (Γ)} = ν_{{TO}_{1} (Γ)} = ν_{{TO}_{2} (Γ)} \equiv ν_{O (Γ)}, \\ ν_{{TA}_{1} (X)} = ν_{{TA}_{2} (X)} \equiv ν_{TA (X)}, ν_{LA (X)} = ν_{LO (X)} \equiv ν_{L (X)}, \\ ν_{{TO}_{1} (X)} = ν_{{TO}_{2} (X)} \equiv ν_{TO (X)}, ν_{{TA}_{1} (L)} = ν_{{TA}_{2} (L)} \equiv ν_{TA (L)}, \\ ν_{{TO}_{1} (L)} = ν_{{TO}_{2} (L)} \equiv ν_{TO (L)}, ν_{{TA}_{1} (W)} = ν_{{TA}_{2} (W)} \equiv ν_{TA (W)}, \\ ν_{LA (W)} = ν_{LO (W)} \equiv ν_{L (W)}, ν_{{TO}_{1} (W)} = ν_{{TO}_{2} (W)} \equiv ν_{TO (W)} . \end{array}

The thermal coefficient ν_T, the nine degenerate frequencies ν_O(_Γ₎, ν_TA(X), ν_L(X), ν_TO(X), ν_TA(L), ν_TO(L), ν_TA(W), ν_L(W) and ν_TO(W) together with the eight non-degenerate frequencies ν_LA(L), ν_LO(L), ν_TA₁(K), ν_TA₂(K), ν_LA(K), ν_LO(K), ν_TO₁(K) and ν_TO₂(K) are independent fitting parameters determining 60 critical points E_t,A±B of all terms A ± B.

The intrinsic parts of the transition strength function P_A±B(E) are modeled using piecewise smooth functions with possible discontinuities in the critical points. They are expressed using auxiliary variables defined as follows:

\begin{array}{l} X_{I} (E) = \frac{E - E_{0}}{E_{1} - E_{0}} Π_{E_{0}, E_{1}} (E), & Y_{I} (E) = \frac{E_{1} - E}{E_{1} - E_{0}} Π_{E_{0}, E_{1}} (E), \\ X_{II} (E) = \frac{E - E_{1}}{E_{2} - E_{1}} Π_{E_{1}, E_{2}} (E), & Y_{II} (E) = \frac{E_{2} - E}{E_{2} - E_{1}} Π_{E_{1}, E_{2}} (E), \\ X_{III} (E) = \frac{E - E_{2}}{E_{3} - E_{2}} Π_{E_{2}, E_{3}} (E), & Y_{III} (E) = \frac{E_{3} - E}{E_{3} - E_{2}} Π_{E_{2}, E_{3}} (E), \end{array}

where

Π_{E_{min}, E_{max}} (E) = {\begin{array}{l} 1 & E_{min} < E < E_{max} \\ 0 & otherwise . \end{array}

Note that in Eqs. (17) the symbol E_t,A±B is abbreviated to E_t. The same abbreviated notation will be used for the most following functions and parameters. The piecewise smooth function P_A±B(E) is divided into three regions I, II, and III between the critical points as shown in Fig. 2. In each region, the function is composed of contributions L_i(E) corresponding to the critical points at the region boundaries. The combination of functions representing 3D and 2D Van Hove singularities is a good approximation of anisotropic critical points at the boundaries of the Brillouin zone [19]. The forms of the contributions for individual regions are summarized in Table 1. The complete function P_A±B(E) is a linear combination of all the contributions L_i(E):

P_{A \pm B} (E) = H (E) \sum_{i = 0}^{9} A_{i} L_{i} (E),

where A_i are fitting parameters (A_i ≥ 0).

Fig. 2 A schematic diagram of functions L_i(E) modeling 3D and 2D Van Hove singularities in critical points (Arabic digits denote the contribution i).

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Table 1. Mathematical expression of functions L_i(E) modeling 3D and 2D Van Hove singularities in critical points.

