File: SECT05.WM of Tape: Various/ETH/s10-diss
(Source file text)
- 80 - @ka _________________________________________ | | | | | | | | -2| | | fe | | Imp. |A .10 | &DA/A | &Dn/n | &DR | | | cu | | | o | | | | | | | _________________________________________ | | | | | | | Pure | 2.364 | | | | | | | | | | | Sn | 2.515 | .064 | .002 | .04 | | | | | | | | Bi | 2.632 | .113 | .004 | .06 | | | | | | | | Pb | 2.524 | .068 | .002 | .04 | | | | | | | | Tl | 2.345 | -.008 | -.000 | -.00 | | | | | | | | Hg | 2.248 | -.049 | -.002 | -.03 | | | | | | | | Cd | 2.198 | -.070 | -.002 | -.04 | | | | | | | _________________________________________ Table 5.1: Rigid band changes in 1At.% alloys. @ke @ba The calculation of the FS cross-section A was done with a 4-OPW program. The changes in cross-section were calculated in the rigid band approximation, i.e. the number of electrons per atom was changed by taking into account the valency of the impurity. The changes in c/a ratio, due to the alloying, were also taken into account. The results are given in Table 5.1. The fourth column gives the change &Dn/n = (&Dn\h\-&Dn\e\)/n in carrier concentration between the second zone hole surface and the third zone electron surface. The last column shows the corresponding change of the Hall coefficient in the free electron approximation, R\o\^fe^ = &Dn/(n^2^.e) [m^3^/As]. @be - 69 - @ka V. HALL EFFECT MEASUREMENTS IN INDIUM. ______________________________________ 1).Introduction. ________________ @ke @ba Our measurements of the Hall-effect in Indium were initiated by the work of Cooper;Cotti;Rasmussen(1965). These authors studied the influence of the size-effect on the low-field Hall effect in Indium films. They obtained an important contribution of the size-effect , i.e. the sign of the Hall effect could be made to change from the usual positive sign to a negative sign in sufficiently thin samples. @be @ba These results were interpreted with the aid of anisotropic relaxation times in connection with a multi-band scheme for a polyvalent metal such as Indium. These strong lifetime effects showed that it would be interesting to look for other possible scattering mechanisms which would ,presumably, show analogous effects. We decided to look at the effect of alloying, cold working and, to some extent, of the temperature in both the Hall effect and the magnetoresistance of rolled polycrystalline specimens of Indium. @be - 70 - @ka 2). Experimental setup. _______________________ a). Cryogenics. _______________ @ke @ba All measurements of the Hall effect were made in a standard glass Dewar which fitted in a transverse electromagnet. Most measurements were made in liquid Helium and the temperature was determined with the aid of the vapour pressure. Some other selected temperatures were also needed for the temperature dependence measurements.The following temperatures were used: 20K, 77K, 210-300K. The first two temperatures were at the normal pressure boiling points of liquid hydrogen and of liquid nitrogen. The temperatures around 0C were made with a Freon 114 cooling setup. This cooling unit consisted of a standard refri- gerator compressor followed by a Joule-Thomson expansion valve and a vacuum pump which adjusted the pressure after the ex- pansion. The magnetoresistance measurements were only done in the liquid Helium temperature range. @be @ka b). Magnetic Fields. ____________________ @ke @ba The magnet used for the Hall measurements was a conventional iron core electromagnet with a maximal field of .8T in a gap of 100mm. The field-current relationship was calibrated with a flux integrating method and the magnet was always saturated prior to the runs so that the remanent field was the same. The magnetoresistance measurements were made in a split-coil Helmholtz superconducting magnet with a maximum field of 1.5T. The same calibration method was used. @be - 71 - @ka c) Measurements. ________________ @ke @ba The Hall effect was measured with a DC technique. The sample had three electrodes for measuring the Hall voltage; two of these electrodes were connected to a potentiometer of very low resistance which was situated at the same temperature as the sample. This