Scattering treatment of the quasiparticle effective mass

Solid State Communications,

Pergamon

pp. 543-546,

Vol. 98, No. 6,

Copyright

0

1996 Elsevier

Printed in Great Britain.

1996

Science

Ltd

All rights reserved

003%1098/96

$12.00

+ .OO

SOO38-1098(96)00096-S SCATTERING

TREATMENT

OF

THE

QUASIPARTICLE

Andrey

Department

(Received

We obtain dimensional coming

23 June

1995; accepted

the quasiparticle

electron

gas by enforcing

from the Sjijlander-Stott

is a monotonically decreasing are in a reasonable does not explicitly

Keywords:

The

question

of the

been a cornerstone early years. particle

malization

of the Green’s

theory

theory

at the quasiparticle

=

(renormalized) potential

energy,

we will be concerned electron

coming

with T,

and

mass.

(1)

[l], and find the effective

Friedel sum rule. Our results

with the most recent calculations

a =

in the high density (T, <

1) was obtained

The

high

in (2, 31:

+ . ..I-’

and A = (4/9r)‘/“.

XT#/T,

The

separation

(2) Seitz

radius

in the units of

(1 <

8) there has been some controversy

T, <

the behavior

results

= 1/n.

of the effective

of Hedin

1, then increased than

1 for

different

and Ousaka

the

limit.

lic region

a totally

(SS)

particle

>

its minimum

in the metallic

region

3.

behavior.

decreasing

mass ratio as a func-

In the early, essentially

More recent

becoming

results

indicate

mass ratio should be a

function

of T,, thus the quasia bare electron

the Landau interaction

fits based

on the Monte

Carlo

function results

CA3

mass.

using anof Ceperly

and Alder [6] in order to obtain the vertex correction.

m*/m

at

It was argued by Yasuhara

mass is always less than

They analyzed alytic

T,

In the metal-

[4] it assumed

[5] that the effective

monotonously

agreement

mass ratio

gas.

unity

expansion

greater

based on the Fermi Liq-

the effective

approach

(4s/3)(r,ao)3

I.8 z

uid theory.

It is well known that

This

tion of the ground state density.

We will use the

give a reasonable

range, and

radius a):

with the ef-

mass by enforcing

mass

the Bohr

In this

from the Sjclander-Stott

potential

is the average interparticle

RPA,

mass.

data.

of the electron

about

electron

three-

m*/m = (1 - (2 + In a)a/2

as

(l/2)(ac,(k,Wk)/dk)Ik=kp

of the

that the effective

numerical

density

mass m’ given in terms of the

,

surface

the whole metallic

the quasi-

1 - (‘=o(h,wk)/awk)lk=k, 1+

indicate

is less than

since its

to the residue

which is related

effective

communication

effective

has

Z(b)

m is the unrenormalized

fective

parameters

by the renor-

self energy C,(k,wk)

m*Jm

short

Liquid

character

response.

Fermi

sum rule on the effective

Our results

D. dielectric

1995 by E. Mendez)

at the

with the most recent

can be characterized

function

the quasiparticle

where

surface

mass

of v, throughout

use the Fermi Liquid

As is well known from textbooks,

constant

irreducible

Fermi

function

New York, New York 1000s

27 December

the Friedel

theory.

agreement

A. metals;

of the Fermi

excitations

effective

MASS

Krakovsky

New York University,

of Physics,

EFFECTIVE

Be-

SCATTERING

544

yond RPA, the increase

in the quasiparticle

was due to both spin-parallel of the Landau tone decrease

interaction

and Sham

numerical

approach

More results consistency

OF THE QUASIPARTICLE

effective mass

and spin-antiparallel

function

of the effective

Rietschel

TREATMENT

parts

[7]. A similar

mono-

mass ratio was reported

[8]. They

designed

by

physical

behavior

the total

electron

conditions

correction

self-

can be

found in [9].

of small enough impurity

will now be the Sjijlander-Stott plasma

it is essentially

response

of the electron

mation.

It is well known that

purity.

reliable

The

SS theory

of the electron

It uses the correct

gas as an empirical

profile

this theory

equation

where the overscreening one obtains

input

With

the scattering

Friedel

sum rule is a condition

infor-

141. In our case of the jellium

of

between

it takes a familiar

electron

of charge

Here,

purity

2.

density f(q)

density in the linear response put information

ny

the parameterization

This profile

has the correct

which is responsible the Coulomb

The major

= 9-m

If the (negative)

ative.

z

16Xr,

electron

It is an in-

was taken

-(37rq3

wave scattering

density

impurity

dependence at

n

1.

is overscreening

For most of the possible

of the impurity applications

effect [12] because

of the effective

potential

used to in a self-

In our case we take the effective

from the SS theory

dart scattering

calculation

[l].

Then,

po-

we run the stan-

[15] at the Fermi momemtum

p=lcF

where 14) stands for a plane-wave p; Ho = p2/2m’ single-particle

0.01%

= 0)

(4)

several

with effective

scattering

in (7) is adjusted

accuracy.

Procedure

values of 2 within

(3),

within the domain of validity

the region of non-

and effective

V. The effective mass (6) is satisfied (6),

for the effective

within

(7) is repeated

the range

sure that the results

this feature

mass m*;

until

by a hole.

goes neg-

state with momentum

potential

(3) in the

charge is big enough the total

at the location

has an insignificant

tential

phase shifts at the Fermi

sum rule has been routinely

fashion.

