Hall mobility

Measurement of the Hall mobility gives, in principle, insight into the electron transport on a microscopic basis. While the drift mobility may be dominated by trapping 

Different idealized scattering situations can be summarized as in Table 4 (Con-well, 1967)  

Scattering on ionized impurities can be excluded in experiments with nonpolar liquids since the concentration of ions in well-purified samples is very low (see Section 1.8). If temporary localization of the drifting electrons in traps occurs, then the Hall mobility is larger than the drift mobility. 

Measurements on isooctane between 293 and 433 K ( id 7 cm V- s at 295 K) as a function of temperature gave r values between 3.5 and 1.7. The influence of trapping in the transport process is apparent (Itoh et al., 1989). Measurements on low-mobility liquids (Pd 1 cm V s ) would be of considerable interest since large differences between Pn M d would be expected and more insight into the transport mechanisms could be expected (Emin, 1977). 

Measurements of the Hall mobility in liquid argon as a function of density and in solid argon (Ascarelli, 1989) are explained by scattering on phonons and static density fluctuations. A problem exists at the mobility maximum in liquid argon where r values of about 2.5 were measured. The drift mobility at this density is rather high, about 1500 cm V s so that trapping should not be important (Lamp, 1993). 

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Experimental measurement of Hall mobility produces values of the same order of magnitude as the drift mobility their ratio r = jij/l may be called the Hall ratio. If we restrict ourselves to high-mobility electrons in conducting states in which they are occasionally scattered and if we adopt a relaxation time formulation, then it can be shown that (Smith, 1978 Dekker, 1957)... 

It is important to realize that even in the presence of traps, the measured Hall mobility refers to that in the higher conducting state (Munoz, 1991). Thus, a value of r significantly >1.0, and increasing with temperature in a certain interval, has been taken as an evidence in favor of traps in NP near the critical point (Munoz, 1988 Munoz and Ascarelli, 1983). Similarly, a nearly constant value of r near 1.0 in TMS over the temperature interval 22-164°C has been taken to indicate absence of trapping in that liquid. The scattering mechanism in TMS is consistent with that by optical phonons (Doldissen and Schmidt, 1979 Munoz and Holroyd, 1987). 

Munoz and Holroyd (1987) have measured Hall mobility in TMS from 22 to 164° C. This measurement parallels very well the variation of drift mobility with temperature in this liquid, and the Hall ratio remains essentially constant at 1.0 0.1. Both the drift and Hall mobilities in TMS decrease with temperature beyond 100°C, becoming 50 cmV s-1 at 164°C. The overall conclusion is that TMS is essentially trap-free in this temperature range, and the decrease of mobilities is due not to trapping, but to some other scattering mechanism that is more effective at higher temperatures. 

Hall mobility in NP has been measured by Munoz and Ascarelli (1983,1984) as a function of temperature up to the critical point (160°C). It falls relatively slowly from 220 cmV s-1 at 140°C to 170 cmV s-1 at the critical temperature. The drift mobility, however, falls precipitously over that temperature interval to -30 cmV s-1 at the critical temperature. Consequently the Hall ratio r increases sharply from 1.5 at 130° to 5.5 at 160°C. This has been taken as evidence for intrinsic trapping in this liquid. 

Hall and drift mobilities have been measured in mixtures of n-pentane and NP by Itoh et al., (1991) between 20 and 150°C. They found both mobilities to decrease with the addition of n-pentane to the extent that the Hall mobility in a 30% solution was reduced by a factor of about 5 relative to pure NR However the Hall ratio remained in the range 0.9 to 1.5. This indicates that, up to 30% n-pentane solution in NP, the incipient traps are not strong enough to bind an electron permanently. However, they are effective in providing additional scattering mechanism for electrons in the conducting state. 

In comparing the results of the quasi-ballistic model with experiment, generally pq[ = 100 cn v s-1 has been used (Mozumder, 1995a) except in a case such as isooctane (Itoh et al, 1989) where a lower Hall mobility has been determined when that value is used for the quasi-free mobility. There is no obvious reason that the quasi-free mobility should be the same in all liquids, and in fact values in the range 30-400 cmV -1 have been indicated (Berlin et al, 1978). However, in the indicated range, the computed mobility depends sensitively on the trap density and the binding energy, and not so much on the quasi-free mobility if the effective mobility is less than 10 crr v s-1. A partial theoretical justification of 100 cm2 v 1s 1 for the quasi-free mobility has been advanced by Davis and Brown (1975). Experimentally, it is the measured mobility in TMS, which is considered to be trap-free (vide supra). 

