A first principles pseudopotential in the calculation of the phonon limited resistivity of sodium and potassium

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A first principles pseudopotential in the calculation of

the phonon limited resistivity of sodium and potassium

E.A. Mendoza, G.J. Vázquez, L.F. Magaña

To cite this version:

A

first

_{principles pseudopotential}

in the

calculation of

the

phonon

limited

_resistivity

of sodium and

_potassium

E. A.

Mendoza,

G. J.

_Vázquez

and L. F.

_Magaña

Instituto de

Fisica,

Universidad Nacional Autónoma de _México,

_{apartado postal}

_20-364, México

D. F. _01000, México

(Reçu

le 13

_juillet

1989,

accepté

le 8

_septembre

₁₉₈₉₎

Résumé. 2014 Nous

avons calculé la résistivité de

_phonons

à volume constant et à

_pression

constante

dans le sodium et le

_potassium

en utilisant un

_{pseudopotentiel}

local déterminé à

_partir

des

premiers principes.

Ce _type de

_{pseudopotentiel}

s’est révélé utile dans le calcul des

_propriétés

de l’aluminium et du lithium. Il s’obtient à

_partir

de la densité

_{électronique}

induite autour d’un ion dans le gaz d’électrons

correspondant.

A

_partir

de ce

_{pseudopotentiel,}

nous avons obtenu le

potentiel interionique,

les

_phonons

_(que

nous calculons dans

_{l’approximation}

_harmonique)

et,

finalement, la résistivité de

phonons.

Les résultats sont en bon accord avec

_{l’expérience.}

Abstract. 2014 We have calculated the constant volume and the constant

pressure

phonon

limited

resistivity

of sodium and

_{potassium using}

a _local, first

_{principles pseudopotential.}

This kind of

pseudopotential

has been useful in the calculation of

_properties

of aluminum and lithium. It is obtained from the induced electron

density

around an ion in the

_{corresponding}

_{electron gas. From} this

_{pseudopotential}

we obtained the interionic

_potential,

the

_phonons

(which

are calculated

_by

the harmonic

_{approximation)}

and

_finally

the

_phonon

limited

_resistivity.

The results are in

_good

agreement with

experimental

results. Classification

Physics

Abstracts 72.15E

1. Introduction.

It is clear at

_present

that a

_{pseudopotential}

determined in an

_empirical

_way can not be

considered as weak

_always

_[1],

so that its use in the calculation of the interionic

_potential

and from

this,

the

_phonons

to calculate the

_{electron-phonon}

interaction to

_predict

the

_resistivity

and other

_properties

of metals is not

_justified.

We

_{employ a}

_local,

first

_{principles pseudopotential}

constructed

_following

a method

proposed by

Manninen et al.

_[2],

who followed the

_spirit

of the method of Rasolt et al.

_[3].

In the method we have used the

_{starting point}

is the

_displaced

electronic

_density

around an

impurity

in an _{electron gas, which has} an

_{equilibrium density equal}

to that of the

corresponding

metal. This calculation is made

_by

non-linear

_{screening theory,}

and

_considering

the

_screening

of the ion within the model of the nucleus embedded in

_{a jellium}

_vacancy

_[2].

The

_{pseudodensity}

is obtained

_{by smoothing}

the non-linear

_density

in a small

_region

close to

3490

pseudopotential

form factor is

_given

in terms of the Fourier transform of the

_{pseudodensity}

and the dielectric function. The latter

satisfies,

by

construction,

the

_{compressibility}

theorem which is

_important

in connection with the interionic

_potential

_{[2, 4].}

With this definition of the

pseudopotential,

some of the non-linear

_screening

effects are included into the

_pair

potential

obtained from the

_{pseudopotential.}

In the

_approach

of Rasolt et al.

_[2]

the

_displaced

electronic

_density

around in

_impurity

in an

electron gas is also

previously

calculated

_by

non-linear

_{screening theory.}

Then a non-local

pseudopotential

is defined in order to

_reproduce,

as close as

_possible,

the non-linear

_displaced

electronic

_{density by}

linear response

theory,

except

in a

_region

close to _{the ion. In this way,}

the non-linear effects are also

_partially

included.

In

_previous

work we have

_employed

the same kind of

_{pseudopotential}

we used in this

work,

and also within the model of the nucleus embedded in a

_jellium

_{vacancy, with} success in the calculation of the lattice

_specific

heat of lithium

_[5],

and aluminum

_[6],

and _{of the pressure}

dependence

of the lattice

_specific

heat of lithium

_[7],

and aluminum

_[6],

and also in the calculation of the pressure

dependence

of the elastic constants of aluminum and lithium

_[8].