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The function H(E) modifies the shape of P_A±B(E), which was found necessary for precise modeling. It is also piecewise smooth except for the critical points where it is only continuous. The pieces are formed by linear and linear rational functions as follows:

\begin{array}{l} H (E) = κ_{0} Y_{I} (E) + \frac{κ_{1} X_{I} (E)}{(κ_{1} - 1) X_{I} (E) + 1} + \frac{κ_{2} Y_{II} (E)}{(κ_{2} - 1) Y_{II} (E) + 1} \\ + \frac{κ_{3} X_{II} (E)}{(κ_{3} - 1) X_{II} (E) + 1} + \frac{κ_{4} X_{III} (E)}{(κ_{4} - 1) Y_{III} (E) + 1} + κ_{5} X_{III} (E) \end{array}

where κ_j are additional fitting parameters (κ_j > 0). The function H(E) has the following properties:

\begin{matrix} H (E) = κ_{0}, H (E_{1}) = 1, \\ H (E_{2}) = 1, H (E_{3}) = κ_{5}, \\ H (E) = Π_{E_{0}, E_{3}} (E) for κ_{j} = 1 . \end{matrix}

Several examples of possible courses of H(E) are plotted in Fig. 3.

Fig. 3 A schematic diagram of function H(E) modifying the resulting shape of the transition strength function of the individual absorption bands F_A±B(E, T).

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The fundamental temperature dependance follows directly from Eqs. (10)

f_{A + B} (E, T) = 1 + f^{BE} [E_{A} (E)] + f^{BE} [E_{B} (E)]

f_{A - B} (E, T) = f^{BE} [E_{B} (E)] - f^{BE} [E_{A} (E)]

where phonon energies are approximated by linear interpolation:

\begin{matrix} E_{A} (E) = h {ν_{A (M_{0})} Y_{I} (E) + ν_{A (M_{1})} [X_{I} (E) + Y_{II} (E)] + ν_{A (M_{2})} [X_{II} (E) + Y_{III} (E)] \\ + ν_{A (M_{3})} X_{III} (E)} [1 + ν_{T} (T - 300)] ν_{iso} . \end{matrix}

The same linear interpolation is used to approximate E_B(E). This temperature dependence does not require the introduction of any new fitting parameters.

Since it is not possible to normalize the transition strength function of A − B type terms to zero temperature, normalization is done for the room temperature (300 K):

F_{A \pm B}^{0} (E) = \frac{1}{𝒞_{N}} f_{A \pm B} (E, T) P_{A \pm B} (E),

where the normalization constant 𝒞_N is

𝒞_{N} = \int_{E_{0}}^{E_{3}} f_{A \pm B} (E, 300) P_{A \pm B} (E) d E .

The contribution of a two-phonon absorption term to the dielectric function is then obtained by ε-broadening with Gauss–Dawson broadening function β̂:

\hat{β} (x) = - \frac{\sqrt{2}}{π B} D (\frac{x}{\sqrt{2} B}) + i \frac{1}{\sqrt{2 π} B} exp (- \frac{x^{2}}{2 B^{2}}),

where D denotes the Dawson integral [20, 21] and B is the broadening parameter [16] equal to the rms of the Gaussian. Since the Lorentzian and Voigt profiles do not have finite variance, peak widths are usually expressed as FWHM. For Gaussians, these are related to the rms as follows:

β = \frac{2 \sqrt{2 ln 2}}{h c} B .

Therefore β are used as fitting parameters instead of B. The contribution to the normalized dielectric function is then

{\hat{ε}}_{A \pm B}^{0} (E) = \hat{β} * \frac{F_{A \pm B}^{0}}{E},

where symbol * denotes convolution [16].

The TO+TA^(j) terms have the same broadening for all j, therefore, a common normalized TO+TA dielectric function is defined as their linear combination:

{\hat{ε}}_{TO + TA}^{0} (E) = \frac{\sum_{j = 1}^{3} C_{TO + {TA}^{(j)}} {\hat{ε}}_{TO + {TA}^{(j)}}^{0} (E)}{\sum_{j = 1}^{3} C_{TO + {TA}^{(j)}}}

where C_TO+TA^(j) are the fitting parameters determining the strengths of individual terms. The common normalized LA+TA dielectric function is introduced analogously.

