Investigations of free electrons in doped silicon crystals derived from Fourier transformed infrared measurements and ab initio calculations

1. Faculty of Electronics and Computer Sciences, Koszalin University of Technology, ul. Śniadeckich 2, 75-453 Koszalin, Poland
2. Institute for Electronic Structure Dynamics, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Albert-Einstein-Strasse 15, 12489 Berlin, Germany
3. Department of Chemistry, Humboldt-Universität zu Berlin, Brook-Taylor-Strasse 2, 12489 Berlin, Germany
4. Leibniz-Institut für Kristallzüchtung (IKZ), Max-Born-Strasse 2, 12489 Berlin, Germany
5. Department of General Physics, Lviv Polytechnic National University, 12 Stepan Bandera St., 79013 Lviv, Ukraine

Abstract

The reflection spectra of n- and p-type silicon crystals doped with phosphorus and boron were measured for the free carrier concentrations of 1.1 · 10¹⁵ cm^-3 − 1.2 · 10²⁰ cm^-3 in the far- and mid-infrared range between 20–3000 cm^-1 using synchrotron radiation and Fourier transformed infrared technique. Transmittance spectra could be measured for lower sample carrier concentrations from the range studied. The measured reflection spectra were fitted by using the Drude relation and the parameters of free electron conductivity (electron effective mass m* and momentum scattering time τ) were obtained for the n- and p-typedoped silicon. Additionally, the calculations of the band electronic structure and the electric conductivity σ of the crystals were performed in the framework of the density functional theory for different carrier concentrations and temperatures. The study main findings are (1) the substantial decrease of the momentum scattering time τ and (2) the clear increase of the electron effective mass m* with an increase of the carrier concentrations N_c for both n- and p-type-doped silicon crystals.

Introduction

Doping plays a key role in all electronic devices, as all basic parameters of such devices are affected by dopant distributions. Doping refers to introducing impurity atoms in a controlled way to define the electrical properties in semiconductor materials (usually III and V group species). The donors and acceptors modify the electron and hole concentration from 10¹³ to 10²¹ cm^-3. The carrier concentration can also be spatially distributed to create pn-junctions and produce built-in electric fields. Most semiconductor devices use impurities (dopants) as a crucial ingredient to modify the electronic and optical properties, as well as the device structure [1].

Doping of semiconductors allows the manipulation of their electrical conductivity by exploiting the doping properties of the chemical elements in the main crystal matrix. By introducing a limited number of doping elements into the nominal crystal matrix, the conductivity of semiconductors can be significantly increased. The main challenges in this field are (1) ensuring the ability to control electrical conductivity by controlling the bandgap E_g and (2) ensuring the maximum mobility of free carriers (electrons or holes) [2]. Doping is critical to semiconductor devices, from transistors to thermoelectric energy converters. The interaction between electrons and impurities plays a key role in the transport of carriers. It turns out that high mobility of charge carriers can be achieved by properly selecting dopant elements to obtain high mobility in electronic, photonic, and energy conversion applications [3]. The n- and p-doped silicon crystals are widely used to fabricate different optoelectronic devices. That is why deepening the knowledge of their electronic structure is of interest. The commonly used substitutional doping of different elements, such as carbon, chalcogens, and phosphorus, induces various effects on silicon crystal structural, electronic, and optical properties [4–7].

It is known that the character of the temperature dependence of the electric conductivity (metal-like or semiconductor-like) of an extrinsic semiconductor depends on the concentration of residual carriers. A metal‑like character occurs for relatively high carrier concentrations, while a semiconductor‑like type is observed for relatively low carrier concentrations. This substantiates an approach in which the residual carriers may be divided into two parts: (1) the quasi-free and (2) the weakly bonded carriers. The quasi-free carriers are responsible for the metal-like electric conductivity, while the weakly bonded carriers correspond to the semiconductor-like one. The binding energy (eigenfrequency) of the quasi-free carriers is taken to be close to zero (ω₀₁ = 0), whereas the bonded ones may be characterised by one non-zero binding energy (ω₀₂ ≠ 0). In this case, the Drude-Lorentz model of the frequency dependence of dielectric permittivity ε(ω) for electrons in a doped semiconductor may be presented in the following form:

\( \varepsilon(\omega)=\varepsilon_{\infty}-\frac{\omega_{p 1}^2}{\omega^2+i \gamma_1 \omega}+\frac{\omega_{p 2}^2}{\omega_{02}^2-\omega^2-i \gamma_2 \omega} \) (1)

where ε∞ is the high-frequency dielectric permittivity, ω_p1 and ω_p2 are the plasma frequencies, ω₀₂ is the eigenfrequency of the bonded electrons/holes, and γ₁ and γ₂ are the damping coefficients [8].

