- Email: [email protected]
Excitonic instabilities in semiconductor superlattices in the lowest Landau level
Physica E 13 (2002) 353 – 356
www.elsevier.com/locate/physe
Excitonic instabilities in semiconductor superlattices in the lowest Landau level D.C. Marinescua; ∗ , J.J. Quinnb a Department
b Department
of Physics, Clemson University, Clemson, SC 29634, USA of Physics, University of Tennessee, Knoxville, TN 37996, USA
Abstract In a superlattice whose unit cell contains two identical quantum wells distanced by a length b, weak tunneling is allowed to occur both intra- and inter-cell. The intra-cell tunneling leads to the splitting of the electron energy levels in two minibands separated by S AS = ES − EAS , the symmetric–antisymmetric energy gap. The inter-cell tunneling broadens these minibands to a width proportional to the tunneling probability. In the presence of a magnetic 3eld, when additional splitting of the energy levels occurs, a degeneracy between minibands of opposite spins can be realized by tuning the intensity of the magnetic 3eld and the tunneling probability. Within the Hartree–Fock approximation, we investigate the existence of excitonic instabilities that can arise when, in the lowest Landau level (n = 0), the |S ↑ and |A ↓ minibands are almost degenerate. ? 2002 Published by Elsevier Science B.V. PACS: 73.21.Fg; 73.15.Lh; 73.40.Gk Keywords: Superlattice; Exciton; Spin density wave
The existence of excitonic instabilities in electron systems is suspected every time a degeneracy between energy levels is realized. Quite generally, the system can lower its total energy by transferring electrons between the two degenerate energy levels when the electron–hole correlation energy overcomes the loss of exchange energy. Such situations have been predicted to exist in single and multiple-layer systems and can involve both charge and spin transfers [1,2]. In a two-dimensional (2D) single electron layer, in the presence of the magnetic 3eld, when only the 3rst two Landau levels are occupied ( = 2), it was demonstrated that an excitonic spin instability between ∗ Corresponding author. Tel.: +864-656-5315; fax: +864656-0805. E-mail address: [email protected] (D.C. Marinescu).
|0 ↑ and |1 ↓ energy levels occurs even for positive values of ’, the diDerence between the cyclotron energy, ˜!c , and the Zeeman splitting, 2 B. The collective mode associated with this excitonic instability is soft, the system undergoing an abrupt ferromagnetic to paramagnetic magnetic ordering transition [1]. Recent experimental evidence was found to support these results [3,4]. A diDerent situation is encountered in a superlattice of identical quantum wells at = 2 coupled by tunneling. The energy levels of the electrons in the magnetic 3eld are broadened into minibands of width , proportional to the tunneling probability. The initial energy gap between |0 ↑ and |1 ↓ is further reduced by , favoring the existence of magnetic excitonic instabilities. Within the Hartree–Fock approximation it is shown that at tunneling values above a critical
1386-9477/02/$ - see front matter ? 2002 Published by Elsevier Science B.V. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 5 5 6 - 2
354
D.C. Marinescu, J.J. Quinn / Physica E 13 (2002) 353 – 356
level, the system exhibits a spin density wave (SDW) instability [2]. In this paper we analyze the existence of such spin instabilities, in the presence of a magnetic 3eld B, in a quantum-well superlattice structure in which the unit cell consists of two identical quantum wells separated by a distance b, equally spaced along the z-axis ˆ at distance A, the superlattice constant. Weak tunneling is assumed to occur inside the unit cell and between the 3rst-order neighbors in the superlattice. In a simple approximation, the superlattice is described as a periodic array of N attractive -function potentials, i.e. V (z) = − l [(z + b=2 − lA) + (z − b=2 − lA)], with l an integer. For each spin, the single energy level corresponding to the zˆ degree of freedom in the isolated well, 0 = −m2 =2˜2 , is split when weak tunneling is permitted inside the unit cell. The two possible states are described by a symmetric (S) and an antisymmetric (A) wave functions: ±(1 ± e−b )e−b(z+b=2) AS (z) = e−(z−b=2) ± e−(z+b=2) 2[1 ± e−b (1 + b)] (1 ± e−b )e−(z−b=2)
, is, in the same approximation,