potentiometer was used to compensate for the vol- tage due to the misalignement of the Hall-electrodes and was made out of a material similar to the sample, thus the effect of magnetoresistance on the offset was largely compensated. @be @ba This compensated voltage was then fed to a null-compensating bridge. The null detection was done with a "Tinsley" galvano- meter amplifier and a final standard galvanometer. The sample and compensation currents could be switched simultaneously in order to obtain a better null criterion. @be @ba Any remaining misalignement voltage was eliminated by reversing the magnetic field for every data-point. Our setup had an ul- timate sensitivity of approximately 2nV in the range of sample currents used (.5-5A). The resistivities were measured with the same circuit connected to two other electrodes on the sample. @be @ba The incremental magnetoresistance was measured by having two wires of the same material constitute the two arms of a Wheatstone bridge, one arm being in the magnetic field. The off-null voltage of the bridge, which was proportional to &D&r/&r, was directly fed to an X-Y recorder with a sensitivity of 50&mV. @be - 72 - @ka 3). Alloys and samples. _______________________ @ke @ba The alloys were prepared from 99.999% pure Indium of Johnson and Matthey Co. and 99.99% pure solutes of different sources. They were molten and thoroughly mixed in sealed glass ampoules, usually under vacuum, but, in the case of solutes with high vapour pressure under helium gas. We first prepared master alloys of a relatively high concentration (~5At.%), which were then further diluted to the required atomic percent concen- tration. The characteristics of the different alloys were already discussed in sect.II: we did not make any alloys with Mg,Zn,Li for the reasons indicated therein. The ingots so obtained were then tempered at 100^o^C for several days and at room temperature for several weeks. @be @ba As we wanted to measure only polycrystalline substances, the Hall samples were rolled to the required thickness (.1-.25mm as dictated by size effect considerations) and the magnetoresis- tance samples were extruded in the form of wires of .6mm. The rolled Hall film was then cut to size on a template and its thickness determined from its weight. The dimensions were: 1cm wide by 4cm long with electrodes of .5mm by 5mm projecting from the sample. This shape should reduce the systematic errors due to the shape of the film to less than 1%.(see Hurd(1972) p.184) For the measurements of cold working, the sample was twisted through a three pronged fork before being cooled down. During the experiment this fork was then moved from the top to the bottom of the sample and in this way it was deformed twice through an angle of approximately 30^o^. @be - 73 - @ka 4). Results. ____________ @ke @ba We will now give the results of our measurements and give a short discussion of several special details. The results can be divided in three different topics. @be @ka a) Residual resistivity results. ________________________________ @ke @ba In Fig 5.1 we give the residual resistivity (RRR = &r\4.2\/ &r\300\) of the indium films of .1mm thickness which we investigated. Also given are the size effect results (Samples 5 and 6) of Cooper et al.(1965) and the results on cold worked (CW) samples. The scales given for the last two effects are arbitrary and have no connection with the At.% impurity scale. The cold work RRR's make a jump at the first deformation and then only increase slowly with further cold work cycles; this probably comes from the fact that in the first deformation a crystallite size corresponding to the 30^o^ bending is established and this size is not changed much thereafter. @be @ba The value indicated for 99.999% pure indium is already size- effect limited at the thickness of .1mm, because, as Cooper et al.(1965) have shown, the material used had a bulk mean free path of ~.27mm. We used an extra thick sample (.25mm) for the Hall effect result on pure indium; this film had a RRR of 9.10^-5^. @be - 74 - @ba As can be seen from Fig. 5.1, there were no significant devia- tions of a linear relationship between the At.