Investigations of the optical reflection and transmittance spectra of the pure and doped silicon crystals and associated properties of the quasi-free carriers (electrons and holes) are widely presented in the literature. Among these studies, systematic investigations of the carrier momentum scattering times/damping coefficients and the electron effective mass in the relatively wide range of free carrier concentrations, 10¹⁵ − 10²¹ cm^-1, are not numerous. This is partly caused by a relatively small intensity of the frequently used globar light sources in the far-infrared range of λ > 15 μm. In the present study, the optical reflection and transmittance spectra were obtained using synchrotron radiation, which was much brighter in the farinfrared range of λ > 15 μm than the globar sources. This allowed for more valid reflection spectra of doped silicon crystals with relatively small carrier concentrations, 10¹⁵ − 10¹⁷ cm^-1.

The present study aims to deepen the knowledge of the electron/hole parameters of the doped silicon crystals in the relatively wide range of the free carrier concentrations, 10¹⁵ − 10²¹ cm^-1, by using the reflection/transmittance spectra in the far-infrared range of 20–3000 cm^-1 obtained by using the synchrotron radiation. In this range, electrons are excited from doping states into conduction bands or from valence bands into acceptor states of the semiconductor. The results may be useful for practical applications of silicon crystals in microelectronics and optoelectronics.

Fourier transformed infrared (FTIR) spectroscopy is an accurate tool for precise measurements of the reflection and transmittance spectra. Assuming that reflection and transmittance of the sample obey the Fresnel’s formula, several electron- and phonon-related values of the crystals under study can be obtained, which will be helpful in the following discussion and conclusions

Experimental

The samples used in this study were 15 × 15 × 0.45 mm³. The thin plates were cut from the crystals mostly grown by the floating zone (FZ) technique (see Table 1 for details) and doped during the growth process using Phosphine for n-type and Diboran for p-type crystals. Some crystals under study with higher dopant concentrations were grown using the Czochralski technique (CZ). The surface orientation of all samples was (100). Four probe measurements at room temperature determined the samples resistivity.

The specular reflection of the doped silicon samples was taken at an incidence of 8° close to normal and with an opening angle of about 50 msr. A custom-built motorized reflection set-up for the spectrometer sample compartment was used and spectra were taken with a Vertex 70/v FTIR (Bruker) spectrometer in vacuum in the 20–3000 cm^-1 frequency range. A highly intense and brilliant synchrotron radiation at the IRIS beamline [9] from the BESSY II electron storage ring of the Helmholtz-Zentrum Berlin (HZB) was applied as a source.

According to the Drude model of conductivity, the plasma frequency ω_p = ω_p1 (1) of the doped semiconductors depends strongly on the free-carrier (electron or holes) concentration N_c:

\( \omega_p^2=\frac{N_c e^2}{\varepsilon_0 \varepsilon_{\infty} m^*} \) (2)

where e is the electron charge, ε0 is the dielectric constant, ε∞ is the dielectric permittivity caused by the bonded electrons, m* is the effective mass of the charge carriers in a semiconductor [10]. To see the quantitative values, Table 1 presents a numerical correspondence between the plasma frequency ω_p and the electron concentration N_c, according to (2), taking the free electron mass, m₀, as the effective mass, m* = m₀. Plasma frequency for freeelectron concentration below 1 · 10¹⁵ cm^-3 can reach the lower limit of the spectral range of the FTIR spectrometer and therefore is not experimentally accessible by the spectroscopic method applied in this study.

By fitting the measured reflection spectrum R(ω) of the doped silicon crystals by (1) using the RefFIT software [11], the plasma frequency ω_p1 and the dissipation coefficient γ₁ can be obtained. Afterward, the effective mass of electrons/holes m* in the doped silicon crystals of the known carrier concentration N_c can be calculated from the fitted plasma frequency ω_p using (2) [8, 12].

First-principle calculations

Calculations of the electronic properties of the silicon crystals studied were performed in the framework of the density functional theory (DFT) using the Vienna ab initio simulation package (VASP) code [13–18]. Based on the generalized gradient approximation, the standard Perdew- Burke-Ernzerhof (PBE) exchange-and-correlation function [19] has been used.

To calculate silicon crystals electronic properties, a silicon crystal structure with cubic space group of symmetry Fd-3m (no. 227) was used.