where = m=˜2 . (The + sign corresponds to the symmetric solution and − sign corresponds to the antisymmetric one.) The associated energy eigenvalues are AS = 0 (1 ± e−b ). The diDerence S AS = A − S = −20 e−b de3nes the width of the gap between the two energy levels and is a parameter of the problem. When tunneling between the 3rst-order neighbors is allowed in the superlattice, these energy levels broaden into minibands that acquire a width proportional to the tunneling probability, e−a (a = A − b). In a tight binding approximation, the single electron wave function in one of the minibands is: |n; ky ; kz ; SA = eikz z uAS (kz ; z) nky (x; y) with nky (x; y) = L−1=2 eiky y Hn (x + ky ‘), where ‘ = ˜c=eB is the magnetic length, Hn (x) is the nth simple-harmonic oscillator function, and represents the spin eigenfunction. u(kz ; z) is the appropriate periodic wave function, uAS (kz ; z)= √1N l eikz (z−lA) AS (z− lA). The allowed values of ky are 2%j=L, with j an integer and L the length for periodic boundary conditions in the y direction. kz =2%j=Na, where j is an integer in the range ±N=2. The single electron energy for a spin orientation along the direction of the magnetic 3eld,
is in a paramagnetic phase. For S AS ¡ 2 B the lowest energy levels are |S ↓ and |A ↓ and the system is in a ferromagnetic ground state. We assume now that = S AS − 2 B ¿ 0. In this situation, the lowest occupied minibands are |S ↓ and |S ↑. The existence of a diDerential magnetic instability between |S ↑ and |A ↓, the closest miniband completely unoccupied, can be inferred from very simple considerations. When an electron from the |S ↑ miniband moves into the |A ↓ miniband, the energy of the transition can be seen as made up of three parts: the “kinetic energy”, which here is simply the energy gap between the two states in the absence of the electron– electron interactions, the exchange energy of the electron with all other electrons with the same spin, and the binding energy of the electron and the hole which is left in the initial miniband. The matrix elements vii ;jj (kz ; q; Q) of the Coulomb interaction between two electrons, are given by
AS (n; ky ; kz ; ) =˜!c (n + 1=2) +
˜2 ky2 2m
+ AS [1 ± 2e−a cos kz A] + sgn() B;
(2)
where is the gyromagnetic factor that describes the interaction of the electron with the magnetic 3eld. In the following considerations we will focus on the electrons in the minibands that originate in the lowest Landau level, and henceforth set n = 0. Furthermore, we consider that only the lowest two minibands are occupied. The magnetic con3guration of the ground state is determined by the relationship between the various energies introduced in the problem. For S AS ¿ 2 B, the occupied levels are |S ↓ and |S ↑ and the system for z 6 − b=2; for − b=2 6 z 6 b=2;
(1)
for b=2 6 z;
vii ;jj (kz ; q; Q) =
e2 Fii ;jj (kz ; qz ; Qz ) ‘
D.C. Marinescu, J.J. Quinn / Physica E 13 (2002) 353 – 356 ∞
dqx −‘2 =2(qy2 +qx2 −2iqx Qy ) e 2 −∞ q
‘2 2 2 × n; 0 m; 0 + 1 − (qy + qx ) n; 0 m; 1 2
2 ‘2 2 2 (3) + 1 − (qy + qx ) n; 1 m; 1 2
14
13
W' = 0
11
2
with q = k − k and n; m the Kronecker delta. i; j are subband indices for electron states before the interaction and stand for S and A, while i ; j are subband indices for electron states after the interaction. The matrix element diDers from its 2D counterpart obtained in Ref. [1] by the form factor Fii ;jj (kz ; qz ; Qz ) that describes the exchange of momentum along the z direction. We note that Fi; i ;jj (kz ; qz ; Qz )= Fii (kz )Fjj∗ (kz + Qz ). To 3rst order in inter-cell tunneling probability, we readily obtain that
FSS 1 42 (qz ; kz ) = 42 + q2 1 ± e−b (1 + b) FAA
qz b 2 qz b qz b −b ±e + cos sin × cos 2 2 qz 2 qz a 2 qz a qz × ±e−a cos ; cos kz + A sin + 2 qz 2 2 (4)
FERRO
12
hωs(e / l)
×
355
10
SDW
9
8
PARA W=0
7
6 0.02
0.04
0.06
0.08
0.10
t
0.12
0.14
0.16
0.18
tc
Fig. 1. The magnetic phase diagram of a superlattice with a=0:75l and b = 0:5l (l = e=˜B) is represented in the ˜!S − t plane. W = 0 and W = 0 are the excitonic instability curves for the paramagnetic and ferromagnetic ground states, respectively. The intermediate region, at small tunneling values, corresponds to a spatially varying magnetic moment, as in SDW. At tunneling amplitudes larger than a critical value tc , the system undergoes an abrupt paramagnetic (P) to ferromagnetic (F) transition.
