k. s. yi and j. j. quinn- charge and spin response of the spin-polarized electron gas

8/3/2019 K. S. Yi and J. J. Quinn- Charge and spin response of the spin-polarized electron gas

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Charge and spin response of the spin-polarized electron gas

K. S. Yi Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996

and Department of Physics, Pusan National University, Pusan 609-735, Korea

J. J. Quinn Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996

and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831Received 29 March 1996; revised manuscript received 24 July 1996

The charge and spin response of a spin-polarized electron gas is investigated including terms beyond the

random-phase approximation. We evaluate the charge response, the longitudinal and transverse spin response,

and the mixed spin-charge response self-consistently in terms of the susceptibility functions of a noninteracting

system. Exchange-correlation effects between electrons of spin and are included following Kukkonen and

Overhauser, by using spin-polarization-dependent generalized Hubbard local-field factors G and G

. The

general condition for charge-density and spin-density-wave excitations of the system is discussed.

S0163-18299608040-X

Response functions relate the induced charge and spindensities to the strength of an external disturbance and play

an important role in the understanding of many-body sys-

tems. The spin-polarized electron gas SPEG is an

n-electron system with n electrons of spin and n elec-

trons of spin embedded in a uniform positive charge back-

ground. The volume of the system is taken to be unity in

this work. Previous investigations of the response of the

spin-polarized electron system were limited in scope. Some

focused on the paramagnetic response1 or on the charge and

spin-density fluctuations of a ferromagnetic electron gas

within the Hartree-Fock HF approximation.2 Others usedthe random-phase approximation RPA,3 local spin-density-

functional theory,4 or were limited to the infinitesimally po-larized electron liquid.5 However, the role of correlationsbeyond the RPA in the charge-spin response has never beenexamined explicitly for the case of arbitrary spin polarization01, where (nn)/( nn).

The purpose of this paper is to present a treatment ofcharge and spin response in a unified way. The self-consistent effective potential experienced by an electron of

spin s is expressed in terms of the charge-density fluctuation

n and the spin-density fluctuation m . The exchange-correlation interactions between electrons of the same spin(ss ) or of opposite spins (ss ) are included by employing

Hubbard-type spin-dependent local-field factors G s and

G s . The self-consistent linear response method of Kuk-

konen and Overhauser6 is extended to a SPEG by generaliz-ing the local-field factor. The charge and spin response to anarbitrary electric and magnetic disturbance is derived andcompared with the existing theories.

We consider an electron gas in the presence of a uniformpositive charge background. The imbalance in the popula-tions of up and down spins forming a system of SPEG iscaused by an effective dc magnetic field, the origin of whichneed not be specified in detail. Any degree of spin polariza-tion can be obtained by adjusting the value of the

effective magnetic field B . We assume the SPEG is disturbedby an infinitesimal external electric potential v0

ext(r) and

magnetic field b0(r, t). In response to these external electricand magnetic disturbances, charge and spin fluctuations areset up in the system, and the Hamiltonian for an electron

with spin s can be approximated as

HH0H1s , 1

where H0 is the Hamiltonian of a single quasiparticle of the

SPEG in the absence of the external disturbance. H1s is the

spin-dependent self-consistent effective perturbation. The

eigenstates and eigenvalues of H0 are given by k, and

(k). In this work we assume the spin splitting is muchgreater than the Landau level splitting and ignore any degreeof orbital quantization. Since the most general disturbancecan be decomposed into its Fourier components, we choose

the disturbances v0ext , b0, and H1

s to vary as e itiqr. The

self-consistent magnetic disturbance b is the sum of b0 and

4m , where m is the induced magnetization. The Fourier

component of the most general H1s (r,t) can be written as7

H1sq ,0sbv0

extvq

n1G ssm G s

. 2

In Eq. 2, for the sake of brevity, the q and dependence ofthe local fields, fluctuations, and disturbances has not beendisplayed. The parameter 0 is given by 0

