File: SECT03.WM of Tape: Various/ETH/s10-diss
(Source file text)
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III. ANISOTROPIC RELAXATION TIMES .
___________________________________
1) Structural and free electron data.
____________________________________
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Several general quantities always crop up in calculations
concerning Fermi-surfaces or involving pseudo-potentials.
The proper low-temperature values are not always used and we
find it convenient to include these constants for future
reference.
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The periodic boundary conditions in a crystal show that the
density in k-space is given by: 2 V(cryst)/(2&p)^3^. From this
it follows that the Fermi-wavenumber k\F0\, corresponding to
the free electron sphere is:
@be
@ea
3&p^2^ Z s 3&p^2^ Z
(3.1) k\F0\ = [---------]^1/3^ = [-------]^1/3^
V(cryst) &O\o\
@ee
@ba
where Z is the valence of the considered atom, s the number of
atoms per unit cell, V(Cryst) the volume of the unit cell and
&O\o\ the atomic volume.
@be
@ba
Because all calculations are done in k-space, it is of
advantage to work in so called computational units and one
choses one of the unit reciprocal lattice vectors 2 &p/a as the
new unit length. With x(vol)= &O\o\s/a^3^ we get:
@be
@ea
cu 3 Z s
(3.2) k\F\ = (------------)^1/3^ in comp. units
8 &p x(vol)
@ee
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The x(vol) constant is a characteristic of each crystal
structure, i.e. for FCT x=c/a, HEX: x=SQRT(3)c/(2a), HCP:
x=SQRT(2), RHOMB: x=(1-COS(&a)) SQRT(1+2COS(&a))
@be
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One would like to avoid the use of atomic units, but it just
happens that most of the literature abounds in Rydbergs, Bohr
radii, etc. For reasons of completeness we give the definitions
and conversion factors:(au=atomic units,cu=computational units)
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Quantity Unit Factor Unit
______________________________________________________
Mass m\e\ 9.1091 10^-28^ grams
Length h'^2^/me^2^ .52917 Angstroem
=a0
=Bohr radius
Time h'^3^/me^4^ 2.419 10^-17^ seconds
Energy(a.u.) me^4^/h'^2^ 4.3594 10^-18^ Joule
27.2105 eV
" (Rydberg) me^4^/2h'^2^ 2.1797 10^-18^ Joule
13.6052 eV
Resistivity at. units 21.74 &m&O-cm/100At%
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We now have:
@be
@ea
au au cu
k\F\ = ( 2 &p / a ) k\F\
@ee
@ba
and the Fermi energy in computational units is usually defined
as:
@be
@ea
cu cu
E\F\ = (k\F\ )^2^
@ee
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Table 3.1 gives a collection of lattice spacings and atomic
volumes in a.u. for the solvents and solutes of interest (no
values are given for the transition metal solutes because we
can not treat them with the pseudopotential formalism ). Where
available the low temperature lattice spacings have been used,
otherwise we extrapolated the room temperature l.c.'s to 0K
with the aid of thermal expansion data.
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_________________________________________________________
| | | | | | | |
| Metal | Struct| a | c | b | | |
| and | and | in | in | or | c/a | &O |
|valence| states| a.u. | a.u. | Angle | | o |
| | | | | | | |
_________________________________________________________
| | | | | | | |
| Bi/5 | RH/2 | 8.9319| | 57.314| | 236.36|
| | | | | | | |
|*Si/4 | DIA/8 |10.2628| | | | 135.11|
| | | | | | | |
|*Ge/4 | DIA/8 |10.6910| | | | 152.74|
| | | | | | | |
| Sn/4 | BCT/4 | 10.985| 5.9636| | .5429 | 179.91|
| | | | | | | |
| Pb/4 | FCC/4 | 9.2874| | | | 200.27|
| | | | | | | |
| Al/3 | FCC/4 | 7.6192| | | | 110.58|
| | | | | | | |
| Ga/3 | ORH/8 | 8.5324| 14.422| 8.4814| 1.6903| 130.46|
| | | | | | | |
| In/3 | FCT/4 | 8.6038| 9.3254| | 1.0839| 172.58|
| | | | | | | |
| Tl/3 | HEX/2 | 6.4970|10.3521| | 1.5934| 189.22|
| | | | | | | |
| Mg/2 | HEX/2 | 6.0361| 9.7959| | 1.6229| 154.55|
| | | | | | | |
| Zn/2 | HEX/2 | 5.0275| 9.1868| | 1.8273| 100.54|
| | | | | | | |
| Cd/2 | HEX/2 | 5.6103|10.4482| | 1.8624| 142.40|
| | | | | | | |
| Hg/2 | RH/1 | 5.6434| |70.7433| | 155.18|
| | | | | | | |
| Li/1 | FCC/4 | 8.3338| | | | 144.70|
| | | | | | | |
| Cu/1 | FCC/4 | 6.8093| | | | 78.93|
| | | | | | | |
| Ag/1 | FCC/4 | 7.6892| | | | 113.65|
| | | | | | | |
_________________________________________________________
Table 3.1: Low temperature structural data
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Table 3.2 gives the followig constants computed with the values
of Table 3.1: the free electron Fermi-vector (in c.u.), the
Fermi energy in a.u., the reciprocal lattice constant in a.u.
and the factor A(cu), which is used to convert dHvA frequencies
F to computational areas : Area=A(cu) F[Gigagauss]. The
references indicated in Table 3.2 also apply to Table 3.1.