where as before, the upper sign corresponds to FSS and the lower to FAA . Additionally, form factors of the type FAS are also considered as
the states |S; ky ; kz ; ↑ and |A; ky + Qy ; kz + Qz ; ↓ is given by
FAS (qz ; kz )
ex W = − 20 [cos(kz + Qz )A + cos kz A] − SS (kz )
=
4i2 1 42 + q2 1 − e−2b (1 + b)2
qz a 2 qz a −a cos × sin qz b=2 + e sin + 2 qz 2 qz ×sin kz + A : (5) 2
By summing over the exchanged momentum q in Eq. (3), one obtains the exchange energy and the electron–hole binding energy. The latter is ij (kz ; Qy ; Qz ) = − qy ;qz vii;jj (k; q; Q), where i = j, whereas the former is simply ijex (kz ) = − qy ;qz vij;ji (k; q; 0). The exciton energy involving
ex + AS (kz ) + AS (kz ; Qy ; Qz ):
(6)
The paramagnetic ground state becomes diDerentially unstable for W 6 0. The 3rst electronic states to experience this instability are those with kz = ±%=A, where the maximum energy in the lower band coincides with the minimum in the upper band, so that Qz = 0. Using these values of kz and Qz , and by setting ‘Qy = ‘Qyc = 1:25 (the value for which the 2D instability occurs) we plot the curve W = 0 in a plane de3ned by ˜!S = 2 B as the y-axis and t = exp(−l) as the x-axis, in Fig. 1. For the purpose of illustration we have also chosen a = 0:75l and b = 0:5l (in our approximation, a ¡ 2b). Below this curve, at low values of ˜!S , the paramagnetic occupancy (of |S ↓ and
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D.C. Marinescu, J.J. Quinn / Physica E 13 (2002) 353 – 356
|S ↑) is a stable Hartree–Fock solution for the interacting system. For large negative values of and for negligible tunneling, electrons occupy |A ↓ and |S ↓ minibands and the ground state is ferromagnetic. The energy of an exciton involving the states |S; ky ; kz ; ↑ and |A; ky + Qy ; kz + Qz ; ↓ consists of the “kinetic energy” − + 20 [cos(kz + Qz )A + cos kz A, the lost ex ex exchange −AA (kz ) − AS (kz ), and the electron–hole binding energy, AS (kz ; Qy ; Qz ). Clearly, the ferromagnetic ground state becomes diDerentially unstable when the exciton energy ex W = − + 20 [cos(kz + Qz )A + cos kz A] − AA (kz ) ex − AS (kz ) + AS (kz ; Qy ; Qz )
(7)
becomes negative. This 3rst occurs for values of the momentum at the center of the Brillouin zone, kz = 0 and for a momentum transfer Qz = 0. In Fig. 1, the curve W = 0 delimits the region above which the ferromagnetic occupancy (of |S ↓ and |A ↓) is a stable Hartree–Fock solution. The region of simultaneous diDerential instability, obtained in our 3gure for values of the magnetic 3eld for which 2 B is in between W and W , corresponds to a collective-type excitation, characterized by a continuous spatial variation of the magnetic moment. A possible solution to the Hartree–Fock approximation of the ground state is a SDW. This conclusion can be reached by following the general argument presented in Ref. [5]. This regime appears at weak tunneling,