The normalization procedure may seem unnecessarily complicated, however, it is necessary to express the strengths of the individual contributions using quantities directly related to the polarizability of the material in the IR region. The contribution to the dielectric function is expressed relative to the total transition strength corresponding to silicon nuclei N_Si [15] in the following manner

{\hat{ε}}_{A \pm B} (E) = α_{A \pm B} N_{Si} {\hat{ε}}_{A \pm B}^{0} .

The quantities α_A±B are the fitting parameters representing the relative transition strengths. The two-phonon polarizability, which is usually described by the effective charge e^*, is then equal to

e_{2 ph}^{*} = Z_{Si} e \sqrt{\sum_{A \pm B} α_{A \pm B}} .

4. Experiment

The analysis of the two-phonon absorption utilized two sets of experimental data:

Room-temperature transmittance measurements on two types of float-zone c-Si samples were performed in the spectral range 70–7500 cm⁻¹ using a Bruker Vertex 80v spectrophotometer with a parallel beam transmittance accessory. The double-side polished c-Si, 580 μm thick, was suitable for the characterization of relatively strong absorption structures of A + B type in the spectral range 450–1000 cm⁻¹. The double-side polished c-Si, 14 mm thick, was suitable for the characterization of weak absorption structures of type A − B and TA+TA in the spectral range 70–450 cm⁻¹.
Ellipsometric measurements at five temperatures from 300–500 K were performed from 300 to 6000 cm⁻¹ at five angles of incidence 55–75° using Woollam IR-VASE ellipsometer equipped with heating stage. The float-zone double-side polished c-Si, 580 μm in thickness, with aluminum-covered back side was used to enhance the influence of weak two-phonon structures in the ellipsometric spectra in the reflected light.

5. Results and discussion

Although the dispersion model contains a number of simplifications, it allowed an independent determination of phonon frequencies in the points of symmetry. The values of these frequencies at room temperature (300 K) determined from the fit of experimental transmittance data are summarized in Table 2. These values can be compared to the theoretically calculated frequencies in Fig. 1, where they are displayed as small filled circles.

Table 2. Phonon frequencies in points of symmetry determined using the optical characterization of c-Si based on the transmittance spectra at 300 K. The values are in THz units and dagger means fixed value.

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The identification of the critical points M_t for individual terms A ± B, crucial for the presented model, is presented in Tables 3 and 4. The tables contain the phonon frequencies as well as the photon wavenumbers that can be used to identify the corresponding features in the experimental spectra. The parameters α_A±B, describing the relative transition strengths, the broadening parameter β_A±B and remaining parameters describing the shape of the transition strength function of individual terms are summarized in Table 5. Due to the normalization introduced in Eq. (25), one of the parameters A_i had to be fixed at an arbitrary non-zero value, therefore, we fixed A₅ = 1. The introduction of the common relative transition strengths of TO+TA and LA+TA terms required similar fixing C_A±B⁽¹⁾ = 1. Furthermore, a number of parameters describing the shape of the transition strength function corresponding to forbidden transitions or weaker absorption structures which correspond to points of lower symmetry were also fixed. For example, κ₀ and κ₅ were set to zero for those critical points M₀ and M₃ that correspond to the Γ point of symmetry. An example of an individual two-phonon absorption contribution is plotted in Fig. 4 for the TO+TA term. To illustrate the temperature dependence, the dielectric function corresponding to this contribution is plotted for 300, 400 and 500 K. The complete two-phonon contribution to the dielectric function is shown for 300 K in Fig. 5, together with an identification of the bands corresponding to the individual two-phonon term A ± B. Figure 5 also contains a fit of experimental transmittance data. It should be noted that the spectra were measured using different detectors and beam splitters in the two regions, 70–700 cm⁻¹ and 400–7500 cm⁻¹. Even with the parallel beam accessory, the spectra did not agree perfectly in the overlapping region, especially for the thick sample. This is caused mainly by the change in the optical path in the case of thick sample. To avoid artifacts at wavenumbers, where one of the spectra ends (400 and 700 cm⁻¹), it was necessary to include this effect into the model using an empirical correction.

