Landau interaction function and effective mass of an electron liquid

~Solid State Communications, Vol.64,No.5, pp.673-676, 1987. Printed in Great Britain.

LANDAU

INTERACTION

FUNCTION

AND EFFECTIVE

0038-1098/87 $3.00 + .00 ©1987 Pergamon Journals Ltd.

MASS OF AN E L E C T R O N

LIQUID

H. Y A S U H A R A College

of General

Education,

Tohoku University,

Sendai

980,

Japan

and Y. OUSAKA Sendai

National

College

of Technology,

(Received

29 June

The Landau i n t e r a c t i o n function investigated. It is shown that is a slowly d e c r e a s i n g function than unity t h r o u g h o u t the whole

I = ~rs/~

)I /2+--,

]-I

~ =(4/9~


, the

S

.

•

spin-susceptibility

Japan

of an e l e c t r o n liquid is the effective mass ratio m * / m of r and remains smaller region of m e t a l l i c densities.

system. The e f f e c t i v e mass ratio m * / m cannot be an exception. Why does not the c a l c u l a t e d value of m * / m obey the above general trend? The answer is that the correct e v a l u a t i o n of m * / m at m e t a l l i c densities needs a more d e t a i l e d knowledge of e l e c t r o n correlations, c o m p a r e d w i t h the e v a l u a t i o n of all the o t h e r p h y s i c a l quantities. The effect of e l e c t r o n i n t e r a c t i o n s on m * / m can be divided into the H a r t r e e Fock and the c o r r e l a t i o n c o n t r i b u t i o n s . Note that the two c o n t r i b u t i o n s shift m * / m in o p p o s i t e directions. The H a r t r e e - F o c k a p p r o x i m a t i o n makes m * / m vanish. The e x t r e m e l y e x a g g e r a t e d reduction of m * / m caused by the H a r t r e e Fock a p p r o x i m a t i o n is r e c t i f i e d by the c o r r e l a t i o n c o r r e c t i o n and the high density e x p r e s s i o n for m*/m, (i), can be obtained. It is well known that the e s s e n t i a l effect of c o r r e l a t i o n in the high density region is to screen the long-range part of the Coulomb interaction. In the m e t a l l i c density region however, it is still u n s e t t l e d w h e t h e r m * / m is smaller than unity or not since the precise e v a l u a t i o n of the correlatiol c o n t r i b u t i o n is very difficult. In the p r e s e n t c o m m u n i c a t i o n we shall i n v e s t i g a t e the s p i n - p a r a l l e l and spina n t i p a r a l l e l parts of the Landau interaction function f(p,p') of an e l e c t r o n liquid and show that m * / m is a slowly d e c r e a s i n g function of r t h r o u g h o u t the w h o l e region of m e t a l l i c densities. The Landau i n t e r a c t i o n function[3] which is the 2nd functional d e r i v a t i v e of the total e n e r g y with respect to quasi-particle's distribution function is given as

(i) )1/3

where the density p a r a m e t e r r is related to the a v e r a g e d e l ~ c t ~ n density n by (4~ /3) (r a~)~=n -~ (a0:Bohr radius) [i]. s UAs for the e v a l u a t i o n of m * / m in the m e t a l l i c density region, however, there still remain some questions. In almost all c a l c u l a t i o n s of m * / m a t t e m p t e d hitherto, it takes a m i n i m u m about r =1.0 and • s becomes larger than u n i t y ( o r approaches unity) when r goes b e y o n d a certain value in the m e t a l l i c denslty region[2]. From the following general c o n s i d e r a t i o n s we think that such a b e h a v i o u r of m*/m at m e t a l l i c densities is incorrect. The electron gas model is a o n e - p a r a m e t e r system; r is a measure of strength of electr o n S i n t e r a c t i o n s relative to the free e l e c t r o n gas. Let us consider the ratio of a physical q u a n t i t y of the intera c t i n g system to that of the free system, such as the energy ratio S(rs)/Sf(rs) , the c o m p r e s s i b i l i t y ratio .