% concentration and the RRR, up to the highest concentration measured (1- 2At.%). The numerical values will be discussed further in a later paragraph. @be @fa FIGURE 5.1 Figure 5.1: Residual resistivities of Indium alloys ( the results on cold-working and size- effect are also indicated ) @fe - 75 - @ka b) Magnetoresistance. _____________________ @ke @ba Some representative results are shown in Fig. 5.2 , where &D&r/&r\o\ is plotted against the reduced field H\red\ = B&r\&J\/&r\o\. These results were used, along with the transverse Hall voltages, to compute the Hall angle &j. They also show the deviations from Kohler's rule which are to be expected if the relaxation time distribution is changing from sample to sample. (&J\D\=100K, &r\&J\=2.91 &m&O-cm) @be @fa FIGURE 5.2 Figure 5.2: Magnetoresistance in Indium alloys. @fe - 76 - @fa FIGURE 5.3 Figure 5.3: Hall angle in Indium alloys. @fe @ka c) Hall effect. _______________ @ke @ba In Fig. 5.3 we have plotted tan(Hall-angle)/ Red.Field in function of the reduced field instead of the Hall coefficient R\o\, for several reasons: The Hall angle is defined by, @be @ea j\y\ E\Hall\ E\Hall\(B) (5.1) tan( &j ) = -- = ----- = -------------- j\x\ E\long\ &r\long\(B) j\long\ @ee - 77 - @ba This angle is the only quantity which can be directly compared to calculations of j\y\ and j\x\. (j\y\ and j\x\ are not acces- sible in the usual experimental situations.) In low fields, the quantity plotted in Fig. 5.3 is directly related to the low field Hall coefficient R\o\: @be @ea tan(&j) &r\o\ E\Hall\(B) R\o\ (5.2) ---------- = ---------- = -- B &r\&J\ j\long\ B &r\&J\ &r\&J\ @ee @ba This quantity is also known under the name Hall mobility &m\H\. In intermediate field ranges it seems to change in an almost linear way and this fact will allow us to extrapolate to zero field. The data shows a departure from linearity at low fields; this is partly a real effect (one expects the Hall mobility to saturate eventually) and partly is caused by the loss of precision at low fields. The high field value of pure indium is not shown, but we obtained values of 14.9-15.2.10^-11^m^3^/As. This saturation value agrees well with other results (Hurd (1972) p.312; the theoretical value for one electron is 15.9.10^-11^m^3^/As). One can also see that Kohler's rule is not obeyed. This is to be expected if the relaxation times are changing. @be @ba The zero-field extrapolated values of R\o\ are given in Fig. 5.4 in function of the RRR's. These values were obtained by extrapolating the Hall angle to zero field , disregarding the small deviations from linearity at low fields. The two size- effect points are from Cooper et al.(1965). @be - 78 - @fa FIGURE 5.4 Figure 5.4: Extrapolated low field R\o\'s. @fe @ba The points can be roughly separated in three groups: Ga, Cd, Tl, Size saturate at about -5.10^-11^, Pb at -2.5.10^-11^ and Hg, Sn, Bi, cold-work stay at +1.10^-11^m^3^/As. In Fig. 5.5 we show some selected data on the temperature dependence of the Hall coefficient. For comparison the values of Al are also shown. (from Shiozaki; Sato(1967)) @be @ba The symbols appearing in Fig.5.4 and some of the symbols in Fig.5.5 are defined in Fig.5.1. @be - 79 - @fa FIGURE 5.5 Figure 5.5: Temperature dependence of R\o\. (the numbers in this figure refer to the At% concentration of the impurity.) @fe @ka 5). Rigid band effects on the Hall effect. __________________________________________ @ke @ba At the values quoted, the concentration of the impurities is always less than 1At.%. One might ask oneself if these concen- trations are big enough to have a direct influence on the electron concentration and thereby change the Hall coefficient. In order to check this influence we have computed changes in the [110] cross-section A of the third zone for additions of 1At.% impurities. This is shown in Table 5.1. @be - 80 - @ka _________________________________________ | | | | | | | | -2| | | fe | | Imp. |A .10 | &DA/A | &Dn/n | &DR | | | cu | | | o | | | | | | | _________________________________________ | | | | | | | Pure | 2.364 | | | | | | | | | | | Sn | 2.515 | .064 | .002 | .04 | | | | | | | | Bi | 2.632 | .113 | .004 | .06 | | | | | | | | Pb | 2.524 | .068 | .002 | .04 | | | | | | | | Tl | 2.345 | -.008 | -.000 | -.00 | | | | | | | | Hg | 2.248 | -.049 | -.002 | -.03 | | | | | | | | Cd | 2.198 | -.070 | -.002 | -.04 | | | | | | | _________________________________________ Table 5.1: Rigid band changes in 1At.