Dynamic response of two-dimensional electron liquids in the local-density-functional theory

PHYSICAL REVIEW

B

VOLUME 46, NUMBER 3 15JULY 1992-I

Dynamic

response

of

thoro-dimensional

electron

liquids

in

the

local-density-functional

theory

B.

Tanatar

Department ofPhysics, Bilkent University, Bilkent, 06533Ankara, Turkey (Received 3 January 1991)

The exchange-correlation potentials,

f„,

(q

=

0,u), to be used in the local-density approximation

to the density-functional theory for two-dimensional homogeneous electron liquids in the normal and

fully spin-polarized phases are calculated. We make use of accurate Monte Carlo energies and sum

rules in the local-field correction, G(q,ur), to construct

f„,

(q,u) which isvalid in the long-wavelength limit. Our results are qualitatively similar to the three-dimensional case.

I.

INTRODUCTION

The density-functional theory proposed by Hohenberg and Kohni and Kohn and Sham2 provides an excellent tool

to

study the ground-state properties

of

condensed-matter systems. The success

of

local-density approxima-tion (LDA) in time-independent problems is well known, soit is very important

to

develop and test its counterpart for time-dependent ones, especially in connection with re-sponse theory. Recently, Gross and Kohns extended the density-functional formalism

to

treat the dynamic (lin-ear) response

of

electron liquids within the LDA. They constructed an exchange-correlation functional that sat-isfied some exact conditions, including the compressibil-ity and third-frequency moment

((its))

sum rules. In their application

of

the formalism to three-dimensional

(3D)

electron liquids, using an approximate form for the

(iv ₎sum rule, correlation effects were partially neglected. Iwamoto and Gross corrected the condition based on the

(iv ₎ sum rule and obtained

a

strong frequency depen-dence

of

the exchange-correlation functional even though it was evaluated in the long-wavelength limit. A de-tailed review

of

the subject

of

time-dependent density-functional theory and its applications is recently given by Gross and Kohn.s

In the method

of

Gross and Kohn, contact is made with the homogeneous electron gas,

of

which the theo-retical formulation is based on the dynamic local-field correction

G(k,

u).

There have been several attempts to construct

a

dynamic _G(q,ur) for the elecron gas in vari-ous approximations. In relation

to

the time-dependent LDA, Dabrowski proposed

a

simple parametrized form for _G(q,

_u)

of

a

3D electron liquid, which satisfies the low- and high-frequency limits exactly, and generalizes the Gross and Kohn results

to

finite-q values. In their study

of

the high-frequency damping

of

collective exci-tations in fermion systems, Holas and Singwis also pro-posed amodel local-field _{correction G(q,}

_u),

and formally constructed the exchange-correlation potential

_f„,

_(q,

_u)

for

a

two-dimensional

_(2D)

_{electron liquid. At the time,} the static structure factor

_S(k)

and a parametrized form

of

the correlation energy

s,

(r,

)from Monte Carlo (MC) simulations were not available

to

construct G(q,io) ex-plicitly.

The purpose

of

this article istoprovide explicit expres-sions forthe exchange-correlation potentials

_f«(q

=

0,io) that enter the local-density approximation for 2D elec-tron liquids both in the normal and fully spin-polarized phases. We use the results

of a

recent MC calculations

of

2D electron gas

to

construct the model local-field cor-rection G(q,

~)

proposed by Holas and Singwis that

sat-isfies the sum rules. Note that we are interested in the dynamic correlations in the density fluctuations

of

the electron liquid both in the normal (paramagnetic) and fully spin-polarized (ferromagnetic) cases. In particular,

we donot investigate the dynamic response

of

the system due

to

spin fluctuations. The dynamic spin susceptibility within the framework

of

the time-dependent spin density-functional theory has recently been studied by Liu and Vosko. Inthe rest

of

this article, we first discuss the fre-quency dependence

of

the exchange-correlation potential,

f«(q, u),

within the linear-response density-functional theory. Our results for

_{f«(q, ~)}

in the normal and fully spin-polarized cases are presented in

Sec.

III,

and we conclude with

a

brief summary.

II.