below a critical value tc , when S AS is small. This is quite opposite to the result obtained in the case of a superlattice with a single well per unit cell [2]. Just as in the single-layer case, which of these possible solutions is more stable should be determined by a direct comparison of the corresponding total Hartree–Fock energies. The SDW solution describes a lower-energy state of the interacting system. This can be surmized from Overhauser’s original paper on SDW [6], where it was shown that, for a three-dimensional electron gas in the presence of a magnetic 3eld, a linear SDW solution has lower energy than the paramagnetic state (when many Landau levels are assumed to be occupied) independently of the speci3c form of the repulsion potential. JJQ would like to acknowledge support from the Materials Sciences Program, Basic Energy Sciences of the US Department of Energy. References [1] G.F. Giuliani, J.J. Quinn, Phys. Rev. B 31 (1985) 6228. [2] D.C. Marinescu, J.J. Quinn, G.F. Giuliani, Phys. Rev. B 61 (2000) 7245. [3] A.J. Daneshvar, C.J.B. Ford, M.Y. Simmons, A.V. Khaetskii, A.R. Hamilton, M. Pepper, D.A. Ritchie, Phys. Rev. Lett. 79 (1997) 4449. [4] V. Piazza, V. Pellegrini, F. Beltram, W. Wegscheider, T. Jungwirth, A.H. McDonald, Nature 402 (1999) 638. [5] D.C. Marinescu, J.J. Quinn, G.F. Giuliani, Physica E 6 (2000) 807. [6] A.W. Overhauser, Phys. Rev. 128 (1962) 1437.
www.elsevier.com/locate/physe
Excitonic instabilities in semiconductor superlattices in the lowest Landau level D.C. Marinescua; ∗ , J.J. Quinnb a Department
b Department
of Physics, Clemson University, Clemson, SC 29634, USA of Physics, University of Tennessee, Knoxville, TN 37996, USA
Abstract In a superlattice whose unit cell contains two identical quantum wells distanced by a length b, weak tunneling is allowed to occur both intra- and inter-cell. The intra-cell tunneling leads to the splitting of the electron energy levels in two minibands separated by S AS = ES − EAS , the symmetric–antisymmetric energy gap. The inter-cell tunneling broadens these minibands to a width proportional to the tunneling probability. In the presence of a magnetic 3eld, when additional splitting of the energy levels occurs, a degeneracy between minibands of opposite spins can be realized by tuning the intensity of the magnetic 3eld and the tunneling probability. Within the Hartree–Fock approximation, we investigate the existence of excitonic instabilities that can arise when, in the lowest Landau level (n = 0), the |S ↑ and |A ↓ minibands are almost degenerate. ? 2002 Published by Elsevier Science B.V. PACS: 73.21.Fg; 73.15.Lh; 73.40.Gk Keywords: Superlattice; Exciton; Spin density wave
The existence of excitonic instabilities in electron systems is suspected every time a degeneracy between energy levels is realized. Quite generally, the system can lower its total energy by transferring electrons between the two degenerate energy levels when the electron–hole correlation energy overcomes the loss of exchange energy. Such situations have been predicted to exist in single and multiple-layer systems and can involve both charge and spin transfers [1,2]. In a two-dimensional (2D) single electron layer, in the presence of the magnetic 3eld, when only the 3rst two Landau levels are occupied ( = 2), it was demonstrated that an excitonic spin instability between ∗ Corresponding author. Tel.: +864-656-5315; fax: +864656-0805. E-mail address: [email protected] (D.C. Marinescu).