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_________________________________________
| | | | | |
| | 2&p | cu | au | |
| Metal | ---- | k | E | A(cu) |
| | a(au) | F | F | |
| | | | | |
_________________________________________
| | | | | |
| Bi | .7035 | 1.2163| .3661 | 5.4020| Barret (1960)
| | | | | |
| *Si | .6122 | 1.5632| .4580 | 7.1313| Ben et al.(1962)
| | | | | |
| *Ge | .5877 | 1.5632| .4220 | 7.7389| Cooper (1960)
| | | | | |
| Sn | .5720 | 1.5209| .3784 | 8.1703| Lee;Raynor(1954) a)
| | | | | |
| Pb | .6765 | 1.2407| .3595 | 5.8402| Pearson (1964) b)
| | | | | |
| Al | .8246 | 1.1273| .4321 | 3.9306| Figgins (1956) c)
| | | | | |
| Ga | .7364 | 1.1946| .3870 | 4.9293| Barret (1962)
| | | | | |
| In | .7303 | 1.0974| .3211 | 5.0120| Graham (1955) d)
| | | | | |
| Tl | .9671 | 0.8036| .3020 | 2.8580| Barret (1958)
| | | | | |
| Mg |1.0409 | 0.6978| .2638 | 2.4669| Pearson (1964) e)
| | | | | |
| Zn |1.2498 | 0.6707| .3513 | 1.7114| Lynch (1965) e)
| | | | | |
| Cd |1.1199 | 0.6665| .2786 | 2.1311| Lynch (1965) e)
| | | | | |
| Hg |1.1133 | 0.6515| .2631 | 2.1564| Barret (1957)
| | | | | |
| Li | .7539 | 0.7816| .1736 | 4.7023| Khatkevic (1952)
| | | | | |
| Cu | .9227 | 0.7816| .2601 | 3.1394| Frohnmeyer (1953) f)
| | | | | |
| Ag | .8171 | 0.7816| .2040 | 4.0031| Smakula (1955) b)
| | | | | |
_________________________________________
Table 3.2: Free electron constants.
(*: Room temperature values)
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Thermal expansion references:
a) White (1964)
b) Nix;McNair (1942)
c) Extrapolated.
d) Collins et al.(1967)
e) McCammon;White (1965)
f) Nix;McNair (1941)
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2) Pseudopotentials.
___________________
a). Introduction.
_________________
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The problem of determining the band structure of a metal is an
old one and there are several , more or less succesful, ways of
solving it.
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What one always has to do, is to solve the Schroedinger
equation:
@be
@ea
(3.3) H &Q\i\ = ( T + V(r) ) &Q\i\ = E\i\ &Q\i\
@ee
@ba
where T is the kinetic energy, V(r) the self-consistent
potential seen by an electron and E\i\ the energy of the i'th
state. Now it happens that the potential is "strong" in the
sense that it creates low lying bound states, called "core
states". If one solves the Schroedinger equation in the
k-representation:
@be
@ea
(3.4) det| [ 1/2 ( k - g )^2^ - E ] &d\gg'\ + V\gg'\ | = 0
@ee
@ba
(here the V\gg'\'s are the Fourier components of V(r) ), we
have to go to very large k's (small wavelengths) in order to
make the calculation convergent. ( For Al the secular equation
would have to be of order 10^6^ x 10^6^ ). The lowest
eigenvalue of this solution will correspond to the lowest core
state and will not be particularly interesting for the
properties of a metal. It is quite clear that such a
calculation would be senseless and some means are needed to
eliminate the core states.
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The fact that the V(r) has a deep potential well, leading to
core states, entails that the wave functions have several nodes
in the core region. One can now prescribe a core radius R\c\,
such that the core states (wiggles) are all included in a
sphere of radius R\c\. One can calculate the wave function at
R\c\ for the free atom and join these solutions to the solution
of the Schroedinger equation outside this sphere. One can now
show that the original V(r) can be replaced by another poten-
tial V\ps\(r) called the pseudopotential (PP). The solutions of
the Schroedinger equation using this potential are much
smoother and correspond to the outer valence states. Now the
lowest eigenvalue belongs to the lowest valence state.
@be
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Another way of looking at this problem is by using the
scattering theory approach. An incident wave |k> falling on an
atom will emerge with an angular distribution:
@be
@ea
(3.5) 4&p/k $S (2l+1) [ exp(2i&h\l\) - 1] P\l\(cos &j)
@ee
@ba
The phase angle can be written:
@be
@ea
(3.6) &h\l\ = p\l\ &p + &d\l\ |&d\l\| <= &p/2
@ee
@ba
The constant p\l\ corresponds to the number of bound states,
but does not have any influence on the scattered wave. A
pseudopotential is now a potential which produces only phase
shifts of &d\l\; the bound states have been removed. We shall
be using these two aspects in our further discussion: the
second aspect will be helpful in discussing scattering
properties in relation with the character of the wave packets
(s,p,d, character)
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b). The empty core or Ashcroft(1966) potential.