More

_recently

we also

_explored,

with

_good

results,

its

_application

in the calculation of the

phonon

limited

_resistivity

of aluminum

_[9].

In this work we calculated the

_{corresponding}

local,

first

_{principles pseudopotentials}

for sodium and

_potassium

and

_employed

them in the calculation of the constant volume and

constant _pressure

_phonon

limited

_resistivity

of these materials.

Shukla and

_Taylor

_[15]

have calculated the constant volume and the constant _pressure

resistivity

of sodium and

_{potassium using}

the non-local first

_{principles pseudopotential}

defined in reference

_[3],

_{with very}

_good

results. The results we have obtained in this work

using

local,

first

_{principles pseudopotentials}

are _{very similar} to theirs.

In the second section we describe

_briefly

the method to construct the

_{pseudopotential}

from the

_displaced

electron

_density.

We also exhibit the dielectric function we have used in this

work,

and the vertex correction for the screened

_{pseudopotential}

form

factor,

which is

important

in the calculation of the

_phonon

limited

_resistivity.

We use the third section to describe the calculation of

_phonons

and to discuss the

application

of the

_expression

for the

_phonon

limited

_resistivity

we have

_employed.

Section 4 is

for results and conclusions.

We have used atomic units

_(i.e.

_magnitude

of the electron

_charge

= electron

mass = Ir =

1).

The energy is

given

in double

Rydbergs.

2. The

_{pseudopotential}

and interionic

_potential.

The first

_step

was to

_calculate,

_using

the

_density

functional formalism

_{[10, 11],}

the

_displaced

electron densities around a nucleus embedded in a

_jellium

_{vacancy for sodium}

_(Na)

and

potassium

(K).

Taking

into account that in the

_{pseudopotential}

formulation the

_{pseudodensity}

must not contain

_wiggles

near the

ion,

these

_wiggles

in the calculated

_density

had to be removed.

From

_{pseudopotential}

_theory

and linear response

theory

[12],

the interionic

_potential

is

given by :

where r is the

_separation

between the two

_ions,

Z is the

_charge

of the metal

_ion,

e (q ),

is the dielectric _{response function of the electron gas and 8n (q )} is the Fourier transform

For the model of the nucleus embedded in a

_jellium

_{vacancy, the induced} electronic

_density

is calculated

_by

_taking

the difference

_{[2] :}

where

_{n (r) is}

calculated with the total

_{charge density corresponding}

to a nucleus located at the

center 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. Bound} states are

_represented

by t/J b.

The unscreened

_{pseudopotential}

form

factor,

v(q),

is related to the Fourier transform of the induced

_{charge pseudodensity,}

_6n(q),

_{by :}

We calculated

_8n(q)

_using

the induced electronic

_density,

_8n(r),

which was

_computed

_by

the

_density

functional

_formalism,

_{[10], [11],}

with a

_smoothing

in a

_region

near the

_origin

_[2].

In this

_smoothing,

the conditions that the electronic

_charge

is conserved and that

Sn(r),

and

_(alar)

_[8n(r)]

are

_continuous,

is

_imposed

_[2].

Then,

equation

(3)

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 this

pseudopoten-tial. 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

_{orthogonalization}

of conduction states to core orbitals.

The dielectric function we

used,

as we have

_already

said,

satisfies

_by

construction,

the

compressibility

theorem which is

_important

in connection with the interionic

_potential

_{[2, 4].}

The dielectric function is

_{[2, 13] :}

where :

and

_{Go (q )}

is the usual Lindhard

polarizability, kTF

is the Fermi-Thomas

_screening

constant,

and L is the ratio :

In

_equation

_{(6) u}

is the chemical

_{potential, EF}

is the Fermi energy and

where

_{g,,, (r,)}

is the

_{exchange-correlation}

contribution to the chemical

_potential.

3492

is

_important

in the calculation of the

_resistivity.

The vertex correction is

_{C (q )}

which,

following

the work of Rasolt

_[14]

and the work of Shukla and

Taylor

[15]

is

_{given by :}

where

_H(EF)

is the

_{quasiparticle}

renormalization constant at the Fermi level

_{given by}

Hedin

[16]

and it is a function of rs,

(B/Bo )

is the ratio of the electron gas

_{compressibility}

(B )

with that of the

_{non-interacting}

electron gas

(Bo ),

and the

_{quasiparticle}

electron mass has been taken

_equal

to the electron mass. In table 1 we

reproduce

the tabulated values of

H(EF),

from the work

_by

Hedin

_[16].