% alloys. @ke @ba The calculation of the FS cross-section A was done with a 4-OPW program. The changes in cross-section were calculated in the rigid band approximation, i.e. the number of electrons per atom was changed by taking into account the valency of the impurity. The changes in c/a ratio, due to the alloying, were also taken into account. The results are given in Table 5.1. The fourth column gives the change &Dn/n = (&Dn\h\-&Dn\e\)/n in carrier concentration between the second zone hole surface and the third zone electron surface. The last column shows the corresponding change of the Hall coefficient in the free electron approximation, R\o\^fe^ = &Dn/(n^2^.e) [m^3^/As]. @be - 81 - @ba We see that these rigid band changes are very small and even if one assumes that this rough evaluation is wrong by a factor 10, our observed R\o\'s cannot be explained by the electron-hole changes. (Even the systematics is wrong!). We are thus led to conclude that anisotropic relaxation times must be responsible for the changes in R\o\ on alloying. @be - 82 - @ka 6). Discussion. _______________ a). Resistivity: ________________ @ke @ba We will now discuss our results in the light of the results obtained earlier in sect. IV. Eq.4.3 obtained there, was an expression for the inverse relaxation time representing a lifetime broadening. This means that all scattering events are weighted equally and this expression is appropriate for calculating Dingle temperatures (see sect.VI). In the context of resistivity calculations one must start by solving the Boltzmann equation. As shown in sect.IV, eq.4.6 will now take the form: (see also Sorbello(1974) @be @ea 1 v\z\(k')&t\z\(k') (5.3) ----- = $I [1 - ------------] P\kk'\ dk' &t\z\(k) v\z\(k) &t\z\(k) @ee @ba In the limit of a spherical Fermi surface and isotropic scattering this leads to (see e.g. Ziman(1964)) an expression containing only the scattering potential and a term which takes into account the loss of velocity in the direction of current flow. By inserting eq. 3.31 in eq. 5.3 one obtains, @be @ea 1 1 6&pZ (5.4) - = --- $I |<k+q|V|k>|^2^ x^3^ dx &t E\F\ 0 @ee - 83 - @ba here x=q/2k\F\ and all quantities are in atomic units. One can also express the resistivity directly with the aid of the formula, &r=m\e\/(ne^2^&t), and then the constant in eq. 5.4 will be: 6&p&O\o\/E\F\ and the resistivity &r will be expressed in atomic units (as defined in sect. III.1). @be @ba This same expression for the resistivity can be written in function of the phase shifts we calculated in sect.III and one can show that the correct formula is: @be @ea l=4 4 &p (5.5) &r^au^ = --- $S l sin^2^ (&d\l-1\ - &d\l\) Zk\F\ l=1 @ee @ba In calculating the resistivities with these formulas and our scattering potentials (or phase shifts) we first got values which were sensibly lower than the experimental values of the indium alloys. This can be understood by looking at the scatte- ring potentials (see Fig.3.3) and noting that they are usually peaked near x=0. It is just this region which is ignored by the x^3^ term in eq.5.4. Because these formulas were derived for a free electron surface, it is not astonishing that they do not work in a metal like indium which has a rather strongly dis- torted Fermi surface. @be - 84 - @ba As we noted in sect.IV we should do the full calculation in the framework of an OPW band structure. This is a rather costly and complicated calculation and we have taken the approximate way out. Because the trivalent metals have a Fermi surface con- sisting of several bands, the scattering angle necessary to annihilate practically all the current contribution of a state &f\k\ is quite small (approximately 30^o^ i.e. 1/3 of the angle in the free electron case). This fact led us to separate the Fermi surface in 3 bands each with a relaxation time given by eq.4.3. Thus, we say that the intraband scattering events are angle independent and that interband