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Phonon-limited resistivity of aluminium using a first-principles pseudopotential
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1990 J. Phys.: Condens. Matter 2 623
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.I.
Phys.: Condens. Matter
2
(1990) 623-630. Printed in the
UK
Phonon-limited resistivity
of
aluminium using
a
first-principles pseudopotential
G
J
Vgzquez and
L
F
Magaiia
Instituto de Fisica, Universidad Nacional Aut6noma de MCxico, apartado postal 20-364,
MCxico
D
F
01000, Mexico
Received 9 May 1989
Abstract.
The phonon-limited resistivity
of
aluminium has been calculated using a local,
first-principles pseudopotential which has been useful in the calculation
of
other properties
of aluminium. This pseudopotential is obtained from the induced electron density around
an aluminium ion in an electron gas. From this pseudopotential, the interionic potential, the
phonons (which are calculated by the self-consistent harmonic approximation) and finally
the phonon-limited resistivity have been obtained. The results are very similar to those
obtained using a phenomenological, Heine-Abarenkov pseudopotential for aluminium.
1.
Introduction
The pseudopotential formulation has been very useful in the calculation of the properties
of solids. Pseudopotentials may be phenomenological or may be obtained from first
principles.
In
the former case the parameters used to define the pseudopotential are
usually determined by fitting some electronic properties, predicted with the pseudo-
potential, to experimental information such as the electrical resistivity of the metal,
the shape
of
the Fermi surface, or spectroscopic data. It is clear at present that a
pseudopotential determined in an empirical way cannot always be regarded as weak
[l],
so
that its use in obtaining the interionic potential and, from this, the phonons to be used
in the calculation of the resistivity is not justified.
In
this work we use a first-principles, local pseudopotential which is constructed
following a method proposed by Manninen
et
a1
[2],
who followed the method of Rasolt
and Taylor
[3],
with some differences.
In the approach of Rasolt and Taylor, the displaced electronic density around an ion
in an electron gas is calculated using a non-linear screening theory and the full electron-
ion pseudopotential. Then
,
a non-local pseudopotential is selected in order to reproduce,
as close as possible, the non-linear displaced electronic density by linear response theory,
except in the vicinity of the ion.
In
this way the non-linear effects are partly included in
the pseudopotential. The interionic potentials calculated using these pseudopotentials
have been used with success to calculate phonon dispersion curves in simple metals
[4,5],
and the resistivity for Rb and
Cs
[6],
for low temperatures.
In
the method of Manninen
et
al,
the starting point is also the displaced electronic
density around an impurity in
an
electron gas, which has been calculated by non-linear
screening theory. Then a local pseudopotential is defined in such a way that, in linear
response theory, this displaced electronic density is reproduced exactly, except in a small
0953-8984/90/030623
+
08
$03.50
@
1990
IOP
Publishing Ltd
623
624
G
J
Vazquez
and
L
F
Magaiia
region close to the ion. In this region a modelling of the electronic density is done in
order to remove all the wiggles of the density. This modelled density plays the role
of
the pseudodensity. The pseudopotential form factor is given in terms of the Fourier
transform of this pseudodensity and a dielectric function which satisfies the com-
pressibility theorem. In this way, some of the non-linear screening effects are also
included into the pair potential calculated from this pseudopotential. Manninen
et
a1
obtained the pseudopotential for aluminium, considering two models to calculate the
displaced electron density using non-linear screening theory. In the first model they
calculated the screening
of
an aluminium nucleus in an electron gas. In the second they
considered the nucleus embedded in a jellium vacancy. This second model gave much
better results for the cohesion energy of the metal and for the equilibrium lattice
constant, bulk modulus, vacancy formation energy and electrical resistivity of the liquid
phase. In a more recent work, Jena
et
a1
[7]
made a calculation of the phonon dispersion
curve of aluminium using the interionic potential for this material reported by Manninen
er
a1
[2], for the model of the nucleus embedded in a jellium vacancy, obtaining good
agreement with the experimental results.
In previous work we obtained this kind of pseudopotential, with the model of the
nucleus embedded in a jellium vacancy, and used it with success to calculate the lattice
specific heat of lithium
[8]
and aluminium
[9],
the pressure dependence of the lattice
specific heat of lithium
[lo]
and aluminium
[9],
and also to calculate the pressure
dependence
of
the elastic constants of aluminium and lithium
[ll].