Table I.

_{- Quasiparticle}

renormalization constant at the Fermi level as

_{given by}

Hedin

_[16].

It is a

_{function of}

_rs.

In the derivation of this

_expression

for C

_{(q ),}

a non local electron-electron

interaction,

and

scattering

on the Fermi surface were considered

_{[14, 15].}

Using

the

_expression

of Gunnarson and

_Lundquist

_[17],

for

_{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.

The

_expression

for the

_resistivity,

p

( T ),

as function of the

_{temperature, T,}

we use in this work has been derived and discussed

by

several authors

_[18,

_{19] :}

where

_{W (q )}

is the screened

_{pseudopotential}

form factor.

_{e(, À )}

is the

_polarization

vector of the lattice vibration with wave vector _q and

_frequency

_{to (q, À ),}

/3

is

_{1 /kB}

T,

kB being

the Boltzmann constant and A is a constant

_{given by :}

where M is the ion mass,

_VF

and

kF

are the electron

_velocity

and wave vector at the Fermi

level,

respectively.

The

_integral

in

_equation

₍₉₎

is over a

_sphere

of radius 2

_kF.

The

_{pseudopotential describing}

electron

_scattering

at the Fermi surface is assumed to

_{depend only}

on momentum transfer q.

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

[18, 19].

In sodium and

_potassium

the Fermi surface is free electron-like and

_only

_very

_slightly

distorted. We should

_expect

that

_{multiple plane}

wave effects are not _very

_important.

At low

temperatures

this same

_{pseudopotential}

could be used but we believe that it is necessary to

consider the

_anisotropy

in the

_{electron-scattering}

_probability.

This

_anisotropy

can be considerable as a result of the

_strong

orientational

_dependence

of the

_{electron-phonon}

Umklapp

interaction at low

_temperatures

_{[20] (i.e.}

_temperatures

below

_(JD/5,

where

OD

is the

_Debye

_{temperature).}

For

_high

_temperatures

_(for

_{example, larger}

than the

_{corresponding Debye}

_{temperatures)}

we

_expect

that anharmonic effects become more

_important,

as it

happens

with the

_specific

heat

_[21,

_22].

We believe that our calculation will be sufficient to

_explore

the

_{applicability}

of our

pseudopotential

in the calculation of the

_phonon

limited

_resistivity

of sodium and

_potassium.

A careful calculation of this

_property,

for low

_{temperatures,}

_using

our

_{pseudopotential,}

can

be

_{performed following}

the method

_given

in reference

_[20]

for

_potassium,

where different

pseudopotentials

are

_employed.

It is

clear,

from

_{equation (9),}

that we need information about the

_phonon

_frequencies

and

polarization

vectors and that we also need the screened

_{pseudopotential}

form factor.

From the induced

_{pseudodensity}

and the dielectric

function,

we obtained the interionic

potential, given by equation

(1).

From this interionic

_potential

we calculated the

_phonons

to

be

_employed

in the

_expression

for the

_resistivity.

The force constants associated to our

interionic

_potential

were calculated

_using

the harmonic

_{approximation.}

To calculate all the

_phonon

_frequencies

and

_polarization

vectors

_entering

in 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

[23].

This method consists of

solving

the secular

_equations

associated with the

_dynamical

matrix

_only

at a

_relatively

small

number of

_points

₍₃₀₀₀₎

in the irreducible first Brillouin zone.

_Then,

_by

means of linear

extrapolation

the other

_phonon

_{eigenfrequencies}

are extracted from within small

cubes,

each centered 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 vectors _q are used to

_complete

the

_{integration region}

_up to 2

_kF.

4. Results and discussion.

In order to calculate the resistivities we started

_{by obtaining}

the induced densities

_according

to

equation

(2)

and

_using

the

_density

functional formalism. For this is necessary to calculate the

displaced

electronic densities around a nucleus embedded into a

_jellium

_{vacancy and also the}

displaced

electronic

_density

around a _{vacancy alone. We} made the calculations for nucleus of

sodium and

_{potassium respectively,}

and

_jelliums

_{corresponding}

to sodium and

_potassium.

After

this,

a

_smoothing

of the densities near the ions is done in order to construct the

displaced

electronic

_{pseudodensities.}

In

_figures

1 and 2 we show the

_displaced

electronic densities calculated

_{using equation}

₍₂₎

and the

_{corresponding}

smoothed densities for sodium and

_potassium.