scattering is neglected. @be @ba In order to evaluate the resistivity we recast the eq.4.11 expression for the conductivity, @be @ea 1 (5.6) &s\xx\ = ---- $I &t v dS 12&p^3^ @ee FS @ba this expression is in atomic units and if one inserts compu- tational units under the integral and defines the relaxation time &t as (see eq. 4.3): @be @ea l\m\ 1 (5.7) ----- = $S (2l+1) F\l\(k) sin^2^ (&d\l\) &t\o\(k) l=0 @ee - 85 - @ba we get for our 3 band model, @be @ea 24&p (5.8) &r\xx\ = --- 1/ [ &t\1\v\1\S\1\ + &t\2\v\2\S\2\ + &t\3\v\3\S\3\ ] k\F\ @ee @ba We use the values for v and S in the different bands as given in the last paragraph in Table 4.2. If we insert these band parameters in eq. 5.8 and use the &t's obtained from eq. 5.7, we obtain theoretical values for the resistivity of 1At% solutions of the different impurities under consideration. We compare these theoretical values with our experimental results in Table 5.2. @be @ba In the same framework, we have also calculated the resistivi- ties of our selection of impurities in aluminum. These results are compared, in Table 5.3, to experimental data taken from Blatt (1968). For a further discussion of the resistivities of the samples used in our de Haas- van Alphen experiments, see Wejgaard (1975). @be - 86 - @ka _________________________________ | | | | | | | | fe | mb | | Solute| &r | &r | &r | | | exp | calc| calc| | | | | | _________________________________ | | | | | | Hg | .17 | .08 | .25 | | | | | | | Cd | .33 | .08 | .35 | | | | | | | Tl | .24 | .04 | .05 | | | | | | | Ga | .20 | .07 | .13 | | | | | | | Pb | .57 | .16 | .43 | | | | | | | Sn | .50 | .11 | .53 | | | | | | | Bi | 1.38 | .49 | 1.59 | | | | | | _________________________________ Table 5.2: Resistivities for Indium alloys (&m&O-cm/At%). &r\exp\ is taken from Fig. 5.1 with &r\300\= 9.01 &m&O-cm &r^fe^\calc\ is from eq. 5.4 or 5.5. &r^mb^\calc\ is from the multi-band eq. 5.8. @ke @ba We see that the agreement is encouraging, in a simple model like ours, for the charged impurities. The homovalent solutes Tl and Ga do not give the correct values of &r. There can be different reasons for this; the scattering potential is mainly given by the distortions of the lattice and, as we discussed in paragraph 3.3.b, this is the most doubtful part of the calcu- lation. Another reason might be that these scatterers have a strong s-like character and that the 3-band model breaks down. We have tried to use the Ashcroft potential of paragraph 3.2.b for these solutes, but the results were not sensibly different and we shall not give these results here. @be - 87 - @ka ________________________________ | | | | | | | | fe | mb | | Solute| &r | &r | &r | | | exp | calc| calc| | | | | | _________________________________ | | | | | | Cd | .50 | .07 | .14 | | | | | | | Zn | .22 | .08 | .27 | | | | | | | Mg | .45 | .15 | .55 | | | | | | | Ga | .30 | .05 | .05 | | | | | | | Ge | .80 | .24 | .43 | | | | | | | Si | .70 | .22 | .59 | | | | | | _________________________________ Table 5.3: Resistivities for aluminum alloys. in units of &m&O-cm/At%. The experimental data is from Blatt(1968) @ke @ba The same general remarks, as given for Indium, apply to the results of the Aluminum impurities. Here, also, the worst agreement is observed for the homovalent solutes. @be - 88 - @ka b). Low field Hall coefficient. _______________________________ @ke @ba In sect.IV we obtained the expressions 4.11-13 for the con- ductivity tensor. For a typical transverse arrangement we can take H along the z-axis and put E\z\=0, J\y\= J\z\= 0 and we get in second order in H: @be @ea (5.9) &s\xx\ = &s^o^\xx\ + &s^2^\xx\ H^2^, &s\xy\ = &s^1^\xy\ H, &s\zz\ = 0 @ee @ba In order to compare these quantities with experiment one has to obtain the corresponding resistivity expressions by calculating the expression &r\ce\= [ J.E ] / [ J^2^ ] and we get, @be @ea &s\xy\ &s^1^\xy\ H &s^1^\xy\ (5.10) tan( &j ) = --- = ---------------- , R\o\ = ------- &s\xx\ &s^o^\xx\ + &s^2^\xx\ H^2^ [&s^o^\xx\]^2^ @ee @ba for the Hall angle and coefficient respectivily. It should be noted that the sign of &s^2^\xx\ is implicitly negative due to the twofold differentiation of a trigonometric function in eq.4.13. Thus, one would expect the quantity tan(&j)/H\red\, plotted in Fig.5.3, to vary as R\o\(1+|&s^2^\xx\|.H^2^+...) in the limit of low fields. This seems not to be the case, unless the quadratic behaviour is located at very low fields. @be @ba The expression tan(&j) = &o\H\.