GENERATION-RECOMBINATION NOISE AND OTHER FEATURES OF DOPED SILICON IN A WIDE TEMPERATURE RANGE

1. Introduction

Although the processes of free charge carrier generation (emission) and recombination (capture) have been investigated over the last seventy years [1–3], many experiments have been performed with various semiconductor materials [4–7] and devices [8–15], and explanations are provided. During the  generation-recombination (g-r) process, the free charge carrier number fluctuations keep the  charge neutrality in the  total sample. This term is usually used for describing the free charge carrier number fluctuations caused by donor or acceptor levels. The  free charge carrier number fluctuations in equilibrium also produce fluctuations in the resistance of the sample. These resistance fluctuations can be simply measured by using direct current because they produce the  flowing current fluctuations which are proportional to the  square of the  magnitude of the direct current. In the case of the deep defect levels, the  generation-recombination process is usually described as a charge carrier emission and capture, or retrapping process. In the latter case, the free charge carrier fluctuations are caused by localized defect states with different energy levels in the band gap and are characterized by various free carrier relaxation times [16]. The free charge carrier emission-capture process is also thermally activated and does not change the neutrality condition in the sample. According to Refs. [6, 17], the relaxation times of the g-r process for silicon monocrystals are distributed in the time interval from 0.1 s to 10 μs. In Ref. [18], it is demonstrated that the charge carrier capture-emission process is the main source of the low-frequency noise in homogeneous semiconductors and this process can produce the free charge carrier mobility fluctuations in particular cases [19].

In this work, important properties of the generation-recombination process in a  silicon sample doped with shallow donors will be presented. The main focus here will be on the temperature dependence of the generation-recombination noise.

2. Temperature dependence of the generation-recombination noise

2.1. Silicon sample with one partially ionized donor level

Due to the effect of the thermal lattice vibration in donor-type semiconductors some electrons are excited from donor energy levels into the conduction band (generation process), and at the  same time there happens the reverse process (recombination): some electrons are captured by ionized donors. To estimate the characteristics of the g-r noise, it is sufficient to find the  generation g(n) and recombination r(n) rate expressions from the physical model.

Consider the  g-r noise temperature properties for a  donor-type silicon sample with the  volume V = 0.01 cm³ and the donor density n_d = 10¹⁵ cm^–3 at the energy level E_d = 0.1 eV below the conduction band (Fig. 1). In the general case, the average free electron density n₀ is described by the following equation [20]:

where ε_d = E_d/kT, η = E_F/kT (here E_F is the Fermi level energy), and n⁰_d is the neutral donor density, for donors β^–1 = 2.

For silicon $N_{\text{c}}\text{ = 2}{\text{.746·10}}^{\text{19}}{\text{(}T\text{/295)}}^{\text{3/2}}{\text{ cm}}^{\text{-3}\text{.}}$

Dependences of the free electron density n₀ and the Fermi energy E_F on temperature for silicon with n_d = 10¹⁵ cm^–3 at the energy level E_d = 0.1 eV are presented in Fig. 2. When the Fermi energy level coincides with the donor energy level, the free electron density in the conduction band is n₀ = n_d/3. As Fig. 2 shows, all donors are ionized at temperature T > 200 K. In the very low temperature range, Eq. (1) can be approximated as

The rate of free electron generation g(n) is proportional to the non-ionized donor density n_d – n_d⁺, i.e. to the density of electrons in the donor energy level E_d:

Here a is the  proportionality coefficient, and n = n₀ + ∆n is a fluctuating quantity. And the rate of electron recombination r(n) is proportional to the density of the free electrons and to the empty (ionized) donor centre density $n_{\text{d}}^{+}$  = n:

Here γ is the recombination coefficient: γ = v_Tσ_s, v_T is the electron thermal velocity, and σ_s is the electron capture cross-section. The coefficient a can be found from the equilibrium condition g(n₀) = r(n₀):

The relaxation time of free electrons is estimated in the following way [2],

and the variance of the free electron density fluctuations is calculated as

where V = 0.01 cm³. Including the fact that relaxation time is approximately inversely proportional to n₀, and γ ~ T^1/2, dependences of the relaxation time τ₀ and τ₀ multiplied by a factor (T/295)^5/4 on reciprocal temperature (1000/T) are depicted in Fig. 3. When 1000/T > 7 (or T < 40 K), the relaxation time exponentially increases with temperature.