In
this work we want to explore the applicability of the same kind of pseudopotential
to the calculation of transport properties. In particular, we are interested in comparing
the prediction of the phonon-limited resistivity of aluminium using this pseudopotential
with the prediction made using a phenomenological, Heine-Abarenkov pseudo-
potential and experimental results.
In $2 we describe briefly the method of Manninen
er
a1
to construct the local,
first-principles pseudopotential from the displaced electron density. We also show
the dielectric function used in this work and the vertex correction for the screened
pseudopotential form factor used for the calculation of the phonon-limited resistivity.
Section
3
is used to describe the phonons and the expression employed for the
phonon-limited resistivity. Our results and conclusions are given in
§
4.
We have used
atomic units (i.e. the magnitude of the electronic charge
=
electron mass
=
h
=
1).
The
energy is given in double Rydbergs.
2. The pseudopotential
The unscreened pseudopotential form factor,
v(q),
is related to the Fourier transform
of the induced charge pseudodensity,
6n(q),
by
44)
=
4nwq)E(q)/q2[1
-
4q)l
(1)
where
~(q)
is the dielectric response function of the electron gas.
6n(q)
was calculated using the induced electronic density,
6n(r),
which was com-
puted by the density functional formalism [12, 131 with a smoothing in a region near the
origin [2]. In this smoothing, the conditions that the electronic charge is conserved and
that
6n(r)
and
(a/ar)[6n(r)]
are continuous are imposed [2]. Then, equation
(1)
is used
to obtain an effective local pseudopotential, which in linear response will give the exact
induced displaced electronic density outside the region of smoothing. In this way some
of the non-linear screening effects are included into the pair potential calculated from
Phonon-limited resistivity of aluminium
625
this pseudopotential. It should be remarked that in the pseudopotential formulation,
the pseudodensity must not contain wiggles near the ion, and the induced density
calculated from density functional theory contains those wiggles in that region due to
the orthogonalisation of conduction states to core orbitals.
From pseudopotential theory and linear response theory
[
141, the interionic potential
is given by
where
r
is the separation between the two ions and
Z
is the charge of the metal ion.
density is calculated by taking the difference [2]
For the model of the nucleus embedded
in
a jellium vacancy, the induced electronic
an(r)
=
n(r)
-
nv(r>
-
2x
/qb(r)l2
(3)
b
where
n(r)
is calculated with the total charge density corresponding to a nucleus located
at the centre of a vacancy in jellium and
nv(r)
is the electron density around a jellium
vacancy alone. Charge neutrality of the metal is a necessary condition. The bound states
are represented by
qb.
The dielectric function used satisfies, by construction, the compressibility theorem
which is important in connection with the interionic potential
[2,
151.
The dielectric
function is [2,16]
where
G,(q)
is the usual Lindhard polarisability,
kTF
is the Fermi-Thomas screening constant
and
L
is the ratio
In equation (6)
p
is the chemical potential,
EF
is the Fermi energy and
where
pxc(rs)
is the exchange-correlation contribution to the chemical potential.
4q)
=
1
+
(4Jc/q2)G(q)
G(q)
=
GO(q)/[l
-
(4n/k%F>c0(q)(1
-
L)l.
(4)
(5)
=
(ap/drs)/(dEFdrs).
(6)
p(rs>
=
EF(rs)
+
~xc(~s)
On
the other hand, the screened pseudopotential form factor,
W(q),
given by
Wq)
=
(44)/&(4))W
(7)
is important in the calculation of the resistivity. The vertex correction is
C(q)
which, in
the simplest approximation for a local pseudopotential, is [16,
171
Using the expression of Gunnarson and Lundqvist
[
181
for the exchange-correlation
(which is the one we used in the calculation
of
the induced electronic density), the
corresponding value of
L
is
3.
Resistivity
and
phonons
c(q)
=
-
(4n/k$F)GO(q)(1
-
L)l.
L
=
1
-
(4/q~c~)'/~r,(l
+
[0.6213/(rS
+
11.4)]rs}.
(8)
The expression used in this work for the resistivity,
p(
T),
as function of the temperature,
T,
has been derived and discussed by several authors [19,20]:
where
W(q)
is the screened pseudopotential form factor.
E(q,
A)
is the polarisation
626
G
J
Vazquez
and
L
F
Magaiia
vector of the lattice vibration with wavevector
q
and frequency
w(q,
A),
p
is
l/kBT,
kB
being the Boltzmann constant, and
A
is a constant given by
where
M
is the ion mass and
VF
and
kF
are the electron velocity and wavevector at the
Fermi level, respectively.
The integral in equation
(9)
is over a sphere of radius
2kF.
