File: SECT04.WM of Tape: Various/ETH/s10-diss
(Source file text)
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IV. REAL FERMI SURFACES AND GALVANOMAGNETIC EFFECTS.
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1). Multiple wave scattering.
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a). Introduction.
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The transition probabilities treated in the preceeding section
were all written in function of plane wave states |k>. In a
real metal the state of an electron on some specific part of
the Fermi-surface is surely described by a more complicated
wave function. One always has to solve a Schroedinger equation
like the one given in eq.3.3. The important point is that one
has to include some previous knowledge, i.e. use the symmetry
properties of the metal whose Fermi-surface we want to deter-
mine. This is done by substituting |k> by a wave function
having a basis containing at least the essential elements of
the metals symmetry and then expanding the pseudowavefunction
&f\k\:
@be
@ea
1
(4.1) &f\k\(r) = --- $S a\n\(k) exp(i(k-q\n\).r)
$R &O
n
@ee
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where the set <q\n\> is the reciprocal lattice vector basis.
The vector a\n\(k) is the eigenvector corresponding to the
eigenvalue k(E\F\) when the following equation is solved,
@be
@ea
h'^2^
(4.2) a\n\(k)[--(k-q\n\)^2^ - E\F\]&d + $Sa\l\(k)<k-q\n\|V|k-q\l\> = 0
2m q\n\q\l\
l
@ee
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Having now expressed our solution as a pseudowavefunction
&f\k\, we must also use these in the golden rule of eq.3.31.
This leaves us with a double sum over matrix elements weighted
by products of a\n\(k). We shall call this the k representation
for the scattering probability P\kk'\. This is the represen-
tation most suited for numerical calculations and it is the one
we used in our OPW calculations.
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b). Amplitude factors.
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Sorbello(1974) has shown that it is advantageous to change over
to what we shall call the l representation. This representation
in angular momentum and phase shift has the advantage that one
can truncate the l summation at a small l\m\. By using the
T-matrix approach and the optical theorem he obtains an
expression for the scattering time &t:
@be
@ea l\m\
1 8E\F\
(4.3) ----- = ---- $S (2l+1) F\l\(k) sin^2^(&d\l\)
&t\o\(k) 3&ph'Z
l=0
@ee
@ea
where,
m=+l
4&p
(4.4) F\l\(k) = ---- $S | $S a\n\(k) Y^*^\lm\(k-q\n\) |^2^
2l+1
m=-l n
@ee
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The constant in eq.4.3 is: 0.85E\F\/Z when &t is to be
expressed in atomic units.
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The factor F\l\(k), which Sorbello called "amplitude factor",
measures the character of the wave function in the l
representation at the point k on the Fermi surface. It is very
convenient that this expression for &t(k) separates in two
contributions having to do with scattering and Fermi surface
properties separately.
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In view of the fact that in both the Hall effect and the dHvA
effect one is interested in averages of the relaxation time
over some representative orbit, the amplitude factors can be
averaged over these same orbits and this average <F\l\> will
then be interpreted as the effective character of the wave
function. As will be shown later on, the hole-like edges of the
second zone are very important for the Hall effect. This has
led us to calculate the <F\l\> for this part of the Fermi
surface and we give these, along with some values of Sorbello
(1974), in Table 4.1
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The local F\l\ were obtained with a 4 OPW program and then
averaged over the orbit in question. It is interesting to note
the different character of the <F\l\> distribution when
comparing Al and In. In Al the d-character is constant over
most of the surface, whereas in In the p-character is very
smoothly distributed. The greatest ratio of <F\l\> is for the
s-part between second zone holes and third zone electrons: .18
for Al and 5.2 for In. For the definition of the various bands
and zones refer to figure 4.1.