Temperature dependences of the  variance of free electron density fluctuations and the  Fermi energy for silicon with the donor impurity density 10¹⁵ cm⁻³ at E_d = 0.1 eV are represented in Fig. 4(a). The maximum of the free carrier fluctuation variance is observed at the  temperature T₁  =  136  K when E_F/E_d  =  E₁/E_d  ≈  1.1. However, E_F  =  E_d at the temperature T = T_F = 116.5 K, i.e. the maximum of the fluctuation variance appears when the Fermi level energy is below the donor level energy.

From Eq. (9) it follows that the relative variance of free electron number N fluctuations in the silicon sample with the number of donors N_d can be expressed as

This relation 〈∆N²〉/N₀ = 0.5 when N₀ ≪ N_d. The dependence of the relative variance of free electron number fluctuations on temperature is shown in Fig.  4(b), and it clearly illustrates that 〈∆N²〉/N₀ = 0.5 when the Fermi level energy is over the donor level energy. For random particles with Pois-son’s law of distribution, this ratio is equal to one. The  smaller ratio for the  electrons in the  donor silicon is due to the degeneracy factor β = 1/2. For a silicon sample with one donor level E_d = 0.1 eV at temperature T < 120 K, the Fermi level energy is always over the donor level energy (Fig. 2). There the  free electron density exponentially decreases with temperature decreasing (Fig. 2), while the relaxation time exponentially increases (Fig. 3). In this temperature range, the relative variance of free electron number fluctuations 〈ΔN²〉/N₀ = 0.5. The maximum of the variance of free electron density fluctuations 〈Δn²〉 occurs when the Fermi level energy is about 10% below the donor level energy (Fig. 4(a)).

For the g-r noise spectrum calculation, it is convenient to present the results as a normalized power spectral density (multiplied by frequency):

The frequency dependences of the normalized power spectral density of free electron number fluctuations for a silicon sample at different temperatures are presented in Fig. 5.

The maximum of the normalized spectral density of free electron number fluctuations decreases and shifts to higher frequencies with increasing temperature (Fig. 5). The peak values are observed at frequencies f₀ = 1/(2πτ₀). By presenting the measurement results in this way, it is very easy to determine the relaxation time τ₀.

2.2. G-r properties in silicon with various shallow donor densities

At high shallow donor densities, donor ions interact with each other, causing their wave functions to overlap and the donor energy level to split [21]. This results in a decreased ionization energy and the formation of the band tail at the bottom of the conduction band rather than a well-defined band edge. Taking into account that the donor atoms replace the silicon atoms in the lattice [22], the donor levels cannot be considered as additional levels to the levels of the conduction band. These donor levels partly overlap with the energy levels in the conduction band, and when E_d = 0, the donor energy states completely overlap the conduction band, and all valence electrons are in the conduction band. The decrease of the shallow donor level energy (E_d0 = 0.05 eV) with the increase in donor density for the silicon sample is presented in Fig. 6(a) and can be described as [20]

Temperature dependences of the  free electron density and the Fermi level energy at three different shallow donor densities, 10¹³, 10¹⁵ and 10¹⁷  cm⁻³, in silicon are presented in Fig.  6(b). The dots A₁, A₂ and A₃ represent the free electron densities at temperatures T_F1, T_F2 and T_F3, respectively, when the Fermi level energy coincides with the donor level energy and Eq. (12) is taken into account. Similarly, the dots B₁, B₂ and B₃ denote the Fermi level energies when they coincide with the donor level energies and Eq. (12) is also taken into account. Therefore, it is clear that the relation n₀ = (1/3) n_d is perfectly fulfilled for various presented donor densities when the Fermi level energy coincides with the donor level energy.

Temperature dependences of the relative free electron density n₀/n_d for three different donor densities are shown in Fig.  7(a). Usually, it is considered that all donors with a shallow energy level of 0.05  eV are completely ionized at temperatures above liquid nitrogen. As Fig. 7(a) indicates, the requirement for 95% of donors to be ionized at temperatures above liquid nitrogen is only fulfilled at low (≤10¹³ cm^–3) donor densities. This ionization level for donor density of 10¹⁵ cm^–3 is only achieved at temperatures above 114 K, and for 10¹⁷ cm^–3 only at temperatures above 252 K. Thus, the  statement that all shallow donors are ionized at temperatures above liquid nitrogen should be used with extreme caution. What determines that not all donors are ionized at higher shallow donor density? The  higher the  shallow donor density, the  lower the  relative free electron density (Fig. 7)(a): only 20% of donors are ionized at the  temperature of liquid nitrogen when donor density is 10¹⁷  cm^–3. The  dependence of the  conduction band states filling func tion g(E) f (E) on energy at T = 100 K is presented in Fig. 7(b). At low donor densities, electrons fill energy states near the bottom of the conduction band. However, for higher donor densities, some electrons must be excited to the higher energy levels of the conduction band because the energy levels near the bottom of the conduction band are occupied.