DYNAMIC

RESPONSE

IN

THE

DENSITY-FUNCTIONAL

THEORY

To make this note self-contained we outline the basic steps

of

density-functional formalism (following closely the original work

of

Gross and Kohns) for

a

2D elec-tron liquid, and later explicitly construct the exchange-correlation potentials required in the LDA. In the density-functional theory, 3one considers an unperturbed homogeneous electronic system with density no(r) in the ground

state of

the static external potential

vo(r).

For

a

small perturbing potential

vi(r,

t)

and correspond-ing density

_{ni(r, t),}

the associated Fourier components

vi(r,

~)

and

_{ni(r, u)}

are related by

n,

(r,

~)

=

f

d'r'

y(~r

—

r'~;

~)v, (r',

~),

where y(~r

—

r'~;

~)

isthe exact density-density response function.

If

we now assume that the density

_no(r)

₊

ni(r, t)

can be reproduced by the noninteracting system in an appropriate _{potential vP(r)+v~P}

_(r,

_t),

we can write

1348

B.

TANATAR 46

Dl I',(d d

r'

_yp(~r

—

r'~;

_~)

v',

(r',

~),

₍₂₎

G(q,cu) is proposed that interpolates between the small-u

and large-~ behaviors, where yp denotes dynamic response

of

a

noninteract-ing system,

i.e.

, the Lindhard function. Defining the

exchange-correlation part

of

v& by

v,eff

(r,

~)

=

vi(r,

&u)

+

d2 I

r,

fll

I

ibad

+

vi

„,

(r,

~),

/r

—

r'

f

we introduce the functional relation

G(q,u))

=

G(q,

oo)+

i~i(q)

cu+iu2

q

(10)

where cur(q) and _~q(q) are q-dependent functions to be determined. They are obtained by taking certain limits of G(q,

~),

viz.,

lim

[~lmG(q,

~)]

=

~i(q),

vi

„,

(r, u)

=

d

r'

f„,

(~r

—

r'~;u)ni(r',

cu),

(4)

G(q, 0)

=

G(q, oo)

+

~i(q)

~2(q)

where the exchange-correlation potential

f„,

depends on the unperturbed ground-state density np(r). From the above definition

of

_f„„we

can formally write (in Fourier space),

f-(q,

~)

=

X

'(q,

~) —

X

'(q,

~) —

v(q)

Hence

_~i(q)

and u2(q) are given by llew

~i(q)

=

₃₂

q,

( )

~i(q)

G(q, 0)

—

G(q, oo) '

(14)

where v(q) is the Coulomb potential in 2D, and yp(q,

u)

is the dynamic susceptibility

of

the noninteracting sys-tem (Lindhard function). To obtain an approximation for vi

«,

the LDA is invoked, np(r) and

_ni(r,

_u)

are as-sumed

to

be slowly varying so that

_f„,

is evaluated for the local density np(r), and

ni(r',

~)

in

Eq. (4)

isreplaced by

ni(r,

u).

This yields

The high- and low-frequency limits

of

the local-field cor-rection G(q,

~)

at

long wavelengths were calculated by

Iwamoto' consistent with the compressibility and

(~

) sum rules,

limG(q, oo)

=

5 err,

+

—

—

7,

n

_{r, s,}

,

₊

—

19,

n

_r,

3d~,

_q,

vi

«(r,

u)

f„,

(q

=

0,4);

np(r))ni(r,

~)

.

₍₆₎

Hence we need

to

find an expression for the exchange-correlation potential

_f„,

of

the homogeneous electron liq-uid in the long-wavelength limit _(q

~

_0).

Using the defining equation for the local-field correction G(q,

~)

x(q,

~)

=

₁₊

v(q)G(q,

~)gp(q,

~)

'

we can make contact with the exchange-correlation po-tential

(8)

where the Coulomb potential reads v(q)

=

4rr/q (in

a.

u.

_).

The above relation holds for homogeneous systems within the linear-response theory.

Holas and Singwis derived an asymptotic expres-sion for the imaginary part

of

the dielectric func-tion, Ime(q,

u),

considering a selected set of diagrams (particle-hole diagrams) in the second-order perturbation theory. They obtained the corresponding high-frequency limit for the local-field _{correction G(q,}

_~)

in 2D

(Ref.