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_________________________________________________
| | | | | |
| Band and orbit <F\0\(k)> <F\1\(k)> <F\2\(k)> <F\3\(k)>
| | | | | |
| | | | | |
_________________________________________________
Aluminum
--------
_________________________________________________
| | | | | |
| Second band | | | | |
| ----------- | | | | |
| total orbit | .77 | .97 | 1.21 | .86 |
| | | | | |
| electron-like | .83 | .95 | 1.20 | .88 |
| | | | | |
| hole-like | .37 | 1.13 | 1.30 | .75 |
| | | | | |
| Third band | | | | |
| ---------- | | | | |
| total orbit | 2.02 | .33 | 1.13 | 1.24 |
| | | | | |
_________________________________________________
Indium
------
_________________________________________________
| | | | | |
| Second band | | | | |
| ----------- | | | | |
| total orbit | 1.14 | 1.22 | .68 | 1.06 |
| | | | | |
| electron-like | 1.09 | 1.23 | .70 | 1.06 |
| | | | | |
| hole-like | 1.46 | 1.17 | .51 | 1.03 |
| | | | | |
| Third band | | | | |
| ---------- | | | | |
| total orbit | .28 | 1.29 | 1.24 | .63 |
| | | | | |
_________________________________________________
Table 4.1: <F\l\(k)> for orbits in [110] plane on the
Fermi surfaces of Aluminum and Indium.
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2). Conduction in a magnetic field.
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a). General expression.
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In discussing transport phenomena in magnetic fields it is
imperative to reconsider the Sorbello(1974b) treatment of
relaxation times, but this time in a magnetic field. In fol-
lowing his assumption that the definition of the scattering
term (df\k\/dt)\scatt\ in the Boltzmann equation is given by
-g\k\/&t(k), we can write,
@be
@ea
df^o^ g\k\ e dg\k\
(4.5) e E.v\k\ (- ---) = ---- + - (v\k\ x H).---
dU &t(k) h' dk
@ee
@ba
where E and H are,respectively, the electric and magnetic field
and U the energy. Our &t(k), appropriate to electronic conduc-
tion in a magnetic field, also obeys, for elastic scattering,
@be
@ea
1 g\k'\
(4.6) ---- = $I (1 - ---) P\kk'\ dk'
&t(k) g\k\
@ee
@ba
This integral equation self-consistently defines the relaxation
time appropriate to the situation in a magnetic field, provided
a solution to eq. 4.5 can be found. This way of defining &t(k)
allows the separation of the problem in the solution of a
linear differential equation and an integral equation. Turning
to the solution of eq. 4.5 and noting that due to the vectorial
character of the second term on the right hand side, the
differentiation d/dk takes place in a plane perpendicular to H
and parallel to the Fermi-surface defined by the delta-function
df^o^/dU, we can write,(see Fig. 4.1)
@be
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@ea
g(&f,h) e v\p\(&f,h) dg(&f,h)
(4.7) eE v\e\(&f,h) &d(U-U\F\) = ------ + - ------- ------- |H|
&t(&f,h) h' &r(&f,h) d&f
@ee
@ba
In the last equation the variables are now &f and h, where &f
is a phase angle in the plane perpendicular to H and h stands
for k\H\: the part of the k-vector parallel to H. v\p\ is the
velocity perpendicular to the Fermi-surface, v\e\ is the
component of v\p\ parallel to the electric field E, and &r the
curvature radius of the surface at the point &f,h, which
appeared due to the transformation dk\para\ = &rd&f. In the
case of a circular orbit the term in H can be written:
&o\H\.dg/d&f, where &o\H\ = eH/m* is the cyclotron frequency.
The quantity m*, the cyclotron effective mass, can be described
in a local fashion, with our variables, by m* = h'&r/v\p\. We
now see that eq. 4.7 is a linear inhomogeneous differential
equation for g in &f and in our steady state situation the
particular integral can be shown to be,
@be
@ea
0 &f
h'E &r''v\e\'' h' &r'd&f'
(4.8) g(&f,h) = -- &d(U-U\F\) $I ------- exp(- -- $I -----) d&f''
H v\p\'' eH &t'v\p\'
-$~ &f''
@ee
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In this integral, the ' or '' quantities are defined by, for
example, v\p\''=v\p\(&f'',h). The integration only concerns &f
directly; h is a parameter here.
@be
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This is our generalized form (for any odd shaped Fermi surface)
of the type of equation appearing for instance in Ziman(1964)
p.258 and we quote his description of the formula: "The
displacement of the Fermi surface at the point whose phase
angle is &f is the sum of the displacements created by the
electric field at other points on the orbit, which are then
driven round the orbit by the magnetic field, decaying with the
(integrated) relaxation time &t".