Temperature dependence of the  free electron density for high shallow donor densities in silicon, which is estimated by Eq. (1), is depicted in Fig. 8(a). The Hall effect measurement data show that all donors are ionized when donor density is 10¹⁹ cm^–3 [20]. Thus, Eq. (1) incorrectly describes the free electron density at high donor densities in silicon. Then, the expression for the free electron density can be estimated as

where the probability multiplier [1 – exp(–ε_d)] shows that at E_d = 0, i.e. the donor energy states overlap with the conduction band levels. Considering that every donor changes the silicon atom (substitutional impurity) [22] and that donor levels are formed as decoupled levels from the conduction band, the total density of states due to doping changes weakly [20]. It is confirmed by electronic heat measurements of silicon with a high donor doping [23].

The temperature dependence of the  free electron density for high donor densities estimated by Eq. (13) is presented in Fig. 8(b) and it is in good agreement with the experimental data [24]. Equation (13) is valid for any donor density with a low or high donor level energy. Equations (1) and (13) completely coincide at shallow donor densities ≤10¹⁷ cm^–3. To evaluate the density of the free electrons for any donor density, it is convenient to have an analogous expression for the function F_1/2(η),

where the parameter

Dependences of the functions exp(η), F^1/2(η) and α on η are presented in Fig. 9(a). The difference between the  approximation curve (Eq.  (15)) and the  ratio exp(η)/F_1/2(η) does not exceed 5% in the whole η range (η ≤ 15). At η values below –3, the ratio exp(η)/F_1/2(η) = 1. Temperature depend-ences of the  free electron density calculated by Eq. (13) for five different donor densities are presented in Fig. 9(b). The obtained data are in good agreement with the experimental Hall effect measurement results [20]. The Fermi level energy for all donor densities has negative values, but for n_d = 10¹⁹ cm^–3 it becomes positive at temperatures below 150 K.

Temperature dependences of the  variance of electron density fluctuations and the relative variance of free electron number fluctuations 〈∆N²〉/N₀ at three different densities of shallow donors are depicted in Fig. 10. The temperature dependence of the variance of free electron density fluctuations is characterized by a  maximum, which strongly increases (about four orders of magnitude) with increasing donor density and shifts in the temperature range from 50 to 125 K (Fig. 10(a)). The relative variance of free electron number fluctuations 〈∆N²〉/N₀ at low temperatures is equal to 0.5, and this value does not depend on the  donor density but decreases strongly at higher temperatures (Fig. 10(b)). It can be pointed out that the value 0.5 reflects a situation where the free electron fluctuations obey Poisson’s law of distribution.

2.3. Temperature properties of the generation-recombination noise in silicon with two donor levels in the presence of acceptors

A donor doped silicon sample of volume V  =  0.01  cm³ with the  shallow donor density n_1d = 2·10¹³ cm^–3 at the energy level E_1d = 0.05 eV and with the deep donor density n_2d = 3·10¹⁴ cm⁻³ at the level E_2d = 0.25 eV is partially compensated by acceptors with the  density n_a  =  1.5⋅10¹³  cm^–3 (Fig. 11).

In the general case, the electron generation and recombination from the shallow donor level E_1d in the silicon partially compensated by acceptors can be described as

where n⁰_1d is additionally decreased due to the capture of electrons by acceptors; the parameter a₁ can be found from the equilibrium condition g₁(n₀) = r₁(n₀):

Then, the electron relaxation time of the g-r process between the shallow donor level and conduction band can be estimated as

and the free electron variance produced by the shallow donors is

The electron generation and recombination from the deep donor level E_2d can be described in a similar way:

Considering that the effective density of donors n1d decreases by the value na due to the number of electrons captured by acceptors, the average free electron density can be calculated by the following expression:

Temperature dependences of the average density of free electrons are shown in Fig. 12(a). Dependences of the relaxation times from the donor levels E_1d and E_2d on the reciprocal temperature (1000/T) are presented in Fig. 12(b). The relaxation time from the shallow donor level exponentially increases when temperature T < 50 K (or 1000/T > 20). The excitation of free electrons from the deep donor level E_2d is noticeable only at T > 100 K (or at 1000/T < 10). Due to the compensation by acceptors, the free electron density decreases by the amount of the acceptor density (plateau region in Fig. 12(a)). However, as Fig. 12(b) indicates, the relaxation time of free electrons experiences small changes due to the compensation by acceptors.