989-31,

1987 by T.Tsuzuki)

It is well known that the e f f e c t i v e mass ratio m * / m of the q u a s i - p a r t i c l e in an e l e c t r o n liquid is smaller than unity for a finite value of r in the high density r e g i o n ( r s ~ l ) li~e m*/m=[l-(2+inl

Miyagi-Cho

•

ratio

Xf/X , q u a s i - p a r t i c l e ' s r e n o r m a l i z a t i o n constant z and the s p i n - a n t i p a r a l l e l pair d i s t r i b u t i o n function at zero separation g~ ~(0) . All these ratios are reduced to unity in the limit of r ~ 0. As r s increases, they all decrease m o n o t o n o u s l y t h r o u g h o u t the whole region of m e t a l l i c densities. Such a general trend can easily be u n d e r s t o o d from the fact that the electron gas model is a o n e - p a r a m e t e r

f(p,p')=2fo(p,p')+fe(p,p') 673

,

(2)

674

LANDAU INTERACTION FUNCTION AND EFFECTIVE MASS OF AN ELECTRON LIQUID

w h e r e f0 and f are the s p i n - i n d e p e n d e n t a n d s p i n - d e p e n d e n t parts, r e s p e c t i v e l y . A c c o r d i n g to L a n d a u ' s f o r m u l a s , m*/m,
of f0 and fe as follows:

m/m e = 1 i

< f/K= m/m*£ f ~ d x f

f ldxf

Xf/X=m/m*+

J-i

(x) ,

e

w h e r e x is the d i r e c t i o n a n g l e b e t w e e n p and p'. and fe(X) reduced

denote

(x)

(3)

'

c o s i n e of the f(x), f0(x)

the c o r r e s p o n d i n g

functions

#(0) has b e e n d e t e r m i n e d such that P--plim'h(P-P''0)=
like

formula

for
12 + 0 . 1 7 1

,

13 . (7)

This e x p r e s s i o n v a n i s h e s at r =5.3 in a c c o r d a n c e w i t h <{/< o b t a i n e d s t h e r m o d y n a m i c a l l y from the total e n e r g y e v a l u a t e d by the M o n t e Carlo m e t h o d [ 4 ] . The ist t e r m in e q u a t i o n (4) is the e x c h a n g e c o n t r i b u t i o n from all the d i a g r a m s that can be s e p a r a t e d into two parts by c u t t i n g only one C o u l o m b line. T h e r e are no d i r e c t terms c o r r e s p o n d i n g to this b e c a u s e of c h a r g e n e u t r a l i t y of the system( an e l e c t r o n gas plus a u n i f o r m p o s i t i v e b a c k g r o u n d ) ; this is g u a r a n t e e d by the r e l a t i o n v ( q ) = 0 at q=0. The 2nd t e r m is due to all the particle-hole reducible interactions that come from b o t h f0 and fe In F i g . l we have p l o t t e d ft t(x) as a f u n c t i o n of x for s e v e r a l v a l u e s of r . For very small r s ,the ist t e r m of s

(3N/8ef) f ( p , p ' ) = f ( x ) , w h e r e N is the t o t a l e l e c t r o n n u m b e r and sf the F e r m i energy. We s h a l l first c o n s i d e r the spinp a r a l l e l part of f ( p , p ' ) , i.e., f~ ~ ( p , p ' ) = f 0 ( p , p ' ) + f e ( p , p ' ) . In the case of p=p', there o c c u r s a c o m p l e t e c a n c e l l a t i o n b e t w e e n the p a r t i c l e - h o l e i r r e d u c i b l e part of f0 and its e x c h a n g e

ft #(x) is d o m i n a n t and is n e a r l y equal to the a n a l o g u e in the RPA since h(p-p',0)~l. In the limit of r ~ 0 the v a l u e of ff t(x) at x=l (i.e. p~p') tends to -o.25. At m e t a l l i c d e n s i t i e s , three v e r t e x c o r r e c t i o n s e n t e r i n g in the ist t e r m (two in the n u m e r a t o r and one

c o u n t e r p a r t of re; a v e r y l a r g e a m o u n t of c a n c e l l a t i o n can also be e x p e c t e d for p#p' and we have o m i t t e d t h e s e t e r m s from the first. An a p p r o x i m a t e e x p r e s s i o n for ft t(p,p,) is then g i v e n as, ftt (p,p,) = _ _ ~1_ . A (p-p',~ 0)v(p-p' (p-p' ,0) )h (p-p' ,0)

+_!l

fitting


dxxf(x) ,

Vol. 64, No. 5

-I 0

0

I

-I

¢(p-p') ~(p-p' ,0) ¢ (p-p') i-~ (p-p') ~ (p-p', 0)