|0 ↑ and |1 ↓ energy levels occurs even for positive values of ’, the diDerence between the cyclotron energy, ˜!c , and the Zeeman splitting, 2 B. The collective mode associated with this excitonic instability is soft, the system undergoing an abrupt ferromagnetic to paramagnetic magnetic ordering transition [1]. Recent experimental evidence was found to support these results [3,4]. A diDerent situation is encountered in a superlattice of identical quantum wells at = 2 coupled by tunneling. The energy levels of the electrons in the magnetic 3eld are broadened into minibands of width , proportional to the tunneling probability. The initial energy gap between |0 ↑ and |1 ↓ is further reduced by , favoring the existence of magnetic excitonic instabilities. Within the Hartree–Fock approximation it is shown that at tunneling values above a critical
1386-9477/02/$ - see front matter ? 2002 Published by Elsevier Science B.V. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 5 5 6 - 2
354
D.C. Marinescu, J.J. Quinn / Physica E 13 (2002) 353 – 356
level, the system exhibits a spin density wave (SDW) instability [2]. In this paper we analyze the existence of such spin instabilities, in the presence of a magnetic 3eld B, in a quantum-well superlattice structure in which the unit cell consists of two identical quantum wells separated by a distance b, equally spaced along the z-axis ˆ at distance A, the superlattice constant. Weak tunneling is assumed to occur inside the unit cell and between the 3rst-order neighbors in the superlattice. In a simple approximation, the superlattice is described as a periodic array of N attractive -function potentials, i.e. V (z) = − l [(z + b=2 − lA) + (z − b=2 − lA)], with l an integer. For each spin, the single energy level corresponding to the zˆ degree of freedom in the isolated well, 0 = −m2 =2˜2 , is split when weak tunneling is permitted inside the unit cell. The two possible states are described by a symmetric (S) and an antisymmetric (A) wave functions: ±(1 ± e−b )e−b(z+b=2) AS (z) = e−(z−b=2) ± e−(z+b=2) 2[1 ± e−b (1 + b)] (1 ± e−b )e−(z−b=2)
, is, in the same approximation,
where = m=˜2 . (The + sign corresponds to the symmetric solution and − sign corresponds to the antisymmetric one.) The associated energy eigenvalues are AS = 0 (1 ± e−b ). The diDerence S AS = A − S = −20 e−b de3nes the width of the gap between the two energy levels and is a parameter of the problem. When tunneling between the 3rst-order neighbors is allowed in the superlattice, these energy levels broaden into minibands that acquire a width proportional to the tunneling probability, e−a (a = A − b). In a tight binding approximation, the single electron wave function in one of the minibands is: |n; ky ; kz ; SA = eikz z uAS (kz ; z) nky (x; y) with nky (x; y) = L−1=2 eiky y Hn (x + ky ‘), where ‘ = ˜c=eB is the magnetic length, Hn (x) is the nth simple-harmonic oscillator function, and represents the spin eigenfunction. u(kz ; z) is the appropriate periodic wave function, uAS (kz ; z)= √1N l eikz (z−lA) AS (z− lA). The allowed values of ky are 2%j=L, with j an integer and L the length for periodic boundary conditions in the y direction. kz =2%j=Na, where j is an integer in the range ±N=2. The single electron energy for a spin orientation along the direction of the magnetic 3eld,