Table 3. The identification of the critical points of the two-phonon absorption bands in mid-infrared region (above 450 cm⁻¹).

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Table 4. The identification of the critical points of the two-phonon absorption bands in far-infrared region (below 450 cm⁻¹).

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Table 5. The parameters of the thermal independent part of the dispersion model. The daggers mean fixed values.

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Fig. 4 Spectral dependencies of dielectric function for three different temperatures of TO+TA two-phonon absorption band. The position of critical points are plotted for 300 K.

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Fig. 5 Two-phonon absorption spectra: (a) transmittances of 580 μm and 14 mm thick float-zone silicon samples; (b) full line – modeled dielectric function of c-Si; dotted line –individual bands of two-phonon absorption.

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Ellipsometry in the reflected light is not suitable for measuring weak absorption occurring in the transparent region. For this reason, the back of the silicon sample was covered by a sufficiently thick aluminum film. The film increased the sensitivity of ellipsometry in the reflection configuration to the weak absorption. Another advantage of this back cover was that the back side reflection became well-defined. This would otherwise be difficult to achieve because the heating stage does not have a well-defined surface from the optical point of view. In spite of the improvement in the measurement sensitivity, the ellipsometric data were not as good as the transmittance data. Therefore, the ellipsometric measurements were used only to verify the simplifications of the dispersion model and to determine the thermal coefficient ν_T. This coefficient was the only free parameter of the dispersion model in the ellipsometric data processing.

Its value found by fitting was:

ν_{T} = - 3.4 \times 10^{- 5} (K^{- 1}) .

The comparison of the measured and fitted spectral dependencies of the associated ellipsometric parameter I_c,III are plotted for three selected temperatures and one angle of incidence in Fig. 6. It can be seen that the agreement at 300 K is excellent. At higher temperatures, the agreement is good except for the region 750–850 cm⁻¹ where the experimentally observed quantities change with temperature faster than the model predicts. This is probably caused by a non-negligible contribution of three-phonon absorption. This contribution grows faster with temperature then two-phonon absorption. At present, the model includes three-phonon processes only above 1000 cm⁻¹ [9], where they are dominant. It is evident that they will have to be taken into account also in the region 750–850 cm⁻¹ to describe the thermal effects with higher precision. Note that it was not necessary to introduce any dependence of β on temperature to fit the experimental data in range 300–500 K which indicates that thermal broadening is not the dominant broadening effect. Therefore, the only temperature dependent quantities in the presented model of two-phonon absorption are the critical point energies given by Eq. (15) and thermal factors in Eqs. (22) and (23) given by the Bose–Einstein statistics in Eq. (11). Even though the temperature dependence of these factors is relatively complicated in general, at sufficiently high temperatures it is very close to linear as reported in a previous paper [9].

Fig. 6 Associated ellipsometric parameter I_{c_III} = cos2Ψ illustrating the temperature dependence of two-phonon absorption of float-zone silicon for 70 °.

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6. Conclusion

A dispersion model describing two-phonon absorption was developed and applied to crystalline silicon. The model was based on the simplification of the quasiparticle approach to two-phonon absorption. The dielectric response was constructed from the absorption bands corresponding to individual additive and subtractive combinations of phonon branches. These bands were modeled using a linear combinations of 3D and 2D Van Hove singularities. These approximate the anisotropic 3D critical points corresponding to points of symmetry lying at the boundary of the Brillouin zone. Gauss-Dawson broadening of these bands led to spectral dependencies of optical constants that were in an excellent agreement with experimental transmittance data for the float-zone silicon in the spectral range 70–1000 cm⁻¹. In other words, the model was able to describe all features in the spectra observable at 300 K. Furthermore, this optical characterization provided independently the phonon frequencies in the points of symmetry which were in a good agreement with the ab initio calculations. The model also included a thermal dependence which consisted of two effects. Firstly, it is changes of the transition strength with temperature originating in Bose-Einstein statistics and secondly the shift of phonon frequencies accompanying thermal expansion. The model of the thermal effects was verified using the ellipsometric measurements in the temperature range 300–500 K. The agreement between the modeled and experimental data was good, except the spectral range 750–850 cm⁻¹, in which better agreement at temperatures above 300 K would require inclusion of the three-phonon absorption. The analysis of the temperature-dependent ellipsometric data provided a reliable value of the thermal coefficient describing the phonon frequency shift and proved that changes of structure broadening with temperature were negligible within the range 300–500 K.