(4)

-2

v (p-p')=4~e2/(p-p' )2 I

w h e r e ~ is the v o l u m e of the s y s t e m and z(p-p',0) the L i n d h a r d function. The d i e l e c t r i c f u n c t i o n e(p-p',0) and the p r o p e r v e r t e x p a r t A(p-p',0) are g i v e n as

--3

t

~ (p-p' ,0) =l+v(p-p')z(p-p',0)h(p-p',0)

-.4 , (5)

1 h(p-p',0)=

.(6) i-~ (p-p') ~ (p-p', 0)

In e q u a t i o n (4) we have n e g l e c t e d the w a v e n u m b e r d e p e n d e n c e of the e x c h a n g e and c o r r e l a t i o n i n t e r a c t i o n ~(p-p') and u s e d its v a l u e at p=p' since such an a p p r o x i m a t i o n is s i m p l e and t o l e r a b l e for the p r e s e n t p r o b l e m where Ip-p'l ~ 2pf (pf:Fermi w a v e n u m b e r )

-5

Fig. 1. The s p i n - p a r a l l e l p a r t of the Landau i n t e r a c t i o n function f/l(x) as a function

of

x

for

several

values

of

r

. s

Vol. 64, No. 5

LANDAU INTERACTION FUNCTION AND EFFECTIVE MASS OF AN ELECTRON LIQUID

in the d i e l e c t r i c function) have an i m p o r t a n t i n f l u e n c e on x - d e p e n d e n t f e a t u r e s of ft t(x). Their main effect is to l o w e r the v a l u e of ft t(x) in the n e i g h b o u r h o o d of x=l, c o m p a r e d w i t h the a n a l o g u e in the RPA, and to i n c l i n e f~ ~(x) to the r i g h t w i t h i n c r e a s i n g r . In the a n a l o g u e in the RPA, on the s o t h e r hand, the v a l u e of ft t(x) at x=l is i n d e p e n d e n t of r and f i x e d to -0.25. S The 2nd t e r m of e q u a t l o n (4) (due to particle-hole reducible interactions) is p o s i t i v e and r e d u c e s the l o w e r i n g of ft ~(x) c a u s e d by the ist term. As can be seen from the figure, f# ~(x) as a w h o l e is m o r e l o w e r e d and m o r e i n c l i n e d to the r i g h t w i t h i n c r e a s i n g r . This i m p l i e s t h a t f~ ~(x) m a k e s m * / m s d e c r e a s e m o n o t o n o u s l y as a f u n c t i o n of r . S u c h an e f f e c t of fit (x) at m ~ t a l l i c d e n s i t i e s has b e e n o v e r l o o k e d b e c a u s e of i n c o m p l e t e t r e a t m e n t of the vertex correction. When r r e a c h e s the c r i t i c a l v a l u e of 5.3 w h ~ r e the v e r t e x p a r t is d i v e r g e n t , the l o w e r i n g of f ~ ( x ) at x=l is t w i c e as m u c h as that in the limit of r ~ 0 . If we o m i t the 2nd term, the v a l u ~ of fiT(x) at x=l is l o w e r e d m u c h f a s t e r with increasing r and d i v e r g e n t at rs=5.3 due to t h e S f a c t o r
675

