A calculation of correlation functions based on band structure

Physica B 172 (1991) North-Holland

271-278

A calculation

of correlation

functions based on band structure

H. Winter”, Z. Szotekb and W.M. Temmerman” “Kernforsckur2~.~zenrrum Karlsruhe. INFP, P. 0. Box 3640, W-7_SOO Karlsruhe. “SERC Dawshury Laboratory. Warrington, WA 44A D. UK

We discuss the evaluation of functions are constructed using equations are approximated by reasonable results for a wide class the case of ceramic materials.

Germany

correlation functions in the framework of the RPA. Whereas the one-particle Green the calculated band structures, the K-matrices of the corresponding Bethc-Salpeter functional derivatives of the exchange-correlation potentials. This approach leads to of transition metals and compounds. We point out extensions of the theory. necessary in

1. Formulation In addition to one-particle quantities that are directly obtained from band structure calculations the knowledge of correlation functions is vital for the understanding of both equilibrium and nonequilibrium properties of solids. This is obvious for the spin-density-spin-density correlation function, with magnetic phenomena. However, it turns out x“, and the orbital susceptibility, x “, in connection that x’ together with the density correlation function, x”, has also to be known in order to account for superconducting properties. To calculate any correlation function, x, we need two ingredients: an approximation for the “non-interacting” part xp and the particle-hole irreducible K-matrix. The results of a satisfactory theory for x should both compare favourably to scattering experiments that probe its spectral density directly (magnetic neutron scattering in the case of x’ and x0, ELS in the case of x”) and lead to reasonable values for the derived quantities like effective masses, superconducting transition temperafirst-principles theory is desirable. A tures, etc. To fulfill these requirements a parameter-free, reasonable first approach worthwhile to be tested is the RPA-LDA approximation, where xp is approximated by the “bubble” diagram consisting of the one-particle Green function of the LDA response of HLDA to an appropriate Hamiltonian, fILDA, and K is derived from the linear unretarded external field. The Bethe-Salpeter-equation for the lattice Fourier transform x,(p~, ~‘7’; w) of x, which contains all the information, then reads [l]:

restricted to the Wigner-Seitz cells of the sites T, T’, . . , Here p, p’, . . . , are the local coordinates within one unit cell. The expression for the lattice Fourier transform of K, K,, depends on the kind of correlation function in question, in particular, KJ pi, ~‘7’)

forx,

= - ; [8V,,(n,

m) /6m]6(p

- p’)S,,,.

= K;,

(24

,

KJ PT, ~‘7’) = 0 , 0921.3526/91/$03.50

0

1991 ~ Elsevier

(2b) Science

Publishers

B.V. (North-Holland)

tor x,, It is ;I massive piece of work to evaluate k i ah the l’ollowing formula show\. quantity in terms of the angular momentum-dccomposecl Bloch state coefticienh. I?;( p. F). and the hand cncrgiea tA (. part> of the single-site wave functions,,

I‘0

obtain

Im

,,y“

we

perform

careful

Brillouin

/.onc

integration4

o\‘cr

thaw

~rcgion\

which cxprcac\ ( i , , . the

whcl-e

the\ tratli;ll

tt-an41tion4

~~1.
2. Results for the spin density

correlation

function

We have calculated x” for some systems differing in their spin tluctuation amplitudes 141. Vanadium is an example of a moderately Stoner-enhanced system (S -- 2.7). Iii fig. 1 \vc 4hon the diagonal part of the double real-space Fourier transform of Im x ” for some wa\‘c vector\ 9 in the (1. 0, 0) direction and fig. 2 shows the same for the interacting quantity 4 The sharp peaks at small frequencies and the wave vectors are related to intraband transitrona III the vicinity of the Fermi energy. The interaction enhances them and shifts them to smaller frequcncic\ III proportion to S. Because of matrix-element effects. the widths of x” and % ’ shrink to Lcro ;15 q transitions come in and cnhanccmcnt approaches zero. For increasing q more and more interband effects are of minor importance. As an example for a highly Stoner-enhanced system WC considered Pd( S = 0 ). Figure5 -3 and 4 \hov, lm x”( 9, 9’; w) and Im x‘( 9. 9’; CO). respectively. Again, enhancement effcctx arc most import;lnt ;II small wave vectors and frequencies, where interband transitions rclatcd to the scaffolding structure parts of the energy surfaces near the Fermi surface dominate. In those systems the magnitudes of the differential cross sections for magnetic neutron scattering arc too small to allow for ;I measurcmcnt of Im x‘( 9. 9: w) through the use of contemporary neutron sources. The situation is different for the nearly ferromagnetic compound Ni,Ga 151. the Stoner enh;tncemc~nt of which is estimated to be near 100. from the cxpcrimcnts of ref. [6]. Th c Grnilar compound Ni :A1 i\ ferromagnetic at low temperatures as refs. 171 and [X] have shown. Our results for Im ,y’( 9. 9: CO)in the long-wavelength limit arc displayed in tig. 5 and the quantity Im x’( 9. 9: W);OJ folded with the experimental resolution function (fig. 6) allows for :I direct comparison with the results of ref. (61

ofcorrelationfunctions

H. Winter et al. I Calculation

9

.06

--I

273

---

0.80

--

0.60

q

0.40

_ z ' S'

04

-

0.20

~--

0.10

-

”

0.80

--

0.60

-

0.20

---.