is in a paramagnetic phase. For S AS ¡ 2 B the lowest energy levels are |S ↓ and |A ↓ and the system is in a ferromagnetic ground state. We assume now that = S AS − 2 B ¿ 0. In this situation, the lowest occupied minibands are |S ↓ and |S ↑. The existence of a diDerential magnetic instability between |S ↑ and |A ↓, the closest miniband completely unoccupied, can be inferred from very simple considerations. When an electron from the |S ↑ miniband moves into the |A ↓ miniband, the energy of the transition can be seen as made up of three parts: the “kinetic energy”, which here is simply the energy gap between the two states in the absence of the electron– electron interactions, the exchange energy of the electron with all other electrons with the same spin, and the binding energy of the electron and the hole which is left in the initial miniband. The matrix elements vii ;jj (kz ; q; Q) of the Coulomb interaction between two electrons, are given by
AS (n; ky ; kz ; ) =˜!c (n + 1=2) +
˜2 ky2 2m
+ AS [1 ± 2e−a cos kz A] + sgn() B;
(2)
where is the gyromagnetic factor that describes the interaction of the electron with the magnetic 3eld. In the following considerations we will focus on the electrons in the minibands that originate in the lowest Landau level, and henceforth set n = 0. Furthermore, we consider that only the lowest two minibands are occupied. The magnetic con3guration of the ground state is determined by the relationship between the various energies introduced in the problem. For S AS ¿ 2 B, the occupied levels are |S ↓ and |S ↑ and the system for z 6 − b=2; for − b=2 6 z 6 b=2;
(1)
for b=2 6 z;
vii ;jj (kz ; q; Q) =
e2 Fii ;jj (kz ; qz ; Qz ) ‘
D.C. Marinescu, J.J. Quinn / Physica E 13 (2002) 353 – 356 ∞
dqx −‘2 =2(qy2 +qx2 −2iqx Qy ) e 2 −∞ q
‘2 2 2 × n; 0 m; 0 + 1 − (qy + qx ) n; 0 m; 1 2
2 ‘2 2 2 (3) + 1 − (qy + qx ) n; 1 m; 1 2
14
13
W' = 0
11
2
with q = k − k and n; m the Kronecker delta. i; j are subband indices for electron states before the interaction and stand for S and A, while i ; j are subband indices for electron states after the interaction. The matrix element diDers from its 2D counterpart obtained in Ref. [1] by the form factor Fii ;jj (kz ; qz ; Qz ) that describes the exchange of momentum along the z direction. We note that Fi; i ;jj (kz ; qz ; Qz )= Fii (kz )Fjj∗ (kz + Qz ). To 3rst order in inter-cell tunneling probability, we readily obtain that
FSS 1 42 (qz ; kz ) = 42 + q2 1 ± e−b (1 + b) FAA
qz b 2 qz b qz b −b ±e + cos sin × cos 2 2 qz 2 qz a 2 qz a qz × ±e−a cos ; cos kz + A sin + 2 qz 2 2 (4)
FERRO
12
hωs(e / l)
×
355
10
SDW
9
8
PARA W=0
7
6 0.02
0.04
0.06
0.08
0.10
t
0.12
0.14
0.16
0.18
tc
Fig. 1. The magnetic phase diagram of a superlattice with a=0:75l and b = 0:5l (l = e=˜B) is represented in the ˜!S − t plane. W = 0 and W = 0 are the excitonic instability curves for the paramagnetic and ferromagnetic ground states, respectively. The intermediate region, at small tunneling values, corresponds to a spatially varying magnetic moment, as in SDW. At tunneling amplitudes larger than a critical value tc , the system undergoes an abrupt paramagnetic (P) to ferromagnetic (F) transition.