8)

[we use atomic units

_(a.

u.

_),

so that energies are in Ryd-bergs and lengths are in units

of

Bohr radius

_a~],

11m_q lim Im_G(q,

_u)

=

M~OO ₃₂ 4)

Strictly speaking the above expression is only valid at

high densities,

i.e.

,

r,

~

0, but it is assumed here as in

the 3Dcase to hold forawide range

of

densities. Then a simple form (following the work ofGross and I&ohn ₎for

lim G(q, 0)

=

q~p Or n2r3 dZ,

0'r4

d'Z vr 8

dr,

8 dr2 q)

(1

1rrz/8) 0

It

is clear that the exchange-correlation potential con-structed above obeys all the conditions and has proper-ties set out by Gross and Kohns for the 3D case, and in particular

f„,

satisfies the Kramers-Kronig relations, Vlz.,

(18)

d~'

_Re[f„,

_(q,

_~')]

—

_f

Im xc q (d I i

7r (d

—

4J

where

P

denotes the principal part

of

the integral. We have also checked the relation _fp

(

f

(

0 (for all

r,

) for the MC data and observed that it is fulfilled for both the normal and the fully spin-polarized liquid phases.

III.

RESULTS

We present our results

of

the frequency-dependent exchange-correlation pot,ential in the long-wavelength where n

=

2 i _and

c,

_{is the} correlation _energy.

If

we further define

_f~

=

—

limz pv(q)G(q, oo) and fp ——

DYNAMIC RESPONSE OFTWO-DIMENSIONAL ELECTRON.

.

.

1349 0 I I I i I I I ] I I I [ I I I i I 1 I I I I J I I I l I I I J I I I [ 1 I I —₅ —10 3 0' _— 15 Q —₂₀ normal 25 I I I l i

»

t

«

i I i i i I 0 2 4 6 8 10 n) (a.u.₎

FIG. 1.

Frequency dependence of the real part of the model exchange-correlation potential

f„,

_(q

=

0,ur) for a 2D electron system in the normal fluid phase for

r,

=

1 (solid line), 2 (dashed line), and 5 (dotted line).

—5 a5 —₁₀ 3 —₁₅ N 07 —20 polarized 25 I I I l I [ I l I I I I I I I I I I I 0 2 4 6 8 10 ~ (a.u.₎

FIG.

3.

Frequency dependence of the real part of the model exchange-correlation potential

f„,

(q

=

0,

~)

for a 2D electron system in the spin-polarized Quid phase for

r,

=

1

(solid line), 2 (dashed line), and 5(dotted line).

limit,

_f«(q

=

0,

~),

in the following. We display in

Fig.

1

the frequency dependence

of

the real part

of

_f„,

_(q

=

0,

u)

for

r,

=

1, 2, and 5, indicated by solid, dashed, and dotted lines, respectively. Also shown is

_{f~(r, =}

₅₎

by the dotted-dashed line. We may estimate the error that would ensue in the adiabatic approximation, when

f(q

=

0,

~)

is replaced by its value

at

~

=

0,

i.

e.

, by

fo For

r.

,

=

5 the maximum error

6

=

(fp

—

f~)/fo,

would be 70Fo. In practice, there exists

a

character-istic frequency

of

the system 0

(

a'

(

oo for which

f„,

is dominant and the errors

of

using

f«(u')

rather than the full frequency range is considerably less than our es-timate

L.

In

Fig.

2 we show the frequency dependence

of

the imaginary part

of f„~(q

=

0,

u)

for

r,

=

1, 2, and 5 indicated by solid, dashed, and dotted lines, respec-tively. The behavior

of

_f«

isqualitatively similar

to

the 3D case, but since Im

f«1/u

as

u

~

oo, it approaches zero slower than its counterpart in

3D.