The pseudopotential
describing electron scattering at the Fermi surface is assumed to depend only on momen-
tum transfer
q.
The Fermi surface is taken as spherical
so
that the two surface integrals
describing transitions from an initial to a final state
on
the Fermi surface can be converted
to a three-dimensional integral over
q.
A one-phonon approximation is considered
when equation
(9)
is derived
[19,20].
Since much of the aluminium Fermi surface is
free-electron-like,
it
is expected that multiple-plane-wave effects might not be very
important. In fact, one-plane-wave calculations with a spherical Fermi surface give a
reasonable description of the experimental data of the resistivity of aluminium at high
temperatures (between
-70
and
140
K)
using a Heine-Abarenkov pseudopotential
[20,21].
At low temperatures this same pseudopotential can be used but it is necessary
to consider the Fermi surface and the electron-phonon matrix elements in greater
detail
[21,22].
For temperatures above
140
K
we should expect that anharmonic effects
become more important.
On
the other hand, it is not our aim in this work to make a precise calculation of
the phonon-limited resistivity
of
aluminium for the whole range of temperatures. For
simplicity, we are interested in the temperature range (see
[21])
for which the expression
for the resistivity, given by equation
(9),
is applicable for comparison with experimental
data. We are also interested in a comparison of the prediction made using our first-
principles pseudopotential, and equation
(9),
with the results obtained with a phenom-
enological, Heine-Abarenkov pseudopotential, and the same expression for the res-
istivity. We believe that this will be sufficient to explore the applicability of our
pseudopotential to the calculation of the phonon-limited resistivity of aluminium. A
careful calculation of this property, for low temperatures, using our pseudopotential
can be performed following the method given in
[21],
where a Heine-Abarenkov
pseudopotential is employed.
It is clear from equation
(9)
that we need information about the phonon frequencies
and polarisation vectors, and that we also need the screened pseudopotential form
factor.
The interionic potential, given by equation
(2),
was obtained from the induced
pseudodensity and the dielectric function from this interionic potential we calculated
the phonons to be employed in the expression for the resistivity. The force constants
associated with our interionic potential were calculated using the self-consistent har-
monic approximation
[23-251,
The resulting set of self-consistent equations which must be solved in order to obtain
the phonon dispersion curve and force constants for this approximation is as follows:
A
=
3Qo/16MV~k~
(10)
4(k)Gw
x
D,p(k)4
(k)
(11)
P
where
is
is the component aof the polarisation vector
en(k)
and the dynamical matrix
with
Phonon-limited
resistivity
of aluminium
627
where
p/
is the vector describing the displacement of atom
1
from its equilibrium position
RI, and @.np(RI
+
p/)
is the tensor derivative of the interatomic potential evaluated at
RI
+
P/*
Finally,
where
N
is the number of ions and the sum is over the first Brillouin zone.
To solve the set of self-consistent equations (ll), (12), (13) and (14), we start with
the frequencies generated by the harmonic approximation as the first trial. Then the
convergence procedure is followed.
To calculate all the phonon frequencies and polarisation vectors entering the
expression for the resistivity (equation (9)) from the force constants obtained in the
phonon dispersion curve, we followed the method of Gilat and Raubenheimer [26]. This
method consists of solving the secular equations associated with the dynamical matrix
only at a relatively small number of points in the irreducible first Brillouin zone. Then,
by means of linear extrapolation the other phonon eigen-frequencies are extracted from
within small cubes, each centred at one point. These cubes can be arranged to fill
the entire irreducible first Brillouin zone and thus can yield the complete frequency
distribution of the crystal. Simple translations of the vectors
q
are used to complete the
integration region up to 2kF.
4.
Results
and
discussion
In order to calculate the resistivity we started by obtaining the induced density shown in
equation (3) using the density functional formalism. For this it is necessary to calculate
the displaced electronic density around an aluminium nucleus embedded into a jellium
vacancy and also the displaced electronic density around a vacancy alone. After this a
smoothing of the density near the ion is done in order to construct the displaced electronic
pseudodensity. In figure
1
we show the displaced electronic density calculated using
equation (3) and the corresponding smoothed density.
The next step was to calculate the Fourier transform of the pseudodensity. This was
achieved using the asymptotic form for
&(r)
given by
&(r)
=
B
cos(2kFr
+
q)/r3
where the constants
B
and
q~
were obtained using the last points in our calculation of
6n(r).
This asymptotic form was taken for distances larger than
R,,,
=
15.04ao,
where
a.
is the Bohr radius
(ao
=
0.529
A).
