Effect of phosphorus dopant concentration on the carrier mobility in crystalline silicon

This study investigated the effect of phosphorus dopant concentration on mobility of crystalline silicon (c-Si). It considers different temperature ranges, from 100 K to 500 K, and dopant concentration from 1012 cm-3 to 1020 cm-3 in relation to its effect on the mobility of the crystalline silicon. This study indicates that the mobility of phosphorus doped silicon, n-type silicon, at different dopant concentration, tends to reduce as the temperature is increased. On the other hand, the mobility of the doped semiconductor, at different temperatures, showed different trends as the dopant concentration increases: I) mobility decreased in between 1015 to 1017 cm-3, II) mobility saturates doping concentrations less than 1014 cm-3, and III) mobility is not significantly affected, by increasing the temperature for high doping concentration 1018 to 1020 cm-3. The two issues, lattice and impurity, dominate one another depending on the doping concentration and temperature, and thus contributed to dependence of mobility on temperature, in different trend, while being dependent on the fundamental theory of doping in semiconductors. Based on the study, as the temperature gets higher for higher doping concentration, mobility by the impurity scattering increases while it decreases by the lattice scattering, the two cases balance one another, and as a result mobility becomes almost constant, that is, the rate of change of mobility is relatively insignificant.


INTRODUCTION
Silicon is by far the dominant semiconductor material used in electronics and photonic devices.
Since the birth of semiconductor industry, it has been the key semiconductor material, and it is seen as the backbone of electronics and photovoltaic industry (Lukasiak and Jakubowski, 2010).
Its application in electronic devices, mainly in transistor, as well as in photonic and photovoltaic materials has entirely revolutionized our life style (Seto, 1985;Fourmond et al., 2011). In relation to its application, the electronic property is deeply investigated in many studies, where many of the studies are based on doping. The studies has made significant progress in detailed understanding of the material and played a key role in advanced device application. The studies in electronic property of the material are mainly based on pand n-type doped crystalline silicon, and it mainly focuses on conductivity/resistivity, mobility and related parameters. The mobility can refer the majority and/or the minority carrier mobility, that is, electron and hole mobility (Cardona, 2010;2011). The carrier mobility is affected by scattering, and theoretically the scattering can be of the following type: i) phonon (lattice) scattering, ii) ionized impurity scattering, iii) scattering by neutral impurity atoms and defects, iv) carriercarrier scattering, and (v) piezoelectric scattering (Pierret, 2003). Of these, the charge carrier mobility is dominantly affected by the lattice scattering and impurity scattering (Bulusu and *Corresponding author: amarebenor@yahoo.com Walker, 2008 Apparently, a higher mobility results in better device performance (Chan,1994;Yacobi, 2003;Doering and Nishi, 2014). As a result, mobility is particularly a key factor in the performance of electronic devices (Watanabe, 1999 (Arora et al., 1982) which takes into account the anisotropic scattering effects and the impurity scattering mobility, at a given temperature T, is given by: (1) where N I is the number of ionized impurity atoms and G(b) is a function given by: (2) where b is given by: where n ' = n [2-(n/N)] and it assumes the acceptor concentration to be zero, n being electron density per cubic centimeter. Besides, a detailed review work was made on the charge transport properties of silicon (Jacoboni et al., 1977). Latter, Arora et al. (1982) derived analytical relation on the electron and hole mobility in silicon as a function of concentration and temperature. In his study, he noted that the lattice scattering mobility can be fitted very well by the following equation: (4) Thus, after taking into account both effects m L and m I , the total charge carrier mobility can be calculated. These former studies made significant contribution to the charge carrier transport. However, the very depth insight of charge transport is still somehow not well explained and theoretical modeling has been improved in time so as to match with the experimental results, as an example, Klassen mobility modeling. In particular, this modeling is not yet fully used in detailed study of carrier mobility of phosphorus doped silicon. Thus, the effect of dopant concentration, to make p-or n-type Si, on the mobility of crystalline silicon is one key issue; and the case with a phosphorus dopant, Klassen mobility modeling, is our scientific issue. Consequently, we focus our attention on the characteristics of semiconductor materials that can be altered significantly by the addition of the impurity or doping into the crystalline silicon, which is an n-type c-Si doped by phosphorus. In other words, the study intends on extended insight of the effect of phosphorus dopant concentration on the carrier mobility of crystalline silicon. The study will see the effect of temperature and doping concentration on the mobility of the n-type semiconductor. Here, the study will consider the combined effect of both temperature and doping concentration and related trend in the electronic property of the material.
Additionally, the two important related factors affecting the mobility, lattice and impurity scattering, will be discussed in detail in relation to the temperature and doping concentration. The study gen-erally indicates that lattice and impurity scattering dominate one another, in different rends, depending on the doping concentration and the temperature; and such a trend is well investigated in this study.  Figure 1, shows the electron and hole mobility as a function of dopant concentration, with a range of 10 12 cm -3 to 10 20 cm -3 , for phosphorus doped crystalline silicon at room temperature or 300K. As seen from the figure, the charge carrier mobility of holes and electrons tends to decline as the doping concentration increases. Besides, the charge carrier motilities, of electrons and holes, tend to be constant below 10 15 cm -3 and above 10 19 cm -3 dopant concentrations. Furthermore, it is evident that the carrier mobility of electrons is higher than that of holes. In intrinsic crystalline semiconductor, e.g., crystalline silicon, the only factor that affects mobility is the temperature or phonon effect, where the mobility decline as the temperature rises. However, in extrinsic semiconductor, like phosphorus doped semiconductor c-Si, there are two factors or two contributions affecting the charge carrier mobility, namely impurity scattering and lattice scattering (Beadle et al., 1995).