Acknowledgments

We would like to thank Adam Dubroka for providing us the thick float-zone c-Si slab. This work was supported by projects “R&D center for low-cost plasma and nanotechnology surface modifications” ( CZ.1.05/2.1.00/03.0086) and “CEITEC – Central European Institute of Technology” ( CZ.1.05/1.1.00/02.0068) from European Regional Development Fund and project TA02010784 of Technological Agency of Czech Republic.

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i	type	region	expression L_i(E)
0	3D M₀	I	$\sqrt{X_{I} (E)}$
1	3D M₁	I	$1 - \sqrt{Y_{I} (E)}$
2	2D M₁	I	−lnY_I(E)
3	2D M₁	II	−lnX_II(E)
4	3D M₁	II	Y_II(E)
5	3D M₂	II	X_II(E)
6	2D M₁	II	−lnY_II(E)
7	2D M₁	III	−lnX_III(E)
8	3D M₂	III	$1 - \sqrt{X_{III} (E)}$
9	3D M₃	III	$\sqrt{Y_{III} (E)}$

	ν_TA₁	ν_TA₂	ν_LA	ν_LO	ν_TO₁	ν_TO₂
Γ		0^†			15.63
X	4.71		11.98		14.09
L	3.14		11.05	12.93	14.99
W	6.39		10.63		14.33
K	4.82	6.70	10.88	11.30	13.90	14.91

Type	Combination	ν_TO (THz)	ν_TO (THz)	ν_TO+TO (THz)	ν̃_TO+TO (cm⁻¹)

M₀	2TO(X)	14.09	14.09	28.18	939.9
M₁	2TO(W)	14.33	14.33	28.66	955.9
M₂	2TO(L)	14.99	14.99	29.98	1000.1
M₃	2O(Γ)	15.63	15.63	31.27	1042.9

Type	Combination	ν_TO (THz)	ν_LO (THz)	ν_TO+LO (THz)	ν̃_TO+LO (cm⁻¹)

M₀	TO(W)+L(W)	14.33	10.63	24.96	832.6
M₁	TO(X)+L(X)	14.09	11.98	26.07	869.7
M₁	TO(L)+LO(L)	14.99	12.93	27.92	931.4
M₃	2O(Γ)	15.63	15.63	31.27	1042.9

Type	Combination	ν_TO (THz)	ν_LA (THz)	ν_TO+LA (THz)	ν̃_TO+LA (cm⁻¹)

M₀	O(Γ)+A(Γ)	15.63	0	15.63	521.5
M₁	TO₁(K)+LA(K)	13.90	10.88	24.78	826.5
M₂	TO(W)+L(W)	14.33	10.63	24.96	832.6
M₃	TO(L)+LA(L)	14.99	11.05	26.05	868.8

Type	Combination	ν_LO (THz)	ν_LA (THz)	ν_LO+LA (THz)	ν̃_LO+LA (cm⁻¹)

M₀	O(Γ)+A(Γ)	15.63	0	15.63	521.5
M₁	2L(W)	10.63	10.63	21.27	709.4
M₂	LO(K)+LA(K)	11.30	10.88	22.18	739.9
M₃	LO(L)+LA(L)	12.93	11.05	23.99	800.1

Type	Combination	ν_TO (THz)	ν_TA (THz)	ν_TO+TA (THz)	ν̃_TO+TA (cm⁻¹)

