spin-susceptibility and spin-density excitations in the correlated quasi-one-dimensional electron...

*Corresponding author. Fax: #33 5 6225 7999; e-mail:[email protected].

Physica E 2 (1998) 426—430

Spin-susceptibility and spin-density excitations in the correlatedquasi-one-dimensional electron gas

A. Gold*, L. CalmelsLaboratoire de Physique des Solides, Universite& Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse, France

Abstract

For the quasi-one-dimensional electron gas with long-range Coulomb interaction we calculate the ground-state energyas a function of the spin polarization. An oscillator potential is used for the confinement. We describe the density andwidth dependence of the spin susceptibility. Analytical and numerical results are presented within a one-subband model.Using the spin susceptibility sum-rule we calculate spin-density excitations in quantum wires. ( 1998 Elsevier ScienceB.V. All rights reserved.

Keywords: Quantum wires; Spin susceptibility; Spin-density modes; Many-body Effects

1. Introduction

The transition from an unpolarized to a polar-ized three-dimensional electron gas at low elec-tron density induced by many body effects is calledthe Bloch transition [1]. Recently, we have arguedthat in a quasi-one-dimensional electron gas thistransition should occur at relatively high electrondensity N described by the critical (c) density para-meter r

4#[2,3]. A polarized state is expected for

r4'r

4#, where r

4J1/N is the density parameter.

Experimental indications for this instability foundin ballistic transport experiments with point con-tacts in GaAs/Al

xGa

1~xAs heterostructures have

already been reported [4,5]. For high density

r4(r

4#, spin-density excitations have been ob-

served experimentally in quasi-one-dimensionalquantum wires as realized in dopedGaAs/Al

xGa

1~xAs heterostructures [6—8]. In this

paper, we describe the spin-susceptibility and spin-density modes in quantum wires for 0(r

4(r

4#.

In the high-density limit many-body effects in theinteracting electron gas can be described by therandom-phase approximation (RPA) [9]. The RPAmust be improved for intermediate and low densit-ies. A very powerful approach to study many-bodyeffects beyond the RPA was developed by Singwi,Tosi, Land, and Sjolander (STLS), for a review,see Ref. [10]. This theory provides results for theground-state energy in very good agreement withquasi-exact Monte-Carlo calculations. We have re-cently applied this theory to unpolarized quasi-one-dimensional systems [11,12]. In fact, theground-state energy determines the compressibility

1386-9477/98/$19.00 ( 1998 Elsevier Science B.V. All rights reservedPII: S 1 3 8 6 - 9 4 7 7 ( 9 8 ) 0 0 0 8 8 - 5

Fig. 1. Ground-state energy e0(r4, f) versus spin polarization

parameter f for b"a* and (a) r4"1, (b) r

4"r

4#"2.98, (c)

r4"5. The dotted lines represent the e

0(r4, f)!e

0(r4, f"0)Jf2

behavior for fP0.

and the long-wavelength charge-density excita-tions. We generalized the STLS-approach to polar-ized electron systems [13]. Within this theory, wecan calculate the spin susceptibility. Spin densityexcitations are determined by the spin suscept-ibility.

2. Model

We study quantum wires with the wire axisin z-direction. The electron density defines theFermi wave number k

Fvia N"2k

F/p. The spin-

polarization parameter f with 0)f)1 is definedas f"(N

`!N

~)/N with N"N

`#N

~and the

electron densities NB

are given byN

B"N(1$f)/2 with Fermi wave numbers

kFB

"kF(1$f)"pN

B. The density parameter

r4

is given by r4"1/2Na*, where a*"e

L/m*e2 is

the effective Bohr radius defined with the effectiveelectron mass m* and the background dielectricconstant e

L. We study a one-subband model at zero

temperature. Therefore, the Fermi energy k2F`

/2m*is assumed to be smaller than the intersubbandenergy distance *E

12/Ry*"a*2/b2, which leads

for the oscillator confinement to r4'r*

4"

pb(1#f)/4a*. b is the width parameter of the wire.The energy scale is the effective Rydberg defined byRy*"1/2m*a*2. For the Planck’s constant we useh/2p"1. For r

4(r*

4at least two subbands are

occupied. The Fourier transform of the interactionpotential »(q) is given by »(q)"e2f (qb)/2e

L. In this

paper we study an oscillator confinement withf (x)"2E

1(x2)exp(x2) [14].