The formalism we have set out

to

construct the dy-namic local-field correction and the exchange-correlation

potential may readily be extended

to

calculate these quantities for the fully spin-polarized electron liquid. The correlation energy

s,

(r,

)

(which enters fo and

_f~)

for the spin-polarized system obtained from

a

MC calcu-lation has

a

parametrized form. Note that we are not evaluating the spin-density response functions, hence the exchange-correlation potential is still given by

Eq. (5)

but the Fermi wave _{vector qF} that enters _y(q,

_u)

is the one appropriate for the fully spin-polarized liquid. Since the Fermi wave vectors

of a

spin-polarized and normal liq-uids (in 2D) are related by qFl

—

~2qF,

the expressions

for the local-field factor G(q,

~)

given in Eqs.

(15)

and

(16)

are accordingly scaled. In Figs.3 and 4we show the real and imaginary parts

of

the exchange-correlation po-tential

_f„c(q

=

O,

u)

for the fully spin-polarized electron liquid. We have plotted Figs.3 and 4 on the same scale as Figs.1 and 2, for easy comparison. We observe that Re

f«has

a

somewhat stronger frequency dependence in the normal fluid phase than that in the spin-polarized case. The same istrue forIm

f«and

the difference in the

0.0 0,0 , I I I I I I I —

₁₀

-'. _—1.

0—

—2.0 0' st 2

₃₀

—4.0 0 normal 2 4 6 8 n) (a.u.₎ 10 —2.0 Q' N 8

g0

40 I i i I 0 2 polarized I i i i I I i i I 4 6 8 ~ (a.u. ) 10

FIG. 2. Frequency dependence ofthe imaginary part of the model exchange-correlation potential

_f„,

_(q

=

0,u) for a 2D electron system in the normal fluid phase for

r,

=

1 (solid line), 2 (dashed line), and 5(dotted line).

FIG. 4. Frequency dependence of the imaginary part of the model exchange-correlation potential

_f„,

_(q

=

O,u) for a

2D electron system in the spin-polarized fluid phase for

r,

=

1 (solid line), 2 (dashed line), and 5 (dotted line).

1350

B.

TANATAR 46 high-frequency behavior

of

Imf„,

for normal and

spin-polarized electron liquids diminishes with increasing

_r,

(see Figs.2 and

4).

IV.

SUMMARY

In summary, we have presented explicit expressions for the exchange-correlation potentials

_f„,

_(q

=

O,

u)

for 2D

electron liquids in normal and spin-polarized phases as proposed by Holas and Singwis and the results are qual-itatively similar

to

the 3D case. They should be useful in future applications

of

the density-functional theory to 2D homogeneous electron liquids. The time-dependent density-functional theory in 3D has been successfully applied

to

a

range

of

problems including the photore-sponse

of

atoms and molecules, metallic and semiconduc-tor surfaces, and bulk semiconductors. The exchange-correlation potentials obtained here could beused in sim-ilar problems where the physical system has

a

2D

charac-ter, and to the extent the linear-response theory is appli-cable such as the calculation

of

atomic polarizabilities. In this work we presented

a

model

_f„,

_(q

=

0,

~)»mi-lar

to

the 3D case, using appropriate limiting forms

of

G(q,

~).

It

would be interesting to construct a G(q,cu)

with the MC structure factor

_S(q)

as input or some other scheme, such as the self-consistent field approximation

of

Singwi et al. or the approach advanced by Dabrowski, thereby avoiding the q

~

0approximation.

ACKNOWLEDGMENTS

We would like

to

thank Professor Abdus Salam, the International Atomic Energy Agency and the United Na-tions Educational, Scientific and Cultural Organizations for their hospitality during the author's stay

at

the Inter-national Centre for Theoretical Physics in Trieste where most

of

this work was done. We gratefully acknowledge

Dr.

G.

Pastore for his valuable suggestions.

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136,

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J.

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K. U. Gross, Phys. Rev. B

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S.

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149,

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B.

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B.

Tanatar and D. M. Ceperley, Phys. Rev. B

39,

5005

(1989).

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J.

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30,

3289 (1984). We noticed a typographical error in this paper in Eq.(4.18a). In the second term (the term proportional to

nr,

e,

)the numerator should read (d

—

1)(4d

—

1)instead of 4(d

—

1) (d

+

1).

'

_{See Ref. 5for a complete discussion ofthe applications of}

time-dependent density-functional theory.

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176,

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