&t for the free-electron Hall angle, can be used along with eq.5.2 to give the relation &o\H\&t = R\fe\.H\red\/&r\&J\, where R\fe\ is the free electron Hall constant and H\red\ our reduced field variable. @be - 89 - @ba Numerically, this gives in the case of indium, &o\H\&t =~ 2.10^-3^.H\red\. This means that in a reduced field H\red\ =~ 500 T we expect the transition from the low- to high-field behaviour of the free electron parts of the metal in question. This was the case for our high field values, where we observed a complete saturation of R\o\ at H\red\ = 1000 T. @be @ba We see from our experimental curves in Fig.5.3, that if there is a quadratic behaviour at all, it must be limited to field values H\red\ <^~^ 5 T. This could be explained by assigning an effective lowest cyclotron mass m* =~ .01 or (see sect.IV.2.a) a curvature radius of the smallest part of the FS of &r <^~^ &r\fe\/100. Such a small curvature radius is hard to understand in the light of the results obtained in Table 4.2. On the other hand it must be remembered that the low field condition is really given by &o\H\&t << m*/m\e\, which might explain the discrepancy of a factor of 10. @be @ba For the following numerical discussion we limit ourselves to our simple model of the Fermi surface. If we consider each band to be of a cylindrical shape, it is easy to see that the cur- vature radii &r drop out of eq.4.12 and the two first terms reduce to a sin(&f)^2^, which has an average value of 1/2. For our 3 bands we get the following simple expression, @be - 90 - @ea i=3 e^3^ (5.11) &s^1^\xy\ = ----- $S <&t\i\^2^> <v\pi\^2^> &f\i\ h\i\ 8&p^3^h^2^ i=1 @ee @ba If we now use eq.5.10 to express the Hall coefficient one gets in computational units: @be @ea 3 3 9&O\o\ (5.12) R\o\ = --- $S [ &t\i\^2^ v\pi\^2^ &f\i\ h\i\ ] / [ $S ( &t\i\ v\pi\ S\i\) ]^2^ e 1 1 @ee @ba Here we have dropped the averages and the constant in front of the sum is = 1.43.10^-9^ for In and = 0.92.10^-9^ for Al, when R\o\ is expressed in [m^3^/As]. @be @ba We can now insert the FS model parameters obtained in Table 4.2. The relaxation times used are given by the dimensionless formula 5.7 with the <F\l\> characteristic of each band taken from Table 4.1. We give the results for Indium and Aluminum in Table 5.4. The experimental values of the low field Hall coef- ficient are taken from Fig.5.4 at the RRR value of ~7.10^-3^ where the impurities dominate the phonon scattering. The few experimental values for Aluminum were taken from Boening et al.(1975), for comparison purposes. @be @ba At a first inspection of the data in Table 5.4, we see that the order of magnitude of the predicted R\o\ is correct. We noticed during the computation of these values that the results are extremely sensitive to changes in the scattering potential. @be - 91 - @ka _________________________________________ | | | | | | | | Indium | Aluminum | | Solute| | | | | | | R\calc\ | R\exp\ | R\calc\ | R\exp\ | | | | | | | | | | | | | _________________________________________ | | | | | | | Li | -2.3 | --- | --- | --- | | | | | | | | Hg | +3.1 | +0.5 | --- | --- | | | | | | | | Cd | -1.1 | -5.0 | +0.9 | --- | | | | | | | | Zn | -0.6 | --- | -2.5 | -1.2 | | | | | | | | Mg | -3.0 | --- | +1.7 | -0.5 | | | | | | | | Tl | +0.5 | -5.0 | --- | --- | | | | | | | | Ga | -2.4 | -5.0 | +5.9 | --- | | | | | | | | Pb | -3.6 | -2.5 | --- | --- | | | | | | | | Sn | -0.7 | +0.7 | --- | --- | | | | | | | | Ge | --- | --- | +6.9 | +1.8 | | | | | | | | Si | --- | --- | +4.2 | --- | | | | | | | | Bi | -1.7 | +0.7 | --- | --- | | | | | | | _________________________________________ Table 5.4: Comparison of calculated and experimental values of R\o\ in Indium and Aluminum. ( in units of 10^-11^ m^3^/As) @ke @ba Some values were calculated with the aid of the simple Ashcroft potential (see sect.III) and, although the scattering poten- tials look quite similar, quite substantial