The

_following

step

was to calculate the Fourier transform of the

_{pseudodensities.}

This was

achieved

_using

the

_asymptotic

form for

_8n(r)

_{given by :}

3494

Fig. 1

Fig. 2

Fig.

1. - Electronic densities for sodium. Calculated

displaced

electronic

_{density :}

_{(...) ;}

_displaced

electronic

pseudodensity

which is obtained

by smoothing

the calculated

_displaced

electronic

_{density :}

(2013201320132013).

In this case we are

_{taking rs}

= 3.93 _ao. The

Debye

_temperature of sodium is 156 K.

Fig.

2. - Electronic densities for

potassium.

Calculated

_displaced

electronic

_{density :}

_{(...) ;}

_displaced

electronic

_{pseudodensity}

which is obtained

_{by smoothing}

the calculated

displaced

electronic

_{density :}

(2013201320132013).

In this case we are

_{taking rs}

_{= 4.86 ao. The}

_Debye

_temperature for

_potassium

is 90.6 K.

ao is the Bohr radius

(ao

=

0.529 À ).

The accuracy of the Fourier transform was tested

taking

the inverse Fourier transform of

_{Sn (q)}

and the

_resulting

difference with

_respect

to the

original

values of

_{Sn (r)}

was less than 0.1 % for each

_point.

With

_{6 n (q )}

and the dielectric functions defined in section 3 we could calculate the interionic

_{potential, using equation}

_(1).

From this interionic

_potential

we found the force

constants

_by

the harmonic

_{approximation,}

and

_using

these and the method of Gilat and Raubenheimer

_[23],

we obtained the

_{phonon frequencies}

and

_polarization

vectors to be used in

_equation

₍₉₎

in order to calculate the

_phonon

limited

_resistivity.

The results for the constant volume

_resistivity

are shown in

_figures

3 and

4,

for sodium and

potassium respectively.

In these

_figures

we make a

_comparison

between

_experimental

results

[24, 25],

and the results from our first

_principles

calculation. We considered maximum

temperatures

a little bit above the

_Debye

_temperature,

for each material. It is known that for

sodium exists a martensitic transformation below 35 K. We are not

_considering

_temperatures

below _{this for sodium. The range of}

_temperatures

we are

_considering

for

_{potassium begins}

at

20 K. We can see from these

_figures

a

_good

_agreement

between our

_prediction

and

experimental

results. In

_figures

5 and 6 we show a

_comparison

between the

_experimental

results

_{[24, 25]}

and ours, for the constant _pressure

_resistivity,

for the same _{range of}

temperatures

for both materials.

_Again,

we can see a

_good

_agreement

between our

predictions

and

_experiment.

It is convenient to mention that to calculate the constant _pressure

resistivity

it is necessary to evaluate the interionic

_potential

and the

_{phonon frequencies}

for the

_{corresponding}

lattice

_parameter

at each

_temperature.

We should mention that it is not our

_goal

in this work to

_perform

a

_precise

calculation of the

phonon

limited

_resistivity

of sodium and

_potassium

for the _{whole range of}

temperatures.

We

want

_only

to assess the

_suitability

of our

_{pseudopotential}

for the calculation of this

property

Fig. 3

Fig. 4

Fig.

3. - Constant volume

phonon

limited

resistivity

of sodium.

Experimental

results

_{[24, 25] :}

(-) ;

result of this work :

_{(----) ;}

results of reference

_[15]

are very similar to the results of this work.

Fig.

4. - Constant volume

phonon

limited

_resistivity

of

potassium. Experimental

results

_[24,

25] :

(-) ;

result of this work :

_{(----) ;}

results of reference

_[15]

are _{very similar} to the results of this work.

Fig. 5

Fig. 6

Fig.

5. -

Constant pressure

phonon

limited

_resistivity

of sodium.

_Experimental

results

_{[24, 25] :}

(-) ;

result of this work :

_{(----) ;}

results of reference

_[15]

are _{very similar} to the results of this work.

Fig.

6. Constant pressure

phonon

limited

_resistivity

of

_{potassium. Experimental}

results

_{[24, 25] :}

(-) ;

result of this work :

_{(----) ;}

results of reference

_[15]

are very similar to the results of this work.

From above we can _{say that} our

_{pseudopotential}

is

adequate

for the calculation of the

phonon

limited

_resistivity

of sodium and

_potassium,

and that a

_good

_agreement

with

experimental

results can be seen _{for the range of}

_temperatures

for which our calculations are

3496

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