0.10

-

0.05

0.40

0.05

3

---

r.

02

cr--

A _E

__-_.-.- -------_________

c_-___..

0 0

0.2

0.4

0.6

FREQUENCY

a

0.8

1.0

I .2

(t-V)

I

I

I

1

I

I

0.2

0.4

0.6

0.8

1.0

I.:

a

FREQUENCY

(eV)

0.08 0.04 0.06

---

0.80

--

0.60 0.40

0.04

-

0.20

---

0.10

-

0.05

\ 4 0 2

b

FREQUENCY

(eV)

Fig. 1. Im x”( q. q: W) for C’ with q in the (1.0.0) direction. (a). high-frequency range; (b). low-frequency range. q is in units of 27FIa.

b

4

6 FREQUENCY

Fig. 2. Im

8

10

(eV)

,y‘(q. 9; w) for V

In agreement with the experiment are the small widths of the curves (~0.5 meV at q = 0.04 T/U), the high value of the Stoner enhancement (S = 30 at q = 0.04 T/U), which extrapolates to about 100 for q - 0 and the rapid decay of the fluctuation amplitudes with increasing q. We stress again that it is essential to solve the full Bethe-Salpeter equation (1) for x’ instead of using any scalar version of it. Most important are the matrix element effects, because the so-called generalized susceptibility, which adds up the intraband transitions while leaving the matrix elements out differs markedly from the full susceptibility. Indeed, as the investigations of refs. [9] and [lo] show, x0 behaves quite differently from x’ and xp, though one and the same particle-hole transition contributes to both quantities and this difference is entirely the result of matrix element effects.


0 020

0 015

0.010

0.005

0 0

?

t1

3. Derived

4 FREQUENCY

6

Y

I ,_ 1

(e'i)

quantities

As an example WC consider the effect of the electron-spin tluctuation coupling to the normi self-energy. that is its contribution to the electronic mass cnhanccmcnt. It i\ pwsihle to formulate thi< in close analogv to the electron-phonon coupling case. We obtain the follow~in~ wt of cquntions:

27s

H. Winter et al. I Calculation of correlution functions

q=OO2

-E

)

(du

l.W4~

-

sol 020

0

I

I

I

I

I

I

I

080

loo

Ix)

140

I60

180

I

I

14

16

IE

I

040

063

2

0

(b) qz0.05 Cd u

I 0

Frequency Fig. 5. Im

x‘(q, q; w) of Ni,Ga

II 2

4

for small wave vectors.

I

I

I

II

6

a

IO

12

Frequency

CmeV) Fig. 6. Im ,y‘(q. q; perimental resolution

J

imeV

w)/w for Ni,Ga, function.

21

I

folded

with

the

ex-

7‘he

analogy

between

electron-phonon lattice full

Fourier

dependence

on Our

V and 0.16 It

is

yields

for

enhancement

Pd.

of

The

the

7‘,

by for

coupling

Hecause

~lrouIl‘, put

the

in

/lq. order

arc compatible parameter-free like

from

using

the

Note. 2nd to

one

Grlc

get

and

h’;,

that it i4 exential

the correlation

the

with

:~ncl the

to emplo\

t’unction.

reasonable

magnetic

specific

\aluc5

the

1 ‘. 111-thcll

1hc ITI;IS\

for

band

neutron

No simple

structure

heat mcauremcnt~.

baxccl on band

KPA

\cattcring

\tructurc cros\

\imult:mcou\l!

xxtion.

scalin g hetb\ecn the latter

the fact that WC have taken

the

electron

of x‘

;I

McMillan

the

Green

mechanism

in parallel

(_Y. J’ = F, ).

F~ cc.4:“<“1’1”” which to 0. I3 with

x. the

account

the

function>

instead

of

Stoncl

quantity

of the broken

espre\sccl

transition

and ~lpE’r”“ir-natinf

and alloys.

the

the

WC have formulated and

elect-ran-phonoll

and

.\

translational

reorting

to

an\

LX_

the

(4) In

\uggcsts

that

\+c

prcvlou\

tact.

dvnamic

C‘oulomb

correct

the \ ;iluc’\ oi

a11 Eliashbergh clcctron-spin

I

Inlcl

cquatlor fluctuation

It is

range of x‘ the gap now depends K”‘““‘.

relevant

energy

for

II!

temperature\.

’ . of the order- of 0. 1.3 failccl to predict

Whereas

stands

respect

F

containing

[ I I].

coupling

alone

compounds

frcqucncy

energy-variable

fluctuation

superconducting

mechanism

parameter.