The calculations were performed by VASP using projector-augmented wave PBE (PAW_PBE) pseudopotentials, xc-functional PBE, and a cut-off energy of 500 eV for electron plane waves. One of the calculated values was the dielectric permittivity ε∞ (dielectric permittivity caused by bonding electrons). A 3 × 3 × 3 silicon supercell containing 216 atoms was used to calculate the dielectric permittivity ε∞. The effective electron mass m* and the corresponding electric conductivity of the n- and p-type silicon crystals were calculated within the constant relaxation time approximation using the AMSET software [20] developed based on VASP and BoltzTraP2 codes [13–18, 21]. The authors’ calculations using the AMSET code used eight Si atoms in the unit cell (1 × 1 × 1) of a pure silicon crystal (a = b = c = 5.44 Å).

Results and discussion

The obtained experimental specular reflection spectra of doped silicon crystals are shown in Fig. 1. In these reflection spectra, the authors did not find the features that can be associated with excitations of bonded electrons or phonons. Therefore, the fitting of the experimental reflection spectra was performed using only free electrons, which means using only the first two members in the right part of (1):

\( \varepsilon(\omega)=\varepsilon_{\infty}-\frac{\omega_{p 1}^2}{\omega^2+i \gamma_1 \omega} \) (3)

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In Table 2, the parameters of (2) and (3), ε∞, ω_p, γ, and m*, calculated by the RefFIT package based on the measured reflectance spectra R(1/λ) of the samples, are dependent on the free carrier concentration N_c. A slight decrease of the dielectric constant ε∞ observed with an increase of the free carrier concentration N_c (Table 2) may be caused by the screening of the light wave electric field by free electrons in the doped silicon crystals.

The experimental dependence of the square of plasma frequency ω_p 2 on the carrier concentration N_c reveals the nonlinear character, which can be fitted sufficiently well by the allometric function y = a · xb, with the coefficient b less than unity [b = 0.877, see Fig. 2(a)]. Considering (2) and the fitted coefficient b < 1, one can conclude that the monotonous increase of the effective mass m* occurs at the increase of the carrier concentration N_c.

However, this fitting may not be sufficiently precise in all regions of the wide carrier concentration range of 10¹⁵ − 10²¹ cm^-3. To overcome this drawback, one can (1) use the double logarithmic scales for N_c and ω_p and (2) use a more complicated function containing more than two fitting parameters. Taking into account the latter comments, the fitting of the dependence ω_p(N_c) is realized in the double logarithmic scales using the logistic function containing four fitting parameters, y = A2 + (A1 − A2)/(1 + (x/x0)p) [Fig. 2(b)].

Having obtained the dependence ω_p(N_c) [Fig. 2(b)], the corresponding dependence of the effective electron masses m* on the free carrier concentration N_c is calculated using (2) and presented in Table 2 and Fig. 3. In Fig. 3, the point corresponding to the electron carrier concentration N_c = 5.6 · 10¹⁷ cm^-3 (Table 2) is omitted because of the substantial difference between the experimental and fitted reflection spectra in Fig. 1(a). The latter circumstance could cause a large deviation of the effective mass m* from the fitted linear dependence m*(log(N_c)) in Fig. 3. In the relatively wide range of the carrier concentration of 10¹⁵−10²⁰ cm^-3, the dependence m*(log(N_c)) in Fig. 3 is very close to linear. The second order polynomial fitting of the experimental dependence m*(log(N_c)) (points in Fig. 3) , allows the estimation of the effective electron mass m* of the carriers in doped silicon crystals in a relatively wide range of carrier concentration of 10¹⁵−10²⁰ cm^-3.

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\( m^*=-2.4172+0.1754 \log \left(N_{\mathrm{c}}\right)-5.5101 \cdot 10^{-4}\left(\log \left(N_{\mathrm{c}}\right)\right)^2 \) (4)

The values of the effective electron mass of the doped silicon m* in the carrier concentration range of 1 · 10¹⁸ − 5 · 10²⁰ cm^-3 obtained from the authors’ experimental reflection spectra by using the Drude model (2), m* = 0.7 m₀ (Table 2 and Fig. 3), are larger than the reference data m* = 0.3 m₀ [23] for the corresponding concentration N_c. One of the possible reasons for this effective electron mass difference may be a different quality of the corresponding silicon samples, which can result in different values of the dielectric permittivity ε ∞. However, the latter point is not considered in some cases [24–26]. The authors have performed DFT calculations for pure silicon crystal and heavily phosphorus-doped silicon crystals (N_c = 5 · 10²² cm^-3) corresponding to one phosphorus and 215 silicon atoms in the supercell and they have found the increase of the dielectric permittivity ε∞ from ε∞ = 11.46 for the pure silicon crystal to ε∞ = 13.09 for the phosphorus- doped one. Consequently, the effective electron mass m* calculated using (2) will be more negligible for a more significant value of the dielectric permittivity ε∞. The increased dielectric permittivity ε∞ in the phosphorusdoped silicon crystal is probably caused by donor energy states formed in the forbidden energy gap of the phosphorus-doped silicon crystal. This occurs because, according to the Kramers-Kronig relation, the input to the real part of the dielectric function ε1 is inversely proportional to the energy difference between the ground and exited electron states. Similar DFT calculations of the heavily boron-doped silicon crystals (N_c = 5 · 10²² cm^-3) were performed and a decrease of the dielectric permittivity ε∞ was detected (ε∞ = 10.38) in comparison to the similar value for the pure silicon crystal, ε∞ = 11.46.