12g*B with

g*, B , s, and v(q) being the effective g factor, the Bohrmagneton, the Pauli spin operator, and the Fourier transformof the bare Coulombic potential, respectively. Equation 2is the generalization of the effective interaction Hamiltonianof the SPEG in the presence of infinitesimal magnetic and

electric disturbances. The local fields G s and G s

are re-

sponsible for charge- and spin-induced correlation effects onan electron of spin s Ref. 6

PHYSICAL REVIEW B 15 NOVEMBER 1996-IVOLUME 54, NUMBER 19

540163-1829/96/5419/133984 /$10.00 13 398 1996 The American Physical Society


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G sG ss

xcG s s

c,G s G s s

xcG s s

c, 3

where G ssxc (G s s

xc) and G s s

c(G s s

c) account for the parallel-

spin exchange-correlation and the antiparallel-spin correla-tion effects in linear response theory.

The charge- and spin-density fluctuations n(q,),m i(q ,) (iz ,, and ) are given, in terms of the self-

consistent effective perturbation H1s

(q ,), using the equa-tion of motion of the density matrix.8

By taking the matrix element of the effective perturbation,

Eq. 2, with respect to eigenstates k,s , then Fourier trans-forming the resulting expressions and combining the resultswith the definitions of charge- and spin-density fluctuations,

we obtain the coupled equations for s 1H1s(q,)s 2 in

terms of the external charge and spin disturbances. We then

solve the coupled equations for s 1H1s s2 and substitute the

corresponding matrix elements back into the expressions ofthe fluctuations. The charge-density fluctuation n , longitu-dinal spin-density fluctuation mz , and transverse spin-density fluctuations m and m can then be expressed in

terms of a susceptibility matrix as

en0mz

ee em

me mm 0

ext

b 0z 4

and

0m0m

mm 0

0

mm b 0

b 0 . 5

Here b 0b0xi b0y , and 0

ext denotes the external electricpotential corresponding to the external disturbance

v0

ext

(

e0ext

). The various susceptibilities are written as

eeq,e 2

D

0

0160

0 0

2v0

0G

G

, 6

emq ,e0

D

0

02v

0 0G

G

,

7

meq ,e0

D

0

02v

0 0G

G

,

8

mmq ,

02

D

0

02v

0 02G

G

, 9

mmq ,

12 0

20

1 12 vG40

0, 10

and

mmq,

12 0

20

1 12 vG40

0. 11

In Eqs. 611, s1s20 is the Lindhard-type electric

(s 1s2) or spin (s1s 2) susceptibility.

s1s20 q,

k

n0

s1kqn

0

s2k

s 1kqs2k

i, 12

where n 0s(k) denotes the equilibrium distribution of qua-siparticles having spin s . The D is given by

Dq ,12 12v

01G

xcG

c

12v0G

xcG

c80

0

12 12v

01G

xcG

c

12v0G

xcG

c80

0. 13

The various terms containing factors proportional to 0 inthe expressions of susceptibilities have their origin in the useof the self-consistent magnetic disturbance. If we neglect the

induced magnetization m , those terms disappear from theexpressions for various susceptibilities. The spin-polarizationdependent Fermi wave number of the majority minorityelectrons with spin () is given by

kFkF01

1/3, 14

where kF0 is the Fermi wave number for the unpolarized

case. The expression of 0

is obtained by replacing the

quantities of spin indices in the expression of0 by that

of. The 0

and 0

appearing in Eqs. 10 and 11 are

the susceptibility functions of the spin-flip processes.In the absence of the perturbation, the charge densities

associated with the majority- and minority-spin electrons arespatially uniform but unequal. Hence they have only a non-vanishing q0 Fourier component. However, in the SPEG aspatially varying electric or magnetic disturbance with finitewave number q induces electron density fluctuation of eachspin, n and n, and hence a finite spin-density fluctua-