V and V-based self-energy

energy

spin

on

on the phonon

of the extended

equal

propagator

mass enhancement.

result

anomalous

F and a real Fermi

of

based

effects

the

the

quantities

by

influence

, especially

for

(I ‘b on

is obvious.

&sumptic;ns.

the

calculations action

IOU values that

for

systems

formulation

investigate

‘I‘hcsc

findings

jellium-model-like

function side

q. p and p’

variables

results

These

Eliashberg

on the other

arc

and the electronic

is observed. symmetry

their

to observe

reasonable

the

of the particle-hole

results

gratifying

and

potential

transform

enhancement.

for

I( lo)

coupling

the usual phonon region

of

the s17in-independent

to ;I cut-off

of 50[:,‘:::“.

ti““”

on both the Matsubnra

kernel. ” with

part

is essentially respect

for- I’

to the

to .\ and 1‘ ih ;I tcu

of the (‘oulomb

OUI- results

trcquc‘nclc’x

restricted

inter&on.

ha

CL’

been

;iic‘

I‘, ==20 K. if K“““”

is put

‘f‘, -9.5 if k“‘““” We transition

K

equal

to zero.

and

.

is included. conclude

that

spin

temperatures

in

fluctuations the

ca\c of

are V

indeed

and its

\‘ery

effective

compounds.

in

reducing

Ncverthelcss.

the

the

supcrconductlng

present

thcor!

stiII

H.

Winter

et al.

I Calculation

of correlation

277

functions

So it seems necessary to try a overestimates the value of T, of V (TzxP ~5.4 K) appreciably. as well by coupling the electrons to the density fluctuations. Work first-principles theory for p * “““sp’n along this line, implying the calculation of x” in the LDA-RPA approximation, is in progress.

4. Limitations

of the LDA-RPA

It is interesting to apply the LDA-RPA to the parent material of high-temperature superconductors, LazCuO,, in spite of the fact that other investigations [12-141. show that spin-polarized LSDA to be unable to account for the antiferromagnetic semiconducting ground state of this substance. In this way one learns about the degree of failure and the main causes for that failure. Our calculations of the BZ boundary in the x”(P, p;(j) [I51 s h ow the following features: as the wave vector q approaches (1, 1,O) direction, matrix element effects become more and more effective in depressing the contributions of band 17 crossing the Fermi level (fig. 7). On the other hand, interband contributions become important. As a whole, however, x ’ is too small by a factor of three to five to allow for a magnetic phase transition. This disencouraging result could cause one to abandon the LDA altogether and to resort instead to some kind of Hubbard model-like treatment of this strongly correlated electron system.

5. Summary We have demonstrated in this contribution that band structure theory based on the local spin density approach is not only able to account for the ground state properties of a wide class of materials but also leads to a reasonable description of excited states as it provides an approximation to the one-particle Green function entering the evaluation of correlation functions. In spite of the simple approximation

la)

t

Fig. 7. The contributions matrix-element effects.

of band

17 to Im ~‘(9.

9; w) of La?CuO,

both

neglecting

(hatched

lines) and including

(full

lines)

used

for

the

clcctronic

K-matrix

we

self-energies

dcnsit4’-“orr”lation for

the

obtain

and function

“classical”

;I consistent

the

quantities

along

the

superconductors

LVe conclude that the drawhacks

;trc

not

mainlv

electronic

the

result

Work ceramic

of the

one-particle

states

sp;ic~-averaged instead

of

is undet- way systems

procccdiugs,

nlethods encouraging

References

by

vc2t-v

bjth

tht: results.

that

going

promising for

RPA

and

prck~ious

this

remedy

LDA.

connection

self-interaction

example.

the

hut

solid

the

ot

our

to ;I fully

method

the

A\

gase\

wattcring

failure

of

the

(SIC‘)

[ IS]

and

prcwnt eutcnsivelv

[ lb.

171. which

or some

patxmc~et’. functioti~

cmploving

unrcali~tk

barking

with

t-c‘;tl

function\.

more

implctiicntatic)n

the

theor\

c‘clrrel;ition

;ind

interactic)n

described

is the

of

10

mitt-oscopic

lot.

cot~wqucn~‘c

umkhpp

corrections noble

lexl theorie

arc‘

neglecting

to the

no1-111;11 and the ;IIIOIIWI~~LI~

the estcn\ion

jelliitm-model-liar

eluctt-on-fluctuation

is intended beyond

both ‘I‘hc

~thovc ma> well

used therchv

encrgie4.

tor them.

use of the ad hoc ~aluc~ Ior the McMillan

the

of

thcorl from

sketched

spcc-dqwndent

in

so-called

lines

avoiding

p

unified dcrivcd

approxh

con\:ctitional

tranxition-inctal

111 the

clsewhct-c

htld

IGI\C alread! oxidc\

~‘25~ itI

01

thcx,

~tt~uc‘tut~c~ ytcldcci

[ l(Jj.