In principle, another reason of errors leading to possible inaccuracies in estimating the effective electron mass m* from the authors’ far-infrared studies of doped silicon samples is an inaccurate calculation of the carrier concentration N_c based on the electric conductivity/ resistivity measurements. Measurements of the electric conductivity/resistivity of the samples studied were performed by using the four-terminal sensing [27]. Afterward, the corresponding carrier concentrations N_c were calculated using the known formulae for the borondoped and phosphorus-doped silicon crystals [28]. Thus, there are two sources of the possible inaccuracy in determining the carrier concentration N_c. One derives from the experimental measurements of the samples conductivity/ resistivity and the other from the fixed formulae for calculating the carrier concentration based on the measured sample conductivity/resistivity [28].

To study the likely differences in the electron parameters of n- and p-doped silicon samples, the corresponding experimentally obtained dependences of the plasma frequencies on the carrier concentrations ω_p(N_c) are presented separately in Fig. 4(a). Here, a clear difference between two straight lines corresponding to the linear fitting of the logl-log wp(N_c) dependences for the n- and pdoped silicon samples is observed. Calculated dependences of the effective electron masses on the carrier concentration, m* n(N_c) and m* p(N_c) corresponding to these p(N_c) dependences, are presented in Fig. 4(b). For the carrier concentration of N_c < 2 · 10¹⁷ cm^-3, the effective electron mass is smaller than the effective hole mass, m* n < m* p [Fig. 4(b)]. The latter agrees with the known reference data for the conductivity effective mass in a silicon crystal, m* n = 0.27 and m* p = 0.39 [24]. For the carrier concentration of N_c > 2 · 10¹⁷ cm^-3, the opposite relation between the effective masses of m* n > m* p takes place [Fig. 4(b)]. Such change of the sign of the effective mass difference m* n − m* p with an increase of the carrier concentration, N_c, is the consequence of crossing the straight lines on the dependences log(ω_p) on log(N_c) [Fig. 4(a)]. Taking into account that the DFT calculated dielectric permittivity of the heavily boron-doped silicon crystals, ε∞ = 10.38, is smaller than the similar value for the pure silicon, ε∞ = 11.46, the curve of the dependence m* p(N_c) in Fig. 4(b) will be shifted a bit to the larger magnitudes of the effective mass, m* p. Thus, one may expect the relation of m* n > m* p to take place for the carrier concentration a bit larger than the above-mentioned one N_c > 2 · 10¹⁷ cm^-3.

A clear increase of the damping coefficient γ with an increase in the free carrier concentration N_c observed (Table 2) is caused by a corresponding increase in the free electron-electron interaction/scattering. The observed inverse dependence of logarithm of the momentum scattering time log(τ) (τ = 1/(γc), where c is the light velocity on the logarithm of carrier concentration log(N_c) (Fig. 5), is a result of this interaction. Therefore, the free electron/hole momentum scattering time τ decreases by two orders of magnitude when the free carrier concentration N_c of the doped silicon crystals increases by four orders of magnitude (Fig. 5). For relatively low free carrier concentrations N_c < 10¹⁷ cm^-3, the momentum scattering times of free electrons are a bit larger than the momentum scattering times of the holes (Fig. 5).

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According to the theory of electron conductivity in metals, the direct proportionality between the conductivity σ and the carrier concentration N_c takes place if the momentum scattering time τ and effective mass m* are constant:

\( \sigma=\frac{1}{\rho}=\frac{e^2 \tau N_{\mathrm{c}}}{m^*} \) (5)

This, however, is not realized because a clear nonlinearity of the dependences σn(N_c (n)) and σn(Np (p)) is observed: the relation τ/m* decreases with an increase of N_c [Fig. 6(a)]. This agrees with a relatively strong decrease of the momentum scattering time τ (Fig. 5) and a moderate increase of the effective mass m* [Figs. 3(a), 4(b)]. Thus, the main reason for the observed nonlinearity of the dependences σn(N_c (n)) and σn(N_c (p)) is a decrease of the momentum scattering time τ with an increase in the carrier concentration N_c. The same conclusion can be derived from comparing the experimental and the theoretical dependences σ(N_c) presented in log-log scales [Fig. 6(b)].