tion m . The ee , mm , and

mm are the ordinary chargesusceptibility, longitudinal spin susceptibility, and transversespin susceptibilities, respectively. The off-diagonal mixedsusceptibilities em (me ) correspond to the charge-densityresponse to a magnetic disturbance the longitudinal spin-density response to an electric disturbance. The susceptibili-ties given by Eqs. 611 reduce to appropriate forms forthe unpolarized or infinitesimally polarized limits. If we setall the local fields G ss 0, we obtain the RPA susceptibili-

ties of the spin-polarized system. When 0, 0

0

and GG

. In this case, ee and

mm , respectively, re-

duce to the well-known expressions5

0eeq,

e200q,

1v00q,1G

, 15

54 13 399BRIEF REPORTS


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mm 0q ,

020

0q ,

100q ,vG

40

, 16

where 00

0

02

0 . The mixed responses van-

ish in the unpolarized system.6 But, they become finite andequal to each other in the case of infinitesimally polarized

system,5 in which 0

0but G

G

. However, since

it is not necessary that G

c

G

c

in the system with finitespin polarization, we conjecture from Eqs. 7 and 8 that,for the most general case, the charge-spin cross susceptibili-ties em and me could be different. The inequality of the

cross-correlation local fields Gc

, Gc

is expected from the

fact that the density n(h)(r), associated with the exchange-

correlation hole around a given electron with spin located

at the origin, and n(h)

(r) around an electron with spin , are

given in terms of the corresponding pair correlation func-tions g(r) by

nhrn1grn1gr , 17

n

hrn1g

rn1g

r . 18

Equations 17 and 18 show that the n(h)(r)n

(h)(r) in

the SPEG. Within the HF or RPA-type approximations,

Gc0G

c, hence, we have that emme even in the

spin polarized system.2,3,9 In the local spin-densityapproximation,4 Gunnarsson and Lundqvist observed thatemme by keeping only the diagonal elements of the ma-trix C in their expressions for the off-diagonal charge-spinsusceptibilities Eq. 69 in Ref. 4. The matrix elementsCs ,s in Ref. 4 are directly related with the local fields

G s( s )

in the present work. In the same context, Eqs. 10 and

11 suggest that, in general, the transverse spin response

functions mm (q ,) and mm (q,) could be different in theSPEG. Equations 611 can be considered as definitionsof the wave-number- and frequency-dependent local fields

G and G

, in terms of the corresponding fluctuations, in

the SPEG. Within the HF approximation the local fields sat-

isfy the relation GG

and, hence, the mixed charge-spin

response functions become equal (emme ).Because the divergences in these response functions give

the collective modes in the system, various susceptibilityfunctions obtained here can be used to investigate the collec-tive modes such as charge-density and spin-density wave ex-citations in the SPEG. The coupling of charge-density wavesand spin-density waves is expected in the SPEG, and the

conditions for the spin-flip transverse modes are written,from Eqs. 10 and 11, by

1 12 vG40

00 19

and

1 12 vG40

00. 20

For example, the divergence of

mm leads us, in a long-wavelength limit, to a mode

20Bn

240vG

q2, 21

where the coefficient depends, in general, on the degree ofspin polarization of the system and reduces to

1 2

2m 162

5m0kF

for the case of complete spin polarization, 1. The secondterm of Eq. 21 disappears as the spin polarization of the

system vanishes. On the other hand, the general expressionfor the dispersion relation of the coupled longitudinal modesis given by the zeros of the D(q ,) defined by Eq. 13:

01v012G

xc 1v012G

xc

400

080v

0 02G

G

v2

0 012G

c12G

c. 22

In RPA, the above expression reduces to

1v400

016v0

0 00,

23

where 0 0 . One can expect from Eq. 23 coupled

modes of charge-density and spin-density wave excitations,with a long-wavelength limit dispersion relation given by

q PLPL

2 1 4nnn2 0e2

9

5n 2 n2

q TF2

n2

qTF2 q2, 24

qPL

4nn

n2

0

e 2

9

5n2 n

2

qTF2

n2

q TF2

1/2

q.

