File: SECT05.WM of Tape: Various/ETH/s10-diss
(Source file text)
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_________________________________________
| | | | | |
| | -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
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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].
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V. HALL EFFECT MEASUREMENTS IN INDIUM.
______________________________________
1).Introduction.
________________
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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.
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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.
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2). Experimental setup.
_______________________
a). Cryogenics.
_______________
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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.
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b). Magnetic Fields.
____________________
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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.
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c) Measurements.
________________
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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.
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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.
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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.
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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.
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3). Alloys and samples.
_______________________
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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.
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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^.
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4). Results.
____________
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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.
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a) Residual resistivity results.
________________________________
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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.
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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^.
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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.
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FIGURE 5.1
Figure 5.1: Residual resistivities of Indium alloys
( the results on cold-working and size-
effect are also indicated )
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b) Magnetoresistance.
_____________________
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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)
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FIGURE 5.2
Figure 5.2: Magnetoresistance in Indium alloys.
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FIGURE 5.3
Figure 5.3: Hall angle in Indium alloys.
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c) Hall effect.
_______________
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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
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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).
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FIGURE 5.4
Figure 5.4: Extrapolated low field R\o\'s.
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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))
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The symbols appearing in Fig.5.4 and some of the symbols in
Fig.5.5 are defined in Fig.5.1.
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FIGURE 5.5
Figure 5.5: Temperature dependence of R\o\.
(the numbers in this figure refer to the
At% concentration of the impurity.)
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5). Rigid band effects on the Hall effect.
__________________________________________
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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.
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_________________________________________
| | | | | |
| | -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].
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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.
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6). Discussion.
_______________
a). Resistivity:
________________
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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\(k)
@ee
@ba
In the limit of a spherical Fermi surface and isotropic
scattering this leads to (see e.g. Ziman(1964)) an expression
containing only the scattering potential and a term which takes
into account the loss of velocity in the direction of current
flow. By inserting eq. 3.31 in eq. 5.3 one obtains,
@be
@ea
1
1 6&pZ
(5.4) - = --- $I |<k+q|V|k>|^2^ x^3^ dx
&t E\F\
0
@ee
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here x=q/2k\F\ and all quantities are in atomic units. One can
also express the resistivity directly with the aid of the
formula, &r=m\e\/(ne^2^&t), and then the constant in eq. 5.4
will be: 6&p&O\o\/E\F\ and the resistivity &r will be expressed
in atomic units (as defined in sect. III.1).
@be
@ba
This same expression for the resistivity can be written in
function of the phase shifts we calculated in sect.III and one
can show that the correct formula is:
@be
@ea
l=4
4 &p
(5.5) &r^au^ = --- $S l sin^2^ (&d\l-1\ - &d\l\)
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.
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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
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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).
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_________________________________
| | | | |
| | | 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.
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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.
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________________________________
| | | | |
| | | 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)
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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.
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b). Low field Hall coefficient.
_______________________________
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@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
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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
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_________________________________________
| | | | | |
| | 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