@be
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This expression for g(&f,h) can then be inserted in eq.4.6 and
the relaxation time found by iteration. This is the only place
where the relaxation between different h slices comes into the
determination of &t.
@be
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Once the relaxation time has thus been determined it can be
inserted in the following expression for the current density
tensor,
@be
@ea
e &r
(4.9) J\ce\ = ---- $I $I $I v\c\ g\e\ -- dU d&f dh
4&p^3^h' v\p\
@ee
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here the subscript e denotes the driving electric field
direction and c the current response direction. When inserting
eq.4.8 in eq. 4.9 the delta-function will take out the energy
integration and tell us that we have to stay on the Fermi
surface. We get,
@be
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@ea
2&p 0 &f
eE &r v\c\ &r''v\e\'' -h' &r'd&f'
(4.10) J\ce\ = ---- $I $I ---- $I ------- exp(-- $I -----)d&f''d&fdh
4&p^3^H v\p\ v\p\'' eH &t'v\p\'
h 0 -$~ &f''
@ee
@ba
This is our exact expression for the current tensor, provided
the relaxation time is defined self-consistently by eq.4.6, the
linearized Boltzmann equation is valid (i.e. Ohm's law is
valid) and we are in a non-quantum regime. This equation can be
expanded in the limit of high fields and will lead to the
result that the Hall angle divided by the magnetic field will
tend to a constant proportional to the difference of the
concentration of electron and hole states. (see Ziman(1964) p.
261).
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If the integration over the phase angle &f is done in the
extended zone then the electron and hole states are
characterized by the sign of &r and &r'' in eq.4.10. It should
be remembered that a Brillouin zone intersection requires the
use of the reduced zone scheme when following a specific orbit
over the FS; than the &r's will be given an absolute value and
the &f's will be defined with a + or - sign, depending on the
local character of the FS.
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b). Expansion in powers of H.
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In the case of low fields, of interest to us, we see that the
integral in the exponential will be limited to small &f's when
H is small and the integrand can be expanded and integrated.
The factor in front of the exponential can also be expanded in
&f and than the whole integral in &f'' integrated by parts.
This leads to an expansion in powers of H and the first three
terms of the conductivity tensor are,
@be
@ea
2&p
e^2^ &rv\c\ &rv\e\ &tv\p\
(4.11) &s^o^\ce\ = ---- $I $I --- --- --- d&f dh
4&p^3^h' v\p\ v\p\ &r
h 0
@ee
@ea
e^2^ eH &rv\c\ d &rv\e\ &tv\p\
(4.12) &s^1^\ce\ = ---- (--) $I $I --- -- [---] [---]^2^ d&f dh
4&p^3^h' h' v\p\ d&f v\p\ &r
@ee
@ea
e^2^ eH &rv\c\ d^2^ &rv\e\ &tv\p\
(4.13) &s^2^\ce\ = ---- (--)^2^ $I $I --- --- [---] [---]^3^ d&f dh
4&p^3^h' h' v\p\ d&f^2^ v\p\ &r
@ee
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These integrals are impossible to integrate in a closed form,
because all quantities depend in a complicated way on &f and &t
is to be determined by solving eq.4.6. But there are several
generalities one may remark upon. Due to the definition of &f
it follows that if the FS is made up of several topologically
disconnected bands, each of these bands will have have to be
separately integrated in &f and the expressions for the
different &s\ce\ will then always consist of sums over bands.
(Ignoring the interband scattering effects introduced by
eq.4.6)
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If now this separation in bands also entails that the curvature
radii &r are good parameters of each band (which is not always
completely true), then these &r can be taken out from the &f
differentiation. The ratios v\x\/v\p\ are then really only
trigonometric functions of the angle &f and, in the case of
almost cylindrical bands, the orthogonality of cos and sine
functions leads us to identify &s^o^ with the zero field
conductivity, &s^1^ with the Hall conductivity and &s^2^ with
the magneto- conductivity.