25

Here PL is the plasma frequency corresponding to the totalelectron density nnnand qTF()

is the Thomas-Fermi

wave number10 of the majority minority electrons of spin (). The terms involving 0 have their origin in the self-consistent magnetic responses and, especially, those termscontaining 4nn result from the coupling of the electric and

magnetic responses in Eq. 23. The terms of n2 and n

2are

due to the contribution to the noninteracting susceptibility ofthe extra kinetic energy in the SPEG Fermi sea. The minussign in Eq. 25 indicates that, in the long-wavelength region,

the slope of the dispersion of the mode is suppressed inthe presence of the coupling between the oscillations ofcharge density and spin density of the SPEG. When 0,Eq. 22 reduces as

1v001G

100vG

400, 26

which is the product of the conditions of self-sustaining os-cillations of charge and spin densities in a spin-unpolarized

system given by Eqs. 15 and 16. For the case G0, Eq.

26 reduces to 1v00 1400

00, which is the

RPA result of an unpolarized system. The first factor leads usto the well-known charge-density-wave excitation due toCoulomb interaction,10



4/4

PL 1 910q 2

q TF2 .

On the other hand, the second factor gives us the spin-density-wave excitation, in response to the self-consistentmagnetic disturbance of the spin-unpolarized Fermi sea,

PL

0e2 1

9

5

e 2

0

1

q TF2

1/2

q .

For the case of complete spin polarization, 1, the fre-quency and wave-number dependence of various longitudi-nal susceptibilities given by Eqs. 69 becomes the sameand the condition for the longitudinal collective modes isgiven by

10v12G

xc400. 27

If we set 0

0but G

G

, Eq. 22 becomes

01v0 0 1G

0 0

4v0

01G

vG

40. 28

Taking the external magnetic disturbance b0 as our effective

magnetic disturbance in Eq. 2 instead of the self-consistent

field b makes the factor 40 on the right-hand side of Eq.

28 disappear, and the expression reduces to the result of an

infinitesimally spin-polarized system.5

In summary, a unified treatment of the response of the

spin-polarized electron gas is presented in this paper and

general expressions for various susceptibility functions arederived. The present results reproduce exactly the known

results for several simple situations. We believe that our re-

sults could be useful in understanding electric, magnetic, and

optical properties of a number of spin-polarized systems

such as ferromagnetics and dilute magnetic semiconductors.

This work was supported in part by the Oak Ridge Na-

tional Laboratory, managed by Lockheed Martin Energy Re-search Corp. for the US Department of Energy under Con-tract No. DE-AC05-96OR22464. One of the authorsK.S.Y. appreciates the support by the 1995 program of theKorea Research Foundation and in part by the BSRI-96-2412program of the Ministry of Education, Korea. The authorswould like to thank J. Cooke, G. F. Giuliani, A. W. Over-hauser, and P. Vashishta for helpful comments.

1 P. Vashishta and K. S. Singwi, Solid State Commun. 13, 901

1973.2 D. J. Kim, B. B. Schwartz, and H. C. Praddaude, Phys. Rev. B 7,

205 1973.3 A. K. Rajagopal, J. Rath, and J. C. Kimball, Phys. Rev. B 7, 2657

1973.4 O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274

1976.5 S. Yarlagadda and G. F. Giuliani, Phys. Rev. B 49, 7887 1994.

6 C. A. Kukkonen and A. W. Overhauser, Phys. Rev. B 20, 550

1979.7 X. Zhu and A. W. Overhauser, Phys. Rev. B 33, 925 1986.8 M. P. Greene, H. J. Lee, J. J. Quinn, and S. Rodriguez, Phys. Rev.

177, 1019 1969.9 M. M. Pant and A. K. Rajagopal, Solid State Commun. 10, 1157

1971.10 A. L. Fetter and J. D. Walecka, Quantum Theory of Many-

Particle Systems McGraw-Hill, New York, 1971.


k. s. yi and j. j. quinn- charge and spin response of the spin-polarized electron gas

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