Temperature dependences of the total variance of free electron density fluctuations 〈∆n²〉 (left scale) and the Fermi level energy (right scale) are represented in Fig. 13(a) (solid line is for the silicon without the  compensation by acceptors; dashed line is for the silicon in the presence of acceptors with density n_a = 1.5⋅10¹³ cm^–3). The solid line branch of 〈∆n²〉 below T = 75 K is caused by the shallow donor level E_1d, and when T > 75 K, the  branch of 〈∆n²〉 temperature dependence is due to the deep donor level E_2d. The minimum in the  temperature dependence of 〈∆n²〉 occurs at T  ≈  120  K. The  compensation by acceptors significantly shifts the  minimum to T  ≈  50  K, and the  magnitude of this value decreases by about five orders (Fig. 13(a)). Temperature dependences of the relative variance of the free electron number fluctuations 〈∆N²〉/N₀ are shown in Fig. 13(b) (the meaning of solid and dashed lines is the same as in Fig. 13(a)). The solid line branch of 〈∆N²〉/N₀ at T < 120 K is caused by the fluctuations of the free electron number from the shallow donor level, and the branch of 〈∆N²〉/N₀ at T > 120 K is caused by the fluctuations of the free electron number from the  deep donor level. The  temperature dependence of 〈∆N²〉/N₀ has a minimum at T ≈ 125 K for the silicon without acceptors. The compensation by acceptors also noticeably shifts this minimum to the lower temperature T ≈ 65 K, and the magnitude of the minimum value decreases by about five orders. When the Fermi level energy exceeds E_1d, then 〈∆N²〉/N₀ ≈ 0.5 for the non-compensated silicon as in the  case of silicon with one donor level.

The normalized spectral density of free electron number fluctuations in this case can be described as

The calculation results by Eq. (27) for the silicon without the  compensation by acceptors are presented in Figs. 14–15. The noise maximum in Fig. 14(a) abruptly decreases with temperature and shifts to the higher frequencies due to the electron number fluctuations in the shallow donor level E_1d. The same tendency has been observed in the silicon sample with one donor level (Fig. 5). However, changes in the temperature dependence of the nor-malized spectra of free electron number fluctuations caused by the donor level E_2d (Fig. 14(b)) are different: in the  temperature range from 120 to 200  K, the  noise maximum increases and slowly shifts to the higher frequencies with temperature, and then, at higher temperatures, the  maximum decreases in the same way as in Fig. 14(a). The g-r noise from the shallow donor level prevails at temperatures T < 100 K, while at T > 140 K, the total fluctuations are caused by the donor level with energy E_2d = 0.25 eV.

The comparison of the  data presented in Fig. 14(a, b) reveals that in the latter case the position of the maximum of the normalized resistance fluctuation spectrum very weakly depends on the  temperature in the  range from 140 to 200  K. The  normalized spectra of free electron number fluctuations due to both donor levels when the  contribution to noise spectra of both donor levels is of the same order in the intermediate temperature range (100−150) K are presented in Fig.  15. The  first peak of the  noise spectrum is observed in the frequency interval from 10 to 100 kHz and weakly depends on the temperature in the intermediate temperature range. However, the  position of the  second noise peak changes from 1 to 100  MHz in the  temperature interval from 80 to 140 K.

Consider the  noise spectra of the  same sample including the  compensation by acceptors. The spectral density of the g-r noise in this case can be described as

The calculation results obtained by Eq. (28) are presented in Fig. 16. The  dashed curves are due to the  free electron number fluctuations from the  shallow donor level E_1d. The  dashed arrows show the  changes of the  g-r noise spectra when temperature increases. The solid curves represent the noise spectra caused by the free electron number fluctuations from the  deep donor level E_2d. The  g-r noise spectra due to the  deep level donors always exceed the g-r noise level caused by the shallow donors at temperatures T > 100 K.

The results of the normalized g-r noise spectra calculated by Eq. (27) are presented in Fig. 17.

The comparison of the  calculation results for silicon samples with the same donor densities, but in one case without the compensation by acceptors (Fig. 14) and in another with partial compensation by acceptors, shows that the noise spectra presented in Fig. 17(a) and caused by the shallow donors have noticeable changes due to the influence of acceptors and free electrons from the deep donor level at temperatures T > 80 K. The noise spectra caused by the  deep donors (Fig. 17(b)) have almost the same pattern as in Fig. 14(b).