_______________________________________________
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This is the simplest of all pseudopotentials and uses the
"cancellation theorem" (Heine(1970)) to set the depth of the
well exactly to zero:
@be
@ea
V\ash\^ion^(r) = 0 for r<R\c\
(3.7)
= -Z/r for r>R\c\
@ee
@ba
Here R\c\ is some model radius adjusted for best fit with
experimental data and Z is the valence of the metal. The
Fourier transform of a potential is:
@be
@ea
(3.8) v(q) = 1/&O\o\ $I V^ion^(r) exp(-iqr) d^3^r
@ee
thus,
@ea
(3.9) v\ash\ = - [ 4&pZ/ ( q^2^&e(q) &O\o\] cos( qR\c\)
@ee
@ba
In reduced q's: x=q/2k\F\
@be
@ea
k\F\
(3.10) v\ash\ (x) = - ------------ cos( 2 x R\c\ k\F\)
3 &p x^2^ &e(x)
@ee
@ba
where &e(x) is the dielectric screening function without
exchange:
@be
@ea
k\F\
(3.11) &e(x) = 1 + ---------------- f(x)
3 &p x^2^ (2/3 E\F\)
@ee
@ea
with f(x) the Lindhardt screening function:
1 - x^2^ 1 + x
(3.12) f(x) = 1/2 + ------ ln | ----- |
4 x 1 - x
@ee
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It is seen that in the limit of x going to zero this potential
goes to -2/3E\F\ (the deformation potential), as every local
pseudopotential should.
@be
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We have fitted this PP to experimental Fermi surface data,
where available, and we give the obtained R\c\'s and the
sources used in Table 3.3.
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These pseudopotentials are useful for qualitative discussions,
but it is difficult to see how they are influenced when used
for solute calculations. In the next paragraph we develop a
much more realistic model for this situation.
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_________________________________________________
| |
| Li Mg Al Si |
| 1.73 1.30 1.15 1.02 |
| |
| Cu Zn Ga Ge |
| 0.81 1.11 1.06 0.98 |
| |
| Ag Cd In Sn |
| 1.04 1.25 1.09 1.11 |
| |
| Hg Tl Pb Bi |
| 1.27 1.11 1.00 0.98 |
| |
_________________________________________________
Table 3.3: R\c\'s for Ashcroft PP. (in a.u.)
Zn,Al,Mg,Pb,In Sorbello;Griessen (1973)
Bi,Tl,Li Cohen;Heine (1974)
Cu,Ag Ashcroft;Langreth (1967)
Si,Ge Brust (1964)
Sn Stafleu;de Vroomen (1967)
Ga Reed (1969)
Hg Brandt;Rayne (1966)
Cd Stark;Falicov (1967)
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c). The Shaw potential.
_______________________
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The pseudopotentials of the Heine-Abarenkov-Shaw type are so
called "model potentials" and were first given by Animalu and
Heine(1965) and later on Shaw(1968) showed that it was possible
to optimize them in a certain way. These potentials are called
model potentials to distinguish them from ab initio
pseudopotentials and also because they make use of empirical
information concerning the valence electron energy levels.
@be
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As already mentioned, the problem lies in knowing the
wavefunction of the Z times ionized atom outside some model
radius R\l\.(the R\l\'s may be different for different angular
momenta l=0,1,2;see Fig. 3.1) These wavefunctions will
naturally be energy dependent and one has to evaluate them at
the appropriate energy E\F\^ion^ with respect to the free ion.
Now we know that the solutions to the hydrogenlike Schroedinger
equation are only finite for energies at the term values (n,l)
of the ion in question and can be represented by Whittaker
functions of the first kind for the quantum numbers n,l.
(Shaw(1968b)).
@be
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The logarithmic derivative at the radius R\l\ of the solution
n,l can then be calculated and a solution corresponding to the
interior square well of depth A\x\ can be made to fit this
derivative. The resultant A\x\ will then specify the depth of
the well A\l\ for a certain l.
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FIGURE 1
Figure 3.1:The optimized model potential; the square
wells fit exactly in the Coulomb part
Z/r and are of different depths.
@fe
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Shaw(1968,1968b) has shown that an optimized form of the
potential (optimized in the sense that the wavefunction is as
smooth as possible) results if the condition A\l\=Z/R\l\ is
fulfilled. He also shows that it is not neccessary to model
wave-functions higher than l\0\, where l\0\ is the highest
angular momentum of the core-wavefunctions.(e.g. l\0\=0 for
Li). He has given tables for relating the A\l\ with the term
values E\l\. We have reproduced these tables in a graphical
form in Fig. 3.2, where ln(A\l\/Z^2^) is given in function of
ln(-E\l\/Z^2^).
@be
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@ba
The only, but major, problem remaining is the inter- or
extrapolation of the A\l\(n)'s to the energy E\F\^ion^. Shaw
has shown, empirically, that a linear extrapolation is
reasonable provided that the d-bands are far enough in energy
fron the conduction band. We have found that a linear
interpolation in our ln-ln plot is often superiour and a
logarithmic extrapolation will tend to flatten out at low
energies provided the absolute value of the slope is smaller
than 1. (which is the case for the metals we considered). There
is no a priori reason to make a linear extrapolation and the
fact that our interpolation is often better, makes us feel
confident that our method is comparable or even better than the
linear method. Our interpolation formula is thus as follows:
@be
@ea
(3.13) A/ Z^2^ = &a ( -E / Z^2^ )^&b^
@ee
@ba
Some of the elements we studied did not allow optimization,
viz: Cu,Ag and the transition metals Ti,V,Cr and Mn. In these
metals the d-band is too close to the conduction band and the
usual hydrogen-like term scheme breaks down. Animalu(1973) has
given a whole series of model potentials, which includes the
last mentioned, but the effect of scattering resonances seems
rather difficult to take into account.