3. Theory

The ground-state energy per particle e0(r4, f) can

be expressed in terms of the kinetic, exchange, andcorrelation contributions as [9,11,12]

e0(r4, f)"e

,*/(r4, f)#e

%9(r4, f)#e

#03(r4, f). (1)

The kinetic energy per particle is given bye,*/

(r4, f)/Ry*"p2(1#3f2)/48r2

4. We neglect the

Hartree energy, because we consider a model,where a positive jellium gives rise to a local neutral-ity. The exchange energy can be calculated analyti-

cally [13] and the correlation energy is calculatedwithin the sum-rule version [13] of the STLS-ap-proach.

Our results for the ground-state energy as func-tion of the polarization parameter f are shown inFig. 1 for different values of r

4and b"a*. For

r4'r

4#"2.98, we find that the polarized state has

a lower energy than the unpolarized state. We men-tion that we use the condition e

0(r4#

, f"0)"e0(r4#

,f"1) for the definition of r

4#[2,3]. For fixed width

parameter b, the unpolarized electron gas is stablefor r

4(r

4#, while for fixed density parameter r

4, the

unpolarized system is stable for b'b#.

4. Spin-susceptibility

The spin susceptibility i4

[9] is defined in thelimit of a vanishing magnetic field applied parallelto the wire axis: 1/i

4"Lim[2(Ne

0)/f2]f?0

. Withi0"16r3

4a*/n2 Ry* as the spin susceptibility of the

free electron gas, the spin susceptibility including

A. Gold, L. Calmels / Physica E 2 (1998) 426—430 427

Fig. 2. Inverse spin susceptibility 1/i4

(in units of the inversespin susceptibility of the free electron gas 1/i

0) (a) versus b/a* for

r4"3 and (b) versus r

4for b"2a* in different approximations.

The solid, dashed and dotted lines represent the STLS-ap-proach, the MSA and the HFA, respectively. The solid dotsrepresent the VWN-approach. The critical confinement widthb#"1.02a* and the critical density parameter r

4#"4.25 for the

Bloch instability are shown as arrows.



0) versus r

4for

different b/a* and within the STLS-approach (solid dots). Thesolid lines represent Eq. (6) with p"0.55.

exchange and correlation is given by

i0

i4

"1#8r2

4n2 Ry*

Limf?0

C2(e

%9#e

#03)

f2 D"1#

io

i4,%9

#

8r24

n2a4. (2)

a4is the spin stiffness. i

4defines the magnetic sus-

ceptibility XM"k2

BN2i

4, where k

Bis the Bohr mag-

neton.Within the Hartree—Fock approximation (HFA),

which takes into account exchange effects, the spinsusceptibility is given by [13]

i0

i4,HFA

"1!2

p2r4f (2k

Fb) (3)

i0/i

4,HFAbecomes negative at low electron densit-

ies.Analytical results for the spin susceptibility can

also be obtained within the mean-spherical approx-imation (MSA) [15], where the LFC is set to zeroand an analytical expression for the static structurefactor is used. This approximation is very similar tothe RPA, where the static structure factor is givenby an integral over the frequency. We obtained theanalytical expression [13]

i0

i4,MSA

"

1

[1#4r4f (2k

Fb)/p2]1@2

(4)

We have also calculated the spin stiffness follow-ing a proposal given by Vosko, Wilk, and Nusair(VWN) [16] for the three-dimensional electron gas.a4is determined by

a4"a

4,MSA

e#03

(r4, f"1)!e

#03(r4, f"0)

e#03, MSA

(r4,f"1)!e

#03, MSA(r4, f"0)

.

(5)

a4,MSA

is the spin-stiffness calculated using theMSA. Our numerical results for i

0/i

4versus b/a*

and versus r4are shown in Fig. 2a and b, respective-

ly. For b'b#

and r4(r

4#, we find a very good

agreement of our numerical results calculated with-in the STLS-approach and the VWN-approach.With Fig. 2, we conclude that 1/i

4,HFA(1/i

4(

1/i4,MSA

.

An analytical (A) expression i4,A

for the spin-susceptibility is given by

i0

i4,A

"

1

2 C(1!p)i0

i4,HFA

#(1#p)i0

i4,MSA

D. (6)

We obtain a very good fit of our data with Eq. (6)for p"0.55; see Fig. 3 for i

0/i

4versus r

4and Fig. 4

for i0/i

4versus b/a*. Note the large validity range

of i0/i

4,A: r

4)r

4#and b*b

#.

