Local-field correction of superlattices: exchange and correlation

Physica C 235-240 (1994) 2315-2316 North-Holland

PHYSICA

Local-field correction of superlattices: exchange and correlation A. Gold Laboratoire de Physique des Solides (URA 74), Universit6 Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse, France The local-field correction of electron superlattices is calculated by taking into account (intraplane) exchange and (interplane and intraplane) correlation effects. An analytical expression is derived. We find that the stability region of the normal Fermi liquid in superlattices is reduced by correlation effects. I. INTRODUCTION Many-body effects beyond the randomphase approximation (RPA) are described by the local-field correction (LFC) [1]. The Singwi-TosiLand-Sjtilander (STLS)-approach [2] can be used to describe the LFC in the homogeneous electron gas. In a recent paper [3] we used a twosum-rule approximation of the STLS-approach to calculate the LFC in analytical form w!'.h density dependent coefficients. Our analytical form of the LFC corresponds to a generalized Hubbard approximation and takes exchange and correlation effects into account. In this paper we study many-body effects in superlattices via the LFC. An analytical expression for the LFC is derived. The calculation extents the discussion of many-body effects in superlattices as performed in [4], where only exchange effects were taken into account. 2. MODEL AND THEORY As the model we consider planes in the xy-direction and layered in z-direction. The interplane distance in z-direction is d. The electron density in a plane is N, which defines the RPA-parameter rs as rs=(1/=Na'2)l/2. In a superlattice the electron-electron Coulomb interaction in the Fourier space is described by the potential V(q,qz). q is the two-dimensional wave number and qz is the wave vector in zdirection (<
(I) ~Lq [ch(qd)-c°s(qzd)]

~L is the dielectric constant of the background material. In [6] we have studied exchange and correlation effects within the four-sum-rule approximation (q--e0, q___>o<,_qzd=M2 , and qzd==) of the STLS-approach and we presented detailed results for d=5a* with a* as the effective Bohr radius. In this paper we present results for d=10a*. In high-Tc superconductors the parameter d/a* is large: 5
3. RESULTS AND DISCUSSION The four-sum-rule approximation for the LFC G(q,qz ) is written as C4 q [cll(qd)-cos(qzd)] F(q,qz) G(q,qz) =

~,2a) 2 [C42kF2+C12q2] 1/2 sh(qd)

with kF as Fermi's wave number and F(q,qz)=[q2+qz2] -1 [C lq2+C2qz 2 (23.

_C3arcctg[(q2d2 +kF2d 2) 1/2/C3]sin(qzd)qz/d ] . The four coefficients C i are determined by nonlinear integral equations [6]. The C(s depend on d and rs. C1, C2 and C 4 determine G(q_~0,qzd=0), G(q-->0,qzd=~ ) and G(q-->oo,qz), respectively. Within the Hubbard approximation (HA) with Ci=l and the Hart~ee-Fock approximation (HFA) with C1=0.848, C2=1.696, C3=1.769 and C4=1, only exchange effects are taken into

0921-4534D4/S07.00 © 1994 - [-Iscvicr Scicncc B.V. All rights reserved. SSDI 0921-4534(94)01723-9

A. Gold/Physica C 235-240 (1994) 2315-2316

2316

Table 1. Parameters C i for d/a*=10 and different values of rs.

Table 2. rsc and parameters C i for different values of d/a*.

d/a*

rs

C1

C2

C3

C4

d/a*

rsc

C1

C2

C3

C4

10 10 10 10 10 10 10 10

0.1 0.4 1 2 3 4 5 6

0.89 0.98 1.08 1.17 1.22 1.25 1.26 1.26

1.79 1.97 2.17 2.34 2.43 2.47 2.47 2.41

1.91

1.13 1.40 1.69 1.90 1.98 2.03 2.07 2.20

0.05 0.1 0.4 1 2 3 5 10 20

0.66 0.70 0.94 1.35 1.98 2.57 3.70 6.30 1 1.0

1.64 1.43 1.21 1.16 1.17 1.19 1.22 1.30 1.40

2.24 2.21 2.10 2 09 214 2 20 2 29 2 47 271

2.14 2.06 1.84 1.77 1.76 1.77 1.80 1.82 1.85

2.00 2.00 1.75 1.73 1.81 1.87 1.95 2.00 2.00

1.91

1.90 1.89 1.89 1.88 1.87 1.83

account and the coefficients are independent of d and rs [4]. Exchange effects dominate the manybody effects for rs--e0. With increasing r s correlation effects become increasingly important and the coefficients C i increase. =or d>a* the correlation effects are mainly intraplane correlation effects and interplane correlation effects are small. For d.0 [!] we conclude that C4<_2. At large rs we found within the four-sum-rule approximation C4>2, see Table 1. In that case we fixed C4=2 and used a three-sum-rule approximation for C 1, C2 and C3, see Table 2. The static suscept~bd~ty X(q,qz ) of the layered electron gas is given in terms of the susceptibility Xo(q) of the free electron gas and the LFC by X{q,qz) = Xo[q)/[l*v[q,qz)[l-Gi, q,qz)JXo{q) ] . (3) The ,,,o,~,-,,,,,x ,~u,,,,e of the super,a[~,~c corresponds to X0(q)/X(q,q~)<0 [4] From eq(3) we conclude that X o [ q = 0 ) , , X t q = 0 , q z d = ~ ) becomes negative for rs>rsc (N
e'l,

,-~ h i

I ;.I.~,

,. , ~ ,..,, ; ,-.,-,

,

. ~ . • ,., _

Corre!ation effects are small for d/a*=1 but increase to be important for d/a*>>1 (intraplane correlation) and for d/a*<<1 (interplane qorrelation). The critical density Nc can be written .,S

Ncd2 = C22/2=(1+a*/d) 2

(4)

Because of C2>C2,HFA=I .696 we conclude that Nc>Nc, HFA. We believe that the many-body induced instability, as discussed in this paper, is very important in high-Tc superconductors and a theory based on th~s ettect was proposed in [7j. Anyway, the analytical form of the LFC is an important quantity in layered systems as realized in artificial superlattices (semiconductors) or natural superlattices (high Tc superconductors).

REFERENCES 1. G.D. Mahan, Many-particle physics, Plenum, New York, 1991. 2. K.S. Singwi, M.P. Tosi, R.H. Land and A. Sjolander, Phys. Rev.,176 (1968)589. 3. A Gold and L. Ca!me!s, Phys. Rev. B, 48 (1993) 11622. 4. A. Gold, Z. Phys. B, 86 (1992) 193: Z. Phys. B, 90 11993) 173. 5. D. Grecu, Phys. Rev. B, 8 (1973) 1958 6. A. Gold, Z. Phys. B, 95 (1994) irl press. 7. A. Gold, Phil. Mag. Lett., 69 1,1994) 77; Z. Phys B, 94 (1994) 373.