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The metals Zn,Cd and Hg are an intermediate case, where the
influence of the d-bands just starts to be noticeable. This
series of elements has been studied by Evans(1970) by
incorporating the quantum defect method in Shaw's formalism. We
have determined our constants &a and &b from his values of A\l\
and $dA\l\/$dE at E\F\^ion^ for Zn,Cd,Hg and also for Pb,In,Tl.
In Table 3.4 we give our fitted values of &a and &b for future
reference.
@be
@fa
FIGURE 3.2
careful: 0 1 2 regions not indicated
Figure 3.2: ln-ln plot of Shaw's tables. Region 1
corresponds to the usual one node so-
lution, region 0 to exponentially de-
caying solutions.
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d). Fermi energy for solutes.
_____________________________
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The problem remains of the E\F\^ion^ at which one should
evaluate the A\l\'s. This problem has been discussed by Animalu
and Heine(1965) and their method has been adopted Shaw,Evans,
etc. In solving the Schroedinger equation (3) for an ion in a
metal the potential energy term can be written as follows:
@be
@ea
(3.14) V(r) = V\ion\(r) + V\rest\(r)
@ee
@ba
where V\ion\ is the potential for a single isolated ion and
V\rest\ a potential due to the rest of the system, principally
the conduction electrons in the neighbourhood of the ion when
immersed in the metal. If V\rest\ can be considered more or
less constant in the model sphere, the energy at which to solve
for the free ion will be:
@be
@ea
(3.15) E\F\^ion^ = E\F\^vac^ - V\rest\
@ee
@ba
The value of the absolute energy E\F\^vac^ can be obtained by
considering the following expression for the cohesive energy:
(Kittel(1963)p.93,Seitz(1940),Animalu;Heine(1965))
@be
@ea
(3.16) -|B| = |I| + E\HF\ + E\c\ + E\WS\ - E\xc\^cb^ - E\se\^cb^
@ee
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_________________________________________________________
| | | | | | | |
| Metal | s | p | d |
| | &a | &b | &a | &b | &a | &b |
| | | | | | | |
_________________________________________________________
| | | | | | | |
| Bi | .2548 | .3640 | .2623 | .3508 | .0563 |-.0872 |
| | | | | | | |
| Si | .2125 | .2022 | .1681 | .0211 | -- | -- |
| | | | | | | |
| Ge | .2296 | .2286 | .1837 | .0728 | .0927 |-.1462 |
| | | | | | | |
| Sn | .2377 | .2918 | .1915 | .1546 | .1141 | .0271 |
| | | | | | | |
| Pb | .2602 | .3130 | .1751 | .1373 | .0435 |-.2786 |
| | | | | | | |
| Al | .2243 | .1703 | .1961 | .0289 | -- | -- |
| | | | | | | |
| Ga | .2605 | .1992 | .2125 | .0313 | .1464 |-.0421 |
| | | | | | | |
| In | .2640 | .2499 | .1860 | .0544 | .0438 |-.4070 |
| | | | | | | |
| Tl | .3021 | .2833 | .2434 | .1588 | .0455 |-.3445 |
| | | | | | | |
| Mg | .2534 | .1318 | .2427 | .0241 | -- | -- |
| | | | | | | |
| Zn | .3544 | .1884 | .4832 | .1820 | .1904 | .0084 |
| | | | | | | |
| Cd | .3511 | .2288 | .3725 | .1560 | .1903 | .0076 |
| | | | | | | |
| Hg | .4185 | .2528 | .4584 | .1991 | .3507 | .1935 |
| | | | | | | |
| Li | .3707 | .0726 | -- | -- | -- | -- |
| | | | | | | |
_________________________________________________________
Table 3.4: Values of &a and &b.
@ke
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This equation is set up per electron (not per atom!) and the
quantities have the following meaning:
@be
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B = absolute value of cohesive energy per electron
I = absolute value of (first Z ionization energies)/Z
E\HF\ = Hartree-Fock energy of free-electron gas
= (3/5) E\F\ + E\x\ (E\x\=exchange energy)
E\c\ = correlation energy of free-electron gas
E\WS\ = Wigner-Seitz bottom of band energy
E\xc\^cb^ = exchange/correlation of conduction band
E\se\^cb^ = self-energy of conduction band electrons
@ke
@ba
The last two terms had to be subtracted, otherwise they would
be counted twice in E\HF\ and E\WS\.The quantity E\F\^vac^ we
want is the energy from the vacuum to the bottom of the
conduction band(=E\WS\) plus the energy from the bottom to the
top of the conduction band(=E\F\).Rewriting the above equation
16 we get:
@be
@ea
(3.17) E\F\^vac^ = -|B| - |I| + (2/5)E\F\+ E\se\^cb^+ E\xc\^cb^- E\x\- E\c\
@ee
@ba
The term V\rest\ can be expressed as: V\rest\=V\pot\+E\xc\^cb^,
where V\pot\ is the position dependent potential of an electron
interacting with Z electrons in the cell with radius r\o\ and
E\xc\^cb^ the exchange and correlation interaction with these
same electrons.The potential V\pot\ can be obtained by
integrating over the cell and then averaging over a sphere of
radius R\l\:
@be
@ea
Z 3Z R\l\
(3.18) V\pot\ = ---- [ 3 - (r/r\o\)^2^] =^~^ ---- [ 1 - (----)^2^]
2 r\o\ 2 r\o\ 2 r\o\
@ee
- 41 -
@ba
We can now write a final expression for E\F\^ion^ which will be
used as the energy to be used for extrapolating for the A\l\'s.