RESULTS AND DISCUSSION
Thus, Phosphorus doped semiconductor c-Si, mobility is affected by these two factors: lattice and impurity scattering. As shown in Figure 1, the charge carrier mobility of electrons exceeds that of holes is due to the fact that the effective mass, in c-Si, of electrons is less than that of holes.
Besides, the decline of charge carrier mobility, by increasing the doping concentration, appears to be related to the increased impurity scattering where it induce more carrier or electron-electron interaction and tends to reduce the charge carrier mobility. The mobility of electrons is about three times greater than that of holes for low doping concentration, e.g., 10 16 cm -3 , while it is less than two fold at higher concentration after 10 19 cm -3 . For low dopant concentrations, less than 10 16 cm -3 , the electron mobility declines by temperature or phonon/lattice scattering, and the lattice scattering has dominant effect particularly at low temperature.
The relatively high carrier mobility, in the first regime, or concentrations less than 10 15 cm -3 and mainly for low temperatures, is originated by the relatively low doping concentration.
However, in the same regime and at the same doping concentration, the decrease of mobility by increasing the temperature is due to the increase of lattice scattering while the impurity scattering is relatively the same. Besides, as the temperature gets higher for higher doping concentration, mobility by the impurity scattering increases while it decreases by the lattice scattering, the two cases balance one another, and as result mobility as a function of temperature becomes relatively constant and the rate of change of mobility is relatively insignificant (Figure 3). Additionally, for a doping concentration less than 10 15 cm -3 the mobility of electron is almost constant and is primarily limited by phonon scattering, that is, the rate of change of electron mobility is insignificant ( Figure 3). However, for higher doping concentration, above 10 17 cm -3 , the carrier mobility is dominantly hindered by the impurity scattering and electron mobility is significantly influenced even for the cases with low temperature.
Here, it is important to note that the very high doping concentration, dominated by impurity scattering, the semiconductor almost reaches to its lowest mobility ranges and it attains to a phase that temperature no more bring a relatively significant change in charge carrier mobility, and thus mobility seems to be constant. The effect of temperature on carrier mobility is insignificant at higher doping concentration, and correspondingly the effect of doping concentration is also insignificant at higher temperature. Generally, as the temperature gets higher, the electron mobility decrease, since lattice vibration increases with increasing temperature,  1976). At lower temperatures, impurity scattering dominates and it is governed as T -3/2 (Masettiet al., 1983). As a result, as the temperature increases, impurity scattering increases and the mobility decreases.
To get further uderstanding, the electron mobility as a function of temperature was made at different doping concentration ranging from 10 12 cm -3 to 10 20 cm -3 . Correspondingly, the temperature range was made from 100 K to 500 K. At a given temperature, the electron mobility tends to decreases as the doping concentration increases (Figure 4). This effect is particularly noticeable at low temperature, mainly for a temperature less than 300 K. However, at higher temperatures, greater than 350 K, the effect of concentration on the charge carrier mobility is insignificant.
The phenomena, mentioned above, witnessed the fact that the effect of doping concentration on carrier mobility is relatively insignificant at higher temperature and it is in consistent with the finding in Figure 2. It is evident that lattice/phonon and impurity scattering is dominating somehow above and below 400 K, respectively. In the first case,  of electron mobility as a function temperature is studied and presented in Figure 5.

CONCLUSION
In this study, we investigated that the temperature massive change with short ranges and finally saturates at low doping concentrations. The mobility of crystalline silicon decreases as dopant concentration increases, and this effect is pronounced at low temperature, particularly at 100 K. Furthermore, as the temperature increases the carrier mobility decreases and such a trend is more observable at low doping concentration, particularly at 10 12 cm -3 . The study generally indicates that lattice and impurity scattering dominate one another, with different trends, depending on the doping concentration and the temperature.