3. Conclusions

The generation-recombination processes in silicon have been analyzed in the temperature range from 20 to 360 K. It is demonstrated that the electron density strongly depends on temperature: the  shallow donors with a  density smaller than 10¹³ cm^–3 are completely ionized at liquid nitrogen and higher temperatures, while only 20% of donors are ionized when the  density of shallow donors is about 10¹⁷ cm^–3 at liquid nitrogen temperature. The  electrons fill the  energy states near the  bottom of the conduction band when donor density is low, but for the higher donor densities, some electrons must be excited to the higher energy levels of the conduction band, because the energy levels near the  bottom of the  conduction band are occupied, and the higher temperature is needed for excitation. It is also shown that in the transient region to the degenerate state, the standard Eq. (1) does not fully accurately describe the free electron density dependence on the shallow donor density. The revised formula (Eq. (13)), which is valid at any shallow donor density, is presented.

The maximum of the variance of free electron density fluctuations for a silicon sample with shallow donors strongly increases (about four orders of magnitude) with increasing donor density and shifts in the temperature range from 50 to 125 K. The  relative variance of free electron number fluctuations (〈∆N²〉/N₀) is always equal to 0.5 at low temperatures and reflects the  situation when the free electron fluctuations obey the Poisson’s law.

The normalized spectra of the  generation-recombination noise in silicon with shallow and deep donor levels are presented in the frequency interval from 1 to 10¹⁰ Hz and the temperature range from 25 to 360 K. The comparison between the g-r noise spectra in silicon with and without the partial compensation by acceptors is also presented.

References

V. Palenskis et al. / Lith. J. Phys. 62, 148–160 (2022)

GENERACINIS-REKOMBINACINIS TRIUKŠMAS IR KITOS LEGIRUOTO SILICIO YPATYBĖS PLAČIAME TEMPERATŪROS INTERVALE

V. Palenskis, J. Glemža, J. Matukas

Vilniaus universiteto Taikomosios elektrodinamikos ir telekomunikacijų institutas, Vilnius, Lietuva

Santrauka

Silicio generacinio-rekombinacinio triukšmo charakteristikos yra ištirtos temperatūros intervale nuo 25 iki 360  K. Ypač didelis dėmesys skirtas siliciui su sekliomis donorinėmis priemaišomis, esant labai plačiam donorų tankiui. Parodyta, kad laisvųjų elektronų tankis stipriai priklauso nuo temperatūros net sekliųjų donorų (donorinio lygmens energija lygi 0,05 eV) atveju. Seklieji donorai, esant jų tankiui mažesniam nei 10¹³ cm^–3, skysto azoto temperatūroje yra visiškai jonizuoti, o sekliųjų donorų, kurių tankis yra apie 10¹⁷ cm^–3, tik apie 20 % yra jonizuoti. Esant mažam sekliųjų donorų tankiui, laisvieji elektronai užpildo energijos lygmenis, esančius arti laidumo juostos dugno. Didėjant donorų tankiui, dalis elektronų turi būti sužadinami į aukštesnius laidumo juostos energijos lygmenis, todėl jiems sužadinti reikalinga aukštesnė temperatūra. Taip pat parodyta, kad pereinamojoje link išsigimusių elektroninių dujų srityje standartinis sąryšis (1) ne visai tiksliai parodo laidumo juostos elektronų tankio priklausomybę nuo sekliųjų donorų tankio. Pateikta patikslinta išraiška (13), kuri galioja esant bet kokiam sekliųjų donorų tankiui.

Minėtos sekliųjų donorų ypatybės pasireiškia ir generacinio-rekombinacinio triukšmo charakteristikose. Generacinio-rekombinacinio triukšmo dispersijos maksimumas dėl laisvųjų elektronų tankio fliuktuacijų silicyje, kylant sekliųjų donorų tankiui nuo 10¹³ iki 10¹⁷ cm^–3, stipriai didėja (apie 4 eiles), ir šis maksimumas pasislenka temperatūros skalėje nuo 50 iki 125 K. Parodyta, kad laisvųjų elektronų skaičiaus N fliuktuacijų dispersija (〈∆N²〉/N₀) žemoje temperatūroje visada lygi 0,5, ir tai rodo, kad žemoje temperatūroje laisvųjų elektronų skaičiaus fliuktuacijos apibūdinamos Puasono (Poisson) tikimybių pasiskirstymo dėsniu.

Normuoto generacinio-rekombinacinio triukšmo spekt rų [S_N(f)/N ₀² × f] dažninės priklausomybės yra pateiktos in ter vale nuo 1 iki 10¹⁰ Hz temperatūros srityje nuo 25 iki 360 K. Taip pat pateiktos silicio generacinio-rekombinacinio triukšmo charakteristikos, esant dviem donoriniams lygmenims, įskaitant kompensavimą akceptoriais.