This energy will, itself, depend on A\l\ through the term
V\pot\ and we will have to solve self-consistently for each
A\l\.
@be
@ea
(3.19) E\F\^ion^ = -|B| - |I| + .2 k\F\^2^ + .6 Z/r\o\ + .239 k\F\
+.0575 - .0155 ln(1.9192/k\F\) - V\pot\
@ee
@ba
In this equation the explicit expressions for the different
energies have been inserted, k\F\ is the usual Fermi-momentum
and r\o\ the radius of the Wigner-Seitz sphere.All the energies
are given in atomic units.It is to be noted that the term
E\xc\^cb^ dropped out in subtracting E\F\^vac^ and V\rest\. We
calculated the energies E\F\^ion^ self-consistently, for the
elements in question, for each l. The average of the energies
for s,p,d was then used to extrapolate the A\l\'s with the aid
of equation 3.13.The different quantities of interest, for the
pure elements, are given in Table 3.5. (The quantity E\F\^*^ is
the constant part of E\F\^ion^, i.e. E\F\^ion^+ V\pot\)
@be
@ba
The situation of an impurity in an Al or In matrix, in the
context of a Shaw potential, has not been treated elsewhere.
Our assumptions are, first of all, that the Fermi energy of the
host matrix is unchanged for small concentrations of the
impurity atoms, thus, the term E\F\^*^ will have to be taken
for the host matrix. Secondly, the terms in V\rest\ will change
in a way which will depend on the properties of the immersed
impurity atom. We propose the following reasonable assumptions:
@be
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a) the radius r\o\ of the Wigner-Seitz cell, used in the
calculation of V\pot\, will be taken equal to the radius of the
impurity including any changes of volume due to distortion,b)
the the valence Z used in V\pot\ will be the impurity valence
(the screening effectively neutralizes the impurity), c) the
term E\xc\^cb^ will be given the same value as that of the
host.
@be
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_________________________________________________________
| | | | | | | |
| | * | ion | | | | |
| Metal | E | -E | A | A | A | Z |
| | F | F | 0 | 1 | 2 | |
| | | | | | | |
_________________________________________________________
| | | | | | | |
| Bi | .053 | 1.727 | 2.41 | 2.57 | 1.78 | 5 |
| | | | | | | |
| Si | .219 | 1.526 | 2.11 | 2.56 | --- | 4 |
| | | | | | | |
| Ge | .169 | 1.507 | 2.14 | 2.47 | 2.09 | 4 |
| | | | | | | |
| Sn | .205 | 1.348 | 1.85 | 2.09 | 1.71 | 4 |
| | | | | | | |
| Pb | .140 | 1.343 | 1.92 | 1.99 | 1.39 | 4 |
| | | | | | | |
| Al | .352 | .988 | 1.39 | 1.66 | --- | 3 |
| | | | | | | |
| Ga | .247 | 1.051 | 1.53 | 1.79 | 1.44 | 3 |
| | | | | | | |
| In | .207 | .936 | 1.35 | 1.48 | .99 | 3 |
| | | | | | | |
| Tl | .142 | .968 | 1.45 | 1.54 | .88 | 3 |
| | | | | | | |
| Mg | .237 | .550 | .78 | .93 | --- | 2 |
| | | | | | | |
| Zn | .273 | .630 | 1.00 | 1.38 | .75 | 2 |
| | | | | | | |
| Cd | .206 | .610 | .91 | 1.11 | .75 | 2 |
| | | | | | | |
| Hg | .131 | .702 | 1.08 | 1.30 | 1.00 | 2 |
| | | | | | | |
| Li | .175 | .185 | .33 | --- | --- | 1 |
| | | | | | | |
_________________________________________________________
Table 3.5: Final parameters for optimized potential
in pure elements.
@ke
- 43 -
@ka
e). Results: Shaw potential parameters for solutes in Al and In.
_______________________________________________________________
@ke
@ba
The results for impurities in Al and In are given in Tables 3.6
and 3.7., where the values of A\l\ and R\l\ are the input
parameters for the final calculation of the pseudopotential.
@be
@ba
The values for the cohesive energy B were taken from a recent
compilation which appears in Kittel (1971). The ionization
energies I came from the tables of Moore (1949), the same
source was used for the atomic energy levels needed for the
determination of our constants &a and &b.
@be
@ka
_________________________________________________
| | | | | | |
| | ion | | | | Evans|
| Solute| -E\F\ | A\0\ | A\1\ | A\2\ | R\2\ |
| | | | | | |
_________________________________________________
| | | | | | |
| Si | 1.542 | 2.12 | 2.56 | --- | |
| | | | | | |
| Ge | 1.438 | 2.12 | 2.47 | 2.11 | |
| | | | | | |
| Ga | .979 | 1.51 | 1.78 | 1.45 | |
| | | | | | |
| Mg | .464 | .76 | .92 | --- | |
| | | | | | |
| Zn | .535 | .97 | 1.34 | .75 | 3.0 |
| | | | | | |
| Cd | .550 | .89 | 1.09 | .75 | 3.0 |
| | | | | | |
_________________________________________________
Table 3.6: Final parameters for Al solutes.