changes in the Hall coefficient resulted. We do not include these results here because they would only confuse the issue and, surely, the Shaw potentials are closer to the reality. @be - 92 - @ba On the average the R\o\ for In are more negative than those for Al. This is just the opposite behaviour to the one one would expect by looking only at the Fermi surfaces. Remember that the third zone of In should contribute much less to the R\o\ than the Al "monster". It seems that the influence of the strong differences in s-p character, reflected in the values of F\l\, is responsible for this abnormal behaviour. @be @ba The calculated values for Tl and Bi in In, have the worst fit to the experimental values. Tl in In was also the worst can- didate for the resistivity calculation and it seems plausible to assume that some aspect of the potential calculation is in error; most probably the distortion effect, because homovalent impurities are prone to these kind of errors. Bi has a valency difference of 2 with In and it's s-phase shift is already quite large, which might partially invalidate the Born-approximation. @be @ba The agreement for the Al solutes is not too good, the calcu- lated values are too large in absolute value, whereas the trend in polarity seems to be better. It is possible that due to the limited solubility in Al the experimental values did not attain their saturation level and that the phonon contribution to the Hall coefficient is still predominant. @be - 93 - @ka c). Temperature dependence of the Hall coefficient. ___________________________________________________ @ke @ba The effect of temperature on the Hall effect in selected alloys and pure indium and aluminum was shown in Fig.5.5 and we shall now discuss these results here. @be @ba Let us first note that the Hall coefficient of In has a rather strong variation between it's Debye temmperature &J\D\ of 100K and the melting point of 157C, where it even makes an abrupt jump to the value of three free electrons. It is very likely that this change comes from the change in c/a ratio as shown in Fig.2.4. We saw there that the c/a ratio stayed approximately constant up to the temperature of 150K. At higher temperatures this ratio diminishes quite rapidly toward the cubic ratio of 1. This temperature range coincides with the range where R\o\ is dropping rapidly. If we assume that the final jump in R\o\ corresponds to the change in c/a ratio from the extrapolated value of 1.069 to 1, than, assuming the change in R\o\ is linear in c/a-1, we can assign a value of ~-.7.10^-11^ to the high temperature R\o\, had there been no change in c/a ratio. Then we can see that the temperature dependence of Al and In are very similar, the only difference being a constant shift toward the positive side of ~5.10^-11^m^3^/As for In. @be - 94 - @ba Now, in a more quantitative description of the influence of the phonons on the value of R\o\, we must emphasize a very prac- tical property concerning the calculation of R\o\. As we saw in eq.5.12 the expression giving R\o\ contains the factors &t^2^ both in the numerator and the denominator. This means that the calculation is reduced to purely geometric problem weighted with the relative influence of the relaxation time. This is very important in discussing temperature effects, because, as is well known, the assignement of absolute values to e.g. resistivity calculations, is quite difficult. @be @ba The most difficult part in the discussion of the influence of phonon scattering is the fact that, in contrast to impurity scattering, the phonon carries a momentum q of it's own which leads to so called "Umklapp" scattering. This Umklapp scat- tering only occurs if a reciprocal lattice vector g can be found so that k\initial\ + q = g + k\final\. This condition is always satisfied on those parts of the FS where a Brillouin zone intersects it, but when these parts are reassembled in a reduced zone scheme as shown for instance in Fig.4.1, it can be seen that all these Umklapp processes are still intraband effects in our definition of these bands and in the limit of low temperatures. @be - 95 - @ba The limit of low temperatures is given by the fact