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We want to stress here again that our newly obtained expres-
sions 4.11,4.12 and 4.13 are exact in the limit of low fields
provided the relaxation times are determined with eq.4.6 for
zero magnetic field. (Sorbello's &t\z\). We have to go to the
second order expansion because, due to symmetry considerations,
it is possible that the first order term is exactly zero.
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The low field condition is characterized by the fact that in
expanding the exponential factor in eq.4.10 we assumed that
&o\H\^eff^&t = eH&tv\p\/(h&r) << 1 for all parts of the FS.
Because the curvature radius &r is in the denominator, we see
that the low field limit is given by those parts of the FS with
the smallest curvature radius. In the case of Al or In we can
see in Fig.4.1 that this condition is realized by the hatched
parts, representing the edges of the second zone and that the
low field regime &o\H\^eff^&t << 1 can be located at consi-
derably lower fields than the low field condition &o\H\^fe^&t
<< 1 of the free electron case.
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3). Definition of 3 band model of the Fermi surface.
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In the preceeding paragraph we indicated that one should define
a model which will represent the Fermi surfaces of Indium and
Aluminium. As can be seen from Figs.2.1 and 2.5, it makes sense
to separate them in three bands, each of them with character-
istic, and more or less constant, dimensions. The bands are:
Band 1: The nearly free electron part of the second zone, Band
2: The strongly curved parts of the second zone near to the
Brillioun zones (this is not really a topologically separated
band but we assume we can treat it as such in the limit of low
fields), Band 3: The third zone "monster".(Only this band is
radically different in Al and In)
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We assume that these bands are of a pseudo-cylindrical shape;
naturally this is a very rough approximation, but it is the
only way to obtain an almost quantitative picture of the
behaviour of the galvanomagnetic effects. Then the formulas
4.11,12,13 will be reduced, for each band, to a multiplication
of appropriate averaged quantities <&r\i\>, <&f\i\>, <h\i\>,
<v\i\> and <S\i\>. (Here S is the surface to be used in the
usual resistivity formula and h is the length of the pseudo-
cylinder. We will drop the averaging brackets from now on and
assume that our quantities are always averaged values).
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FIGURE 4.1
Figure 4.1: 3 Band model of Fermi surface.
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Fig.4.1 represents a [110] section of the FS of Al and shows
the three aforementioned bands. The picture of this section in
In would not be radically different; the main differences can
be seen by comparing Fig.2.2 and 2.6. The limiting points of
the hole like band 2 are fixed by the inflection points in
curvature and we assign a total value for the angle: &f\2\ =
4&f\2a\ + 2&f\2b\ and also &f\1\ = &f\2\ - 2&p and clearly
&f\3\ = 4&p.
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The evaluation of the lengths h\i\ is critical because it
implies an averaging procedure; we ignored this complication
and directly took the extremal dimensions of the FS. The second
zones of Al and In are quite similar, but the third zone of In
has only 1/3 of the Al length when averaged over all
directions. We give the results for this model in Table 4.2.
All dimensions and the average velocities were calculated with
a 4-OPW program and are given in computational units and
radians. In these units a FCC trivalent metal free electron
sphere has a volume of 6 and a surface of 18. The sign of the
total phase angle shows if the band is electron or hole like.
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_________________________________________________
| | | | | | |
| Band | 1/&r\i\ | &f\i\ | h\i\ | S\i\ | v\i\ |
| | | | | | |
| | | | | | |
_________________________________________________
Indium
______
_________________________________________________
| | | | | | |
| 1 | .90 | -2.45 | 1.65 | 10.0 | 1.05 |
| | | | | | |
| 2 | 11.50 | 8.75 | 1.65 | 1.4 | 0.75 |
| | | | | | |
| 3 | 9.50 | -6.30 | 0.35 | 2.1 | 0.90 |
| | | | | | |
_________________________________________________
Aluminum
________
_________________________________________________
| | | | | | |
| 1 | .90 | -3.15 | 1.75 | 10.5 | 1.10 |
| | | | | | |
| 2 | 9.00 | 9.45 | 1.60 | 1.9 | 0.90 |
| | | | | | |
| 3 | 18.00 | -6.30 | 2.30 | 3.30 | 0.75 |
| | | | | | |
_________________________________________________
Table 4.2: Parameters of 3 band model of Fermi
surfaces of In and Al.
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