The accuracy of the Fourier transform was tested
by taking the inverse Fourier transform of
bn(q),
and the resulting difference with
respect to the original values of
Sn(r)
was less than 0.1% for each point.
With
6n(q)
and the dielectric function defined in
0
3 we could calculate the interionic
potential using equation (2). From this interionic potential we found the force constants
by the self-consistent harmonic approximation [23-251. Using these together with the
method
of
Gilat and Raubenheimer [26] we obtained the phonon frequencies and
polarisation vectors to be used in equation (9) to calculate the phonon-limited resistivity.
The results are shown in figures 2 and 3.
628
G
J
Vazquez
and
L
F
Magaria
.^
i
0
2.5
50
1.5
10.0
12.5
r
(aul
Figure
1.
Displaced electronic density
(,
.
.)
and displaced electronic pseudodensity obtained
by smoothing the displaced electronic density(-).
c
0.1
1
1
1
1 1
I
70
80
90
100
110
120
130
1LO
T
(K)
Figure
2.
Resistivity
of
aluminium
for
temperatures between 70 and
140
K:
experimental
results [27]
(-);
result of this work
(---);
result from equation
(9)
using a Heine-
Abarenkov pseudopotential (taken from [21])
(.
.
.).
As we have already mentioned, the expression used for the resistivity is adequate
for a comparison with experimental data for temperatures between
70
and
140
K
(see
[21]).
In figure
2
we show the results from our first-principles calculation compared with
experimental results
[27]
and with the calculation of
[21]
from a Heine-Abarenkov
pseudopotential for the same range of temperatures, using the same expression for the
resistivity given in equation
(9).
This figure shows a reasonable agreement between our
prediction and the experimental results and the results from the Heine-Abarenkov
pseudopotential. We have already said that it is not our aim in this work to make a
precise calculation of the phonon-limited resistivity
of
aluminium for the whole range
of temperatures. We want only to assess the suitability
of
our pseudopotential for the
calculation
of
this property of aluminium.
Phonon-limited resistivity
of
aluminium
629
0.001
I
I
I
20
LO
60
80
100 120
1LO
T
IKI
Figure
3.
Resistivity
of
aluminium for temperatures between
20
and
140
K:
experimental
results
[27]
(-);
result obtained from equation
(9)
using
our
pseudopotential
(*
*
*).
The
chain curve is calculated by equation
(9)
and a Heine-Abarenkov pseudopotential [21]. The
dotted curve is the result obtained using the scattering-time approximation,
15
plane waves
for the electron-phonon interaction, the Heine-Abarenkov pseudopotential and a
non-
spherical Fermi surface
[21].
Finally, the broken curve refers to a calculation similar to that
of
the dotted curve; the
only
difference is that the expression for the resistivity is from a
variational approximation
[21].
In figure
3
we show experimental values for the resistivity of aluminium for tem-
peratures between 20 and
140
K,
taken from [27]. We also show the predictions made
using equation (9) with our pseudopotential and with a Heine-Abarenkov pseudo-
potential (the latter is taken from [21]), for the same range of temperatures. We can see
that these two are very similar. Also in figure
3
we show a more careful calculation, from
[21], of the resistivity. In this calculation, the approximation of a spherical Fermi surface
is no longer taken and, for the electron-phonon matrix element,
15
plane waves have
been used to describe the electronic states. For the solution to the Boltzmann transport
equation two approaches were taken. One was the scattering time approximation and
the other was obtained from a variational principle
[21].
The Heine-Abarenkov pseu-
dopotential is again used in both approximations. The scattering-time approximation
gave better results for temperatures below
70
K.
However, for higher temperatures they
are practically identical and both results are very similar to ours.
From the foregoing we can say that our pseudopotential is adequate for the cal-
culation of the phonon-limited resistivity of aluminium.
A
reasonable agreement with
experimental results can be seen for the range of temperatures for which equation (9) is
expected to be adequate (between 70 and
140
K)
[21].
On the other hand, within the approximations implicit in equation (9) (which are not
valid at low temperatures), the results obtained with our first-principles pseudopotential
630
G
J
Vazquez
and
L
F
Magaria
and
a
Heine-Abarenkov pseudopotential are very similar and the latter were used
successfully in
[21]
for a careful calculation of the phonon-limited resistivity
of
aluminium.
Finally, it is reasonable
to
expect that our first-principles pseudopotential could
also be used successfully in
a
careful calculation
of
the phonon-limited resistivity
of
aluminium for a much wider range
of
temperatures.
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