that all phonon q's should be smaller than q\min\ as defined in Fig.4.1. Assuming a Debye model we find that q\D\ = 1 c.u. and q\min\/ q\D\=0.05. This corresponds to a temperature of .05 &J\D\ = 5K for both Al and In (compare Figs. 2.2 and 2.6). Thus, in our liquid helium temperature range, it is a good approximation to ignore Umklapp processes and treat the scattering as a normal quasi-elastic event. In order to calculate the corresponding phase shifts one still has to introduce the Bose statistics of the phonon distribution. This leads to a replacement of the usual matrix element <k+q|w|k> by an expression proportional to, @be @ea q/c(T) T (5.13) <k+q|w|k> ----------------- , c(T) = q\D\ - exp[ q/c(T) ] - 1 &J\D\ @ee @ba here c(T) is a cutoff variable which has the value of 20 in our low temperature range. We see that this effective matrix element is very sharply peaked at q=0 and has a width of of approximately .04 in q/(2k\F\). The formula 3.45, giving the phase shifts, being in some sense a Fourier transform in sphe- rical coordinates, we find that the phase shifts resulting are all equal up to at least l=4. @be @ba In the opposite case of very high temperatures the Bose factor in 5.13 is reduced to a constant and the scattering of the phonons is the same as those produced by vacancies of the pure metal. We assume that the neglect of Umklapp processes is not too serious because all the normal process q vectors are already interconnecting the totality of the FS. @be - 96 - @ba Invoking the argument given on the top of the preceding page we use the uniform value of 1 for the phase shifts at low tem- peratures and the values given under Al in Table 3.8 for the high temperatures. We can now calculate the R\o\'s in Al and In and compare them with experiment in Table 5.5. @be @ka _________________________________ | | | | Low temp. | High temp. | | | | | | | | | | R\calc\ | R\exp\ | R\calc\ | R\exp\ | | | | | | | | | | | _________________________________ | | | | | | | | | | Al | -0.3 | -1.5 | -2.3 | -4.5 | -- | | | | | | | | | | In | +2.2 | +2.5 | +0.2 | -0.7 | -- | | | | | | | | | | _________________________________ Table 5.5: Temperature dependence of R\o\ in Al and In. [10^-11^m^3^/As] @ke @ba We see that the results are encouraging in the case of indium and less so in the case of aluminum. The positive trend in the calculated values of Al was already noticeable in the impurity results. It may be that some aspect of the FS was neglected and that the 3-band parameters are somewhat biased. @be @ba The results on the temperature dependence of the impure speci- mens are clearly a manifestation of the transition of an impurity dominated regime to a phonon dominated one. @be - 97 - @ba The low concentration specimen with Cd has an RRR which is already very close to the RRR of pure In and its temperature dependence is very strong, whereas the specimen with Pb has an RRR of .063 and phonons with a higher q vector are needed to bring R\o\ back to the pure In case. @be @ba Up to here we have not yet discussed the results of cold work and size effect. We left these aspects out on purpose because they are difficult to handle in the framework of atomic scattering potentials. @be @ba The cold work introduces additional grain boudaries in the normally large grained samples. These grain boundaries are very large objects compared to the electron wavelength, when we assume that their principal scattering effect is due to their associated strain field. This leads to the conclusion that they should behave in a similar way as the low temperature long wavelength phonons. The experimental result that the value of R\o\ stays roughly constant when going from the pure to the cold worked samples (see Fig.5.4), confirms this. @be @ba It is difficult to see how the size effect can be incorporated in our model. Let us assume that the scattering is isotropic on the sample boundary and that this leads to a predominant s-like phase shift. Our previous results showed that the impurities with large s-like phase shifts tended to have a negative R\o\. Thus, isotropic surface scattering might explain the negative sign of R\o\ for the size effect experiments. @be