@ke
- 44 -
@ka
_________________________________________________
| | | | | | |
| | ion | | | | Evans|
| Solute| -E\F\ | A\0\ | A\1\ | A\2\ | R\2\ |
| | | | | | |
_________________________________________________
| | | | | | |
| Bi | 1.606 | 2.34 | 2.50 | 1.79 | |
| | | | | | |
| Sn | 1.347 | 1.85 | 2.09 | 1.71 | |
| | | | | | |
| Pb | 1.272 | 1.88 | 1.98 | 1.41 | |
| | | | | | |
| Ga | 1.065 | 1.53 | 1.79 | 1.44 | |
| | | | | | |
| Tl | .912 | 1.42 | 1.52 | .90 | |
| | | | | | |
| Mg | .577 | .78 | .93 | --- | |
| | | | | | |
| Zn | .661 | 1.01 | 1.39 | .75 | 3.0 |
| | | | | | |
| Cd | .589 | .91 | 1.10 | .75 | 3.0 |
| | | | | | |
| Hg | .623 | 1.05 | 1.27 | .98 | 3.4 |
| | | | | | |
| Li | .147 | .32 | --- | --- | |
| | | | | | |
_________________________________________________
Table 3.7: Final parameters for In solutes.
@ke
@ba
The values of R\2\ indicated for some elements in the last two
tables are the values given by Evans(1970) for those elements
he was not able to optimize. As he indicated, this failure was
due to the inclusion of the d-core states in the quantum-defect
method. All other R\l\'s can be obtained by the relation:
R\l\=Z/A\l\.
@be
- 45 -
@ka
f). Calculation of Shaw potential.
__________________________________
@ke
@ba
From these values of the A\l\'s we have calculated the
pseudopotentials, for our selection of impurities, in Al and
In. Shaw (1968,1968B) shows that his optimized pseudopotential
can be written in the following form:
@be
@ea
l\o\
Z Z
(3.20) V\o\ = - - - $S &J(R\l\ - r) [A\l\ - -] Pr\l\
r r
l=0
@ee
@ba
where &J is a step function and Pr\l\ an operator which
projects out the l'th component of the wave function. The fact
that we are only interested in potentials representing the
scattering effects of impurities allows us to neglect the
refinements of non-local screening etc., which are necessary
for calculations involving either the total energy or the
effective ion-ion interaction. We thus use essentially the
screening function as indicated in equations (3.11) and (3.12).
The unscreened potential in function of x=q/(2k\F\) can be
split up in two parts, the first is a local contribution B(q),
the second a non-local term f(k\F\,k\F\+q); the sum of these is
then locally screened.
@be
@ea
4&pZg 4&p &m\c\R\c\^3^ Zh&a\eff\
(3.21) B(q) = - ---- - -- [ ----- + ------ ] [j\o\(qR\c\)+j\2\(qR\c\)]
&O\o\q^2^ &O\o\ 3 q^2^
@ee
- 46 -
@ba
the expression for the non-local part is:
@be
@ea
l\o\
4&p
(3.22) f(k\F\,k\F\+q) = - -- $S (2l+1)P\l\(cos&j)A\l\R\l\^3^ I(R\l\,k,q)
&O\o\
l=0
@ee
@ba
with
@be
@ea
1
(3.23) I (R\l\,k,q) = $I x (x-1) j\l\((k\F\+q)R\l\x) j\l\(k\F\R\l\x) dx
0
@ee
@ba
In the equation for B(q) the first term comes from the pure
Coulomb part of the potential, the second term contains
contributions from the correlation and orthogonalization with
the conduction band of the host material. Zg and Zh are the
valences of the guest and host material, &m\c\ a quantity
analogous to the term E\c\ appearing in eqs. 3.16-19, R\c\ is
the ionic radius, &a\eff\ the value defined by Animalu; Heine
(1965) for the orthogonalization hole (note that we used the
R\c\ of the guest atom, but Zh for the valence), j\l\ and P\l\
the Bessel and Legendre functions of order l.The expression for
the non-local part is obtained by expanding the plane waves in
spherical coordinates and then evaluating the matrix elements.
@be
@ba
The screening is done with the following expression:
@be
@ea
(3.24) &e^~^ (x) = [ &e(x) - 1 ] [ 1 - f(x) ] + 1
f (x) = x^2^/ [ 2x^2^+ .5 + 1/&pk\F\]
@ee
- 47 -
@ba
where &e(x) is from eq. 3.11 and f(x) is a correction for
exchange and correlation due to Sham(1965).
@be
@ba
We hope that our new evaluation of scattering potentials will
give a more reliable estimate of resistivities, Dingle
temperatures, etc. We will use these results further on when
discussing the results of Hall effect measurements and
Wejgaard(1975) has also used them for comparison to our de
Haas-van Alphen experiments.
@be
@ka
3) Scattering.
______________
a). Transition probabilities.
_____________________________
@ke
@ba
As indicated in the preceding paragraph, the potential felt by
an electron in a metal can be represented by a relativily weak
pseudopotential. Let v(r) be the PP of a single ion, then the
potential energy of an electron at the position r is given by:
@be
@ea
(3.25) V(r) = $S v ( | r - r\j\ | )
j
@ee
@ba
where the r\j\'s are the positions of the individual ions. We
do not require any particular arrangement of the ions for the
moment. If the self-consistent screening of the electrons is
already included in the v(r)'s, then the V(r) will also be a
screened potential.