M₀	O(Γ)+A(Γ)	15.63	0	15.63	521.5
$M_{1}^{(1)}$	TO(L)+TA(L)	14.99	3.14	18.13	604.7
$M_{1}^{(2)}$	TO₁(K)+TA₁(K)	13.90	4.82	18.72	624.5
$M_{2}^{(1)}$	TO(X)+TA(X)	14.09	4.71	18.80	627.0
$M_{1}^{(3)}$	TO₂(K)+TA₁(K)	14.91	4.82	19.73	658.2
$M_{2}^{(3)}$	TO₁(K)+TA₂(K)	13.90	6.70	20.60	687.1
$M_{2}^{(2)}$	TO(W)+TA(W)	14.33	6.39	20.72	691.2
M₃	TO₂(K)+TA₂(K)	14.91	6.70	21.61	720.8

Type	Combination	ν_LO (THz)	ν_TA (THz)	ν_LO+TA (THz)	ν̃_LO+TA (cm⁻¹)

M₀	O(Γ)+A(Γ)	15.63	0	15.63	521.5
M₁	L(X)+TA(X)	11.98	4.71	16.69	556.8
M₂	L(W)+TA(W)	10.63	6.39	17.03	568.0
M₃	LO(K)+TA₂(K)	11.30	6.70	18.00	600.4

Type	Combination	ν_LA (THz)	ν_TA (THz)	ν_LA+TA (THz)	ν̃_LA+TA (cm⁻¹)

M₀	2A(Γ)	0	0	0	0
M₁	LA(L)+TA(L)	11.05	3.14	14.19	473.4
$M_{2}^{(2)}$	LA(K)+TA₁(K)	10.88	4.82	15.70	523.7
$M_{2}^{(1)}$	L(X)+TA(X)	11.98	4.71	16.69	556.8
M₃	LA(K)+TA₂(K)	10.88	6.70	17.58	586.3

Type	Combination	ν_TA (THz)	ν_TA (THz)	ν_TA+TA (THz)	ν̃_TA+TA (cm⁻¹)

M₀	2A(Γ)	0	0	0	0
M₁	2TA(L)	3.14	3.14	6.27	209.2
M₂	TA₂(K)+TA₁(K)	6.70	4.82	11.52	384.3
M₃	2TA(W)	6.39	6.39	12.79	426.5

Type	Combination	ν_TO (THz)	ν_TA (THz)	ν_TO−TA (THz)	ν̃_TO−TA (cm⁻¹)

M₀	TO₁(K)−TA₂(K)	13.90	6.70	7.20	240.3
M₁	TO(X)−TA(X)	14.09	4.71	9.38	312.9
M₂	TO(L)−TA(L)	14.99	3.14	11.85	395.4
M₃	O(Γ)−A(Γ)	15.63	0	15.63	521.5

Type	Combination	ν_LO (THz)	ν_TA (THz)	ν_LO−TA (THz)	ν̃_LO−TA (cm⁻¹)

M₀	L(W)−TA(W)	10.63	6.39	4.24	141.4
M₁	LO(K)−TA₂(K)	11.30	6.70	4.60	153.6
M₂	LO(L)−TA(L)	12.93	3.14	9.79	326.7
M₃	O(Γ)−A(Γ)	15.63	0	15.63	521.5

Type	Combination	ν_LA (THz)	ν_TA (THz)	ν_LA−TA (THz)	ν̃_LA−TA (cm⁻¹)

M₀	A(Γ)−A(Γ)	0	0	0	0
M₁	L(W)−TA(W)	10.63	6.39	4.24	141.4
M₁	L(X)−TA(X)	11.98	4.71	7.28	242.7
M₂	LA(L)−TA(L)	11.05	3.14	7.92	264.1

Type	Combination	ν_TO (THz)	ν_LA (THz)	ν_TO−LA (THz)	ν̃_TO−LA (cm⁻¹)