6i(b)

in [ll].

IEF = (3x2n)1/3.

of the profile relation

impurities

gas

from

the cusp condition

1 + nhd(r

model of the electron

form

This

parameter

shortcoming

domain of repulsive

electron

for satisfying

an im-

source:

nind

4

around

high-q

of the band [13,

of 2 comes from the spin degeneracy,

momentum.

of the local field correction

relation

scattering

(6)

are the partial

approximation.

Ic,q are in the units of the Fermi radius

of the single particle

of

from the

is the induced

and in our calculations

on the difference

im-

consistent gives induced

the

using the SS

we can check the

the top and the bottom

adjust free parameters

This

We can extract

potential

potential

matrix

f(n)(1+Qp2+

=

profile.

scattering

and

sum rule.

The factor

4

density

linear

is capable

resulting

Friedel

a reliable

single particle

(5)

by a hole does not occur

the trace of the logarithm

is

#.d

to the range

charges

struc-

density profiles around a repulsive

density

ourselves

However,

shown

employ the microscopic

ture of the Fermi Liquid theory.

producing

theory

[l]. As was recently

a fluid description

gas, and does not directly

to 0 in this region.

put

- 0.3 < 2 < -0.03,

theory.

[lo],

density

to be on the safe side, we restrict

effective

of the two-component

Vol. 98, No. 6

a self-consistent

with the different

on the vertex

MASS

is very small and one can simply

based on the RPA scheme of Hedin.

of the approaches

Our objective

EFFECTIVE

for

(5) in order to inmass are consistent

of the SS treatment

of non-

linear screening. Our results gether suhara

with

for the effective

the results

and Ousaka

mass

of Hedin (black

(boxes),

and

ratio

(disks)

triangles),

Rietschel

and

toYa-

Sham

SCATTERING

Vol. 98, No. 6

TREATMENT

OF THE QUASIPARnCLE

1.05,

I

EFFECTIVE

MASS

54.5

title mass is always less than the bare electron mass, and decreases

as one goes further into the metallic

region.

A few words about our numerical procedure appropriate.

In the solution

approximate remaining

of (3) we could use (4) to

the high Q behavior, domain

would be

and after solving in the

match the solutions.

This procedure

is, however, very tedious and ineffective.

We found that

the solution is not affected if we solve the integral equation on the whole semiinfinite

0.85 L_T__T--7-74

Gauss-rational scattering

Fig.

Quasiparticle

of the Seitz radius.

effective

Rietschel

and Sham

[8].

indicated

by a dashed line.

ratio is a monotonously whole metallic

rule for discretizing

calculation

[O,oo) using the

the integral.

we used 12 partial waves.

black triangles

-

and Ousaka [5], triangles

-

The high density Contrary

decreasing

limit

to the RPA,

function

(1) is this

of T, in the

range.

In conclusion,

we used the SS theory

effective single particle potential. calculated

by enforcing

a monotonically

The effective mass was

the Friedel sum rule. We report

the whole metallic range. A similar cal-

can be done for the two-dimensional

is in progress now, and will be reported triangles)

are shown on Fig.

the high density limit (T, < expansion

(1) indicated

1) all results converge to the

by a dashed line. In the metallic

of the effective mass ratio.

electron This work

elsewhere.

As we can see, in

regime our results confirm the monotonously character

the

decreasing behavior of the effective mass

ratio throughout culation

to extract

gas, but requires much more numerical efforts.

(empty

In the

mass ratio as a function

Disks - present,

Hedin [4], boxes - Yasuhara

domain

decreasing

Thus, the quasipar-

The author would like to thank Professor cus for extremely the Physics gratefully

helpful discussions.

Department

J. K. Per-

Partial support

of

at the New York University

is

acknowledged.

References PI A.

Sjiilander and M. S. Stott,

Phys. Rev. B 5, 2109

M. Gel&Mann,

The Theory of Interacting

tems, Benjamin,

(1972).

PI

[71 P. Nozieres,

Phys. Rev. 106, 369 (1957).

181 H. Rietschel

Fermi Sys-

New York (1969).

and L. J. Sham, Whys. Rev. B 28,510O

(1983). [31 D. F. DuBois, Ann. Phys. 7, 1’74 (1959). 141L. Hedin, Phys. Rev. 139, A796 (1965). 151H. Yasuhara and Y. Ousaka, Solid State Comm. 64, 673 (1987). 161P. M. Ceperly and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).

PI S.

Ichimaru, Statistical

Plasma Physics,

Addison-

Wesley (1994).

1101A. Krakovsky

and J. K. Percus,

Phys. Rev. B 52,

7901 (1995).

WI B.

Farid, V. Heine, G. E. Engel, and I. J. Robertson,

Phys. Rev. B 48, 11602 (1993).

546

SCATTERING TREATMENT OF THE QUASIPARTICLE EFFECTIVE MASS

[12] A. Krakovsky and J. K. Percus, Phys. Rev. B 52, R2305 (1995). [13] J. Friedel, F’hilos. Mug. 43, 153 (1952).

Vol. 98, No. 6

[14] J. S. Langer and V. Ambegaokar Phys. Rev. 121, 1090 (1961). 1151J.

J.

Sakurai,

Modem

Addison-Wesley (1985).

Quantum

Mechanics,