where as before, the upper sign corresponds to FSS and the lower to FAA . Additionally, form factors of the type FAS are also considered as
the states |S; ky ; kz ; ↑ and |A; ky + Qy ; kz + Qz ; ↓ is given by
FAS (qz ; kz )
ex W = − 20 [cos(kz + Qz )A + cos kz A] − SS (kz )
=
4i2 1 42 + q2 1 − e−2b (1 + b)2
qz a 2 qz a −a cos × sin qz b=2 + e sin + 2 qz 2 qz ×sin kz + A : (5) 2
By summing over the exchanged momentum q in Eq. (3), one obtains the exchange energy and the electron–hole binding energy. The latter is ij (kz ; Qy ; Qz ) = − qy ;qz vii;jj (k; q; Q), where i = j, whereas the former is simply ijex (kz ) = − qy ;qz vij;ji (k; q; 0). The exciton energy involving
ex + AS (kz ) + AS (kz ; Qy ; Qz ):
(6)
The paramagnetic ground state becomes diDerentially unstable for W 6 0. The 3rst electronic states to experience this instability are those with kz = ±%=A, where the maximum energy in the lower band coincides with the minimum in the upper band, so that Qz = 0. Using these values of kz and Qz , and by setting ‘Qy = ‘Qyc = 1:25 (the value for which the 2D instability occurs) we plot the curve W = 0 in a plane de3ned by ˜!S = 2 B as the y-axis and t = exp(−l) as the x-axis, in Fig. 1. For the purpose of illustration we have also chosen a = 0:75l and b = 0:5l (in our approximation, a ¡ 2b). Below this curve, at low values of ˜!S , the paramagnetic occupancy (of |S ↓ and
356
D.C. Marinescu, J.J. Quinn / Physica E 13 (2002) 353 – 356
|S ↑) is a stable Hartree–Fock solution for the interacting system. For large negative values of and for negligible tunneling, electrons occupy |A ↓ and |S ↓ minibands and the ground state is ferromagnetic. The energy of an exciton involving the states |S; ky ; kz ; ↑ and |A; ky + Qy ; kz + Qz ; ↓ consists of the “kinetic energy” − + 20 [cos(kz + Qz )A + cos kz A, the lost ex ex exchange −AA (kz ) − AS (kz ), and the electron–hole binding energy, AS (kz ; Qy ; Qz ). Clearly, the ferromagnetic ground state becomes diDerentially unstable when the exciton energy ex W = − + 20 [cos(kz + Qz )A + cos kz A] − AA (kz ) ex − AS (kz ) + AS (kz ; Qy ; Qz )
(7)
becomes negative. This 3rst occurs for values of the momentum at the center of the Brillouin zone, kz = 0 and for a momentum transfer Qz = 0. In Fig. 1, the curve W = 0 delimits the region above which the ferromagnetic occupancy (of |S ↓ and |A ↓) is a stable Hartree–Fock solution. The region of simultaneous diDerential instability, obtained in our 3gure for values of the magnetic 3eld for which 2 B is in between W and W , corresponds to a collective-type excitation, characterized by a continuous spatial variation of the magnetic moment. A possible solution to the Hartree–Fock approximation of the ground state is a SDW. This conclusion can be reached by following the general argument presented in Ref. [5]. This regime appears at weak tunneling,
below a critical value tc , when S AS is small. This is quite opposite to the result obtained in the case of a superlattice with a single well per unit cell [2]. Just as in the single-layer case, which of these possible solutions is more stable should be determined by a direct comparison of the corresponding total Hartree–Fock energies. The SDW solution describes a lower-energy state of the interacting system. This can be surmized from Overhauser’s original paper on SDW [6], where it was shown that, for a three-dimensional electron gas in the presence of a magnetic 3eld, a linear SDW solution has lower energy than the paramagnetic state (when many Landau levels are assumed to be occupied) independently of the speci3c form of the repulsion potential. JJQ would like to acknowledge support from the Materials Sciences Program, Basic Energy Sciences of the US Department of Energy. References [1] G.F. Giuliani, J.J. Quinn, Phys. Rev. B 31 (1985) 6228. [2] D.C. Marinescu, J.J. Quinn, G.F. Giuliani, Phys. Rev. B 61 (2000) 7245. [3] A.J. Daneshvar, C.J.B. Ford, M.Y. Simmons, A.V. Khaetskii, A.R. Hamilton, M. Pepper, D.A. Ritchie, Phys. Rev. Lett. 79 (1997) 4449. [4] V. Piazza, V. Pellegrini, F. Beltram, W. Wegscheider, T. Jungwirth, A.H. McDonald, Nature 402 (1999) 638. [5] D.C. Marinescu, J.J. Quinn, G.F. Giuliani, Physica E 6 (2000) 807. [6] A.W. Overhauser, Phys. Rev. 128 (1962) 1437.