@be
@ba
To obtain the wave-function representing this electron one has
to solve the time dependent Schroedinger equation:
@be
- 48 -
@ea
(3.26) ( T + V(r) ) &F = ih' $d&F / $dt
@ee
@ba
For the scattering problem one asks for the evolution of an
electron in state |k> into a state |k+q>. Expanding &F in plane
waves:
@be
@ea
(3.27) &F(r,t) = $S a\q\(r) |k+q> exp( i&o\k+q\t )
@ee
@ba
where &o\k+q\ is the frequency corresponding to the energy of
the state |k+q>. Setting a\0\ =1, a\q\=0 for q =/ 0 at the
time 0, and substituting in eq. 3.26, on obtains in first order
perturbation:
@be
@ea
.
(3.28) i h' a\q\ exp( -i&o\k+q\t ) = <k+q|V|k> exp( -i&o\k\t )
@ee
@ba
where the matrix elements are defined as:
@be
@ea
(3.29)
<k+q|V|k> = 1/&O $I exp(-i(k+q)r) V(r) exp(ikr) d&t
&O
@ee
@ba
Integrating this with respect to time and using a\q\(0)=0, one
obtains the probability that an electron will be found in the
state |k+q> at the time t:
@be
@ea
|<k+q|V|k>|^2^ sin^2^[(&o\k\-&o\k+q\) t/2]
(3.30) a\q\^*^a\q\ = 4 ------------ -------------------
h'^2^ (&o\k\-&o\k+q\)^2^
@ee
@ba
Recognizing that the factor sin^2^(&at)/&pt&a^2^ behaves like a
&d function for long times, we obtain the usual Golden Rule
no:1 for the transition probability:
@be
- 49 -
@ea
(3.31) P\k,k+q\ = (2&p/h') * |<k+q|V|k>|^2^ * n(E)
@ee
@ba
where n(E) is the density of final states. It is to be noted
that this derivation is a first order perturbation result and
the transitions are supposed to be due to a quasi-periodic
perturbation. This will not be exactly the case for isolated
impurities which are randomly distributed in an otherwise
regular host crystal.
@be
@ba
Now we have to find out more about V(r): consider a binary
solid solution containing host ions h and guest ions g. Then
(see eq.3.25) the potential W of the dilute alloy will be:
@be
@ea
(3.32) W(r) = $S v^h^(r-r\h\) + $S v^g^(r-r\g\)
h g
@ee
@ba
for the matrix elements we get using the definition (3.29):
@be
@ea
(3.33) <k+q|W|k> = 1/N $S exp(-iqr\h\)<k+q|v^h^|k>
h
+ 1/N $S exp(-iqr\g\)<k+q|v^g^|k>
g
@ee
@ba
If we assume that the impurity atoms are substitutional, i.e.
if they are located on host atom sites, we can write:
@be
@ea
(3.34) <k+q|W|k> = 1/N $S exp(-iqr\h,g\)<k+q|v^h^|k>
h,g
+ 1/N $S exp(-iqr\g\)<k+q|v^g^-v^h^|k>
g
@ee
- 50 -
@ea
(3.35) <k+q|W|k> = S(q)<k+q|v^h^|k> + S'(q)<k+q|v^g^-v^h^|k>
@ee
@ba
The S(q) is the structure factor of the average lattice, which,
in the case of substitutional impurities, is equal to the pure
structure factor. This structure factor is non-zero only for
reciprocal lattice vector values q\o\ of q.This is an
infinitesimal set of all the q's playing a role in scattering
processes and all it does is determine the shape of the Fermi-
surface and the character of the wave functions on this same
surface. When q=/q\o\, S(q) is zero and we are left with:
@be
@ea
(3.36) <k+q|W|k> = S'\i\(q) <k+q|v^g^|k> + S'\v\(q) <k+q|v^h^|k>
@ee
@ba
Here S'(q) is the structure factor of the randomly distributed
guest atoms. It is easily seen that S'\v\(q)=-exp(-iqr\o\)/N
for a vacancy and S'\i\(q)=exp(-iqr\o\)/N for an interstitial
at the position r\o\.
@be
@ka
b). Lattice distortions.
________________________
@ke
@ba
There is one complication in giving an exact expression for
S'\v\(q): an impurity introduced in the host lattice induces a
local distortion of its surrounding. This effect is difficult
to take into account, but several authors have given a
treatment of this problem. Eshelby(1954,1956) has shown that in
the limit of continuum elasticity the distortion can be written
as
@be
@ea _
(3.37) u(r) = A r / r^3^
@ee
- 51 -
@ba
and that the constant A can be obtained from X-ray data in the
following way:
@be
@ea
&Da 4
(3.38) -- = - &p&hA&g
a 3
@ee
@ba
where &Da/a is the change in X-ray lattice constant, &h the
concentration of defects per unit volume and &g=3(1-&s)/(1+&s),
where &s is the Poisson ratio for the host material. (&s=.353
for Al, &s=.425 for In) Hence, the relative change in volume
around an impurity is:
@be
@ea
(1+&s) 1 da 4&pA
(3.39) ----- - -- = ---
(1-&s) a dC &O\o\
@ee
@ba
where c is the relative atomic concentration and &O\o\ the host
atomic volume. Now S'\v\(q) becomes:
@be
@ea
(3.40) S'\v\(q) = 1/N $S exp(-iq(r\i\+u\i\)
@ee
@ea
__
4&pA &O\o\qr\i\
(3.41) = 1/N $S exp(-iqr\i\) - --- $S ----- exp(-iqr\i\)
&O\o\ 4&pr\i\^3^
i
@ee
@ba
The first term evaluated for a vacancy, gives exp(-iqr\o\)/N
and the second sum can be shown to give (Harrison,1966):
@be
@ea
q&O\o\ j\1\(q|r\i\-r\o\|)
(3.42) g(q) exp(-iqr\o\)/N , g(q) = --- $S ------------
4&p |r\i\-r\o\|^2^
@ee
- 52 -
@ba
Hence we are left with:
@be
@ea
4&pA
(3.43) S'\v\(q) = - (1 + --- g(q) ) exp(-iqr\o\)/N
&O\o\
@ee
@ba
The term in brackets represents the effective volume of the
vacancy in the limit of long wavelengths. This change in volume
has been used by Blatt(1957) to define an effective valence Z
(1+&DV/&O\o\) for the host. The function g(q) was evaluated and
tabulated by Shaw(1968b).