Baird and Rehfeld express A ° in terms of the trap concentration and the chemical potentials of the empty trap and of the electron in the quasi-free and trapped states. Further, they indicate a statistical-mechanical procedure to calculate these chemical potentials. Although straightforward in principle, their actual evaluation is hampered by the paucity of experimental data. Nevertheless, Eq. (10.13) is of great importance in determining the relative stability of the quasi-free versus the trapped states of the electron if data on time-of-flight and Hall mobilities are available. 

Fio. 3. Dependence on hydrogenation temperature of the free-electron concentration (a) and the electron Hall mobility (b) in phosphorus-implanted n-type silicon (Johnson et al., 1987c). 

CuInS2 CuCl InCl3 H2S XRD, XPS, SEM, UV, conductivity, Hall mobility, DSC,TG 135-137... 

FIGURE 6.3 Carrier concentration and Hall mobility of ITO films as functions of hydrogen partial pressure. 

Electrons have not been detected by optical absorption in alkanes in which the mobility is greater than 10 cm /Vs. For example, Gillis et al. [82] report seeing no infrared absorption in pulse-irradiated liquid methane at 93 K. This is not surprising since the electron mobility in methane is 500 cm /Vs [81] and trapping does not occur. Geminately recombining electrons have, however, been detected by IR absorption in 2,2,4-trimethyl-pentane in a subpicosecond laser pulse experiment [83]. The drift mobility in this alkane is 6.5 cm /Vs, and the quasi-free mobility, as measured by the Hall mobility, is 22 cm /Vs (see Sec. 6). Thus the electron is trapped two-thirds of the time. 

The drift mobility of electrons in nonpolar liquids ranges from high values such as that for liquid xenon of 2000 cm /Vs to low values like that for tetradecane of 0.02 cm /Vs. It has often been suggested that the mobility is high for symmetrical molecules and low for straight chain molecules like -alkanes. Inspection of Table 2 shows that liquids with symmetrical molecules are indeed at the top of the list. However, other less symmetrical molecules like A-trimethylsilylmethane and 2,2,4,4-tetramethylpentane also show high drift mobility. A more important factor may be the existence of many methyl groups in the molecule. In any case, for liquids for which 10 cm /Vs, the electron is considered to be quasi-free. This is supported by the Hall mobility studies, as discussed below. 

This theory has also been used to predict mobility for molecular liquids. Neopentane and TMS are liquids that exhibit maxima in the electron mobility at intermediate densities [46]. These maxima occur at the same densities at which Vq minimizes, in accordance with the Basak Cohen theory. The drift mobility in TMS has been measured as a function of pressure to 2500 bar [150]. The observed relative experimental changes of mobility with pressure are predicted quite well by the Basak-Cohen theory however, the predicted value of /i ) is 2.5 times the experimental value at 1 bar and 295 K. In this calculation, the authors used xt to evaluate the mobility. This is reasonable in this case since for liquids, there is little dilference between the adiabatic and isothermal compressibilities. A similar calculation for neopentane showed that the Basak-Cohen theory predicted the Hall mobility of the electron quite well for temperatures between 295 and 400 K [151]. Itoh... 

The main experimental elfects are accounted for with this model. Some approximations have been made a higher-level calculation is needed which takes into account the fact that the charge distribution of the trapped electron may extend outside the cavity into the liquid. A significant unknown is the value of the quasi-free mobility in low mobility liquids. In principle, Hall mobility measurements (see Sec. 6.3) could provide an answer but so far have not. Berlin et al. [144] estimated a value of = 27 cm /Vs for hexane. Recently, terahertz (THz) time-domain spectroscopy has been utilized which is sensitive to the transport of quasi-free electrons [161]. For hexane, this technique gave a value of qf = 470 cm /Vs. Mozumder [162] introduced the modification that motion of the electron in the quasi-free state may be in part ballistic that is, there is very little scattering of the electron while in the quasi-free state. 

On lowering the temperature through Ty, a bandgap Eg = 0.1 eV appears in the FeB-ai(l) conduction band of Fig. 3 at Ep. The Hall coefficient increases as Rh exp(Ty/T), indicating that the charge-carrier density increases exponentially with T" , as in a normal semiconductor, and the Hall mobility increases from about 0.1 to 0.4 cm /Vs on lowering the temperature from Ty = 120 K to 77 K ... 

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