In Fig. 5 we have plotted i0/i

4versus r

4/r

4#for

different values of the confinement parameter b.

428 A. Gold, L. Calmels / Physica E 2 (1998) 426—430



0) versus b/a* for

different r4and within the STLS-approach (solid dots). The solid

lines represent Eq. (6) with p"0.55.



0) versus r

4/r

4#for

b"a*/5, b"a* and b"4a* within the STLS-approach.

This shows that i0/i

4is nearly a universal function

of r4/r

4#.

5. Charge-density and spin-density modes

The dynamical response functions for charge-density fluctuations X

#(q, u) and spin-density fluc-

tuations X4(q, u) are given by X

#,4(q, u)"

X0(q, u)/[1#»

#,4(q)X

0(q, u)] [17]. X

0(q, u) is the

response function (Lindhard function) of the freeelectron gas and »

#(q)"»(q)[1!G

#(q)] and

»4(q)"!»(q)G

4(q) are the effective potentials in-

cluding the local-field corrections for charge-den-sity fluctuations G

#(q) and spin-density fluctuations

G4(q). The collective modes are given by

1#»#,4

(q)X0(q, u

#,4(q))"0. Collective charge-den-

sity modes u#(q) are determined by »

#(q), while

collective spin-density modes u4(q) are determined

by »4(q). For small wave numbers, we find

u#,4

(qP0)"vFDqD [1#o

F»

#,4(qP0)]1@2, where

oF"2m*/pk

Fis the density of states of the free

one-dimensional electron gas and vF"k

F/m*. Note

that »#(qP0)'0 and »

4(qP0)(0, which implies

that u4(qP0)(v

FDqD(u

#(qP0).

The long-wavelength behavior of the effectivepotentials is determined by the compressibilityi#and the spin-susceptibility i

4[17,18]: i

0/i

#,4"

1!oF»(qP0)G

#,4(qP0). These sum-rules allow

us to express the collective modes as a function ofi#,4

as

u#(qP0)"v

FDqD [o

F»(qP0)#i

0/i

#]1@2, (7)

u4(qP0)"v

FDqD [i

0/i

4]1@2. (8)

From our results for the spin-susceptibility, we obtainfor r

4)r

4#u

4the relation (qP0)"u

%)(qP0)[i

0/

i4]1@2(u

%)(qP0), where the electron—hole (eh)

excitation spectrum is defined by u%)

(qP0)"vFDqD.

For r4@r

4#, we derive u

4(qP0)+u

%)(qP0), which

is in agreement with experiments [6—8]. Wemention that spin-density excitations cannot bedescribed within the RPA where G

4(q)"0. We

conclude that measurements of spin-density excita-tions allow to study many-body effects in one-dimensional systems.

References

[1] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Hold,Rinehart & Winston, New York, 1976, Chap. 32.

[2] A. Gold, L. Calmels, Phil. Mag. Lett. 74 (1996) 33.[3] A. Gold, L. Calmels, Solid State Commun. 100 (1996) 137.[4] R.D. Tscheuschner, A.D. Wieck, Superlatt. Microstruct. 20

(1996) 615.[5] K.J. Thomas et al., Phys. Rev. Lett. 77 (1996) 135.[6] A.R. Gon8 i et al., Phys. Rev. Lett. 67 (1991) 3298.[7] A. Schmeller et al., Phys. Rev. B 49 (1994) 14778.[8] C. Schuller et al., Phys. Rev. B 54 (1996) R17304.[9] G.D. Mahan, Many-Particle Physics, Plenum, New York,

1991.

A. Gold, L. Calmels / Physica E 2 (1998) 426—430 429

[10] K.S. Singwi, M.P. Tosi, Solid State Phys. 36 (1981) 177.[11] L. Calmels, A. Gold, Phys. Rev. B 52 (1995) 10841.[12] L. Calmels, A. Gold, Phys. Rev. B 56 (1997) 1762.[13] L. Calmels, A. Gold, unpublished.[14] W.I. Friesen, B. Bergersen, J. Phys. C 13 (1980) 6627.

[15] J.G. Zabolitzky, Phys. Rev. B 22 (1980) 2353.[16] H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980)

1200.[17] N. Iwamoto, D. Pines, Phys. Rev. B 29 (1984) 3924.[18] A. Gold, L. Calmels, unpublished.

430 A. Gold, L. Calmels / Physica E 2 (1998) 426—430

spin-susceptibility and spin-density excitations in the correlated quasi-one-dimensional electron...

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