M₀	TO(X)−L(X)	14.09	11.98	2.11	70.2
M₁	TO(W)−L(W)	14.33	10.63	3.69	123.2
M₂	TO(L)−LA(L)	14.99	11.05	3.94	131.3
M₃	O(Γ)−A(Γ)	15.63	0	15.63	521.5

par.	TO+TO	TO+LO	TO+LA	LO+LA	TO+TA⁽¹⁾	⁽²⁾	⁽³⁾
α	1.24 × 10⁻⁶	4.57 × 10⁻⁶	3.95 × 10⁻⁶	2.51 × 10⁻⁶	7.50 × 10⁻⁶	–	–
β (cm⁻¹)	16.12	14.96	14.59	4.58	6.58	–	–
C	–	–	–	–	1^†	0.0537	0.0846

A₀	1.18	0.817	0.0806	6.94 × 10⁻⁵	0.948	0^†	0^†
A₁	0.344	0.0516	1.45	0.000249	1.49	0^†	0.876
A₂	0.0138	0.422	5.05 × 10⁻⁵	0.0323	2.07 × 10⁻⁵	0^†	0^†
A₃	0^†	0.000367	0.00127	0.0107	6.67 × 10⁻⁵	0^†	0^†
A₄	1.23	1.65	0.00072	0.0832	1.66	1.3	0.846
A₅	1^†	1^†	1^†	1^†	1^†	1^†	1^†
A₆	0.000725	0.0266	0.258	0.000179	0.515	0^†	0^†
A₇	0.144	0.234	0.000271	0.0726	0.029	0^†	0^†
A₈	0.0177	0.000226	0.00174	0.00193	0.119	2.24	1.43
A₉	0.842	0.626	0.321	0.782	0.264	0^†	0^†

κ₀	1.08	1.76	0^†	0^†	0^†	0^†	0^†
κ₁	0.747	0.315	0.312	0.1	0.11	1^†	1^†
κ₂	3.67	3.8	1.24	0.811	3.59	1^†	1^†
κ₃	4.22	2.06	4.97	9.97	9.78	2.84	1^†
κ₄	9.46	0.723	3.66	0.679	0.43	1^†	1^†
κ₅	0^†	0^†	0.809	0.895	5.05 × 10⁻⁵	1^†	1^†

Dispersion model of two-phonon absorption: application to c-Si

Abstract

1. Introduction

2. Two-phonon absorption

3. Construction of the dispersion model

4. Experiment

5. Results and discussion

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (5)

Equations (33)

Optical Materials Express

LO+TA	LA+TA⁽¹⁾	⁽²⁾	TA+TA	TO−TA	LO−TA	LA−TA	TO−LA
6.90 × 10⁻⁷	5.06 × 10⁻⁶	–	1.96 × 10⁻⁷	1.59 × 10⁻⁷	3.94 × 10⁻⁷	1.85 × 10⁻⁷	3.16 × 10⁻⁷
7.98	8.98	–	8.63	6.40	7.30	7.45	11.18
–	1^†	0.0755	–	–	–	–	–

0^†	0.0154	0^†	0^†	4.26	3.64	34.6	8.42
0.0345	0.214	0^†	0^†	20.1	17.5	139	15.7
0.0181	0.0481	0^†	0^†	0.511	26.2	40.1	17.1
0.000201	2.39 × 10⁻⁵	0^†	0^†	6.41	8.74	77.8	16.3
29	0.42	0^†	0^†	1.9	27.3	218	38.2
1^†	1^†	1^†	1^†	1^†	1^†	1^†	1^†
24.7	0.0725	0^†	0.00901	0.329	0.171	0.513	25.1
3	0.024	0^†	1.9 × 10⁻⁵	0.00061	5.84 × 10⁻⁵	21.5	7.28
6.84	0.000503	1^†	0.187	1.36	3.77	0.0417	7.24
60.9	0.533	0^†	0^†	0.00811	0.00146	0.245	0.00539

0^†	0^†	0^†	0^†	1.35	0.186	0^†	0.876
7.82	0.256	1^†	1^†	0.785	0.933	1.74	0.195
4	0.248	1^†	1^†	1.07	2.89	9.89	0.259
7.53	10	1^†	0.183	1.8	9.99	9.88	0.152
0.191	2.08	1^†	9.98	1.1	0.483	0.1	0.378
1.23	0.953	1^†	0^†	0^†	0^†	0.00079	0^†