parallel particle-hole reducible interaction. In e q u a t i o n (8) w e h a v e a p p r o x i m a t e d the i n t e r a c t i o n ~' (p-p') by its v a l u e at p=p' and d e t e r m i n e d ~' (0) from the r e l a t i o n p~plim~l-~' ( 0 ) ~ ( p - p ' , 0 ) ) = X f / X ; we have u s e d n u m e r i c a l v a l u e s of Xf/X w h i c h seems to be the m o s t improved[8] . f#~(x) in the RPA comes from the particle-particle and p a r t i c l e - h o l e l a d d e r i n t e r a c t i o n s of 2nd o r d e r in the s c r e e n e d i n t e r a c t i o n and is d i v e r g e n t like in(l+x) at x=-l. Let the d e n o m i n a t o r s of the ist and 2nd terms in e q u a t i o n (8) be unity, then we have an a p p r o x i m a t e e x p r e s sion for the 2nd o r d e r p a r t i c l e p a r t i c l e and p a r t i c l e - h o l e i n t e r actions. The c o n t r i b u t i o n of m / m * - i from f~$(x) in the RPA can e a s i l y be e s t i m a t e d by d i f f e r e n t i a t i n g the s e l f - e n e r g y of ist o r d e r in the screened interaction. We have first d e t e r m i n e d a(r ) in e q u a t i o n (8) so that the RP~ l i m i t of our e x p r e s sion r e p r o d u c e s the s p i n - a n t i p a r a l l e l c o n t r i b u t i o n of m / m * - i in the RPA for 0 ~ r s ~ 5 . 3 ; in this case m* is i n d e p e n d e n t of C(rs). We have d e t e r m i n e d c(r ) in e q u a t i o n (9) so t h a t our e x p r e s s l o n s for ft7 (x) and fT$(x) ( e q u a t i o n s (4) and (8)), w h e n i n t e g r a t e d o v e r x by L a n d a u ' s f o r m u l a s , r e p r o d u c e the v a l u e of K~/< g i v e n by e q u a t i o n (7) Ifi Fig.2 we have p l o t t e d f ~~( x ) as a f u n c t i o n of x for s e v e r a l v a l u e s of

×

0

-.1

Y

. _

J

f

f~$(p,p' ) =(3N/8£f)

-i

( a ( r s ) ) 2 l n ( l + x ) / c ( r s)

J

/

/ l - a ( r s ) i n ( l + x ) / c ( r s)

-.2 + - -1

~' ( p - p ' ) ~ ( p - p ' , 0 ) ~ ' ( p - p ' ) i-~' (p-p') ~ (p-p' , 0)

(s) The f i r s t t e r m of e q u a t i o n (8) is the c o n t r i b u t i o n f r o m an i n f i n i t e series of p a r t i c l e - p a r t i c l e ladder i n t e r a c t i o n s and the 2nd t e r m the c o n t r i b u t i o n from the s p i n - a n t i -

Fig.

2.

the as r

a .

S

The

Landau function

spin-antiparallel

interaction of

x

function for

several

part

of

fT$(x) values

of

676

LANDAU INTERACTION FUNCTION AND EFFECTIVE MASS OF AN ELECTRON LIQUID

r . N o t e t h a t ft$(x) has a f i n i t e v S l u e at x=-l; the s l o p e is l o g a r i t h mically divergent, f~%(x) is i n c l i n e d to the left and e n h a n c e s m*/m. The e n h a n c e m e n t in the RPA is m u c h overestimated at m e t a l l i c d e n s i t i e s , for i n s t a n c e , by a f a c t o r of 6 at r =5.0. This is b e c a u s e ffi(x) in t~e R P A i n v o l v e s a s i n g l e l o g a r i t h m i c t e r m m u l t i p l i e d by two s c r e e n e d i n t e r a c t i o n s and is m u c h m o r e i n c l i n e d to the left w i t h i n c r e a s i n g r . In F i g . 3 we h a v e p l o t t e ~ the e f f e c t i v e m a s s r a t i o m * / m e v a l u a t e d in the p r e s e n t a p p r o x i m a t i o n as a f u n c t i o n of r . For c o m p a r i s o n the r a t i o in the RPA iSs also d r a w n in the figure. As can be seen from the figure, the p r e s e n t r a t i o m * / m is a s l o w l y d e c r e a s ing f u n c t i o n of r and r e m a i n s s m a l l e r s t h a n u n i t y t h r o u g h o u t the w h o l e r e g i o n of m e t a l l i c d e n s i t i e s . F i n a l l y we add that the s p i n - s u s c e p t i b i l i t y r a t i o Xf/X e v a l u a t e d f r o m our e x p r e s s i o n s for f#@(x) and ft&(x) u s i n g Landau~s f o r m u l a s is in f a i r l y g o o d a g r e e m e n t w i t h the most improved value[8].

Vol. 64, No. 5

/

1.05

,

2/

'~/i

1.0

3

4

5

I

I

rs

'

E .95

PRESENT

Fig.

3.

The

effective of

r

present

mass

value

ratio

of

m*/m as

the a

function

. s

Acknowledgement -We w o u l d like to t h a n k Dr. G V i g n a l e for i n f o r m i n g us t h a t the d i v e r g e n c e of the ist t e r m of e q u a t i o n (4) at x=l w h e n r approaches 5.3 can be s u p p r e s s e d by t~e 2nd term.

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