@be
@ba
The assumption that the distortion can be represented by an
elastic law like in eq.3.37, has been questioned by several
authors: Benedek; Baratoff(1971), Flocken; Hardy(1970), Beal-
Monod; Kohn(1968). They calculate the ion displacement from the
force constants between the ions and, thus, obtain anisotropic
displacements. Although the treatment of anisotropic scattering
centers seems rather difficult to take into account, it would
be interesting to include these nearest neighbour displacements
in defining an effective scattering potential of the distortion
cluster. We will see in the next sections that the homovalent
impurities give rather bad results, when compared to the
experiments. It is just in these cases that the influence of
distortions could be very important.
@be
- 53 -
@ka
c). Phase shifts.(Results)
__________________________
@ke
@ba
The final form of the scattering potential W of an impurity in
a host lattice is as follows: (see eqs. 3.36 and 3.43)
@be
@ea
(3.44) <k+q|W|k> = 1/N exp(-iqr\i\) <k+q|w|k>
4&pA
<k+q|w|k> = <k+q|v^g^|k> - [1 + --- g(q) ] <k+q|v^h^|k>
&O\o\
@ee
@ba
Some typical results of our computed <k+q|w|k> are shown in
Fig.3.3. For later discussions it will be convenient to
parametrize these potentials with the phase shifts obtained in
the Born approximation and calculated with the help of eq.3.45
(see Harrison(1970) p.186)
@be
@ea
3&pZh
(3.45) &d\l\^BA^ = - ---- $I P\l\(cos &j) <k+q|w|k> d(cos &j)
8E\F\
@ee
@ba
In Tables 3.8 and 3.9 we give the resulting phase shifts for Al
and In (including the pure metal and the vacancy impurity)
along with the effective valence Z^*^ of the impurity obtained
from the Friedel sum rule:
@be
@ea
4
* 2
(3.46) Z = - $S (2l + 1) &d\l\
&p
l=0
@ee
- 54 -
@fa
FIGURE 3.3
Figure 3.3:Selected scattering potentials <k+q|w|k>
for impurities in Al and In.
@fe
- 55 -
@ka
_________________________________________________________
| | | | | | | |
| | | | | | | * |
| | &d | &d | &d | &d | &d | Z |
| | 0 | 1 | 2 | 3 | 4 | |
| | | | | | | |
_________________________________________________________
| | | | | | | |
| Al | .773 | .769 | .241 | .052 | .008 | 3.01 |
| | | | | | | |
| AlVac | .042 | .108 | .075 | .062 | .032 | 0.93 |
| | | | | | | |
| AlSi | .629 | .266 | .063 | .027 | .011 | 1.29 |
| | | | | | | |
| AlGe | .618 | .197 | .011 | -.004 | -.007 | 0.75 |
| | | | | | | |
| AlGa | .177 | -.027 | -.049 | -.014 | -.007 | -0.19 |
| | | | | | | |
| AlMg | -.515 | -.296 | -.076 | -.037 | -.016 | -1.39 |
| | | | | | | |
| AlZn | -.145 | -.163 | -.157 | .007 | .004 | -0.85 |
| | | | | | | |
| AlCd | -.196 | -.103 | -.090 | .047 | .022 | -0.27 |
| | | | | | | |
_________________________________________________________
@ke Table 3.8: Phase shifts and Z^*^ for Al.
@ka
_________________________________________________________
| | | | | | | |
| | | | | | | * |
| | &d | &d | &d | &d | &d | Z |
| | 0 | 1 | 2 | 3 | 4 | |
| | | | | | | |
_________________________________________________________
| | | | | | | |
| In | 1.116 | .788 | .158 | .055 | .006 | 3.00 |
| | | | | | | |
| InBi | .819 | .383 | .054 | -.032 | -.014 | 1.20 |
| | | | | | | |
| InSn | .395 | .220 | .082 | -.010 | -.004 | 0.87 |
| | | | | | | |
| InPb | .465 | .158 | .018 | -.027 | -.010 | 0.48 |
| | | | | | | |
| InGa | .247 | .029 | .070 | .024 | .010 | 0.60 |
| | | | | | | |
| InTl | .123 | -.016 | -.059 | -.007 | -.003 | -0.19 |
| | | | | | | |
| InMg | -.606 | -.239 | .006 | -.024 | -.003 | -0.95 |
| | | | | | | |
| InZn | -.179 | -.123 | .017 | .057 | .021 | 0.08 |
| | | | | | | |
| InCd | -.325 | -.187 | -.031 | .004 | .004 | -0.62 |
| | | | | | | |
| InHg | -.097 | -.173 | -.114 | .022 | .007 | -0.62 |
| | | | | | | |
_________________________________________________________
Table 3.9: Phase shifts and Z^*^ for In.
@ke